Fault Tolerant Control for Uncertain Time-Delay Systems Based on Sliding Mode Control

Kybernetika

Jun Sheng Wu; Zhengxin Weng; Zuo Hua Tian; Song Jiao Shi Fault tolerant control for uncertain time-delay systems based on sliding mode control

Kybernetika, Vol. 44 (2008), No. 5, 617--632

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This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz KYBERNETIKA—VOLUME 4 4 (2008), NUMBER 5, PAGES 617–632

FAULT TOLERANT CONTROL FOR UNCERTAIN TIME–DELAY SYSTEMS BASED ON SLIDING MODE CONTROL

Junsheng Wu, Zhengxin Weng, Zuohua Tian and Songjiao Shi

Fault tolerant control for uncertain systems with time varying state-delay is studied in this paper. Based on sliding mode controller design, a fault tolerant control method is proposed. By means of the feasibility of some linear matrix inequalities (LMIs), delay dependent sufficient condition is derived for the existence of a linear sliding surface which guarantees quadratic stability of the reduced-order equivalent system restricted to the sliding surface. A reaching motion controller, which can be seen as a fault tolerant controller, can retain the stability of the closed loop system in the present of uncertainties, disturbances and actuator fault is designed. A numerical simulation shows the effectiveness of the approach. Keywords: fault tolerant control, sliding mode control, time varying state-delay, uncer- tainty, linear matrix inequality (LMI) AMS Subject Classification: 34D20, 93D09, 15A39, 37L45

1. INTRODUCTION

To increase overall system reliability and safety in safety-critical systems, Fault- Tolerant Control (FTC) of the systems become extremely important. Therefore, FTC has been an important topic of research. Generally speaking, FTC systems can be categorized into two main classes: passive and active. Passive FTC systems are designed with the consideration of a set of presumed failure modes without the need to detect their presence. No alternation or adaptation is made to the control law so these controllers may be thought of as a specific class of robust controllers. The resulting control system performance tends to be conservative. It also has the limitation to deal with unanticipated faults. But the design of the controller is easy. In contrast, active FTC systems react to the occurrence of system faults on-line in an attempt to maintain the overall system stability and performance. They can deal with unanticipated faults with Fault Diagnosis (FD) and Controller Reconfiguration (CR); and therefore can achieve better performance than the passive methods. But the active FTC depends on the fast and accurate information provided by FD. In 618 J. WU, Z. WENG, Z. TIAN AND S. SHI other words, the active FTC system would be useless without an effective FD scheme. What’s more, the overall system becomes more complicated and costly. This paper is concerned with the use of sliding mode idea to design FTC system for uncertain time-delay systems. Recent work has explored how sliding mode ideas can be used in FTC system design. The methods can be also divided into two cate- gories: a) First, a sliding mode observer is used for fault estimation and then based on which a CR is adopted; b) Design a sliding mode controller directly to retain the stability and some performances of the closed-loop system. That’s because a sliding mode controller which can drive the system onto the sliding surface in the present of fault input can be looked on as a fault tolerant controller. The former ones have been widely used, see [2, 7, 15, 16, 18, 19] and the references therein for some new results of this field. The latter ones have been studied, too. But unfor- tunately, few works of them concerned with time-delay systems. As we all known, time delays are commonly encountered in practical applications and might lead to poor system performance or even instability. Therefore, over the past decades, the analysis and synthesis of time-delay systems have been one of the most active research areas in system sciences. The recent developments on sliding mode control involving time-delay systems can be found in [3, 4, 5, 8, 11, 12, 17]. [11] considers sliding mode control for nonlinear state-delay systems based on neural network. It’s an approximate calculation method. And the theoretical analysis of the influence of the approximation error to the control performance should be presented. The other methods deal with state-delay systems and input-delay systems. Most of them re- quire the states of the time-delay system are available, and in these SMC schemes, the control laws usually utilize full-state feedback. But in practice, this is often not possible. To overcome this, [12] adopt state observer to obtain the unknown states, and then synthesize a sliding mode control law based on state estimates. To the best of our knowledge, the results of them are delay independent and thus are conservative. In this paper, the problem of designing fault tolerant control based on sliding mode control for a class of uncertain systems with time-varying state-delay has been considered. The time-delay is not a constant but defined with some restrictions. So the result is less conservative than the results mentioned before and is more close to real life. Both parametric uncertainties and disturbance are considered. In terms of LMI, delay dependent sufficient condition is derived for the existence of a linear sliding surface which guarantees quadratic stability of the reduced-order equivalent system restricted to the sliding surface. And a reaching motion controller is proposed. The sliding motion and the reaching motion are robust against the mismatched uncertainties matched disturbance and actuator fault. The paper is organized as follows. Section 2 gives the problem formulation. The main results are given in Section 3 which is composed of 2 parts. Subsection 3.1 shows how to design the sliding surface and 3.2 gives the design method of the reaching motion control law. The effectiveness of the approach proposed will be demonstrated via an example in Section 4, which will be followed by some concluding remarks in Section 5. Fault Tolerant Control for Uncertain Time-Delay Systems . . . 619

2. PROBLEM FORMULATION

Consider the time-delay system with the form

x˙ (t) = (A + ∆A)(t) + (Ad + ∆Ad(t))x(t τ(t)) + Bu(t) + Hw(t) − (1) x(t) = φ(t), t [ τ,¯ 0) ∈ − where x(t) Rn, u(t) Rm are the state vector and control input respectively. l ∈ ∈ w(t) R is the disturbance. A, Ad, B and H are real constant matrices with appropriate∈ dimensions and rank(B) = m; τ(t) is the variable time-delay function which satisﬁes 0 τ(t) τ,¯ τ˙(t) µ, t 0. ∆A(t), ∆Ad(t) are real-unknown but norm-bounded≤ matrix≤ functions| | ≤ representing∀ ≥ time-varying uncertainties. The admissible uncertainties are deﬁned as:

[∆A ∆Ad] = DF(t)[E a E d] (2) where D, E a, E d are real-known constant matrices and F(t) is a real-unknown time-varying matrix with Lebesgue measurable elements satisfying F T(t)F(t) I . ≤ Assumption 1. Hw(t) satisﬁes the matching condition Hw(t) range B, thus ∈ there exists d(t) Rm such that Hw(t) = Bd(t). And each component of d(t) is bounded by the known∈ d¯ (t), i. e. d (t) d¯ (t). i i ≤ i

Assumption 2. The actuator faults can be denoted by B f = BL, where

1 `1 0 0 − ···. 0 1 ` .. 0 L =  2  . −. . .  ......     0 0 1 `   ··· − m    and ì, i = 1, . . . , m denote the control effectiveness factors. ì = 0, i = 1, . . . , m denote the healthy ith actuator and ì = 1 corresponds to total failure of the ith actuator. By the assumption rank(B) = m, one can get the singular value decomposition (SVD) of B: T m m B =[U 1 U 2] V , where R × is a diagonal positive-definite 0(n m) m ∈ » −P × – matrix and V , [U 1 U 2] are unitary matrices.P T U 1 B 1 Define T = T then we can have TB = . By the state transfor- U 2 0(n m) m » – » − × – mation z = Tx,(1) has the form (with Assumptions 1 and 2):

B 1 z˙ (t) = (A¯ +∆A¯ (t))z (t)+(A¯ d +∆A¯ d(t))z (t τ(t))+ L (u(t)+d(t)) − 0(n m) m · − × ¸ z (t) = φ¯(t), t [ τ,¯ 0) ∈ − (3) 620 J. WU, Z. WENG, Z. TIAN AND S. SHI

1 1 1 1 where A¯ = TAT − , A¯ d = TAdT − , ∆A¯ = T∆AT − , ∆A¯ d = T∆AdT − , and φ¯(t) = Tφ(t). Furthermore, similarly to [17] one can get z˙ (t) = (A¯ + ∆A¯ (t))z (t) + (A¯ + ∆A¯ (t))z (t τ(t)) 1 11 11 1 d11 d11 1 − +(A¯ + ∆A¯ (t))z (t) + (A¯ + ∆A¯ (t))z (t τ(t)) 12 12 2 d12 d12 2 − +B 1L (u(t) + d(t)) (4) z˙ (t) = (A¯ + ∆A¯ (t))z (t) + (A¯ + ∆A¯ (t))z (t τ(t)) 2 21 21 1 d21 d21 1 − +(A¯ + ∆A¯ (t))z (t) + (A¯ + ∆A¯ (t))z (t τ(t)) 22 22 2 d22 d22 2 − z (t) = φ¯ (t), t [ τ,¯ 0), z (t) = φ¯ (t), t [ τ,¯ 0) 1 1 ∈ − 2 2 ∈ − where m n m T ¯ T z 1(t) R , z 2(t) R − , B 1 = V , A11 = U AU 1, ∈ ∈ 1 ¯ T ¯ T ¯ T A12 = U 1 AU 2, A21 = U 2 AU 1, PA22 = U 2 AU 2 and the other parameter matrices can be got in the same way. Then one can see that the second equation of (4) represents the sliding motion dynamics of (3), and the sliding mode surface can be chosen as follows:

S = IC z = z 1 + Cz 2 = 0 (5)

n m where C R − . Substituting£ (5) into¤ (4) shows the sliding motion: ∈ z˙ (t) = (A¯ + ∆A¯ (t) A¯ C ∆A¯ (t)C )z (t) 2 22 22 − 21 − 21 2 + (A¯ + ∆A¯ (t) A¯ C ∆A¯ (t)C )z (t τ(t)) (6) d22 d22 − d21 − d21 2 − z (t) = φ¯ (t), t [ τ,¯ 0). 2 2 ∈ − The objective of this paper is to design constant gain C and control law u(t) such that: 1. the sliding motion (6) is quadratically stable; 2. system (4) is asymptotically stable with the reaching control law u(t) in the present of actuator fault and disturbance d(t).

3. MAIN RESULTS

3.1. Sliding surface design Let us consider the nominal time-delay system that can be obtained from (6) by setting ∆A 0, ∆A (t) 0. ≡ d ≡ z˙ 2(t) = (A¯ 22 A¯ 21C )z 2(t) + (A¯ d22 A¯ d21C )z 2(t τ(t)) − − − (7) z (t) = φ¯ (t), t [ τ,¯ 0). 2 2 ∈ − Fault Tolerant Control for Uncertain Time-Delay Systems . . . 621

Using descriptor type model transformation, one can represent (7) in descriptor form as:

z˙ 2(t) = y(t) (8) y(t) = (A¯ A¯ C )z (t) + (A¯ A¯ C )z (t τ(t)) 22 − 21 2 d22 − d21 2 − where y(t) is the ‘fast varying’ state variable.

Theorem 1. Consider the linear time-delay sliding motion (7). Then given the scalarsτ ¯ > 0 and µ, this sliding motion is quadratically stable for any time-delay satisfying 0 τ(t) τ,¯ τ˙(t) µ, t 0 if there exist matrix C , symmetric ≤ ≤ | | ≤ ∀ ≥ positive deﬁnite (s.p.d.) matrices P 1, P 22, Q11, Q22, R, S, T and arbitrary 1 matrices P 12, Q12, P i (i = 2,..., 11) with appropriate dimensions satisfying

P P Q Q 1 12 0, 11 12 0 P 22 ≥ Q ≥ · ∗ ¸ · ∗ 22 ¸ and Π11 Π12 Π13 Π14 Π15 0 Π22 Π23 Π24 Π25 0  ∗ Π Π Π 0  Π = ∗ ∗ 33 34 35 < 0 (9)  Π Π Π   ∗ ∗ ∗ 44 45 46   Π 0   ∗ ∗ ∗ ∗ 55   Π   ∗ ∗ ∗ ∗ ∗ 66    where T ¯ ¯ ¯ T T ¯ T T 1 T 2 Π11 = P 2 (A22 A21C )+(A22 C A21)P 2+P 7+P 7 +µP 12S − P 12+R+¯τ Q11, − T − T Π = P P T + A¯ C TA¯ P + P +τ ¯2Q , 12 1 − 2 22 − 21 3 8 12 T ³ T ´ T T T Π = P (A¯ A¯ C ) + A¯ C A¯ P P + P , 13 2 d22 − 21 22 − 21 4 − 7 9 T T T ³ ´ T T T T Π = A¯ C A¯ P +P , Π = A¯ C A¯ P +P +P P , 14 22 − 21 5 10 15 22 − 21 6 11 12− 7 ³2 ´ T T ¯³ ¯ ´ T Π22 =τ ¯ Q22 P 3 P 3 , Π23 = P 3 (Ad22 Ad21C ) P 8 P 4, − − T − − − Π24 = P 12 P 5, Π25 = P 6 P 8 , − −T − T Π = P T(A¯ A¯ C )+ A¯ C TA¯ P P P T+µ(S+T)+(µ 1)R, 33 4 d22− d21 d22 − d21 4− 9− 9 − T T T ³ ´ T T T T Π = A¯ C A¯ P P , Π = A¯ C A¯ P P P , 34 d22 − d21 5 − 10 35 d22 − d21 6 − 11 − 9 Π = ³ Q , Π =´P P T Q , ³ Π = µP , ´ 44 − 11 45 22 − 10 − 12 46 22 Π = P P T Q , Π = µT. 55 − 11 − 11 − 22 66 −

And the sliding surface can be chosen as S = IC z = z 1 + Cz 2 = 0. £ ¤ 1 denotes the symmetricity of the matrix. ∗ 622 J. WU, Z. WENG, Z. TIAN AND S. SHI

P r o o f . Choosing a candidate Lyapunov–Krasovskii functional as in [13]: T T z 2(t) 0 P 12 z 2(t) V (z (t), t) = Z (t)EPZ (t)+ t t 2 z (s) ds z (s) ds " t τ(t) 2 # P 22 " t τ(t) 2 # − · ∗ ¸ − T 0 t z (s) R Q Q z (s) R +τ ¯ 2 11 12 2 ds dθ τ¯ t+θ y(s) Q22 y(s) Z− Z · ¸ · ∗ ¸ · ¸ t T + z 2 (s)Rz 2(s) ds t τ(t) (10) Z − where T T T Z (t) = T T t t , z 2 (t) y(t) z 2 (t τ(t)) t τ(t) z (s) ds t τ(t) y(s) ds − − − · ³ ´ ³ ´ ¸ I 0 0 R R 0 0 0 P 1 0 0 0 0 E =  0 0 0  , P = P P P P P .  2 3 4 5 6   0 0 0  P P P P P   7 8 9 10 11  0 0 0      Then the time derivative of (9) along the trajectory (8) is T d T z 2(t) 0 P 12 V˙ (z (t), t) = 2 Z (t) EPZ (t) + 2 t 2 dt z (s) ds " t τ(t) 2 # P 22 ³ ´ − · ∗ ¸ y(t) R t y(s) ds +τ ˙(t)z (t τ(t)) × " t τ(t) 2 # − − T (11) R z (t) Q Q z (t) +τ ¯2 2 11 12 2 y(t) Q y(t) · ¸ · ∗ 22 ¸ · ¸ T t z (s) Q Q z (s) τ¯ 2 11 12 2 ds + z T(t)Rz (t) − t τ¯ y(s) Q y(s) 2 2 − · ¸ · ∗ 22 ¸ · ¸ [1R τ˙(t)] z T(t τ(t))Rz (t τ(t)). − − 2 − 2 − t Using (8) and Newton–Leibniz formula t τ(t) y(s) ds = z 2(t) z 2(t τ(t)), one can see: − − − R d T 2 dt Z EPZ (t) ³ ´ = 2 z˙ 2(t) y˙ (t) z˙ 2(t τ(t)) z 2(t) z 2(t τ(t)) y(t) y(t τ(t)) EPZ (t) − y(t) − − − − £ ¤ = 2 z˙ 2(t) 0 0 PZ (t) = 0  0  £ ¤   y(t) T T ¯ ¯ ¯ ¯ = 2Z (t)P y(t) + (A22 A21C)z 2(t) + (Ad22 Ad21C )z 2(t τ) − − − −  z (t) z (t τ(t)) t y(s) ds  2 − 2 − − t τ(t) T T  −  = Z (t) Ω0 + Ω0 Z (t) R (12) ¡ ¢ Fault Tolerant Control for Uncertain Time-Delay Systems . . . 623

0 I 0 0 0 T where Ω = P A¯ 22 A¯ C I A¯ d22 A¯ C 0 0 . 0 2 − 21 − − d21 3 I 0 I 0 I − − Moreover, 4 5

T z 2(t) 0 P 12 y(t) 2 t t z (s) ds y(s) ds +τ ˙(t)z (t τ(t)) " t τ(t) 2 # P 22 " t τ(t) 2 # − · ∗ ¸ − − R T R t z 2(t) P 12 t τ(t) y(s) ds +τ ˙(t)z 2(t τ(t)) − − = 2 t t z 2(s) ds  T ³ ´  " t τ(t) # P 12y(t) + PR 22 t τ(t) y(s) ds +τ ˙(t)z 2(t τ(t)) − − − R  ³ ´  T t R = 2z 2 (t)P 12 t τ(t) y(s) ds +τ ˙(t)z 2(t τ(t)) − T − t ³R T t ´ +2 t τ(t) z 2(s) ds P 12y(t) + P 22 t τ(t) y(s) ds +τ ˙(t)z 2(t τ(t)) − − − T³ ´ ³ ³ ´´ Z (Rt)Ω1Z (t) R ≤ (13) 1 T µP 12S − P 12 0 0 0 P 12 0 0 P 12 0 2 ∗ 3 where Ω1 = µ(S + T) 0 0 , S, T are some ∗ ∗ 1 6 µP 22T − P 22 P 22 7 6 ∗ ∗ ∗ 7 6 0 7 6 ∗ ∗ ∗ ∗ 7 s.p.d. matrices4 and denotes the symmetricity of the matrix. 5 ∗ The remainder of (10) can be dealt with as:

T z (t) Q Q z (t) τ¯2 2 11 12 2 y(t) Q y(t) · ¸ · ∗ 22 ¸ · ¸ T t z (s) Q Q z (s) τ¯ 2 11 12 2 ds − t τ¯ y(s) Q y(s) − · ¸ · ∗ 22 ¸ · ¸ + z TR(t)Rz (t) [1 τ˙(t)] z T(t τ(t))Rz (t τ(t)) 2 2 − − 2 − 2 − T z (t) Q Q z (t) τ¯2 2 11 12 2 ≤ y(t) Q y(t) · ¸ · ∗ 22 ¸ · ¸ T t z (s) Q Q z (s) τ(t) 2 11 12 2 ds (14) − t τ(t) y(s) Q y(s) − · ¸ · ∗ 22 ¸ · ¸ + z T(tR)Rz (t) z T(t τ(t)) (1 µ) Rz (t τ(t)) 2 2 − 2 − − 2 − T z (t) Q Q z (t) τ¯2 2 11 12 2 ≤ y(t) Q y(t) · ¸ · ∗ 22 ¸ · ¸ t T t t τ(t) z 2(s) ds Q11 Q12 t τ(t) z 2(s) ds − − − t Q t " R t τ(t) y(s) ds # 22 " R t τ(t) y(s) ds # − · ∗ ¸ − + z T(t)RzR (t) z T(t τ(t)) (1 µ) Rz (t τR(t)) = Z T(t)Ω Z (t) 2 2 − 2 − − 2 − 2 624 J. WU, Z. WENG, Z. TIAN AND S. SHI where 2 2 R +τ ¯ Q11 τ¯ Q12 0 0 0 τ¯2Q 0 0 0 ∗ 22 Ω =  (µ 1)R 0 0  . 2 ∗ ∗ −  Q Q   ∗ ∗ ∗ − 11 − 12   Q   ∗ ∗ ∗ ∗ − 22    From (12) – (14), one can have:

V˙ (z (t), t) Z T(t)ΩZ (t) where Ω = Ω + ΩT + Ω + Ω . 2 ≤ 0 0 1 2

According to Lyapunov–Krasovskii theorem, system (7) is asymptotically stable when V˙ (z 2(t), t) < 0. In order to guarantee V˙ (z 2(t), t) < 0, one needs Ω < 0. It also implies that there exists a suﬃciently small ξ > 0 such that:

ξI 0 0 0 0 0 0 0 0 0 Ω +  0 0 0 0 0  < 0. (15)  0 0 0 0 0     0 0 0 0 0      2 From the inequality above, one can get V˙ (z 2(t), t) < ξ z 2 . Thus the sliding motion (7) is quadratically stable (cf. [10]) with the matricesk deﬁnedk in Theorem 1. 1 Note that in the term Ω44 = µP 22T − P 22 Q11, P 22 and T are both unknown variables and in a multiplicative form, so the− inequality Ω < 0 is a bilinear one which maybe not dealt with easily. In fact, by applying Shur’s complement (cf. [6]) to Ω < 0, the resulting inequality (9) is a linear matrix inequality (LMI) that can be solved eﬀectively by using Matlab LMI toolbox. This completes the proof. ¤

The uncertain sliding motion (6) is considered in the next part of this subsection. The following theorem provides the suﬃcient conditions for the robust stability of sliding motion (6).

Theorem 2. Consider the uncertain time-delay sliding motion (6). Then given the scalarsτ ¯ > 0 and µ, this sliding motion is quadratically stable for any time- delay satisfying 0 τ(t) τ,¯ τ˙(t) µ, t 0 if there exist matrix C , positive ≤ ≤ | | ≤ ∀ ≥ scalars ε, γ, symmetric positive deﬁnite (s.p.d.) matrices P 1, P 22, Q11, Q22, R, S, T and arbitrary matrices P 12, Q12, P i (i = 2,..., 11) with appropriate dimensions satisfying P P Q Q 1 12 0, 11 12 0 P 22 ≥ Q ≥ · ∗ ¸ · ∗ 22 ¸ Fault Tolerant Control for Uncertain Time-Delay Systems . . . 625 and

Π11 + εE aU 2E aU 2 Π12 Π13 + εE aU 2E dU 2 Π14 Π15 Π22 Π23 Π24 Π25  ∗ Π + εE U E U Π Π ∗ ∗ 33 d 2 d 2 34 35  Π Π  ∗ ∗ ∗ 44 45 Π =  Π u  ∗ ∗ ∗ ∗ 55   ∗ ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗ ∗  (16) 0 P TU TDC TU TE T P TU TD 2 2 1 a − 2 2 0 P TU TD 0 P TU TD 3 2 − 3 2  0 P TU TDC TU TE T P TU TD 4 2 1 d − 4 2 Π P TU TD 0 P TU TD  46 5 2 5 2  T T − T T  < 0 0 P 6 U 2 D 0 P 6 U 2 D  −  Π66 0 0 0   εI 0 0  ∗ − 1  γ− I 0  ∗ ∗ −  γI  ∗ ∗ ∗ −  where Π , i, j = 1,..., 6 are deﬁned as in (9). ij { } And the sliding surface is S = IC z = z 1 + Cz 2 = 0.

P r o o f . The proof follows from£ the proof¤ of Theorem 1. Consider LMI (9) and replace A¯ 22, A¯ 21, A¯ d22, A¯ d21 with A¯ 22 + ∆A¯ 22(t), A¯ 21 + ∆A¯ 21(t), A¯ d22 + ∆A¯ d22(t), A¯ d21 + ∆A¯ d21(t), respectively. One can have:

T T Πn + Πd22 + Πd22 + Πd21 + Πd21 < 0 (17) where Πn = Π in (9) and T ¯ T ¯ P 2 ∆A22 0 P 2 ∆Ad22 0 0 0 T T P ∆A¯ 22 0 P ∆A¯ d22 0 0 0  3 3  P T∆A¯ 0 P T∆A¯ 0 0 0 Π = 4 22 4 d22 d22  P T∆A¯ 0 P T∆A¯ 0 0 0   5 22 5 d22   T ¯ T ¯   P 6 ∆A22 0 P 6 ∆Ad22 0 0 0     0 0 0 0 0 0    T ¯ T ¯ P 2 ∆A21C 0 P 2 ∆Ad21C 0 0 0 − T ¯ − T ¯ P 3 ∆A21C 0 P 3 ∆Ad21C 0 0 0  − T ¯ − T ¯  P 4 ∆A21C 0 P 4 ∆Ad21C 0 0 0 Πd21 = − − .  P T∆A¯ C 0 P T∆A¯ C 0 0 0   5 21 5 d21   − T ¯ − T ¯   P 6 ∆A21C 0 P 6 ∆Ad21C 0 0 0   − −   0 0 0 0 0 0    626 J. WU, Z. WENG, Z. TIAN AND S. SHI

T ¯ By substituting U 2 DF(t)E aU 2 for ∆A22, one can decompose Πd22 as Πd22 = H 2F(t)E 2, where T T P 2 U 2 D P TU TD  3 2  P TU TD H = 4 2 and E = E U 0 E U 0 0 0 . 2  P TU TD  2 a 2 d 2  5 2   T T  £ ¤  P 6 U 2 D     0  For further analysis, the following lemma is needed.

Lemma 1. (Petersen [14]) Let E, F and H be real matrices of appropriate dimensions, with F TF I , then we have that for any scalar ε > 0, ≤ T T T T 1 T EF(t)H + H F (t)E εE E + ε− HH . ≤

According to Lemma 1, with a given positive scalar ε, one can have T T 1 T Π + Π εE E + ε− H H (18) d22 d22 ≤ 2 2 2 2 In the same way T T 1 T Π + Π γE E + γ− H H (19) d21 d21 ≤ 1 1 1 1 where γ is a given positive scalar and T T P 2 U 2 D P TU TD  3 2  P TU TD H = 4 2 , E = E U C 0 E U C 0 0 0 . 1 −  P TU TD  1 a 1 d 1  5 2   T T  £ ¤  P 6 U 2 D     0    Substituting (18) – (19) into (16) and applying Schur’s complement, the LMI given in (15) is obtained. Thus sliding motion (6) is robustly quadratically stable. This completes the proof. It is worth to be pointed out that the results presented in Theorem 1 and Theo- rem 2 are both delay dependent. Comparing with the results given in [3, 4, 8, 17], our results are less conservative. For example, in [17], the Lyapunov–Krasovskii functional been adopted can be seen as a special case of the one introduced in this paper.

3.2. Fault tolerant controller design As mentioned in the introduction, a sliding mode controller which can drive the system onto the sliding surface in the present of actuator fault together with the matched disturbances can be looked on as a fault tolerant controller. So based on Subsection 3.1, we should consider how to realize the design of the reaching motion controller. Fault Tolerant Control for Uncertain Time-Delay Systems . . . 627

Theorem 3. Suppose LMI (15) is feasible and the sliding motion is given by (5). Then the trajectory of the closed-loop system with fault input can be driven onto the sliding surface in limited time and keep it there for all subsequent time with the control:

u = (B L)† KS + α sign(S) + C¯ Az¯ (t) + C¯ A¯ z (t τ(t)) − 1 d − (20) + diag (sign(£ s1) sign(s2) sign(sm))(N 1 + N 2 + N 3)] where ··· T S = s s s , C¯ = IC , 1 2 ··· m T T sign(£S) = sign(s1) sign(¤ s2) £ sign(s¤m) , B = b1 b2 bm , ··· 1 ··· K = diag(k£ i), α = diag(αi), ki ¤ £ ¤ and αi are positive constants. 1 1 N = C¯ TDE T − z (t) , N = C¯ TDE T − z (t τ(t)) , N = b d¯ are the 1i i a 2i i d − i | i| ith row¯ of N 1, N 2 and N¯ 3, respectively.¯ ( )† denotes the¯ Pseudo-inverse of the argument¯ . ¯ ¯ × ¯ × P r o o f . From the sliding surface S = IC z (t), one can have S˙ = IC z˙ (t) £ ¤ (21) = C£¯ (A¯ + ∆¤A¯ (t))z (t) + C¯ (A¯ d + ∆A¯ d(t))z (t τ(t)) + B 1L(u(t) + d(t)). − Substituting (19) for u(t) in (20) S˙ = KS α sign(S) − − (N 1diag (sign(s1) sign(s2) sign(sm)) C¯ ∆A¯ (t)z (t)) − ··· − (22) (N diag (sign(s ) sign(s ) sign(s )) C¯ ∆A¯ (t)z (t τ(t))) − 2 1 2 ··· m − d − (N diag (sign(s ) sign(s ) sign(s )) B Ld(t)). − 3 1 2 ··· m − 1 Then each element of S˙ is s˙ = k s α sign(s ) (N sign(s ) C¯ ∆A¯ (t)z (t)) i − i i − i i − 1i i − i (N sign(s ) C¯ ∆A¯ (t)z (t τ(t))) (23) − 2i i − i d − (N sign(s ) b Ld(t)). − 3i i − i Note that 1 1 C¯ ∆A¯ (t)z (t) C¯ TDF(t)E T − z (t) C¯ TDE T − z (t) = N i ≤ i a ≤ i a 1i 1 1 C¯ ∆A¯ (t)z (t τ(t)) C¯¯ TDF(t)E T − z (t τ¯(t))¯ C¯ TDE T − z¯(t τ(t)) =N d − ≤ ¯ i d − ¯ ¯ ≤ i d ¯ − 2i ¯ ¯ ¯ biLd(t) bi L d bi d¯ =¯ N 3i. ¯ ¯ ≤ | | | | ≤ | | ¯ ¯ ¯ s˙i < 0, if si > 0 So one can deduce that , which implies that the trajectory ( s˙i > 0, if si < 0 of the closed-loop system with actuator fault can be driven onto the sliding surface in limited time and keep it there for all subsequent time with the control law (19). This completes the proof. ¤ 628 J. WU, Z. WENG, Z. TIAN AND S. SHI

Remark 1. In this paper, if L is ﬁxed a prior for the worst situation can be known, then the fault tolerant controller is a passive one. On the other hand, if L is estimated in real time, then the fault tolerant controller can be designed adaptively according to the fault. From this point of view, it’s an active fault tolerant controller. The fault situations L can be getting use the method proposed in [9] and [20]. The interested reader can refer to these articles for details.

Remark 2. It is worth to be pointed out that, in this paper, quadratically stable of the close-loop system is chosen as the performance index. For further research, we can choose a more complex performance index such as an LQ one which is often used in design a passive fault tolerant controller, for example, in [1]. The proofs of the theorems will be a bit complex but in a similar way with the proofs of the corresponding theorems in this paper.

4. NUMERICAL EXAMPLE

Consider an approximate linear model of the lateral dynamics of an aircraft. The state space representation is given as follows:

1.6689 0.0759 0.0100 1 2 0 2 0 − − − A = 1.6920 22.3750 0.1712 , Ad = 0.03 0.26 0 , B = 1 0 .  −5.6971 − 0.0615 6.6168   0.1 0.23 5.36   4 1  − − −       T The state x = δβ δp δr , where δβ is the incremental sideslip angle, T δp and δr are incremental£ roll rate¤ and yaw rate, respectively; u = p˙c r˙c are the generalized roll acceleration and yaw acceleration commands respectively. Disturbance is assumed as sine wave, which satisﬁes d(t) 1. The£ time-delay¤ function can be chosen as sin(0.5t) , withτ ¯ = 1 and |µ =| 0. ≤5 denoting the upper bound on the time-varying| delay and| its derivative, respectively. Assume that the model uncertainties in this example are as follows:

0.1 D = 0 , E a = 0.1 0.2 0.3 , E d = 0.3 0 0.1 .  0.32  £ ¤ £ ¤   The initial condition of the states is 2, 0, 1 with the actuator fault de- − 0.2 0 noted as L = which means the ﬁrst actuator is 80 % fail and the second 0 1 £ ¤ actuator is healthy.» – Here rank(B) = 2 and the SVD of B is:

0.4209 0.7892 0.4472 −0.2105 −0.3946 −0.8944 4.6659 0   0.9820 0.1891 B = −0.8824− 0.4706 0 0 0.4792 − − . −   0.1891 0.9820   0 0 · − ¸  U 1 U 2       | {z } | {z }  Fault Tolerant Control for Uncertain Time-Delay Systems . . . 629

U T The state transformation is z = Tx with T = 1 . U T » 2 – 4.5817 0.8824 So that B = − − and as shown in Section 2, we can get A¯ 1 0.0906 0.4706 22 » − – and other parameter matrices in (6). 2.0691 It follows from Theorem 2 in Section 3 that C = and the sliding mode 1.6239 surface is: » – 1.3462 1.6402 0.8824 S = IC z (t) = IC Tx(t) = x(t) = 0. −1.5154 1.0579− 0.4706 · − ¸ £ ¤ £ ¤ With x 0.0417 0.0834 0.1251 1 N = x , 1 0.0001 0.0002 0.0003 ¯ 2 ¯ · ¸ ¯ x 3 ¯ ¯ ¯ x ¯(t τ(t))¯ 0.1251 0 0.0417 1¯ − ¯ N = x ¯(t τ(t))¯ 2 0.0003 0 0.0001 ¯ 2 − ¯ · ¸ ¯ x 3(t τ(t)) ¯ ¯ − ¯ 4.5817 0.8824 ¯ ¯ N = sin¯ t , ¯ 3 0.0906 0.4706 · | ¯ | ¯ · ¸ the reaching control law can be get from Theorem 3:

0.9174 0.8824 5.5553 36.7475 5.5441 u = − − KS +α sign(S)+ − − − x(t) − 0.0181 0.4706 3.4202 23.8136 3.3101 · − ¸ · · − ¸ 1.3853 3.3219 4.7294 + − x(t τ(t))+diag(sign(s1) sign(s2))(N 1 +N 2 +N 3) . 1.4366 3.1977 2.5224 − · − − ¸ ¸ Figures 1 – 3 illustrate the simulation results. It can be seen that the present fault tolerant control scheme eﬀectively eliminated the eﬀects of parameter uncertainties and fault input, and guaranteed the asymptotic stability of the closed-loop system.

Remark 3. The reaching control law u is an exponential one and the parameters K and α can be tuned to reduce the chattering on the sliding surface. In order to reduce the chattering and guarantee a relatively far reaching speed simultaneously, one should increase K and decrease α at the same time. Here we select K = 18 and α = 1.

5. CONCLUSION

In this paper, the problem of fault tolerant control for uncertain systems with time varying state-delay has been considered. Based on sliding mode control, a fault tolerant control design scheme is proposed. Delay dependent suﬃcient condition is derived for the existence of the sliding surface which guarantees the system restricted to the sliding surface. And a reaching motion controller that can be considered as a fault tolerant controller is proposed. The sliding motion and the reaching motion 630 J. WU, Z. WENG, Z. TIAN AND S. SHI

1 x

−1

−2 0 2 4 6 8 10 t/s Fig. 1. State response.

s 0

−2

−4 0 2 4 6 8 10 t/s Fig. 2. Sliding surfaces.

are robust against the mismatched uncertainties, actuator fault and disturbance. The time-delay considered in this paper is the time-varying one, the results are delay dependent, which is less conservative than the delay-independent ones in the references. At last, a numerical example has been included to demonstrate the eﬀectiveness of the presented method. Fault Tolerant Control for Uncertain Time-Delay Systems . . . 631

120

100

u 20

−20

−40

−60

−80 0 2 4 6 8 10 t/s Fig. 3. Control input.

ACKNOWLEDGEMENT

The authors would like to thank the anonymous reviewers for their detailed comments which helped to improve the quality of this paper. This work is supported by National Nature Science Foundation of China under Grant 60574081.

(Received May 28, 2007.)

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Junsheng Wu, Zhengxin Weng, Zuohua Tian and Songjiao Shi, Department of Automa- tion, Shanghai Jiaotong University, Don chuan Road 800, 200240 Shanghai. China. e-mails: [email protected], zxweng, zhtian, [email protected]