Reliable robust flight tracking control: an lmi approach ... - IEEE Xplore

76

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY 2002

Reliable Robust Flight Tracking Control: An LMI Approach Fang Liao, Jian Liang Wang, Senior Member, IEEE, and Guang-Hong Yang

Abstract—This paper studies the reliable robust tracking controller design problem against actuator faults and control surface impairment for aircraft. First, models of actuator faults and control surface impairment are presented. Then a reliable robust tracking controller design method is developed. This method tracking is based on the mixed linear quadratic (LQ)/ performance indexes and multiobjective optimization in terms of linear matrix inequalities. Flight control examples are given, and both linear and nonlinear simulations are given. Index Terms—Actuator faults, control surface impairment, flight control, linear matrix inequality (LMI), reliable control, tracking.

I. INTRODUCTION

I

N THE design of the advanced tactical fighter (ATF), reliability, maintainability, and survivability are three basic design requirements. A fighter aircraft with reliability should be able to maintain a satisfactory closed-loop system performance in the presence of sensor/actuator faults or control surface impairment in the battle environment. Aircraft has certain inherent redundancy built in. For example, conventional aircraft has mainly three sets of control surfaces: the elevator (horizontal stabilator) for pitch control; the differential aileron for roll control; and the rudder for yaw (directional) control. For ATF, there are usually also canard and flaps for pitch and yaw control. If the movement of the left and right elevators and ailerons are made independent of each other, the inherent redundancy of aircraft control surfaces can be made use of to tolerate the loss of certain control surfaces. To a certain extent, the left and right elevators provide roll (roll and pitch) control if they deflect the same (different) amount toward different direction, and the left and right ailerons also provide pitch (pitch and roll) control if they deflect the same (different) amount toward the same direction. Currently there are several fault-tolerant/reliable flight control system design conceptions, such as self-repairing flight control systems (SRFCS) [1], [17], reliable flight control systems [18], [26], and so on. SRFCS design conception involves such procedures as real-time fault detection, isolation (FDI),

Manuscript received May 29, 2001. Manuscript received in final form October 29, 2001. Recommended by Associate Editor A. Ray. This work was supported in part by the Academic Research Fund of the Ministry of Education, Singapore, under Grant MID-ARC 3/97 and DSO National Laboratories, Singapore, under Grand DSOCL01144. F. Liao and J. L. Wang are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]). G.-H. Yang is with the Temasek Laboratories, National University of Singapore, Singapore 119260. Publisher Item Identifier S 1063-6536(02)00339-1.

and flight control system reconfiguration. This approach is considered as an active approach as reconfiguration is required of the system in the event of faults. Due to the flexibility offered by the flight control system reconfiguration procedures, the controllers in such systems may not be in a fixed form. However, time is needed for FDI to identify the fault and to execute reconfiguration. Another approach to reliable flight control system design is to exploit the inherent redundancy of aircraft control surfaces to design a fixed flight control system so as to achieve a tolerable system performance degradation in the event of component faults. The designed fixed controller guarantees satisfactory system performance not only during normal operations but also under various component faults without the need for FDI and system reconfiguration. This is very important in the case that the available reaction time for the system is very little after the occurrence of a severe fault. As no system reconfiguration is needed, this approach is passive. This passive approach to reliable control system design has attracted considerable attention recently, and several approaches have been developed, for example, ARE-based approach [15], [16], [21], LMI-based approach [6], [20], [22], [23], Pole region assignment technique [27], QFT method [18], etc. In many flight control system designs, robust tracking performance is an important requirement. Currently there are several approaches developed to solve tracking problems. The classical approach for LTI systems has been to design a closed-loop system that achieves the desired transfer function as close as possible [4]. The inherent shortcoming is over-design. The linear quadratic (LQ) control theory method [7], [14] requires a priori knowledge of the dynamics of the reference signal. optimal tracking solution [5], [19] is suitable for cases The where the tracking signal is measured on-line, and it can hardly deal with the case where a priori knowledge on this signal is available or when it can be previewed. The major disadvantage of the solution is the fact that the reference signal is assumed to be an arbitrary signal, hence taken as a disturbance signal so as to minimize the performance for a worst-case reference signal. Game theory [13] is another method for solving tracking problem. This approach is most suitable to finite time control of time-varying systems. In this paper, we study the reliable robust flight tracking controller design problem for a fighter aircraft in the presence of actuator faults and/or control surface impairment. The multiobjective optimization methodology is used to ensure that the designed flight tracking controller guarantee the stability of the closed-loop system and optimal tracking performance during normal system operation and maintaining an acceptable lower

1063–6536/02$17.00 © 2002 IEEE

LIAO et al.: RELIABLE ROBUST FLIGHT TRACKING CONTROL

77

level of tracking performance in the event of actuator faults and/or control surface impairment. This paper is organized as follows. Section II gives the fault models of actuators and control surfaces impairment. Section III discusses the problem formulation. Section IV presents the design method of the reliable robust tracking controller. In Section V, a flight control system example is presented, followed by some concluding remarks in Section VI. II. FAULT MODEL Consider a LTI aircraft model described by (1) is the state, where is the output, is the control input and is the bounded input disturbance. To investigate the reliable flight control and tracking problem in the event of actuator faults or control surface impairment, the represent the fault model must be established first. Let control surface input vector after failures have occurred. Then the following actuator fault model is adopted for this study:

with the uncertain constant parameter vector satisfying

(5) vertices, of which, one vertex corresponds Here, there are to the normal case (no fault), and the remaining vertices correspond to the fault cases. Uncertain system with polytopic uncertainty has been studied by many researchers [9], [11], [12]. In this paper, we use it to model control surface impairment faults. are known The matrices , , , and constant matrices of appropriate dimensions, which represent the vertices of possible control surface impairment (including the nominal case of no control surface impairment). Without corresponds to the nomloss of generality, suppose that , , , . inal case, i.e., Note that there is a certain degree of over-design in adopting this polytopic uncertainty model. However, this ensures that the designed performance can be achieved for intermediate control surface impairment faults (as will be demonstrated by the nonlinear simulation results in Section V-B). Hence, the aircraft dynamics with both actuator faults (2) and control surface impairment (4) is described by

(2) (6)

where the scaling factor satisfy

or

(3)

for , the fault model Obviously when (2) corresponds to the case of the th actuator outage. When , it corresponds to the case of no fault in the th actuator. , namely, Without loss of generality, we assume that corresponds to the normal control input vector . The usual control surface fault is the control surface impairment. Unlike actuator faults of the form (2), aircraft control surface impairment will change the aerodynamic characteristics of the aircraft (i.e., , , , and matrices). Control surface impairment can be characterized by the percentage loss of the total control surface area. Models of the aircraft can be obtained for a number of percentage losses of certain control surfaces. By using these models, aircraft models corresponding to intermediate percentage losses of such control surfaces can be obtained approximately by linear interpolation. To ensure that the design method can handle the inherent modeling errors in these interpolated models, we adopt the following polytopic uncertainties in the matrices , , , and :

It is noted that, in the faulty system description (3)–(6), when , we have , , and , namely, without control surface im, the model (6) covers the pairment fault. When case with control surface impairment faults. The parameter provides interpolation in-between the vertices (models) corresponding to the selected percentage losses of the control surfaces and the no-fault case. This provides approximate models and for the intermediate faults. Obviously, when in (2), the aircraft model (6) represents the nominal aircraft model (1) of having neither outage faults nor control surface impairment faults. III. PROBLEM FORMULATION Consider the aircraft dynamics (6) with both actuator faults (2) and control surface impairment faults (4). The design problem considered in this paper is to find a controller such that: • The closed-loop system is robustly stable for all ( ) and all . tracks the refer• In the normal operation, the output without steady-state error, that is ence signal (7)

(4)

and with optimal closed-loop performance. [See the peris a known formance definition in (13).] Here

78


Fig. 1. Block diagram of fight control system with independent control surfaces.

constant matrix used to form the output required to track the reference signals. • In the event of actuator faults or control surface impairtracks the reference signal ment, the output without steady-state error and with an acceptable lower level of tracking performance. [See the performance definition in (13).] It is well known that the tracking error integral action of a controller can effectively eliminate the steady-state tracking error. In order to obtain a robust tracking controller with state feedback plus tracking error integral, we introduce the following augmented state-space description of the aircraft model (6)

It is easy to see from (4) and (10) that this augmented system can be expressed in the polytopic form as follows:

(11) where

(8) (12) Fig. 1 shows the block diagram of the flight control system. De, fine the augmented state vector and disturbance vector , . The augmented system (8) can be rewritten as

If we obtain a controller robustly stabilizing the augmented system (9), then the controller also stabilizes the original system . (6) and achieves the output regulation In the following design, we choose the performance indexes of normal and fault cases as

(9) where

, and (13)

(10)

and are symmetric positive where is symmetric positive definite. semidefinite and of the Our objective is to minimize the performance index nominal system (1), while simultaneously guarantees that the ) meet certain upper performance indexes . bounds for all probable actuator faults and for all


79

IV. RELIABLE ROBUST TRACKING CONTROLLER DESIGN First, to ensure that the augmented system (9) corresponding to the normal operation and each of the probable faults be all stabilizable via state feedback, assume that is stabilizable for all and [H1] . , i.e., full row rank, for all [H2] , and . Consider the augmented system (9) with the following statefeedback tracking controller:

Proof: According to [2, Lemma 3.1], under assumptions [H1] and [H2], if (16) and (17) are satisfied, then and

(21) are satisfied. By Schur complement, (21) is equivalent to

(14)

where system is given by, for

. The closed-loop augmented

(15) Adopting the multiconvexity concept in [2], the following Lemma 1 is easily derived. Lemma 1: Assume that the hypotheses [H1] and [H2] hold. ( ; For every given upper bound ), if there exist a matrix and with , satissymmetric matrices and fying, for all

(22) , Since (22) we have that

,

and

, from

(23) affinely quadratically stabilizes Hence controller the augmented system (9). Furthermore, substituting into (13) yields, for all

(16)

(17) where

(18) then the closed-loop system (15) achieves the upper bounds of performance indexes

(19) where

This completes the proof. It is noted that the above Lemma 1 gives a sufficient condition guaranteeing the closed-loop system (15) be robustly stable and ( ) be the quadratic performance indexes ( ) correspond bounded as in (19). Here norm of the transfer function from the input to the in (15) to the performance output

is as in (5) and

(20)

It is also noted that in Lemma 1, we adopt different Lyapunov where represent the actuator function

80


outage faults and represent the vertices of control surface impairment faults. Hence the conservativeness of Lemma 1 is smaller compared to using a common Lyapunov function . The (17) is used to guarantee that the upper bounds of performance indexes of the intermediate faults (i.e., faults between the vertices of control surface impairment faults) lie among those of the vertices of control surface impairment faults. Now the reliable tracking controller design problem can be redefined as Minimize

(24)

( ; subject to (16), (17), and ) as well as ( ; ; ) where are performance bounds in the fault cases. However, the matrix inequalities (16) and (17) in Lemma 1 are not jointly convex. In order to get around this problem and reduce the conservativeness, an iterative method is adopted here. Theorem 1: Consider the closed-loop system (15). Assume that the hypotheses [H1] and [H2] hold. For given upper , and a bounds given sufficiently small positive constant as well as a given and given symmetric positive-definite matrix , suppose that there exist a matrices and symmetric matrices feedback gain matrix satisfying, for all and ,

where , , and are as in (18). Then, the controller (14) robustly stabilizes the closed-loop system of the augmented system (9). Furthermore, the upper bounds of perare given by formance indexes (27) , and and as in (20). Proof: By the Schur complement formula, (25) and (26) are equivalent to

with

(28) and

(29) (25)

(26)

respectively. Since is positive definite and is a sufficiently small positive constant, it is obvious that (16) and (17) are satisfied if (28) and (29) are satisfied. The subsequent proof is similar to that of Lemma 1 and is omitted here. To reduce the conservativeness, Theorem 1 leads to the following iterative algorithm for designing reliable tracking controllers. Algorithm 1: ( ; ), 1) Choose proper , subject to and and minimize

with

(30) (31) Let

.


2) Minimize and

81

axis roll rate , and sideslip angle must be stabilized and should be tracked perfectly. In this section, two examples of tracking control for a linear F-16 aircraft model are given to demonstrate the proposed methods. The nonlinear dynamic model for this F-16 aircraft can be found in [10] and [25]. The trimmed values of the F-16 aircraft equations are

) subject to

and

(32)

m

(33) m/s

. then we obtain and performance 3) Choose a small error tolerance ( ; bounds in the fault cases ; ). Let ( ; ) and . ), minimize 4) At the th iteration ( subject to the LMI constraints and for and , and for and , . Here and are defined by (25) and (26), respectively. , set and stop. 5) If and for Otherwise, let and , and go back to Step 4. , , It should be noted that the existence of solutions and in (25) and (26) is equivalent to the existence of and in (16) and (17). Therefore, Theorem 1 solutions does not introduce any additional conservativeness other than and , (25) and those in Lemma 1. However, for given , which can be solved by using the (26) are LMIs for and LMI Toolbox in MATLAB environment. Hence, in this sense, Theorem 1 converts a nonconvex problem into a convex problem tends to and its conservativeness can be minimized when ( ; ) and tends to . ( ; In Algorithm 1, the initial values of ) in (25) and (26) are obtained by using the common LMI solution. be defined in Step 4 Theorem 2: Let is converof Algorithm 1. Then the sequence gent. , is a feasible soProof: For a given integer ) in step 4, and is lution for the th optimization ( . the optimal solution. So must be convergent. Hence, the sequence Theorem 2 shows that Algorithm 1 is convergent. It should also be noted that Algorithm 1 is guaranteed to achieve a local optimum only. There is no guarantee for achieving the global optimum. V. APPLICATION EXAMPLES In modern tactical fighter aircraft design, the control system must be able to perform challenging maneuvers such as high flight under a high angle of attack flight and high roll rate low speed condition. Under these conditions, there are significant kinematics and inertial couplings, so the variables stability

is the total where is the engine throttle, is the altitude, is the reference genter of gravity location, and airspeed, and are, respectively, the horizontal stabilizer, the , and aileron, and the rudder. Let

where the matrix determines the outputs required to track, i.e., , and . The tracking commands of , , and are unit steps. A. Example 1 (Actuator Outage) First, we consider actuator outage faults only. After the left and right elevator and aileron are made independent [24], the aircraft equation with the actuator outage fault becomes (34) , , , represents the vertical gust disturbance and , , and are given in the Appendix. In this example, the following possible actuator faults are considered: —Left elevator actuator 1) outage; —Right elevator actuator 2) outage; —Right aileron actuator 3) outage; —Left aileron actuator outage; 4) —Left elevator actuator and 5) right aileron actuator outage; —Right elevator actuator and 6) right aileron actuator outage; —Right elevator actuator and 7) left aileron actuator outage; —Left elevator actuator and 8) left aileron actuator outage.

where

82


TABLE I PERFORMANCE INDICES COMPARISON FOR FAULTS 1–8 (! –! ) WITH

Since we do not consider the control surface impairment in , and . In the example, then order to maintain the conventional control surface movements (i.e., symmetric motion for left and right elevators, and antisymmetric motion for left and right ailerons) under normal operwith , ation, we force , . Consider simultaneously all outage faults ( to ) to guarantee the performance requirement when any of these eight outage faults occurs. By solving the convex multiobjective optimization problem using the proposed iterative and ) are obAlgorithm 1, the reliable trackers ( tained under different design requirement of the fault performance indexes. The design results are tallied in Table I. The and are 1) ( design bounds used for ); 2) ( ) as indicated in Table I. For comparison purpose, the standard tracker design (without using considering the actuator fault) and a reliable tracker the common LMI solution (i.e., Steps 1 and 2 in Algorithm 1 before the iterations in Steps 3–5) are also included in Table I. The corresponding controller parameters are given in the Appendix. From Table I, we can see that, in this particular example, the standard controller (which considers no fault and achieves an optimal nominal performance of 16.7024) can still stabilize the system when any one of the eight actuator outage faults occurs. But the performance becomes very bad, especially for and . For the reliable controller , the fault cases of by imposing the performance index of 99 on all fault cases, our method gives an optimal guaranteed nominal performance of 16.9275 and an actually achieved nominal performance of 16.9251. The guaranteed and the actually achieved nominal performance values are very close, and both are just slightly (1%) larger than that achieved by the standard controller. This indicates that the conservativeness of our method is very small. In , by sacrificing the nominal the other reliable controller performance slightly (6.4% from optimal value of 16.7024 to 17.7757), the performance indexes for the eight fault cases can

1 = 0:01

all be improved significantly (ranging from 29.1% to 55.8%), except for cases of and where about the same level of performance as that of the standard controller is achieved. Also, where the same Lyapunov compared to the controller function is used for both the nominal and the fault cases, the is improved by nominal closed-loop performance using 30% (from 25.4500 to 17.7757) while maintaining about the same level of performance when any of the eight outage faults occurs. In the flight control system shown in Fig. 1, the following actuator dynamics are assumed [10]. 1) The left and right elevator actuators are modeled as 0.0495-s first-order lags with rate limits of 60 /s and the surface deflection limits of 25 . 2) The left and right aileron actuators are modeled as 0.0495-s first-order lags with rate limits of 80 /s and the surface deflection limits of 21.5 . 3) The rudder actuator is modeled as a first-order lag of 0.0495 s with a rate limit of 120 /s and the rudder deflection limit of 30 . Simulation results using the standard controller are given in Fig. 2, and the results using the are given in Fig. 3. Results for both reliable controller without and with fault are given. Each curve corresponds to one of the eight outage faults. It can be seen from the simulation results that, while the standard and the reliable controllers give about the same response in the normal case, the reliable controllers perform much better when any of the eight outage faults occurs. In the above simulation studies, a vertical gust disturbance of 5 m/s is considered. B. Example 2 (Control Surface Impairment) In this example, we consider control surface impairment faults. Due to the limitation of the aerodynamic data of the aircraft model, we consider the case where the horizontal stabilators are symmetrically impaired up to 75%. Since the aircraft model may not change linearly between normal operation and the operation with 75% loss of the horizontal stabilators,


83

Fig. 2. Response curves of the nominal and fault 1–8 cases with Standard tracker

we consider two additional design points of 25% and 50% symmetric losses of the horizontal stabilators. The trimmed values of the F-16 aircraft equations in normal operation are same as (33). The retrimmed values and the system matrices of the corresponding linear equation with the horizontal stabilator impairment are given in the Appendix. The system model with control surface impairment can be written as (35) where , disturbance and

,

, represents the vertical gust

(36) , and , , , and ( ) being as with given in the Appendix. By using the iterative algorithm, a reliable tracking controller is designed to tolerate the above surface impairment faults in the

K.

horizontal stabilator. The design bounds adopted are: 1) , and 2) , . The design results are given in Table II. The corresponding controller parameters are given in the Appendix. For comparison purpose, two other designs are also carried out. The first one is a standard controller with the design bound (without considering any fault), and the second one is a reliable controller using common LMI solutions (i.e., based on steps 1 and 2 in Algorithm 1 before the iterations in Steps 3–5). The design results are also listed in Table II. It can be seen from Table II that, in the nominal case, an optimal performance of 14.1553 is achieved by the standard . However, the performance deteriorates greatly to tracker 264.5050 when there is a 75% loss of the horizontal stabilator, even though stability can still be maintained by the standard . tracker improved On the other hand, our reliable controller significantly the performance for the 75% loss fault. Similar to the outage fault case in Example 5.1, this improvement is achieved by sacrificing slightly the nominal case performance. Also, compared to the common LMI solution results, both the designed bound and the achieved performance can be improved by using the iteration procedure in Algorithm 1. Simulation studies are also carried out to verify the effectiveness of the designed controller. First, simulations results using linear aircraft model are given in Figs. 4 and 5. The tracking

84


Fig. 3.

Response curves of the nominal and fault 1–8 cases with Reliable tracker

TABLE II PERFORMANCE COMPARISON THE HORIZONTAL STABILATOR IMPAIRMENT WITH :

1 = 0 01

K

.

loss at 20 s. It can be seen that the standard controller can not stabilize the closed-loop system when the horizontal stabilator loss is 75%, while our reliable controller can. The control surface deflections are also given in Figs. 6 and 7. Summarizing the two examples, it is noted that the reliable tracker design method can significantly improve the system performance in the event of fault cases compared to the standard design method, under minimal sacrifice of nominal performance. It is easy to see that we can obtain various reliable trackers depending on the upper bound requirements of fault case performance indexes in the design of the reliable robust tracker. This provides the designer with the flexibility of trading off between the nominal performance and the performance when faults occur. VI. CONCLUSION

command for , , and are all set as one. It can be seen that the response in Fig. 4 using the standard controller becomes very oscillatory and is almost unstable when a 75% loss of the horizontal stabilator occurs. But for our reliable controller, response remains well damped. To verify the robustness of our reliable tracking controller, simulation results using the original nonlinear aircraft model and data are carried out, and the results are given in Figs. 6 and 7. In these simulations, a 25% loss of the horizontal stabilator is introduced at 12 s, followed by a 50% loss at 16 s and a 75%

In this paper, we have investigated a reliable robust tracking controller design method for an airplane in the presence of actuator faults and control surface impairment. Based on the multiobjective robust performance analysis of the system in the nominal case and the faulty cases using an LMI method, a reliable robust tracking controller with state feedback plus tracking errors integral has been developed. This method is applied to an F-16 aircraft model. Both linear and nonlinear six degree-of-freedom simulations are performed, and the effectiveness of the proposed method is demonstrated.


Fig. 4. Response curves of the nominal and horizontal stabilator impairment cases with the standard tracker.

Fig. 5. Response curves of the nominal and horizontal stabilator impairment cases with the reliable tracker.

Fig. 6. Simulation result for the standard tracker using nonlinear aircraft model.

85

86


Fig. 7. Simulation result for the reliable tracker using nonlinear aircraft model.

APPENDIX System Matrices in Example 1

System Matrices in Example 2 1) The system matrices of the aircraft without the horizontal stabilator impairment fault

2) The trimmed values of the aircraft with 25% loss of the horizontal stabilator are

m m/s


87

4) The trimmed values of the aircraft with 75% loss of the horizontal stabilator are

m m/s

3) The trimmed values of the aircraft with 50% loss of the horizontal stabilator are: m m/s

Controller Parameters in Example 1

88


Controller Parameters in Example 2

REFERENCES [1] A. K. Caglayan, S. M. Allen, and K. Wehmuller, “Evaluation of a second generation reconfiguration strategy for aircraft flight control systems subjected to actuator failure/surface damage,” in Proc. IEEE 1988 Nat. Aerospace Electron. Conf. (NAECON 1988), vol. 2, 1988, pp. 520–529. [2] P. Gahinet, P. Apkarian, and M. Chilali, “Affine parameter-dependent Lyapunov functions and real parametric uncertainty,” IEEE Trans. Automat. Contr., vol. 41, pp. 436–442, Mar. 1996. [3] P. Gahinet, A. Nemirovskii, A. J. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: The MathWorks, Inc., 1994. [4] I. M. Horowitz, Synthesis Feedback Systems. New York: Academic, 1963. [5] H. Kwakernaak, “A polynomial approach to H optimization of control systems,” in NATO Workshop on Modeling, Robustness and Sensitivity Reduction in Control Systems, Groningen, Dec. 1986. [6] J. Lam and Y. Y. Cao, “Simultaneous linear-quadratic optimal control design via static output feedback,” Int. J. Robust Nonlinear Contr., vol. 9, pp. 551–558, 1999. [7] J. W. Lee, J. H. Lee, and W. H. Kwon, “Robust tracking control for uncertain linear systems without matching conditions,” in Proc. 33rd IEEE Conf. Decision Contr., Lake Buena Vista, FL, 1994, pp. 4181–4186. [8] F. Liao, J. L. Wang, and G.H. Yang, “LMI-based reliable robust preview tracking control against actuator faults,” in Proc. Amer. Contr. Conf., Arlington, VA, 2001, pp. 1047–1052. [9] T. Mori and H. Kokame, “A parameter-dependent Lyapunov function for a polytope of matrices,” IEEE Trans. Automat. Contr., vol. 45, pp. 1516–1519, Aug. 2000. [10] L. T. Nguyen and M. E. Ogburn et al., “Simulator study of stall/poststall characteristics of a fighter airplane with relaxed longitudinal static stability,” NASA Tech. Paper 1538, 1979. [11] T. Pancake, M. Corless, and M. Brockman, “Analysis and control of polytopic uncertain/nonlinear systems in the presence of bounded disturbance inputs,” in Proc. Amer. Contr. Conf., Chicago, IL, June 2000, pp. 159–163. [12] U. Shaked, “Improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty,” IEEE Trans. Automat. Contr., vol. 46, pp. 652–656, Apr. 2001.

[13] U. Shaked and C. E. de Souza, “Continuous-time tracking problems in an H setting: A game theory approach,” IEEE Trans. Automat. Contr., vol. 40, pp. 841–852, May 1995. [14] K. Takaba, “Robust preview tracking control for polytopic uncertain systems,” in Proc. 37th IEEE Conf. Decision Contr., Tampa, FL, 1998, pp. 1765–1770. [15] R. J. Veillette, “Reliable linear-quadratic state-feedback control,” Automatica, vol. 31, no. 1, pp. 137–143, 1995. [16] R. J. Veillette, J. V. Medanic´, and W. R. Perkins, “Design of reliable control systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 290–304, Mar. 1992. [17] Y. Wang, S. S. Hu, and Y. Xiong, “Some basic problems about self-repairing flight control,” in Proc. 3rd World Congr. Intell. Contr. Automat., Hefei, P.R. China, June 28–July 2 2000. [18] S. F. Wu, M. J. Grimble, and W. Wei, “QFT based robust/fault tolerant flight control design for a remote pilotless vehicle,” in Proc. 1999 IEEE Int. Conf. Contr. Applicat., Kohala Coast Island, HI, Aug. 22–27, 1999. [19] I. Yaesh and U. Shaked, “Two degree of freedom H optimization of multivariable feedback systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1272–1276, 1991. [20] G. H. Yang, J. L. Wang, and K.-Y. Lou, “Reliable state feedback control synthesis for uncertain linear systems,” in Proc. 6th Int. Conf. Contr., Automat., Robot. Vision (ICARCV’2000), Singapore, Dec. 6–8, 2000. Session WA2.3. [21] G. H. Yang, J. L. Wang, and Y. C. Soh, “Reliable H controller design for linear systems,” Automatica, vol. 37, no. 5, pp. 717–725, 2001. [22] G. H. Yang, J. S. Yee, and J. L. Wang, “Reliable mixed H =H statefeedback controller design for discrete-time linear systems,” in Proc. IASTED Conf. Contr. Applicat., Cancun, Mexico, 2000, pp. 177–181. [23] J. S. Yee, G. H. Yang, and J. L. Wang, “Reliable output-feedback controller design for discrete-time linear systems: An iterative LMI approach,” in Proc. Amer. Contr. Conf., Arlington, VA, 2001, pp. 1035–1040. [24] P. Zhang et al., “On modeling of aircraft with independent control surface movements,” Research Report, Beijing University of Aeronautics and Astronautics, 1998. [25] W. Q. Zhang, “Control of a high-performance fighter aircraft by using H theory and neural-fuzzy concepts,” Master’s thesis, Nanyang Technol. Univ., Singapore, 1997. [26] Q. Zhao and J. Jiang, “Reliable control system design against feedback failures with applications to aircraft control,” in Proc. 1996 IEEE Int. Conf. on Control Applications, Dearborn, MI, Sept. 1996, pp. 994–999. [27] , “Reliable tracking control system design against actuator failures,” in Proc. 1997 SICE, Tokushima, July 1997, pp. 1019–1024.

Fang Liao received the B.E. degree in automatic control and navigation in 1992 and the M.E. degree in flight control, guidance, and simulation in 1995, both from Beijing University of Aeronautics and Astronautics (BUAA). Currently, she is pursuing the Ph.D. degree in the School of Electrical and Electronic Engineering at Nanyang Technological University. From 1995 to 1999, she was an Engineer with the Research Institute of Unmanned Air Vehicles at BUAA. Her research interest includes robust and reliable control and tracking as well as genetic algorithms and their applications to aircraft flight control system design.

Jian Liang Wang (M’91–SM’98) received the B.E. degree in electrical engineering from Beijing Institute of Technology, China, in 1982. He received the M.S.E. and Ph.D. degrees in electrical engineering from The Johns Hopkins University, Baltimore, MD, in 1985 and 1988, respectively. From 1988 to 1990, he was a Lecturer with the Department of Automatic Control at Beijing University of Aeronautics and Astronautics, China. Since 1990, he has been with the School of Electrical and Electronic Engineering at Nanyang Technological University, Singapore, where he is currently an Associate Professor. His current research interest includes robust and reliable control and filtering, nonlinear control, two-time-scale systems, and their application to flight control system design. Dr. Wang was the Technical Program Chairman for International Conference on Control, Automation, Robotics and Vision, Singapore, December 2000.


Guang-Hong Yang was born in Jilin, China, on September 2, 1963. He received the B.S. and M.S. degrees in mathematics from Northeast University of Technology, China in 1983 and 1986, respectively, and the Ph.D. degree in control engineering from Northeastern University (formerly Northeast University of Technology) in 1994. From 1986 to 1995, he was with the Northeastern University. From 1996 to July 2001, he was a Postdoctoral/Research Fellow with the School of Electric and Electronic Engineering, Nanyang Technological University, Singapore. From August 2001, he joint the Tamasek Laboratories at National University of Singapore as a Research Scientist. His research interests include decentralized control, symmetric systems, fault-tolerant control, nonlinear control, and nonfragile controller design with applications to aircraft flight control.

89