Fuzzy Logic Non-Minimum Phase Autopilot Design

AIAA 2003-5550

AIAA Guidance, Navigation, and Control Conference and Exhibit 11-14 August 2003, Austin, Texas

Fuzzy Logic Non-Minimum Phase Autopilot Design Dr. KELLY COHEN, Lt Col, Israeli Ministry of Defense % Visiting Researcher, US Air Force Academy

Downloaded by UNIVERSITY OF CINCINNATI on November 24, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2003-5550

Dr. DAVID E. BOSSERT, Lt Col, USAF* US Air Force Academy

Abstract This paper describes the design of a fuzzy logic autopilot for a non-minimum phase autopilot application – an altitude hold system for a Tower Trainer 60 unmanned aerial vehicle operating at a nominal cruise condition. The response is compared to both a conventional fixed gain compensator designed through classical root-locus techniques as well as to PID control. This design presents an interesting challenge due to the non-minimum phase characteristic of the altitude to elevator transfer function. In addition, the design is tested with a 50% reduction in both longitudinal stability and pitch damping. The different case allows analysis of the off-design performance characteristics, or fault-tolerant capabilities, for the fuzzy logic and fixed gain controllers. Results show that the fuzzy logic controller outperforms the PID controller in terms of rise time and lower overshoot while both have comparable settling times for the on design case. The classical compensator performs better than the fuzzy logic design in the design case. For the off-design case of degraded dynamics, the fuzzy controller performance is roughly the same as the on-design case, while the PID and classical compensator designs have severe degradation. The effectiveness of a fuzzy logic controller.

______________________________________ % Visiting Researcher, US Air Force Academy * Assoc Prof of Aeronautics, US Air Force Academy; Senior Member AIAA

shows potential for application to other autopilot control modes in lieu of conventional designs, especially in the presence of model uncertainty or changing dynamics Introduction Many modern aircraft such as the F-22 possess fly-by-wire flight control systems, which allows for implementation of not only stability augmentation for dynamic performance but also implementation of autopilot modes. In addition, with the proliferation of autonomous unmanned aerial vehicles (UAVs), autopilot modes have become very important. Modern control laws allow for the design of robust controllers which have the potential to be effective for multiple flight conditions, including degraded performance due to partial aircraft failure. Of particular difficulty is the design of autopilots for non-minimum phase systems where closing the feedback loop can be very challenging due to the tendency of closed loop roots to move to the right half plane, causing instability. There has been some research on modern control for aircraft and UAVs. Atkins at the University of Michigan has done work on developing a fixed wing autonomous UAV called the Solus with a focus toward fault detection, isolation, and recovery [1]. In addition, Ozimina at the Naval Research Lab has implemented a variable proportional and rate feedback scheme for small unmanned aircraft [2]. Snell at the University of California has investigated robust longitudinal control design using Quantitative Feedback Theory (QFT) [3] and Bossert at the US Air Force Academy has examined QFT for pitch attitude hold systems [4]. Bossert and Cohen have examined Fuzzy Logic Pitch Attitude Hold systems for fighter jets as well [5,6]. Also,

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.


Steinberg did an excellent comparison of intelligent, adaptive and non-linear flight control laws [7]. Other work by Fujimoro at Shizuoka University analyzed gain-scheduled control using fuzzy logic based on linear matrix inequalities to implement a full-order observer-based flight control [8]. Another example for modern flight control is the design of automatic landing systems using mixed H2/Hinf control [9]. Other modern control techniques applied to flight control are nonlinear control [10] and neural networks [11]. Steinberg [7] compares in simulation six different nonlinear control laws for multi-axis control of a high performance aircraft and reported that the fuzzy logic control system had remarkably stable performance across the envelope for a fixed controller. Wijetunge et. al [12] have shown the effectiveness of a fuzzy controller, designed using the MATLAB Fuzzy Logic Toolbox, as compared with a PID controller for the nonminimum phase diesel engine control system. The focus of this effort [12] was to consider the interactive gas management behavior of the engine concentrating on the control of exhaust gas recirculation and variable geometry turbocharging parameters. Seng et. al. [13] successfully demonstrate the robustness of a neuro-fuzzy control strategy for stabilization of a robotic manipulator whose plant has an unstable pole with a non-minimum phase. The main aim of this research effort is to examine the effectiveness of a fuzzy logic based altitude hold system for a Tower Trainer 60 (TT60) Unmanned Aerial Vehicle (UAV). The design represents an interesting challenge due to the non-minimum phase characteristics of the altitude to elevator transfer function. Fuzzy logic, which is the logic on which fuzzy control is based, is a convenient way to map an input space into an output space. The experience of the past decade, with the successful marketing of a wide variety of products based on the Fuzzy Logic Control (FLC), has shown that for certain applications using FLC can lead to lower development costs, superior features, and better end product performance. One of the inherent properties of fuzzy logic systems is that it has the capability of being a universal approximator as shown by the Stone-Weirstrass theorem [14]. This implies that by using enough inputs, a number of rules and a number of fuzzy sets for

each input variable, a fuzzy based system can approximate any real continuous nonlinear function to an arbitrary degree of accuracy. The implementation of a variable damping strategy requires such a universal approximator that can successfully emulate the bang-bang type of minimum-time control. The reasons for selection of fuzzy logic control for the altitude hold system of a TT60 UAV are the relative ease and simplicity of implementation and the robustness characteristics. The application is a UAV autopilot with the interface shown in Figure 1. The ground station interfaces with an RF uplink/downlink for both command/telemetry and video data. The approach is to build a generic interface and change the flight control for a specific UAV. The UAV autopilot is implemented with an altitude hold, airspeed hold, and heading hold. Since the altitude hold transfer function is nonminimum phase, it is the most challenging aspect of the design and is the focus of this paper. . Current Interdisciplinary Program for autonomous UAVs

Camera

RF Uplink/ Downlink

On-Board Flight Manager

Ground Station

GPS/IMU

Flight Control System

Figure 1 – UAV Autopilot Interface The layout of this paper is to define the TT60 plant models, and then the altitude hold control scheme. Next, the classical compensator and PID designs for the baseline case of a TT60 are presented. Then, the design of the fuzzy logic controller is presented and compared to the compensator and the PID cases for the on and off design cases. Finally, conclusions are presented.

angle of attack, and airspeed to elevator deflection transfer functions).

h

Since the

e

PLANT MODELS


A TT60 is chosen since it is a stable platform to develop UAV applications at the USAF Academy. The baseline condition is cruise at 7000 ft and a speed of 40 mph (58.6 ft/s). Figure 2 shows a picture of the TT60 aircraft.

altitude to elevator transfer function is needed it must be obtained from two of the three basic transfer functions. This is accomplished through the following equation [16]:

h( s ) U 1 = s e (s)

( s) e ( s)

(s) e ( s) ( 1)

where U1 is the steady state or trim velocity (40 mph for the TT60 in cruise). This gives a general

format of:

Figure 2 – Tower Trainer 60 Aircraft To develop the plant model, physical parameters such as planform area, span, and inertias are measured. Next, stability and control derivatives are calculated using analytic techniques documented in several sources [15, 16]. The

majority of the major stability and control derivatives are calculated this way, specifically:

h( s ) ( s zero1)( s + zero 2)( s + zero3) = s ( Phugoid )( Short _ Period ) e (s) ( 2) This shows the variation in altitude from a trim altitudefor an elevator input. All plant models include actuator dynamics modeled as 10/(s+10), or an actuator with a time constant of 0.1 seconds. To investigate fault tolerance, several degraded variants of this basic flight condition were examined as well. Specifically, a 50% reduction in the following derivatives: static longitudinal stability derivative, Cm , and the pitch damping derivative, Cmq. The plant transfer functions without the actuators are summarized in Table 1.

C L , C m , C n , C mq , C m & , C l p , C nr , C m e , C l a , C n r . For those stability derivatives which can not be easily calculated, values for the Cessna 182 (which is very similar to the TT60 but at a larger scale) in cruise are used [15]. Finally, stability parameters and transfer functions are calculated [16]. The three transfer functions generated in

,

this manner are: e

, e

u e

(Pitch angle,

To show the variation of the plant dynamics, a pole-zero map of the two plant cases are plotted in Figure 4. Note the variation in plant dynamics for the case where both Cmq and Cm are degraded, most notably due to the non-minimum phase zero moving further to the right. Additionally, a Bode plot is shown in Figure 4.

Table 1 – h/delta_e Transfer Function

Flight Condition


Design Condition

Degraded Condition, 50% Reduction in both Cm & C mq

Transfer Function 3

2

- 34.16 s - 144.4 s + 7047 s + 557.2

------------------------------------------------------s 5 + 13.18 s 4 + 95.93 s 3 + 14.61 s 2 + 31.94 s

-34.16 s 3 - 62.64 s 2 + 8252 s + 715.9 ---------------------------------------------------------s 5 + 10.79 s 4 + 48.61 s 3 + 7.852 s 2 + 15.96 s

Figure 4 – Bode Plot of plant cases

ALTITUDE HOLD SCHEME An altitude hold system is an autopilot mode which tries to maintain altitude by using the elevator to correct for variations in altitude. The altitude input can either be from a barometric

altimeter, radar altimeter, or a Global Positioning System output. The block diagram for an altitude hold system is shown in Figure 5 [16].

+ K hc

Gc

10 ( s + 10)

h

h e

-


Figure 5 – Altitude Hold Block Diagram The “compensator” in the above system, Gc, can be a simple gain, an adaptive gain, a controller with poles and zeros (possessing dynamics), or any of the many robust controllers available. This paper examines three of the above compensators – a fixed gain controller with poles and zeros designed through root-locus techniques, a proportional-integral-derivative (PID) controller, and a fuzzy logic controller.

FIXED COMPENSATOR DESIGN The baseline design was made to introduce senior level undergraduate students to the difficulties in closing the loop when a nonminimum phase system. The simplest

compensator is a straight gain. The gain is chosen to get all the roots as far from the origin as possible to speed up the response while keeping the system stable. The root locus showing this is provided in Figure 6. A root locus zoomed in around the origin is provided in Figure 7. Closing the loop with a Gc = .00192 results in a very undesirable time response, as shown in Figure 8. Notice the oscillations and length of time required to reach steady state. A fixed compensator is now needed to improve the response. This is accomplished by moving the phugoid roots to the left and the zero closest to the origin at -.07895 much further to the left.

Figure 6 – Root Locus, Basic Altitude Hold Systems


Gc=K (s+new_zero2)(phugoid

Figure 7 – Zoomed-in Root Locus, Basic Altitude Hold With Fixed Gain Compensator

.

Figure 8 – Time Response, Fixed Gain Altitude Hold

Gc=K (s+new_zero2)(phugoid roots)/(s+zero2)(new phugoid roots)

(3)

The specific values chosen for the compensator are:


Gc =

0.00842(s + 7.895)(s 2 + 0.108s + 0.3393) (s + 0.07895)(s 2 + 4s + 8) (4)

The root locus for this new compensator is shown in Figure 9. The greatly improved time response is shown in Figure 10.

PID DESIGN

The next attempt at designing an altitude hold controller is to use a proportional-integralderivative (PID) controller. The block diagram for this is shown in Figure 11. The chosen gains are Kp=0.017, Ki=0.008, and Kd=0.025. These values were obtained by tuning the gains to achieve a time response with fast rise and settling times and minimum undershoot and overshoot. This results in the time response shown in Figure 12. Notice that there is very little undershoot due to non-minimum phase, and there is a peak value of 1.24 for a step input and a 5% settling time of 10 seconds. This is a much better response than the straight gain design response, but not as good as the compensator.

Figure 9 - Altitude Hold – Fixed Compensator


Figure 10 – Fixed Compensator Altitude Hold Time Response

Figure 11 – Altitude Hold With PID


CONCLUSIONS

Figure 12 – PID Altitude Hold Step Input Time Response

Why did the PID controller provide this response? Examination of the root locus with the PID controller can provide valuable insight. A PID controller can be represented as a transfer function for purposes of analysis.

adds two zeros and one pole, and this changes the shape of the root locus.

K d s + K p s + Ki K = PID _ TF ( s ) = K d s + K p + i = s s 2

For the altitude hold PID controller, this corresponds to a transfer function of:

PID _ TF ( s ) =

Kd s2 +

Kp Kd

s+

Ki Kd

s

0.025(s 2 + 0.068s + 0.32) 0.025(s + 0.34 + / i 0.45) = s s

The root locus plot of Figure 13 shows why this is successful. The non-minimum phase characteristic alters the root locus construction assumption that the real locus lies between an odd number of poles/zeros. A PID effectively

(5)

Figure 14 shows a close up of the PID root locus near the origin. Notice how the zeros associated with the PID controller actually draw the complex poles near the origin (phugoid) away from the origin, this speeding up the response. Also, the damping on the short period roots is

(6)


increased. Thus, there is an improvement to both modes

Figure 13 – PID Altitude Hold Step Root Locus

Figure 14 – PID Altitude Hold Step Root Locus, Zoomed-in


FUZZY LOGIC CONTROL DESIGN

The altitude hold system of the Tower Trainer 60 (TT60) UAV exhibits dynamics of a secondorder system . Cohen, Weller and Ben-Asher [17] propose an effective means of controlling such systems by introducing a variable damping strategy, which is realized by a fuzzy logic algorithm. The main advantages of using a fuzzy approach are the relative ease and simplicity of implementation and the robustness characteristics. The parameters of the fuzzy controller may be adapted to provide fairly fast control for large deviations, of the measured state of the plant from the desired state, and a minor amount of control for small deviations. The successful implementation of a fuzzy logic controller depends, among other design aspects, on the heuristic rule base from which control actions are derived. In order to obtain the required heuristic physically-based insight, a single DOF system based on optimal control theory was analytically examined, to observe the characteristics of a minimum time solution [17]. Based on this analysis, Cohen, Weller and BenAsher [17] introduced a fuzzy logic non-linear mapping function, which has the potential of being a universal approximator to emulate the above minimum time solution. The resulting rule base is the core of the control law that is applied to all the current application. This approach has been applied to vibration suppression of flexible structures [18] and for active suppression of aircraft cabin noise that is induced by structure borne vibration [19]. Furthermore, the above method has been demonstrated experimentally on smart structures at the Technion [20]. The Fuzzy Logic design was accomplished using MATLAB. A typical layout of a fuzzy logic controller is shown in Figure 15 and Figure 16. The fuzzy controller is implemented as a 25-rule Mamdani Fuzzy system with 2 inputs and 1 output. The two inputs are Altitude_error and Altitude_Integral, and the output is elevator deflection. Five membership functions are used to describe each of the input and output parameters, namely, POSITIVE, SMALL POSITIVE, ZERO, SMALL NEGATIVE and NEGATIVE. The respective membership functions for the inputs / output parameters are obtained after a tuning process. The fuzzy

adaptation strategy is based on rules of the form "if...then..." that convert inputs to a single output, i.e. conversion of one fuzzy set into another. Heuristic rules based on previous experience [17-20] are coupled with fuzzy reasoning whereby large values of the inputs require a lightly damped system, which would provide quick rise times. However, when the plant state is in the vicinity of the desired state, the damping factor is large to reduce the overshoot and steady state error. In the next step, all the output values, obtained by clipping or scaling, are then brought together to form the final output membership function. After evaluation of the propositions, the output values represented are unified to produce a fuzzy set incorporating the solution variable. This unification of outputs of each rule, referred to as aggregation, occurs only once for each output variable. The aggregation process is always comprised of a commutative method. In this effort, the method applied is the Bounded SUM (simply the sum of each rule's output set having an upper bound of 1). Applying the sum to the rule base, the union of the fuzzy sets for the same output variable is taken to reach the respective aggregation of the output. Finally, in order to reach a practical controller a control action comprising of a single numerical value is required. Therefore, the space of the fuzzy damping factor, obtained using the method described in the previous section, is mapped into a non-fuzzy space (crisp) in a process known as defuzzification. There are various strategies aimed at producing a crisp value. Herein, the center of area (COA) scheme is adapted. Further details of the fuzzy controller may be found in Cohen et al. [17-20].

Rule Base

Inference Engine


Defuzzifier

Fuzzifier

y1, y 2

F Plant

Figure 15 – Fuzzy Controller Components

Figure 16 – Fuzzy Controller Block Diagram


Figure 17 – Fuzzy Controller Step Input Time Response

Comparison It should be apparent by now that for the design condition, the Fuzzy Logic and PID controllers provided good results, while the compensator performed exceptionally. A comparison of all three plots is shown in Figure 18, and shows that to be the case. The Fuzzy controller has a slightly lower overshoot, so has a slight advantage over the PID controller at the design condition. The compensator is the clear winner at the design condition.

How does the control effort between the controllers compare? Many times, high performance controllers achieve the results with a penalty of high control effort. Figure 19 shows that in this case, the PID and compensator controllers have slightly less control effort than the Fuzzy controller. After the first 4 seconds, the control efforts are nearly identical.


Figure 18 – Controller Comparison at the Design Condition for a Step Input

Figure 19 – Elevator Control Input – Design Condition


The real test of the robustness of a controller is coping with changing dynamics or uncertainty in a plant. Thus, all the controllers were tested in the plant case where there is a 50% reduction in both static longitudinal stability and pitch damping. One would expect the Fuzzy controller to shine in this case due to its inherently robust nature. In fact, Fuzzy does perform extremely well in this case, while the PID controller has severe degradation and the

the same as the design case! This is very important because it means the aircraft response is nearly the same for the design condition and the degraded condition. The penalty for robust performance is not actuator saturation, as is the case for many robust controllers. Table 2 summarizes all of the responses in terms of settling time, rise time, peak time, peak overshoot, and aggregate elevator control effort. For the design condition, the PID and Fuzzy controllers are comparable, while the

Figure 20 – Time Response of Degraded System to a Step Input fixed compensator exhibits degraded performance. This is shown in Figure 20. The peak value for the Fuzzy controller is only slightly higher than the design condition at 1.22, and the settling time increases to about 15 seconds. The PID overshoot goes up to 1.35 while the settling time jumps up to 80 seconds with many oscillations. The compensator has by far the highest peak response of any of the controllers at the off-design condition with a value of 1.52 The corresponding control effort is shown in Figure 21. The amazing feature of this plot is that the Fuzzy controller control effort is roughly

compensator has the best performance. For the degraded flight condition, the Fuzzy controller has a huge advantage over the PID and a smoother response than the compensator.


Figure 21 – Elevator Response for Degraded System with a Step Input

Table 2 – Time Response Summary Case

Ts (s)

Tr (s)

Tp (s)

Mp (ft)

Control Effort*

Compensator (Design Condition)

3.31

1.81

6.94

1.0

58.8

PID (Design Condition)

10

1.45

3.4

1.24

60.6

Fuzzy (Design Condition)

10

1.11

3.5

1.1

71.7

Compensator 50% Reduction in Cm and Cmq

13.2

0.758

2.3

1.52

39.5

PID 50% Reduction in Cm and Cmq

80

1.12

4.4

1.35

363.2

Fuzzy 50% Reduction in Cm and Cmq

15

2.09

4.6

1.22

72.8


CONCLUSIONS AND RECOMMENDATIONS

The results show that compensators designed to move undesirable roots to the left, Fuzzy logic controllers, and PID controllers are successful of tackling a non-minimum phase flight control problem, while straight gains have limited use. In this case, the application was a non-minimum phase altitude hold system for a TT60 UAV. If robustness is an issue, Fuzzy Logic can provide good performance for degraded fight conditions with roughly the same control effort as the design case. Future work will implement this flight control law on a TT60 UAV. This is part of an effort at the US Air Force Academy to push UAVs to a higher level of recognition both in the curriculum and to cadets at the US Air Force Academy. Eventually, some of the UAVs will be used a flying test beds for robust flight control law comparison.

Conference, Montreal, Quebec, Canada, AIAA2001-4084, 6-9 August 2001. [6] Bossert, D.E., and Cohen K., “PID and Fuzzy Logic Pitch Attitude Hold Systems for a Fighter Jet”, AIAA Guidance, Navigation, and Control Conference, Monterey, California, AIAA-2002-4646, 5-8 August 2002. [7] Steinberg, M. L., “A Comparison of Intelligent, Adaptive, and Nonlinear Flight Control Laws”, Naval Air Warfare Center Aircraft Div., Patuxent River, MD, ADA368768, 4 June 1999, pp. 1-11. [8] Fujimoro, A., and Tsunetomo, H., “GainScheduled Control Using Fuzzy Logic and Its Application to Flight Control”, Journal of Guidance, Navigation, and Control, Vol. 22, No. 1, Jan-Feb 1999, pp.175-177. [9] Shue, S., and Agrawal, R., “Design of Automatic Landing Systems Using H2/Hinf Control”, Journal of Guidance, Control, and Dynamics, Vol 22, No. 1, Jan-Feb 1999, pp. 103-114.

References

[10] Lin, Y., and Liu, S., “Integrated Projection Gain and Nonlinear Control of Unmanned Air Vehicle”, Proceedings of the AIAA Guidance, Navigation, and Control Conference, 1998, pp. 437-446.

[1] Atkins, E., et. al., “Solus: An Automatic Aircraft for Flight Control and Trajectory Planning Research”, Proceedings of the 1998 American Control Conference, 1998, pp. 689693.

[11] Kim, B., and Calise, A., “Nonlinear Flight Control Using Neural Networks”, Journal of Guidance, Control, and Dynamics, Vol 20, No. 1, Jan-Feb 1997, pp. 26-33.

[2] Ozimina, C. et. al., “Flight Control Systems Design for a Small Unmanned Aerial Vehicle”, Proceedings of the 1995 American Control Conference, 1995, pp. 2694-2669.

[12] R.S.Wijetunge, R.S., Brace, Hawley, J.G., Vaughan, N.D., “Fuzzy Logic Control of Diesel Engine Turbocharging and Exhaust Gas Recirculation”, CONTROL 2000 UKACC International Conference on Control, University of Cambridge, 4-7 September 2000 MiniSymposium on Engine Control Systems.

[3] Snell, A. and Stout, P., “Robust Longitudinal Control Design Using Dynamic Inversion and Quantitative Feedback Theory”, Journal of Guidance, Navigation, and Control, Vol 20, No. 5, Sep-Oct 1997, pp. 933-940. [4] Bossert, D. et. al., “Design of Robust Quantitative Feedback Theory Controllers for Pitch Attitude Hold Systems”, Journal of Guidance, Navigation, and Control, Vol 17., No. 1, Jan-Feb 1994, 217-219. [5] Bossert, D.E., and Cohen K., “Design of Fuzzy Pitch Attitude Hold Systems for a Fighter Jet”, AIAA Guidance, Navigation, and Control

[13] Seng T. L., Khalid, M., Yusof R., and Omatu, S., “Adaptive Neuro-Fuzzy System by RBF and GRNN Neural Networks”, Journal of Intelligent and Robotic Systems, Kluwer Academic, Vol. 23, 1998, pp. 267289. [14] Mendel J.M., 1995, "Fuzzy Logic Systems for Engineering: A Tutorial", Proceedings of the IEEE, Vol.83, No.3, 345377.

[15] Roskam J., Airplane Flight Dynamics and Automatic Flight Controls Part 1, DARCorp, Lawrence, KS, 1995


[16] Yechout, T., Morris, S., Bossert, D., and Hallgren, W., An Introduction to Aircraft Flight Mechanics, AIAA Education Series, 2003. [17] Cohen, K., Weller T., and Ben-Asher J., “Control of Linear Second-Order Systems by a Fuzzy Logic Based Algorithm”, Journal of Guidance, Control, and Dynamics, Vol. 24, No. 3, May-June 2001. [20] Cohen, K., Weller T., and Ben-Asher J., “Active Control of Flexible Structures based on Fuzzy Logic”, 22nd International

Congress of Aeronautical Sciences – Harrogate, United Kingdom, August 27September 1, 2000, pp.493.1-493.9. [21] Cohen, K., Weller T., and Ben-Asher J., “Fuzzy Logic Control of Aircraft Cabin Noise”, 41st Israel Annual Conference on Aerospace Sciences, Tel-Aviv and Haifa, Israel, February 21-22, 2001. [22] Cohen, K., Yaffe R., Weller T., and Ben-Asher J., “Experimental Studies on Piezo-Ceramic Material Control Using Fuzzy Logic”, 10th International Conference on Adaptive Structures and Technologies, Paris, France, October 11-13, 1999,

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Fuzzy Logic Non-Minimum Phase Autopilot Design

Fuzzy Logic Non-Minimum Phase Autopilot Design

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