Multivariable identification and controller design of an integrated ...

Applied Mathematical Modelling 31 (2007) 2733–2743 www.elsevier.com/locate/apm

Multivariable identification and controller design of an integrated flight control system D.T.W. Yau a, E.H.K. Fung a

a,*

, Y.K. Wong b, H.H.T. Liu

c

Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China c Institute for Aerospace Studies, University of Toronto, Toronto, Ont., Canada

b

Received 1 February 2006; received in revised form 1 September 2006; accepted 24 October 2006 Available online 26 January 2007

Abstract This paper investigates the multivariable identification and controller design for the longitudinal channel of a Boeing 747 transport. The transfer function matrix of the system is identified using the prediction error (PE) identification method with multivariable ARX model. An ellipsoidal parametric uncertainty set is constructed from the covariance matrix of the identified parameters. It contains the parameters of actual system at a certain probability level. The identified models and the associated uncertainty sets are validated by measuring the worst-case m-gap and then compared with the maximum value of the generalized stability margin. In automatic flight control system or autopilots, multiple specifications criteria are needed to be satisfied concurrently, such as good holding (small static altitude holding error), fast response, smooth transition (less oscillation, overshoot). The design of a Multiple Simultaneous Specifications (MSS) controller effectively and practically is a very significant and challenging job. Liu and Mills [H.H.T. Liu, J.K. Mills, Multiple specification design in flight control system, in: Proceedings of the American Control Conference, Chicago, Illinois, 2000, pp. 1365– 1369] proposed a MSS controller design method using a convex combination approach. In this paper, we apply the method [H.H.T. Liu, J.K. Mills, Multiple specification design in flight control system, in: Proceedings of the American Control Conference, Chicago, Illinois, 2000, pp. 1365–1369; H.H.T. Liu, Design combination in integrated flight control, in: Proceedings of the American Control Conference, Arlington, Virginia, 2001, pp. 494–499; H.H.T. Liu, Multi-objective design for an integrated flight control system: a combination with model reduction approach, in: Proceedings of IEEE International Symposium on Computer Aided Control System Design, Glasgow, 2002, pp. 21–26] to design a MSS controller based on the identified models of the Boeing 747 transport aircraft longitudinal channel. The controllers are also validated by simulation using the true plant transfer functions.  2006 Elsevier Inc. All rights reserved.

1. Introduction The problem of modeling and identification for control design has received substantial attention over the past years [1–10]. There is a need to design a robust controller based on the identified model in many *

Corresponding author. Tel.: +852 27666647; fax: +852 23654703. E-mail address: [email protected] (E.H.K. Fung).

0307-904X/$ - see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.10.027

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applications. However, it is often necessary to assess the quality of the model and tune the uncertainty region associated to it before it is used for control design. There are different ways to validate an identified model. Ljung [4–6] proposed model validation and model error modeling that are based on correlation analysis of residuals. Garulli and Reinelt [7] applied the model error modeling to set membership identification techniques in order to highlight the separation between unmodeled dynamics and noise. Wang and Goodwin [8] developed a robust control algorithm that utilized the statistical confidence bounds of the identified plant. Smith and Doyle [9] addressed the gap between the models used in control synthesis and those obtained from identification experiments by considering the connection between uncertain models and data. Recently, Bombois and Date [1,2] developed a model validation and controller validation procedure in a prediction error identification framework. A parametric uncertainty ellipsoid set and its corresponding set of parameterized transfer functions are constructed. They validated the model for control by measuring the size of the uncertainty set that is directly connected to the size of a set of controllers that stabilize all systems in the model uncertainty set. They also presented the necessary and sufficient conditions for the controller to stabilize and/or to achieve a given level of performance with all plants in the uncertainty set. In actual flight situation, the parameters of the system are varying due to the change in flight conditions. There is a need to perform identification under different flight conditions in order to update the aircraft model and also the controllers. Since flight control is a multivariable system hence multivariable identification is required. Fung et al. [11] performed multivariable identification for the pitch angle and speed control channels separately of a Boeing 747 transport aircraft. On the other hand, in automatic flight control system or autopilots, multiple specifications criteria are needed to be satisfied concurrently, such as good holding (small static altitude holding error), fast response, smooth transition (less oscillation, overshoot). Liu and Mills [12] and Liu [13] presented a convex combination approach to design the MSS (Multiple Simultaneous Specifications) controller, which can satisfy all the multiple specifications simultaneously. It is also realized that both the pitch and speed share the same control effectors: elevator and throttle. In order to account for the pitch/ speed interaction effect, Liu [14] proposed the design combination methods for an integrated pitch attitude/ speed control system. Fung et al. [15] applied the method [12–14] to design the longitudinal control system of a F-16 fighter. The objective of this paper is to extend the work of [11] to perform multivariable identification and to design an integrated pitch/speed control system for the longitudinal channel of a commercial Boeing 747 transport aircraft flying at a particular flight condition. In the first part of the paper, multivariable identification is performed, where the input–output data samples are collected by numerical simulation of the true system represented by the mathematical model of the aircraft. The discrete time transfer function matrix of the system are identified using prediction error (PE) method [16] with multivariable ARX model. An ellipsoidal parameter uncertainty set containing the parameters of the true system at a certain probability level is constructed. The identified plants are validated by using the model validation procedures developed by Bombois and Date [1] in which the worst-case m-gap is compared with the maximum value of the generalized stability margin. In the second part of the paper, an integrated pitch/speed MSS controller is designed based on the identified models of the Boeing 747 transport aircraft longitudinal channel using the method developed in [12–14] . The controllers are validated by simulation using the true plant transfer functions. 2. Multivariable identification of flight control system The plant transfer functions of a Boeing 747 transport aircraft [17] flying at 40,000 ft at Mach number 0.8 are selected for multivariable identification. The linearized longitudinal state-space equations of motion are given by 

x_ ¼ Ax þ Bu y ¼ Cx þ Du

ð1Þ

where x = [U W q h]T, u = [de dp]T, y = [h U]T. U and W are components of velocity; q and h are the pitch rate and pitch angle, respectively; de and dp are the deflections of the elevator and throttle, respectively. The system matrices A and B are all given in English units [17] and D = 0.

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Eq. (1) is transformed into transfer function matrix as: y ¼ CðsI  AÞ1 Bu ¼ G0 u;

ð2Þ

where G0 represents the true system, which is the mathematical model of the aircraft. Eq. (2) can be expressed as two MISO systems:  h ¼ G0;11 de þ G0;12 dp ; ð3Þ U ¼ G0;21 de þ G0;22 dp ; where G0,11, G0,12, G0,21 and G0,22 are the true plant transfer functions. 2.1. Pitch angle control The continuous time s domain transfer functions for the pitch angle in Eq. (3) are transformed into discrete time z domain transfer functions assuming a zero-order hold (ZOH) and unit sampling interval. The pitch angle MISO true system is expressed in the multivariable ARX structure: hðtÞ ¼ Gðz; d0 Þ½ u1

T

u2  þ

eðtÞ ; 1 þ Z D d0

ð4Þ

where [u1u2]T is input signal, e(t) is white noise, d0 is the true parameter vector of the transfer functions, and ½ Z N 1 d0 Z N 2 d0  Gðz; d0 Þ ¼ 1 þ Z D d0   Z D ðzÞ ¼ z1 z2 z3 z4 0 0 0 0 0 0 0 0   Z N 1 ðzÞ ¼ 0 0 0 0 z1 z2 z3 z4 0 0 0 0   Z N 2 ðzÞ ¼ 0 0 0 0 0 0 0 0 z1 z2 z3 z4 d0 for pitch angle

¼ ½ 2:859

3:2 1:81

0:2963 0:0001973

0:4721

0:4723

0:0007376

0:4162

3:911e  5

0:35 7:666e  5 T :

Numerical simulation of this true system G0 = G(z, d0) is performed using an input signal, whose two components are white noise with variance 1 and e(t) with variance 1. 20,000 input–output data samples are collected. Prediction error (PE) identification method [16] is used to identify a model G mod ¼ Gðz; ^dÞ and an uncertainty region D containing the true system G0 at a certain probability level a. The uncertainty region is centered at Gmod and has the following general form [1]: 8 9   f1 þ Z N 1 d f2 þ Z N 2 d < = Gðz; dÞjGðz; dÞ ¼ and 1þZ D d D¼ ; ð5Þ : ; T d 2 W ¼ fdjðd  ^ dÞ Rðd  ^ dÞ < v2 g where ^ d is the estimated parameter vector, R is a symmetric positive definite matrix that is equal to the inverse of the covariance matrix of ^ d. v2 is determined by the desired probability level a. f1 and f2 are known transfer functions. The robust stability measure for the uncertainty region D can be measured by computing the worst-casemgap dWC(Gmod, D) between the identified model Gmod and all plants in the uncertainty region D. The worstcase m-gap is an extension of the m-gap, introduced by [18], which is a global measure of distance between two transfer functions. It is defined as dWC ðGmod ; DÞ ¼ sup jWC ðGmod ; DÞ;

ð6Þ

x

where jWC(Gmod, D) is the worst case chordal distance. jWC(Gmod, D) is a frequency function defined as the maximum chordal distance between Gmod and the frequency responses of all plants in D [10,18]. It is

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equal to problem:

pffiffiffiffiffiffiffi copt , where copt(x) is the optimal value of c(x) in the following standard convex optimization

min c c;s

ð7Þ

subject to s P 0

and a set of linear matrix inequalities (LMIs) [1] with decision variables c and s. The worst-case m-gap is compared with the maximum value of the generalized stability margin given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð8Þ bopt ðGmod Þ ¼ 1  k½ N M kH ; where kAkH is the Hankel norm of the operator A [10,19] and [N M] is the normalized left coprime factorization of Gmod. If dWC is smaller than bopt, then the worst-case m-gap between Gmod and any plant in D is small and the set of Gmod-based controllers that are guaranteed by the m-gap theory [18] to robustly stabilize all systems in D is large. The smaller the dWC, the larger is the robustly stabilizing controller set. 2.2. Speed control The s domain transfer functions for the speed in Eq. (3) are transformed into z domain transfer functions assuming a zero-order hold (ZOH) and unit sampling interval. The speed control MISO true system is expressed in the multivariable ARX structure: U ðtÞ ¼ Gðz; d0 Þ½ u1

u2  T þ

eðtÞ ; 1 þ Z D d0

ð9Þ

where [u1 u2]T is input signal, e(t) is white noise; ZD(z), Z N 1 ðzÞ, Z N 2 ðzÞ and G(z, d0) are defined in Eq. (4), d0 is the true parameter vector of the transfer functions d0 for speed ¼ ½ 2:859 5:151

3:2

1:81

0:4721

2:076 9:624

3:596

10:89

17:99 12:94

T

4:576  :

A PE identification experiment [16] is performed by numerical simulation of this true system using an input signal, whose two components are white noise with variance 1 and e(t) with variance 1. 10,000 input–output data samples are collected. The procedure described in the ‘pitch angle control’ Section 2.1 for computing the worst-case m-gap and the maximum value of generalized stability margin is repeated for the speed control. 3. Integrated pitch/speed controller design A general framework for control system includes the plant represented by a transfer matrix P, an exogenous input w and actuator input u, a controller represented by a transfer matrix K, and a regulated output z and sensor output y, as shown in Fig. 1 [12]. We partition the plant transfer matrix P as   P zw P zu P¼ : ð10Þ P yw P yu z

w P

u

K

y

Fig. 1. Control system framework.

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Hence z ¼ P zw w þ P zu u; y ¼ P yw w þ P yu u

ð11Þ

where Pij is the transfer matrix from j to i for i = z, y; j = w, u. Now suppose the controller is operating, so that we have u ¼ Ky:

ð12Þ

We can solve for z in terms of w to get 1

z ¼ ðP zw þ P zu KðI  P yu KÞ P yw Þw

ð13Þ

that is, the closed-loop transfer matrix H can be represented as 1

H ¼ P zw þ P zu KðI  P yu KÞ P yw

ð14Þ

In our problem, U T ; y ¼ ½ eh eU T ; w ¼ ½ hc     0 0 G11 G12 ¼ ; P zu ¼ ; 0 0 G21 G22     1 0 G11 G12 ¼ ; P yu ¼ ; 0 1 G21 G22

z ¼ ½h P zw P yw

U c T ;

u ¼ ½ de

dp  T ;

ð15Þ

where G11, G12, G21 and G22 are the identified plant transfer functions. When selecting the de ! h channel for pitch control and dp ! U channel for speed control, the centralized control has the following decentralized implementation:   Kh 0 K¼ : ð16Þ 0 KU Many control design specifications are convex functions with respect to the closed-loop transfer matrices H [12], that is, all performance specifications can be considered simultaneously as functions in terms of H, which are evaluated under every different controller K. If there are n convex specifications required to be satisfied simultaneously, denoted as /1 ðH Þ 6 a1 ; /2 ðH Þ 6 a2 ; 

ð17Þ

/n ðH Þ 6 an ; where ai(i = 1, 2, . . . ,n) denote the expected specification value, then a MSS control problem can be formalized as: design a controller K such that all the specifications hold simultaneously. We call such a controller a satisfactory controller. Liu and Mills proposed a two-stage design procedure [12] to obtain the MSS controller: sample control laws are designed to meet individual specifications, respectively, then based on those sample control laws, a so-called convex combination approach is applied to find a new controller to achieve multiple simultaneous specifications (MSS). The integrated longitudinal control system of aircraft is shown in Fig. 2. Assume that the overall multiple performance requirements are: the pitch attitude and speed control both have good design criteria in term of tracking (small steady state error and fast setting time) and safety (acceptable overshoot). The cross effect is represented by the simulation stop time value under the cross step command.

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θc

+

eθ

–

δe

Kθ

θ

+

G11

G21

G12 Uc

eU

+

KU

+

G22

U

δp

–

Fig. 2. Integrated longitudinal control system.

/1 ðH Þ ¼ /Uovershoot ¼ ðmaxjU ðtÞjtP0;U c ¼1ðtÞ  1Þ /2 ðH Þ ¼ /3 ðH Þ ¼

/Usettle ¼ jU ðT Þ  1j 6 a2 ; /hovershoot ¼ ðmaxjhðtÞjtP0;hc ¼1ðtÞ

/5 ðH Þ ¼

/hsettle /Ucross

¼ jU ðtÞjhc ¼1ðtÞ

6 a5 ;

/6 ðH Þ ¼

/hcross

¼ jhðT ÞjU c ¼1ðtÞ

6 a6 ;

/4 ðH Þ ¼

¼ jhðT Þ  1j

 1Þ

6 a1 ; 6 a3 ;

ð18Þ

6 a4 ;

where T is the simulation end time. The desired specification values of the Boeing 747 transport aircraft longitudinal control system are defined by [13]: a1 ¼ 4  102 fps;

a2 ¼ 3  103 fps;

a4 ¼ 1  102 rad;

a5 ¼ 2:5  102 fps;

a3 ¼ 4  101 rad; a6 ¼ 1  105 rad:

In this paper, we apply the open-loop combination method [14] to design the proper integrated controller. First, design the individual controller by MSS controller design methods [12] and then integrate the individual controllers to meet the total specifications. 4. Results 4.1. Identified model validation 4.1.1. Pitch angle control The parameter vector ^ d is identified using the prediction error (PE) identification method with multivariable ARX model [16] of orders [4 [4 4] [1 1]]. The uncertainty region D containing G(z, d0) is defined by Eq. (5) with a = 0.95, v = 21.026 and ^ d for pitch angle ¼ ½ 2:8611 3:2024 0:3466

1:8088

0:3027 0:0114

0:4707 0:0037

0:4625 0:0147

0:4149 T

0:0027  :

The maximum value of the generalized stability margin is computed using Eq. (8) and is found to be bopt (Gmod) = 0.5557. The worst case chordal distance jWC(Gmod, D) at each frequency is computed using Eq. (7) and is shown in Fig. 3. The worst case m-gap dWC(Gmod, D) is equal to the maximum value of jWC(Gmod, D), which is found to be 1.0. The worst-case m-gap dWC(Gmod, D) is hence greater than the stability margin bopt(Gmod). It can be concluded that the set of Gmod-based controllers that are guaranteed by the m-gap theory

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Fig. 3. Pitch angle control: worst case chordal distance jWC(Gmod, D) at each frequency.

to robustly stabilize all systems in D is relatively small. However in this case we still keep the pair {GmodD} in order to design a Gmod-based controller for G0. Validation experiments [10] can be used to see if the Gmodbased controller stabilises G0. In this problem the controller is validated by simulation with G0. 4.1.2. Speed control The parameter vector ^ d is identified using the prediction error (PE) identification method with multivariable ARX model [16] of orders [4 [4 4] [1 1]]. The uncertainty region D containing G(z, d0) is defined by Eq. (5) with a = 0.95, v = 21.026 and ^ d for speed ¼ ½ 2:8603

3:2031 1:8124

5:1599

2:0661 9:6311

0:4727 3:5908 18:0119

10:8830

12:9402

T

4:5966 

The maximum value of the generalized stability margin is computed using Eq. (8) and is found to be bopt(Gmod) = 0.0417. The worst case chordal distance jWC(Gmod, D) at each frequency is computed using Eq. (7) and is shown in Fig. 4. The worst case m-gap dWC(Gmod, D) is equal to the maximum value of jWC(Gmod, D), which is found to be 0.0128. The worst-case m-gap dWC(Gmod, D) is hence less than the stability margin bopt(Gmod). It can therefore be concluded that the set of Gmod-based controllers that are guaranteed by the m-gap theory to robustly stabilize all systems in D is relatively large. The pair {GmodD} is used to design a Gmod-based controller for G0 and is then validated by simulation with G0. 4.2. Integrated pitch/speed MSS control design For the speed control loop, we need to satisfy the specifications /1 and /2. Using the MSS controller design method [12], two sample controllers J 1U and J 2U are designed to satisfy one specification at one time [20] 1 J 1U ¼ 5 þ ; s

J 2U ¼ 10 þ

5 s

then /1 ðH U1 Þ ¼ 0:0038241;

/2 ðH U1 Þ ¼ 0:00097127;

/1 ðH U2 Þ ¼ 0:0048616;

/2 ðH U2 Þ ¼ 0:00019801:

ð19Þ

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Fig. 4. Speed control: worst case chordal distance jWC (Gmod, D) at each frequency.

From the MSS controller design method [12], we need to solve the inequality " #    a1 /1 ðH U1 Þ /1 ðH U2 Þ k1 6 ; a2 /2 ðH U1 Þ /2 ðH U2 Þ k2 ki P 0;

ð20Þ

k1 þ k2 ¼ 1:

Using the linear programming optimization routine in MATLAB, k1 = 0.5, k2 = 0.5 are found. The final MSS controller is derived as: JU ¼

7:5002ðs þ 0:2004Þ s

ð21Þ

The simulation results of the controller satisfy the two specifications i.e. /1 = 0.0025793; /2 = 0.00064642. For the pitch attitude control loop, the same method is used. Two sample controllers J 1h and J 2h are designed to satisfy one specification at a time [20]: J 1h ¼ 1:5 

0:4 ; s

J 2h ¼ 1 

0:3 : s

The following specification matrix is obtained, i.e. " #   0:373730 0:2540100 /3 ðH h1 Þ /3 ðH h2 Þ ¼ : 0:007419 0:0098751 /4 ðH h1 Þ /4 ðH h2 Þ Since

"

/3 ðH h1 Þ /3 ðH h2 Þ /4 ðH h1 Þ /4 ðH h2 Þ

#

ð22Þ

ð23Þ

     0:4308 0:305590 a3 ; ¼ 6 a4 0:5692 0:008817

we have k1 = 0.4308, k2 = 0.5692. The MSS controller is then given by: Jh ¼

1:2154ðs þ 0:2824Þ : s

ð24Þ

The simulation results show that the control objectives can be satisfied successfully. From the above simulation results, it is obvious that the MSS controller satisfies the required objectives of the respective loop. The individual loops can be integrated through either open-loop or closed-loop combination method [14] to form the integrated control system, where the total specifications are evaluated. To reduce the order of the

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controller, the open-loop combination method is used in this paper. According to the open-loop combination method, the integrated controllers Kh and KU are K h ¼ bJ h ;

 U; K U ¼ bJ

ð25Þ

 are constant coefficients. Proper selection of the coefficients b and b  depends on the designer’s where b and b  experience [14]. For simplicity, we select b ¼ b ¼ 1 to be the proper coefficients for the controllers. In order to validate the stability and performance of the MSS controller, simulation is performed using the true plant integrated with the MSS controller. The simulation results of the integrated speed/pitch autopilot for T = 100 s are presented in Figs. 5 and 6, respectively. It can be found from Fig. 5b that the performance under the integrated control system for speed U is /1 = 0.0025956 fps and /2 = 7.5668 · 106 fps, respectively. It means that overshoot of the integrated system becomes larger than individual loop but it is still at the satisfactory level. On the other hand, the settling time value becomes smaller than before probably due to the influence of the other channel. The cross effect are very

Fig. 5. Integrated system simulation with true plant transfer functions for h and U of Uc input. (a) h of Uc input, /6 = 1.6719 · 107 rad, (b) U of Uc input, /1 = 0.0025956 fps, /2 = 7.5668 · 106 fps.

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Fig. 6. Integrated system simulation with true plant transfer functions for h and U of hc input. (a) h of hc input, /3 = 0.33267 rad, /4 = 9.0214 · 104 rad (b) U of hc input, /5 = 0.02077 fps.

small, where /5 = 0.02077 fps and /6 = 1.6719 · 107rad can be seen from Figs. 6b and 5a, respectively. Similar conclusions may be drawn from Fig. 6a, in which the following performance for pitch h is obtained: /3 = 0.33267 rad, /4 = 9.0214 · 104 rad. It is obvious that the MSS controller can be applied to the linear system of the Boeing 747 transport aircraft. 5. Conclusions In this paper, multivariable identification has been successfully performed on the logitudinal pitch angle and speed control channels of a Boeing 747 transport aircraft. The identified models have been validated using the procedure developed by Bombois and Date [1]. An integrated pitch/speed MSS controller is designed based on the identified models using the method developed in [12–14]. The controllers are validated by simulation using the true plant transfer functions. The simulation results verify the effectiveness of the above design method in Boeing 747 transport aircraft longitudinal control system. In the future work, we will continue to study the

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robustness of MSS controller design method and its application in the more complicated aircraft autopilots such as the design of the fault tolerant control systems [21] under changing flight conditions. Acknowledgement The authors thank The Hong Kong Polytechnic University for the financial support (Project Number A-PE77) towards this work. References [1] X. Bombois, P. Date, Connecting PE identification and robust control theory: the multiple-input single-output case, Part I: uncertainty region validation, in: 13th IFAC Symposium on System Identification, Paper WeA01-02, Rotterdam, August 2003. [2] X. Bombois, P. Date, Connecting PE identification and robust control theory: the multiple-input single-output case, Part II: Controller validation, in: 13th IFAC Symposium on System Identification, paper WeA01-03, Rotterdam, August 2003. [3] M. Gevers, B.D.O. Anderson, B. Codrons, Issues in modeling for control, in: Proceedings of the American Control Conference, Philadelphia, Pennsylvania, 1998, pp. 1615–1619. [4] L. Ljung, Identification, model validation and control, Plenary lecture, in: 36th IEEE Conference on Decision and Control, San Diego, California, USA, 1997. [5] L. Ljung, Identification for control – what is there to learn? in: Workshop on Learning, Control and Hybrid Systems, Bangalore, India, 1998. [6] L. Ljung, Model error modeling and control design, in: Proceedings of IFAC Symposium on System Identification, Paper WeAM1-3, Santa Barbara, CA, 2000. [7] A. Garulli, W. Reinelt, On model error modeling in set membership identification, in: Proceedings of the IFAC Symposium on System Identification, SYSID, Paper No. WeMD1-3, 2000. [8] L. Wang, G.C. Goodwin, Integrating identification with robust control: a mixed H2/H1 approach, in: Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, 2000, pp. 3341–3346. [9] R.S. Smith, J.C. Doyle, Model validation: a connection between robust control and identification, IEEE Transactions on Automatic Control 37 (7) (1992) 942–952. [10] B. Codrons, Process Modelling for Control: A Unified Framework Using Standard Black-Box Techniques, Springer, 2005. [11] E.H.K. Fung, Y.K. Wong, D.T.W. Yau, Multivariable identification and controller validation of a flight control system, in: ASME International Mechanical Engineering Congress & Exposition, Orlando, Florida, USA, Paper IMECE2005-80182, 2005. [12] H.H.T. Liu, J.K. Mills, Multiple specification design in flight control system, in: Proceedings of the American Control Conference, Chicago, Illinois, 2000, pp. 1365–1369. [13] H.H.T. Liu, Multi-objective design for an integrated flight control system: a combination with model reduction approach, in: Proceedings of IEEE International Symposium on Computer Aided Control System Design, Glasgow, 2002, pp. 21–26. [14] H.H.T. Liu, Design combination in integrated flight control, in: Proceedings of the American Control Conference, Arlington, Virginia, 2001, pp. 494–499. [15] E.H.K. Fung, Y.K. Wong, H.H.T. Liu, Y.C. Li, Design of longitudinal control system for a nonlinear F-16 fighter using MSS method, in: Proceedings of 16th IFAC World Congress, Prague, Czech Republic, Paper No. Mo-A02-T014, 2005. [16] L. Ljung, System Identification: Theory for the User, 2nd ed., Prentice-Hall, Upper Saddle, NJ, 1999. [17] B. Etkin, L.R. Reid, Dynamics of Flight: Stability and Control, John Wiley, 1996. [18] G. Vinnicombe, Frequency domain uncertainty and the graph topology, IEEE Transactions on Automatic Control 38 (9) (1993) 1371–1383. [19] K. Zhou, J.C. Doyle, K. Glover, Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ, 1996. [20] G.C. Goodwin, S.F. Graebe, M.E. Salgado, Control System Design, Prentice Hall, New Jersey, USA, 2001. [21] S. Simani, C. Fantuzzi, R.J. Patton, Model-based Fault Diagnosis in Dynamic Systems Using Identification Techniques, Springer, 2003.