Soft Computing Methods in Flight Control System Design

Soft Computing Methods in Flight Control System Design

Marcel Oosterom

Soft Computing Methods in Flight Control System Design

Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College van Promoties, in het openbaar te verdedigen op maandag 13 juni 2005, om 15.30 uur door

Marcel Laurens Jean OOSTEROM ingenieur luchtvaart en ruimtevaart geboren te Deurne

Dit proefschrift is goedgekeurd door de promotor: Prof. dr. R. Babuˇska, M.Sc.

Samenstelling promotiecommissie: Rector Magnificus Prof. dr. R. Babuˇska, M.Sc. Prof. dr. ir. J.A. Mulder Prof. ir. H.B. Verbruggen Prof. dr. ir. M. Verhaegen Prof. R.J. Patton, Ph.D. Dr. ir. G. Schram K. Rosenberg, M.Sc.

voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft Technische Universiteit Delft Technische Universiteit Delft The University of Hull SKF Reliability Systems BAE SYSTEMS Avionics Limited

ISBN 90-8559-060-4

c 2005 by M. Oosterom. Copyright
No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the author.

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Contents

Summary

xi

Samenvatting

xvii

0 Notations and Abbreviations

1

0.1

List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

0.2

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . .

2

0.3

Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1 Introduction

7

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.2

Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3

New developments . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.4

Research aims and motivation of the methods used . . . . . . . . .

11

1.5

Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . .

14

2 Digital Fly-By-Wire Flight Control Systems

17

2.1

Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2

Description of the Digital Fly-By-Wire Flight Control System . . .

19

2.3

Advantages and Disadvantages of Digital Fly-By-Wire . . . . . . .

21

3 Fuzzy Clustering for Partitioning of the Flight Envelope

25

3.1

Nonlinear Control

. . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.2

Fuzzy Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.3

Fuzzy Partitioning of the Flight Envelope . . . . . . . . . . . . . .

28

3.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4 Scheduled Classical Control

43

4.1

Stability and Control Augmentation System . . . . . . . . . . . . .

43

4.2

Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

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Contents

4.3

Automatic Tuning Procedure . . . . . . . . . . . . . . . . . . . . .

46

4.4

The Scheduler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.5

Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

5 Scheduled Robust Multivariable Control

55

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

5.2

Overview of Scheduled Robust MV Control . . . . . . . . . . . . .

56

5.3

General Description of the Robust Control Problem . . . . . . . .

57

5.4

Robust Multivariable Flight Control Design . . . . . . . . . . . . .

59

5.5

Partition of the Flight Envelope . . . . . . . . . . . . . . . . . . . .

66

5.6

Parameter Scheduled Robust Multivariable Control . . . . . . . . .

67

5.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

6 Virtual Angle-of-Attack Sensor

77

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

6.2

Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . .

78

6.3

Structure of the Virtual Angle-of-Attack Sensor . . . . . . . . . . .

79

6.4

Design of the TS Fuzzy Model and the NN Model . . . . . . . . .

83

6.5

Validation of the Virtual AoA Sensor . . . . . . . . . . . . . . . . .

86

6.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

7 Soft Sensor Management and Virtual Sensors for FDIR

91

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

7.2

Conventional Sensor Management and FCL Reconfiguration . . . .

92

7.3

Sensor Management and FCL Reconfiguration Based on Soft Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

7.4

Virtual Sensor for FDIR . . . . . . . . . . . . . . . . . . . . . . . . 104

7.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8 Conclusions

111

8.1

Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.2

Efficiency Improvement of the Design of the System . . . . . . . . 114

8.3

Enhancement of Flight Safety . . . . . . . . . . . . . . . . . . . . . 115

8.4

Recommendations for Further Research . . . . . . . . . . . . . . . 115

A Synthetic Environment and Real-Time Code Generation

119

A.1 Synthetic Environment . . . . . . . . . . . . . . . . . . . . . . . . . 119

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A.2 Real-Time Code Generation . . . . . . . . . . . . . . . . . . . . . . 124 B Short-Period Approximation

125

B.1 Derivation of the Short-Period Approximation . . . . . . . . . . . . 125 B.2 Derivation of Short-Period Related Equations . . . . . . . . . . . . 126 C Performance Measures and Cross Validation

129

C.1 Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . 129 C.2 Cross Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 D Soft Computing Techniques

131

D.1 Fuzzy Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 D.2 Identification via Fuzzy Clustering . . . . . . . . . . . . . . . . . . 135 D.3 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 E Genetic Algorithms

143

E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 E.2 How Do They Work? . . . . . . . . . . . . . . . . . . . . . . . . . . 143 E.3 Theoretical Foundation . . . . . . . . . . . . . . . . . . . . . . . . 151 E.4 Application Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 F Linear Matrix Inequalities for Control

153

F.1 Output-feedback H∞ Control Problem . . . . . . . . . . . . . . . . 153 F.2 LMI Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 F.3 Pole Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Acknowledgements

159

Curriculum Vitae

161

List of Publications

163

Bibliography

165

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Contents

Summary Soft Computing Methods in Flight Control System Design Marcel Oosterom

The Digital Fly-By-Wire (DFBW) flight control system has many advantages over the mechanical Flight Control System (FCS) in terms of weight, fuel consumption, flexibility in the configuration of the bare airframe, etc. The most powerful feature of the DFBW FCS is the wide range of capabilities that can be programmed into the Flight Control Computer (FCC). The major drawback is the additional cost associated with the design and initial acquisition of the system. Without a mechanical back-up, the DFBW FCS is a safety-critical system. Stringent requirements therefore apply with respect to the integrity and availability of the system. Fault tolerance is achieved through hardware redundancy and dissimilarity among the redundant hardware (and software) components is used to avoid common mode failures. For military (fighter) aircraft and large commercial aircraft, the advantages of the DFBW FCS justify the additional cost of the system. However, this is not so obvious for small commercial aircraft. The (relative) additional cost of the DFBW FCS is much higher, while the advantages are not so evident as for large commercial aircraft. A radical change in current practices is required in order to bring DFBW FCS technologies to the small commercial aircraft market at affordable cost whilst maintaining the stringent safety requirements. The possibilities for a cost-effective application of this technology to small commercial aircraft have been investigated within the European project “Affordable Digital Fly-By-Wire Flight Control Systems for Small Commercial Aircraft” (Phases I and II). Partners were involved from industry, research institutes, and universities. The work described in this thesis is for a large part performed within this project. The aim of the research described in this thesis is to improve the efficiency of flight control system design and/or to improve the flight safety. The improvement of the efficiency of the FCS design results in reduced development cost, which is a key factor in making DFBW technology affordable for small commercial aircraft. xi

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Summary

Besides the economical aspect, the improvement of flight safety is also an important argument to justify bringing DFBW to the small commercial aircraft market. In this thesis the focus has been put on two topics, namely: 1. Automated design techniques. 2. Sensor management, including virtual sensors. The first item is specifically of interest with respect to nonlinear flight control law design, while the second item focuses mainly on sensor management (voting/monitoring). These two application areas are discussed in more detail in the remainder of this section. The applications described in this thesis are developed in the Synthetic Environment (SE), which is defined as an ultra-high fidelity simulation tool that is structurally representative of the practical implementation of a DFBW aircraft. The SE has been developed within the ADFCS project. Automated design techniques Gain scheduling has been perhaps the most common systematic approach to control of nonlinear systems in practice. In the aerospace industry, e.g. for the design of the FCLs for commercial DFBW aircraft, and for a wide range of other application areas. Even with the introduction of powerful control strategies such as model predictive control and feedback linearization, gain scheduling remains an attractive control strategy because of its simplicity and practical use. The design of the scheduler is an iterative process, where each iteration consists of the identification of the operating points, the tuning of the controller parameters, the design of the scheduler and the evaluation of the global performance of the system. This is a slow and costly procedure, however, despite this drawback almost no effort has been spent on the development of a systematic approach to identify the operating points and to design the corresponding scheduler. In Chapter 3, an automated procedure is proposed for the identification of the operating points for which the local flight control law parameters need to be tuned. This procedure is based on the application of fuzzy clustering to a data set which represents the variation of the dominant aircraft dynamics over the flight envelope. The resulting cluster centers serve as the operating points. A singleton TS fuzzy model is constructed to provide the interpolation mechanism for the Flight Control Law (FCL) parameters in order to obtain a global nonlinear controller. The main advantage of this approach is that the operating points are identified simultaneously, in contrast to the iterative trail-and-error approach that is carried out by the flight control engineer. Besides the reduction in the design effort with respect to identifying the operating points, the fuzzy clustering approach results in fewer operating points. This reduces the design effort for the tuning of the FCL parameters. By using the same scheduling variables and operating points for all FCL parameters that require scheduling, the scheduler becomes more transparent

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and the (local) effects of parameter modifications are more predictable. This does not necessarily mean that the best performance is achieved by scheduling all FCL parameters in exactly the same way. The fuzzy clustering approach should be considered as an interactive tool for the flight control engineer. It supports the flight control engineer in deciding on the number and the location of the operating points. The iterations in the design procedure are automated, which simplifies the design procedure. It should be noted that the identification of the operating points through fuzzy clustering does not put any restrictions on the scheduling mechanism to be used. The application of the automated design procedure to the classical FCLs that are available in the SE is described in Chapter 4. The scheduler of the six most relevant controller parameters is replaced by the scheduler designed by using the fuzzy clustering approach. The selected scheduling variables are the Mach number and the dynamic pressure. Eight operating points are identified for the flight envelope in clean configuration and two for the flight envelope in landing configuration. Pilot-in-the-loop simulations demonstrated that the performance of the FCLs designed by using the automated design procedure is equivalent to the performance of the default FCLs. Even though the fuzzy gain scheduler used fewer operating points than in the conventional approach, it can be concluded that its performance is comparable. A significant contribution in reducing the DFBW development costs, without compromising the safety requirements, can be obtained by replacing the classical single-loop frequency response and root-locus design techniques by advanced MultiVariable (MV) control design techniques for the development of the flight control laws. The MV design approach has the advantage of reducing time and cost for flight control design and refinement, and at the same time of a priori taking into account robustness with respect to model uncertainties. In Chapter 5 the local controllers are designed using robust MV control techniques. The challenge is to combine the resulting MV controllers with gain scheduling, because of the complexity and the opaque structure of such controllers. The combination of fuzzy clustering and robust multivariable control is new and potentially very effective in reducing the design effort since both the identification of the operating points and the scheduler as well as the design of the local linear controllers are highly automated. The local H∞ controllers are designed using a model-matching approach. The design is performed in continuous-time using Linear Matrix Inequalities (LMIs). Order reduction is performed to reduce the number of parameters to be scheduled. First a Hankel model reduction is applied and then the high-frequency modes are removed from the controller. After Tustin discretization, the local controllers are transformed to the δ-operator form to reduce their coefficient-pole sensitivity. The latter is important with respect to gain scheduling. The gain-scheduled robust MV controller has been evaluated off-line (linear and nonlinear simulations, stability analysis) and through pilot-in-the-loop simulations. The results of the off-line evaluation are satisfactory, although additional

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Summary

tuning is required to further improve the performance and stability characteristics. The test pilots gave a Cooper-Harper (CH) rating of 1 in all flight conditions concerning the aircraft dynamics. However, due to the high control force needed to maneuver the aircraft, especially in the low dynamic pressure region, the overall CH ratings were between 2 and 3. This can be corrected by adjusting the reference model. In conclusion, the contribution of the fuzzy clustering approach to improving the efficiency of the design of flight control laws is significant. It is a global approach, all the operating points are identified simultaneously, which results in a transparent scheduling scheme with fewer operating points. Moreover, it is a model-based approach that uses the nonlinear dynamics in the design phase and not only in the evaluation phase. In combination with modern MV control techniques for the design of the local controllers, the reduction in the design effort for the design of the flight control laws is even more evident.

Sensor Management Sensor management based on majority voting and point consolidation of like signals, i.e. signals from different sensors measuring the same variable, is a proven technology in modern fly-by-wire flight control systems. The assumption is that the majority of like signals represents the truth and that any single dissimilar signal is the result of a failure. Such a signal must be disconnected as soon as the failure is detected. This principle fails in the event that there are two like signals left (duplex operation), since there is no longer a majority. In the conventional approach, the decision whether a sensor has failed or not is crisp. In order to reduce the sensitivity of this decision to uncertainties like quantization and measurement noise, a properly adjusted threshold is used. Two opportunities have been identified to improve the flight safety (and comfort). The introduction of a virtual sensor that makes use of non-like signals (analytical redundancy) and can be used as an arbitrator during duplex operation. Furthermore, an alternative sensor management procedure which improves the consolidated signal and reduces failure-induced transients is introduced. Many applications of analytical redundancy for FDI in flight control systems are reported. Most frequently applied are observer-based techniques, parity-space methods, and parameter-estimation schemes. The use of virtual sensors in aerospace applications has not been widely investigated yet, although this technique has been successfully applied in other fields like process control and engine control. The approach investigated in Chapter 6 of this thesis is based on a combination of white-box and black-box modelling and is demonstrated through the design of a virtual angle-of-attack (AoA) sensor. The philosophy is to first achieve the maximum performance out of a white-box modelling approach, using the well known relations of the linearized aircraft dynamics. In the second step, the remaining estimation error is further reduced by adding a black-box model that is designed to fit the estimation error of the white-box model. The inputs of this

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nonlinear, black-box model are determined using a nonlinear input selection approach. The virtual AoA sensor makes use of another virtual sensor that estimates the aircraft weight and the position of the center of gravity. This virtual sensor does not make use of AoA sensor readings. The performance of the virtual sensor is demonstrated by a large number of nonlinear simulations for which the flight conditions and maneuvers are selected randomly. The performance of the virtual sensor is good, with maximum estimation errors for the angle-of-attack of less than 0.8 degrees. The conventional sensor management system makes use of crisp thresholds. The consolidated signal is computed by taking the (weighted) average of each sensor reading. The mid-value signal is taken as a reference and the two extreme-value signals are limited in their deviation from the mid-value (crisp threshold). The monitor compares each sensor signal with the consolidated signal. If the absolute difference exceeds a predefined (crisp) threshold, the corresponding monitor count is increased. If the count value has reached the failure declaration value, a failure is declared and the signal is latched. The locations of the thresholds is a compromise between the minimization of false alarms and minimization of failure-induced transients. While the first dictates large thresholds, the latter dictates small thresholds. This compromise can be circumvented by introducing soft thresholds instead of crisp thresholds, increasing flight safety and/or comfort. Fuzzy logic is an excellent technique to implement this concept. Although these techniques have been implemented in other application domains, such as the process industry, their application in flight control systems has not been extensively investigated yet. In Chapter 7 of this thesis a sensor management procedure based on soft computing is proposed. In this soft sensor management system a weight between (and including) zero and one (soft threshold) is assigned to each sensor reading. This weight is computed based on the smallest absolute difference between the other sensor readings. The consolidated signal is the weighted average of the signals. When the weight of a sensor reading is equal to zero, the corresponding signal is not contributing to the consolidated signal and the corresponding monitor count is increased. The main difference from the conventional monitoring scheme is that the monitor count rate is not a function of the difference between the ith sensor reading and the consolidated signal, but a function of the difference between the ith sensor reading and the other like sensor readings. The soft sensor management system including a normal acceleration virtual sensor has been demonstrated by means of closed-loop simulation examples in the SE and on the flight simulator with pilot-in-the-loop simulations. In the conventional approach, signal consolidation and signal monitoring are performed separately. With the introduction of soft thresholds, signal consolidation and sensor monitoring are integrated. This results in a more accurate consolidated signal and reduces the failure induced transients, which contributes not only to flight safety, but also to passenger comfort. Furthermore, it is demonstrated how virtual sensors can be used to identify the faulty sensor in the case of a discrepancy between two like sensor signals. When also the last sensor fails, the signal is no longer available and the FCLs reconfigure to not using this signal. Furthermore, the FCL reconfiguration is smoother as a direct result of the soft sensor management strategy.

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Summary

In conclusion, the contribution of the soft computing to increase flight safety is significant. With a virtual sensor it is possible to identify the faulty signal when a discrepancy is detected between the signals from two physical like sensors. Moreover, it is possible to detect a failure on the last remaining physical sensor. The virtual sensor increases the integrity of the FCS and therefore contributes to flight safety. Clearly virtual sensors can be designed using other techniques as well. However, soft computing allows the designer to use well-known linear techniques to create a global nonlinear system, which adds to the accuracy and reliability of the virtual sensor. The contribution of soft sensor management to flight safety is less significant than for the virtual sensor, but it does improve the system at no extra cost. Future research is suggested in the direction of optimization of the scheduling mechanism for the FCL parameters making use of genetic algorithms, after the operating points have been identified through fuzzy clustering. Furthermore, a more integrated design of the scheduled robust MV controller is suggested, where the tuning of the weighting functions and the scheduling mechanism are optimized in an iterative manner taking into account global stability and performance requirements. Still an open issue is the FCL reconfiguration when using robust MV controllers. With respect to virtual sensors, a topic that needs further attention is how to integrate the virtual sensor in the sensor management system and how to deal with the limitations of a virtual sensor.

Samenvatting Soft Computing Methods in Flight Control System Design Marcel Oosterom

Het Digitale Fly-By-Wire vliegtuigbesturingssysteem (DFBW) heeft vele voordelen ten opzichte van mechanische besturing met betrekking tot gewicht, brandstofverbruik, flexibiliteit in de vliegtuigconfiguratie, enzovoort. De sterkste eigenschap van DFBW is de uitgebreide functionaliteit die in de besturingscomputer geprogrammeerd kan worden. Het voornaamste nadeel van DFBW wordt gevormd door de extra kosten die zijn gemoeid met het ontwerp en de initi¨ele aankoop. Zonder de aanwezigheid van een mechanisch back-up systeem is de beschikbaarheid en goede werking van DFBW essentieel voor de veilige operatie van het vliegtuig. Strenge eisen gelden derhalve voor de integriteit en beschikbaarheid van het systeem. Tolerantie ten aanzien van defecten (fault tolerance) wordt bereikt door middel van hardware redundantie. Om zogenaamde common mode storingen te voorkomen, wordt ongelijkheid tussen redundante hardware (en software) toegepast. Voor militaire (jacht)vliegtuigen en grote passagiersvliegtuigen zijn de voordelen van DFBW dermate groot, dat de extra kosten van het systeem gerechtvaardigd zijn. Dit is echter niet zo duidelijk het geval voor kleine passagiersvliegtuigen. De (relatieve) extra kosten van DFBW zijn vele malen groter voor kleine passagiersvliegtuigen, terwijl aan de andere kant de voordelen niet zo overtuigend zijn als voor grote passagiersvliegtuigen. Bij militaire (jacht)vliegtuigen spelen de kosten een veel kleinere rol en gaat het voornamelijk om de extra functionaliteit. Om DFBW besturingstechnologie tegen redelijke kosten op de markt voor kleine passagiersvliegtuigen te kunnen brengen, waarbij de strenge veiligheidseisen worden gehandhaafd, is een fundamentele verandering in het huidige ontwerpproces noodzakelijk. In het Europese project genaamd Affordable Digital Fly-By-Wire Flight Control Systems for Small Commercial Aircraft (ADFCS) zijn de mogelijkheden voor een rendabele toepassing van DFBW in kleine passagiersvliegtuigen onderzocht. Partners uit de industrie, onderzoekscentra en universiteiten hebben aan dit project deelgenomen.

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Samenvatting

Het werk dat in dit proefschrift is beschreven is voor een groot deel uitgevoerd binnen dit project. Het had als doel mogelijkheiden te onderzoeken om het ontwerpen van vliegtuigbesturingssystemen effici¨ enter te maken en/of de vliegveiligheid te verbeteren. Een grotere effici¨entie bij het ontwerpen van het vliegtuigbesturingssysteem resulteert in lagere ontwikkelingskosten, hetgeen een belangrijk aspect is met betrekking tot het economisch haalbaar maken van DFBW voor kleine passagiersvliegtuigen. Naast het economische aspect is de verbetering van de vliegveiligheid een belangrijk argument om de extra kosten te rechtvaardigen. In dit proefschrift is de nadruk gelegd op twee onderwerpen, namelijk: 1. Geautomatiseerde ontwerptechnieken. 2. Sensor management, inclusief virtuele sensoren. Het eerste onderwerp is specifiek van belang bij het ontwerpen van de niet-lineaire vliegtuigregelaar, terwijl bij het tweede onderwerp de aandacht voornamelijk wordt gericht op voting en monitoring van sensorsignalen. Deze twee toepassingsgebieden worden in de rest van deze samenvatting nader besproken. De toepassingen die in dit proefschrift zijn beschreven, zijn ontwikkeld in het zogenaamde Synthetic Environment (SE), welke is gedefinieerd als een zeer natuurgetrouwe simulatie tool die structureel representatief is voor de praktische implementatie van een DFBW vliegtuig. Het SE is ontwikkeld binnen het ADFCS project. Geautomatiseerde ontwerptechnieken Gain scheduling is wellicht de meest toegepaste systematische methode in de praktijk voor het regelen van niet-lineaire systemen. In de luchtvaartindustrie wordt het bijvoorbeeld gebruikt voor het ontwerpen van de vliegtuigregelaar, maar er zijn vele andere toepassingsgebieden. Zelfs met de introduktie van krachtige regelontwerptechnieken zoals model predictive control en feedback linearisatie, blijft gain scheduling een aantrekkelijke regelstrategie omdat het simpel is en praktisch in het gebruik. Het ontwerp van de scheduler, met andere woorden het interpolatiemechanisme, is een iteratief proces, waarbij iedere iteratie bestaat uit het identificeren van de ontwerppunten, het afstellen van de regelparameters, het ontwerpen van de scheduler en het evalueren van de globale prestaties van het systeem. Dit een tijdrovend en kostbaar proces, maar ondanks dit nadeel is er tot op heden nauwelijks aandacht besteed aan het ontwikkelen van een systematische methode voor het identificeren van de ontwerppunten en het ontwerpen van de bijbehorende scheduler. In Hoofdstuk 3 wordt een geautomatiseerde procedure voor de identificatie van ontwerppunten ge¨ıntroduceerd, waarin de parameters van de vliegtuigregelaar moeten worden afgesteld. Deze procedure is gebaseerd op het toepassen van fuzzy clustering op een data set die de verandering van de dominante vliegtuigdynamica in het operationele vlieggebied representeert. De centra van de daaruit voort-

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komende clusters dienen als ontwerppunten. Een singleton TS fuzzy model wordt geconstrueerd voor de interpolatie van de parameters van de vliegtuigregelaar om zo een globaal niet-lineaire regelaar te verkrijgen. Het voornaamste voordeel van deze methode is dat de ontwerppunten gelijktijdig ge¨ıdentificeerd worden, hetgeen niet het geval is bij de iteratieve trial-anderror methode die meestal wordt toegepast door vliegtuigregeltechnici. Naast de vermindering van de ontwerpinspanning met betrekking tot het identificeren van de ontwerppunten, resulteert de fuzzy clustering methode ook in een kleiner aantal ontwerppunten. Dit resulteert wederom in minder inspanning voor het afstellen van de parameters van de vliegtuigregelaar. Door dezelfde scheduling variabelen en ontwerppunten te gebruiken voor alle parameters van de vliegtuigregelaar die onderhevig zijn aan scheduling, wordt de scheduler transparanter en zijn de locale effecten van parameter modificaties voorspelbaarder. Dit betekent overigens niet dat de beste prestaties worden behaald door alle parameters op exact dezelfde manier te interpoleren. De fuzzy clustering methode moet gezien worden als een interactief gereedschap. Het ondersteunt vliegtuigregeltechnici bij het bepalen van het aantal en de locatie van de ontwerppunten. De iteraties in de ontwerpprocedure zijn geautomatiseerd, hetgeen de ontwerpprocedure versimpelt. De identificatie van de ontwerppunten met behulp van fuzzy clustering legt geen restricties op met betrekking tot het te gebruiken scheduling mechanisme. De toepassing van de geautomatiseerde ontwerpprocedure op de klassieke vliegtuigregelaar die beschikbaar is in het SE wordt beschreven in Hoofdstuk 4. De scheduler van de zes meest relevante regelparameters is vervangen door de scheduler die is ontworpen door middel van de fuzzy clustering methode. De geselecteerde scheduling variabelen zijn het Mach getal en de dynamische druk. Acht ontwerppunten zijn ge¨ıdentificeerd voor het operationele vlieggebied in de kruisvluchtconfiguratie en twee ontwerppunten voor het operationele vlieggebied in de landingsconfiguratie. Simulaties met de piloot in de lus tonen aan dat de prestaties van de vliegtuigregelaar ontworpen met de geautomatiseerde ontwerpmethode equivalent zijn aan de prestaties van de vliegtuigregelaar die als standaard dient in het ADFCS project. Dit ondanks het feit dat de vliegtuigregelaar die is ontworpen met fuzzy clustering minder ontwerppunten gebruikt. Een significante bijdrage aan de kostenreductie van een DFBW vliegtuigbesturingssysteem, zonder daarbij de veiligheidseisen te schenden, kan worden bereikt door de klassieke single-loop frequentie respons en root-locus ontwerptechnieken te vervangen voor geavanceerde Multi-Variabele (MV) technieken voor het ontwerpen van de vliegtuigregelaar. De MV regelontwerptechniek heeft als voordeel het reduceren van tijd en kosten bij het ontwerpen en verfijnen van de vliegtuigregelaar, terwijl tegelijkertijd rekening wordt gehouden met de robuustheid met betrekking tot model onzekerheden. In Hoofdstuk 5 zijn de lokale regelaars ontworpen met behulp van een robuuste MV regelontwerptechniek. De uitdaging is om gain scheduling toe te passen op de resulterende MV regelaars, omdat dergelijke regelaars geen fysisch interpreteerbare en/of duidelijke structuur hebben. De

xx

Samenvatting

combinatie van fuzzy clustering en robuuste MV regelontwerptechnieken is nieuw en heeft de potentie om zeer effectief de ontwerpinspanning te reduceren doordat zowel de identificatie van de ontwerppunten als het ontwerpen van de lokale lineaire regelaars in grote mate geautomatiseerd zijn. De lokale H∞ regelaars zijn ontworpen met behulp van de model-matching methode. Het ontwerpen is uitgevoerd in continue tijd, gebruik makend van Linear Matrix Inequalities (LMIs). De orde van de lokale regelaars wordt gereduceerd om het aantal parameters voor het scheduling mechanisme te reduceren. Eerst wordt er een Hankel model reductie uitgevoerd en vervolgens worden de hoogfrequente polen en nulpunten verwijderd uit de regelaars. Na Tustin discretisatie wordt de regelaar omgezet in de δ-operator vorm om de zogenaamde pool-coeffic¨ıent gevoeligheid van de regelaar verder te reduceren. Dat laatste is erg belangrijk in verband met gain scheduling. De gain-scheduled robuuste MV regelaar is zowel off-line (lineaire en nietlineaire simulaties en stabiliteitsanalyse) en on-line ge¨evalueerd door middel van simulaties met de piloot in de lus. De resultaten van de off-line analyse zijn bevredigend, alhoewel aanvullende afstelling nodig is om de prestaties en de stabiliteitseigenschappen verder te verbeteren. De testpiloten gaven een Cooper-Harper (CH) rating van 1 in alle vliegcondities voor de vliegtuigdynamica. Echter, door de hoge stuurkrachten die nodig waren om het vliegtuig te manoeuvreren, vooral in het vlieggebied met lage dynamische druk, vari¨eerden de uiteindelijke CH ratings tussen de 2 en 3. De benodigde stuurkracht kan eenvoudig gecorrigeerd worden door het referentiemodel aan te passen. De bijdrage van de fuzzy clustering methode om de effic¨ıentie van het ontwerp van vliegtuigregelaars te verbeteren is significant. Het is een globale techniek, alle ontwerppunten worden gelijktijdig ge¨ıdentificeerd, hetgeen resulteert in een transparant scheduling mechanisme met minder ontwerppunten. Bovendien is het een modelgebaseerde techniek die de niet-lineaire dynamica gebruikt in de ontwerpfase en niet alleen tijdens de evaluatie. In combinatie met moderne MV regelontwerptechnieken voor het ontwerpen van de lokale regelaars is de reductie in de ontwerpinspanning voor het ontwerpen van vliegtuigregelaars nog groter. Sensor Management Sensor management gebaseerd op majority voting en consolidatie van gelijksoortige signalen, dat wil zeggen signalen van verschillende sensoren die dezelfde variabele meten, is een bewezen technologie in moderne DFBW vliegtuigbesturingssystemen. De aanname die daarbij wordt gemaakt, is dat de meerderheid van de gelijksoortige signalen de waarheid representeren en dat een enkele afwijkend signaal het gevolg is van een storing. Een dergelijk signaal moet ontkoppeld worden zodra de storing gedetecteerd is. Dit principe faalt in het geval dat er nog maar twee gelijksoortige signalen beschikbaar zijn (duplex operatie), omdat er in dat geval geen sprake van een meerderheid kan zijn. In de conventionele methode is de beslissing of er wel of niet een storing is opgetreden eenduidig, met andere woorden er is w`el of geen storing. Om de gevoeligheid van deze beslissing te verminderen met het oog op onzekerheden zoals quantisatie en meetruis, wordt een zorgvuldig afgestelde

xxi

drempelwaarde gebruikt. Twee mogelijkheden zijn ge¨ıdentificeerd om de vliegveiligheid (en het vliegcomfort) te verbeteren. Ten eerste is er de introductie van een virtuele sensor die gebruik maakt van ongelijksoortige signalen (analytische redundantie) en die gebruikt kan worden als een arbitrator tijdens duplex operatie. Daarnaast is een alternatieve sensor management procedure ge¨ıntroduceerd die het geconsolideerde signaal verbetert en storingsge¨ınduceerde overgangsbewegingen verkleint. Vele toepassingen van analytische redundantie voor Fout Detectie en Isolatie (FDI) in vliegtuigbesturingssystemen zijn gerapporteerd. Meestal worden observergebaseerde technieken, parity-space methoden en parameterschattingsschema’s gebruikt. De toepassing van virtuele sensoren in de luchtvaartindustrie is nog niet goed onderzocht, alhoewel deze techniek succesvol is gebleken in andere toepassingsgebieden zoals procesbesturing en motorbesturing. De methode die is onderzocht in Hoofdstuk 6 van dit proefschrift is gebaseerd op de combinatie van white-box en black-box modellering en wordt gedemonstreerd aan de hand van het ontwerp van een virtuele invalshoek sensor. De werkwijze die hierbij is gebruikt is om eerst de maximale prestatie uit de white-box modelleringsmethode te halen, gebruik makend van bekende, lineaire vliegtuigdynamica. In de tweede stap wordt de resterende schattingsfout verkleind door een black-box model toe te voegen dat is ontworpen om de schattingsfout van het white-box model te modelleren. De ingangen van dit niet-lineaire, black-box model worden bepaald met behulp van een niet-lineaire ingangsselectie procedure. De virtuele invalshoeksensor maakt gebruik van een andere virtuele sensor die het vliegtuiggewicht en de positie van het zwaartepunt van het vliegtuig schat. Deze virtuele sensor maakt geen gebruik van het invalshoeksignaal zelf. De prestatie van de virtuele sensor is gedemonstreerd door middel van een groot aantal niet-lineaire simulaties waarvan de vliegcondities en manoeuvres voor iedere simulatie opnieuw willekeurig werden geselecteerd. De prestatie van de virtuele sensor is goed, met een maximale schattingsfout van de invalshoek van minder dan 0,8 graden. Het conventionele sensor management systeem maakt gebruik van harde drempelwaarden. Het geconsolideerde signaal wordt berekend door het (gewogen) gemiddelde te nemen van alle gelijksoortige sensor signalen. De middelste waarde wordt als referentie genomen en de afwijking van de twee uiterste signalen met betrekking tot de referentie waarde wordt gelimiteerd (harde drempelwaarde). De monitor vergelijkt elk signaal met het geconsolideerde signaal. Als het absolute verschil een bepaalde van tevoren gedefinieerde drempelwaarde overschrijdt, wordt de bijbehorende monitor teller bij elke computer cyclus verhoogd. Als de teller de failure declaration value heeft bereikt (bijvoorbeeld na tien cycli), wordt er een storingsmelding gegeven en wordt het betreffende signaal afgesloten. De posities van de drempelwaarden is een compromis tussen het willen minimaliseren van het aantal valse storingsmeldingen en het minimaliseren van de storingsge¨ınduceerde overgangsbeweging. Het eerste vereist een grote drempelwaarde, terwijl het laatste juist een kleine drempelwaarde voorschrijft. Dit compromis kan worden omzeild

xxii

Samenvatting

door vage drempelwaarden in plaats van harde drempelwaarden te gebruiken en op die manier de vliegveiligheid en/of het comfort te verhogen. Fuzzy Logic (FL) is een uitstekende techniek om dit concept te implementeren. Alhoewel FL technieken in andere gebieden veelvuldig zijn toegepast, zoals in de procesindustrie, is de toepassing ervan in de vliegtuigindustrie nog niet uitgebreid bestudeerd. In Hoofdstuk 7 van dit proefschrift is een sensor management procedure voorgesteld die is gebaseerd op soft computing. In dit zogenaamde soft sensor management systeem krijgt elk sensor signaal een gewicht tussen (en inclusief) nul en ´e´en (vage drempelwaarde) die wordt berekend met behulp van het kleinste absolute verschil met betrekking tot de andere gelijksoortige sensor signalen. Wanneer het gewicht van een sensor signaal gelijk aan nul is, wordt het betreffende signaal niet meer meegenomen bij het berekenen van het geconsolideerde signaal en wordt de betreffende monitor teller bij elke computer cyclus verhoogd. Het verschil tussen het conventionele en het soft sensor management systeem is dat bij het conventionele systeem de monitor teller wordt geactiveerd op basis van het verschil tussen het ide sensor signaal en het geconsolideerde signaal, terwijl dit bij het soft sensor management systeem gebeurt op basis van het verschil tussen het ide sensor signaal en de andere gelijksoortige sensor signalen. Het soft sensor management systeem, inclusief een virtuele normaalversnellingssensor, is gedemonstreerd door middel van closed-loop simulatie voorbeelden in het SE en door middel van vliegsimulaties met de piloot in de lus. Bij de conventionele methode worden signaal consolidatie en signaal monitoring gescheiden uitgevoerd. Met de introductie van vage drempelwaarden zijn deze twee functies ge¨ıntegreerd. Dit resulteert in een nauwkeuriger geconsolideerd signaal en reduceert de storingsge¨ınduceerde overgangsbewegingen, hetgeen niet alleen bijdraagt aan de vliegveiligheid, maar ook aan het passagierscomfort. Verder is getoond hoe virtuele sensoren gebruikt kunnen worden om foutieve sensoren te identificeren voor het geval dat er een discrepantie is tussen twee gelijksoortige sensor signalen. Als er ook een fout optreedt in de laatste sensor, is het signaal niet langer beschikbaar en wordt de vliegtuigregelaar zodanig gereconfigureerd dat het betreffende signaal niet langer nodig is. Ook de reconfiguratie van de vliegtuigregelaar verloopt soepeler ten gevolge van de soft sensor management strategie. In conclusie, de bijdrage van soft computing voor het verbeteren van de vliegveiligheid is significant. Met een virtuele sensor is het mogelijk om foutieve signalen te identificeren wanneer er een discrepantie is tussen twee signalen van twee fysieke gelijksoortige sensoren. Bovendien is het mogelijk om een fout te detecteren bij de laatste overgebleven fysieke sensor. De virtuele sensor verhoogd de betrouwbaarheid van het vliegtuigbesturingssysteem en derhalve de vliegveiligheid. Virtuele sensoren kunnen ook met andere technieken ontworpen worden, echter, met behulp van FL is het mogelijk om lineaire technieken te gebruiken voor het ontwerpen van een globaal niet-lineair systeem, hetgeen bijdraagt aan de accuratesse en betrouwbaarheid van de virtuele sensor. De bijdrage van soft sensor management voor de vliegveiligheid is minder groot dan de bijdrage van de virtuele sensor, maar het levert een verbetering van het systeem op meerprijs.

xxiii

In Hoofdstuk 8 wordt onderzoek voorgesteld in de richting van het optimaliseren van het scheduling mechanisme voor de parameters van de vliegtuigregelaar, nadat de ontwerppunten zijn ge¨ıdentificeerd met behulp van fuzzy clusteren. Verder wordt voorgesteld om de mogelijkheid van een meer ge¨ıntegreerd ontwerp van de gain-scheduled robuuste multivariabele regelaar te onderzoeken, waarbij het afstellen van de gewichten en het scheduling mechanisme worden geoptimaliseerd met behulp van een (geautomatiseerd) iteratief proces en waarbij globale stabiliteit en prestatievoorschriften worden meegenomen. De reconfiguratie van de vliegtuigregelaar bij het gebruik van robuuste multivariabele regelsystemen vereist nog veel onderzoek. Ten slotte dient meer aandacht besteed te worden aan de vraag hoe de virtuele sensor het best ge¨ıntegreerd kan worden in het (soft) sensor management systeem en hoe het beste omgegaan kan worden met de beperkingen van de virtuele sensor.

xxiv

Samenvatting

0 Notations and Abbreviations 0.1 0.1.1 c g m n ni no Nr s w

List of Notations General Parameters number of clusters gravitational acceleration fuzziness exponent number of states number of inputs number of outputs number of rules Laplace operator weight factor

[m s−2 ]

Greek symbols β ∆ ∆ µ ρ ω ζ

firing degree threshold / difference indication model uncertainty block membership degree air density natural frequency natural damping

[kg m−3 ] [rad s−1 ]

Superscripts T 0.1.2 b c c h

transposed Aircraft Related Parameters wing span rate-of-climb mean aerodynamic chord altitude

[m] [ft min−1 ] [m] [ft] 1

2

Chapter 0. Notations and Abbreviations

m M ny nz p q q r S u V VC VT w W

aircraft mass Mach number lateral acceleration normal acceleration roll rate pitch rate dynamic pressure roll rate surface area forward speed velocity calibrated airspeed true airspeed downward speed aircraft weight

[lbs] [m s−2 ] [m s−2 ] [deg s−1 ] [deg s−1 ] [mbar] [deg s−1 ] [m2 ] [knots] [knots] [knots] [knots] [m s−1 ] [lbs]

Greek symbols α β δ γ φ θ ψ χ

angle-of-attack sideslip angle (control surface) deflection flight-path angle bank angle pitch attitude heading tracking angle

Subscripts a b c e fl ph r s sl sp th v

aileron body-axes system column elevator flap phugoid motion rudder stabilator slat short-period motion throttle vehicle-carried axes system

0.2

List of Abbreviations

ACE ADC ADFCS AIM

Actuator Control Electronics Air Data Computer Affordable Digital fly-by-wire Flight Control System Aircraft Inertial Matrix

[deg] [deg] [deg] [deg] [deg] [deg] [deg] [deg]

0.2. List of Abbreviations

ANN AoA CAP CC CG CGS CH DEL DFBW DOF EFCS EGPWS EVM FAR FC FCC FCL FCS FDI FDIR FEPS FGS FHV FL GA GK GM GS HIRF HW HQR ISA JAR LC LG LMI LPV LS mac MF MOSAIC MV NLR NN PM PROSIM RFS

Artificial Neural Network Angle-of-Attack Control Anticipation Parameter Clean Configuration Center-of-Gravity Conventional Gain Scheduler Cooper-Harper Direct Electrical Link Digital Fly-By-Wire Degree-Of-Freedom Electronic Flight Control System Enhanced Ground Proximity Warning System Error Validity Measure Federal Aviation Regulations Flight Condition Flight Control Computer Flight Control Law Flight Control System Fault Detection and Isolation Fault Detection, Isolation and Reconfiguration Flight Envelope Protection System Fuzzy Gain Scheduling Fuzzy Hyper-Volume Fuzzy Logic Genetic Algorithm Gustafson-Kessel Gain Margin Gain-Scheduled High Intensity Radiated Field HardWare Handling Quality Requirement International Standard Atmosphere Joint Aviation Requirements Landing Configuration Landing Gear Linear Matrix Inequality Linear Parameter-Varying Least Squares mean aerodynamic chord Membership Function Model-Oriented Software Automatic Interface Converter MultiVariable National Aerospace Laboratory Neural Network Phase Margin Programme and Real-time Operations SIMulation Research Flight Simulator

3

4

Chapter 0. Notations and Abbreviations

RMSE ROC SCA SCAS SE SFENA SP SW SXB TS VAF VS VSTOL WCD

0.3

Root Mean-Squared Error Rate Of Climb Small Commercial Aircraft Stability and Control Augmentation System Synthetic Environment ´ Soci´et´e Fran¸caise d’Equipement pour la Navigation A´erienne Short-Period SoftWare Xie-Beni validity measure Takagi-Sugeno Variance Accounted For Virtual Sensor Very Short Take-Off and Landing Within-Cluster Distance

Coordinate Systems

Several reference systems are being used in aeronautics. One of the most important is the Earth axis system (XE , YE , ZE ), whose origin is fixed at the center of the Earth. The XE is pointing north, the YE axis is pointing east and the orthogonal triad is completed with the ZE pointing down. This reference system is primarily used to express gravitational effects, altitude, horizontal distance and the orientation of the aircraft. Furthermore, it serves and as a basic frame of reference, to which any other axis frames are referred. Also the aircraft itself must have a suitable axis system and the choice of this axis system dictates the from taken by the equations of motion. Parallel to the Earth reference system, but with the origin in the center-of-gravity of the aircraft is the vehicle-carried reference system (XV , YV , ZV ), see also Figures 0.1 to 0.3.

Figure 0.1: Side view of the SCA model. However, by using a system of axes fixed in the aircraft the inertia terms, which

0.3. Coordinate Systems

5

Figure 0.2: Top view of the SCA model. appear in the equations of motion, can be considered constant. The body axis system (XB , YB , ZB ) is such an axis system and is therefore used to govern the equations of motion. The origin of the body axis system coincides with the centerof-gravity of the aircraft and has the XB axis pointing forward out of the nose of the aircraft, the YB axis pointing out through starboard and the ZB axis pointing down (see Figure 0.1 to 0.3). In this axis system the aerodynamic forces and moments depend only upon the angles α and β, which orient the vector of the true airspeed VT in relation to the body axis system.

6

Chapter 0. Notations and Abbreviations

Figure 0.3: Front view of the SCA model.

1 Introduction In this chapter a brief description of digital fly-by-wire flight control systems is given, together with its advantages and disadvantages with respect to the mechanical flight control system. The problem statement is described and a range of potential solutions are addressed. Some of those are investigated in this thesis and are therefore discussed in more detail. The chapter concludes with an outline of the remainder of the thesis.

This chapter is organized as follows: A brief background in the history of flight control is given in Section 1.1 (see Chapter 2 for a more elaborate discussion on digital fly-by-wire flight control systems). In Section 1.2 the problems involved with the introduction of Digital Fly-By-Wire (DFBW) to small commercial aircraft are discussed. Section 1.3 describes the new developments that are currently taking place to reduce the cost of DFBW. The research aim and the motivation of the methods used in this thesis are described in Section 1.4. In Section 1.5 an overview of the thesis is given.

1.1

Background

The first powered flight took place on December 17, 1903 at Kitty Hawk, North Carolina. In the first decades of aviation the pilot controlled the aircraft directly through the application of manual force. As engine power and speeds increased, more force was needed to move the control surfaces and hydraulically-boosted control systems emerged. As the electronic era grew in the 1950’s, so did the idea of aircraft with Electronic Flight Control Systems (EFCSs). Full authority electronic flight control systems forced the issues of safety, availability and integrity to be addressed. The EFCS now had the capability to hazard the aircraft under fault or failure conditions and became subject to the airworthiness requirements. The first generation systems were based on analog technology but, more importantly, retained the hydraulic-mechanical system as a back-up. Later generations abandoned the hydraulic-mechanical back-up to become full-time fly-by-wire. The introduction of DFBW in the primary Flight Control System (FCS) was a landmark development in the aerospace industry. The possibility to decouple the 7

8

Chapter 1. Introduction

Figure 1.1: Digital fly-by-wire flight control system (Source: (Collinson 1999)). requirements for flight stability from the basic airframe configuration is especially useful for military (fighter) aircraft. The first test of a DFBW system in an aircraft took place in 1972 on a modified F-8 Crusader at the Dryden Flight Research Center. Further advantages of DFBW technology are increased functionality, safety and maintainability. These features make this technology also attractive for commercial aircraft. The Airbus 320 is the first commercial aircraft with DFBW for the primary FCS and is certified in 1988. The DFBW FCS is illustrated in Figure 1.1 in simple diagrammatic form. The core of the system is the Flight Control Computer (FCC). The output of the FCC is the commanded control surface deflection, which is computed based on the pilot command and the aircraft motion and air data sensor information. The actuator drives the control surface to the commanded position through the actuator control electronics. The corresponding aircraft response is again sensed by the motion and air data sensors and is fed back to the FCC. The DFBW FCS has many advantages over the mechanical FCS in terms of weight, fuel consumption, flexibility in the configuration of the bare airframe, etc. The most powerful feature of the DFBW FCS is the wide range of capabilities that can be programmed into the FCC. Examples are stability and control augmentation, flight envelope protection and flight control law reconfiguration. However, the design, testing and certification of such systems is costly. It is up to the manufacturer to determine which of these possible capabilities have enough added value to be included in the FCS.

1.2

Problem statement

The major drawback of DFBW is the additional cost associated with design and initial acquisition of the system compared to its mechanical counterpart. The additional cost is for a large part related to (dissimilar) hardware replication, development of the flight control system and the generation of flight safety critical software. The DFBW FCS without mechanical back-up is a safety critical system, there-

1.3. New developments

9

fore, stringent requirements are put on the integrity and availability of this system. Fault tolerance is achieved by installing redundant sensors, flight control computers, actuators and power supplies. Moreover, dissimilarity among the redundant hardware and software is used to avoid common mode failures. Flight safety critical software has to be developed and tested extensively while following strict and elaborate procedures. In order to reduce the risk of generic failures, typically the SW is developed separately for each FCC by different companies. The development of the stability and control augmentation system, sensor and actuator monitoring system, control reconfiguration system, flight envelope protection system, etc., which are all incorporated in the flight control computer, accounts for a large portion of the development cost of the aircraft. The reason for this is the increased complexity of these systems, while the design and validation methods that are used in the aerospace industry lag behind with the state-of-theart technologies. For military (fighter) aircraft and large commercial aircraft, the advantages of the DFBW FCS justify the additional cost of the system. However, this is not so obvious for small commercial aircraft. The (relative) additional cost of the DFBW FCS is much higher for small commercial aircraft, while the advantages are not so evident as for large commercial aircraft.

1.3

New developments

A radical challenge to current practices is required in order to bring DFBW FCS technologies to the small commercial aircraft market at affordable cost whilst maintaining the stringent safety requirements. The possibilities for a cost-effective application of this technology to small commercial aircraft have been investigated within the European project “Affordable Digital Fly-by-wire Flight Control Systems for Small Commercial Aircraft” (ADFCS). Partners were involved from industry, research institutes, and universities. The following topics have been identified as having potential to reduce the cost of fly-by-wire flight control systems for small commercial aircraft. Novel design techniques Classical design techniques are typically based on a single-loop iterative procedure which is labor-intensive and therefore expensive. The application of novel, more automated, design and validation techniques have great potential to reduce the design effort of, for example, the flight control laws. In this respect a well defined set of design requirements, such as the handling qualities requirements, is essential. Handling qualities requirements The military set of Handling Quality Requirements (HQRs) clearly distinguish between the handling qualities for large transport aircraft and small fighter aircraft. The civil equivalent is not in existence. There are HQRs that have been developed for large civil aircraft but these have not been validated or re-derived

10

Chapter 1. Introduction

for small commercial aircraft applications. In order to ensure the efficiency of the control law generation process it is important to have a clear and complete set of requirements against which the process can be invoked. Synthetic Environment There is an increasing trend towards the systems engineering approach to testing and validating top level conceptual designs through simulation. The purpose is to identify potential problems at an early stage such that solutions can be found prior to manufacture, avoiding time and cost penalties associated with rework later in the life-cycle. Wherever possible reliance will be placed on the use of the Synthetic Environments (SE) to design, develop, and assess options. This is considered to be extremely cost-effective in the development and integration of large, multi-component systems. Pilot monitor Although more related to safety than affordability, a relatively large number of safety-related incidents recorded for small aircraft applications are attributed to pilot error or omission. The feasibility of applying an advanced technology solution to the task of monitoring the pilot actions and the aircraft state has great potential in improving the flight safety record. Such a system could provide a warning in the event that the pilot makes an unexpected action or fails to make an expected action System integration The current practice mitigates against high levels of integration with equipment suppliers only considering that equipment which is under their direct control. Specifically, sensors for flight control purposes tend to be dedicated and no use is made of information that can be provided by other equipment such as navigation systems. This leads to additional, costly replication of equipment and also limits the scope for novel cross-equipment synergistic savings to be realized. In order to realize the potential benefits and cost savings of integrated systems, a close cooperation between equipment suppliers is essential. More integrated systems, with the corresponding fault-tolerant strategy, will allow for a leaner hardware structure without decreasing the safety of the system. For example, in current DFBW FCS each computer is dedicated to perform a specific task. This is especially true for flight safety critical applications. The challenge is to guarantee that each application can never interfere with the other applications that are running on the same computer. Also the sharing of sensor information can lead to a smaller number of hardware components. This can be achieved by introducing Fault Detection and Isolation (FDI) technologies that also use information from non-like sensors to monitor the signal of interest. Mixed actuation The DFBW FCS puts stringent requirements on the availability of electrical and hydraulic power to ensure continued control capability following loss of power generation capabilities from both engines. The ultimate fall back is to retain the mechanical control links as a reversion, but this negates one of the primary cost

1.4. Research aims and motivation of the methods used

11

saving benefits of installing a DFBW system. A more effective solution is to consider alternative power sources, removing the reliance on hydraulic power supplies. A further option, which is to be investigated, is to redistribute the primary control surface function to other non-hydraulic dependent secondary control effectors such as surface tabs.

1.4

Research aims and motivation of the methods used

The research aims are to investigate the possibilities to improve the efficiency of flight control system design and/or to improve the flight safety. The improvement of the efficiency of the FCS design results in reduced development cost, which is a key factor in making DFBW technology affordable for small commercial aircraft. On the other hand, besides the economical aspect, the improvement of flight safety is also an important argument to justify bringing DFBW to the small commercial aircraft market. In order to achieve the research aims, the focus has been put on two topics: 1. Automated design techniques. 2. Replacement of hardware redundancy by analytical redundancy. The first point is specifically of interest with respect to nonlinear flight control law design, while the second point focusses mainly on sensor management (voting/monitoring). These two application areas are discussed in more detail in the remainder of this section. 1.4.1

Automated design techniques

Gain scheduling has been perhaps the most common systematic approach to control of nonlinear systems in practice (Stengel et al. 1978, Shamma and Athans 1990, Shamma and Athans 1992, Murray-Smith and Johansen 1997). Even with the introduction of powerful control strategies such as model predictive control (Garcia et al. 1989) and feedback linearization (Isidori 1989), gain scheduling remains an attractive control strategy because of its simplicity and practical use. Applications of fuzzy logic for gain scheduling in flight control demonstrate the feasibility and flexibility of the approach to provide adequate control performance across the flight envelope (Gonsalves and Zacharias 1994, Fujimori et al. 1997, Schram 1998). The gain-scheduled controller consists of two elements, namely the scheduler and the parameters obtained from tuning the local linear controllers in the operating points. Automation of the design process for both elements and being able to combine the results, are key factors in reducing the design effort. Fuzzy clustering for gain scheduling In spite of the fact that gain scheduling has been around for a long time, almost no effort has been put in the development of a systematic approach to identify the operating points and to design the corresponding scheduler. The design of the

12

Chapter 1. Introduction

scheduler is an iterative process, where each iteration consists of the identification of the operating points, tuning of the controller parameters, design of the scheduler and evaluation of the global performance of the system. This is a slow and costly procedure due to the lack of automation. Moreover, different scheduling variables and operating points are identified for different parameters, which makes the overall system very complex and the design process difficult to manage. The more complex the system to be controlled, the more time-consuming the design of the scheduler. In this thesis it is proposed to apply fuzzy clustering for the identification of the operating points. Clustering is a technique to identify subsets in a data set. With crisp clustering each data point belongs to a single subset, while with fuzzy clustering a data point can belong to several subsets. In the latter case a membership degree is assigned to each data point with respect to all subsets. The total membership degree is equal to one. In this thesis fuzzy clustering is applied to partition the flight envelope in a number of flight regimes. The objective is to automatically identify a (small) number of operating points for which the flight control law parameters are tuned. Fuzzy clustering is used to make sure that there is a smooth transition from one operating regime to the other. The application of fuzzy clustering for the partitioning of the flight envelope automates the design procedure, which significantly reduces the number of iterations that need to be performed. Moreover, the resulting controller is more transparent, since the same scheduling variables and operating points are used for all controller parameters. Scheduled robust multivariable control For most DFBW aircraft flying today the control laws have been developed by using essentially classical single-loop frequency response and root-locus design techniques. A significant contribution in reducing the DFBW development costs, without compromising the efficiency and safety requirements, can be obtained by using advanced multivariable control design techniques for the development of control laws (Magni et al. 1997, Amato et al. 2001). The multivariable design approach has the advantage of reducing the time and cost for flight control design and refinement, and at the same time of a priori taking into account robustness with respect to model uncertainties. The challenge is to combine the resulting multivariable controllers with gain scheduling. One of the difficulties with implementing gain-scheduled multivariable control laws is the complexity of such control laws (Ly et al. 1985, Nichols et al. 1993, Hyde and Glover 1993). Multivariable control design techniques have been developed that either attempt to design a robust global controller which can operate over a wide range of the plant operation without scheduling, see for example (Perez and Nwokah 1991), or directly synthesize a gain-scheduled controller using multiple models of the plant in the control design (Ostroff 1992, Reichert 1992, Apkarian and Gahinet 1995). Due to the large operating range and model uncertainties in the case of an aircraft, a single multivariable controller will not satisfy the design requirements. Two categories of gain-scheduled robust multivariable controllers can be found

1.4. Research aims and motivation of the methods used

13

in the literature, namely scheduling of the controller output matrix and scheduling of both the controller dynamics and the controller output matrix. An example of the first category can be found in (Garg 1997). A nominal controller is designed that gives a stable closed-loop system in the entire operating range. The parameters of the output matrix are optimized such that the closedloop system at the off-design points closely matches the closed-loop system in the design point. Many examples of the second category can be found in the literature, showing a wide variety of design approaches (Nichols et al. 1993, Hyde and Glover 1993, Pellanda et al. 2000, Lin and Khammash 2001). In these approaches an attempt is made to reduce the order of the controller and/or to impose a certain structure on the controller in order to simplify the scheduling problem. The modifications that need to be made to the local linear controllers in order to improve their “schedulability” impairs the performance of the local controllers and therefore also that of the global controller. The combination of fuzzy clustering and robust multivariable control is new and potentially very effective in reducing the design effort. In this case both the identification of the operating points and the scheduler as well as the design of the local linear controllers are highly automated. Also here it is important to find a suitable method to schedule the parameters of the local linear models. 1.4.2

Replacement of hardware redundancy by analytical redundancy

Sensor management based on majority voting and point consolidation of like signals is a proven technology in modern fly-by-wire flight control systems (Rosenberg 1998). The assumption is that the majority of like signals represents the truth and that any single dissimilar signal is the result of a failure. Such a signal must be disconnected as soon as the failure is detected. This principle fails in the event that there are only two like signals left (duplex operation), since there is no longer a majority. In the conventional approach, the computation of the consolidated signal (voting) and the monitoring of the sensor signals are performed separately. In order to minimize the sensitivity of the sensor monitor to uncertainties like quantization and measurement noise, a properly adjusted (crisp) threshold is used. Two opportunities have been identified to improve the flight safety (and comfort). First of all the introduction of a virtual sensor that makes use of non-like signals (analytical redundancy) and can be used as an arbitrator during duplex and simplex operation. Furthermore, the integration of the voting and monitoring functions and the introduction of soft thresholds have the potential to improve the consolidated signal and to increase the robustness of the sensor management system with respect to uncertainties. Virtual sensors In the literature, many applications of analytical redundancy for FDI in flight control systems are reported. Most frequently applied are observer-based techniques (Menke and Maybeck 1995, Patton et al. 1989), parity-space methods (Chow

14

Chapter 1. Introduction

and Willsky 1984, Patton and Chen 1992, Gopisetty and Stengel 1998, Schram et al. 1998), and parameter-estimation schemes (Isermann 1984, Frank 1990). The use of virtual sensors in aerospace applications has not been widely investigated yet, although this technique has been successfully applied in other fields like process control and engine control (Leal et al. 1997, Hanzevack et al. 1997). In this thesis the design of a virtual angle-of-attack is described making use of fuzzy logic and neural networks. Fuzzy logic is used to compute the parameters of a Linear Parameter-Varying (LPV) model. The neural network is used to account for those dynamics that cannot be explained with the LPV model. Voting/monitoring Conventional sensor management is based on cross comparison of sensor signals. The computation of the consolidated signal and the monitoring of the signals is performed separately. In the first step the consolidated signal is computed using all the sensor signals and in the second step the monitoring of the sensor signals is performed based on the consolidated signal. This approach has the effect of a loworder filter. In a more direct approach, where the monitoring of the sensor signals is performed using the sensor signals themselves, sensor faults can be detected faster. The location of the (crisp) threshold is a compromise between minimization of false alarms and minimization of failure-induced transients. The first dictates large thresholds, while the latter dictates small thresholds. This compromise can be circumvented by introducing soft thresholds instead of crisp thresholds, increasing flight safety and/or comfort. Fuzzy logic is an excellent technique to implement this concept. Although FL techniques have been implemented in other application domains, such as the process industry (Schneider and Frank 1996, Frank and Marcu 1999), their application in flight control systems has not been extensively investigated yet.

1.5

Overview of the thesis

The outline of this thesis is as follows: In Chapter 2 several aspects of the digital fly-by-wire flight control systems are described. An historic overview of flight control systems is given after which the DFBW FCS is addressed from a functionality point of view. The latter is important in order for the reader to be able to understand the context of the applications described in this thesis with respect to the entire flight control system. The advantages and disadvantages of the DFBW FCS are described in more detail. With Chapter 3 the description of the control part of the performed research starts. In this chapter the partitioning of the flight envelope using fuzzy clustering is introduced, which serves as the basis for gain scheduling. Two applications of gain scheduling are described in this thesis for which this approach is used. A classical aircraft control example is described in Chapter 4. Here the baseline structure of the flight control laws in the SE is kept, where the gain scheduling mechanism is replaced by the scheduling system introduced in Chapter 3. A robust multivariable control problem with gain scheduling is described in Chapter 5.

1.5. Overview of the thesis

15

With Chapter 6 the sensor fault detection and isolation part of the thesis starts. In this chapter the design of a virtual sensor for angle-of-attack is described. The virtual sensor can serve either as an additional sensor or as an arbitrator in case of sensor failures. The voting/monitoring system based on soft computing is described in Chapter 7. Also the use of a virtual sensor as part of the voting/monitoring system is introduced in this chapter. In Chapter 8 the concluding remarks and recommendations for further research are discussed. Six appendices are added to provide the reader with background information on the synthetic environment and the generation of the real-time code (Appendix A), the short-period motion approximation (Appendix B), performance measures (Appendix C), soft computing techniques (Appendix D), genetic algorithms (Appendix E) and linear matrix inequalities for control (Appendix F).

16

Chapter 1. Introduction

2 Digital Fly-By-Wire Flight Control Systems In this chapter the digital fly-by-wire flight control system concept is described. The introduction of DFBW brought about an immense shift in the design philosophy for the bare airframe and the flight control system. Stability and control requirements can now be met independently of the bare airframe and the DFBW FCS enables a range of capabilities that were not possible before. This chapter provides the background information that enables the reader to understand the context in which the applications described in this thesis should be placed.

This chapter is organized as follows: An historical overview of the development of flight control systems is given in Section 2.1. In Section 2.2 the basic architecture of the digital fly-by-wire flight control system is addressed and the functionality of its modules is described. Section 2.3 discusses the advantages and disadvantages of the DFBW FCS compared to its mechanical counterpart.

2.1

Historical Overview

The first powered flight took place on December 17, 1903 at Kitty Hawk, North Carolina. The Flyer, built by Wilbur and Orville Wright, flew for 120 feet in 12 seconds, see Figure 2.1. The Wright brothers flew three more times that day, the longest flight being over 852 feet in 59 seconds. In the first decades of aviation the pilot controlled the aircraft directly through the application of manual force, moving control sticks and rudder pedals linked by cables and pushrods that pivoted control surfaces on the wings and tails. As engine power and speeds increased, more force was needed to move the control surfaces and hydraulically-boosted control systems emerged. Soon, all highperformance and large aircraft had hydraulic-mechanical flight control systems. This provided the means to include augmentation and automation systems that 17

18

Chapter 2. Digital Fly-By-Wire Flight Control Systems

Figure 2.1: Kitty Hawk, NC, December 17, 1903. Orville Wright’s famous first airplane flight. (Source: http://www.wam.umd.edu/∼stwright/WrBr/wrights/ 1903.html). provided inputs additional to those of the pilot. In these early systems safety was assured by severely restricting the control authority such that total failure could not hazard the aircraft. These flight control systems restricted designers in the configuration and design of aircraft because of the need for inherent flight stability. As the electronic era grew in the 1950’s, so did the idea of aircraft with Electronic Flight Control Systems (EFCSs). Full authority electronic flight control systems forced the issues of safety, availability and integrity to be addressed. The EFCS now had the capability to hazard the aircraft under fault or failure conditions and became subject to the safety requirements of the airworthiness authorities. The first generation systems were based on analog technology but, importantly, retained the hydraulic-mechanical system as a back-up. Later generations abandoned the hydraulic-mechanical back-up to become full-time fly-bywire. This represented a watershed achievement that to a large extent decoupled the requirements for flight stability from the basic airframe configuration (Schmitt et al. 1998, Collinson 1999). In parallel, the possibilities of digital host computing were being investigated under the expectation of increased flexibility and reduced cost by decoupling the flight control system functionality (software) from the host computer. Neither expectation was fully achieved, but current generation FCS implementations are almost universally based on digital technology. The first test of a digital fly-by-wire system in an aircraft was in 1972 on a modified F-8 Crusader at the Dryden Flight Research Center. It was the forerunner of the DFBW flight control systems now used on the space shuttles and modern military and civil aircraft to make them safer, more manoeuvrable, and more efficient. The first electrical flight control system for a civil aircraft was subcontracted by A´erospatiale to Elliot Brothers (London) Ltd (now BAE SYSTEMS) in Britain and SFENA (now Sextant Avionique) in France and was installed on Concorde. This is an analog, full-authority system for all control surfaces. The first application of electrical signalling of secondary flight controls with digital technology appeared on several civil aircraft at the start of the 1980s with the Airbus A310 program (Briere et al. 1995). These systems control the slats, flaps and spoilers.

2.2. Description of the Digital Fly-By-Wire Flight Control System

19

The Airbus A320 (certified in early 1988) is the first example of a second generation of civil fly-by-wire aircraft, rapidly followed by the A340 aircraft (certified at the end of 1992). The first DFBW aircraft of Boeing, the B777, is certified in April 1995.

2.2

Description of the Digital Fly-By-Wire Flight Control System

In this section the digital fly-by-wire flight control system is discussed in more detail. First the architecture of the flight control system is described and then the functionality of the flight control computer is addressed. Flight control system architecture The digital fly-by-wire flight control system is illustrated in Figure 1.1 in simple diagrammatic form. The core of the system is the flight control computer. The output of the FCC is the commanded control surface deflection, which is computed based on the pilot command and the aircraft motion and air data sensor information. The actuator drives the control surface to the commanded position through the actuator control electronics. The corresponding aircraft response is again sensed by the motion and air data sensors and fed back to the FCC. Safety is defined in terms of the probability of the occurrence of a catastrophic failure per flight hour. According to the airworthiness requirements for civil transport aircraft (JAR-25 and FAR-25), the probability of a catastrophic failure should be less than 10−9 per flight hour. In general, the reliability of single components (computers, sensors, actuators, etc) is not sufficient to meet these requirements. In order to be able to guarantee the integrity of the DFBW FCS, redundant components are installed in parallel (hardware redundancy) together with a fault detection and isolation system. The required level of redundancy is dependent of the reliability of the single components and the safety requirement of the corresponding system. Safety critical systems are subject to the failure probability of 10−9 per flight hour, since this corresponds to a catastrophic failure. In a triplex redundant FCS each input signal of the FCC is measured with three independent sensors, see Figure 2.2. These three signals are consolidated into one signal, the consolidated or voted signal, such that all three flight control computers receive the same input values. Through cross-comparison of these three like-signals a possible sensor failure can be detected. The failure detection scheme is based on the assumption that the probability of the failure of two like-sensors at the same time is extremely remote. The sensor with an output that is dissimilar from the output of its like-sensors, is therefore assumed to be the failed sensor. When all three sensors are working properly, the consolidated signal is, for example, the average of the three sensor signals. The three FCCs have exactly the same functionality and their outputs are subject to cross-comparison and consolidation in the same way as the sensor signals. The consolidated values of the computer outputs, the consolidated control surface angle commands, are fed to

20

Chapter 2. Digital Fly-By-Wire Flight Control Systems

Figure 2.2: Lane processing scheme of a triplex flight control system (Adopted from: (Collinson 1999)). the actuator control electronics, see also Figure 1.1. For a triplex system, the system continues to operate after the first like-sensor failure. However, in case of a second like-sensor failure, it is not possible to identify the failed sensor by crosscomparison since there is no longer a “majority”. In this case the whole system must be isolated and, if the corresponding system is essential for the safe operation of the aircraft, the second like-sensor failure could have catastrophic consequences. This implies that the probability of a second like-sensor failure should be lower or equal to the allowed probability of a catastrophic failure. The failure survival philosophy goes even further. Besides the hardware redundancy, also dissimilar redundancy is built in the flight control system in order to avoid generic or common-mode failures. When all three flight control computers are exactly the same, it is possible that they will exhibit undesirable behavior at the same time due to for example a design or coding error. In order to minimize the risk of generic faults, dissimilar software and hardware are implemented in the flight control system. It should be noted that dissimilar hardware and software will not prevent generic faults that are introduced in the design specifications themselves. Functionality of the flight control computer The number of functionalities that can be implemented in the FCC of the DFBW FCS is virtually unlimited. The three most important and/or common functionalities are: 1. Sensor Management. 2. Stability and Control Augmentation. 3. Flight Envelope Protection. The functioning of the corresponding systems will be discussed in more detail below.

2.3. Advantages and Disadvantages of Digital Fly-By-Wire

21

Sensor Management The sensor management system serves as the portal of the FCC and is essential for the safe operation of the aircraft. It has to make sure that the sensor information that is used by the aircraft systems is correct. Therefore the outputs of the (redundant) sensors are continuously checked through cross-comparison. In case of a discrepancy amongst like-sensor signals, the outlier is identified and a failure is declared on the corresponding sensor. The signals from the healthy sensors are used to compute the consolidated signals. The consolidated signals are then used to perform the other tasks of the FCC and are transmitted to all other aircraft systems that require sensor information. Stability and control augmentation With a mechanical flight control system the stability is inherently provided by the airframe or, if it is lacking or insufficient, by a dedicated Stability and Control Augmentation System (SCAS). Through the SCAS, the flight control engineer is able to modify the dynamics of the bare airframe (open-loop aircraft) such that they comply with the design and/or handling quality requirements. Flight envelope protection The Flight Envelope Protection System (FEPS) is designed such that the operational and structural limits, e.g. maximum angle-of-attack, maximum bank angle, maximum speed or Mach number, maximum load factor, etc., will not be exceeded. In case the pilot is about to exceed one of the limits, compromising the safe operation of the aircraft, the control surface deflections commanded by the SCAS are modified to prevent this from happening. This system increases the safety and allows the pilot to react rapidly and/or strongly if necessary, without having to worry about exceeding the operational/structural limits of the aircraft (care-free handling). For example, when the Enhanced Ground Proximity Warning System (EGPWS) issues a warning, the pilot should climb as fast as possible. With the FEPS the pilot can pull back the column to the maximum position without exceeding the maximum attitude and apply full throttle.

2.3

Advantages and Disadvantages of Digital Fly-By-Wire

The digital fly-by-wire flight control system has many advantages over the mechanical flight control system, both in terms of implementation as well as functionality. The disadvantages will be discussed at the end of this section. Advantages of the DFBW FCS system From the implementation point of view, the DFBW FCS has replaced the cables, pushrods, springs, bob-weights, etc. by sensors, electrical wires, and computers (Schmitt et al. 1998). This change had a revolutionary impact on the flight control system design compared to its mechanical counterpart:

22

Chapter 2. Digital Fly-By-Wire Flight Control Systems

• Wires replacing cables and pushrods give designers greater flexibility in configuration and size and placement of components such as tail surfaces and wings. • The DFBW system is smaller, more reliable, and in military aircraft the system is less vulnerable to battle damage and jamming. • DFBW permits the pilot’s command inputs (signals) to be transmitted electrically over considerable distance without distortion. • Temperature changes have little effect on electrical signaling designs, whereas mechanical systems, especially those using cables and operating over a wide temperature range, are likely to encounter problems. • The system is easy to install and maintain; it requires less adjustments or devices to ensure proper operation once installed. • A DFBW system is more responsive to pilot control inputs. The result is more efficient, safer aircraft with improved performance and design. Furthermore, a DFBW system enables easier tuning of aircraft response to pilot inputs. It therefore simplifies obtaining the same or similar handling characteristics in a family of aircraft with the same DFBW system, like in the Airbus family A320, A330 and A340, thus enabling reduced pilot training effort. From the functionality point of view the DFBW FCS offers potential capabilities that were not possible before with mechanical flight control systems: • The SCAS provides additional design options to tailor the aircraft dynamics. Besides the advantages mentioned earlier for military fighters, also commercial aircraft benefit from the use of relaxed stability airframes. Less stable means less negative lift and smaller fins and therefore less fuel consumption (at the price of increased control activity). When it is still possible for the pilot to fly the aircraft in the “conventional” way, a Direct Electrical Link (DEL) could be used as a back-up. • The FEPS increases the safety of the aircraft operations, both in the sense of correcting undesired excursions outside the normal flight envelope as well as carefree handling in emergency situations and thus obtaining maximum performance more easily and consistently. • Since the FCLs are embedded in the SW, the FCS is very flexible with respect to changing its architecture. This property can be used for reconfiguration purposes, both with respect to the loss of a signal as well as the loss of a control surface. In both cases the architecture of the FCLs can be modified to minimize the effect of the corresponding failure. In this way the effect of the loss of a signal or control surface can be minimized. Also adaptive techniques could be used to compensate for structural damage. This capability can either be used to increase the safety or to reduce the number of hardware components.

2.3. Advantages and Disadvantages of Digital Fly-By-Wire

23

The FCC allows for a number of features that are not possible in a mechanical flight control system. A major design issue is deciding which features to include, which offer the greatest return in terms of improved safety and cost efficiency, and what the design and performance requirements are in order to bound the design activity.

Disadvantages of the DFBW FCS system The most important disadvantage of the fly-by-wire flight control system is the initial cost of procurement of the system and the increased complexity. The digital fly-by-wire flight control system is completely dependent on electric and hydraulic power, which makes these systems critical for the safe operation of the aircraft. This is already the case for (large) commercial aircraft with hydraulically-boosted controls without manual reversion. Small commercial aircraft typically have manual controls or hydraulically-boosted controls with manual reversion. The introduction of a DFBW system in a small commercial aircraft would therefore mean that additional systems need to be installed in order to ensure the availability of electric and hydraulic power. The design of a mechanical flight control system is relatively simple, because of the limited tools available. For the digital fly-by-wire flight control systems the design freedom is unlimited so to speak. The flight control laws are now purely embedded in software and the only hard boundary is the structural limit of the airframe itself. It is now possible to add features to the FCLs such as pilot command shaping, flight envelope protection, etc. In order to ensure the availability of the system, redundant sensors, actuators, flight control computers, etc. are implemented and their performance is crosschecked continuously. The software of each flight control computer may have to be written by a different team and compiled with different compilers. Dissimilarity in software and hardware reduces the risk of generic failures, i.e. failures that occur in all similar redundant components at the same time. Fault detection and identification schemes need to be added to the system in order to manage all the redundant hardware. This adds to the cost of the DFBW FCS compared to the mechanical flight control system. Another disadvantage of the DFBW system is the sensitivity for High Intensity Radiated Fields (HIRF). The electrically transmitted signals could be distorted by radiation fields. In order to prevent this, the electrical wires need to be protected. Conclusion For large commercial aircraft the advantages outweigh the disadvantages. For small commercial aircraft this is not so obvious and the architecture of the FCS and the design methodologies need to be optimized in order to reach an acceptable compromise.

24

Chapter 2. Digital Fly-By-Wire Flight Control Systems

3 Fuzzy Clustering for Partitioning of the Flight Envelope The use of a linear design technique in combination with gain scheduling is the most common systematic approach to the design of nonlinear flight control laws. However, the selection of the operating points and the design of the interpolation scheme remains a time-consuming procedure. In order to reduce the design effort, an automated procedure has been developed and applied to the design of a longitudinal flight control law in a fly-by-wire flight control system. The number of operating points and their locations are determined automatically by using fuzzy clustering on the basis of the changes in the aerodynamic characteristics over the flight envelope. This approach also directly provides the interpolation mechanism (membership functions) for the local flight control law parameters.

This chapter is organized as follows: A brief introduction into nonlinear control is given in Section 3.1. In Section 3.2 the basics of fuzzy clustering are discussed (see Appendix D for a more detailed description on fuzzy clustering and related subjects). The partitioning of the flight envelope using fuzzy clustering is described in Section 3.3. Concluding remarks are given in Section 3.4.

3.1

Nonlinear Control

The general function of the flight control laws in the fly-by-wire flight control system is to improve the handling qualities of the bare aircraft, in particular with respect to stability, control and flight envelope protection (Tischler 1996). The design of the flight control laws is a nonlinear control problem due to the nonlinear aircraft dynamics, which also vary with the flight condition and aircraft configuration. As a single linear controller cannot meet the design requirements over the entire operating range, there are two principle solutions to this nonlinear control problem: 1. Nonlinear design techniques, such as feedback linearization (Isidori 1989), 25

26

Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

provide a single nonlinear controller that meets the design requirements over the entire operating range. The application of nonlinear design techniques is often complicated due to the restrictive conditions that have to be met by the system to be controlled. In many cases these conditions are too restrictive and the corresponding nonlinear design technique cannot be applied at all. Moreover, stability and robustness analysis still requires the use of linear techniques. 2. The use of a linear design technique in combination with gain scheduling is perhaps the most common systematic approach to the control of nonlinear systems (Stengel et al. 1978, Shamma and Athans 1990, Shamma and Athans 1992, Murray-Smith and Johansen 1997). Linear controllers are designed for a number of design points and their parameters are subject to interpolation for the flight conditions in between the design points. In this way a global nonlinear controller is obtained. Despite recent advances in nonlinear control, gain scheduling remains an attractive control strategy because of its simplicity and practical usefulness. In this thesis gain scheduling is considered. The scheduler design problem consists of two tasks: selection of the scheduling variables and design of the parameter scheduling algorithm (Stengel et al. 1978). Scheduling variables are usually selected on the basis of the following two heuristics (Shamma and Athans 1990, Shamma and Athans 1992): • The scheduling variables should capture the nonlinearities of the system to be controlled. • The scheduling variables should be slowly time-varying compared to the desired bandwidth of the closed-loop system. Although many tools are available for the design of the local linear controllers, the selection of the operating points and the design of the scheduling mechanism are less straightforward. Typically, it results from an iterative procedure based on past experience of the control engineer. During each iteration step, the following tasks are performed: 1. Selection of a set of operating points. 2. Tuning of the flight control law parameters. 3. Design of the scheduler. 4. Validation by linear analysis and nonlinear simulation. This process, which is completed when the closed-loop system dynamics are satisfactory over the entire operating range, is time-consuming and often leads to more operating points than necessary. In order to reduce the design effort, an automated procedure for the design of gain scheduled controllers has been developed using fuzzy logic techniques. The

3.2. Fuzzy Clustering

27

potential of fuzzy gain-scheduled flight control has been demonstrated in a number of articles, see (Fujimori et al. 1997, Gonsalves and Zacharias 1994, Schram 1998) among others. In these works however, the operating points and/or membership functions are selected quite arbitrarily. The approach proposed in this chapter goes one step further: the number of operating points and their locations are determined automatically by using fuzzy clustering on the basis of changes in the system dynamics over the operating range.

3.2

Fuzzy Clustering

An effective approach to the identification of complex nonlinear systems is to partition the available data into subsets and approximate each subset by a linear model. Fuzzy clustering can be used as a tool to obtain a partition of data where the transitions between the subsets are gradual rather than abrupt. Fuzzy partitions can be seen as a generalization of hard partitions, which is formulated in terms of classical subsets.

3.2.1

Clustering algorithms

Clustering techniques are unsupervised methods that can be used to organize data into groups based on similarities among the individual data items. Most clustering algorithms do not rely on assumptions common to conventional statistical methods, such as the underlying statistical distribution of data, and therefore they are useful in situations where little prior knowledge exists. A large family of fuzzy clustering algorithms is based on the minimization of an objective function J formulated as:

J(Z; U, V, Ai ) =

c
N


2 (µik )m Dik , A i

(3.1)

i=1 k=1

where Z is the n × N data matrix, U is the c × N partition matrix, and V the n × c matrix of cluster prototypes (centers). The number of variables in the data set is denoted by n, N is the number of data samples, and c is the number of clusters. DikAi is the distance measure, which is computed as follows: 2 Dik = (zk − vi )T Ai (zk − vi ), A i

(3.2)

where zk is the kth data point and vi the ith cluster center. The n × n norminducing matrix of the ith cluster is denoted by Ai , µik is the membership degree of the kth data point with respect to the ith cluster and m is the ‘fuzziness’ exponent. Iteratively solving the minimization problem results in c operating points. Each operating point represents the center of an operating regime in which the data are close to each other with respect to the objective function.

28

3.2.2

Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

Tunable parameters and validation measures

The clustering parameters influence the clustering results. The two basic parameters that need to be tuned are the number of clusters c and the fuzziness exponent m. Number of clusters In order to determine the optimal number of clusters, the fuzzy clustering results are evaluated for different values of c. Clustering algorithms generally aim at locating well-separated and compact clusters. When the number of clusters is chosen equal to the number of groups that actually exist in the data, it can be expected that the clustering algorithm will identify them correctly. In (Gath and Geva 1989), it is suggested to assess the goodness of the obtained partition by evaluating the separation between the clusters, the volume of the clusters, and the number of data points concentrated in the vicinity of the cluster prototype. There are many cluster validity measures and in this case we use the Fuzzy Hyper-Volume (FHV) (Gath and Geva 1989), the Within-Cluster Distance (WCD) (Krishnapuram 1994), and the Xie-Beni validity measure (SXB) (Xie and Beni 1991). Good partitions are indicated by small values of all three validity measures. Fuzziness exponent The fuzziness exponent influences the fuzziness of the resulting partition. As m approaches one from above, the partition becomes hard rather than fuzzy. The higher the fuzziness exponent, the fuzzier the partition, and hence more overlap between the clusters. Usually, m = 2 is initially chosen.

3.3

Fuzzy Partitioning of the Flight Envelope

Few examples of an automated approach to the identification of operating points can be found in literature. In (Garduno-Ramirez and Lee 2000) the eigenvalue with the largest magnitude variation is used to determine the operating points. Each partition covers 20% of the range of the corresponding eigenvalue. Two articles have been found that describe an automated procedure for the design of a gainscheduled controller (Wada and Osuka 1997, McNichols and Fadali 2003). Here it concerns an iterative procedure of scheduling, controller design and closed-loop evaluation. The fuzzy partitioning of the flight envelope for gain-scheduled flight control using fuzzy clustering is described step by step in this section. An example of the design of a gain-scheduled state feedback controller is given to illustrate the design procedure.

3.3. Fuzzy Partitioning of the Flight Envelope

29

45 40 35 M

MAX

Altitude [ft]

30 25

= 0.85

VC = 150 Kts

20 15

VC = 375 Kts

10 5 0 0.2

0.3

0.4

0.5 0.6 0.7 Mach number [−]

0.8

0.9

Figure 3.1: The operating range of the SCA in clean configuration, defined as a function of Mach number, altitude and calibrated airspeed. 3.3.1

Generation of the data set

The data set used for fuzzy clustering must capture those aircraft dynamics that are important from the control point of view. Figure 3.1 illustrates the flight envelope of the Small Commercial Aircraft (SCA) model in clear configuration, which is defined in terms of Mach number, altitude and calibrated airspeed. The SCA model is modified from the model of an existing business jet aircraft and is implemented in the SE that is used in the ADFCS project, see Appendix A. When considering the longitudinal controller, the phugoid motion dynamics are of minor importance. This has long been recognized, and trivial or no specifications are placed on the phugoid mode in handling quality requirements. However, the Short-Period (SP) motion dynamics are of major importance. It is their deficiencies or those of their equivalents in DFBW aircraft, that have caused many difficulties and even accidents (Gibson 1999). The linearized model for longitudinal motion is described as follows (see also Appendix A):        u˙ Xu Xw Xq Xθ Xδe u w˙   Zu Zw Zq Zθ  w  Zδe       = (3.3) ˜u M ˜w M ˜q M ˜ θ   q  + M ˜ δ  δe ,  q˙  M e θ 0 0 1 0 0 θ˙ where u denotes the forward speed, w denotes the downward speed, q denotes the pitch rate, θ denotes the pitch attitude and δe denotes the elevator deflection. For the short-period approximation, only the downward velocity w and the pitch rate q are considered, while the forward speed is assumed to be constant (u˙ = 0):     Zδe Zw Zq w w˙ (3.4) + = ˜ ˜q q ˜ δe δe . q˙ Mw M M

30

Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

27 26 Altitude [kft]

25

Mδ

e

−0.045

24

−0.05

23

−0.055 −0.5

22

−2 −0.6

21 0.44

0.46 0.48 Mach number [−]

0.5

(a) Grid points as a function of Mach number and altitude.

M

q

−3 −4

M

α

˜q M ˜ δ ] cor˜α M (b) Data points zk = [M e responding to the grid points.

Figure 3.2: Construction of the data set. For the grid points in the 2-dimensional space defined by Mach number and altitude (left), points in the 3-dimensional space ˜ q and M ˜ δ are obtained via linearization of the nonlinear aircraft ˜ α, M defined by M e model (right). The aim of fuzzy clustering is to identify regions in the flight envelope where the short-period motion dynamics can be approximated by a single linear model. Therefore, the variables selected for the data set should contain information with respect to the short-period motion dynamics. Example The most important aerodynamic derivatives with respect to the short˜ q , and M ˜ δ (McLean 1990), where α denotes the ˜ α, M period motion are M e ˜ w , where U0 is the ˜ α = U0 M angle-of-attack. It should be noted that M forward speed in the corresponding trimmed condition. The data set is defined as: Z = {zk |k = 1, 2, ..., N }, where N is the number of data samples. To generate the data, a grid is defined in the operating range with steps of 0.01 in Mach number and steps of 1000 ft in altitude. At each grid point the nonlinear aircraft model is ˜ α, trimmed and linearized and subsequently the aerodynamic derivatives M ˜ ˜ Mq , and Mδe are obtained from the linear model (see also Figure 3.2): ˜ α (Mk , hk ) M ˜ q (Mk , hk ) M ˜ δ (Mk , hk )]T , zk = [M e where (Mk , hk ) denotes the kth grid point in terms of Mach number and altitude, respectively. This data set is generated automatically and the total number of data samples is equal to N = 1805.
In the example only the aerodynamic derivatives are used for clustering. One can also choose to add variables that describe the corresponding flight condition, for

3.3. Fuzzy Partitioning of the Flight Envelope

31

example Mach number (M ) and angle-of-attack (α), to the data set, in which case the data set would be defined as: ˜ α (Mk , hk ) M ˜ q (Mk , hk ) M ˜ δ (Mk , hk )]T . zk = [Mk hk M e

(3.5)

Explicitly including such variables increases the probability of obtaining convex clusters with respect to the flight condition. If there are for example two operating regimes with more or less the same aerodynamic derivatives but well separated in terms of Mach number, two separate clusters will be identified. In case Mach number is not part of the data set, it is likely that one cluster is identified that results in a non-convex Membership Function (MF) as a function of Mach number. However, the fuzzy partition is no longer solely based on the aircraft dynamics. Convex clusters can always be obtained by sufficiently increasing the number of clusters to be identified. 3.3.2

Fuzzy clustering

In this case, the objective of fuzzy clustering is to partition the available data into subsets and to approximate each subset by a single linear model as given in Equation 3.4. The rationale being that regimes for which the aircraft dynamics can be approximated by a single linear model, are also suitable to control the aircraft using a single linear controller. The most relevant elements of the linear model, i.e. the most relevant aerodynamic derivatives, are used for fuzzy clustering.

3.3.3

The number of clusters and fuzziness exponent

Besides the standard validity measures FHV, WCD and SXB, a fourth validity measure is introduced in order to find the suitable number of clusters. This is the Error Validity Measure (EVM), which is a modelling performance measure. To evaluate the EVM, the fuzzy partition is used to construct a singleton TakagiSugeno fuzzy model (see Appendix D) of each of the aerodynamic derivatives in the data set as a function of the scheduling variables: Ri :

If Z1 is Z1i and Z2 is Z2i then M = Mi

i = 1, . . . , Nr ,

(3.6)

where in this case Z1 and Z2 denote the scheduling variables, Z1i and Z2i denote the corresponding membership functions, Nr denotes the number of rules and Mi denotes the vector of aerodynamic derivatives. The degree of fulfillment of the ith rule is computed by the product of the membership degrees of each statement in that rule: (3.7) βi = µZ1i µZ2i . The weight wi of each rule Ri is computed as follows: βi wi = Nr j=1

βj

.

(3.8)

32

Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

5000 WCD

FHV

0.05 0.04 0.03 2

4000 3000 2000

4 6 8 10 12 Number of clusters

2

4 6 8 10 12 Number of clusters 4 x 10

3

2

ERR

SXB

2.5

1.5

1

1 2

2

4 6 8 10 12 Number of clusters

2

4 6 8 10 12 Number of clusters

Figure 3.3: Performance of the validity measures as a function of the number of clusters. ˆ of the singleton model is then: The output M ˆ = M

Nr


wi Mi .

(3.9)

i=1

The reason to use a singleton model is that eventually we are going to use the TS fuzzy model for gain scheduling. The resulting scheduler is in fact a singleton model and therefore it is decided to use the singleton model for the error validity measure as well. The error validity measure is evaluated by summation of the squared difference between the outputs of the TS fuzzy models at each grid point ˆ k ) and their corresponding true values in the data set (Mk ): (M  N 1
ˆ k )2 . (Mk − M (3.10) EVM = N k=1

The validity measures are used to determine the number of clusters in the partition. When the number of clusters is fixed, the fuzzy partition is evaluated for different values of the fuzziness exponent m. The choice of the fuzziness exponent is made after visual inspection of the partition for different values of m. Of most importance is that the clusters are convex in terms of the potential scheduling variables, e.g. Mach number and altitude. Example, continued In this example the Gustafson-Kessel (GK) clustering algorithm is used. The advantage of this algorithm is that the norm-inducing matrices Ai are subject to the optimization. This clustering algorithm is able to identify clusters of different shape and orientation, and is therefore more likely

3.3. Fuzzy Partitioning of the Flight Envelope

33

to discover the true partition of the data set. The fuzzy c-means algorithm for example is limited to identifying circular clusters of equal volume (Bezdek 1980), which makes the algorithm less flexible in correctly identifying the clusters that are present in the data set. In Figure 3.3 the validation measures are illustrated as a function of the number of clusters. Good performance is indicated by small values of all four validity measures. In this case both the Xie-Beni validity measure and the modelling error show a local minimum for c = 8, while both the fuzzy hyper-volume and within-cluster distance show a local minimum for c = 9. However, the rate of descent of the within-cluster distance has significantly decreased for c > 8. The fuzzy hyper-volume shows a similar behavior. Moreover, the local minimum of the Xie-Beni validity measure is most pronounced compared to those of the other three validity measures and the number of clusters is therefore set to c = 8. It should be noted that in all these cases the fuzziness exponent was set to m = 2. While keeping the number of clusters fixed to c = 8, the fuzzy partition is evaluated for different values of the fuzziness exponent m. Finally m = 1.8 is selected to be used here.


3.3.4

Partition and membership functions

The resulting fuzzy partition is defined by the fuzzy partition matrix U , see also Appendix D.1. In the fuzzy partition matrix each data point is assigned a membership degree with respect to each cluster. The sum of the membership degrees per data point is equal to one. In order to be able to construct the scheduler as a TS fuzzy model, the multidimensional fuzzy partition is approximated by the Cartesian product of onedimensional membership functions. This is obtained through the following procedure: Selection of the scheduling variables. The fuzzy partition is approximated by the membership functions and the product operator. Taking into account that the membership functions are obtained by orthogonal projection of the clusters onto the axes of the scheduling variables, the approximation of the fuzzy partition is most accurate when the orientation of the fuzzy clusters are aligned with the scheduling variables. Projection of the clusters onto the axes of the scheduling variables. After selecting the scheduling variables, the orthogonal projections of the clusters onto each of their axes are obtained.

34

Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

(a) Mach number vs altitude.

(b) Mach number vs dynamic pressure.

Figure 3.4: Fuzzy clustering result for c = 8 and m = 1.8. Approximation of the projections. The membership functions are obtained by fitting the orthogonal projections with a parametric membership function. Examples of shapes of membership functions are triangular, trapezoidal and sigmoidal. A posteriori modification of the membership functions. It is necessary to modify the membership functions such that each operating point fully and exclusively belongs to its corresponding cluster. In this way it is made sure that in each operating point the scheduler reproduces those FCL parameters as tuned for the corresponding operating point. The membership degree of each operating point is thus equal to one for its corresponding cluster and zero for all other clusters. Finally the kernel of each membership function is reduced to the corresponding operating point only. Example, continued In Figure 3.4 the maximum membership degree for each data point is plotted. Light areas indicate a maximum membership degree close to one, while dark areas indicate lower maximum membership degree. The latter areas indicate those regimes that belong to more than one cluster. Since each data point is obtained by linearizing the nonlinear aircraft model, their corresponding Mach number and altitude or dynamic pressure can be used to construct Figures 3.4a and 3.4b, respectively. In Figure 3.4 it can be seen that the orientation of the fuzzy clusters is diagonal with respect to Mach number and altitude and (more or less) orthogonal with respect to Mach number and dynamic pressure. The approximation of the fuzzy partition is most accurate when the orientation of the fuzzy clusters are aligned with the scheduling variables. This makes

Membership degree

3.3. Fuzzy Partitioning of the Flight Envelope

35

MN

1

5

DP5 0.5

0 250 200

1 150

0.8 0.6

100 0.4

50

Dynamic pressure [mbar]

Mach number [−]

0.2

0.5

0

0.3

0.4

0.5 0.6 Mach number [−]

0.7

0.8

1

0.5

0

50

100 150 200 Dynamic pressure [mbar]

(a) Membership functions.

250

Membership degree

1

Membership degree

Membership degree

Membership degree

Figure 3.5: Projection of the fuzzy clusters onto the axes of the scheduling variables (dashed-dotted) and their corresponding approximations by sigmoidal membership functions (continuous). 1

0.5

0

0.3

0.4

0.5 0.6 0.7 Mach number [−]

0.8

1

0.5

0

50

100 150 200 Dynamic pressure [mbar]

250

(b) Modified membership functions.

Figure 3.6: Membership functions.

Mach number and dynamic pressure the most suitable scheduling variables. An example of the orthogonal projection of the clusters onto the axes of the scheduling variables is denoted in Figure 3.5 with the dash-dotted lines. For example, the dash-dotted line as a function of Mach number is constructed by evaluating the maximum membership degree of all dynamic pressures for each Mach number. In Figure 3.5 the sigmoidal membership functions are denoted with the continuous line. There are eight membership functions for each scheduling variable resulting from the projection of the eight fuzzy clusters, see Figure 3.5a.

36

Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

Figure 3.7: Decomposition of the flight envelope into eight operating regimes. Table 3.1: The operating points obtained through fuzzy clustering. FC Mach Altitude Dynamic number [kft] pressure [-] [mbar] 1 0.83 38.0 118 2 0.59 32.3 72 3 0.65 11.0 220 4 0.39 5.9 90 5 0.33 15.4 44 6 0.59 23.2 108 7 0.55 9.8 160 8 0.79 25.8 185

The modified membership functions are illustrated in Figure 3.5b. The partition resulting from the membership functions is illustrated in Figure 3.7. The corresponding operating points are given in Table 3.1.


3.3.5

The local controller design

When the singleton model of Equation 3.6 is established, the rule-base and membership functions can be used for gain scheduling. Each membership function along Mach number is connected to one membership function along dynamic pressure, see also Figure 3.5, which results in a so-called diagonal rule-base. The local linear controller parameters are tuned in the operating points and putting these param-

3.3. Fuzzy Partitioning of the Flight Envelope

37

260 240 Dynamic pressure [mbar]

220 200 8

180

13

7,12

160

9

140 120 10

100

1 11

6

80 60 40 0.3

0.4

0.5 0.6 0.7 Mach number [−]

0.8

0.9

Figure 3.8: Test flight regime. The gray dots denote the test flight conditions. eters in the singleton TS fuzzy model results in the global nonlinear controller: Ri : If Z1 is Z1i and Z2 is Z2i then K = Ki ,

i = 1, . . . , Nr ,

(3.11)

where in this case Z1 denotes Mach number, Z2 denotes dynamic pressure and Ki denotes the local controller. The fuzzy gain scheduling approach does not put any restrictions on the structure of the linear (flight) control laws. In Chapters 4 and 5 the classical and the robust multivariable control techniques are applied, respectively. Example, continued The flight regime for which the gain-scheduled state feedback controller will be designed is enclosed by the four operating points 1, 6, 7 and 8, see Figure 3.8. The set of grid points that are within the flight regime of interest are considered. From this set, only those grid points that have a nonzero membership degree corresponding to one or more of the four operating points 1, 6, 7 and 8 are selected. The grid points that have a nonzero membership degree corresponding to one of the other operating points are removed from the set. The resulting set of Flight Conditions (FCs) is denoted by gray dots in Figure 3.8. These flight conditions will be referred to as the test flight conditions. A fixed controller and a second gain-scheduled controller are designed for comparison. The fixed controller is designed in FC9 and the operating points for the second gain-scheduled controller are FC10 to FC13. These operating points are chosen arbitrarily and the corresponding membership functions are determined accordingly. The details on the design flight conditions are given in Table 3.2. In each of the nine design flight conditions a state feedback controller is de-

Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

4

4

3

3

2

2 Imaginary axis

Imaginary axis

38

1 0 −1

1 0 −1

−2

−2

−3

−3

−4 −4

−3

−2 Real axis

−1

0

−4 −4

−3

(a) Fixed controller.

−2 Real axis

−1

0

(b) FGS controller.

Figure 3.9: Closed-loop poles in the test flight conditions. The black x-marks denote the closed-loop poles in the test flight conditions. The gray +-marks denote the desired closed-loop poles. Table 3.2: Design points and state feedback controller gains. FC

1 6 7 8 9 10 11 12 13

Mach nr. [-] 0.83 0.59 0.55 0.79 0.70 0.55 0.83 0.55 0.83

Alt. [kft] 38.0 23.2 9.8 25.8 25.9 19.6 39.8 9.8 31.6

Dyn. press. [mbar] 118 108 160 185 140 108 108 160 160

K

[ 0.251 [ 0.084 [ 0.030 [ 0.017 [ 0.099 [ 0.066 [ 0.281 [ 0.030 [ 0.085

−0.124 −0.008 −0.194 −0.344 −0.149 −0.008 −0.085 −0.194 −0.379

−75.10 −57.86 −30.79 −35.74 −46.58 −55.04 −83.76 −30.79 −48.14

−53.26 ] −36.13 ] −15.69 ] −17.23 ] −26.25 ] −33.78 ] −63.07 ] −15.69 ] −24.51 ]

 2 signed such that the closed-loop poles are p1,2 = −ωsp ζsp ± j ωsp 1 − ζsp  −1 2 , where ζ and p3,4 = −ωph ζph ± j ωph 1 − ζph , sp = 0.8, ωsp = 3.6 rad s ζph = 0.9 and ωph = 0.1 rad s−1 . The subscripts sp and ph denote the shortperiod and phugoid motion, respectively. The resulting controllers are given in Table 3.2. The fuzzy gain-scheduled state feedback controller for which the operating points are identified through fuzzy clustering is validated in the test flight conditions and compared with the fixed state feedback controller designed

3.3. Fuzzy Partitioning of the Flight Envelope

39

0.3

0.15

K2

K1

0.2

0.1

0.2 0.1

0.05 0 180

180 160

160

0.8 140

Dynamic pressur [mbar]

0.6

Mach number [−] Dynamic pressur [mbar]

0.7 120

0.6

Mach number [−]

−20

−40

−30

−50

K4

K3

0.8 140

0.7 120

−60

−40

−70

−50

180

180 160

0.8 140

Dynamic pressur [mbar]

0.7 120

0.6

160

0.8 140

Mach number [−] Dynamic pressur [mbar]

0.7 120

0.6

Mach number [−]

Figure 3.10: The four gains of the scheduled state feedback matrix K = [K1 K2 K3 K4 ] as a function of Mach number and dynamic pressure. in FC9. In Figure 3.9 the closed-loop poles in the test flight conditions are illustrated, both using the fixed (Fig 3.9a) as well as the fuzzy gainscheduled state feedback controller (Fig 3.9b). The spread of the closed-loop poles using the fixed state feedback controller is much larger than of the closed-loop using the scheduled state feedback controller. It should be noted that using the fuzzy gain-scheduled state feedback controller, the damping of the closed-loop poles is in most test flight conditions better (higher) than in the design flight conditions. The reason for this phenomenon is not clear, but it cannot be expected that this is generally the case. For the fuzzy gain-scheduled state feedback controllers, the rule-base to compute the state feedback matrix as a function of Mach number and dynamic pressure becomes as in Equation 3.11 for Nr = 4. The state feedback matrix is computed in the same way as illustrated by Equations 3.7 to 3.9. The four gains of the state feedback matrix are illustrated in Figure 3.10 as a function of Mach number and dynamic pressure. In Figure 3.11 the time histories of the pitch rate resulting from the initial condition response of the closed-loop system are illustrated. The light gray area denotes the simulations using the fixed controller. The black area denotes the simulations using the gain-scheduled controller which uses the flight conditions FC10 to FC13 as operating points. The dark gray area denotes the simulations using the gain-scheduled controller for which the

40

Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

1.2 Fixed controller Scheduled Controller (square) Scheduled controller (fuzzy)

−1

Pitch rate [deg ⋅ sec ]

1 0.8 0.6 0.4 0.2 0 −0.2 0

1

2

3

4

5

Time [s]

Figure 3.11: Time histories of the initial condition response in pitch rate of the closed-loop system. The gray area denotes the simulations using the fixed controller, the black area denotes the simulations using the scheduled controller and the dark gray area denotes the simulations using the scheduled controller for which the operating points are identified through fuzzy clustering. operating points are identified through fuzzy clustering. As expected, the simulations using the scheduled state feedback controllers are much more consistent than the simulations using the fixed state feedback controller. Moreover, the scheduled state feedback controller using the operating points obtained through fuzzy clustering outperforms the scheduled state feedback controller with manual selected operating points. The difference between the two gain-scheduled controllers is not big, which is partly due to the fact that the arbitrarily selected operating points are close to the operating points identified through fuzzy clustering. 

3.4

Conclusions

Fuzzy clustering is applied to a data set that is selected based on the flight control law design objectives and the handling quality requirements. Subsequently the parameters of the clustering algorithm are selected and the obtained fuzzy partition is used to determine the operating points and membership functions. The scheduling variables are selected based on the orientation of the fuzzy clusters. A simple example demonstrates the effectiveness of the proposed methodology. It can be concluded that the scheduled state feedback controller using the operating points identified through fuzzy clustering slightly outperforms the scheduled state feedback controller with arbitrarily selected operating points. However, the main objective of the automated approach based on fuzzy clustering is not to improve

3.4. Conclusions

41

the performance, but is focussed towards reducing the design effort with respect to locating the operating points. The intended reduction of the design effort can be achieved by the automated identification of the operating points, circumventing a time-consuming, iterative procedure for the identification of the operating points and the tuning of the FCL parameters. Moreover, this model-based approach is likely to result in fewer operating points, which further reduces the design effort. As can be seen in Figure 3.7, three regimes can be found as a function of Mach number, namely a regime for M  0.46, a regime for 0.46  M  0.70 and a regime for M  0.70. It turns out that the aerodynamic model is structured in exactly the same way. All the operating points are more or less in the center of each of these regimes. When an operating point is placed on, or close to, the boundary between two regimes, the corresponding operating regime would be relatively small because of the nonlinear dynamics in these parts of the flight envelope and would therefore result in an unnecessary high number of operating points. This can be prevented by using a model-based approach to identify the operating points. The transparency of the resulting scheduler, using the same operating points and scheduling variables for all longitudinal FCL parameters, makes it easier to predict the effect of small adjustments made to the FCL parameters in the later stages of the design process. This will also contribute to the reduction of the design effort.

42

Chapter 3. Fuzzy Clustering for Partitioning of the Flight Envelope

4 Scheduled Classical Control The problem addressed in this chapter is the design of a parameter scheduling scheme for the longitudinal stability and control augmentation system of the SCA model. The baseline flight control laws that are available within the synthetic environment are used to illustrate the fuzzy gain scheduling approach. The six most relevant gains of the longitudinal controller are scheduled using this approach, while for all the other gains, including those of the lateral controller, the baseline scheduling scheme is preserved.

This chapter is organized as follows: In Section 4.1 the longitudinal stability and control augmentation system is described and the gain scheduling problem is addressed. The partitioning of the flight envelope in operating regimes is discussed in Section 4.2. Section 4.3 describes the design procedure that is used to tune the local flight control law parameters. The resulting rule-base and implementation of the fuzzy gain scheduled controller is discussed in Section 4.4. In Section 4.5 the validation of the gain-scheduled flight control laws is described, which includes both linear analysis as well as pilot-in-the-loop simulation of the complete nonlinear aircraft model. Concluding remarks are given in Section 4.6. In Appendix A a brief description the longitudinal equations of motion and the longitudinal aerodynamic model of the SCA model is given.

4.1

Stability and Control Augmentation System

The simplified structure of the “classical” longitudinal FCLs of the SCA model is illustrated in Figure 4.1. Refer to Appendix A for a brief description of the longitudinal flight dynamics of the SCA model. The functionality of the longitudinal FCLs can be divided into the following parts: Pitch damper. Proportional feedback of the pitch rate q is the most basic approach for augmentation of Cmq , which is the aerodynamic coefficient that represents the change in the moment around the Y-axis m due to a change in q. This aerodynamic coefficient is directly related to the natural short-period damping ζsp of the aircraft. The ideal value for the gain GQ is given by the 43

44

Chapter 4. Scheduled Classical Control

Figure 4.1: Simplified representation of the longitudinal FCLs. desired augmentation of Cmq , i.e. ∆Cmq , and the elevator effectiveness Cmδ : GQ =

∆Cmq , Cmδ

(4.1)

where Cmδ is the aerodynamic coefficient that represents the change in the moment around the Y-axis m due to a change in δe . Feedback path. Proportional feedback of the angle-of-attack α is the most basic approach for augmentation of Cmα , which is the aerodynamic coefficient that represents the change in the moment around the Y-axis m due to a change in α. This aerodynamic coefficient is directly related to stiffness and the frequency of the short-period motion ωsp . The ideal feedback gain Gα is given by the desired augmentation of Cmα , i.e. ∆Cmα , and the elevator effectiveness: ∆Cmα . (4.2) Gα = Cmδ Since the AoA signal is not available for control, the normal acceleration and pitch rate are used with appropriate scheduling. The short-period approximation transfer function from normal acceleration to angle-of-attack α, ignoring the effect of the elevator deflection, is described by the following expression: α mg = 1 2 , (4.3) nz 2 ρVT SCNα

4.1. Stability and Control Augmentation System

45

Figure 4.2: Definition of the membership degrees µδc and µVC . Left: µδc as a function of the absolute value of the column deflection |δc | in degrees. Right: µVC as a function of the calibrated airspeed VC in knots. where m is the mass of the aircraft, g the gravity acceleration, ρ the air density, VT the true airspeed, S the wing reference area, and CNα is the aerodynamic coefficient that represents the change in normal force N due to a change in AoA. The short-period approximation transfer function from pitch rate to normal acceleration, ignoring the effect of the elevator deflection, is described by the following expression: VT 1 nz = , q g τs + 1

(4.4)

U0 where τ = N in the trimmed condition. The latter equation is reconstructed α in the feedback path illustrated in Figure 4.1. The derivation of the relations given in Equations 4.3 and 4.4 is given in Appendix B.2.

The normal acceleration that is fed back is a blending of the normal acceleration measurement and the normal acceleration signal obtained through the pitch rate measurement (see Equation 4.4). The blending is a function of the column deflection δc and the calibrated airspeed VC :   VT 1 nz,f b = w nz + (1 − w) q, (4.5) g τs + 1 where nz,f b denotes the resulting feedback signal. The weight w is computed as follows: (4.6) w = µδc µVC , where µδc is the membership degree computed from the column deflection and µVC is the membership degree computed from the calibrated airspeed, see also Figure 4.2. With the column centered (w = 0), only the pitch rate signal is contributing to the feedback signal. The same holds for large stick deflections at low calibrated airspeed. The normal acceleration signal is contributing to the feedback signal for large stick deflections at medium and high calibrated airspeed (w > 0). The ideal proportional gain is obtained as follows: mg , KP = Gα 1 2 2 ρVT SCNα

(4.7)

46

Chapter 4. Scheduled Classical Control

where the ideal value for Gα is given in Equation 4.2. The integral action KI s is added in the feedback path in order to reduce the steady-state error of the achieved pitch rate compared to the commanded pitch rate and to increase the robustness of the controller with respect to the performance in off-design flight conditions and disturbances. Feedforward path. The feedforward path feeds the shaped pilot command, multiplied by the feedforward gain, directly to the elevator. This augments the control of the aircraft without compromising its stability. The shortperiod approximation transfer function from the elevator deflection to the normal acceleration nz is: nz (s) VT Kq = . s 2 δe (s) g ( ωsp ) + 2ζsp ( ωssp ) + 1

(4.8)

With the feedforward gain this relation is modified as follows: nz (s) Kq VT = GF F . δe (s) g ( ωssp )2 + 2ζsp ( ωssp ) + 1

(4.9)

Command shaping filter. This lead-lag filter shapes the pilot command such that the desired attitude (and flight path) response is obtained (Gibson 1999). The short-period approximation transfer function from elevator deflection δe to the pitch rate q is as follows: Kq (Tθ2 s + 1) q(s) = s 2 . δe (s) ( ωsp ) + 2ζsp ( ωssp ) + 1

(4.10)

The command shaping is achieved by cancelling the (stable) zero corresponding to the short-period motion and place a new (stable) zero at the desired location: Kq (Tθ2,new s + 1) τLEAD s + 1 Kq (Tθ2 s + 1) = , ( ωssp )2 + 2ζsp ( ωssp ) + 1 τLAG s + 1 ( ωssp )2 + 2ζsp ( ωssp ) + 1

(4.11)

where τLAG = Tθ2 and τLEAD = Tθ2,new .

4.2

Partition

The procedure to obtain a fuzzy partition of the flight envelope using fuzzy clustering is described in more detail in Chapter 3. The fuzzy partition that is derived in Chapter 3 is used in this chapter to design the gain-scheduled classical FCLs. The fuzzy partition computed by using the modified membership functions (see Figure 3.5b) is illustrated in Figure 4.3.

4.3

Automatic Tuning Procedure

Usually, the tuning of the FCLs is performed using different objectives for different operating points. The operating points represent the centers of the clusters, see

4.3. Automatic Tuning Procedure

47

Figure 4.3: Decomposition of the flight envelope into eight operating regimes. The dots denote the operating points, while the asterisks denote the test flight conditions. Each operating point and test flight condition is labelled with a number. Section 3.3. For instance, an operating point for the landing phase flight condition requires high precision attitude control, while in an operating point representing the cruise flight condition the requirements are less restrictive. The tuning of the gains of the classical FCLs, the default FCLs in the SE, is a time-consuming procedure. In order to be able to evaluate the performance of the scheduler in an early stage of the design process, before implementation in the nonlinear FCLs, the simplified FCLs (see Figure 4.1) are used in combination with an automatic tuning procedure. This procedure makes use of a fourth order linear longitudinal model. When the gains are set to zero, except for the feedforward gain which is set to one, the closed-loop system is equal to the bare aircraft model. In each iteration in the automatic tuning procedure, the Matlab/Simulink TM model is trimmed and linearized and the damping and frequency of the short-period and phugoid motion are evaluated. The automatic tuning strategy, which is developed in cooperation with experienced flight control engineers from industry, consists of four steps: 1. Tuning GQ to obtain the required short-period damping. The main effect of increasing GQ is the increase of the short-period damping ζsp . The pitch damper gain GQ is set such that the short-period damping is ζsp = 0.80, using an optimization algorithm.

48

Chapter 4. Scheduled Classical Control

Table 4.1: The operating points and their corresponding FCL parameters resulting from the automated tuning procedure on the simplified longitudinal FCL structure. FC

1 2 3 4 5 6 7 8

Mach nr. [-] 0.83 0.59 0.65 0.39 0.33 0.59 0.55 0.79

Alt. [kft] 38.0 32.3 11.0 5.9 15.4 23.2 9.8 25.8

Dyn. press. [mbar] 118 72 220 90 44 108 160 185

GQ

KP

KI

GF F

τLEAD

τLAG

1.09 0.90 0.53 0.63 0.61 0.77 0.56 0.78

0.22 0.63 0.22 0.66 1.87 0.42 0.29 0.28

0.59 1.74 0.63 2.06 3.60 1.34 0.92 0.97

0.16 0.14 0.07 0.12 0.21 0.12 0.08 0.09

0.42 0.58 0.29 0.48 0.88 0.44 0.35 0.30

1.49 1.68 0.67 1.09 3.28 1.21 0.81 0.81

2. Tuning KI and KP to obtain the required phugoid damping and maintain the required short-period damping. The main effect of increasing KI is the increase of the phugoid damping ζph . However, an undesirable side-effect is the decrease of the short-period damping. Although the main effect of increasing KP is the increase of the short-period frequency (see Section 4.1), it also results in an increase of the short-period damping. However, an undesirable side-effect is the decrease of the phugoid damping. The tuning KP and KI is therefore an iterative process. The objective is to tune KI and KP such that the phugoid damping is equal to ζph = 0.90, while the short-period damping stays at ζsp = 0.80. 3. Tuning GF F to provide optimal direct input to the elevator. The dynamics of the aircraft are augmented and fixed at this point. The purpose of the feedforward path is to improve handling performance. The feedforward gain GF F should be such that the direct input to the elevators brings the aircraft close to the commanded set point. The pilot input is in fact a commanded normal acceleration. The feedforward gain is set equal to the inverse of the gain of the short-period approximation of the transfer function from elevator deflection δe to normal acceleration nz . 4. Tuning the lead-lag filter to obtain zero dropback. The lead-lag filter denominator time constant is chosen to cancel the zero of the short-period motion Tθ2 . The lead-lag filter numerator time constant is chosen to obtain zero dropback in pitch attitude. The operating points and their corresponding FCL parameters resulting from the automatic tuning procedure are given in Table 4.1.

4.4

The Scheduler

The scheduler for the gains of the FCLs is implemented as a TS fuzzy model. The rule-base of the TS fuzzy model is equivalent to the rule-base given in Equation 3.11

4.4. The Scheduler

49

Figure 4.4: Hierarchical structure of the scheduler. Table 4.2: Config. Flaps [deg] 1 0 2 0 3 12 4 20 5 40

Aircraft configurations. Slats Remarks [deg] 0 clean configuration 25 25 25 take-off configuration 25 landing configuration

and the string of local FCL parameter is defined as Ki = [GQi , KPi , KIi , GF Fi , τLEADi , τLAGi ]. The gain scheduler that is described above is designed for clean configuration. However, during take-off and landing the configuration of the aircraft is typically modified to optimize its performance with respect to the corresponding flight task. This is achieved by deploying the flaps and slats that are mounted on the trailing and leading edge of the wing, respectively. The five configurations that can be selected in the SCA model, in terms of flaps and slats deflection, are described in Table 4.2. Since the aircraft dynamics change significantly with aircraft configuration, the FCL parameters need to be scheduled accordingly. The landing gear is not taken into account because it has only minor effects on the aircraft dynamics when it is extended. Besides the gain scheduler that is described above, a second gain scheduler is designed for the landing configuration using the same method. Since the flight envelope for landing configuration is much smaller, fewer operating points are needed compared to the clean configuration. The number of operating points for the landing configuration in this case is two. The rule-base for the landing configuration therefore has two rules. The schedulers for Clean Configuration (CC) and Landing Configuration (LC) run in parallel and their respective outputs, KCC and KLC , are scheduled as a function of the flap deflection: R1 : R2 :

If δf l = SM ALL then K = KCC If δf l = BIG then K = KLC

50

Chapter 4. Scheduled Classical Control

1

8

G

K

P

Q

6 0.5 0.8

2 0.6

0.4

200

150

100

0.8

50

Dynamic pressure [mbar]

Mach number [−]

150

100

50

Dynamic pressure [mbar]

FF

G

KI

5

5 0.6

0.4

200

150

100

0.8

50

Dynamic pressure [mbar]

Mach number [−]

0.6

0.4

200

150

100

50

Dynamic pressure [mbar]

Mach number [−]

1 τLAG

LEAD

0.35 0.3

τ

200

10

10

0.2 0.15 0.8

0.4

Mach number [−]

15

0.8

0.6

0.5

0.6

0.4

Mach number [−]

200

150

100

50

Dynamic pressure [mbar]

0.8

0.6

0.4

Mach number [−]

200

150

100

50

Dynamic pressure [mbar]

Figure 4.5: The gain scheduled FCL parameters GQ , KP , KI , GF F , τLEAD , and τLAG as a function of Mach number and dynamic pressure (clean configuration).

Configurations 1 through 3 in Table 4.2 correspond to small flap deflections, while configurations 4 and 5 correspond to big flap deflections. A schematic interpretation of the resulting scheduler is given in Figure 4.4. The rule-bases for clean and landing configuration could be combined in a single rule-base with flap deflection as a third antecedent or scheduling variable. However, this would result in more rules and it would obscure the underlying physical structure.

4.5

Evaluation

As mentioned before, the FCL parameters given in Table 4.1, which are obtained through the automated tuning procedure, are only used to evaluate the scheduler in the early stages of the design. The FCL parameters that are implemented in the full nonlinear FCLs, see Table 4.3, are taken from the FCLs that serve as the default of the SE. Figure 4.5 shows the corresponding FCL parameters as a function of Mach number and dynamic pressure. In this section the closed-loop system is evaluated in a number of test flight conditions selected by flight control engineers from industry. This evaluation considers straight and level flight conditions in clean configuration. The GS FCLs are eval-

4.5. Evaluation

51

Table 4.3: The operating points and their corresponding FCL parameters resulting from manual tuning using the full longitudinal FCL structure. FC

1 2 3 4 5 6 7 8

Mach nr. [-] 0.83 0.59 0.65 0.39 0.33 0.59 0.55 0.79

Alt. [kft] 38.0 32.3 11.0 5.9 15.4 23.2 9.8 25.8

Dyn. press. [mbar] 118 72 220 90 44 108 160 185

GQ

KP

KI

GF F

τLEAD

τLAG

1.13 0.90 0.57 0.64 0.59 0.79 0.58 0.82

0.14 0.62 0.11 0.62 1.99 0.36 0.20 0.18

0.58 1.74 0.62 2.06 3.60 1.35 0.92 0.96

0.16 0.14 0.07 0.12 0.21 0.12 0.08 0.09

0.42 0.58 0.29 0.48 0.88 0.44 0.35 0.30

1.49 1.68 0.67 1.09 3.28 1.21 0.81 0.81

Table 4.4: Test flight conditions. FC Mach Altitude Dynamic number [kft] pressure [-] [mbar] 14 0.35 15.0 51 15 0.30 5.0 54 16 0.70 35.0 92 17 0.60 25.0 104 18 0.50 5.0 157 19 0.70 15.0 221 20 0.75 40.0 85

uated by linear analysis and nonlinear simulation. The test flight conditions are given in Table 4.4 and illustrated in Figure 4.3. 4.5.1

Linear off-line evaluation

The linear analysis consists of evaluating the gain margin GM , the phase margin P M , the short-period frequency ωsp and the short-period damping ζsp . Ideally the GM exceeds 12 dB and the P M exceeds 60◦ . It can be seen in Table 4.5 that these criteria are amply met in all the test flight conditions. Furthermore a number of handling qualities are evaluated: the Control Anticipation Parameter (CAP) criterion, the pitch rate time history criteria (PR), the pitch axis equivalent time delay criterion (ETD), and the dropback criterion (DB). These handling qualities are considered to be the most significant (Gibson 1999, Alony et al. 1998). In order to comply with the design requirements for normal operation the CAP should be between 0.085 and 3.6, the pitch rate transient peak ratio should be less than 0.05, Trise VT should be between 5.33 and 296.2, the ETD should be less than 0.10 sec, q ∆θ and for the DB the point ( peak qss , qss ) should be below the line that runs through

52

Chapter 4. Scheduled Classical Control

q [deg⋅s−1]

4 2 0 −2

5

10

15

0

5

10

15

0

5

10

15

0

5

10

15

1

0.5 θ [deg]

15 10 5 0 0 −5

e

δ [deg]

0

1.5

z

n [−]

2

−10

Time [s]

Figure 4.6: Linear simulation results with fuzzy gain scheduling in the cruise flight condition: pitch rate q, normal acceleration nz , attitude θ, and elevator deflection δe as a function of time. the points (0, 3) and (1, 2.35). These criteria are met in all the test flight conditions, see Table 4.5. In Figure 4.6 the results of a linear simulation example with fuzzy gain scheduling are illustrated for test condition FC20. This flight condition corresponds to the cruise flight condition. During the simulation a block-shaped pilot input is used, which starts at t = 1 sec and terminates at t = 7 sec. It can be seen from the time history of θ that the dropback in pitch attitude is indeed negligible.

Table 4.5: Evaluation results in the test flight conditions: gain margin GM , phase margin P M , short-period frequency ωsp , short-period damping ζsp , control anticipation parameter criterion (CAP), pitch rate time history criteria (PR), pitch axis equivalent time delay criterion (ETD), and the dropback criterion (DB). FC

GM PM ζsp ωsp CAP [dB] [deg] [-] [rad s−1 ]

14 15 16 17 18 19 20

23.6 23.3 26.4 26.8 28.5 30.4 26.5

147.7 137.2 179.9 138.9 124.5 134.0 179.5

0.78 0.81 0.80 0.79 0.82 0.77 0.79

2.22 2.31 3.22 3.57 4.51 5.53 3.03

0.44 0.44 0.37 0.43 0.48 0.47 0.37

PR

∆q2 ∆q1

0.020 0.012 0.015 0.019 0.011 0.022 0.017

DP PR ETD , qmax ) ( DB Trise VT qss qss 50.6 42.6 65.6 54.0 38.7 45.0 74.3

0.03 0.03 0.03 0.03 0.03 0.03 0.03

(0.19,1.19) (0.18,1.18) (0.13,1.18) (0.12,1.18) (0.09,1.17) (0.08,1.19) (0.14,1.18)

53

1.5

20

1 KP

GQ

0.4

0.3

0.5 0.2 0

20

40

0 0

60

100

15

50

10

20

40

0 0

60

20

40

60

20

40

60

40

60

20

GFF 0 0

20

40

5 0

60

20

40

0.7 0.6

0.65

τLAG

τLEAD

30

0.4

20 10 0

10 5 0

60

40 Flap deflection [deg]

10

CGS FGS

15 KI

Dynamic pressure [mbar]

Mach number [−]

4.5. Evaluation

0.6

0.2 20

40

60

0 0

20

Time [s]

(a) Inputs.

40

60

0.55 0

Time [s]

20 Time [s]

(b) Outputs.

Figure 4.7: (a) Inputs of the Fuzzy Gain Scheduler. (b) Outputs of the conventional (CGS) and fuzzy (FGS) gain scheduler. The solid line denotes the conventional gain scheduler (CGS), while the dash-dotted line denotes the fuzzy gain scheduler (FGS).

4.5.2

Pilot-in-the-loop evaluation

Pilot-in-the-loop tests were performed in the Research Flight Simulator (RFS) of the National Aerospace Laboratory (NLR) in the Netherlands. In Figure 4.7 the results of a flight simulator test with the Fuzzy Gain Scheduler (FGS) are illustrated. This specific test is to verify the transient behavior due to the transition from landing configuration to clean configuration while the aircraft is accelerating. During this test the aircraft crossed four operating regimes, namely two for landing configuration and two for clean configuration. The inputs to the FGS are shown in Figure 4.7a. The Mach number and the dynamic pressure are continuously increasing (which indicates the acceleration), while the flaps are retracted from 40 degrees to 0 degrees. The latter is performed in three stages according to the following settings: flaps/slats 40/25 (landing configuration), 20/25 (take-off configuration), 12/25 (intermediate configuration) and 00/00 (clean configuration) degrees. The transition from intermediate to clean configuration is not shown in Figure 4.7. The parameters of the Conventional Gain Scheduler (CGS) are added using the same input data in order to give an indication of the similarities and differences. The CGS parameters are therefore not resulting directly from pilotin-the-loop simulation. The differences between the parameters of the FGS and those of the CGS are due to the differences in the scheduling mechanism. In the operating points of the FGS the parameters are the same for both controllers. As can be seen in Figure 4.4, the scheduling as a function of flap deflection takes place between 12 and 20 degrees. This transition occurs at about 46 sec. Comparing the SCAS parameters from the CGS with those of the FGS it is clear that the gains of the FGS are much smoother, especially during the transition from flaps/slats

54

Chapter 4. Scheduled Classical Control

20/25 to 12/25 degrees. In the CGS the scheduling as a function of flap deflection is a switch at 15 degrees, which explains the discontinuous behavior of the CGS gains in this region. During this flight simulator test of the FGS the pilot commented that he could feel no transients due to flaps/slats retraction. During the evaluation of the FGS, covering about 80% of the flight envelope, the two test pilots agreed that the FGS performed as good as the CGS. The FGS was not evaluated in the upper left corner of the flight envelope, see Figure 4.3, due to the lack of available simulator time and because this region is uninteresting from an operational point of view. The fact that the FGS is designed in a more automated fashion indicates the improvement of this approach. Although it should be taken into account that the FGS approach is implemented only for the most relevant parameters of the longitudinal SCAS, we conclude that the flight simulator test results are very encouraging and that they serve as a proof-of-principle of the proposed fuzzy gain scheduling design approach.

4.6

Conclusions

The presented procedure shows good potential for automated design of gain scheduled flight control laws. The objective was to reduce the design effort while at least maintaining performance. The results were evaluated and discussed extensively with the industrial partners of the ADFCS project. The experienced test pilots could not detect any significant difference between the conventional FCLs and those implemented with FGS for the SCAS of the longitudinal axis. Even though the FGS used fewer operating points than in the conventional approach, it can be concluded that its performance is comparable. The reduction of the design effort seems evident because of the more automated approach resulting in fewer operating points, but this has not been evaluated in detail. When the flight control engineer designs the gain scheduler for a classical controller, this is typically performed separately for each FCL parameter that requires scheduling. Moreover, each FCL parameter is not necessarily scheduled with the same scheduling variable(s). In other words, each FCL parameter has its own set of operating points, which makes the structure of the gain scheduler opaque. This iterative, single-loop approach of tuning the (scheduled) FCL parameters is timeconsuming and contributes therefore significantly to the total design cost. Due to this opaque structure, the mutual dependencies of the FCL parameters are hard to identify. This makes it difficult to understand what needs to be changed when it turns out that the performance is not as expected in a certain flight condition. By using the same scheduling variables and operating points for all related FCL parameters that require scheduling, the mutual dependencies of the FCL parameters are clearer and corrections are easier to make. However, this does not necessarily mean that the best performance is achieved by scheduling all FCL parameters in exactly the same way. To get the same closed-loop dynamics over the entire operating range, each FCL parameter should have a dedicated scheduling mechanism. Moreover, it makes sense to use different operating points for the longitudinal and lateral FCL parameters, since they correspond to different aircraft dynamics.

5 Scheduled Robust Multivariable Control The main contribution of this chapter is the combination of the gain scheduling technique based on fuzzy clustering, introduced in Chapter 3, and an H∞ design approach. The design objective is to have the closed-loop system match a predefined reference model, taking into account disturbances and uncertainties. The H∞ controllers are designed locally in a number of operating points and the interpolation between them (scheduling) takes place through fuzzy membership functions. The H∞ design approach requires less design effort than the classical design approach, but it does not necessarily result in a better controller in terms of stability and performance. The performance using the gain-scheduled H∞ controller is significantly improved compared to the performance using a fixed H∞ controller.

This chapter is organized as follows: An overview of scheduled robust multivariable control is given in Section 5.2. In Section 5.3, the H∞ design problem is described. Section 5.4 outlines the design process of the local H∞ controller, while in Section 5.5 the partition of the flight envelope and the design points are briefly described. The parameter scheduling approach for H∞ controllers is described in Section 5.6. Concluding remarks are given in Section 5.7.

5.1

Introduction

The main advantage of using robust multivariable (MV) control techniques over classical control techniques is that they replace the single-loop design approach by a multi-loop design approach. This reduces the required design effort, while simultaneously taking into account model uncertainties and disturbances. In the case of an aircraft, examples of model uncertainties are: 1. Variation in the aircraft dynamics as a function of flight condition. 2. Variations in the aircraft dynamics as a function of the weight and the centerof-gravity. 55

56

Chapter 5. Scheduled Robust Multivariable Control

3. Aerodynamic uncertainties, e.g., due to variations in the production or model mismatch. 4. Unmodelled higher-order dynamics. The latter is not considered in this thesis, since the controller validation is performed using the same model as used for the controller design. The SCA model does facilitate the option to modify the aerodynamic characteristics. It is unlikely that the design specifications can be met over the entire operating range using a single robust multivariable controller. Typically a compromise has to be found between performance and robustness. Two types of robustness are considered, namely robust stability and robust performance. Robust stability implies that the closed-loop system is stable for all possible plants as described by the uncertainty description. Robust performance implies that the performance objective is achieved under all possible plants as described by the uncertainty description. Expanding the uncertainty, i.e. expanding the operating regime, reduces the probability that robust stability and robust performance can be achieved. Scheduling is a common approach to overcome this trade-off, however, scheduling of multivariable controllers is a difficult task.

5.2

Overview of Scheduled Robust MV Control

One of the difficulties with implementing gain scheduled multivariable control laws is the complexity of such control laws. For a controller of order n with ni inputs and no outputs, there are n(1 + ni no ) + ni no controller parameters to schedule (Ly et al. 1985, Nichols et al. 1993, Hyde and Glover 1993). Two categories of gainscheduling robust multivariable controllers can be found in the literature: 1. Scheduling of the controller output matrix. 2. Scheduling of both the controller dynamics and the controller output matrix. An example of the first category can be found in (Garg 1997). A nominal controller is designed that gives a stable closed-loop system in the entire operating range. The parameters of the output matrix are optimized such that the closed-loop system at the off-design points closely matches the closed-loop system in the design point. The controller eigenvalues are not affected. It is possible that this simplified scheduling scheme is not sufficient to account for significant variations in the plant poles and zeros. Many examples of the second category can be found in the literature, showing a wide variety of design approaches. However, in all approaches an attempt is made to reduce the order of the controller and/or to impose a certain structure on the controller in order to simplify the scheduling problem. In (Nichols et al. 1993) a gain-scheduled robust multivariable controller is designed for the autopilot function of a missile. The system has two inputs and one output. The order of each controller is reduced from seven to four. Moreover, after the order reduction, two poles and two zeros that do not vary significantly from one operating condition to the next are replaced by their average values. This

5.3. General Description of the Robust Control Problem

57

modification yields fourth-order linearized controller transfer functions containing an identical second-order factor at each operating point. This significantly reduces the number of parameters to be scheduled. Scheduling takes place directly on the gains, poles and zeros of the fourth order controllers. This is possible if the poles and zeros are identifiable or recognizable, i.e. if it is possible to identify the pole(s) and/or zero(s) of a certain mode in each local controller. Scheduling is performed through linear interpolation as a function of Mach number and angle-of-attack. A similar approach is taken in (Lin and Khammash 2001). Moreover, in this paper the parameters are modified such that they monotonically increase/decrease with the indicated airspeed. The scheduling of H∞ controllers would be simplified if it is possible to write the controller as a plant observer plus state feedback: x ˆ˙ = Aˆ x + H(y − C x ˆ) + Bu u

= Fx ˆ.

The clear structure lends itself to gain scheduling of the F and H matrices. In general, it is not clear that H∞ controllers can be written as exact plant state observers as there will be a worst disturbance term entering the observer state equation as shown in (Doyle et al. 1989). However, in (Hyde and Glover 1993) it is shown that the normalized coprime factor robust stabilization approach produces a controller which can be written as a plant observer plus state feedback. The application describes the design of a controller for a Very Short Take-Off and Landing (VSTOL) aircraft, which has significant variations in airspeed. The scheduler is based on linear interpolation of the matrices F and H as a function of true airspeed and angle-of-attack. A similar design approach can be found in (Pellanda et al. 2000). An alternative to the approaches described above is to design all the controllers simultaneously, inherently solving the stability problems that can occur when scheduling multivariable controllers. This can be done by using Linear Matrix Inequalities (LMIs) and is described in (Apkarian et al. 1995, Apkarian and Gahinet 1995, Apkarian et al. 2000). The nonlinear model of the plant is transformed into the Linear Parameter-Varying (LPV) format, where the parameters vary as a function of the scheduling variables. The resulting controller is also described in the LPV format. By incorporating the scheduling variables, the controller adjusts to the variations in the plant dynamics in order to maintain stability and high performance along all trajectories of the scheduling variables. This approach is valid only when a single Lyapunov function is used over the entire operating range. The design approach is demonstrated on a second order LPV model of a missile (Apkarian et al. 1995).

5.3

General Description of the Robust Control Problem

Figure 5.1 illustrates the robust control problem (Zhou et al. 1995). The control objective is to design the dynamic controller K such that the closed-loop system

58

Chapter 5. Scheduled Robust Multivariable Control

Figure 5.1: Robust Control Problem. meets the design requirements, while taking into account certain disturbances d and model uncertainties. The block ∆ is a matrix of which the elements δi,j vary between -1 and 1, δi,j ∈ [−1, 1]. The model uncertainty can be defined by appropriately choosing the vector signal z and the structure of the matrix ∆. An illustrative example can be found in (Balas et al. 1991). For a detailed description of the robust control problem, the reader is referred to (Zhou et al. 1995). 5.3.1

The generalized plant

The generalized plant is a linear model of the nominal open-loop system (plant model, actuator dynamics, sensor model, anti-aliasing filters, etc), including the reference model and weighting filters for the design of the controller. Since the generalized plant is a linear approximation of a nonlinear system, it will vary with the operating point. The three outputs of the generalized plant are the vectors z, e and y: z

inputs of the uncertainty block ∆ (variations from the nominal model).

e

performance measures: tracking error (e1 ) and control activity penalty (e2 ).

y

feedback signals that are used as input to the controller K.

The three inputs of the generalized plant are the vectors w, d and u: w

outputs of the uncertainty block ∆.

d

disturbances, to be separated in the command signal (d1 ), sensor noise (d2 ) and disturbances (d3 ).

y

control signal (output of the controller).

The reference model that is used for the design of the robust multivariable controller represents the desired closed-loop system dynamics. The reference model should be chosen such that it fulfills the stability and performance requirements.

5.4. Robust Multivariable Flight Control Design

59

Each measurement is corrupted with sensor noise which becomes more severe with increasing frequency. This is phenomenon is realized through the noise filter. The tracking and actuator filters are (dynamic) weighting filters that need to be tuned. 5.3.2

Model uncertainties

The model uncertainties can be divided into two categories: 1. Model uncertainties due to scheduling. 2. Model uncertainties due to changes in variables which are not scheduling variables, due to parameter uncertainties and due to unmodelled (higherorder) dynamics. The first category of model uncertainties are due to nonlinearities in the aircraft model. In order to effectively deal with the nonlinearities, a scheduling mechanism for the controller is introduced. However, in general, the scheduled controller at off-design flight conditions is not equal to the controller designed for that specific flight condition (Babuˇska and Oosterom 2003). This mismatch between model and controller can be interpreted as a model uncertainty. Examples of uncertainties of the second category are uncertainties due to variations of the weight and the position of the center-of-gravity, parameter uncertainties, and uncertainties due to the (false) assumption of a rigid body. It should be noted that the second category of uncertainties are present both in the design flight condition as well as in the off-design flight conditions. 5.3.3

Controller

The controller needs to be robust against the model uncertainties and the disturbances, meaning that it should perform within the specifications under all uncertainties represented by the block ∆ and the disturbances represented by the vector d. There are several techniques to design the controller, such as the µ-synthesis approach or the LMI approach. In this chapter the controller results from design process using LMIs (Doyle et al. 1989). This is discussed in more detail in Section 5.4 and Appendix F.

5.4

Robust Multivariable Flight Control Design

The objective is to design a fuzzy gain-scheduled H∞ controller for the full flight envelope of the SCA model. Furthermore, the FGS H∞ controller should maintain robust stability and robust performance under the above described uncertainties. The generalized plant illustrated in Figure 5.2, has seven inputs and six outputs. The seven inputs are (the dimensions are given in brackets): • w[1]: scalar output from the uncertainty block (representing model uncertainties)

60

Chapter 5. Scheduled Robust Multivariable Control

Tracking Weight Col [deg] q [deg/s]

num(s) den(s)

Reference Model

G2 Column Dynamics Fs [lb] Col [deg] 2 d1: pilot command

4 y1: q_c 5 y2: q_fb 6 y3: nz_fb

K

1 z1

∆

1 w1

tau.s+1 7 u1: control signal

G1

tau.s

5 d4: u_gust

u_gust

6 d5: w_gust

w_gust sens em

de de_c de_dot

Elevator Actuator

2 e1: perf_q

de

Linear AC de_dot perf_de

Control Activity Weight

3 e2: perf_de

q [deg/s] q VT/g theta [deg] nz [g] nz_comp VT [m/s]

Anti−aliasing Filters and Computation q_noise nz_noise

Noise Filter

3 d2: q_noise 4 d3: nz_noise

Figure 5.2: The generalized plant with input uncertainty. • d[5]: the pitch reference signal (d1 ), two sensor noise sources (d2 and d3 ), and two atmospheric turbulence sources (d4 and d5 ) • u[1]: the control input (commanded elevator deflection). while the six outputs are: • z[1]: scalar input of the uncertainty block • e[2]: the pitch rate error signal (e1 ) and the control activity signal (e2 ) • y[3]: the command path signal (y1 , commanded pitch rate) and two feedback signals (y2 and y3 , pitch rate and normal acceleration ). The controller is a multi-input, single-output system with the input vector y and with the output scalar u. The pitch rate error signal penalizes the difference between the true pitch rate q and the output of the reference model qref . The control activity signal penalizes the time derivative of the elevator deflection, δ˙e . 5.4.1

Reference model

The following longitudinal handling quality requirements are used as a guideline for the reference model: CAP criterion, pitch attitude bandwidth criterion, dropback criterion and the pitch rate time history criteria (Hodgkinson 1999, Gibson 1999). The controller is of the pitch rate command/attitude hold type, which implies that the commanded pitch rate is a function of the column deflection and that the pitch attitude is kept constant at the current value when the column is at the center position. The required attitude hold capability implies that the phugoid motion should be cancelled out and for this reason the phugoid motion is not included in the reference model. The reference model therefore represents only the

5.4. Robust Multivariable Flight Control Design

61

short-period dynamics. For the clean configuration the reference model is: Kq (Tθ2 s + 1) q(s) = s 2 , δe (s) ( ωsp ) + 2ζsp ( ωssp ) + 1 where Kq = 7.84, Tθ2 = 0.56, ζsp = 0.9 and ωsp = 3.2. The short-period damping ζsp and natural frequency ωsp are chosen such that they comply with the handling qualities for normal operation. The time-constant Tθ2 is selected such that zero dropback is achieved (Gibson 1999). The gain Kq is determined based on the pilot comments concerning the required stick force to maneuver the aircraft. To further improve the attitude hold performance, the output of the controller is fed through a low-frequency integrator H(s) = τ s+1 τ s , see Figure 5.2. 5.4.2

Model uncertainty

Two types of model uncertainty can be manipulated in the synthetic environment, namely uncertainty in the weight and balance and uncertainty in the aerodynamic model. The ranges of these uncertainties are described below. Weight and Balance Envelope In the SCA model the Aircraft Inertia Matrix (AIM) depends on the aircraft weight and the position of the Center-of-Gravity (CG) along the X-axis. The position of the CG along the Y -axis and Z-axis has no impact on the AIM and they are therefore not taken into account. The weight and balance envelope, or centogramme, is illustrated in Figure 5.3. The centogramme is defined in cooperation with the manufacturer of the aircraft on which the SCA model is based. Aerodynamic Uncertainties One can select from four pre-defined aerodynamic models (see Appendix A for more details on the aerodynamic coefficients): 1. Nominal model (default). 2. Increased control effectiveness (15% increase of ∆Clδe , ∆Cmδe , ∆Clδs , ∆Cmδs , ∆CYδr , ∆CRδa , ∆CRδr , ∆CNδa and ∆CNδr ). 3. Decreased control effectiveness (15% decrease of ∆Clδe , ∆Cmδe , ∆Clδs , ∆Cmδs , ∆CYδr , ∆CRδa , ∆CRδr , ∆CNδa and ∆CNδr ). 4. Reduced stability (30% decrease of ∆Cmq and ∆CRp and 10% decrease of ∆Cm , ∆Cn and ∆CNr ). Furthermore it is possible to modify the (longitudinal) aerodynamic derivatives Cmα , Cmq , Cmδe , Cmδs , Clδe , and Clδs . Only the four pre-defined aerodynamic models are considered. In this chapter the model uncertainties with respect to the weight and balance and the aerodynamic model are not implicitly described in the uncertainty block ∆. These uncertainties are only used for the validation of the controller. The model

62

Chapter 5. Scheduled Robust Multivariable Control

45 2

3

43

maximum take−off weight

weight [103 lbs]

41 39 6 37

maximum flight weight

35

maximum landing weight

33

1

31 29 7 27 25 18

minimum weight

5

20

22

24

26

28 X

CG

4

30 32 [% mac]

34

36

38

40

Figure 5.3: The centogramme of the SCA model as a function of aircraft weight and the position of the center-of-gravity along the X-axis. uncertainty description that is used for the design of the local H∞ controllers is an uncertainty on the input gain, see Figure 5.2. The gains G1 and G2 determine the level of the input uncertainty. The block ∆ is in this case a scalar δ and varies between -1 and 1. When the input u should vary between cmin u and cmax u, the gains G1 and G2 have the following values: G1 =

cmax + cmin , 2

G2 =

cmax − cmin . 2

Taking into account the uncertainty scalar δ, the uncertain input becomes: uunc = (G1 + δG2 ) u. 5.4.3

Weight functions

In the generalized plant, as presented in Figure 5.2, there are two weight functions that need to be tuned, the tracking weight function and the control activity weight function. The proportion between these weights determines the relative importance of the following two objectives: 1. Design the controller such that the closed-loop system matches the reference model. 2. Limit the required actuator control activity. These weights can either be constant or frequency dependent. These weights influence the achievable γ, which is defined as ||Tzw ||∞ (see also Figure 5.1).

5.4. Robust Multivariable Flight Control Design

63

30 20

Gain [dB]

10 0 −10 −20 −30

Reference model Tracking weight Control Activity Weight Local Model (Mach = 0.55, Alt = 12 kft)

−40

−2

10

0

10 −1 Frequency [rad ⋅ sec ]

2

10

Figure 5.4: Gain of the reference model, tracking weight and control activity weight as a function of the frequency. The tracking weight is frequency dependent: Wtrack =

432 (s + 1) . (s + 0.1)(s + 12)2

The tracking weight has a high DC-gain of 30 dB to suppress steady state errors on the pitch rate. In the frequency range of interest with respect to the reference model, between 0.01 and 10 rad sec−1 , the gain should be lower. For this reason the pole p1 = −0.1 is in the denominator of the reference model. To prevent the gain of the tracking weight to become too small in the frequency range of interest of the reference model, the zero z1 = −1 is added to the numerator. Finally the two poles p2,3 = −12 are added to the denominator to make sure the emphasis is on the DC-gain and the frequency range of interest of the reference model. The gain of the reference model, the tracking weight, the control activity weight are illustrated in Figure 5.4 as a function of frequency. The gain of the open-loop longitudinal model in one of design flight conditions (FC27) as a function of frequency is added as an example, see also Section 5.5. The weight corresponding to the actuator control activity is frequency independent and is set equal to one. In this way the control activity weight is inferior to the tracking weight in the frequency range of interest with respect to the reference model, see also Figure 5.4. At the same time the control activity weight is superior to the tracking weight in the high end of the frequency range of the elevator actuator. The maximum speed of the elevator is 50 rad/sec. 5.4.4

Controller Design

The output-feedback H∞ controller is designed in continuous-time using the LMI approach, see Appendix F for a brief description. This approach is further moti-

64

Chapter 5. Scheduled Robust Multivariable Control

Figure 5.5: Possible ways to tackle the robust multivariable control problem.

vated below. The objective is to design a low-order discrete-time H∞ controller. There are a number of arguments for low-order controllers, e.g., to reduce the required computational power or the required storage space. However, in this case the main reason to reduce the order of the controller is to minimize the number of parameters that need to be scheduled, which greatly simplifies the scheduling problem for robust multivariable controllers. The order of the controller is equal to the order of the generalized plant for the LMI approach (Boyd et al. 1994, Apkarian et al. 1995) or equal to the order of the generalized plant plus twice the order of the scaling filter D for the µ-synthesis approach (Zhou et al. 1995). The LMI approach is selected over the µ-synthesis approach because the resulting controller is of significant lower order. Additional order reduction is required to further simplify the scheduling problem. There are a number of approaches to the design of a low-order, discretetime controller through the LMI approach (see Figure 5.5). The design of the H∞ controller can be performed in continuous-time and in discrete-time. In this case it is chosen to perform the design of the controller in continuous-time, using the full-order generalized plant. Experiments show that order reduction on the generalized plant has a negative impact on the performance and robustness of the resulting controller. Performing order reduction on the controller is much less sensitive in this respect. For this reason it was chosen not to perform order reduction on the generalized plant. The main argument to perform the design in continuous-time is that in Matlab TM there are tools readily available for order reduction of continuous-time systems. Since the order reduction is performed after the design of the controller, this means that the design itself should be performed in continuous-time. The order reduction and discretization are discussed in more detail in the following paragraphs.

5.4. Robust Multivariable Flight Control Design

5.4.5

65

Order reduction

As mentioned before, the order of the controller is equal to the order of the generalized plant for the LMI approach. This typically leads to a controller of higher order than is strictly necessary to comply with the design specifications. Consequently, the excess of poles and zeros are pushed to the high frequency range in order to minimize their (unwanted) influence. A low-order controller is obtained in two steps, namely: 1. Hankel order reduction The minimum realization of the generalized plant, see Figure 5.2, is of 18th order. The controller is of the same order. Using Hankel order reduction the order of the controller is reduced to eleven, while the Bode diagram does not change in the frequency range of interest ([0.01, 10] rad sec−1 ). The frequency range of interest is such that it includes both the phugoid motion as well as the short-period motion frequency, see also Figure 5.4. However, the Hankel order reduction still leaves some high frequency modes. 2. Remove high-frequency modes The 11th order controller has modes with a frequency higher than the Nyquist frequency, ωN = fs · π, where fs denotes the sample frequency. The sample frequency of the flight control system of the SCA model is equal to fs = 50 Hz and the Nyquist frequency is therefore equal to ωN = 157.1 rad sec−1 . Removing these modes results in a ninth order controller. 5.4.6

Discretization

Since the controller is to be used in a digital system, it needs to be discretized. A straightforward Tustin discretization is used to obtain the discrete-time controller, since this method results in a discrete-time controller that best matched the continuous-time controller in terms of the frequency response. 5.4.7

Transformation to the δ-operator form

First attempts showed that parameter scheduling using the discrete-time transfer function form resulted in unstable controllers in a number of off-design flight conditions. A method approach to reduce the parameter-pole sensitivity is to transform the controller to the δ-operator form (Astr¨ om and Wittenmark 1997):     a1 1 0 . . . 0 b1   . .   . . b  a2 1 1 . .  2        .   ..  , C = 1 0 ... 0 , D = d . A =  a3 0 1 . . . 0 , B =      . .  .. . . . .  ..   .. . . 1 . bn an 0 . . . 0 1 In this form the parameter-scheduled controller turned out to be stable for all flight conditions. An example is illustrated in Figure 5.6. The poles of the scheduled

66

Chapter 5. Scheduled Robust Multivariable Control

−3

x 10

Transfer function

−3

x 10 2 Imaginary Axis

Imaginary Axis

2

δ−operator

1 0 −1 −2

1 0 −1 −2

0.998

0.999 1 Real Axis

1.001

0.998

0.999 1 Real Axis

1.001

Figure 5.6: Coefficient-pole sensitivity. Comparison between the discrete-time transfer function form (left) and the δ-operator form (right). The black x-marks denote the poles of the five controllers. The grey x-marks denote the poles of scheduled controllers.

controllers using the transfer function form are scattered, moreover, they cross the unit circle. The poles of the scheduled controllers using the δ-operator move around in a much more predictable manner. It should be noted that during the transformation from discrete-time transfer function form to the δ-operator form, the poles do not stay exactly in the same position due to the finite computer accuracy (compare the black x-marks in the left and right plot of Figure 5.6).

5.5

Partition of the Flight Envelope

The partitioning of the flight envelope is obtained by fuzzy clustering of the relevant aerodynamic derivatives. The approach is the same as in Section 3.3, however, due to some modifications in the aerodynamic model the resulting partition of the flight envelope is not equivalent to the partition presented in Chapter 4. Based on validation and performance criteria, the optimal number of clusters is found to be nine. Since in this case the aerodynamic database is smoother, due to a discontinuity in the aerodynamic database that has been fixed, the number of clusters has increased. The resulting partition of the flight envelope is illustrated in Figure 5.7 as a function of Mach number and altitude. Note that the membership functions are defined in terms of Mach number and dynamic pressure (see also Chapter 3). The nine design points are denoted by black dots, see Table 5.1 for more details. The light areas around the design points denote large membership degrees, i.e., areas where one controller is dominating, the dark areas denote the flight conditions where the controller parameters of neighboring design points are interpolated. Interpolation takes place through the membership functions, which are defined as functions of the two scheduling variables: Mach number and dynamic pressure. The membership functions are shown in Figure 5.8.

5.6. Parameter Scheduled Robust Multivariable Control

67

Figure 5.7: Partitioning of the flight envelope of the aircraft for clean configuration as a function of Mach number and altitude.

5.6

Parameter Scheduled Robust Multivariable Control

The concept of the parameter scheduling approach is illustrated in Figure 5.9. There is a single linear dynamic controller and the scheduling is performed directly on the parameters of this controller. In contrast to output scheduling, the parameter scheduling concept imposes a number of restrictions on the local controllers. In order to make sure that the scheduling takes place amongst like parameters, the local controllers need to be of the same order and need to have the same structure, e.g., the observable canonical form. In order to design local H∞ controllers that

Table 5.1: The nine design points resulting from fuzzy clustering, denoted by FC26 to FC34. FC

26 27 28 29 30 31 32 33 34

Mach number [-] 0.60 0.55 0.77 0.56 0.53 0.24 0.83 0.75 0.34

Altitude [kft] 7.2 12.0 24.1 20.5 30.2 1.7 28.0 35.0 8.3

Dynamic pressure [mbar] 214 147 188 108 63 39 188 108 62

Chapter 5. Scheduled Robust Multivariable Control

Membership degree

Membership degree

68

1

0.5

0 0.2

0.3

0.4

0.5 0.6 Mach number [−]

0.7

0.8

1

0.5

0

50

100 150 200 Dynamic pressure [mbar]

250

Figure 5.8: Membership functions for the flight envelope in clean configuration.

Figure 5.9: Schematic representation of the parameter scheduling concept. are suitable for gain scheduling, the following three issues are of importance: 1. The controllers should be stable (preferably also with stable zero dynamics). 2. The controllers should be of low-order (designed through low-order generalized plant and order reduction on the H∞ controller, see Section 5.4). 3. The controllers should have equivalent structures (the same number of poles and zeros). It is known that a necessary and sufficient condition of the existence of a stable stabilizing controller (strong stabilization) is the so-called parity interlacing property (Youla et al. 1974). Since the H∞ controller is in general not unique, it is reasonable to expect that even if the H∞ central controller is unstable, there might still be a stable controller that could satisfy the H∞ norm bound. In (Zeren and ¨ Ozbay 1999, Cao and Lam 2000) an approach for designing such high order stable H∞ controller has been suggested based on the parametrization of all suboptimal H∞ controllers. This approach conservatively converts the stable H∞ controller design problem into another 2-block standard H∞ problem. In addition, the order of the controller may be twice as high as the order of the central controller.

5.6. Parameter Scheduled Robust Multivariable Control

69

45 40 35 FC30

30

Altitude [kft]

FC33 MMAX = 0.85 FC32

VC = 150 Kts

25

FC36 FC29

20 15

V = 375 Kts C

FC27

10 5 0 0.1

FC28

FC34

FC26

FC37 FC31

0.2

0.3

0.4 0.5 0.6 Mach number [−]

0.7

0.8

Figure 5.10: The black dashed continuous line denotes the flight envelope for clean configuration (FC26 → FC34). The black dots denote the design flight conditions for clean configuration, which are obtained through fuzzy clustering. The light gray dashed line denotes the flight envelope for landing configuration. The light gray dot denotes the design flight condition for landing configuration (FC37). The black asterisk denotes a test flight condition. The dark gray continuous line denotes in which parts of the flight envelope the evaluation of the parameter scheduling concept has taken place. In (Campos-Delgado and Zhou 2001) a weighting function is introduced to alleviate the conservativeness of this approach. Fortunately the central controller is stable for this application, which is also influenced by the reference model and weight functions, and the above mentioned method is therefore not applied in this thesis.

5.6.1

Scheduling of the H∞ controllers

The ideal situation would be to schedule the poles and zeros of the local controllers directly. However, this only makes sense when there is an unambiguous relation between the poles and zeros of each of the H∞ controllers. If this relation is not present, the parameters of the controllers need to be scheduled instead. In that case the local controllers should be written in a format with low coefficientpole sensitivity in order to improve their schedulability. It is well known that the coefficient-pole sensitivity increases with the order of the system (Astr¨om and Wittenmark 1997). For this reason the order of the H∞ controller should be as low as possible. Rewriting the local controllers in the δ-operator form further reduces the coefficient-pole sensitivity.

70

Chapter 5. Scheduled Robust Multivariable Control

Figure 5.11: Hierarchical structure of the parameter scheduler. The design procedure as described in Section 5.4 is performed for 10 flight conditions, see Figure 5.10. One design point represents the landing configuration and nine design points represent the clean configuration. The order of all 10 controllers is reduced to nine. For the parameter scheduling the controller is rewritten from the discrete-time description into the observable canonical δ-operator form. Since the basic controller is a three input single output system, the controller is defined as follows:   B1 On×1 On×1 A On×n On×n  On×n A On×n On×1 B2 On×1  , K=  On×n On×n A On×1 On×1 B3  C C C D1 D2 D3 where



a1

  a2   A =  a3  .  ..

1

0

1

1

0 .. .

1 .. . an 0 . . .   C = 1 0 ... 0 ,

... .. . .. . .. . 0

 0 ..  .   , 0   1 1

 bi1  bi2     ..   Bi =   . ,  .   ..  bin   Di = di , i = 1, 2, 3. 

The number of parameters that are scheduled is equal to (1 + ni ) n + ni for the dynamic controller, where n denotes the order of the controller and ni denotes the number of inputs, plus the time-constant of the low-frequency integrator. In this case the controllers are of 9th order, with three inputs and one output. In total there are 40 parameters subject to scheduling. These parameters are tuned in the design flight conditions illustrated in Figure 5.10. The flight condition FC37 denotes the design flight condition for landing configuration and the flight conditions FC26 through FC34 denote the nine design flight conditions for clean configuration. The parameter scheduler is illustrated in Figure 5.11. The landing configuration is covered by a single operating point, and therefore also by a single parameter set for the controller K = KLC and the time-constant τ = τLC . The clean configuration

5.6. Parameter Scheduled Robust Multivariable Control

Table 5.2: Gain and phase margin of the closed-loop system conditions and an additional test flight condition (FC36). FC Mach Alt. Dyn. δf l δsl LG nr. [kft] press. [deg] [deg] [0/1] [-] [mbar] 26 0.60 7.2 214 0 0 0 27 0.55 12.0 147 0 0 0 28 0.77 24.1 188 0 0 0 29 0.56 20.5 108 0 0 0 30 0.53 30.2 63 0 0 0 31 0.27 2.0 49 0 0 0 32 0.83 28.0 188 0 0 0 33 0.75 35.0 108 0 0 0 34 0.34 8.3 62 0 0 0 36 0.67 23.0 144 0 0 0 37 0.22 2.0 32 40 25 1

71

in the 10 design flight GM [dB]

PM [deg]

6.5 7.2 7.1 8.0 9.6 10.4 7.5 8.3 9.4 6.7 12.4

23.3 26.4 26.0 30.5 37.2 55.6 28.1 32.4 37.2 27.1 50.3

is covered by nine operating points, and therefore also by nine parameter sets for the controller K = KCC . The parameters for K = KCC are a function of Mach number and dynamic pressure and are computed using the following rule-base: Ri :

If M is MNi and q is DPi then KCC = Ki

The membership functions corresponding to M is MNi and M is DPi are illustrated in Figure 5.8. In all nine operating points for clean configuration, the same time-constant τ = τCC is used. The controller parameters for landing and clean configuration are then scheduled as a function of flaps deflection to obtain K and τ : R1 : R2 :

5.6.2

If δf l is Extended then K = KLC and τ = τLC If δf l is Retracted then K = KCC and τ = τCC

Stability analysis

In Table 5.2, the gain margin and phase margin of the closed-loop system in the 10 design points and an additional test point (FC36) are given. In all cases the gain margin exceeds 6 dB, however, the 30 degrees phase margin is not always achieved. It should be noted that in the aerospace industry often a minimum gain margin of 12 dB and a minimum phase margin of 60 degrees is taken as a reference. In order to achieve this, the local designs have to be improved. The stability margins can be influenced through the reference model, the weight functions, the uncertainty model description, etc. However, here we have concentrated on the scheduling aspects of the local controllers and not on the optimization of the local controllers

72

Chapter 5. Scheduled Robust Multivariable Control

45 40 FC33

35 FC30

30

Altitude [kft]

MMAX = 0.85 FC32

V = 150 Kts

25

C

FC28 FC29

20 15

FC27

10 5 0 0.2

FC34

V = 375 Kts C

FC26

FC31

0.3

0.4

0.5 0.6 0.7 Mach number [−]

0.8

0.9

Figure 5.12: Randomized stability analysis. The gray dots denote the flight conditions that have been used for stability analysis. The black dots denote the two design flight conditions, while the black o-marks denote the flight conditions for which the output gains are tuned. The black x-marks denote the 9 flight conditions for which the gain margin is smaller that 3 dB and/or the phase margin is smaller than 15 degrees.

to meet industrial standards. 5.6.3

Controller validation

The stability margins have been evaluated at 200 randomly defined flight conditions for clean configuration. Not only the Mach number and altitude were selected randomly, but also the aircraft weight and center-of-gravity as well as the aerodynamic uncertainty. Nine flight conditions resulted in a closed-loop system with a gain margin lower than 3 dB and/or a phase margin lower than 15 degrees, see Figure 5.12. All these flight conditions correspond to a low dynamic pressure, lower than any of the nine design points. This indicates that either the design points should be closer to the edge of the flight envelope or the robust stability of the local controllers needs to be improved. The parameter-scheduled controller was successfully tested during pilot-in-the-loop simulations in the NLR Research Flight Simulator. The pilot performs longitudinal maneuvers to evaluate the dynamics and performance of the closed-loop system at several flight conditions scattered over the flight envelope and rates the system using the Cooper-Harper (CH) rating scale (Cooper and Harper Jr. 1969), see Figure 5.13. As can be seen in Figure 5.10, a large part of the flight envelope was covered. In all flight conditions where the longitudinal dynamics were evaluated the pilot gave the system CH-1. With respect to the required stick force when performing longitudinal maneuvers, the pilot gave ratings ranging from CH-2 to

5.6. Parameter Scheduled Robust Multivariable Control

73

Figure 5.13: Cooper-Harper rating scale (Source: (Cooper and Harper Jr. 1969)).

CH-4. There is a clear correlation between the dynamic pressure and the pilot rating, namely the lower the dynamic pressure, the higher the pilot rating. Note that a high CH rating means a poor performance. The reason for this is that the same reference model was used for all nine design flight conditions for clean configuration. The pilot expects a more responsive system for low dynamic pressure. This can be easily solved by modifying the reference model accordingly. Figures 5.14 and 5.15 show the time histories of an approach and landing. In Figure 5.14 the position of the aircraft in terms of the deviation from the glide-slope is illustrated as a function of time. In Figure 5.15 the variables that are interesting with respect to the scheduling mechanism and/or configuration changes are illustrated. The main interest of this exercise is to evaluate the scheduling mechanism while going through a number of configuration changes and covering a fair part of the range in terms of Mach number and dynamic pressure. No anomalies were found. In Figure 5.16 the time history of a push-pull to pitch attitude is illustrated. The data are taken from the RFS recordings. The pilot controls the aircraft to -5, 0 and 5 degrees of pitch attitude while evaluating the performance and dynamics. It can be seen that the control in pitch attitude is fairly accurate.

Chapter 5. Scheduled Robust Multivariable Control

Altitude [kft]

Lateral deviation [m]

74

100

0 Flight simulator data Glide slope −100 −15

−10

−5

0

−10

−5 Downrange [km]

0

2 1 0 −15

Landing gear [−]

Flaps/Slats [deg] Dynamic pres. [mbar] Mach number [−]

Figure 5.14: Flight simulator time histories of the final approach and landing. The lateral deviations are the result of intentional pilot inputs to evaluate the lateral controller.

0.25 0.2 0

20

40

60

80

100

120

140

160

180

200

220

0

20

40

60

80

100

120

140

160

180

200

220

60

40

20 40 25 20 12

Flaps deflection Slats deflection

0 0

20

40

60

80

100

120

140

160

180

200

220

0

20

40

60

80

100 120 Time [s]

140

160

180

200

220

1 0.5 0

Figure 5.15: Flight simulator time histories of the final approach and landing. The illustrated variables are the scheduling variables and/or denote configuration changes.

Pitch attitude [deg] Column position [deg]

5.7. Conclusions

75

4 2 0 −2 −4

0

5

10

15

20

25

30

0

5

10

15 Time [s]

20

25

30

5 0 −5

Figure 5.16: Flight simulator time histories of a push/pull maneuver to ± 5 degrees pitch attitude.

5.7

Conclusions

A gain scheduling concept for multivariable H∞ controllers is presented. The design points and the scheduling mechanism are obtained through fuzzy clustering of relevant aerodynamic derivatives. The local H∞ controllers are designed in continuous-time using LMIs and order reduction is performed before the transformation to discrete-time. The gain scheduling concept has been evaluated off-line (linear and nonlinear simulations, stability analysis) and through pilot-in-the-loop simulations. The results are satisfactory, although additional tuning is required to further improve the performance and stability characteristics. There are still stability problems in the operating regimes between the outer operating points and the edge of the flight envelope. These problems occurred in particular in the low dynamic pressure region. It is recommended to force the outer operating points closer to the edge of the flight envelope, such that these stability problems no longer occur. Since the fuzzy clustering algorithm uses the Euclidean distance measure, one possible approach is to weight the distance for data points close to the edge of the flight envelope more than for data points in the center of the flight envelope. Another option is to use the operating points obtained through fuzzy clustering as an initial condition for a similar procedure as described in (McNichols and Fadali 2003). In this way the operating points and the parameters of the scheduler are tuned based on the performance of the global nonlinear closed-loop system.

76

Chapter 5. Scheduled Robust Multivariable Control

6 Virtual Angle-of-Attack Sensor An aircraft carries many (redundant) hardware sensors on board, measuring a wide variety of variables. Due to the relations between the measured signals, a lot of redundant information is available. This redundant information can be used to estimate a certain variable through a number of available signals that represent other variables, i.e. non-like signals. In this chapter, the design of a virtual angle-of-attack sensor is described. The virtual sensor consists of a linear parameter varying model, whose parameters are determined by a Takagi-Sugeno fuzzy model, plus a nonlinear black-box model. The TakagiSugeno fuzzy model is designed using data from linear models. The black-box model consists of a neural network that is trained to reduce the estimation error of the linear parameter varying model. The inputs of the neural network are selected using a genetic search algorithm followed by a backward elimination procedure. The neural network is designed and trained using data from nonlinear simulations.

This chapter is organized as follows: A brief introduction is given in Section 6.1 followed by the description of the design requirements for the virtual angle-ofattack sensor in Section 6.2. The structure of the virtual sensor is addressed in Section 6.3. In Section 6.4 the design and parameter optimization of the TakagiSugeno fuzzy model and the neural network are discussed. The validation of the virtual sensor is described in Section 6.5. Concluding remarks and suggestions for future work are presented in Section 6.6.

6.1

Introduction

Voting/monitoring systems that are based on cross-comparison of like-signals are unable to isolate a sensor failure when there are two physical sensors in operation. A virtual sensor can in this case be used as a discriminator, which may also be used to identify a fault in the last available physical sensor. In terms of reliability or flight safety, with the virtual sensor it is possible to get more performance out of the same number of physical sensors. A second option is to replace a physical sensor by a virtual sensor, which results in fewer hardware components and the 77

78

Chapter 6. Virtual Angle-of-Attack Sensor

45 40 35 M

MAX

Altitude [kft]

30

= 0.85

VC = 150 Kts

25 20 15

VC = 375 Kts

10 5 0

0.2

0.3

0.4 0.5 0.6 Mach number [−]

0.7

0.8

0.9

Figure 6.1: Flight envelope of the SCA. The black solid line defines the flight envelope in clean configuration. The grey line defines the flight envelope in landing configuration.

associated cost benefits. Although the integration of sensor information has its advantages, it should be noted that it introduces mutual dependencies amongst non-like signals. This is true for all model-based FDI methods. In particular for the physical Angle-of-Attack (AoA) sensor, there are more issues to consider that motivate the use of a virtual sensor. First of all, these sensors are mounted on the outside of the fuselage. Due to its vicinity to the fuselage, the physical AoA sensor is not situated in a free airflow and the measured AoA signal is therefore corrupted and noisy. Moreover, these sensors are also vulnerable to damage because of their location and harsh operating environment. Because of these problems that are encountered with the use of physical AoA sensors, a virtual AoA sensor can make a significant contribution. As the research described in this chapter serves as a proof-of-principle for the proposed methodology, only the flight envelope for the landing phase is considered rather than the entire flight envelope. This is typically the flight regime where the AoA is high and therefore most critical with respect to potential stall. In view of the AoA protection system, the potential added value of a virtual AoA sensor is higher for the landing configuration than for the clean configuration.

6.2

Design Requirements

The estimation error of the virtual AoA sensor has to be less than one degree. This level of accuracy makes the virtual sensor suitable for monitoring the physical angle-of-attack sensors. The virtual AoA sensor should be designed such that this accuracy is met under the following circumstances:

6.3. Structure of the Virtual Angle-of-Attack Sensor

79

Dynamic maneuvering (both longitudinal as well as lateral). Variations of flight condition within the landing flight envelope. The flight envelope for the landing configuration is denoted by the grey line in Figure 6.1. The calibrated airspeed varies between 110 and 195 knots, while the altitude up to 10 kft is considered. Variations of aircraft weight and center-of-gravity. The allowed variation in aircraft weight and the variation in the position of the CG along the X-axis is illustrated in Figure 5.3. Aerodynamic uncertainties. The aerodynamics of each aircraft of the same type are different, for example due to production variations, but also weather conditions have an impact on the aerodynamics. In this chapter, the four pre-defined aerodynamic models that are available in the SE are considered, see Section 5.4.

6.3

Structure of the Virtual Angle-of-Attack Sensor

The main difference of the approach proposed in this section with respect to using, for example, a Neural Network (NN) or NARX-model to estimate the AoA, is that in this case only those parts that are unknown and/or (highly) nonlinear are estimated by a black-box model. The aircraft dynamics are well-known and it does not make sense to use a (black-box) model with a completely different structure to represent them. The proposed virtual AoA sensor consists of two parts, a Linear Parameter Varying (LPV) model and a neural network model: α ˆ = αLP V + αN N

(6.1)

The NN model is a black-box model, while the LPV model has an interpretable structure with varying parameters (white-box structure). The latter consists of the trimmed AoA (α0 ) plus a linear Short-Period (SP) approximation of the AoA (αSP ): (6.2) αLP V = α0 + αSP The structure of these three elements of the virtual AoA sensor, namely α0 , αSP and αN N are discussed below.

6.3.1

The trimmed angle-of-attack α0

The trimmed angle-of-attack α0 is the AoA of the aircraft in an equilibrium condition. To estimate the trimmed angle-of-attack, a Takagi-Sugeno fuzzy model is

80

Chapter 6. Virtual Angle-of-Attack Sensor

Figure 6.2: Architecture of the TS fuzzy model to estimate α0 . used (Takagi and Sugeno 1985). The inputs of the TS fuzzy model, which are selected using a Genetic Algorithm (GA) optimization procedure (see Section 6.4), are Mach number, dynamic pressure, bank angle, position of the CG along the X-axis and aircraft weight: Ri :

If M is ZM,jM and q is Zq,jq and φ is Zφ,jφ and XCG is ZXCG ,jXCG and W is ZW,jW then α0 = α0,i

for i = 1, . . . , Nr

(6.3)

where jM , jq , jφ , jXCG and jW denote the jth membership function of their corresponding variable. It should be noted that two of the inputs of the TS fuzzy model, namely the position of the CG along the X-axis and the aircraft weight, cannot be measured directly. Instead a NN based virtual sensor developed by Idan et al. (2004) is used to provide an estimate of these variables. The inputs of this virtual sensor are Mach number (M ), dynamic pressure (q), pitch attitude (θ), flight path angle (γ) and elevator deflection (δe ). The correlation between the aircraft weight and the angle-of-attack is more or less linear. The aircraft weight is directly related to the required lift, which is mainly dependent on the lift coefficient CL and the dynamic pressure. Not taking into account configuration changes, the lift coefficient is mainly a function of the angle-of-attack. Because of this direct relation between AoA and aircraft weight, the TS fuzzy model to estimate α0 consists of two stages. The first stage consists of a number of TS fuzzy models that each estimate the α0 for a specific aircraft weight, see Figure 6.2. The inputs of these TS fuzzy models are Mach number, dynamic pressure, bank angle and the position of the CG along the X-axis. The corresponding rule-base is as follows: Ri :

If M is ZM,jM and q is Zq,jq and φ is Zφ,jφ and XCG is ZXCG ,jXCG then α0,W 1 = α0,W 1i

for i = 1, . . . , Nr

(6.4)

In the second stage the outputs of these TS fuzzy models are weighted as a function of the aircraft weight using linear interpolation.

6.3. Structure of the Virtual Angle-of-Attack Sensor

6.3.2

81

The short-period approximation of the angle-of-attack αSP

The starting point for the derivation of the expression for αSP is the linear shortperiod approximation (see also Appendix B):     Zδe Zw Zq w w˙ (6.5) = ˜ ˜q q + M ˜ δ δe q˙ Mw M e where the variables w, q and δe denote the variation of the corresponding variables with respect to the trimmed condition. Using the formula: G(s) = C (sI − A)−1 B

(6.6)

where in this case the matrix C is an identity matrix of dimension two, the following two transfer functions can be derived from Equation 6.5: w(s) δe (s)

=

q(s) δe (s)

=

˜ δ − Zδ M ˜ q) Zδe s + (Zq M e e ˜ q − Zq M ˜ w) ˜ q )s + (Zw M s2 − (Zw + M ˜ δ s + (Zδ M ˜ w − Zw M ˜δ ) M e e e ˜ q − Zq M ˜ w) ˜ q )s + (Zw M s2 − (Zw + M

(6.7) (6.8)

The transfer function from the pitch-rate to the downward velocity can be obtained by dividing Equation 6.7 by Equation 6.8: ˜ δ − Zδ M ˜ q) Zδe s + (Zq M w(s) e e = ˜ ˜ ˜ q(s) Mδe s + (Zδe Mw − Zw Mδe )

(6.9)

Using the relation α = w/U0 , Equation 6.9 can be rewritten to the transfer function from pitch-rate to angle-of-attack: ˜ δ − Zδ M ˜ q) α(s) 1 Zδe s + (Zq M e e = ˜ δ s + (Zδ M ˜ w − Zw M ˜δ ) q(s) U0 M e e e

(6.10)

Equation 6.10 can be simplified to: α(s) b1 s + b 0 = q(s) a1 s + a0

(6.11)

˜ δ − Zδ M ˜ q ), a1 = M ˜w − ˜ δ and a0 = (Zδ M where b1 = U10 Zδe , b0 = U10 (Zq M e e e e ˜ δ ). The parameters b1 , b0 , a1 and a0 are obtained from the linearized aircraft Zw M e model. Since the virtual sensor is implemented in the digital flight control system, the model must be discretized. The Tustin discretization is used to obtain the discrete-time short-period approximation. This method results in a discrete-time model that matches best the continuous-time model in terms of the frequency response. The discrete time transfer function becomes: α(k) =

ˆb1 σ + ˆb0 q(k) σ+a ˆ0

(6.12)

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Chapter 6. Virtual Angle-of-Attack Sensor

Figure 6.3: Architecture of the LPV model αLP V . where σ denotes the shift operator. Nonlinear simulation experiments have shown that keeping the parameters ˆb0 and ˆb1 fixed at the center values of their ranges, instead of varying them as a function of flight condition, has no effect on the performance of the virtual sensor. For this reason these parameters are kept constant and only a ˆ0 is varying as a function of flight condition. Besides the TS fuzzy model to estimate α0 , see Equation 6.3, a second TS fuzzy model is designed to approximate a ˆ0 . The complete LPV part of the virtual sensor is as follows: αLP V (k) = α0 (x(k)) +

ˆb1 σ + ˆb0 q(k) σ+a ˆ0 (x(k))

(6.13)

where x = [M, q, φ, XCG , W ]. In order to get the correct pitch rate during lateral maneuvers, the pitch rate measurement needs to be modified as follows: qφ = q −

g tan(φ) sin(φ) VT

(6.14)

The term VgT tan(φ) sin(φ) describes the approximation of the output of the pitch rate sensor during turns with constant pitch rate, constant bank angle and constant turn rate (McLean 1990). This signal is therefore not related to the pitch rate in terms of the rotational velocity of the aircraft around the Y-axis. Substitution of q(k) by qφ (k) in Equation 6.13 results in: ˆ0 (x(k))) qφ (z) αLP V = α0 (x) + H(σ; a where H(σ; a ˆ0 (x(k))) =

ˆb1 σ + ˆb0 . σ+a ˆ0 (x(k))

(6.15)

(6.16)

In the first simulation experiments it turned out that, although the static estimation of α0 and a ˆ0 is good, during dynamic nonlinear simulation the performance of the LPV model was poor. The TS fuzzy model is designed based on data of the linearized SCA model in trimmed condition. However, during maneuvering the aircraft is not trimmed, which means that the estimated trimmed angle-of-attack is leading the true angle-of-attack. This effect is compensated for by a low-pass filter, which lags the estimated α0 during maneuvering. The resulting structure of αLP V is illustrated in Figure 6.3.

6.4. Design of the TS Fuzzy Model and the NN Model

83

Figure 6.4: Architecture of the virtual AoA sensor. 6.3.3

The neural network αN N

Neural networks have been trained to perform complex functions in various fields of application including pattern recognition, identification, classification, speech, vision and control systems. In general a neural network is used when the exact nature of the relationship between inputs and outputs is not known. If this relationship is known, it should be modelled directly. The other key feature of neural networks is that they learn the input/output relationship through training. Neural networks are discussed in more detail in Appendix D.3, where also a number of useful references are given. As mentioned above, neural networks are used when the exact nature of the relationship between inputs and output is not known. In this case the objective is to use a neural network to reduce the estimation error ∆α of the LPV model: ∆α = α − αLP V .

(6.17)

In other words, the neural networks is used to account for that part of the AoA signal that is not accounted for using the LPV model. The structure of the entire virtual AoA sensor is illustrated in Figure 6.4 and is written as follows: α(k) ˆ = α0 (x(k)) + H(σ; a ˆ0 (x(k))) qφ (k) + αN N (xNN (k))

(6.18)

where αN N denotes the output of the NN model and xNN = [θ nz , nz , qφ , cos φ] results from a nonlinear input selection procedure (see Section 6.4).

6.4

Design of the TS Fuzzy Model and the NN Model

The parameters of the LPV model are computed through a TS fuzzy model. The TS fuzzy model is designed using data from linear models in trimmed condition.

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Chapter 6. Virtual Angle-of-Attack Sensor

On the other hand, the NN model is trained using data from (dynamic) nonlinear simulations. 6.4.1

TS fuzzy model design

The TS fuzzy model is designed using data obtained from the nonlinear SCA model. The SCA model is trimmed and linearized for a large number of flight conditions within the flight regime of interest, which in this case is the flight envelope for landing configuration. From each trimmed flight condition, the trimmed angle-of-attack is extracted and put in a data set together with the corresponding input values, i.e. [M, q, φ, XCG ] (see also Figure 6.2). The design of the Takagi-Sugeno fuzzy model is performed in two steps. First the optimal structure of the TS fuzzy model is determined, i.e. the inputs of the model and the number of membership functions for each input. This is done using a GA optimization approach (Goldberg 1989, Michalewicz 1996), see Appendix E. The consequent part of the TS fuzzy model is computed by Least Squares (LS) optimization, with the objective to minimize the Root Mean-Square Error (RMSE), see also Appendix C.1, between the data set and the TS fuzzy model output. The objective function J for the GA optimization is:  N 1
Nr (6.19) (α0,k − α ˆ 0,k )2 · e0.5( 64 ) J = N k=1

where α0,k denotes the kth sample of the data set, α ˆ 0,k denotes the kth output of the TS fuzzy model, N denotes the number of data points and Nr denotes the number of rules. The fitness of each chromosome in the population is evaluated using the 10-fold cross validation method, see Appendix C.2. The secondary objective is to keep the model transparent, for a TS fuzzy model this means to keep the number of rules low. The RMSE of each chromosome is therefore multiplied by the Nr factor e0.5( 64 ) , where the numbers 0.5 and 64 are the result of an iterative tuning process. The more complicated the TS fuzzy models, i.e. the more rules describe the TS fuzzy model, the larger the additional penalty. The resulting inputs of the TS fuzzy model are given in Section 6.3. It should be noted that the structure of the TS fuzzy model is determined for fixed aircraft weight, see also Figure 6.2. In the second step, once the inputs and the number of membership functions per input are determined, the parameters of the membership functions are optimized taking again a GA approach, in combination with the 10-fold cross validation method, and the same data set. Again the objective function in Equation 6.19 is Nr used, except for the penalty term e0.5( 64 ) . 6.4.2

NN model design

The most common type of artificial neural network consists of three groups, or layers, of units: a layer of input units is connected to a layer of hidden units, which is connected to a layer of output units. This type of neural network is also used

6.4. Design of the TS Fuzzy Model and the NN Model

85

in this thesis. The activity of the input units represents the raw information that is fed into the network. The activity of each hidden unit is determined by the activities of the input units and the weights on the connections between the input and the hidden units. The behavior of the output units depends on the activity of the hidden units and the weights between the hidden and output units. The NN model is designed such that it reduces the estimation error (∆α) of the LPV model, see Equation 6.17. In order to obtain data for the design of the NN model, the LPV model is evaluated through a large number of nonlinear simulations (450). The initial condition and the pilot input, in terms of input shape and input force, for each simulation are selected randomly from a set of predefined samples. The initial condition is determined by the Mach number, altitude, aircraft weight, position of the CG along the X-axis and the aerodynamic model. The predefined pilot input shapes are the so-called 3-2-1-1 input in column and in wheel, the block-shaped input in column and in wheel, the pull-up and the push-down maneuver manoeuvre , the wind-up turn maneuver and the combined wind-up turn and pull-up maneuver. From the data that are generated, 60% is used for the training and 40% is used for the validation of the NN model. The NN model is designed in two steps. First the inputs of the NN model are determined through a nonlinear input selection procedure, then the number of neurons in the hidden layer is determined. The objective is to find a compromise between good performance and low complexity (few inputs and few neurons in the hidden layer). The inputs for the NN are determined in two steps. First a set of candidate inputs is selected using a GA based search algorithm. From this set the inputs of the NN are selected through a backward elimination procedure. In the first step a pre-selection is made on the basis of second order polynomial models. In the second step the final selection is made on the basis of NN models. The GA based search algorithm (Maertens et al. 2004) is used to generate candidate inputs for the NN model. The genetic algorithm evolves “in parallel” a large number (e.g. 100) of different structures of the polynomial model. For each given structure, the parameters are determined by the LS method and the fitness value for that model is calculated from the corresponding RMSE. If some of the regressor variables are not present in any of the polynomial terms, they are deleted from the regressor set and the procedure is repeated with a smaller number of regressors until a desired (user-defined) number of variables is reached. Due to the random nature of the GA, the search has to be performed multiple times (e.g. 50) in order to get a statistically sound result. The six most promising variables are selected as candidate variables for the NN model, namely: θ nz , nz , cos φ, qφ , qφ VT and qφ /VT . First the performance of the NN model with these six candidate variables as inputs is evaluated. The neural three-layer feed-forward backpropagation network is trained using the training data set. Both the input layer and the hidden layer have as many neurons as the number of inputs of the NN (in this case six). The output layer has one neuron. The network is trained using the Levenberg-Marquardt

86

Chapter 6. Virtual Angle-of-Attack Sensor

0.17 0.16

RMSE

0.15 0.14 0.13 0.12 0.11 0.1 6

5

4 Number of inputs

3

2

Figure 6.5: Backward elimination procedure, the RMSE as a function of the number of inputs of the NN model. algorithm (Bishop 1995, Shepherd 1997). In order to determine the least valuable candidate variable, six neural networks are trained and validated with five inputs, leaving each time one of the candidate variables out. Again the input layer and hidden layer both have as many neurons as the number of inputs of the NN (in this case five) and the output layer has one neuron. The candidate variable that is left out of the NN model with the best performance of the six NN models, is the least valuable and therefore removed from the set of candidate variables. Typically the performance if this NN model is less than that of the NN model with all six candidate variables as inputs. This procedure, the so-called backward elimination procedure, is terminated when the removal of the latest input results in a relatively large reduction in the performance. This is also illustrated in Figure 6.5, where it can be seen that there is a relatively large reduction in the performance of the NN model when removing the fourth input variable. The number of inputs for the NN model is therefore set to four and these inputs are: θ nz , nz , cos φ and qφ . The optimal number of hidden neurons is determined based on a number of NN models with increasing numbers of neurons. Each extra neuron should have a significant contribution in the performance of the NN model (on the training and on the validation data set). The number of hidden neurons is set to eight.

6.5

Validation of the Virtual AoA Sensor

In this section all the design requirements, as described in Section 6.2, are taken into account, except for the variation in aircraft weight. The aircraft weight varies between W = 31850 and W = 35750 [lbs], which represents 20% of the total range in aircraft weight. Due to the modular structure of the virtual AoA sensor, the total range in aircraft weight can be covered by adding more LPV models designed for different aircraft weights.

6.5. Validation of the Virtual AoA Sensor

87

Table 6.1: Performance of the TS fuzzy models.

RMSE VAF [%]

W = 31850 [lbs] α0 [deg] a ˆ0 [-] 0.1267 2.03 · 10−4 99.86 99.43

W = 35750 [lbs] α0 [deg] a ˆ0 [-] 0.0479 2.07 · 10−4 99.98 99.23

First the TS fuzzy model and LPV model is validated, before the complete virtual AoA sensor is validated. This will enable us to appreciate the added value each element of the virtual sensor. Bank angles larger than 45 degrees are not taken ˆ0 is limited into account, while the bank angle for the estimation of α0 and a to 30 degrees. As mentioned in Section 6.3, the parameters ˆb1 and ˆb0 are kept constant at their center values. For the compensation of the pitch rate signal, see Equation 6.14, the bank angle is limited to 33 degrees.

6.5.1

Validation of the TS fuzzy models

Separate TS fuzzy models have been designed for the estimation of α0 and a ˆ0 . The TS fuzzy model to estimate α0 has seven membership functions, namely two for the Mach number, three for the dynamic pressure and two for the bank angle. The TS fuzzy model uses ten rules, which means that not all the possible 12 combinations of the MFs are used. The two combinations of the MFs that are not used, covered spaces that do not contain data, i.e. the aerodynamic model is not defined there, which means that you can not fly there. The corresponding rule-base is as follows: R1 : R2 : .. .

If M is ZM,1 and q is Zq,1 and φ is Zφ,1 then α0 = α0,1 If M is ZM,1 and q is Zq,1 and φ is Zφ,2 then α0 = α0,2

R10 :

If M is ZM,2 and q is Zq,3 and φ is Zφ,2 then α0 = α0,10

where α0i = c0,i + cM,i M + cq,i q + cφ,i φ + cCG,i XCG . The TS fuzzy model to estimate a ˆ0 has nine membership functions, namely two for the Mach number, the dynamic pressure and the position of the center-of-gravity along the X-axis and three for the bank angle. The TS fuzzy model uses 16 rules out of the 24 possible combinations of the MFs. The performance of the TS fuzzy models to estimate α0 and a ˆ0 for each aircraft weight is summarized in Table 6.1 in terms of RMSE and Variance Accounted For (VAF). These performance measures are defined in Appendix C.1. Performances of the TS fuzzy models are good, although it should be noted that the TS fuzzy model to estimate α0 for aircraft weight W = 35750 [lbs] outperforms the TS fuzzy model to estimate α0 for aircraft weight W = 31850 [lbs].

88

Chapter 6. Virtual Angle-of-Attack Sensor

Table 6.2: Performance of the LPV model in estimating the angle-of-attack in nonlinear simulation.

RMSE [deg] VAF [%]

W = 31850 [lbs] 0.2748 99.38

W = 35750 [lbs] 0.2317 99.52

31850 < W < 35750 [lbs] 0.2323 99.53

15 α [deg]

10 5 0

−5 0

0.5

1

1.5 Sample

2

2.5

3 4 x 10

0.5

1

1.5 Sample

2

2.5

3 4 x 10

∆ α [deg]

1 0.5 0

−0.5 −1 0

Figure 6.6: Performance of the LPV model in nonlinear simulation. 6.5.2

Validation of the LPV model

The performance of the LPV model during nonlinear simulation is evaluated in terms of RMSE and VAF with respect to the true angle-of-attack signal, see Table 6.2. In this table the performance is given for W = 31850 and W = 35750 [lbs], the two aircraft weights for which the TS fuzzy models are designed, and for the range from W = 31850 to W = 35750 [lbs]. Since the performance of the latter is of the same order as for the fixed aircraft weight cases, it can be concluded that the linear interpolation as a function of aircraft weight works well. The RMSE value is below 0.24 degrees while the VAF value is above the 99%. The corresponding time history is shown in Figure 6.6.

6.5.3

Validation of the virtual AoA sensor

The virtual sensor is validated using the validation data set that has not been used in any stage of the design. The performance of the virtual sensor is given in Ta-

6.5. Validation of the Virtual AoA Sensor

89

Table 6.3: Performance of the virtual sensor in estimating the angle-of-attack in nonlinear simulation.

RMSE [deg] VAF [%] max |∆α| [deg]

Training 0.1088 99.90 0.7314

Validation 0.1125 99.90 0.7781

15 α [deg]

10 5 0

−5 0

0.5

1 Sample

1.5

2 4 x 10

0.5

1 Sample

1.5

2 4 x 10

∆ α [deg]

0.8 0.4 0

−0.4 −0.8 0

Figure 6.7: Performance of the virtual sensor in nonlinear simulation on the training data set. ble 6.3. The RMSE values are below 0.12 degrees while the VAF values are again above the 99%. It can be seen that the difference in performance on the training and validation data set is in balance, suggesting that the complexity of the NN is about right. The corresponding simulation results are illustrated in Figures 6.7 and 6.8. In terms of the RMSE, the improvement in the performance of the virtual sensor is significant compared to the LPV model. The absolute estimation error remains below the 0.8 degrees for both the training and the validation data set. In this section the variation in aircraft weight is 3900 lbs. The total allowable variation for the SCA model is 17400 lbs, see also Figure 5.3, which means that five local LPV models would be needed to cover the entire centogramme. In case the performance requirements of the virtual AoA sensor cannot be met with a single NN for the entire range of the aircraft weight, a separate NN model should be designed for each aircraft weight regime (W 1 → W 2, W 2 → W 3, etc). This option will result in more accuracy at the price of a more complicated virtual AoA sensor.

90

Chapter 6. Virtual Angle-of-Attack Sensor

15 α [deg]

10 5 0

−5 0

2000

4000 6000 Sample

8000

10000

2000

4000 6000 Sample

8000

10000

∆ α [deg]

0.8 0.4 0

−0.4 −0.8 0

Figure 6.8: Performance of the virtual sensor in nonlinear simulation on the validation data set.

6.6

Conclusions

The virtual AoA sensor described in this chapter consists of two parts, namely a LPV model and a NN model. In the LPV model part the TS fuzzy model estimates the trimmed angle-of-attack α0 as well as the parameter in the denominator of αSP , while in the (nonlinear) black-box part the NN is designed to fit the estimation error remaining from the LPV model. The design example in this chapter serves as a proof-of-principle for the proposed methodology. The virtual sensor performed well, both for the training and the validation data set. The design requirement of estimation errors less than 1 degree was met. The RMSE was kept well below 0.12 degrees while the VAF was kept well above 99%. It should be noted that the automatically generated data does not reach the maximum angle-of-attack of 16 degrees, because this is prevented by the envelope protection system. Around this angle-of-attack the aircraft model is more nonlinear and it can therefore be expected that improving the data set will result in slightly more complex TS fuzzy models (and neural network) in order to achieve the same performance. It should also be noted that in the simulation results the true aircraft weight or the true position of the center-of-gravity along the X-axis have been used as inputs to the TS fuzzy model and NN. These should in fact be estimates of the virtual sensor developed by IIT (Idan et al. 2004). It is expected that the corresponding decrease in performance of the virtual sensor is negligible, since these variables can be estimated accurately.

7 Soft Sensor Management and Virtual Sensors for FDIR A sensor management system based on soft computing techniques has been developed and implemented in the flight control system of the SCA model. Unlike in the conventional sensor management system, the signals from sensors are assigned weights based on fuzzy membership functions and the consolidated signal is computed as a weighted average. This approach improves the quality of the consolidated signal and reduces transients due to sensor failures. The soft voting scheme serves as a basis for soft flight control law reconfiguration. In addition, it is illustrated how a virtual sensor can serve as an arbitrator which enables the isolation of the failed sensor in the duplex mode and the detection of a sensor failure in the simplex mode. The effectiveness of the proposed methods is demonstrated by using the SCA model, taking into account sensor failures in pitch rate and normal acceleration. The properties of the conventional sensor management system have been retained, with the additional advantage that the quality of the consolidated signal is improved, failure-induced transients are reduced and the consolidated signal remains available up to the last valid sensor.

This chapter is organized as follows: In Section 7.1 a short introduction of the sensor management problem is given. The conventional sensor management system and the flight control law reconfiguration are discussed in Section 7.2. Section 7.3 introduces the sensor management system and flight control law reconfiguration based on soft computing, which is extended to include virtual sensors in Section 7.4. Concluding remarks and future research are discussed in Section 7.5.

7.1

Introduction

Sensor management based on majority voting and point consolidation of like signals is a proven technology in modern fly-by-wire flight control systems (Rosenberg 1998). The assumption is that the majority of like signals represent the truth and that any single dissimilar signal is the result of a failure. Such a signal must be 91

92

Chapter 7. Soft Sensor Management and Virtual Sensors for FDIR

disconnected as soon as the failure is detected. The probability of multiple simultaneous failures is considered to be extremely remote. In the conventional approach, the decision whether a sensor has failed or not is crisp. In order to reduce the sensitivity of this decision to uncertainties like quantization and measurement noise, a properly adjusted threshold is used. This threshold is a compromise between two goals: the absence of false alarms and the ability to detect all possible failures within a certain time frame. This inevitably leads to a transient response during which the consolidated signal temporarily differs from the true value. The soft sensor management system introduced in this chapter maintains the key properties of the conventional sensor management system (majority voting) while improving its performance by applying fuzzy logic techniques. Using fuzzy logic, the decision whether a sensor has failed or not is no longer crisp. In the soft sensor management system the signals are assigned weights based on a cross-comparison of like signals by using fuzzy membership functions. The consolidated signal is then computed as a weighted average. Compared to the conventional management system, the soft management system intervenes at an earlier stage by reducing the weight of the suspected faulty sensor signal, while the failure declaration occurs at a later stage. This approach improves the quality of the consolidated signal with respect to its difference from the true value due to sensor failures. An additional attractive feature of this approach is the reduction of the transients due to sensor failures. Although FL techniques have been implemented in other application domains, such as the process industry (Frank and Marcu 1999, Schneider and Frank 1996), their application for FDI in flight control systems has not been extensively investigated yet. Furthermore, a virtual sensor is introduced in order to be able to identify failed sensors in the duplex mode and to detect a sensor failure in the simplex mode (which is not possible with the current sensor management systems). In the literature, many applications of analytical redundancy for fault detection and fault isolation in flight control systems have been reported (Patton et al. 1989, Patton and Chen 1992, Isermann 1984), however, the use of virtual sensors in aerospace applications is novel.

7.2

Conventional Sensor Management and FCL Reconfiguration

Each signal is measured by a number of independent sensors. The sensor management system has two tasks, namely the computation of a consolidated signal from these measurements (voting) and the validation of each of the sensors (monitoring). The consolidated (or voted) signal is fed to the flight control computer and at the same time serves as a reference for sensor validation.

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Figure 7.1: Conventional triplex sensor management system (Source: (Rosenberg 1998)).

7.2.1

Conventional voting/monitoring scheme

The redundancy level for each signal, i.e. the number of redundant sensors in normal mode, is related to the system architecture, the failure probability of the sensor and the consequence of losing the corresponding signal. For example, the consequence of losing the pitch rate signal is a catastrophic failure. The probability of a catastrophic failure must be less than 10−9 per hour of flight. This level of reliability can not be achieved with a single pitch rate sensor and therefore redundant pitch rate sensors need to be installed. It should be noted that a sensor failure in the duplex mode (two sensor signals available) results in losing the signal, since the conventional voting/monitoring scheme is not able to identify the failed sensor in this case. In order to keep the presentation simple, the triplex sensor system (three sensor signals available) will be used to explain the conventional voting/monitoring philosophy (see Figure 7.1). The three sensor signals are first sorted from the largest value to the smallest one. The mid-value signal is taken as a reference and the two extreme-value signals are limited in their deviation from the mid-value signal. When the limits are not invoked, the voted (consolidated) signal is given by: Svoted = S2 + 0.25 (S1 − S2 ) + 0.25 (S3 − S2 ) = 0.25 (S1 + 2 S2 + S3 ). For two valid signals, a duplex voter is used, and the voted signal is a simple average. The monitor compares each of the three sensor signals Si with the consolidated signal Svoted . If the absolute difference is smaller than a predefined threshold ∆, the monitor count is decreased by one, otherwise it is increased by two:

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If |Si − Svoted | ≤ ∆ then (count rate)i = −1 If |Si − Svoted | > ∆ then (count rate)i = +2. The updated count value is bounded between zero and the failure declaration value. If the count value has reached the failure declaration value, a failure is declared and the signal is latched (see Figure 7.1). Since in this case only two sensor signals are still available, automatically the sensor management system reconverts to duplex mode. The logic in the conventional sensor management system is such that a failed sensor output continues to contribute to the voted signal until it is latched. This implies that when signal is latched, its contribution to the consolidated signal is instantly removed. This discontinuity in the consolidated signal results in a transient in the aircraft motion, which is illustrated by nonlinear, closed-loop simulation examples using the SCA model in Matlab/SimulinkTM . 7.2.2

Simulation examples

The functionality and performance of the sensor management systems is demonstrated using pitch rate and normal acceleration sensor failures during longitudinal maneuvering. The closed-loop transients due to sensor failures are not of the same order for each signal. From Figure 4.1 it can be seen that the pitch rate is fed back through a proportional gain in the pitch damper path and through a proportional and an integral gain in the feedback path. In the feedback path the normal acceleration is fed back in a similar way. However, discontinuities in the voted normal acceleration signal are suppressed by the low-pass filter in the normal acceleration feedback path (Figure 4.1). Closed-loop transients are therefore less evident. For this reason the pitch rate signal is used to demonstrate this additional benefit of the sensor management system based on soft computing techniques. The normal acceleration signal is used only to demonstrate the functionality of the conventional management system in the case of a sensor failure in duplex mode, since this requires a reconfiguration mode which is not available for the loss of the pitch rate signal in the SCA model. As mentioned in Section 7.2.1, the loss of the pitch rate signal is a catastrophic failure. A simplified representation of the longitudinal flight control laws is illustrated in Figure 4.1. In this figure it can be seen that the feedback signal consists of a blending of the pitch rate and the normal acceleration signal. In order to compare the performance of the conventional sensor management system to that of the soft management system, sensor failures are considered in these two signals. The blending function is described in Section 4.1. Two nonlinear simulations are used to illustrated the functionality of the conventional sensor management system. For both simulations, the initial condition is a straight and level flight at a Mach number of M = 0.75 and an altitude of h = 40 kft, which is the cruise flight condition for the SCA model. In this flight condition the calibrated airspeed is equal to VC = 225 knots. The pilot input is a block-shaped input of the maximum positive column deflection starting at

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Figure 7.2: Conventional sensor management: drift failure of a pitch rate sensor. Figures e-g are zoomed in on the failure-induced transients.

t = 1 sec and lasting for 6 sec. During this maneuver, the normal acceleration signal is partly contributing to the feedback signal. This is necessary in order to be able to evaluated failure-induced transients when considering failures in the normal acceleration sensors. The time histories corresponding to the first simulation example are illustrated in Figure 7.2. At t = 1 sec, a drift failure of 1 deg sec−2 occurs in one of the pitch rate sensors (Figure 7.2a). Figure 7.2b shows the difference ∆qvoted between the voted signal and the true pitch rate. Due to the drift failure, the voted signal diverges from the true pitch rate until the contribution of the failed sensor output is limited and the mismatch remains constant. When the difference between the failed sensor (q1 ) and the voted signal (qvoted ) exceeds the monitor threshold, the monitor count rate increases from −1 to +2 (denoted by the first vertical dashdotted line). When the monitor count (Figure 7.2c) reaches the failure declaration value (denoted by the second dash-dotted vertical line) the signal is latched and the number of valid signals reduces to two (Figure 7.2d). The faulty contribution of q1 is omitted instantaneously, which results in an undesirable discontinuity in the consolidated signal (Figure 7.2b). The resulting transients in the elevator deflection (Figure 7.2e), the normal acceleration (Figure 7.2f), and the true pitch rate (Figure 7.2g) signals are evident (solid line) compared to the fault free case (dash-dotted line). The second simulation example illustrates a cut-off sensor fail-

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Flight control law reconfiguration

The voted signal in the duplex mode is computed as the average of the two sensor signals. If a sensor fails in the duplex mode, the majority voting principle can no longer be used to identify the failed sensor. As soon as the difference between these two signals exceeds a certain threshold, both sensors are declared invalid and the FCS reconfigures to not using this particular signal. In Figure 7.4 a drift failure of the second normal acceleration sensor is simulated (Figure 7.4a). The voted signal is the average of the two valid signals. The initial condition is a straight and level flight at a Mach number of M = 0.70 and an altitude of h = 25 kft. This flight condition is selected to increase the contribution of the normal acceleration signal

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Figure 7.4: Conventional sensor management: drift failure of the second normal acceleration sensor. in the feedback path. The pilot input is a block-shaped input of maximum positive column deflection starting at t = 6 sec and lasting for 6 sec. When the difference between the two sensor signals exceeds the threshold, the monitor count of both sensor signals is set to the failure declaration value instantaneously and both input signals are latched (Figure 7.4c). At this point the consolidated signal is no longer available and the FCS reconfigures to not using this signal (Figure 7.4d). This implies that only the pitch rate signal is used in the feedback path for the entire range of the admissible column deflection and calibrated airspeed. The blending of the pitch rate signal and the normal acceleration signal in the feedback path is explained in more detail in Section 4.1.

7.3

Sensor Management and FCL Reconfiguration Based on Soft Computing

The simulation examples in Section 7.2.2 and 7.2.3 have demonstrated the main shortcomings of the conventional sensor management approach, namely the failureinduced discontinuities in the consolidated signal and the inability to identify sensor failures in duplex mode and/or to detect a sensor failure in simplex mode. A fuzzy logic approach can be used to improve the conventional sensor management system without changing the basic concept of majority voting. In this section we focus on reducing, or even eliminating, the transients in the voted signal due to sensor failures.

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Figure 7.5: Soft voting in the triplex mode. The current value of each sensor signal forms the center of its corresponding membership function, which is used to determine the membership degree of this sensor signal. 7.3.1

Soft voting/monitoring scheme

The soft voter is different from the conventional voting scheme in the sense that each input signal is assigned a weight, and the consolidated signal is the weighted average of the input signals: Svoted =

n


wi Si ,

(7.1)

i=1

where wi denotes the weight assigned to the ith input signal Si and n denotes the number of valid sensors. The weight wi is the normalized membership degree µi : µi wi = n j=1

µj

,

(7.2)

where 0 ≤ µi ≤ 1. The computation of the membership degree µi is illustrated in Figure 7.5. The current value of each signal forms the center of its corresponding membership function. The membership degree of the signal is the largest membership degree of the remaining valid signals according to this membership function: µi = max(µi (qj )). i=j

(7.3)

In Figure 7.5a the membership function of q3 is illustrated. With respect to this membership function, q1 has a membership degree of µ3 (q1 ) = 0.35 and q2 has a membership degree of µ3 (q2 ) = 0.65. This implies that q3 has a membership degree of: µ3 = max(µ3 (q1 ), µ3 (q2 )) = 0.65 .

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Figure 7.6: Soft triplex sensor management system.

Clearly, the majority voting concept of the conventional sensor management system is also used in the soft sensor management system. The signal q3 is not in agreement with the signals q1 and q2 , and therefore its weight in the voted signal is reduced. In Figure 7.5b the discrepancy between signal q3 and the signals q1 and q2 is further increased. The corresponding membership degree is now reduced to µ3 = 0. Both the conventional and the soft voting scheme are based on majority voting. The major difference is the way the like sensor signals contribute to the consolidated signal. In the conventional voting scheme, the contribution of a faulty signal is limited, while in the soft voting scheme its weight is reduced. The implementation of the soft voting scheme is illustrated in Figure 7.6 for a triplex sensor system. The vector of like signals is split and sorted. The membership degrees are computed according to Equation 7.3, put back in the original order and combined again in a vector. The voted signal is then computed according to Equations 7.1 and 7.2. In the monitor part, the count rate of the ith signal is the following function of its corresponding membership degree µi : If µi = 1 then (count rate)i = −1 If 0 < µi < 1 then (count rate)i = 0 If µi = 0 then (count rate)i = +2. The main difference from the conventional monitoring scheme is that here the monitor count rate is not a function of the difference between the ith sensor reading and the voted signal, but a function of the difference between the ith sensor reading and the other like sensor readings. The count rate of the ith sensor signal becomes positive when the corresponding weight in the voted signal is equal to zero (wi = 0), therefore no transients occur once the failure is declared on the ith sensor and the

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Simulation examples

The setting is identical to the simulation examples discussed in Section 7.2.2 except for the sensor management system. The time histories of the first simulation example are given in Figure 7.7. At t = 1 sec, a drift failure of 1 deg sec−2 occurs (Figure 7.7a). Figure 7.7b shows the difference between the voted signal and the true pitch rate ∆qvoted . Due to the drift failure the voted signal diverges from the true pitch rate until the weight of the failed sensor output is reduced to zero (Figure 7.7c) and the voted signal is again equal to the true value (not taking into account uncertainties such as quantization, sensor noise, etc.). One can see that the voted signal is smoother than in the conventional sensor management case. By this time the monitor count rate is increased from −1 (µ1 = 1) to 0 (0 < µ1 < 1) and from 0 to +2 (µ1 = 0). When the monitor count reaches the failure declaration value (Figure7.7d) the signal is latched, and the number of valid signals reduces to two (denoted by the second dash-dotted line). As the weight of the corresponding signal is equal to zero at the moment of the failure declaration, no transients occur, see the solid lines of

7.3. Sensor Management and FCL Reconfiguration Based on Soft Computing

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Figure 7.8: Soft sensor management: cut-off failure of a pitch rate sensor. Figures e-g are zoomed in on the failure-induced transients. the elevator deflection (Figure 7.7e), normal acceleration (Figure 7.7f), and the true pitch rate (Figure 7.7g). For comparison, the time histories of the simulations of the fault free case (dash-dotted line) and conventional voting/monitoring case (dotted line) are also included in Figures 7.7e-g. The second simulation example is illustrated in Figure 7.8. At t = 3 sec, a cutoff sensor failure occurs (Figure 7.8a). Due to the abrupt nature of the sensor failure, the weight of the failed signal output becomes zero immediately (Figure 7.8c) and therefore there are no transients in the elevator deflection (Figure 7.8e), the normal acceleration (Figure 7.8f), and the pitch rate (Figure 7.8g) signals.

7.3.3

Flight control law reconfiguration

The soft voting logic is extended to soft flight control law reconfiguration. Also here the voted signal is computed as the average of the two sensor signals. Both normal acceleration sensor signals automatically have the same membership degree, and are therefore equally weighted in the consolidated signal. However, their mutual membership degree is multiplied with the contribution of the normal acceleration signal in the feedback path as well. The blending between the pitch rate and the normal acceleration signals in the feedback path, see also Section 4.1, is now

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Figure 7.9: Soft sensor management: drift failure of the second normal acceleration sensor. a function of the column deflection δc and the calibrated airspeed pressure VC multiplied by the maximum membership degree of the normal acceleration signals. Equation 4.6 then becomes: w = (µδc µVC ) µnz ,

(7.4)

where µnz is equal to the maximum membership degree of all nz signals: µnz = max(µi ).

(7.5)

When the difference between the two signals is such that their mutual membership degree becomes equal to zero, the flight control laws are already reconfigured to not using the normal acceleration signal in the feedback path. This is illustrated in Figure 7.9, where the time histories of a simulation of a drift failure of the second normal acceleration sensor are illustrated (Figure 7.9a). The voted signal is the average of the two input signals. During the maneuver, the signal in the feedback path is for 90% derived from the normal acceleration signal (Figure 7.9d). This is reduced to zero due to the growing discrepancy between the two valid normal acceleration sensor signals. By the time the nz signal is no longer available (Figure 7.9b), the FCLs are reconfigured to not using this signal (Figure 7.9d). 7.3.4

Discussion

In principle the conventional and the soft sensor management system are much alike. The soft sensor management system is a weighted implementation of the

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conventional sensor system, retaining all the benefits of this system. The conventional voting/monitoring system has two separate crisp thresholds, one to limit the contribution of a suspected faulty sensor signal and one for the failure declaration. The selected thresholds are a compromise between two goals: the absence of false alarms and the ability to detect all possible failures within a short time frame. The latter is important to minimize the effect of a sensor failure. This inevitably leads to a transient response during which the consolidated signal temporarily differs from the true value. Although it is possible to reduce transients by introducing filters, the soft sensor management is a more direct solution to this problem. In the soft sensor management system, the compromise between false alarms and the ability to detect sensor failures within a certain time frame is avoided by introducing a soft threshold. Through the soft threshold, the objectives of no false alarms and the minimization of the transient effects are well separated. When the failure declaration procedure is activated, the weight of the corresponding sensor signal is equal to zero and the negative impact of the suspected faulty sensor is avoided. As transients are reduced or even removed, the tuning of the membership function parameters is only driven by the sensor characteristics. For example, expensive sensors are more accurate and may allow for more narrow membership functions than other sensors. The additional computation due to the soft sensor management system is considered to be negligible. The thresholds used in the simulation examples were selected such that the characteristics of both voting/monitoring systems become clear. The crisp threshold in the conventional voting system is in between the upper and lower bound of the soft threshold of the soft voting/monitoring system. The thresholds of the soft sensor management system can always be designed such that the performance is equal to or better than that of the conventional sensor management system, without increasing the probability of false alarms. Performance is expressed in the accuracy of the consolidated, or voted, signal Svoted and in the magnitude of the discontinuities in the voted signal. In Figure 7.10 three different locations of the soft threshold (solid line) are illustrated compared to the crisp threshold (dashed line). The simulation results of a slow drift failure shown in Figure 7.11 correspond to these three locations of the soft threshold. The signal ∆Svoted is a measure of the inaccuracy of the voted signal. In case (c), where the left hand side of the soft threshold coincides with the crisp threshold, the maximum deviation when using the soft threshold is equal to that when using the crisp threshold. Typically case (b) will be applied in practise. An advantage of the soft sensor management system is that the deviation of the voted signal from the true value is reduced when the deviation of the failed sensor signal continues to grow. When using the conventional sensor management system, it remains constant until the corresponding sensor signal is latched. The benefits of the soft sensor management system are most evident for cutoff failures. The worst case sensor failure for the soft sensor management system is a step-like sensor failure that does not result in a membership degree of the corresponding signal that is equal to zero. Only in this case a discontinuity in

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Figure 7.10: Various locations for the soft threshold compared to the crisp threshold. Crisp thresholds (dashed line) versus soft thresholds (solid line). The gray area denotes the range covered by the soft threshold.

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7.4

Virtual Sensor for FDIR

The conventional sensor management system works well down to two signals, where any discrepancy can no longer be related to a majority. In this instance, the system will either reject both signals and reconfigure to not using this information, or, for essential data, a simple average will be used as the best compromise. However,

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Figure 7.12: Soft voting in the duplex mode. The virtual sensor output forms the center of a membership function, which is used to determine the membership degree of the hardware sensor outputs.

there is additional information available that can be used to identify the failed sensor in the duplex mode and to detect a failure in the simplex mode. This additional information can be used to estimate the signal of interest by including a virtual sensor, see also Chapter 6. Monitoring of the hardware sensor(s) in the duplex and simplex mode is then performed by comparison with the virtual sensor output. In this section a virtual normal acceleration sensor is used that is designed using similar techniques as for the LPV part of the virtual AoA sensor described in Chapter 6.

7.4.1

Dynamic thresholds

When using virtual sensors, the residuals resulting from the cross-comparison are more sensitive to uncertainties than in the case when only physical sensors are used, especially with respect to unmodelled dynamics. Dynamic thresholds have been implemented to optimize the performance of the voter/monitor based on the accuracy of the virtual sensor without risking false alarms. Typically the estimation error of the virtual normal acceleration sensor is small during steady-state flight and increases during (aggressive) maneuvering. For this reason, the support of the membership functions widens during maneuvering. In this way, a dynamic threshold is realized (see also Figure 7.12b). For the normal acceleration, the parameters of the membership function are adjusted as follows: bl bu

= =

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Simulation examples

The virtual sensor enables the sensor management system to identify the failed sensor in duplex mode for a drift failure of a second normal acceleration sensor, see Figure 7.13. While signal nz,3 and nz,virtual are in agreement, nz,2 starts to diverge from nz,virtual (Figure 7.13a). In Figure 7.13b the difference between the voted signal and the true normal acceleration ∆nz,voted is shown. Due to the drift, the voted signal diverges from the true normal acceleration until the weight of the failed sensor output is reduced to zero (Figure 7.13c). The absolute differences

7.4. Virtual Sensor for FDIR

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7 8 Time [s]

Figure 7.14: Soft sensor management: second drift failure of a normal acceleration sensor including sensor noise and severe turbulence.

between the sensor signals and the virtual sensor output (∆nz,2 and ∆nz,3 ) are illustrated in Figure 7.13d together with the dynamic lower and upper bounds of the membership function connected to the virtual sensor signal. Here it is also illustrated that the dynamic thresholds indeed correlate with the estimation errors during maneuvering. This scenario is repeated with sensor noise on all signals, including the inputs signals of the virtual sensor, and severe atmospheric turbulence, see Figure 7.14. The soft sensor management system still performs well. Using the virtual sensor, the sensor management system is even capable of identifying a failure of the last available sensor, which is illustrated in Figure 7.15 for a drift failure of the third normal acceleration sensor (Figure 7.15a). The monitor count is disengaged during simplex mode. As soon as the membership degree of nz,3 becomes equal to zero, the monitor count reaches the failure declaration value immediately (Figure 7.15d) and the signal is no longer available. In Figure 7.15c it is shown how the feedback path of the FCLs smoothly reconfigures to not using the normal acceleration signal. It should be noted that even when a sensor failure is detected during simplex mode, it could be both the last available hardware sensor or the virtual sensor. In both cases the best strategy is to reconfigure to not using the normal acceleration signal. The implementation of the virtual sensor is not limited to the soft sensor management system, and could also be implemented in the conventional sensor management system.

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0.8 c) weight

2

1

nz,3 n z,virt n z

0.6

z

a) n [g]

2.5

0.4

1.5

0.2 1

b) threshold

0.6

0.4

11

12

13

14

15

nz,3 11

12

13

14

15

11

12 13 Time [s]

14

15

1 ∆ nz,3

0.8

mf lb mf ub

d) blend

10

0 10

0.6 0.4

0.2

0.2 0 10

11

12 13 Time [s]

14

15

0 10

Figure 7.15: Soft sensor management: third drift failure of a normal acceleration sensor. 7.4.3

Flight simulator results

The soft sensor management system, including the virtual sensor, has been successfully evaluated during pilot-in-the-loop simulations in the Research Flight Simulator of the NLR. Figure 7.16 illustrates the result of one particular flight simulator test. During the simulation, the data was recorded in batches containing mainly the insertion of the sensor failures and the more aggressive maneuvers performed by the pilot. The task of the test pilot was to try to find problems in the system and the virtual sensor, in the latter case by exiting the normal acceleration. Failures in two of the normal acceleration sensors were introduced at t = 95 sec and t = 300 sec. The test pilot took his job very seriously, which can be concluded from the fact that the normal acceleration exceeded the maximum allowed value of nz = 2.5 g (Figure 7.16a) and the bank angle reached a maximum value of φ = 87 deg (Figure 7.16b). Note that in normal flight the maximum bank angle is limited to φ = 66 deg when the pilot is controlling the bank angle and to φ = 33 deg when the wheel is centered. Even in these extreme situations, the soft sensor management system performed as expected. 7.4.4

Discussion

In the conventional sensor management system the consolidated signal is no longer available after a sensor failure in the duplex mode, even if one of the sensors is still healthy. The implementation of a virtual sensor makes it possible to monitor the last available hardware sensor. Since the virtual sensor is implemented in the

b) φ [deg] a) nz,meas [g]

7.5. Conclusions

109

3 2 1

nz,1 n z,2 nz,3

d) ∆ nz,virt [g] c) nz,voted [g]

30 0 −30 −60 −90 3 2 1

0.1 0

e) valid

−0.1

2

0 0

50

100

150

200 Time [s]

250

300

350

Figure 7.16: Flight simulator test results. After 300 sec the aircraft is flying on the last available normal acceleration sensor that is monitored by the virtual normal acceleration sensor. software, the same safety level can be accomplished with less hardware. The cost reduction is more than the cost of the sensor itself, since it requires less supporting equipment and maintenance. Of course the development cost will increase because of the design of the virtual sensor. Virtual sensors can be implemented to increase the capability of the available hardware sensors, or to be able to reduce the number of hardware sensors without compromising the availability of the FCS.

7.5

Conclusions

Fuzzy logic techniques have been applied in the sensor management system, FCL reconfiguration, and the virtual sensor for normal acceleration. The improvement with respect to the conventional sensor management system is in the quality of the consolidated signal and results in a reduction of transients due to sensor failures. Furthermore, the virtual sensor increases the capability of the available hardware sensors, since it adds the ability to identify the failed sensor in the duplex mode and to detect a sensor failure in the simplex mode. This has been demonstrated by means of closed-loop simulation examples using a realistic aircraft model. Final evaluation of the soft sensor management system and the TS fuzzy model

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based virtual sensor has taken place during pilot-in-the-loop simulations in the RFS of the NLR.

8 Conclusions The thesis described two major challenges in the further development of digital fly-by-wire flight control systems: cost reduction by improving the architecture and design efficiency of the flight control system and the enhancement of flight safety. Fuzzy logic techniques and neural networks are used throughout the thesis to investigate novel applications and more automated design procedures that can provide a substantial contribution to solving these problems.

The contributions of this thesis are summarized in Section 8.1. In Section 8.2 an answer is given to the research question whether soft computing techniques can be used to improve the efficiency of the design process. In Section 8.3 an answer is given to the research question whether soft computing can contribute to increasing flight safety. Recommendations and suggestions for future research are given in Section 8.4.

8.1

Thesis Contributions

In Chapter 3, a novel automated procedure is proposed for the identification of the operating points for which the local flight control law parameters need to be tuned. This procedure is based on the application of fuzzy clustering to a set of aerodynamic derivatives that are relevant with respect to the dominant aircraft dynamics. The resulting cluster centers serve as the operating points, while the scheduling variables should capture the nonlinearities of the system to be controlled. A singleton TS fuzzy model structure provides the interpolation mechanism for the Flight Control Law (FCL) parameters in order to obtain a global nonlinear controller. Chapter 4 describes the application of the automated design procedure to the classical FCLs that are available in the synthetic environment. The scheduling mechanism of the six most relevant controller parameters is replaced by the scheduling mechanism designed by using the fuzzy clustering approach. The selected scheduling variables are the Mach number and the dynamic pressure. Eight operating points are identified for the flight envelope in clean configuration and 111

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Chapter 8. Conclusions

two for the flight envelope in landing configuration. Pilot-in-the-loop simulations demonstrated that the performance of the FCLs designed by using the automated design procedure is equivalent to the performance of the default FCLs. The proposed approach uses fewer operating points and requires less design effort. The proposed methodology can be used in any application domain, provided that sufficient information of the system to be controlled is available. This information can either be in the form of a (detailed) nonlinear model or in the form of recorded time histories of relevant system variables. Few examples of an automated approach to the identification of operating points can be found in literature. In these examples an iterative procedure of scheduling, controller design and closed-loop evaluation is described. The model is used to evaluate the closed-loop system and can only be used in combination with an automatic controller design procedure. The controller design procedure described in Chapter 5 goes one step further. Here not only the identification of the operating points and the design of the scheduling mechanism are automated by using the approach of Chapter 3, but also the local controllers are designed using robust MV control techniques. The latter removes the need for the time consuming single-loop iterative tuning of the local flight control law parameters. The problem with robust MV control techniques is the opaque structure of the resulting dynamic controller, which complicates the gain scheduling of the FCL parameters. A design procedure for a gain-scheduled robust MV controller is proposed. The local H∞ controllers are designed in continuous-time using Linear Matrix Inequalities (LMIs) and order reduction is performed before the transformation to the δ-operator form. The gain-scheduled H∞ controller has been evaluated offline (linear and nonlinear simulations, stability analysis) and through pilot-inthe-loop simulations. With respect to the aircraft dynamics, the gain-scheduled H∞ controller was given a Cooper-Harper (CH) rating of 1 in all flight. However, due to the high control force needed to maneuver the aircraft, especially in the low dynamic pressure region, the overall CH ratings were between 2 and 3. The results are satisfactory, however, additional tuning is required to further improve the performance and stability characteristics. Although many examples of scheduled robust multivariable control can be found in literature, in these examples an attempt is made to reduce the order of the controller and/or to impose a certain structure on the controller in order to simplify the scheduling problem. This leads to the undesirable modification of the original controller. In this thesis however, the order reduction of the H∞ controller is done in such a way that its performance is not affected. The use of the δ-operator form for parameter scheduling is proposed to further improve the schedulability of the H∞ controller without modifying their performance. Parameter-scheduled robust multivariable control can in principle be used in any application domain. However, it has been shown that the parameter scheduling of higher-order H∞ controllers becomes problematic. In that case either an alternative scheduling procedure should be chosen, e.g. scheduling of the output matrix only, or an alternative controller design methodology should be selected.

8.1. Thesis Contributions

113

The design procedure of the virtual angle-of-attack sensor is described in detail in Chapter 6. The philosophy of the design procedure is to use as much as possible the well know relations of the linearized aircraft dynamics. The linear, white-box part of the virtual sensor consists of the approximation of the trimmed angle-ofattack plus the linear approximation of the angle-of-attack due to longitudinal maneuvers. The trimmed AoA as well as the parameters for the approximation of the dynamic AoA change as a function of the flight condition. These parameters are estimated by a TS fuzzy model that uses Mach number, dynamic pressure, bank angle, position of the CG along the X-axis and aircraft weight as inputs. The remaining error is estimated by a nonlinear, black-box model. For this a neural network is used. The inputs of the neural network are determined using a nonlinear input selection approach. The performance of the virtual sensor is demonstrated by a large number of nonlinear simulations for which the flight conditions and maneuvers are selected randomly. The performance of the virtual sensor is good, with maximum estimation errors for the angle-of-attack of less than 0.8 degrees. Virtual sensors are used in many application domains, for example chemical industry, nuclear power plants, and medicine. Most virtual sensors are used to estimate variables that can not be measured directly, either because such a sensor does not exist or, if such a sensor does exist, because it can not survive the harsh environments it needs to operate in. In most cases a black-box modelling approach is adopted, making us of artificial neural networks. In the literature, many applications of analytical redundancy for fault detection and fault isolation in flight control systems have been reported, however, the use of virtual sensors in aerospace applications is novel. The virtual angle-of-attack sensor design approach proposed in this thesis distinguishes itself from other approaches due to the combination of white-box and black-box modelling. Sensor management is typically performed by the comparison of like sensor signals. The sensor signal that does not coincide with the other like sensor signals is assumed to be faulty. Crisp thresholds are used to determine whether a sensor signal is faulty or not. An alternative sensor management approach is introduced in Chapter 7. This approach distinguishes itself from the conventional sensor management approach in three ways, namely the integration of the voting and monitoring function, the application of soft thresholds and the use of a virtual sensor for sensor management. This results in a more reliable consolidated signal, while the transients induced by sensor failures are reduced significantly. Moreover, it is demonstrated how a virtual sensor can be used to identify the faulty sensor in the case there is a discrepancy between two like sensor signals. When the last sensor fails, the signal is no longer available and the FCLs reconfigure to not using this signal. The FCL reconfiguration is smooth as a direct result of the soft sensor management strategy. The soft sensor management system, including a normal acceleration virtual sensor, has been demonstrated by means of closed-loop simulation examples using the synthetic environment and by means of pilot-in-the-loop simulations. The proposed sensor management system is an extension to the conventional sensor management system of cross comparison of sensor signals. Although soft

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computing techniques have been implemented in other application domains, such as the process industry, their application for FDI in flight control systems has not been extensively investigated yet. The proposed soft sensor management system can be used in any application domain where hardware redundancy is implemented.

8.2

Efficiency Improvement of the Design of the System

Fuzzy clustering has been applied to partition the flight envelope into operating regimes in an attempt to improve the efficiency of flight control law design. When the flight control engineer designs the gain scheduler for a classical controller, this is typically performed separately for each FCL parameter that requires scheduling. Moreover, each FCL parameter is not necessarily scheduled with the same scheduling variable(s). In other words, each FCL parameter has its own set of operating points, which makes the structure of the gain scheduler opaque. This iterative, single-loop approach of tuning the (scheduled) FCL parameters is timeconsuming and therefore contributes significantly to the total design cost. Due to this opaque structure, the mutual dependencies of the FCL parameters are hard to identify. This makes it difficult to understand what needs to be changed when the performance is not as expected in a certain flight condition. In Chapter 3 it is shown that applying fuzzy clustering to relevant aerodynamic derivatives results in a partition of the flight envelope that makes sense to a flight control engineer. The main advantage of this approach is that the operating points are identified simultaneously, compared to the iterative trial-and-error approach that is carried out by the flight control engineer. Besides the reduction in the design effort with respect to identifying the operating points, the fuzzy clustering approach leads to fewer operating points. Clearly this results in less design effort for tuning the FCL parameters. By using the same scheduling variables and operating points for all related FCL parameters that require scheduling, the mutual dependencies of the FCL parameters are clearer and corrections are easier to make. However, this does not necessarily mean that the best performance is achieved by scheduling all FCL parameters in exactly the same way. In order to get the same closed-loop dynamics over the entire operating range, each FCL parameter should have a dedicated scheduling mechanism. Moreover, it makes sense to use different operating points for the longitudinal and lateral FCL parameters, since they correspond to different aircraft dynamics. This has not been investigated further in this thesis. In Chapter 4 it is illustrated that the automated design approach for the identification of the operating points works well on the classical FCLs that serve as the default in the SE. However, this approach can potentially be used for any linear design approach. In Chapter 5 the automated design approach is used in combination with a robust MV design approach to design the local FCL parameters, which further reduces the required effort for the design of gain-scheduled FCLs. In conclusion, the contribution of the fuzzy clustering approach to improving the

8.3. Enhancement of Flight Safety

115

efficiency of the design of flight control laws is significant. It is a global approach, all the operating points are identified simultaneously, that results in a transparent scheduling scheme with fewer operating points. Moreover, it is a model-based approach that uses the nonlinear dynamics in the design phase and not only in the evaluation phase. In combination with modern MV control techniques for the design of the local controllers, the reduction in the design effort for the design of the flight control laws is even more evident.

8.3

Enhancement of Flight Safety

Two approaches have been proposed to enhance the flight safety by means of soft computing. The first approach is the introduction of a virtual sensor that can be used as an arbitrator or as an additional sensor. The second approach is to apply soft computing for the signal consolidation and sensor monitoring. In Chapter 6, a design procedure for a virtual angle-of-attack sensor is proposed, which strongly relies on soft computing techniques. With a virtual sensor it is possible to identify the faulty signal when a discrepancy is detected between the signals from two like sensors. Moreover, it is possible to detect a failure on the last remaining physical sensor. The virtual sensor increases the integrity of the FCS and therefore contributes to flight safety. In the conventional approach, signal consolidation and signal monitoring are performed separately. Moreover, both applications make use of crisp thresholds. In Chapter 7, signal consolidation and sensor monitoring is integrated with the use of soft thresholds. This results in a more accurate consolidated signal and reduces the failure induced transients, which contributes not only to flight safety, but also to passenger comfort. In conclusion, the contribution of the soft computing to increase flight safety is significant. Clearly virtual sensors can be designed using other techniques, however, the advantage of fuzzy logic techniques is that it is possible to use linear techniques for nonlinear systems. This allows for an easy interpretable structure in combination with accuracy and reliability of the virtual sensor. The contribution of soft sensor management to flight safety is less significant than for the virtual sensor, but it does improve the system at no extra cost.

8.4

Recommendations for Further Research

With respect to the partitioning of the flight envelope using fuzzy clustering, both the cluster centers as well as the membership functions are used for the design of the gain scheduled flight control laws in this thesis. An alternative approach would be to tune the FCL parameter in the operating points that are identified by fuzzy clustering and then design a scheduler that optimizes the performance over the entire flight envelope. One could for example design a (non)linear polynomial that depends on a predefined set of scheduling variables. In this way improved performance can be achieved in off-design flight conditions. It should be noted that the

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scheduling does not have to be the same for all FCL parameters. The use of more advanced clustering algorithms, i.e. cluster algorithms that are more flexible with respect to the size, orientation and volume of the clusters to be identified in the data set, such as the fuzzy maximum likelihood estimates clustering, may result in an improved partition of the flight envelope. As shown in Chapter 5, there are still stability problems in the operating regimes between the outer operating points and the edge of the flight envelope. These problems occurred in particular in the low dynamic pressure region. It is recommended to force the outer operating points closer to the edge of the flight envelope, such that these stability problems no longer occur. Since the fuzzy clustering algorithm uses the Euclidean distance measure, one possible approach is to weight the distance for data points close to the edge of the flight envelope more than for data points in the center of the flight envelope. Although the results obtained with the application of scheduled robust MV control described in this thesis are promising, it is a relatively new subject that requires much additional research. First of all, the design of the local H∞ controllers is performed in a relatively crude manner in the sense of the tuning of the input uncertainty and weighting filters. Most of the design effort for the local H∞ controllers was spent on the tuning of the reference model based on the comments of the test pilots. It is believed that much can be gained by investigating the uncertainty description and weighting filters more closely. This might further improve the robustness of the local H∞ controllers and remove some of the instability problems that were found in the low dynamic pressure region of the flight envelope. The parameters in the robust MV controllers have no apparent physical meaning and are therefore hard to interpret. This is a serious drawback, not only with respect to gain scheduling, but especially with respect to FCL reconfiguration. FCL reconfiguration is typically performed because a sensor signal or actuator is no longer available or, in the case of military aircraft, there is structural damage. It is not expected that the a controller designed for nominal mode has the same inputs and outputs as a controller designed for degraded mode. Therefore these controllers will not have similar structures, which rules out the option of FCL reconfiguration through gain scheduling. Further research is required to investigate this problem. Output scheduling could work, but it means that for each anticipated failure a separate dynamic controller is running in parallel. This will put a heavy burden on the available computing power and storage capacity. Ultimately it would be interesting to investigate the capabilities of a set of scheduled FCLs entirely based on robust MV control techniques, both for the longitudinal and lateral-directional aircraft motion. With respect to interpretability and/or schedulability it is expected that the best possible approach is to design the controllers for the longitudinal and lateral-directional aircraft motion separately and integrate them afterwards. This is how it is done with classical flight control laws as well and is accepted as best practice in the aeronautical world. This would enable the introduction of a highly automated design approach for the primary flight control laws.

8.4. Recommendations for Further Research

117

With respect to virtual sensors, another topic of interest is how they could best be used in the (soft) sensor management system. When there is a whole set of non-like virtual sensors available, they are all largely depending on the same signals (except for the signal they are designed to estimate). This means that there is a high degree of integration and mutual dependency among the physical and virtual sensors. It should be investigated what the most fault tolerant architecture is for the case of multiple non-like virtual sensors, possibly in combination with alternative FDI methodologies.

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A Synthetic Environment and Real-Time Code Generation The Synthetic Environment (SE), developed within the ADFCS project, is a software tool aimed to perform detailed six degrees-of-freedom simulation of a DFBW aircraft. The SE is developed in the Matlab/Simulink TM environment and includes models of sensors, actuators, digital flight control computers, etc. It is defined as an ultra-high fidelity simulation tool that is structurally representative of a practical implementation (Rosenberg 2001). The SE allows to identify potential problems at an early stage by means of extensive and detailed simulation sessions, such that problems can be identified and solved prior to manufacture. In this way time and cost penalties associated with rework later in the life-cycle are avoided. The SE allows for evaluation of new concepts in a realistic simulation environment. Moreover, the SE can be run in real-time and can therefore be used for hardware-in-the-loop testing and demonstration. The content of this appendix is for a large part taken from (Ciniglio 2002).

This appendix is organized as follows: In Section A.1 the architecture of the synthetic environment is discussed, together with a description of the separate modules. This section provides background information for readers who are familiar with the notation used in aerospace engineering. Interested readers who are unfamiliar with the notation that is used in this section, are referred to (McLean 1990). The procedure to generate real-time code starting from the synthetic environment is described in Section A.2.

A.1 A.1.1

Synthetic Environment Synthetic Environment Architecture

The simulation model of the entire augmented SCA model has been structured in several separate blocks, see Figure A.1. Some of the modules in the SE are available with different levels of complexity. The lower the level of complexity, the lower the required computing power to simulate the corresponding module. This 119

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Figure A.1: Synthetic Environment Architecture (Ciniglio 2002). allows the user to modify the SE such that it meets the requirements of each specific simulation test, while minimizing the total required computing power. Each of the blocks illustrated in Figure A.1 is described in more detail below. A.1.2

Bare Airframe

As illustrated in Figure A.1, the bare airframe is defined by the six Degree-OfFreedom (DOF) equations of motion, gravity model, aerodynamic model, hinge moment model and undercarriage model together with the air data and the engines. Equations of motion The six DOF equations of motions are: u˙

=

v˙

=

w˙

=

FX g − qw + rv W FY g − ru + pw W FZ g − pv + qu W

(A.1) (A.2) (A.3)

A.1. Synthetic Environment

p˙

=

q˙

=

r˙

=

121

IZ 2 [L + (IY − IZ )qr + IXZ pq] IX IZ − IXZ IXZ + 2 [N + (IX − IY )pq − IXZ qr] IX IZ − IXZ 1 [M + (IZ − IX )pr − IXZ (p2 − r2 )] IY IXZ 2 [L + (IY − IZ )qr + IXZ pq] IX IZ − IXZ IXZ + 2 [N + (IX − IY )pq − IXZ qr]. IX IZ − IXZ

(A.4) (A.5)

(A.6)

The forces and moments (aerodynamics, engine and gravity) defined in the body axes are: FX

= qSCX + FXE − W sin(θ)

(A.7)

FY = qSCY + FYE + W cos(θ) sin(φ) FZ = qSCZ + FZE + W cos(θ) cos(φ) L = qSCR b + FZE (YE − YCG ) − FYE (−ZE + ZCG ) M N

= qSCM c + FXE (−ZE + ZCG ) − FZE (−XE + XCG ) = qSCN b + FYE (−XE + XCG ) − FXE (YE − YCG ),

(A.8) (A.9) (A.10) (A.11) (A.12)

where CX = CL sin(α) − CD cos(α) cos(β) CZ = −CL cos(α) − CD sin(α) cos(β) FXE = T cos(ψT ) cos(θT ) FYE = T sin(ψT ) FZE = −T cos(ψT ) sin(θT ). Note that Equations A.7 to A.12 hold for the right engine. For the left engine one should substitute FYE = T sin(ψT ) by FYE = −T sin(ψT ) and substitute YE by −YE . The aerodynamic coefficients CL , CD , CY , CR , CM and CN are described in more detail later in this section. The SE provides the capability to trim and linearize the nonlinear aircraft model for the given flight condition and aircraft configuration. The linear longitudinal and lateral aircraft models are given below, respectively:    u˙ Xu w˙   Zu  = ˜u  q˙  M ˙θ 0

Xw Zw ˜w M 0

Xq Zq ˜q M 1

   Xθ Xδe u w  Zδe Zθ     ˜ θ   q  + M ˜δ M e θ 0 0

 Xδs  Zδs   δe ˜ δ  δs M s 0

(A.13)

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   Yv v˙ ˜  p˙   L  = v  r˙  N ˜v ˙ φ 0

Yp ˜p L ˜p N 1

Yr ˜r L ˜r N 0

   Yφ Yδa v ˜ ˜     Lφ   p   L +  ˜δa    ˜ r Nφ Nδa φ 0 0

 Yδr  ˜ δ  δa L r  ˜ δ  δr . N r 0

(A.14)

Aerodynamic model - Longitudinal coefficients CD

=

CD0 + ∆CD (α, δe , δs , δf l ) + CDBIAS (M ) + CDSB (α, δsp , M ) +CDLG (CL , δf l , δsl ) (A.15)

CL

=

CL0 + ∆CL (α, δe , δs , δf l ) + CLSB (α, δsp , M ) c +(CLq q + CLα˙ α) ˙ 2VT CM0 + ∆CM (α, δe , δs , δf l ) + CMSB (α, δsp , M ) c +(CMq q + CMα˙ α) ˙ + CX (ZCG − ZREF ) 2VT −CZ (XCG − XREF ) + CMLG (CL , δf l , δsl ).

CM

=

(A.16)

(A.17)

Aerodynamic model - Lateral coefficients CY

CR

CN

=

=

=

∆CY (α, β, δf l ) + ∆CY (M )β + ∆CY (α, δa , δr , δf l ) b +[CYp (M )p + CYr (M )r] 2VT ∆CR (α, β, δf l ) + ∆CR (M )β + ∆CR (α, δa , δr , δf l ) b +[CRp (M )p + CRr (M )r] 2VT c c −CY (ZCG − ZREF ) − CZ YCG b b ∆CN (α, β, δf l ) + ∆CN (M )β + ∆CN (α, δa , δr , δf l ) b +[CNp (M )p + CNr (M )r] 2VT c c −CY (XCG − XREF ) + CX YCG . b b

(A.18)

(A.19)

(A.20)

Gravity model The gravity model is the standard International Standard Atmosphere (ISA) gravity model (Ruijgrok 1990). Hinge moment model The hinge moments of elevator, aileron and rudder are given below in mbar m3 : HMELEV HMAIL HMRU D

= CELEV (CHαtail (M ) αtail + CHELEV (M ) δe ) q (A.21) = CAIL CHAIL (α, δa , M ) q = CRU D CHRU D (δr , β) q,

(A.22) (A.23)

A.1. Synthetic Environment

123

where αtail = α [1 − α (δf l )] − 0 (δf l ) + δs .

(A.24)

Undercarriage model When the undercarriage is retracted, an additional drag and corresponding moment is modelled. The undercarriage has no influence on the aircraft inertia matrix. Air data model The air data probe and the corresponding Air Data Computer (ADC) are considered to be part of the bare airframe. The air data probe measures static pressure, impact pressure and temperature. Based on these measurements, the ADC computes pressure altitude (Hp ) [ft], rate-of-climb (c) [ft/min], calibrated airspeed (VC ) [kts], Mach number (M ), true airspeed (VT ) [kts], dynamic pressure (q) [mbar] the corrected Angle-of-Attack (AoA) [deg]. Engine model The thrust T is a function of throttle position δth , altitude h and Mach number M: T = f (δth , h, M ). The thrust dynamics are implemented as a simple first order lag. A.1.3

Outside World

The atmosphere model is the standard ISA atmosphere model (Ruijgrok 1990). Furthermore the SE includes two turbulence models. One turbulence model is based on the Dryden theory (Etkin 1972). This turbulence model can be implemented with three levels of intensity (light, medium and severe). A second turbulence model can be used instead of the Dryden model in order to better reproduce the off-line environment that will be used during the simulator sessions, since this turbulence model is implemented in the flight simulator. Finally a windshear model with arbitrary altitude profile has been implemented. A.1.4

Flight Control Computer

The flight control computer model includes the cross-channel data link, the voter/ monitor for sensor/actuator fault detection and identification, the flight control laws to improve stability and control, the autopilot and the flight envelope protection modes. A.1.5

Actuation

The main components of the actuation model are the models of the hydraulic actuators together with the Actuator Control Electronics (ACE) channels, with related

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LVDT sensors. The most complex ACE card model includes sensors, demodulation and actuator loop closure. A.1.6

Sensors, Switches and Controls

The airframe sensor module is implemented in the non-linear dynamic simulation model of the whole DFBW suite. The sensor models allow for modelling of sensor failures (bias, change in the scale factor, hold-on model, etc.). A.1.7

Cockpit

This cockpit model includes generic blocks for pilot command signals generation (or the pilot commands recorded during flight simulator sessions). No displays are available in the SE.

A.2

Real-Time Code Generation

The software environment of the NLR RFS is controlled by the Programme and Real-time Operations SIMulation support tool (PROSIM). Configuring the RFS with the FGS design was ultimately performed by the software tool MOSAIC (Model-Oriented Software Automatic Interface Converter) developed at the NLR (Lammen et al. 1999). The MOSAIC tool automatically transfers the model from Matlab/Simulink TM to PROSIM. The tool takes as input the model source code that has been generated by Real-Time Workshop of Simulink 3.0 and delivers as output the model source code that can run in PROSIM. This automated process of generating real-time code for the flight simulator is only possible if the Simulink model complies with defined software development guidelines (Heesbeen 2001). A more detailed description of this process is given in (Smaili 2001).

B Short-Period Approximation In this appendix the short-period approximation is described. This approximation is a useful tool for longitudinal flight control design, since it greatly simplifies the equations of motion while maintaining the dominant longitudinal aircraft dynamics.

The short-period motion is derived from the linear longitudinal aircraft model in Section B.1. In Section B.2 a few relations are derived from the short-period approximation which are used in Chapter 4 of this thesis.

B.1

Derivation of the Short-Period Approximation

The linear longitudinal aircraft model is derived in Appendix A and is shown again below:        u˙ Xu Xw Xq Xθ Xδe u w˙   Zu Zw Zq Zθ  w  Zδe   =     (B.1) ˜u M ˜w M ˜q M ˜ θ   q  + M ˜ δ  δe .  q˙  M e ˙θ θ 0 0 1 0 0 The longitudinal aircraft motion is divided into two oscillatory eigenmotions, namely the short-period motion and phugoid motion. The phugoid motion has a low eigenfrequency and is typically poorly damped. During the phugoid motion the aircraft exchanges kinetic energy for potential energy and vice versa, which results in variations in airspeed and altitude. The short-period motion describes the aircraft dynamics along the Y -axis in terms of the pitch rate and the angle-of-attack (or downward speed):     Zδe Zw Zq w w˙ (B.2) + = ˜ ˜q q ˜ δ δe . q˙ Mw M M e The short-period approximation follows from the assumption that the forward speed remains constant, i.e. u = 0 (fixed speed assumption). In other words, the 125

126

Appendix B. Short-Period Approximation

short-period approximation assumes that short-period transients are of sufficiently short duration that variations which arise in the forward speed as a result of aircraft motion, control surface deflections and atmospheric turbulence are negligible.

B.2 B.2.1

Derivation of Short-Period Related Equations Pitch rate to angle-of-attack

From Equation B.2 it can be seen that: s w = Zw w + Zq q + Zδe δe ,

(B.3)

where s denotes the Laplace operator. Assuming the elevator deflection to be equal to zero (δe = 0) and with the substitution of w = U0 α, where U0 is the (constant) trimmed forward speed, this equation can be rewritten to: Zq q. U0

s α = Zw α +

(B.4)

α Since Zw = − N U0 and Zq = U0 (McLean 1990) it follows that:

sα=−

Nα α + q, U0

(B.5)

where the aerodynamic derivative Nα is defined as: Nα =

ρU02 SCNα . 2m

(B.6)

The transfer function from pitch rate to angle-of-attack then becomes: α 1 = , α q s+ N U0 which can be rewritten to:

B.2.2

U0 α = q Nα

1 U0 Nα s

+1

(B.7)

.

(B.8)

Normal acceleration to angle-of-attack

From kinematics it follows that during longitudinal motion the normal acceleration can be described by the following relation: nz =

1 (U0 q − s w). g

(B.9)

U0 (q − s α). g

(B.10)

Substitution of w = U0 α results in: nz =

B.2. Derivation of Short-Period Related Equations

127

Substitution of Equation B.5 in Equation B.10 results in: nz =

Nα α. g

(B.11)

The transfer function from the angle-of-attack to the normal acceleration becomes: nz Nα = . α g

(B.12)

The transfer function from the normal acceleration to the angle-of-attack can therefore be written as: α g = . (B.13) nz Nα B.2.3

Pitch rate to normal acceleration

The transfer function from the pitch rate to the normal acceleration can be constructed as follows: α nz nz = . (B.14) q q α Substitution of Equation B.8 and Equation B.12 in Equation B.14 results in: nz U0 Nα = q Nα g

1 U0 Nα s

+1

,

(B.15)

which can be simplified to: U0 nz = q g Introducing the parameter τ = in:

U0 Nα

1 U0 Nα s

+1

.

(B.16)

and substitution of τ in Equation B.16 results

nz U0 1 = . q g τs + 1

(B.17)

128

Appendix B. Short-Period Approximation

C Performance Measures and Cross Validation The performance measures described in this appendix provide a measure of the (mis)match between a system and its model. In order to evaluate the predictive capability of a model, these performance measures should be used in combination with a cross validation procedure.

This appendix is organized as follows: In Section C.1 the performance measures that are used throughout the thesis are defined. The cross validation technique, which is used in Chapter 6, is described in Section C.2.

C.1

Performance Measures

The Variance Accounted For (VAF) index is computed by:   ˆ) var(y − y VAF = 1 − 100%, var(y)

(C.1)

ˆ denotes the model where ‘var’ denotes variance, y denotes the data vector and y output vector. The VAF of 100% means a perfect model prediction (disregarding a constant offset), the VAF of 0% is obtained when the “model” is the mean of the data. Negative VAF values indicate even worse models. The Root-Mean-Squared Error (RMSE) index is computed by:  N 1
(yk − yˆk )2 . RMSE = N

(C.2)

k=1

The RMSE of 0 means a perfect model prediction. Worse models will have larger RMSE values.

129

130

Appendix C. Performance Measures and Cross Validation

The advantage of the VAF index is its insensitivity to the scale of the data.1 This makes it possible to compare results obtained for different variants of one data set (e.g., original and scaled data). The VAF index, however, is not sensitive to a constant offset. The RMSE index is sensitive both to the scale of the data and to offsets. Therefore, the two indices are sometimes used in parallel.

C.2

Cross Validation

Cross validation is a technique that is often used for estimation of the prediction error of a classification or regression function. Estimating prediction error on the same data used for model estimation tends to give underestimates, because the parameter estimates are “fine-tuned” to the peculiarities of the sample. For very flexible methods, e.g. neural networks or tree based models, the error on the training sample can usually be made close to zero. The true error of such a model will usually be much higher however: the model has “overfitted” the training sample. An alternative is to divide the available data into a training sample and a test sample. If the available sample is rather small, this method is not preferred because the test sample may not be used for model estimation in this scenario. By using cross validation all data points are used for training as well as testing. The general V-fold cross validation procedure works as follows: The data set D is randomly split into V mutually exclusive subsets (the folds) D1 , D2 , ..., DV of approximately equal size. The model is trained and tested V times; each time v ∈ 1, 2, ..., V , it is trained on D \ Dv and tested on Dv . The performance measures presented in the previous section can be used in cross validation by defining yˆ as follows:  yˆv . (C.3) yˆ = v=1,2,..,V

1 Note,

however, that modelling methods usually are sensitive to the scale of the data.

D Soft Computing Techniques Soft computing is a generic term for techniques like fuzzy clustering, fuzzy reasoning, neural networks, etc. For example, in fuzzy clustering a certain data point can belong partly to several clusters, hence the term soft. In contrast to crisp clustering, where a certain data point belongs entirely to a certain cluster (hard). Soft computing is used in a wide range of application areas and is especially useful for piecewise linear modelling and when dealing with uncertainties. In this appendix a number of soft computing techniques that are used in this thesis are described.

This appendix is organized as follows: In Section D.1 fuzzy clustering is described in more detail, including the Gustafson-Kessel algorithm. The application of fuzzy clustering for system identification is addressed in Section D.2. Some background information on neural networks is given in Section D.3.

D.1

Fuzzy Clustering

An effective approach to the identification of complex nonlinear systems is to partition the available data into subsets and approximate each subset by a linear model. Fuzzy clustering can be used as a tool to obtain a partition of data where the transitions between the subsets are gradual rather than abrupt. The concept of fuzzy partitioning is essential for cluster analysis and consequently also for fuzzy modelling and identification techniques that are based on fuzzy clustering. Fuzzy partitions can be seen as a generalization of hard partitions, which is formulated in terms of classical subsets. D.1.1

Preliminaries

Clustering techniques are unsupervised methods that can be used to organize data into groups based on similarities among the individual data items. Most clustering algorithms do not rely on assumptions common to conventional statistical methods, such as the underlying statistical distribution of data, and therefore they are 131

132

Appendix D. Soft Computing Techniques

useful in situations where little prior knowledge exists. Clustering techniques can be applied to data that are quantitative (numerical), qualitative (categorical), or a mixture of both. The data are typically observations of some physical process. Each observation consists of n measured variables, grouped into an n-dimensional column vector zk = [z1k . . . znk ], zk ∈ IRn . A set of N observations is denoted by Z = {zk |k = 1, 2, . . . , N} and is represented as an n × N pattern or data matrix:   z11 z12 . . . z1N  z21 z22 . . . z2N    Z= . . ..  . ..  .. .  zn1 zn2 . . . znN Various definitions of a cluster can be formulated, depending on the objective of clustering. Generally, one may accept the view that a cluster is a group of objects that are more similar to one another than to members of other clusters (Bezdek 1981, Dave 1991). The term “similarity” should be understood as mathematical similarity, measured in some well-defined sense. In metric spaces, similarity is often defined by means of a distance norm. Distance can be measured among the data vectors themselves, or as a distance from a data vector to some prototypical object (prototype) of the cluster. Fuzzy clustering methods allow the objects to belong to several clusters simultaneously, with different degrees of membership. Objects on the boundaries between several classes are not forced to fully belong to one of the classes, but rather are assigned membership degrees between zero and one indicating their partial membership. The concept of fuzzy partitioning is essential for cluster analysis and consequently also for the identification techniques that are based on fuzzy clustering. Conditions for the fuzzy partition are given by (Ruspini 1970): µik ∈ [0, 1], c


1 ≤ i ≤ c,

µik = 1,

1≤k≤N

1≤k≤N

(D.1) (D.2)

i=1

0
0. Initialize the partition matrix randomly, such that U (0) ∈ Mf . Repeat for l = 1, 2, . . . 1. Compute the cluster prototypes (means).

n (l−1) (µik )m zk (l) vi = k=1 , (l−1) m n ) k=1 (µik

1 ≤ i ≤ c.

(D.11)

2. Compute the cluster covariance matrices.

N Fi =

(l−1) m (l) ) (zk − vi )(zk k=1 (µik

N (l−1) m ) k=1 (µik

(l)

− vi )T

,

1 ≤ i ≤ c.

(D.12)

3. Compute the distances. 1

D2ikAi = (zk − vi )T [det(Fi ) n F−1 i ](zk − vi ).

(D.13)

4. Update the partition matrix. if DikAi > 0, (l)

1 ≤ i ≤ c, 1 ≤ k ≤ N 1

µik = c

DikAi j=1 ( DjkAi

2

(D.14)

,

(D.15)

) m−1

otherwise (l)

µik = 0 ifDikAi > 0

(l)

and µik ∈ [0, 1] with

c
i=1

(l)

µik = 1

(D.16)

until U (l) − U (l−1)  < .

D.2

Identification via Fuzzy Clustering

An effective approach to the identification of complex nonlinear systems is to partition the available data into subsets and approximate each subset by a linear model. Fuzzy clustering can be used as a tool to obtain a partition of data where the transitions between the subsets are gradual rather than abrupt.

136

Appendix D. Soft Computing Techniques

Figure D.1: Extraction of Takagi-Sugeno rules by fuzzy clustering. The Takagi-Sugeno (TS) model (Takagi and Sugeno 1985) is used to smoothly combine the linear submodels. The TS rules have the following form: Ri :

If x1 is Zi1 and . . . and xp is Zip then yi = aTi x + bi , i = 1, 2, . . . , K .

(D.17)

This is a MISO model with inputs and output y. Zij are linguistic terms (like low, medium, high, etc.), represented by membership functions. Finally, ai and bi are real-valued consequent parameters. TS rules are extracted from data by clustering in the product space of the inputs and outputs. By applying clustering algorithms that are capable of detecting linear substructures in data, a nonlinear regression problem is automatically decomposed it into several local linear subproblems. Each obtained cluster is represented by one rule in the TS fuzzy model. The antecedent membership functions are obtained by projection (Figure D.1) and the consequent parameters can be estimated by various Least Squares (LS) methods (Babuˇska 1998). Illustrative example: Consider a nonlinear function y = f (x) defined piecewise by: y y y

= 0.25x, = (x − 3)2 + 0.75, = 0.25x + 8.25,

for x ≤ 3 for 3 < x ≤ 6 for x > 6.

(D.18)

Figure D.2 shows a plot of this function evaluated in 50 samples uniformly distributed over x ∈ [0, 10]. Zero-mean, uniformly distributed noise with amplitude 0.1 was added to y. The data set {(xi , yi )|i = 1, 2, . . . , 50} was clustered into four clusters. Figure D.3 shows the local linear models obtained through clustering, the bot-

D.2. Identification via Fuzzy Clustering

137

12

y = 0.25 ⋅ x + 8.25

10 8 2

y

6

y = (x−3) + 0.75

4 2

y = 0.25 ⋅ x

0 0

2

4

6

8

10

x

Figure D.2: The nonlinear function D.18 with zero-mean, uniformly distributed noise. 15 y4 = 0.26⋅ x + 8.19

y

10 y3 = 4.45⋅ x − 17.40

5 y1 = 0.25⋅ x + 0.05

0 0

2

y2 = 1.42⋅ x − 3.76

4

6

8

10

Membership degree

x 1

0.5

Z1

0 0

Z2

2

Z3

4

Z4

6

8

10

x

Figure D.3: Cluster prototypes and the corresponding fuzzy sets.

tom plot shows the corresponding membership functions. In terms of TS rules, the fuzzy model is expressed as:

R1 : R2 : R3 : R4 :

If If If If

x x x x

is is is is

Z1 Z2 Z3 Z4

then then then then

y y y y

= 0.25 x + 0.05 = 1.42 x − 3.76 = 4.45 x − 17.40 = 0.26 x + 8.19.

Note that the consequents of R1 and R4 correspond almost exactly to the first and third equation D.18. Consequents of R2 and R3 are approximate local linear models of the parabola defined by the second equation of D.18. 

138

Appendix D. Soft Computing Techniques

D.3

Neural Networks

This section provides a brief introduction into neural networks, their architecture and their training algorithms. For a more detailed description, the interested reader is referred to (Fausett 1994, Gurney 1997, Haykin 1999). D.3.1

Introduction to neural networks

An Artificial Neural Network (ANN) is an information processing paradigm that is inspired by the way biological nervous systems, such as the brain, process information. The key element of this paradigm is the structure of the information processing system. It is composed of a large number of highly interconnected processing elements (neurones) working together to solve specific problems. ANNs, like people, learn by example. Learning in biological systems involves adjustments to the synaptic connections that exist between the neurones. This is true of ANNs as well. Neural networks are applicable in virtually every situation in which a relationship between the predictor variables (independents, inputs) and predicted variables (dependents, outputs) exists, even when that relationship is very complex and not easy to articulate in the usual terms of correlations or differences between groups. A few representative examples of problems to which neural network analysis has been applied successfully are detection of medical phenomena, stock market prediction, credit assignment, monitoring the condition of machinery and engine management. D.3.2

Architecture of neural networks

To capture the essence of biological neural systems, an artificial neuron is defined as follows: • It receives a number of inputs (either from original data, or from the output of other neurons in the neural network). Each input comes via a connection that has a strength (or weight); these weights correspond to synaptic efficacy in a biological neuron. Each neuron also has a single threshold value. The weighted sum of the inputs is formed, and the threshold subtracted, to compose the activation of the neuron. • The activation signal is passed through an activation function (also known as a transfer function) to produce the output of the neuron. If the step activation function is used then the neuron acts just like the biological neuron described earlier, see Figure D.4. Actually, the step function is rarely used in artificial neural networks, as will be discussed later in this section. Note also that weights can be negative, which implies that the synapse has an inhibitory rather than excitatory effect on the neuron: inhibitory neurons are found in the brain. If a network of neurons is to be of any use, there must be inputs (which carry the values of variables of interest in the outside world) and outputs (which form

D.3. Neural Networks

139

Figure D.4: Example of a neuron with step activation. predictions, or control signals). Inputs and outputs correspond to sensory and motor nerves such as those coming from the eyes and leading to the hands. However, there also can be hidden neurons that play an internal role in the network. The input, hidden and output neurons need to be connected together. A simple network has a feedforward structure: signals flow from inputs, forwards through any hidden units, eventually reaching the output units. Such a structure has stable behavior. However, if the network is recurrent (contains connections back from later to earlier neurons) it can be unstable and has very complex dynamics. Recurrent networks are very interesting to researchers in neural networks, but so far it is the feedforward structures that have proved most useful in solving real problems. A typical feedforward network has neurons arranged in a distinct layered topology, see Figure D.5. The input layer is not really neural at all: these units simply serve to introduce the values of the input variables. The hidden and output layer neurons are each connected to all of the units in the preceding layer. It is possible to define networks that are partially-connected to only some units in the preceding layer; however, for most applications fully-connected networks are better. When the network is executed, the input variable values are placed in the input units and then the hidden and output layer units are progressively executed. Each of them calculates its activation value by taking the weighted sum of the outputs of the units in the preceding layer and subtracting the threshold. The activation value is passed through the activation function to produce the output of the neuron. When the entire network has been executed, the outputs of the output layer act as the output of the entire network. D.3.3

Training of the neural network

The best-known example of a neural network training algorithm is back propagation (Fausett 1994, Patterson 1996, Haykin 1999). Modern second-order algorithms such as conjugate gradient descent and Levenberg-Marquardt (Bishop 1995, Shepherd 1997) are substantially faster for many problems, but back propagation still has advantages in some circumstances and is the easiest algorithm to understand. In the back propagation algorithm, the gradient vector of the error surface is cal-

140

Appendix D. Soft Computing Techniques

Figure D.5: Feedforward neural network. culated while propagating backwards through the network (from output layer to input layer). This vector points along the line of steepest descent from the current point. Moving a short distance along this line will decrease the error. A sequence of such moves (slowing when approaching the minimum) will eventually find a minimum of some sort. The difficult part is to decide how large the steps should be. In practice, the step size is proportional to the slope (so that the algorithms settles down in a minimum) and to a special constant: the learning rate. The correct setting for the learning rate is application-dependent, and is typically chosen by experiment; it may also be time-varying, getting smaller as the algorithm progresses. The algorithm is also usually modified by inclusion of a momentum term: this encourages movement in a fixed direction, so that if several steps are taken in the same direction, the algorithm “picks up speed”, which gives it the ability to (sometimes) escape local minimum, and also to move rapidly over flat spots and plateaus. The algorithm progresses iteratively through a number of epochs. On each epoch the training cases are each submitted in turn to the network, the target and actual outputs are compared and the error is calculated. This error, together with the error surface gradient, is used to adjust the weights and then the process repeats. The initial network configuration is random and training stops when a given number of epochs elapses, or when the error reaches an acceptable level, or when the error stops improving (to be selected by the user). The Levenberg-Marquardt algorithm is typically the fastest of the training algorithms. However, it has some important limitations such as: it can only be used on single output networks, it can only be used with the sum squared error function and it has memory requirements proportional to W2 (where W is the number

D.3. Neural Networks

141

of weights in the network). This makes the Levenberg-Marquardt algorithm impractical for reasonably big networks. Conjugate gradient descent algorithms are nearly as good and do not suffer from these restrictions. The second-order training algorithms seem to be prone to stick in local minima in the early phases. To overcome this problem, one could start with a short burst using the back propagation algorithm, before switching to a second-order algorithm.

142

Appendix D. Soft Computing Techniques

E Genetic Algorithms Genetic algorithms are search algorithms that imitate the principles of natural evolution for optimization problems. They combine survival of the fittest among a population of potential solutions with a structured yet randomized information exchange.

This appendix is organized as follows: A brief introduction into genetic algorithms is given in Section E.1. In Section E.2 the basic principles on how genetic algorithms work are described on the basis of an example. The theoretical foundation and application areas of genetic algorithms are discussed in Sections E.3 and E.4, respectively.

E.1

Introduction

Genetic Algorithms (GAs) are randomized optimization algorithms that are based on the principle of survival of the fittest. The population of potential solutions is mani- pulated repeatedly to form a new generation. Each potential solution is defined in a string structure, referred to as a chromosome. A new generation is formed by deleting weak chromosomes from the population, while using strong chromosomes to produce new chromosomes through genetic operators. A more detailed description of how GAs work is offered in Section E.2. Genetic algorithms are especially useful for non-convex optimization problems with large search spaces. For convex optimization problems the use of a gradientbased optimization algorithm is more efficient. For non-convex optimization problems with small search spaces classical exhaustive search methods usually suffice. For larger search spaces a more intelligent search technique must be employed, such as a GA.

E.2

How Do They Work?

Genetic algorithms can be divided into two main groups, namely binary-coded and real-coded GAs. The first group expresses the value of the parameters to be 143

144

Appendix E. Genetic Algorithms

Figure E.1: Flowchart of a genetic algorithm. optimized in a binary code, while the latter group expresses them by their true value. In principle these two implementations are equivalent, however, it has been shown that the real-coded genetic algorithm is faster, more consistent from run to run and provides a higher accuracy (Goldberg 1990, Michalewicz 1996). In this section, the basic elements that are present in all genetic algorithms are discussed, see also Figure E.1. The starting point of the genetic algorithm is an initial population of chromosomes. Each chromosome is evaluated with respect to the fitness function. The selection procedure randomly selects two sets of chromosomes, namely a set of chromosomes to be removed from the population and a set of chromosomes to be used to create new chromosomes. Weaker chromosomes have a higher probability to be removed from the population than stronger chromosomes and stronger chromosomes have a higher probability to be used to create new chromosomes than weaker chromosomes (survival of the fittest). New chromosomes are created using genetic operators (crossover and mutation operators). The fitness of these new chromosomes is evaluated and the selection procedure starts again. The algorithm stops when a certain termination criterion is met. In the remainder of this section, the genetic algorithm that is used in this thesis is explained with a simple example. For this specific genetic algorithm there are five parameters that need to be set by the designer, namely: N Nc Nop pco

number of generations number of chromosomes in the population number of genetic operations per generation percentage of crossover operations

E.2. How Do They Work?

145

0

y

−20 −40 −60 −80 1 1

0.5 0.5

0

0

−0.5 x

−0.5 −1

2

−1

x

1

Figure E.2: The function y = (x1 − 1)5 + (x2 − 1)5 over the domain x1 ∈ [−1, 1] and x2 ∈ [−1, 1]. md

differentiation parameter.

For this example, these parameters are set to be: N = 100, Nc = 20, Nop = 10, pco = 0.80 and md = 4. E.2.1

Description of the optimization problem

The goal is to find the Takagi-Sugeno model that fits the surface illustrated in Figure E.2. The data set that is used for optimizing the TS model is obtained by evaluating y = f (x1 , x2 ) at a number of grid points. The grid is defined by x1 = [−1, −0.98, . . . , 0.98, 1] and x2 = [−1, −0.98, . . . , 0.98, 1]. In this example the structure of the TS fuzzy model is fixed. There are two fuzzy membership functions for each antecedent variable and four rules. The corresponding rule-base is as follows: R1 : R2 : R3 :

If x1 is Z11 and x2 is Z21 then yˆ = c01 + c11 x1 + c21 x2 If x1 is Z11 and x2 is Z22 then yˆ = c02 + c12 x1 + c22 x2 If x1 is Z12 and x2 is Z21 then yˆ = c03 + c13 x1 + c23 x2

(E.1)

R4 : If x1 is Z12 and x2 is z22 then yˆ = c04 + c14 x1 + c24 x2 . For each antecedent variable, the two membership functions are defined by two parameters, see Figure E.3. In total this TS fuzzy model requires 16 parameters, namely four for the membership functions in x1 and x2 and 12 for the local linear models. Each chromosome in the genetic algorithm consists of a string of four realvalued numbers for the membership functions. Once the membership functions are defined, the 12 parameters of the local linear models are determined through least squares optimization. A single chromosome is denoted as vit , where the subscript i denotes the

Appendix E. Genetic Algorithms

Membership degree

146

1

Z

Z

11

12

0.5

0 −1

−0.5 d

1

0 x

1

d

0.5

1

2

Figure E.3: The membership functions Z11 and Z12 , defined by the two parameters d1 and d2 . position of the chromosome in the population and the superscript t denotes the t , where the subgeneration. A single element of this chromosome is denoted by vij t script j denotes the position of the element in the chromosome vi . The constraints t t , . . . , vi4 ) ∈ [−1, 1]. for the parameters of the ith chromosome are (vi1 E.2.2

Initial population

The initial population is the result of an initialization process. In this case the initial population is selected randomly, taking into account the constraints, and consists of 20 chromosomes:   −0.0462 −0.4774 −0.9921 −0.1692 = v10   0.5846 0.4012 0.6715 0.0017 = v20 .. .   0 0.6643 0.0037 −0.5748 0.2749 = v19   0 −0.2857 0.0560 −0.4451 −0.5821 . = v20 E.2.3

Evaluation of the fitness of the chromosomes

Each chromosome represents two sets of membership functions, namely for x1 and for x2 . To evaluate the fitness of the ith chromosome, a TS fuzzy model is created using the corresponding membership functions. The parameters of the consequent part of the rule-base are computed through LS optimization (see also Equation E.1). For each data point the output yˆj of the corresponding TS fuzzy model is evaluated. The corresponding fitness value is the root mean-squared error over the K data points: 

K ˆj )2 j=1 (yj − y t , (E.2) F (vi ) = K where y and yˆ are the true output and the output of the TS fuzzy model using the parameters of chromosome vit , respectively. Each chromosome of the initial population is evaluated using the fitness func-

E.2. How Do They Work?

147

tion. In this case this results in: F (v10 )

F (v20 ) .. . 0 ) F (v19 0 F (v20 )

=

1.5121

=

1.5534

=

4.4841

=

5.0600.

The chromosomes are sorted according to their fitness. Since this is a minimization problem, the stronger the chromosome the lower its fitness value. E.2.4

Selection procedure

Two sets of chromosomes need to be selected. The first set consists of the chromosomes that will be used for reproduction and the second set consists of the chromosomes that will be replaced by the new chromosomes. The basic principle is that chromosomes with a strong fitness value have a higher probability to be selected for reproduction than chromosomes with a weak fitness value. For the selection of chromosomes for deletion this is the other way around. The selection procedure is as follows: 1. Divide the fitness value of the strongest chromosome by the fitness value of all the chromosomes in the population and raise the result to the power md .  Pop (i) =

F (v1t ) F (vit )

md ,

i = 1, . . . , Nc .

(E.3)

2. Normalize the result, such that the sum of the vector is equal to one. Pop (i) , Pop (i) := N c j=1 Pop (j)

i = 1, . . . , Nc .

(E.4)

3. The vector for deletion is the flipped vector for genetic operation, i.e. reproduction. (E.5) Pdel (i) = Pop (Nc + 1 − i), i = 1, . . . , Nc . 4. Compute the cumulative sum for both vectors. Pop (i) :=

i


Pop (j),

i = 1, . . . , Nc

(E.6)

Pdel (j),

i = 1, . . . , Nc .

(E.7)

j=1

Pdel (i) :=

i
j=1

148

Appendix E. Genetic Algorithms

Figure E.4: Roulette wheel.

The higher the value for md , the more the probability is differentiated, i.e. weak chromosomes are less likely to be selected for operation. The vectors Pop and Pdel are shown below: Pop (1) Pop (2) .. .

= =

0.1385 0.2629

Pdel (1) Pdel (2) .. .

= =

0.0011 0.0029

Pop (19) Pop (20)

= =

0.9989 1

Pdel (19) Pdel (20)

= =

0.8615 1.

These two vectors are used to select the chromosomes for operation and deletion respectively. This process is explained using the vector for operation: 1. Generate a random number r ∼ U [0, 1]. 2. If r ≤ Pop (1), then select the first chromosome (v1t ) else select the ith chromosome such that Pop (i − 1) < r ≤ Pop (i). This is repeated Nop times. The same procedure is used to select the chromosomes for deletion, except for the fact that in this case Pop is replaced by Pdel . In both cases the selection procedure is performed with replacement. It can be seen that the probability that chromosome v10 (the strongest chromosome) is selected for operation is higher than the probability that it is selected for deletion, since the random number r is more likely to fall within the range of [0, 0.1385] than within the range of [0, 0.0011]. At the same time the probability 0 (the weakest chromosome) is selected for deletion is higher that chromosome v20 than the probability that it is selected for operation. This can be visualized by the roulette wheel in Figure E.4. When spinning the wheel, the probability that 0 turns up. Since it is likely v10 turns up is higher than the probability that v20 to happen that a single chromosome is selected for operation or deletion more than once, additional coding is required to make sure that the correct number of chromosomes are selected for operation and deletion.

E.2. How Do They Work?

E.2.5

149

Genetic operators

Two kinds of genetic operators are used, namely crossover and mutation. The probability that a crossover operator is selected is determined by the parameter pco . The probability that a mutation operator is selected is equal to 1 − pco . For each genetic operation, three versions are used. When a chromosome is selected for crossover (or mutation) one of the used crossover (or mutation) operators are applied with equal probability. Crossover operators For crossover operations, the chromosomes are treated in pairs (vit , vjt ). 1. Simple arithmetic crossover. vit and vjt are crossed over at the kth position, where k is a random integer between 1 and m. The resulting offsprings are:   t t t t (E.8) vit+1 = vi,1 , . . . , vi,k |vj,k+1 , . . . , vj,m   t t+1 t t t vj (E.9) = vj,1 , . . . , vj,k |vi,k+1 , . . . , vi,m . 2. Whole arithmetic crossover. A linear combination of vit and vjt resulting in: vit+1 vjt+1

= r vit + (1 − r) vjt (1 − r)

(E.10)

vjt ,

(E.11)

= vit + r1 (vit − vjt )

(E.12)

=

vit

+r

where the parameter r ∼ U [0, 1]. 3. Heuristic crossover. vit and vjt are combined such that: vit+1 vjt+1

=

vjt

+

r2 (vjt

−

vit ),

(E.13)

where the parameters r1 ∼ U [0, 1] and r2 ∼ U [0, 1].

Mutation operators For mutation operations, single chromosomes are selected. 1. Uniform mutation. t , k ∈ {1, 2, . . . , m} is replaced by vk , which A randomly selected element vi,k   min max is a random number in the range vi,k , vi,k . The resulting chromosome is: t t t t vit+1 = (vi,1 , . . . , vi,k−1 , vk , vi,k+1 , . . . , vi,m ). (E.14)

150

Appendix E. Genetic Algorithms

1.6

Root Mean−Squared Error

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

20

40 60 Generation

80

100

Figure E.5: Root mean-squared error as function of the generation. 2. Multiple uniform mutation. Uniform mutation of n elements in the chromosome vit , where n is a randomly selected integer between 1 and m. The position of each of the n elements in the chromosome is again determined by a randomly selected integer between 1 and m. 3. Gaussian mutation. All elements of a chromosome are mutated such that:  ), vit+1 = (v1 , . . . , vk , . . . , vm

(E.15)

where vk = vk + fk , k = 1, 2, . . . , m. Here fk is a random number drawn from a Gaussian distribution with zero mean and an adaptive variance σk =

(T − t) (vkmax − vkmin ) , t 3

(E.16)

where T denotes the predefined maximum number of generations. The parameter tuning performed by this operator becomes finer as the generation counter t increases. For an overview of a wide variety of genetic algorithm implementations, selection procedures, genetic operators, etc., the reader is referred to (B¨ ack et al. 2000a, B¨ack et al. 2000b). In this example, it is chosen to let the parameters defining the membership functions be subject of the GA optimization. The parameters of the local linear models are determined through LS optimization (once the membership functions are defined). One could for example also choose to optimize both the parameters defining the membership functions and the parameters of the local linear models using the

Membership degree

E.3. Theoretical Foundation

151

1

Z11

Z12

0.5

0 −1

−0.5

0 x

0.5

1

0 x

0.5

1

Membership degree

1

1

Z21

Z22

0.5

0 −1

−0.5

2

Figure E.6: The optimal membership functions Z11 , Z12 , Z21 and Z22 . GA. However, this increases the number of parameters to be optimized by the GA from 4 to 16 and overcomplicates the optimization process by introducing additional degrees of freedom. E.2.6

Result

Figure E.5 illustrates the convergence of the GA towards the optimal result. The optimal fitness value is equal to root mean-squared error of RMSE = 0.6479. Due to the small number of parameters to be optimized, the optimal value is reached in the 56th generation. The optimal membership functions are illustrated in Figure E.6 and the corresponding rule-base is as follows: R1 R2 R3 R4

: : : :

If If If If

x1 x1 x1 x1

is is is is

Z11 Z11 Z12 Z12

and and and and

x2 x2 x2 x2

is is is is

Z21 Z22 Z21 Z22

then then then then

yˆ = 4.30 + 32.9x1 + 32.9x2 yˆ = 1.02 + 32.9x1 + 1.63x2 yˆ = 1.02 + 1.63x1 + 32.9x2 yˆ = −2.26 + 1.63x1 + 1.63x2 .

The output of the optimal TS fuzzy model is illustrated in Figure E.7a, while the modelling error with respect to the original data (see Figure E.2) is illustrated in Figure E.7b.

E.3

Theoretical Foundation

Although GAs are widely applied, there is a profound lack of hard theory accompanying the empirical results provided by GA researchers (White and Flockton 1995). The description of GAs as schema processing algorithms was first set out by Holland (1975) and forms the foundation of many of the theoretical results. Other methods based on Walsh functions (Goldberg and Rudnick 1990, Forrest

152

Appendix E. Genetic Algorithms

2

0

y

y

0 −50

−2 −4 1

−100 1 1 0 x2

0 −1 −1

x

1

(a) Output of the optimal TS fuzzy model.

1 0 x

2

0 −1 −1

x

1

(b) Modelling error of the optimal TS fuzzy model.

Figure E.7: Output (left) and modelling error (right) of the optimal TS fuzzy model. See Figure E.2 for the original data. and Mitchell 1992, Field 1994), Markov chains (Goldberg and Segrest 1987, Davis and Principe 1993), statistical mechanics (Pr¨ ugel-Bennett and Shapiro 1994) and simulated annealing-like convergence analysis (Goldberg 1987) have since been developed and have attempted, in their turn, to provide a quantitative explanation of the dynamics of the GA. For the class of Elitist GAs it is proven that it converges to the optimum of the fitness function (Jong 1975, Suzuki 1995). However, despite of the vast amount of effort spent, it cannot be proven for other (simple) GAs (Lozano et al. 1999). Other proofs of convergence involve simulated annealing-like genetic algorithms.

E.4

Application Areas

Genetic algorithms are widely applied in planning (routing, scheduling, packing), design, simulation and identification, control and classification (B¨ ack et al. 2000a). Optimization problems in these areas are typically constrained, non-convex optimization problems that involve both continuous as well as binary or integer parameters, i.e. hybrid optimization problems. Unlike gradient-based optimization techniques, GAs are well suited for these kind of optimization problems. The main disadvantage of GAs is the computational effort needed to solve an optimization problem.

F Linear Matrix Inequalities for Control In this appendix it is briefly explained how the robust control problem is defined through linear matrix inequalities. More specifically, the output-feedback robust control problem is considered.

This appendix is organized as follows: The output-feedback H∞ control problem is explained in Section F.1. In Section F.2 it is described how the output-feedback H∞ control problem is transformed into a Linear Matrix Inequality (LMI). How to transform constraints on the location of the controller poles into LMIs is briefly discussed in Section F.3.

F.1

Output-feedback H∞ Control Problem

Suppose a linear time-invariant open-loop system is described by:      B2 x x˙ A B1  z  =  C1 D11 D12  w , C2 D21 D22 u y with state x, exogenous inputs w, control inputs u, controlled outputs z and measured outputs y. The assumptions on the plant parameters are: 1. (A, B2 , C2 ) is stabilizable and detectable. 2. D22 = 0. The first assumption is necessary and sufficient to allow for plant stabilization by dynamic output feedback. The second assumption is considerably simplifying the equations, while it incurs no loss of generality. The output-feedback controller is a finite dimensional linear time-invariant system described as: 153

154

Appendix F. Linear Matrix Inequalities for Control

Figure F.1: Robust control design problem.   AK x˙ K = u CK

BK DK

 xK , y

where xK is the state of the controller. The closed-loop system therefore becomes (see also Figure F.1):   A + BDK C BCK ˙ξ AK BK C = z C1 + EDK C ECK   A B ξ = , w C D

  B1 + BDK F  ξ BK F w D1 + EDK F

T  where ξ = x xK . The suboptimal H∞ control problem of parameter γ consists of finding a controller K such that (Gahinet and Apkarian 1994): 1. The closed-loop system is internally stable. 2. The H∞ norm of Tzw (s) = D + C(sI − A)−1 B (the maximum gain from w to z) is strictly less than γ, i.e. ||Tzw ||∞ < γ.

F.2

(F.1)

LMI Approach

For the output-feedback H∞ control problem, the design objective ||Tzw ||∞ < γ can be rewritten into the following LMI:   B XC T AX + XAT  BT −γI DT  < 0, (F.2) CX D −γI where X is the Lyapunov matrix and A is assumed to be stable. See, for example, Lemma 4.1 in Gahinet and Apkarian (1994). A difficulty in the output-feedback case is that Equation F.2 contains products of the controller parameters and is therefore nonlinear in the parameters

F.2. LMI Approach

155

to be solved. These nonlinearities can be eliminated by an appropriate change of controller variables. This change of variables was introduced in (Gahinet 1996) and is implicitly defined in terms of the (unknown) Lyapunov matrix X. Specifically, partition X and its inverse as:   S N R M −1 X= , X = . (F.3) NT V MT U The new controller variables are defined as: BK CK AK

:= := :=

N BK + SB2 DK CK M T + DK C2 R T

(F.4) T

N AK M + N BK C2 R + SB2 CK M + S(A + B2 DK C2 )R.

The identity XX −1 = I together with Equation F.3 gives: M N T = I − RS.

(F.5)

Thus, M and N have full row rank when I − RS is invertible. The invertibility of I − RS can be assumed without loss of generality, see Lemma 4.2 in Chilali and Gahinet (1996). Theorem F.1 The output-feedback H∞ control problem is solvable if and only if the following system of LMIs is feasible:  R I >0 I S (F.6)   ∗ ∗ ∗ AR + B2 CˆK + (∗)  (B1 + B2 DK D21 )T −γI ∗ ∗    < 0, AˆK + (A + B2 DK C2 )T ˆK D21 ˆK C2 + (∗) SB1 + B SA + B ∗  D11 + D12 DK D21 C1 + D12 DK C2 −γI C1 R + D12 CˆK (F.7) where “ ∗ ” denotes the complex-conjugate transpose. See (Chilali and Gahinet 1996) for the proof of this theorem.  Given any solution of this LMI system: • Compute via singular value decomposition a full-rank factorization M N T = I − RS of the matrix I − RS (M and N are then square and invertible). • Solve the system of linear Equations F.4 for BK , CK and AK (in this order) • Set K(s) := DK + CK (sI − AK )−1 BK . Then K(s) is an nth order controller such that ||Tzw (s)||∞ < γ, where n is the order of A.

156

Appendix F. Linear Matrix Inequalities for Control

Figure F.2: Region S(α, r, θ). Source: (Chilali and Apkarian, 1996). The LMI formulation of constrained H∞ optimization is appealing from a practical standpoint. LMIs can be solved by efficient interior-point optimization algorithms such as those described in for example (Nesterov and Nemirovskii 1993).

F.3

Pole Placement

The transient response of a linear system is related to the location of its poles. By constraining the poles to lie in a prescribed region, a satisfactory transient response can be ensured. In addition, fast controller dynamics can be prevented by prohibiting large closed-loop poles, which is often desirable for digital implementation. One way of simultaneously tuning the H∞ performance and transient behavior is therefore to combine the H∞ and pole placement objectives. Regions of interest include α-stability regions Re(s) ≤ −α, vertical strips, disks and conic sectors. Another interesting region for control purposes is the set S(α, r, θ) of complex numbers x + jy such that: x < −α < 0, |x + jy| < r, tan(θ) x < −|y|,

(F.8)

which describes the intersection of a vertical strip, cone sector and disk (see Figure F.2). Confining the closed-loop poles to this region ensures a minimum decay rate α, a minimum damping ration ζ = cos(θ), and a maximum undamped natural frequency ωd = r sin(θ). This in turn bounds the maximum overshoot, the frequency of oscillatory modes, the delay time, the rise time, and the settling time. Definition F.2 A subset D of the complex plane is called an LMI region if there exist a symmetric

F.3. Pole Placement

157

matrix α = [αkl ] ∈ Rm×m and a matrix β = [βkl ] ∈ Rm×m such that: D = {z ∈ C : fD (z) < 0}

(F.9)

fD (z) := α + zβ + zβ = [αkl + βkl z + βlk z]1≤k,l≤m .

(F.10)

with Note that the characteristic function fD takes values in the space of m × m Hermitian matrices and that “< 0” stands for negative definite.  In other words, an LMI region is a subset of the complex plane that is representable by an LMI in z and z, or equivalently, an LMI in x = Re(z) and y = Im(z). As a result, LMI regions are convex. Moreover, LMI regions are symmetric with respect to the real axis since for any z ∈ D, fD (z) = fD (z) < 0. For more details, the reader is referred to (Chilali and Gahinet 1996).

158

Appendix F. Linear Matrix Inequalities for Control

Acknowledgements In the first place I would like to thank Henk Verbruggen and Gerard Schram for giving me the opportunity to work for the European project named “Affordable Fly-By-Wire Flight Control Systems for Small Commercial Aircraft” (ADFCS). Their preparation work and supervision during the first year made it possible for me to get started immediately and to quickly catch up with the project. I would also like to thank Robert Babuˇska, who became my supervisor after one year and later also my promotor. He gave me the freedom to find my own way, but he was always available for a technical discussion or to provide me with whatever I needed to make my (professional) life easier. I also acknowledge the help of the supporting staff of the department. I would like to thank all the people who worked for ADFCS for their numerous constructive comments and suggestions and for the useful discussions. I valued the personal contact and the atmosphere during the meetings and the flight simulator sessions, both from a professional as well as personal point of view. ¨ During my stay in Orebro, Sweden, in the beginning of 2002 I have worked on scheduled robust multivariable control together with Dimiter Driankov and Pontus Bergsten. The cooperation and discussions greatly improved my understanding on the subject and I would like to thank them for that. Furthermore, I would like to acknowledge the European Union and Marie Curie Foundation for their financial support. Regarding the non-scientific aspects, I am grateful for the social activities at the department, mainly organized by De Biercommissie and friends. I have enjoyed the many (at random) bierages and all the other activities that have taken place. Because of them it was possible for me to find a good balance between working and having fun. I take this opportunity to thank my parents, brothers and other friends for their social support. Although they have not been directly involved in my life as a scientist, they are important to me as a person. In particular I would like to thank Daniela for her love and for withstanding the tension and frustration that haunted the house from time to time during the writing of this thesis. Finally I would like to thank my paranimphen Domenico Bellomo and Hans Roubos for the pleasant working environment when we shared an office and for assisting me during the defence of this thesis.

Marcel Oosterom

Rotterdam, May 2005

159

160

Acknowledgements

Curriculum Vitae Marcel Oosterom was born on March 1, 1974 in Deurne, The Netherlands. In June 1998 he received his M.Sc. degree from the Delft University of Technology, Faculty of Aeronautical Engineering, Department of Control & Simulation. The thesis was entitled “Flight control during final approach in windshear - The MBPC approach”. From September 1998 he worked almost six years for the European project “Affordable Digital Fly-By-Wire Flight Control Systems for Small Commercial Aircraft” on intelligent flight control at the Control Laboratory of the Faculty of Electrical Engineering (now part of the Delft Center for Systems and Control). ¨ During this period he spent three months at the Orebro University, Center for Applied Autonomous Sensor Systems, Sweden. Since February 2005 he is employed at Huisman-Itrec, Schiedam, The Netherlands, working in the field of control system engineering.

161

162

Curriculum Vitae

List of Publications Chapters in books 1. Schram G., M. Oosterom, and H.B. Verbruggen, “Fuzzy Modeling and Control in Avionics”. In “Fuzzy Logic Control - Advances in Applications”, Verbruggen and Babuˇska (Eds.), World Scientific Series in Robotics and Intelligent Systems, Vol. 23, pp. 275-291, 1999. 2. Babuˇska R. and M. Oosterom, “Fuzzy Clustering for Multiple-Model Approaches in System Identification and Control”. In “Granular Computing An Emerging Paradigm”, Physica-Verlag, pp. 306-323, 2001.

Scientific journal publications 3. Oosterom M., R. Babuˇska and H.B. Verbruggen, “Soft Computing Applications in Aircraft Sensor Management and Flight Control Law Reconfiguration”. IEEE Transactions on Systems, Man and Cybernetics - Part C: Applications and Reviews, Vol. 32, No. 2, May 2002. 4. Oosterom M. and R. Babuˇska, “Design of a Gain-Scheduling Mechanism for Flight Control Laws using Fuzzy Clustering”. Accepted for publication in the Control Engineering Practice Journal. 5. Oosterom M. and R. Babuˇska, “Virtual Angle-of-Attack Sensor”. Submitted to the Journal of Aircraft. 6. Oosterom M. and R. Babuˇska, “Scheduled Robust Multivariable Control”. Submitted to the Control Engineering Practice Journal.

Conference publications 7. Oosterom M., G. Schram, H.B. Verbruggen and R. Babuˇska, “Automated Procedure for Gain Scheduled Flight Control Law Design”. In: Proceedings of the AIAA Guidance Navigation and Control Conference, AIAA-2000-4253, Denver, CO, USA, 2000. 8. Oosterom M. and R. Babuˇska, “Virtual Sensor for Fault Detection and Isolation in Flight Control Systems - Fuzzy Modeling Approach”. In: Proceedings of the IEEE Conference on Decision and Control, Sydney, Australia, pp. 2645-2650, 2000. 163

164

Publications

9. Oosterom M. and R. Babuˇska, “Fuzzy Logic Applications in Aircraft Sensor Management Systems”. In: Proceedings of the 40th Israel Annual Conference on Aerospace Sciences, Tel Aviv/Haifa, Israel, pp. 160-167, 2001. 10. Oosterom M. and R. Babuˇska, “Aircraft Sensor Management and Flight Control Law Reconfiguration - Fuzzy Logic Approach”. In: Proceedings of the AIAA Guidance Navigation and Control Conference, AIAA-2001-4358, Montreal, Canada, 2001. 11. Oosterom M. and R. Babuˇska, “Fuzzy Gain Scheduling for Flight Control Laws”. In: Proceedings of the FUZZ’IEEE Conference, Melbourne, Australia, pp. 716-719, 2001. 12. Abonji J., J.A. Roubos, M. Oosterom and F. Szeifert, “Compact TS-Fuzzy Models through Clustering and OLS plus FIS Model Reduction”. In: Proceedings of the FUZZ’IEEE Conference, Melbourne, Australia, pp. 1-4, 2001. 13. Oosterom M. and R. Babuˇska, “Fuzzy Gain-Scheduled H∞ Flight Control Law Design”. In: Proceedings of the AIAA Guidance Navigation and Control Conference, AIAA-2002-4847, Monterey, CA, USA, 2002. 14. Babuˇska R. and M. Oosterom, “Design of Optimal Membership Functions for Fuzzy Gain-Scheduled Control”. In: Proceedings of the FUZZ’IEEE Conference, St. Louis, MO, pp. 476-481, 2003.

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