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Stability analysis of discrete-time fuzzy-model-based control systems with time delay: Time delay-independent approach
Fuzzy Sets and Systems 159 (2008) 990 – 1000 www.elsevier.com/locate/fss
Stability analysis of discrete-time fuzzy-model-based control systems with time delay: Time delay-independent approach H.K. Lama,∗ , F.H.F. Leungb a Division of Engineering, The King’s College London, Strand, London, WC2R 2LS, UK b Centre for Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong Received 10 December 2006; received in revised form 10 July 2007; accepted 8 November 2007 Available online 21 November 2007
Abstract This paper presents the stability analysis of discrete-time fuzzy-model-based control systems with time delays. To facilitate the stability analysis, the T–S fuzzy model is employed to represent the discrete-time nonlinear system. Due to the multiplication property of the fuzzy-model-based control system, the number of LMI-based stability conditions is p(p + 1)/2 where p is the number of rules for the fuzzy model and fuzzy controller. Consequently, the computational demand for solving the solution to the stability conditions is very expensive, especially when the value of p is large. The problem becomes much worse when the relaxed stability conditions are formulated as huge matrices. The huge matrices incorporate the information of all the closed-loop subsystems, which may cause the solution of the LMI-based stability conditions difficult or even impossible to obtain owing to the limited computer power. In this paper, to reduce the computational demand, the fuzzy-model-based control system is divided into parts which the control input will handle separately. As a result, the multiplication property of the fuzzy-model-based control system can be removed, which effectively reduces the number of stability conditions to p and the computational demand for finding the solution can be reduced. LMI-based delay-independent stability conditions are derived using the Lyapunov-based approach. Simulation examples are given to illustrate the effectiveness of the proposed approach. © 2007 Elsevier B.V. All rights reserved. Keywords: Fuzzy control; Stability; Time delay
1. Introduction The dynamics of the nonlinear plant can be systematically represented by the T–S fuzzy model [12,11]. Based on the fuzzy-model-based control approach, the system stability analysis and controller synthesis can be done. Extensive stability analysis results and controller design methods for continuous-time and discrete-time fuzzy-modelbased control systems based on the linear-matrix-inequality (LMI) approach were reported in the past two decades [7–9,13–15,17,18]. In recent years, the stability of the fuzzy-model-based control systems with time delay has drawn a great deal of research attention. To investigate this class of nonlinear systems, the Lyapunov–Krasovskii function [5,10] was ∗ Corresponding author. Tel.: +44 20 7848 1240; fax: +44 20 7848 2932.
E-mail address: [email protected] (H.K. Lam). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.11.012
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employed. Some stability analysis results of the time-delay fuzzy-model-based control systems [1–4,6,16,19] have been reported. However, the stability analysis results on discrete-time time-delay fuzzy-model-based control systems [16,4] are seldom found in the literature when compared with the continuous-time time-delay systems [1–3,6,19]. In general, there are two categories of stability analysis found in the literature: delay independent and delay dependent. For the delay-independent stability analysis approach, the time delay is not considered during the stability analysis. Consequently, the continuous-time [1,2,19] and discrete-time [16] delay-independent stability conditions do not contain the time delays of the nonlinear systems. The system stability of the time-delay fuzzy-model-based control system is guaranteed by the stability conditions for any value of the time delay. On the contrary, for the delay-dependent stability analysis approach, the time delay is considered during the stability analysis. Hence, the continuous-time [6,3] and discrete-time [1,4] delay-dependent stability conditions will contain the time delays of the nonlinear systems. Based on this approach, the upper bound of the time delay can be obtained. Both of the delay-independent and delay-dependent stability analysis results have their own advantages. The delayindependent analysis result is particularly good to deal with the fuzzy-model-based control systems with unknown or inestimable value of time delay, or even time-varying time delay. In the delay-dependent analysis approach, as the time delay is considered during the stability analysis, the stability analysis result is less conservative comparatively, especially when the value of time delay is small. Hence, the delay-dependent analysis result is suitable to deal with time-delay fuzzy-model-based control systems with known or estimable value of time delay. It can be seen that the delay-independent and delay-dependent analysis results cannot replace each other. In this paper, the stability of discrete-time fuzzy-model-based control systems with time delay is investigated. It was shown in [1–4,6,16,19] that the multiplication property of the fuzzy-model-based systems would drastically increase the number of the stability conditions. When the number of rules of the fuzzy model and the fuzzy controller is p, the number of LMI-based stability conditions will at least be p(p+1)/2 [1–4,6,16,19]. It can be seen that the computational demand is very high on finding the solution, particularly when the systems are complicated with a large number of rules. In order to relax the stability analysis, the stability conditions were obtained [1–4,6,16,19] by grouping the closed-loop subsystem matrices together to form huge matrices. The large number of rules and high system order will make the solution to be obtained in a very long time or even unsuccessfully owing to the limited power of the computer. To put the fuzzy-model-based control approach into practice, the computational demand required to solve the solution has to be reduced. As the multiplication property is mainly due to the fuzzy controller interacting with the fuzzy model, it can be removed when the fuzzy controller is considered separately. In this paper, based on a particular system formulation, the LMI-based delay-independent stability conditions can be reduced to p so as to lower the computational demand. The LMI-based delay-independent stability conditions can help to design a stable fuzzy-model-based control system with time delay. This paper is organized as follows. In Section 2, the discrete-time fuzzy model with time delay and the fuzzy controller are briefly introduced. In Section 3, the system stability of the discrete-time fuzzy-model-based control systems with time delay is investigated. LMI-based delay-independent stability conditions are derived. In Section 4, some simulation examples are presented. A conclusion is drawn in Section 5. 2. Discrete-time fuzzy model and fuzzy controller A discrete-time fuzzy-model-based control system comprises a discrete-time nonlinear plant represented by the T–S fuzzy model and a fuzzy controller. The details of the discrete-time fuzzy model and the fuzzy controller are given as follows. 2.1. Discrete-time fuzzy model with time delay Let p be the number of fuzzy rules describing the discrete-time nonlinear plant with time delay. The ith rule is of the following format. i Rule i : IF f1 (x(t)) is M1i AND . . . AND f (x(t)) is M THEN x(t + 1) = Ai x(t) + Adi x(t − d ) + Bi u(t), t = 0, 1, . . . , ∞,
(1)
where Mi is a fuzzy term of rule i corresponding to the function f (x(t)), = 1, 2, . . . , , is a positive integer, i = 1, 2, . . . , p; Ai ∈ n×n , Adi ∈ n×n and Bi ∈ n×m are known constant system, time-delay system and input matrices, respectively; x(t) ∈ n×1 is the system state vector and u(t) ∈ m×1 is the input vector; d is a positive
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integer denoting the constant time delay. The system behavior is described by x(t + 1) =
p
wi (x(t))(Ai x(t) + Adi x(t − d ) + Bi u(t)),
(2)
i=1
where p
wi (x(t)) = 1, wi (x(t)) ∈ [0 1]
for all i,
(3)
i=1
wi (x(t)) = p
Mi (f1 (x(t))) × Mi (f2 (x(t))) × · · · × Mi (f (x(t))) 2 1
k (f (x(t)))) k=1 (M1k (f1 (x(t))) × M2k (f2 (x(t))) × · · · × M
(4)
is a known nonlinear function of f (x(t)). Mi (f (x(t))), = 1, 2, . . . , , is the grade of membership corresponding to the fuzzy term Mi . 2.2. Discrete-time fuzzy controller A discrete-time fuzzy controller with p fuzzy rules is employed to control the nonlinear plant. The jth rule of the fuzzy controller is of the following format. i Rule j : IF f1 (x(t)) is M1i AND . . . AND f (x(t)) is M THEN u(t) = Gj x(t), t = 1, 2, . . . , ∞,
(5)
where Gj ∈ m×n is the feedback gain of rule j. The inferred output of the fuzzy controller is given by u(t) =
p
wj (x(t))Gj x(t).
(6)
j =1
3. Stability analysis The discrete-time fuzzy-model-based control system with time delay, formed by the fuzzy model of (2) and the fuzzy controller of (6) connected a closedloop,is investigated. In the following analysis, wi (x(t)) is written as in p p p wi for short and the equality of i=1 wi = i=1 j =1 wi wj = 1 is applied during the analysis. To investigate the system stability of the fuzzy-model-based control system with time delay, the following representation of the system is considered: x(t + 1) = z(t), z(t) =
p
wi (Ai x(t) + Adi x(t − d ) + Bi u(t)).
(7) (8)
i=1
From (6) and (8), we have the following property which is applied in the stability analysis. ⎤ ⎡ ⎤ ⎡ ⎤⎡ 0 0 0 0 0 x(t) p ⎢ 0 ⎥ ⎢ x(t − d ) ⎥ ⎢ 0 ⎥ 0 0 0 ⎥ ⎢ ⎥ ⎥⎢ wi ⎢ ⎣ Gi 0 −I 0 ⎦ ⎣ u(t) ⎦ = ⎣ 0 ⎦ . i=1 z(t) 0 Ai Adi Bi −I
(9)
To study the system stability, we consider the following discrete-time Lyapunov function candidate: V (t) = V1 (t) + V2 (t),
(10)
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where V1 (t) = x(t)T P1 x(t), V2 (t) =
d
(11)
x(t − )T Sx(t − )
(12)
=1
and P1 = P1T ∈ n×n > 0 and S = ST ∈ n×n > 0. From (7)–(9), (11), we have, V1 (t) = x(t + 1)T P1 x(t + 1) − x(t)T P1 x(t) ⎤T ⎡ ⎤ ⎤⎡ ⎡ −P1 0 0 0 x(t) x(t) ⎥ ⎢ ⎢ x(t − d ) ⎥ ⎢ 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ x(t − d ) ⎥ =⎢ ⎦ ⎣ ⎣ u(t) ⎦ ⎣ 0 u(t) ⎦ 0 0 0 z(t) z(t) 0 0 0 P1 ⎤⎡ ⎤T ⎡ ⎤ ⎡ −P1 0 0 0 x(t) x(t) ⎢ ⎥ ⎢ x(t − d ) ⎥ ⎢ 0 0 0 0 ⎥ ⎥ ⎢ x(t − d ) ⎥ ⎥ ⎢ =⎢ ⎦ ⎣ ⎣ u(t) ⎦ ⎣ 0 0 0 0 u(t) ⎦ 0 0 0 P1 z(t) z(t) ⎛ ⎡ ⎤ ⎞ 0 0 0 0 ⎜ T⎢ 0 ⎟ 0 0 0 ⎥ ⎢ ⎥ ⎟ ⎤T ⎜ ⎤ ⎡ ⎜ P ⎣ Gi 0 −I 0 ⎦ ⎟ ⎡ x(t) x(t) ⎜ ⎟ p ⎟⎢ ⎢ x(t − d ) ⎥ ⎜ Ai Adi Bi −I ⎟ ⎢ x(t − d ) ⎥ ⎥ ⎜ ⎥, + wi ⎢ ⎜ ⎤ ⎡ T ⎟ ⎣ u(t) ⎦ ⎜ ⎟ ⎣ u(t) ⎦ 0 0 0 0 i=1 ⎜ ⎟ z(t) z(t) ⎜ ⎢ 0 ⎟ 0 0 0 ⎥ ⎥ P⎟ ⎜+⎢ ⎝ ⎣ Gi 0 −I 0 ⎦ ⎠ Ai Adi Bi −I ⎡
P1 ⎢ P2 where P = ⎢ ⎣ P4 P7 P6 ∈ m×m and P9
(13)
⎤ 0 0 0 P3 0 0 ⎥ ⎥ ∈ (3n+m)×(3n+m) , Pr ∈ n×n , r = 1, 2, 3, 7, 8, 10, Pk ∈ m×n , k = 4, 5, P5 P6 0 ⎦ P8 P9 P10 ∈ n×m . From (12), we have,
V2 (t) = x(t)T Sx(t) − x(t − d )T Sx(t − d ).
(14)
From (10), (13) and (14), we have, ⎛⎡
⎤ ⎡ ⎤⎞ S − P1 0 0 0 0 0 0 0 ⎢ ⎜⎢ 0 −S 0 0 ⎥ 0 0 0 ⎥ ⎥ + PT ⎢ 0 ⎥⎟ ⎟ ⎡ ⎤T ⎜ ⎢ ⎣ ⎦ ⎣ ⎦ ⎟ ⎡ x(t) ⎤ ⎜ 0 0 0 0 G 0 −I 0 x(t) i ⎜ ⎟ p ⎢ ⎢ x(t − d ) ⎥ ⎜ ⎥ 0 0 0 P1 Ai Adi Bi −I ⎟ ⎜ ⎟ ⎢ x(t − d ) ⎥ ⎢ ⎥ ⎤T ⎡ wi ⎣ V (t) = ⎜ ⎟ ⎣ ⎦ u(t) u(t) ⎦ 0 0 0 0 ⎜ ⎟ i=1 ⎜ ⎟ ⎥ ⎢ z(t) z(t) 0 0 0 ⎥ ⎜+⎢ 0 ⎟ ⎝ ⎣ Gi 0 −I 0 ⎦ P ⎠ Ai Adi Bi −I ⎡ ⎤T ⎡ ⎤ x(t) x(t) p ⎢ x(t − d ) ⎥ ⎢ ⎥ ⎥ i ⎢ x(t − d ) ⎥ , = (15) wi ⎢ ⎣ u(t) ⎦ ⎣ u(t) ⎦ i=1 z(t) z(t)
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where
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⎡
T T T S − P1 + AT i P7 + P7 Ai + Gi P4 + P4 Gi ⎢ P5T Gi + P8T Ai + AT P ⎢ 7 di i = ⎢ P6T Gi + P9T Ai − P4 + BT ⎣ i P7 T A −P P10 7 i
∗ T −S + AT P di 8 + P8 Adi T P9 Adi − P5 + BT i P8 T A −P P10 di 8
∗ ∗ T −P6T − P6 + BT i P9 + P9 Bi T B −P P10 i 9
∗ ∗ ∗
⎤ ⎥ ⎥ ⎥. ⎦
T −P P1 − P10 10
(16) The symbol ‘*’ denotes position. It can be seen from (15) that ⎡ ⎤⎞ ⎛ the transpose of the⎡element at⎤the corresponding 0 x(t) ⎜ ⎢ x(t − d ) ⎥ ⎢ 0 ⎥⎟ ⎥ ⎢ ⎥⎟ ⎢ if i < 0, V (t) 0 ⎜ ⎝equality holds when ⎣ u(t) ⎦ = ⎣ 0 ⎦⎠, which implies the asymptotic stability of the z(t) 0 discrete-time fuzzy-model-based control system with time delay. The analysis results are summarized in the following theorem. Theorem 1. The discrete-time fuzzy-model-based control system with time delay, formed by the nonlinear system in the form of (2) and the fuzzy controller of (6), is asymptotically stable if there exists predefined feedback gains of Gj ∈ m×n and matrices of P1 = P1T ∈ n×n , Pr ∈ n×n , r = 1, 2, 3, 7, 8, 10, Pk ∈ m×n , k = 4, 5, P6 ∈ m×m and P9 ∈ n×m ; i = 1, 2, . . . , p, S = ST ∈ n×n such that the following LMIs are satisfied: P1 > 0, S > 0, and i < 0,
i = 1, 2, . . . , p.
It should be noted that the feedback gains of Gj are needed to be determined prior to applying Theorem 1. In the following, the feedback gains are converted into decision variables. Hence, the values of the feedback gains can be ⎡ ⎤ ⎤ ⎡ x(t) X1 0 0 0 ⎢x(t − d )⎥ ⎢ X2 X 3 0 0 ⎥ ⎥ ⎥ ⎢ determined by using some convex programming techniques. Let v(t)=X−1 ⎢ ⎣ u(t) ⎦, X= ⎣ X4 X5 X6 0 ⎦ X7 X8 X9 X10 z(t) = P−1 ∈ (3n+m)×(3n+m) , X1 = X1T = P1−1 ∈ n×n , Xr ∈ n×n , r = 2, 3, 7, 8, 10, Xk ∈ m×n , k = 4, 5, X6 ∈ m×m and X9 ∈ n×m . To ease the development of the LMI stability conditions, we choose X2 = 0; otherwise, the resultant stability conditions will be in the form of bilinear matrix inequalities which cannot be simply solved by convex programming techniques. However, conservativeness of the stability analysis result will increase. From (15), we have, ⎡ ⎤ ⎡ ⎤ ⎞ ⎛ S − P1 0 0 0 0 0 0 0 ⎢ ⎜ T⎢ −S 0 0 ⎥ 0 0 0 ⎥ ⎥X + ⎢ 0 ⎥X⎟ ⎜X ⎢ 0 ⎟ ⎣ 0 ⎦ ⎣ ⎜ 0 0 0 Gi 0 −I 0 ⎦ ⎟ ⎜ ⎟ p ⎜ ⎟ 0 0 0 P1 Ai Adi Bi −I ⎟ T⎜ V (t) = wi v(t) ⎜ ⎟ v(t) ⎡ ⎤T ⎜ ⎟ 0 0 0 0 i=1 ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ 0 0 0 ⎥ ⎜ +XT ⎢ 0 ⎟ ⎝ ⎠ ⎣ Gi 0 −I 0 ⎦ Ai Adi Bi −I ⎤ ⎡ ⎤ ⎞ ⎛⎡ X1 SX1 − X1 0 0 0 0 0 0 0 ⎢ ⎜⎢ 0 0 0 ⎥ 0 −X3T SX3 0 0 ⎥ ⎥+⎢ 0 ⎥X⎟ ⎟ ⎜⎢ ⎦ ⎣ ⎣ ⎦ ⎟ ⎜ G 0 −I 0 0 0 0 0 i ⎟ ⎜ p ⎟ ⎜ Ai Adi Bi −I 0 0 0 0 ⎟ T⎜ = (17) wi v(t) ⎜ ⎟ v(t), ⎡ ⎤ T ⎟ ⎜ 0 0 0 0 i=1 ⎟ ⎜ ⎟ ⎜ 0 0 0 ⎥ ⎟ ⎜ +XT ⎢ ⎢ 0 ⎥ + s T P1 s ⎠ ⎝ ⎣ Gi 0 −I 0 ⎦ Ai Adi Bi −I
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where s = [X7 X8 X9 X10 ]. It can be seen from (17) that the existence of the term of X3T SX3 will lead to stability conditions not in terms of LMIs. To circumvent this problem, we denote M = MT = S−1 and consider the inequality of (X3 − S−1 )T S(X3 − S−1 ) 0. Expanding it, we obtain the following inequality: M − X3T − X3 − X3T SX3 . From (17) and (18), let Gi = Ni X1−1 where Ni ∈ m×n , i = 1, 2, . . . , p, we have, ⎤ ⎡ ⎤ ⎞ ⎛⎡ X1 SX1 − X1 0 0 0 0 0 0 0 ⎢ ⎜⎢ 0 0 0 ⎥ 0 M − X3T − X3 0 0 ⎥ ⎥+⎢ 0 ⎥X⎟ ⎜⎢ ⎟ ⎦ ⎣ ⎜⎣ Gi 0 −I 0 ⎦ ⎟ 0 0 0 0 ⎜ ⎟ p ⎜ ⎟ Ai Adi Bi −I 0 0 0 0 ⎟ v(t) ⎡ ⎤ wi v(t)T ⎜ V (t) T ⎜ ⎟ 0 0 0 0 ⎜ ⎟ i=1 ⎜ ⎟ ⎢ ⎥ 0 0 0 ⎥ ⎜ +XT ⎢ 0 ⎟ T + s P s 1 ⎝ ⎠ ⎣ Gi 0 −I 0 ⎦ Ai Adi Bi −I p T T T = v(t) wi Vi + s P1 s + r Sr v(t),
(18)
(19)
i=1
⎤ ∗ ∗ ∗ −X1 ⎥ ⎢ 0 M − X3T − X3 ∗ ∗ ⎥, and r = [X1 0 0 0]. It can where Vi = ⎢ T ⎦ ⎣ Ni − X4 −X5 −X6 − X6 ∗ T Ai X1 + Bi X4 − X7 Adi X3 + Bi X5 − X8 Bi X6 − X9 −X10 − X10 be seen from (19) that the discrete-time fuzzy-model-based control system with time delay is asymptotically stable if p T P s + rT Sr < 0. By Schur complement, it is equivalent to the following inequality: V + s w 1 i=1 i i ⎡ ⎤ p ∗ ∗ Vi ∗ ⎦ < 0. wi ⎣ s −X1 (20) r 0 −M i=1 ⎡ ⎤ Vi ∗ ∗ ∗ ⎦ < 0, i = 1, 2, . . . , p, implies V (t) < 0 which implies the From (20), it can be seen that ⎣ s −X1 r 0 −M asymptotic stability of the discrete-time fuzzy-model-based control system with time delay. The stability analysis result is summarized in the following theorem. ⎡
Theorem 2. The discrete-time fuzzy-model-based control system with time delay, formed by the nonlinear system in the form of (2) and the fuzzy controller of (6), is asymptotically stable if there exists M ∈ n×n , Ni ∈ m×n , X1 = X1T ∈ n×n , Xr ∈ n×n , r = 2, 3, 7, 8, 10, Xk ∈ m×n , k = 4, 5, X6 ∈ m×m and X9 ∈ n×m such that the following LMIs are satisfied: ⎤ ⎡ ∗ ∗ Vi ∗ ⎦ < 0, i = 1, 2, . . . , p, X1 > 0, M > 0, and ⎣ s −X1 r 0 −M where
⎤ −X1 ∗ ∗ ∗ ⎥ ⎢ 0 M − X3T − X3 ∗ ∗ ⎥, Vi = ⎢ T ⎦ ⎣ Ni − X4 −X5 −X6 − X6 ∗ T −X Ai X1 + Bi X4 − X7 Adi X3 + Bi X5 − X8 Bi X6 − X9 −X10 10 s = [X7 X8 X9 X10 ], r = [X1 0 0 0] ⎡
and the feedback gains are designed as Gi = Ni X1−1 , i = 1, 2, . . . , p.
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Remark 1. It should be noted that the stability analysis result summarized in Theorem 2 is valid if there exists the inverse of X. Referring to the stability conditions in Theorem 2, if there exists a solution, we have X1 > 0, X3T +X3 > 0, T + X > 0, which are the sufficient conditions that X is a non-singular matrix, i.e. its inverse X6T + X6 > 0 and X10 10 exists. Remark 2. In [7–9,13–15,17,18], due to the multiplication property of the fuzzy-model-based control systems, it can be shown that the number of stability conditions is p(p − 1)/2. In this paper, referring to Theorems 1 and 2, the number of stability conditions is reduced to p only. However, it is due to the introduction of some free matrices, i.e. X3 to X10 ; the computational advantage may not be obvious for a small value of p. Under the case of a large value of p, the influence of introducing free matrices to the computational demand becomes insignificant. Hence, the number of rules plays a crucial role to the computational demand. We conclude that by using the proposed approach, the computational demand on finding the solution can be reduced. The computational advantage is more obvious for fuzzy systems with large values of p and higher system and input matrix dimensions. It will be shown by a simulation example in the following section that the proposed LMI stability conditions are less computationally expensive. 4. Simulation examples Simulation examples are given in this section to illustrate the effectiveness of the proposed approach. 4.1. Simulation Example 1 In this example, the computational demand on solving the solution to the LMI stability conditions is investigated. A discrete-time nonlinear plant with time delay represented by the fuzzy model with the following p(> 3) rules is considered. Rule i : IF x1 (t) is M1i (21) THEN x(t + 1) = Ai x(t) + Adi x(t − d ), i = 1, 2, . . . , p; t = 0, 1, . . . , ∞, 0.1 0.5 0.1 0.25 0.1 0.75 0.15 0 0.25 0 where A1 = , A2 = , A3 = , Ad1 = , Ad2 = , 0.51 −0.1 0.15 −0.5 0.25 −0.8 0.01 0 0.05 0 0.19 0 Ad3 = . The subsystems 4 to p are defined as follows: 0.09 0 Ak =
d1k A1 + d2k A2 + d3k A3 , d1k + d2k + d3k
Adk =
k = 4, 5, . . . , p,
d1k Ad1 + d2k Ad2 + d3k Ad3 , d1k + d2k + d3k
k = 4, 5, . . . , p,
(22)
(23)
where d1k , d2k and d3k , k = 4, 5, . . . , p, are randomly generated scalars in the range of 0–1. The dynamics of the discrete-time nonlinear plant with time delay are defined as follows: x(t + 1) =
p
wi (x1 (t))(Ai x(t) + Adi x(t − d )).
(24)
i=1
It can be checked with the stability conditions in Theorem 1 or [1] that the time-delay system of (24) is open-loop stable for p = 3. With the help of Matlab LMI toolbox, the proposed stability conditions in Theorem 1 are employed to check for the system stability with different values of p. To investigate the computational demand, the averaged time (in seconds) in 10 runs for various values of p is recorded and tabulated in Table 1. The simulation is carried out in a
H.K. Lam, F.H.F. Leung / Fuzzy Sets and Systems 159 (2008) 990 – 1000
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Table 1 Average time (in seconds) required to find the solutions of stability conditions in Theorem 1 and [1] p
Average time (in seconds)
3 5 10 20 50 100 200 300 400 500
Stability conditions in Theorem 1
Stability conditions in [1]
0.0894 0.1517 0.3490 0.8980 3.1560 9.2003 26.5050 50.8900 78.7030 No feasible solution
0.0692 0.1720 1.0032 7.5078 80.1823 No feasible solution No feasible solution No feasible solution No feasible solution No feasible solution
personal computer with 3 GHz CPU and 2 GB RAM. Referring to Table 1, it can be seen that when the value of p is small, the computer takes longer time to find the solution when the proposed stability conditions in Theorem 1 are used instead of the stability conditions in [1]. However, when p 5, the solution to the stability conditions in Theorem 1 can be found in a shorter time comparatively. It can be seen from Table 1 that the computational advantage on finding the solution is more obvious for a large value of p. Furthermore, it is observed that feasible solutions cannot be obtained (LMI solver terminates owing to too slow processes) when the stability conditions in [1] are used for p 100, while those in Theorem 1 are still solvable. From this example, it can be shown that the proposed stability conditions are less computationally expensive. 4.2. Simulation Example 2 In this example, the stability region based on the stability conditions in Theorem 1 is investigated. A discrete-time nonlinear plant with time delay represented by the fuzzy model with the following three rules is considered. Rule i : IF x1 (t) is M1i (25) THEN x(t + 1) = Ai x(t) + Adi x(t − d ) + Bi u(t), t = 0, 1, . . . , ∞, i = 1, 2, 3, 1 0.5 0.1 0.25 a 0.75 0.1 0 0.25 0 where A1 = , A2 = , A3 = ; Ad1 = , Ad2 = , 0.51 −0.1 0.15 −0.5 0.25 −0.8 −0.25 0 0.15 0 0.19 0 0 1 b Ad3 = ; B1 = , B2 = and B3 = where 0 a 1.6 and 0 b 1.8. The inferred 0.06 0 1 −0.5 0.5 dynamics of the discrete-time nonlinear plant with time delay are defined as follows: x(t + 1) =
3
wi (x1 (t))(Ai x(t) + Adi x(t − d ) + Bi u(t)).
(26)
i=1
A three-rule fuzzy controller is employed to control the discrete-time nonlinear plant with time delay. The control action of the fuzzy controller is described by the following rules. Rule j : IF x1 (t) is M1i THEN u(t) = Gj x(t),
t = 0, 1, . . . , ∞, j = 1, 2, 3.
(27)
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1.8 1.6 1.4 1.2
b
1 0.8 0.6 0.4 0.2 0
0
0.2
0.4
0.6
0.8 a
1
1.2
1.4
1.6
Fig. 1. Stability regions based on the stability conditions in Theorem 1 (indicated by “◦”) and [1] (indicated by “×”).
The inferred fuzzy controller is defined as follows: u(t) =
3
wi (x1 (t))Gi x(t).
(28)
i=1
The feedback gains of Gi are designed such that the eigenvalues of Ai + Bi Gi , i = 1, 2, 3, are all placed at −5. On applying the stability conditions in Theorem 1, with the aid of the MATLAB LMI toolbox, the stability region (indicated by “◦”) for 0 a 1.6 and 0 b 1.8 is shown in Fig. 1. For comparison purpose, the stability region (indicated by “×”) on applying the stability conditions in [1] is also shown in Fig. 1. It can be seen that the proposed stability conditions in Theorem 1 offer a larger stability region. 4.3. Simulation Example 3 In this example, the stability conditions in Theorem 2 are employed to obtain the feedback gains of the fuzzy controller to stabilize a time-delayed nonlinear plant. Consider the time-delay nonlinear plant represented by the fuzzy model 2 2 of (26) with a = 1.2 and b = 1.6, where the membership functions are chosen as M1 (x1 (t)) = e−(x1 (t)+5) /2×2 , 1
M1 (x1 (t)) = e−x1 (t) /2×1.5 and M1 (x1 (t)) = e−(x1 (t)−5) /2×2 . 2 3 The fuzzy controller of (28) is employed to control the time-delayed nonlinear plant. By applying the stability conditions in Theorem 2, with the aid of the MATLAB LMI toolbox, the feedback gains are obtained as G1 = [−0.4793 − 0.0436], G2 = [−0.2992 − 0.1695] and G3 = [−0.7459 − 0.1120]. Fig. 2 shows the system state responses and the control signals of the discrete-time fuzzy-model-based control system with time delay under the initial system state condition of x(t) = [2 −1]T for t ∈ [−d 0]. Referring to Fig. 2, it can be seen that the discretetime nonlinear plant with d = 5 and 20 can be stabilized by the fuzzy controller. If the delay-dependent approaches in [6,3,4] are considered, it can be shown that no solution can be obtained when the value of d is sufficiently large. However, the proposed stability conditions guarantee the asymptotic stability of the fuzzy-model-based control system for any value of d . 2
2
2
2
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1 0.5 x2(t)
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10
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30
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70
80
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0.4 0.2 0
u(t)
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40 50 60 Samples
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Fig. 2. System state responses and control signals of the discrete-time fuzzy-model-based control system with time-delay of d = 5 (solid lines) and d = 20 (dotted lines).
5. Conclusion The stability of discrete-time fuzzy-model-based control systems with time delay has been investigated. LMI-based delay-independent stability conditions have been derived using the Lyapunov-based approach. The system stability and feedback gains can be determined with the help of some convex programming techniques. Furthermore, with a particular system formulation, the number of the LMI-based stability conditions has been reduced to alleviate the computational demand for finding the solution. Simulation examples have been given to illustrate the design procedure and the effectiveness of the proposed approach.
Acknowledgements The work described in this paper was substantially supported by Grants from King’s College, London and from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 5218/06E).
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References [1] Y.Y. Cao, P.M. Frank, Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach, IEEE Trans. Fuzzy Systems 8 (2) (2000) 200–211. [2] Y.Y. Cao, P.M. Frank, Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi–Sugeno fuzzy models, Fuzzy Sets and Systems 124 (2) (2001) 213–219. [3] B. Chen, X. Liu, Delay-dependent robust H∞ control for T–S fuzzy systems with time delay, IEEE Trans. Fuzzy Systems 13 (4) (2005) 544–556. [4] C.L. Chen, G. Feng, X.P. Guan, Delay-dependent stability analysis and controller synthesis for discrete-time T–S fuzzy systems with time delays, IEEE Trans. Fuzzy Systems 13 (5) (2005) 630–643. [5] E. Fridman, U. Shaked, Delay-dependent stability and H∞ control: constant and time-varying delays, Internat. J. Control 76 (1) (2003) 48–60. [6] X.P. Guan, C.L. Chen, Delay-dependent guaranteed cost control for T–S fuzzy systems with time delays, IEEE Trans. Fuzzy Systems 12 (2) (2004) 236–249. [7] E. Kim, H. Lee, New approaches to relaxed quadratic stability conditions of fuzzy control systems, IEEE Trans. Fuzzy Systems 8 (5) (2000) 523–534. [8] X. Liu, Q. Zhang, New approaches to H∞ controller designs based on fuzzy observers for T–S fuzzy systems via LMI, Automatica 39 (9) (2003) 1571–1582. [9] X. Liu, Q. Zhang, Approaches to quadratic stability conditions and H∞ control designs for T–S fuzzy systems, IEEE Trans. Fuzzy Systems 11 (6) (2003) 830–839. [10] Y.S. Moon, P. Park, W.H. Kwon, Y.S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, Internat. J. Control 74 (14) (2001) 1447–1455. [11] M. Sugeno, G.T. Kang, Structure identification of fuzzy model, Fuzzy Sets and Systems 28 (1988) 15–33. [12] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Systems Man Cybernet smc-15 (1) (1985) 116–132. [13] K. Tanaka, T. Ikeda, H.O. Wang, Fuzzy regulator and fuzzy observer: relaxed stability conditions and LMI-based designs, IEEE Trans. Fuzzy Systems 6 (2) (1998) 250–265. [14] K. Tanaka, M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems 45 (1992) 135–156. [15] M.C.M. Teixeira, E. Assunção, R.G. Avellar, On relaxed LMI-based designs for fuzzy regulators and fuzzy observers, IEEE Trans. Fuzzy Systems 11 (5) (2003) 613–623. [16] C.S. Tseng, Model reference output feedback fuzzy tracking control design for nonlinear discrete-time systems with time-delay, IEEE Trans. Fuzzy Systems 14 (1) (2006) 630–643. [17] H.O. Wang, K. Tanaka, M.F. Griffin, An approach to fuzzy control of nonlinear systems: stability and the design issues, IEEE Trans. Fuzzy Systems 4 (1) (1996) 14–23. [18] K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, Wiley, New York, USA, 2001. [19] R.J. Wong, W.W. Lin, W.J. Wang, Stabilizability of linear quadratic state feedback for uncertain fuzzy time-delay systems, IEEE Trans. Systems Man Cybernet. Part B Cybernetics 34 (2) (2004) 1288–1292.
Stability analysis of discrete-time fuzzy-model-based control systems with time delay: Time delay-independent approach H.K. Lama,∗ , F.H.F. Leungb a Division of Engineering, The King’s College London, Strand, London, WC2R 2LS, UK b Centre for Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong Received 10 December 2006; received in revised form 10 July 2007; accepted 8 November 2007 Available online 21 November 2007
Abstract This paper presents the stability analysis of discrete-time fuzzy-model-based control systems with time delays. To facilitate the stability analysis, the T–S fuzzy model is employed to represent the discrete-time nonlinear system. Due to the multiplication property of the fuzzy-model-based control system, the number of LMI-based stability conditions is p(p + 1)/2 where p is the number of rules for the fuzzy model and fuzzy controller. Consequently, the computational demand for solving the solution to the stability conditions is very expensive, especially when the value of p is large. The problem becomes much worse when the relaxed stability conditions are formulated as huge matrices. The huge matrices incorporate the information of all the closed-loop subsystems, which may cause the solution of the LMI-based stability conditions difficult or even impossible to obtain owing to the limited computer power. In this paper, to reduce the computational demand, the fuzzy-model-based control system is divided into parts which the control input will handle separately. As a result, the multiplication property of the fuzzy-model-based control system can be removed, which effectively reduces the number of stability conditions to p and the computational demand for finding the solution can be reduced. LMI-based delay-independent stability conditions are derived using the Lyapunov-based approach. Simulation examples are given to illustrate the effectiveness of the proposed approach. © 2007 Elsevier B.V. All rights reserved. Keywords: Fuzzy control; Stability; Time delay
1. Introduction The dynamics of the nonlinear plant can be systematically represented by the T–S fuzzy model [12,11]. Based on the fuzzy-model-based control approach, the system stability analysis and controller synthesis can be done. Extensive stability analysis results and controller design methods for continuous-time and discrete-time fuzzy-modelbased control systems based on the linear-matrix-inequality (LMI) approach were reported in the past two decades [7–9,13–15,17,18]. In recent years, the stability of the fuzzy-model-based control systems with time delay has drawn a great deal of research attention. To investigate this class of nonlinear systems, the Lyapunov–Krasovskii function [5,10] was ∗ Corresponding author. Tel.: +44 20 7848 1240; fax: +44 20 7848 2932.
E-mail address: [email protected] (H.K. Lam). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.11.012
H.K. Lam, F.H.F. Leung / Fuzzy Sets and Systems 159 (2008) 990 – 1000
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employed. Some stability analysis results of the time-delay fuzzy-model-based control systems [1–4,6,16,19] have been reported. However, the stability analysis results on discrete-time time-delay fuzzy-model-based control systems [16,4] are seldom found in the literature when compared with the continuous-time time-delay systems [1–3,6,19]. In general, there are two categories of stability analysis found in the literature: delay independent and delay dependent. For the delay-independent stability analysis approach, the time delay is not considered during the stability analysis. Consequently, the continuous-time [1,2,19] and discrete-time [16] delay-independent stability conditions do not contain the time delays of the nonlinear systems. The system stability of the time-delay fuzzy-model-based control system is guaranteed by the stability conditions for any value of the time delay. On the contrary, for the delay-dependent stability analysis approach, the time delay is considered during the stability analysis. Hence, the continuous-time [6,3] and discrete-time [1,4] delay-dependent stability conditions will contain the time delays of the nonlinear systems. Based on this approach, the upper bound of the time delay can be obtained. Both of the delay-independent and delay-dependent stability analysis results have their own advantages. The delayindependent analysis result is particularly good to deal with the fuzzy-model-based control systems with unknown or inestimable value of time delay, or even time-varying time delay. In the delay-dependent analysis approach, as the time delay is considered during the stability analysis, the stability analysis result is less conservative comparatively, especially when the value of time delay is small. Hence, the delay-dependent analysis result is suitable to deal with time-delay fuzzy-model-based control systems with known or estimable value of time delay. It can be seen that the delay-independent and delay-dependent analysis results cannot replace each other. In this paper, the stability of discrete-time fuzzy-model-based control systems with time delay is investigated. It was shown in [1–4,6,16,19] that the multiplication property of the fuzzy-model-based systems would drastically increase the number of the stability conditions. When the number of rules of the fuzzy model and the fuzzy controller is p, the number of LMI-based stability conditions will at least be p(p+1)/2 [1–4,6,16,19]. It can be seen that the computational demand is very high on finding the solution, particularly when the systems are complicated with a large number of rules. In order to relax the stability analysis, the stability conditions were obtained [1–4,6,16,19] by grouping the closed-loop subsystem matrices together to form huge matrices. The large number of rules and high system order will make the solution to be obtained in a very long time or even unsuccessfully owing to the limited power of the computer. To put the fuzzy-model-based control approach into practice, the computational demand required to solve the solution has to be reduced. As the multiplication property is mainly due to the fuzzy controller interacting with the fuzzy model, it can be removed when the fuzzy controller is considered separately. In this paper, based on a particular system formulation, the LMI-based delay-independent stability conditions can be reduced to p so as to lower the computational demand. The LMI-based delay-independent stability conditions can help to design a stable fuzzy-model-based control system with time delay. This paper is organized as follows. In Section 2, the discrete-time fuzzy model with time delay and the fuzzy controller are briefly introduced. In Section 3, the system stability of the discrete-time fuzzy-model-based control systems with time delay is investigated. LMI-based delay-independent stability conditions are derived. In Section 4, some simulation examples are presented. A conclusion is drawn in Section 5. 2. Discrete-time fuzzy model and fuzzy controller A discrete-time fuzzy-model-based control system comprises a discrete-time nonlinear plant represented by the T–S fuzzy model and a fuzzy controller. The details of the discrete-time fuzzy model and the fuzzy controller are given as follows. 2.1. Discrete-time fuzzy model with time delay Let p be the number of fuzzy rules describing the discrete-time nonlinear plant with time delay. The ith rule is of the following format. i Rule i : IF f1 (x(t)) is M1i AND . . . AND f (x(t)) is M THEN x(t + 1) = Ai x(t) + Adi x(t − d ) + Bi u(t), t = 0, 1, . . . , ∞,
(1)
where Mi is a fuzzy term of rule i corresponding to the function f (x(t)), = 1, 2, . . . , , is a positive integer, i = 1, 2, . . . , p; Ai ∈ n×n , Adi ∈ n×n and Bi ∈ n×m are known constant system, time-delay system and input matrices, respectively; x(t) ∈ n×1 is the system state vector and u(t) ∈ m×1 is the input vector; d is a positive
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integer denoting the constant time delay. The system behavior is described by x(t + 1) =
p
wi (x(t))(Ai x(t) + Adi x(t − d ) + Bi u(t)),
(2)
i=1
where p
wi (x(t)) = 1, wi (x(t)) ∈ [0 1]
for all i,
(3)
i=1
wi (x(t)) = p
Mi (f1 (x(t))) × Mi (f2 (x(t))) × · · · × Mi (f (x(t))) 2 1
k (f (x(t)))) k=1 (M1k (f1 (x(t))) × M2k (f2 (x(t))) × · · · × M
(4)
is a known nonlinear function of f (x(t)). Mi (f (x(t))), = 1, 2, . . . , , is the grade of membership corresponding to the fuzzy term Mi . 2.2. Discrete-time fuzzy controller A discrete-time fuzzy controller with p fuzzy rules is employed to control the nonlinear plant. The jth rule of the fuzzy controller is of the following format. i Rule j : IF f1 (x(t)) is M1i AND . . . AND f (x(t)) is M THEN u(t) = Gj x(t), t = 1, 2, . . . , ∞,
(5)
where Gj ∈ m×n is the feedback gain of rule j. The inferred output of the fuzzy controller is given by u(t) =
p
wj (x(t))Gj x(t).
(6)
j =1
3. Stability analysis The discrete-time fuzzy-model-based control system with time delay, formed by the fuzzy model of (2) and the fuzzy controller of (6) connected a closedloop,is investigated. In the following analysis, wi (x(t)) is written as in p p p wi for short and the equality of i=1 wi = i=1 j =1 wi wj = 1 is applied during the analysis. To investigate the system stability of the fuzzy-model-based control system with time delay, the following representation of the system is considered: x(t + 1) = z(t), z(t) =
p
wi (Ai x(t) + Adi x(t − d ) + Bi u(t)).
(7) (8)
i=1
From (6) and (8), we have the following property which is applied in the stability analysis. ⎤ ⎡ ⎤ ⎡ ⎤⎡ 0 0 0 0 0 x(t) p ⎢ 0 ⎥ ⎢ x(t − d ) ⎥ ⎢ 0 ⎥ 0 0 0 ⎥ ⎢ ⎥ ⎥⎢ wi ⎢ ⎣ Gi 0 −I 0 ⎦ ⎣ u(t) ⎦ = ⎣ 0 ⎦ . i=1 z(t) 0 Ai Adi Bi −I
(9)
To study the system stability, we consider the following discrete-time Lyapunov function candidate: V (t) = V1 (t) + V2 (t),
(10)
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where V1 (t) = x(t)T P1 x(t), V2 (t) =
d
(11)
x(t − )T Sx(t − )
(12)
=1
and P1 = P1T ∈ n×n > 0 and S = ST ∈ n×n > 0. From (7)–(9), (11), we have, V1 (t) = x(t + 1)T P1 x(t + 1) − x(t)T P1 x(t) ⎤T ⎡ ⎤ ⎤⎡ ⎡ −P1 0 0 0 x(t) x(t) ⎥ ⎢ ⎢ x(t − d ) ⎥ ⎢ 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ x(t − d ) ⎥ =⎢ ⎦ ⎣ ⎣ u(t) ⎦ ⎣ 0 u(t) ⎦ 0 0 0 z(t) z(t) 0 0 0 P1 ⎤⎡ ⎤T ⎡ ⎤ ⎡ −P1 0 0 0 x(t) x(t) ⎢ ⎥ ⎢ x(t − d ) ⎥ ⎢ 0 0 0 0 ⎥ ⎥ ⎢ x(t − d ) ⎥ ⎥ ⎢ =⎢ ⎦ ⎣ ⎣ u(t) ⎦ ⎣ 0 0 0 0 u(t) ⎦ 0 0 0 P1 z(t) z(t) ⎛ ⎡ ⎤ ⎞ 0 0 0 0 ⎜ T⎢ 0 ⎟ 0 0 0 ⎥ ⎢ ⎥ ⎟ ⎤T ⎜ ⎤ ⎡ ⎜ P ⎣ Gi 0 −I 0 ⎦ ⎟ ⎡ x(t) x(t) ⎜ ⎟ p ⎟⎢ ⎢ x(t − d ) ⎥ ⎜ Ai Adi Bi −I ⎟ ⎢ x(t − d ) ⎥ ⎥ ⎜ ⎥, + wi ⎢ ⎜ ⎤ ⎡ T ⎟ ⎣ u(t) ⎦ ⎜ ⎟ ⎣ u(t) ⎦ 0 0 0 0 i=1 ⎜ ⎟ z(t) z(t) ⎜ ⎢ 0 ⎟ 0 0 0 ⎥ ⎥ P⎟ ⎜+⎢ ⎝ ⎣ Gi 0 −I 0 ⎦ ⎠ Ai Adi Bi −I ⎡
P1 ⎢ P2 where P = ⎢ ⎣ P4 P7 P6 ∈ m×m and P9
(13)
⎤ 0 0 0 P3 0 0 ⎥ ⎥ ∈ (3n+m)×(3n+m) , Pr ∈ n×n , r = 1, 2, 3, 7, 8, 10, Pk ∈ m×n , k = 4, 5, P5 P6 0 ⎦ P8 P9 P10 ∈ n×m . From (12), we have,
V2 (t) = x(t)T Sx(t) − x(t − d )T Sx(t − d ).
(14)
From (10), (13) and (14), we have, ⎛⎡
⎤ ⎡ ⎤⎞ S − P1 0 0 0 0 0 0 0 ⎢ ⎜⎢ 0 −S 0 0 ⎥ 0 0 0 ⎥ ⎥ + PT ⎢ 0 ⎥⎟ ⎟ ⎡ ⎤T ⎜ ⎢ ⎣ ⎦ ⎣ ⎦ ⎟ ⎡ x(t) ⎤ ⎜ 0 0 0 0 G 0 −I 0 x(t) i ⎜ ⎟ p ⎢ ⎢ x(t − d ) ⎥ ⎜ ⎥ 0 0 0 P1 Ai Adi Bi −I ⎟ ⎜ ⎟ ⎢ x(t − d ) ⎥ ⎢ ⎥ ⎤T ⎡ wi ⎣ V (t) = ⎜ ⎟ ⎣ ⎦ u(t) u(t) ⎦ 0 0 0 0 ⎜ ⎟ i=1 ⎜ ⎟ ⎥ ⎢ z(t) z(t) 0 0 0 ⎥ ⎜+⎢ 0 ⎟ ⎝ ⎣ Gi 0 −I 0 ⎦ P ⎠ Ai Adi Bi −I ⎡ ⎤T ⎡ ⎤ x(t) x(t) p ⎢ x(t − d ) ⎥ ⎢ ⎥ ⎥ i ⎢ x(t − d ) ⎥ , = (15) wi ⎢ ⎣ u(t) ⎦ ⎣ u(t) ⎦ i=1 z(t) z(t)
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where
H.K. Lam, F.H.F. Leung / Fuzzy Sets and Systems 159 (2008) 990 – 1000
⎡
T T T S − P1 + AT i P7 + P7 Ai + Gi P4 + P4 Gi ⎢ P5T Gi + P8T Ai + AT P ⎢ 7 di i = ⎢ P6T Gi + P9T Ai − P4 + BT ⎣ i P7 T A −P P10 7 i
∗ T −S + AT P di 8 + P8 Adi T P9 Adi − P5 + BT i P8 T A −P P10 di 8
∗ ∗ T −P6T − P6 + BT i P9 + P9 Bi T B −P P10 i 9
∗ ∗ ∗
⎤ ⎥ ⎥ ⎥. ⎦
T −P P1 − P10 10
(16) The symbol ‘*’ denotes position. It can be seen from (15) that ⎡ ⎤⎞ ⎛ the transpose of the⎡element at⎤the corresponding 0 x(t) ⎜ ⎢ x(t − d ) ⎥ ⎢ 0 ⎥⎟ ⎥ ⎢ ⎥⎟ ⎢ if i < 0, V (t) 0 ⎜ ⎝equality holds when ⎣ u(t) ⎦ = ⎣ 0 ⎦⎠, which implies the asymptotic stability of the z(t) 0 discrete-time fuzzy-model-based control system with time delay. The analysis results are summarized in the following theorem. Theorem 1. The discrete-time fuzzy-model-based control system with time delay, formed by the nonlinear system in the form of (2) and the fuzzy controller of (6), is asymptotically stable if there exists predefined feedback gains of Gj ∈ m×n and matrices of P1 = P1T ∈ n×n , Pr ∈ n×n , r = 1, 2, 3, 7, 8, 10, Pk ∈ m×n , k = 4, 5, P6 ∈ m×m and P9 ∈ n×m ; i = 1, 2, . . . , p, S = ST ∈ n×n such that the following LMIs are satisfied: P1 > 0, S > 0, and i < 0,
i = 1, 2, . . . , p.
It should be noted that the feedback gains of Gj are needed to be determined prior to applying Theorem 1. In the following, the feedback gains are converted into decision variables. Hence, the values of the feedback gains can be ⎡ ⎤ ⎤ ⎡ x(t) X1 0 0 0 ⎢x(t − d )⎥ ⎢ X2 X 3 0 0 ⎥ ⎥ ⎥ ⎢ determined by using some convex programming techniques. Let v(t)=X−1 ⎢ ⎣ u(t) ⎦, X= ⎣ X4 X5 X6 0 ⎦ X7 X8 X9 X10 z(t) = P−1 ∈ (3n+m)×(3n+m) , X1 = X1T = P1−1 ∈ n×n , Xr ∈ n×n , r = 2, 3, 7, 8, 10, Xk ∈ m×n , k = 4, 5, X6 ∈ m×m and X9 ∈ n×m . To ease the development of the LMI stability conditions, we choose X2 = 0; otherwise, the resultant stability conditions will be in the form of bilinear matrix inequalities which cannot be simply solved by convex programming techniques. However, conservativeness of the stability analysis result will increase. From (15), we have, ⎡ ⎤ ⎡ ⎤ ⎞ ⎛ S − P1 0 0 0 0 0 0 0 ⎢ ⎜ T⎢ −S 0 0 ⎥ 0 0 0 ⎥ ⎥X + ⎢ 0 ⎥X⎟ ⎜X ⎢ 0 ⎟ ⎣ 0 ⎦ ⎣ ⎜ 0 0 0 Gi 0 −I 0 ⎦ ⎟ ⎜ ⎟ p ⎜ ⎟ 0 0 0 P1 Ai Adi Bi −I ⎟ T⎜ V (t) = wi v(t) ⎜ ⎟ v(t) ⎡ ⎤T ⎜ ⎟ 0 0 0 0 i=1 ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ 0 0 0 ⎥ ⎜ +XT ⎢ 0 ⎟ ⎝ ⎠ ⎣ Gi 0 −I 0 ⎦ Ai Adi Bi −I ⎤ ⎡ ⎤ ⎞ ⎛⎡ X1 SX1 − X1 0 0 0 0 0 0 0 ⎢ ⎜⎢ 0 0 0 ⎥ 0 −X3T SX3 0 0 ⎥ ⎥+⎢ 0 ⎥X⎟ ⎟ ⎜⎢ ⎦ ⎣ ⎣ ⎦ ⎟ ⎜ G 0 −I 0 0 0 0 0 i ⎟ ⎜ p ⎟ ⎜ Ai Adi Bi −I 0 0 0 0 ⎟ T⎜ = (17) wi v(t) ⎜ ⎟ v(t), ⎡ ⎤ T ⎟ ⎜ 0 0 0 0 i=1 ⎟ ⎜ ⎟ ⎜ 0 0 0 ⎥ ⎟ ⎜ +XT ⎢ ⎢ 0 ⎥ + s T P1 s ⎠ ⎝ ⎣ Gi 0 −I 0 ⎦ Ai Adi Bi −I
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995
where s = [X7 X8 X9 X10 ]. It can be seen from (17) that the existence of the term of X3T SX3 will lead to stability conditions not in terms of LMIs. To circumvent this problem, we denote M = MT = S−1 and consider the inequality of (X3 − S−1 )T S(X3 − S−1 ) 0. Expanding it, we obtain the following inequality: M − X3T − X3 − X3T SX3 . From (17) and (18), let Gi = Ni X1−1 where Ni ∈ m×n , i = 1, 2, . . . , p, we have, ⎤ ⎡ ⎤ ⎞ ⎛⎡ X1 SX1 − X1 0 0 0 0 0 0 0 ⎢ ⎜⎢ 0 0 0 ⎥ 0 M − X3T − X3 0 0 ⎥ ⎥+⎢ 0 ⎥X⎟ ⎜⎢ ⎟ ⎦ ⎣ ⎜⎣ Gi 0 −I 0 ⎦ ⎟ 0 0 0 0 ⎜ ⎟ p ⎜ ⎟ Ai Adi Bi −I 0 0 0 0 ⎟ v(t) ⎡ ⎤ wi v(t)T ⎜ V (t) T ⎜ ⎟ 0 0 0 0 ⎜ ⎟ i=1 ⎜ ⎟ ⎢ ⎥ 0 0 0 ⎥ ⎜ +XT ⎢ 0 ⎟ T + s P s 1 ⎝ ⎠ ⎣ Gi 0 −I 0 ⎦ Ai Adi Bi −I p T T T = v(t) wi Vi + s P1 s + r Sr v(t),
(18)
(19)
i=1
⎤ ∗ ∗ ∗ −X1 ⎥ ⎢ 0 M − X3T − X3 ∗ ∗ ⎥, and r = [X1 0 0 0]. It can where Vi = ⎢ T ⎦ ⎣ Ni − X4 −X5 −X6 − X6 ∗ T Ai X1 + Bi X4 − X7 Adi X3 + Bi X5 − X8 Bi X6 − X9 −X10 − X10 be seen from (19) that the discrete-time fuzzy-model-based control system with time delay is asymptotically stable if p T P s + rT Sr < 0. By Schur complement, it is equivalent to the following inequality: V + s w 1 i=1 i i ⎡ ⎤ p ∗ ∗ Vi ∗ ⎦ < 0. wi ⎣ s −X1 (20) r 0 −M i=1 ⎡ ⎤ Vi ∗ ∗ ∗ ⎦ < 0, i = 1, 2, . . . , p, implies V (t) < 0 which implies the From (20), it can be seen that ⎣ s −X1 r 0 −M asymptotic stability of the discrete-time fuzzy-model-based control system with time delay. The stability analysis result is summarized in the following theorem. ⎡
Theorem 2. The discrete-time fuzzy-model-based control system with time delay, formed by the nonlinear system in the form of (2) and the fuzzy controller of (6), is asymptotically stable if there exists M ∈ n×n , Ni ∈ m×n , X1 = X1T ∈ n×n , Xr ∈ n×n , r = 2, 3, 7, 8, 10, Xk ∈ m×n , k = 4, 5, X6 ∈ m×m and X9 ∈ n×m such that the following LMIs are satisfied: ⎤ ⎡ ∗ ∗ Vi ∗ ⎦ < 0, i = 1, 2, . . . , p, X1 > 0, M > 0, and ⎣ s −X1 r 0 −M where
⎤ −X1 ∗ ∗ ∗ ⎥ ⎢ 0 M − X3T − X3 ∗ ∗ ⎥, Vi = ⎢ T ⎦ ⎣ Ni − X4 −X5 −X6 − X6 ∗ T −X Ai X1 + Bi X4 − X7 Adi X3 + Bi X5 − X8 Bi X6 − X9 −X10 10 s = [X7 X8 X9 X10 ], r = [X1 0 0 0] ⎡
and the feedback gains are designed as Gi = Ni X1−1 , i = 1, 2, . . . , p.
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Remark 1. It should be noted that the stability analysis result summarized in Theorem 2 is valid if there exists the inverse of X. Referring to the stability conditions in Theorem 2, if there exists a solution, we have X1 > 0, X3T +X3 > 0, T + X > 0, which are the sufficient conditions that X is a non-singular matrix, i.e. its inverse X6T + X6 > 0 and X10 10 exists. Remark 2. In [7–9,13–15,17,18], due to the multiplication property of the fuzzy-model-based control systems, it can be shown that the number of stability conditions is p(p − 1)/2. In this paper, referring to Theorems 1 and 2, the number of stability conditions is reduced to p only. However, it is due to the introduction of some free matrices, i.e. X3 to X10 ; the computational advantage may not be obvious for a small value of p. Under the case of a large value of p, the influence of introducing free matrices to the computational demand becomes insignificant. Hence, the number of rules plays a crucial role to the computational demand. We conclude that by using the proposed approach, the computational demand on finding the solution can be reduced. The computational advantage is more obvious for fuzzy systems with large values of p and higher system and input matrix dimensions. It will be shown by a simulation example in the following section that the proposed LMI stability conditions are less computationally expensive. 4. Simulation examples Simulation examples are given in this section to illustrate the effectiveness of the proposed approach. 4.1. Simulation Example 1 In this example, the computational demand on solving the solution to the LMI stability conditions is investigated. A discrete-time nonlinear plant with time delay represented by the fuzzy model with the following p(> 3) rules is considered. Rule i : IF x1 (t) is M1i (21) THEN x(t + 1) = Ai x(t) + Adi x(t − d ), i = 1, 2, . . . , p; t = 0, 1, . . . , ∞, 0.1 0.5 0.1 0.25 0.1 0.75 0.15 0 0.25 0 where A1 = , A2 = , A3 = , Ad1 = , Ad2 = , 0.51 −0.1 0.15 −0.5 0.25 −0.8 0.01 0 0.05 0 0.19 0 Ad3 = . The subsystems 4 to p are defined as follows: 0.09 0 Ak =
d1k A1 + d2k A2 + d3k A3 , d1k + d2k + d3k
Adk =
k = 4, 5, . . . , p,
d1k Ad1 + d2k Ad2 + d3k Ad3 , d1k + d2k + d3k
k = 4, 5, . . . , p,
(22)
(23)
where d1k , d2k and d3k , k = 4, 5, . . . , p, are randomly generated scalars in the range of 0–1. The dynamics of the discrete-time nonlinear plant with time delay are defined as follows: x(t + 1) =
p
wi (x1 (t))(Ai x(t) + Adi x(t − d )).
(24)
i=1
It can be checked with the stability conditions in Theorem 1 or [1] that the time-delay system of (24) is open-loop stable for p = 3. With the help of Matlab LMI toolbox, the proposed stability conditions in Theorem 1 are employed to check for the system stability with different values of p. To investigate the computational demand, the averaged time (in seconds) in 10 runs for various values of p is recorded and tabulated in Table 1. The simulation is carried out in a
H.K. Lam, F.H.F. Leung / Fuzzy Sets and Systems 159 (2008) 990 – 1000
997
Table 1 Average time (in seconds) required to find the solutions of stability conditions in Theorem 1 and [1] p
Average time (in seconds)
3 5 10 20 50 100 200 300 400 500
Stability conditions in Theorem 1
Stability conditions in [1]
0.0894 0.1517 0.3490 0.8980 3.1560 9.2003 26.5050 50.8900 78.7030 No feasible solution
0.0692 0.1720 1.0032 7.5078 80.1823 No feasible solution No feasible solution No feasible solution No feasible solution No feasible solution
personal computer with 3 GHz CPU and 2 GB RAM. Referring to Table 1, it can be seen that when the value of p is small, the computer takes longer time to find the solution when the proposed stability conditions in Theorem 1 are used instead of the stability conditions in [1]. However, when p 5, the solution to the stability conditions in Theorem 1 can be found in a shorter time comparatively. It can be seen from Table 1 that the computational advantage on finding the solution is more obvious for a large value of p. Furthermore, it is observed that feasible solutions cannot be obtained (LMI solver terminates owing to too slow processes) when the stability conditions in [1] are used for p 100, while those in Theorem 1 are still solvable. From this example, it can be shown that the proposed stability conditions are less computationally expensive. 4.2. Simulation Example 2 In this example, the stability region based on the stability conditions in Theorem 1 is investigated. A discrete-time nonlinear plant with time delay represented by the fuzzy model with the following three rules is considered. Rule i : IF x1 (t) is M1i (25) THEN x(t + 1) = Ai x(t) + Adi x(t − d ) + Bi u(t), t = 0, 1, . . . , ∞, i = 1, 2, 3, 1 0.5 0.1 0.25 a 0.75 0.1 0 0.25 0 where A1 = , A2 = , A3 = ; Ad1 = , Ad2 = , 0.51 −0.1 0.15 −0.5 0.25 −0.8 −0.25 0 0.15 0 0.19 0 0 1 b Ad3 = ; B1 = , B2 = and B3 = where 0 a 1.6 and 0 b 1.8. The inferred 0.06 0 1 −0.5 0.5 dynamics of the discrete-time nonlinear plant with time delay are defined as follows: x(t + 1) =
3
wi (x1 (t))(Ai x(t) + Adi x(t − d ) + Bi u(t)).
(26)
i=1
A three-rule fuzzy controller is employed to control the discrete-time nonlinear plant with time delay. The control action of the fuzzy controller is described by the following rules. Rule j : IF x1 (t) is M1i THEN u(t) = Gj x(t),
t = 0, 1, . . . , ∞, j = 1, 2, 3.
(27)
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1.8 1.6 1.4 1.2
b
1 0.8 0.6 0.4 0.2 0
0
0.2
0.4
0.6
0.8 a
1
1.2
1.4
1.6
Fig. 1. Stability regions based on the stability conditions in Theorem 1 (indicated by “◦”) and [1] (indicated by “×”).
The inferred fuzzy controller is defined as follows: u(t) =
3
wi (x1 (t))Gi x(t).
(28)
i=1
The feedback gains of Gi are designed such that the eigenvalues of Ai + Bi Gi , i = 1, 2, 3, are all placed at −5. On applying the stability conditions in Theorem 1, with the aid of the MATLAB LMI toolbox, the stability region (indicated by “◦”) for 0 a 1.6 and 0 b 1.8 is shown in Fig. 1. For comparison purpose, the stability region (indicated by “×”) on applying the stability conditions in [1] is also shown in Fig. 1. It can be seen that the proposed stability conditions in Theorem 1 offer a larger stability region. 4.3. Simulation Example 3 In this example, the stability conditions in Theorem 2 are employed to obtain the feedback gains of the fuzzy controller to stabilize a time-delayed nonlinear plant. Consider the time-delay nonlinear plant represented by the fuzzy model 2 2 of (26) with a = 1.2 and b = 1.6, where the membership functions are chosen as M1 (x1 (t)) = e−(x1 (t)+5) /2×2 , 1
M1 (x1 (t)) = e−x1 (t) /2×1.5 and M1 (x1 (t)) = e−(x1 (t)−5) /2×2 . 2 3 The fuzzy controller of (28) is employed to control the time-delayed nonlinear plant. By applying the stability conditions in Theorem 2, with the aid of the MATLAB LMI toolbox, the feedback gains are obtained as G1 = [−0.4793 − 0.0436], G2 = [−0.2992 − 0.1695] and G3 = [−0.7459 − 0.1120]. Fig. 2 shows the system state responses and the control signals of the discrete-time fuzzy-model-based control system with time delay under the initial system state condition of x(t) = [2 −1]T for t ∈ [−d 0]. Referring to Fig. 2, it can be seen that the discretetime nonlinear plant with d = 5 and 20 can be stabilized by the fuzzy controller. If the delay-dependent approaches in [6,3,4] are considered, it can be shown that no solution can be obtained when the value of d is sufficiently large. However, the proposed stability conditions guarantee the asymptotic stability of the fuzzy-model-based control system for any value of d . 2
2
2
2
H.K. Lam, F.H.F. Leung / Fuzzy Sets and Systems 159 (2008) 990 – 1000
2
1.5
1.5
1 0.5 x2(t)
x1(t)
1 0.5
0
0
-0.5
-0.5 -1
999
0
10
20
30
40 50 60 Samples
70
80
90 100
-1
0
10
20
30
40 50 60 Samples
70
80
90 100
0.4 0.2 0
u(t)
-0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
0
10
20
30
40 50 60 Samples
70
80
90 100
Fig. 2. System state responses and control signals of the discrete-time fuzzy-model-based control system with time-delay of d = 5 (solid lines) and d = 20 (dotted lines).
5. Conclusion The stability of discrete-time fuzzy-model-based control systems with time delay has been investigated. LMI-based delay-independent stability conditions have been derived using the Lyapunov-based approach. The system stability and feedback gains can be determined with the help of some convex programming techniques. Furthermore, with a particular system formulation, the number of the LMI-based stability conditions has been reduced to alleviate the computational demand for finding the solution. Simulation examples have been given to illustrate the design procedure and the effectiveness of the proposed approach.
Acknowledgements The work described in this paper was substantially supported by Grants from King’s College, London and from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 5218/06E).
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