STABILIZATION FOR A CLASS OF UNCERTAIN MULTI-TIME DELAYS SYSTEM USING SLIDING MODE CONTROLLER. Received April 2010; revised August 2010

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1 International Journal of Innovative Computing, Information and Control ICIC International c 2011 ISSN Volume 7, Number 7(B), July 2011 pp STABILIZATION FOR A CLASS OF UNCERTAIN MULTI-TIME DELAYS SYSTEM USING SLIDING MODE CONTROLLER Shaocheng Qu 1,2, Zhiqiu Lei 1, Quanming Zhu 2 and Hassan Nouri 2 1 Department of Information and Technology Central China Normal University No. 152, Luoyu Road, Wuhan , P. R. China qushaocheng@mail.ccnu.edu.cn; leizhiqiu@mails.ccnu.edu.cn 2 Bristol Institute of Technology University of the West of England, Frenchay Campus Coldharbour Lane, Bristol, BS16 1QY, UK { quan.zhu; Hassan.Nouri }@uwe.ac.uk Received April 2010; revised August 2010 Abstract. The stabilization for uncertain multi-time delays system is studied by using sliding mode control method. The uncertain system is assumed to have unmatched parameter uncertainties as well as matched input uncertainties. A virtual feedback strategy is presented to design the sliding surfaces, and then a relaxed delay-independent sufficient condition of design for sliding mode surfaces is proposed. A sliding mode controller is developed, which can ensure convergence of the system trajectory to the sliding surfaces. The overall asymptotical stability of the closed-loop system is guaranteed. Simulations illustrate the effectiveness of the proposed methodology. Keywords: Time delay system, Sliding mode control, Robust control, LMI 1. Introduction. Time delays frequently occur in many practical systems, such as chemical process, nuclear reactor, manual control, long transmission communication 1. Its existence usually leads to system s poor performance or instability. Hence, stabilization of time delay system has received considerable attention in the past several decades One of the solutions for uncertain time delay system is to use sliding mode control strategy (SMC). SMC has been proved to be an effective robust control strategy for incompletely modelled and uncertain system. An SMC system has various attractive features such as fast response, good transient performance, and robustness with respect to matched parameter uncertainties and external disturbances on the sliding plane. Owing to its robustness and ability to handle nonlinear system, SMC has found wide application to automotive systems, electrical motor control 4. Recently, sliding mode control strategy (SMC) of time delay system has received increasing attentions Many efforts have been made to obtain delay-independent conditions 5-7. Usually, it is difficult to obtain delay-independent conditions of design of controller for many practical systems. In fact, delay-independent conditions of design of controller for some systems could not be obtained. At the same time, delay-independent conditions are too strict for many systems and considered as more conservative than delay-dependent conditions. So more and more attentions are focused on how to obtain delay-dependent conditions of design of sliding mode controller On the other hand, the delay-independent methods, although being conservation, are sometimes more suitable for some practical applications 17. Generally speaking, time 4195

2 4196 S. QU, Z. LEI, Q. ZHU AND H. NOURI delays in some practical system usually contain more than one time delay parameter, and differences among the time delays are considerably large, such as chemical react systems and transmission system. So the delay-dependent conditions are not suitable for those systems. From industry application views, the relaxed delay-independent condition of design controller is more expected for some practical applications owing to adaptable for any long time delay. In this paper, we are concerned about the design of SMC for uncertain multi-time delay system. The uncertain system is assumed to have unmatched parameter uncertainties as well as matched input uncertainty and external disturbance. The sliding surface is defined as a linear function of current system state to avoid the complicated non-singular transformation. More importantly, A virtual feedback strategy is proposed to design the sliding surfaces, A relaxed time-independent sufficient condition of design of the sliding surface is derived in term of LMIs, which can ensure the asymptotical stability of the closed-loop system. Throughout, R n denotes the real n-dimensional linear vector space. denotes the Euclidea norm of a vector or the spectral norm of a matrix. For a real symmetric matrix, M > 0 (< 0) means that M is positively definite (negatively definite). I is used to represent an identity matrix of appropriate dimensions. In addition, if not explicitly claimed, matrices are assumed to have compatible dimensions. 2. Problem Statement and Preliminaries. Consider uncertain time delay system of the form ẋ(t) = A 0 (t)x(t) + A i (t)x(t h i ) + B(t)(u(t) + f(x, t)) (1) x(t) = ϕ(t), t h 0, h = max(h i ) (2) x R n is the system state and u R m is the control input; the ϕ(t) is a continuously differentiable initial function; and A i (t) is appropriate dimension matrix functions with time-varying uncertainties, that is, A i (t) = A i + A i (t), i = 0, 1,..., n (3) A i is known real constant matrices, and A i (t) is unknown time-varying system parameter uncertainty. We assume that the uncertainties are norm-bounded and can be described as A i (t) = D i F i (t)e i (4) D i and E i are known real constant matrices, and F i (t) is an unknown real timevarying matrices with Lebesgue measurable elements bounded by Moreover, uncertain input matrix B(t) can be expressed as F T i (t)f i (t) I, t (5) B(t) = B + B(t) (6) uncertainty B(t) are assumed to be matched and bounded 6,9,10, i.e, B(t) = B B(t), B(t) δb < 1 (7) Here, we assume that external disturbance f(x, t) is the lumped uncertainties given as f(x, t) δ f x(t) (8) Usually, matrix B is assumed to have full column rank, and (A o, B) is stablisable, i.e., there exist matrix K R m n such that A 0 BK is stable.

3 STABILIZATION FOR UNCERTAIN MULTI-TIME DELAYS SYSTEM USING SMC 4197 Without loss of generality, define the following sliding surface S(t) = ΓB T P x(t) = 0 (9) P R n n is a positive definite matrix to be designed. Γ R m m is some nonsingular matrix. For simplicity, Γ is chosen as the identity matrix in this work. The objective in this paper is to design the parameter P of the sliding surface (9) and a reaching motion control law u(t) such that (a) System (1) with the control law u(t) satisfies the reachability of sliding mode system. i.e., system trajectories will reach the sliding surface (9) in finite time and subsequently remain there; and (b) System (1) confined to the sliding surface, which usually called the sliding mode reduced system, is robustly stable. 3. Main Results. This section will present the main resign on designing reaching control law and a sliding surface. At the beginning of this section, we recall the following lemma which will be used in the proof of our main results. Lemma Given D, E and F (t) be real matrices of appropriate dimensions, and F (t) satisfying F T (t)f (t) I, then, for any scalar ε, the following inequality holds: DF (t)e + E T F T (t)d T εdd T + ε 1 E T E Now, we are ready to present our first result in this paper. Theorem 3.1. The trajectory of the closed-loop uncertain system (1) can be driven onto the sliding surface (9) in finite time and subsequently remain there by the sliding mode control law and ρ(x, t) = 1 1 δ B u eq (t) = ( B T P B ) 1 u(t) = u eq (t) + u r (t) (10) B T P A 0 x(t) + B T P A i x(t h i ), u r (t) = ( B T P B ) 1 ρ(x, t) + γ sgn(s) +δ B ( B T P. A 0 x(t) + B T P D 0. E 0 x(t) + B T P D i. E i x(t h i ) ) B T P. A i x(t h i ) + δ f (1 + δ B ) B T P B. x(t) here, the small constant γ > 0, and matrix P > 0 will be chosen later. Proof: Choose a Lyapunov function Differentiating V 1 (x, t) for all S 0, we can obtain (11) V 1 (x, t) = 0.5S T S (12) V 1 (x, t) = S T Ṡ = S T B T P ẋ(t) (13)

4 4198 S. QU, Z. LEI, Q. ZHU AND H. NOURI Substituting (1) to (13) obtains V 1 (x, t) = S T B T P ẋ(t) = S T B T P (A 0 + A 0 (t)) x(t) + + (B + B(t))(u(t) + f(x, t)) (A i + A i (t)) x(t h i ) (14) Substituting control (10) to (14) yields V 1 (x, t) = S T B T P A 0 x(t) + A i (t)x(t h i ) + S T B T P B ( B T P B ) 1 (ρ(x, t) + γ) sgn(s) + S T B T P Bf(x, t) + S T B T P B B(x)u(t) + f(x, t) = S T (ρ(x, t) + γ) + S T B T P Bf(x, t) + S T B T P A 0 x(t) + A i (t)x(t h i ) + S T B T P B B(x)u(t) + f(x, t) Noting the last two terms of (15) and considering (4-8) yield, respectively, S T B T P A 0 x(t) + A i (t)x(t h i ) and = S T B T P D 0 F 0 (t)e 0 x(t) + 2 D i F i (t)e i x(t h i ) S T B T P D 0 E0 x(t) + B T P D i. Ei x(t h i ) S T B T P B B(x)u(t) + f(x, t) δ B S T B T P A0 x(t) + (15) (16) B T P Ai x(t h i ) + ρ(x, t) + γ + δ f B T P B (17) x(t) Substituting (16) and (17) into (15) results in V 1 (x, t) γ(1 δ B ) S T (18) Noting that (7), we can obtain V 1 (x, t) < 0 from (18) for all S 0. So the reachability of sliding mode motion is satisfied. Therefore, the system trajectories under the proposed controller will reach the sliding surface (9) in finite time and subsequently remain there. The proof is completed. The second step is to design the sliding surface (9) such that system (1) confined to the sliding surface (9), i.e., the sliding mode reduced system, is robustly stable. Now, we are ready to present a virtual feedback control idea to help design the relaxed parameter P in the sliding surface (9). Consider the virtual feedback control in following form from (10) u(t) = Kx(t) + ū(t) (19)

5 STABILIZATION FOR UNCERTAIN MULTI-TIME DELAYS SYSTEM USING SMC 4199 ū(t) = u(t) + Kx(t) (20) So we can obtain the following conclusion. Theorem 3.2. Under the assumption (3-8), uncertain system (1-2) confined to the sliding surface (9) is asymptotically stable by the SMC law (10-11) if there exist matrixes X > 0, Q 1 > 0 Q n > 0, matrix L and scalars ε 0 > 0, ε 1 > 0 ε n > 0 such that the LMI (21) are feasible. Moreover, the parameter of the sliding surface (9) is chosen by P = X 1. M11 A 1 X A n X 0 0 XE0 T Q XE1 T Q n 0 0 XEn T 0 < 0 (21) ε 1 I ε n I 0 ε 0 I denotes the entries that are readily inferred by symmetry of a symmetric matrix, and M11 = A 0 X + XA T 0 BL L T B T + n ε i D i Di T + n Q i. i=0 Proof: Define a Lyapunov-Krasovsky function candidate as follows: t V 2 (x, t) = x T (t)p x(t) + x T (s)r i x(s)ds (22) t h i matrices P and R i will be chosen by LMIs (21). Obviously, V (x, t) is positive and bounded. Differentiating V (x, t) yields V 2 (x, t) = 2x T (t)p ẋ(t) + x T (t)r i x(t) x T (t h i )R i x(t h i ) (23) Substituting (1) into system (23) obtains V 2 (x, t) = x T (t)r i x(t) x T (t h i )R i x(t h i ) + 2x T (t)p Ā 0 x(t) + A 0 (t)x(t) + (A i + A i (t))x(x h i ) + 2x T (t)p Bu(t) + Kx(t) + f(x, t) + 2x T (t)p B(t)u(t) + f(x, t) Ā 0 = A BK (25) Noting that definition (7), we obtain from (24) V 2 (x, t) = x T (t)r i x(t) x T (t h i )R i x(t h i ) + 2x T (t)p (Ā0 + A 0 (t) ) x(t) + (A i + A i (t))x(x h i ) + 2x T (t)p B Kx(t) + (I + B(t))(u(t) + f(x, t)) (24) (26)

6 4200 S. QU, Z. LEI, Q. ZHU AND H. NOURI Once on the sliding surface, since S(t) = B T P x(t) = 0, so Equation (26) can be reduced to the following form V 2 (x, t) = x T (t)r i x(t) x T (t h i )R i x(t h i ) + 2x T (t)p = ξ T (t)w ξ(t) (Ā0 + A 0 (t) ) x(t) + (A i + A i (t)) x(x h i ) (27) and W 11 P (A 1 + A 1 (t)) P (A n + A n (t)) R W = (28) R n with ξ T (t) = x(t) x(t h 1 ) x(t h n ) and W 11 = P ( Ā 0 + A 0 (t) ) + ( Ā 0 + A 0 (t) ) T P + R i. Obviously, if matrix W < 0, then V 1 (x, t) < 0. Pre-multiplying and post-multiplying LMIs W by Φ = diag {P 1, P 1,, P 1 }, and defining X = P 1, Q i = XR i X obtains ΦW Φ = Y + Y 0 + Y Y n (29) Ā 0 X + XĀT 0 + n Q i A 1 X A n X Y = Q , Q n A 0 (t)x + X A T 0 (t) Y 0 = Y 1 = A 1 (t)x A n (t)x , Y n = By Lemma 3.1, one can obtain for scalar ε 0 > 0, Y 0 = I 0 0 T D 0 F 0 (t)e 0 XI I 0 0 T XE0 T F0 T (t)d0 T I 0 0 ε 0 D 0 D0 T + ε 1 0 XE0 T E 0 X (30)

7 STABILIZATION FOR UNCERTAIN MULTI-TIME DELAYS SYSTEM USING SMC 4201 Similarly, yields for scalar ε 1 > 0,, ε n > 0, respectively ε 1 D 1 D T ε Y XE1 T E 1 X (31)... ε n D n Dn T Y n... 0 ε 1 n XEn T E n X Substituting (30)-(32) into (29) yields (32) ΦW Φ Ω (33) Ω11 A 1 X 0 A n X Q Ω = 1 + ε 1 1 XE1 T E 1 X Q n + ε 1 n XEn T E n X Ω11 = Ā0X + XĀT 0 + ε 1 0 XE0 T E 0 X + Q i + ε i D i Di T Using the Schur complement theorem, we know Ω < 0 is equivalent to the above LMIs(21). Since Ω < 0, so ΦW Φ < 0 according to (33), then V 2 (x, t) < 0 from (26). Therefore, the proof is thus complete. From Theorem 3.1, it is seen that the reachability of sliding mode with the proposed controller is satisfied, so the system trajectories will approach the sliding surface it in finite time and subsequently remain there. From Theorem 3.2, it is guaranteed that the reduced sliding mode system confined the sliding surface are robust stable. Hence, we can draw the following conclusion. Theorem 3.3. Under the assumption (3-8), the overall the closed-loop system (1-2) with the proposed control (10) is stable if there exist X > 0, Q 1 > 0,, Q n > 0, matrix L and scalars ε 0 > 0, ε 1 > 0,, ε n > 0 such that the LMI (21) are feasible. Remark 3.1. It should be indicated that the feedback control Kx(t) is not indirectly existed according to the SMC law (10-11) and (19-20). The main goal is to help design the relaxed parameter P = X 1 of the sliding surface (9). In following numerical example, we will exhibit that the proposed virtual feedback leads to the more relaxed delay-independent condition. 4. Numerical Example. Situation 1: In order to compare with the previous works, consider the uncertain time delay system proposed by 8 (there, uncertain system only contains external disturbance). A 0 = ẋ(t) = A 0 x(t) + A 1 x(t h 1 ) + Bu(t) + f(t) (34) , A 1 = i= , B = 0 0 1

8 4202 S. QU, Z. LEI, Q. ZHU AND H. NOURI By solving the LMI (21) of Theorem 3.2, we can get the following parameter: X = , Q = L = , ε 0 = , ε 1 = and P = X 1 = It shows that the time delay-independent condition of design of the sliding mode control system is obtained. This implies the proposed sliding mode control is suitable for any long time delay. Remark 4.1. F. Gouaisbaut and J. P. Richard proved that system (34) is asymptotically stable for time delay h in 8. On the other hand, we can obtain the corresponding LMIs conditions similar to LMIs (21) if the proposed virtual feedback controller is not applied, i.e., let BL L T B T = 0 in LMIs (21), but the corresponding LMIs are not feasible. Situation 2: Similarly, we consider the uncertain multi-time delay system in form of (1) proposed by 6, which was general cited by many researchers. A 0 = , A 1 = , B = 0 1, D 0 = D 1 = , E 0 = E 1 = , f(x, t) = sin(2t)x 2 (t) 0, B(x, t) = F 0 (t) = sin(t)i 3 3, F 1 (t) = cos(t)i cos(3t) 0 According to Theorem 3.2, we can obtain P = X 1 = The simulation results are shown in Figures 1-3. The initial conditions for system (1) for t h 1 0 (time delay h 1 = 1) are set as x(t) = T. Obviously, the system in the presence of uncertainties and external disturbances is successfully stabilized. By comparisons with Figure 1 in 8, it is easy to find that system dynamics with the proposed controller are improved, and amplitude of controller is also smaller. Situation 3: In order to demonstrate that the proposed controller is suitable for any long time delay. Based on Situation 2, consider long time delay h 1 = 3 (At Situation 2, time delay h 1 = 1). The simulation results are shown in Figures 4-6. From Figure 4-6, long time delay which occurs at the moment of the 3s bring out deterioration of system response. However, the uncertain long time delay system is also efficiently stabilized.,

9 STABILIZATION FOR UNCERTAIN MULTI-TIME DELAYS SYSTEM USING SMC 4203 Figure 1. System response under the proposed control Figure 2. The proposed control Figure 3. System sliding mode motion

10 4204 S. QU, Z. LEI, Q. ZHU AND H. NOURI Figure 4. System response with long time delay Figure 5. The control for long time delay Figure 6. System sliding mode motion for long time delay

11 STABILIZATION FOR UNCERTAIN MULTI-TIME DELAYS SYSTEM USING SMC Conclusions. In this paper, problem of the design of SMC for uncertain time delay system are considered. A sliding mode controller with time delay-independent condition is introduced according to sliding mode reachability. More importantly, a virtual feedback strategy is proposed to design the sliding surfaces, and a relaxed time delay-independent condition of design of the sliding mode controller is derived, which can guarantee the asymptotically stability of the closed-loop system. Numerical example is given to demonstrate the effectiveness of the proposed techniques. Acknowledgment. This work is partially supported by the National Natural Science Foundation of China under Grant /F and the self-determined research funds of CCNU from the colleges basic research and operation of MOE. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation. REFERENCES 1 J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, vol.39, pp , E. Fridman and U. Shaked, An improved delay dependent H filtering of linear neutral systems, IEEE Trans. on Signal Processing, vol.52, no.3, pp , S. Xu and J. Lam, Improved delay-dependent stability criteria for time-delay systems, IEEE Trans. on Automatic Control, vol.50, no.3, pp , V. I. Utkin, J. Guldner and J. Shi, Sliding Mode Control in Electromechanical Systems, Taylor & Francs Press, K. S. Kim, Y. Park and S. H. Oh, Designing of robust sliding surfaces for parametric uncertain systems: A Riccati approach, Automatica, vol.36, pp , X. Li and R. A. Decarlo, Robust sliding mode control of uncertain time delay systems, International Journal of Control, vol.76, no.13, pp , S. Qu and Y. Wang, Robust control of uncertain time delay system: A novel sliding mode control design via LMI, Journal of System Engineering and Electronics, vol.17, no.3, pp , F. Gouaisbaut, M. Darnbrine and J. P. Richard, Robust control of delay systems: A sliding mode control design via LMI, Systems & Control Letters, vol.46, pp , Y. Xia and Y. Jia, Robust sliding mode control of uncertain time-delay systems: An LMI approach, IEEE Trans. on Automatic Control, vol.48, no.6, pp , Y. Xia, G. Liu, S. Peng and D. Rees, Robust delay-dependent sliding mode control for uncertain time-delays, International Journal of Robust Nonlinear Control, vol.18, pp , F. Castanos and L. Fridman, Analysis and design of integral sliding manifolds for systems with unmatched perturbations, IEEE Trans. on Automatic Control, vol.51, no.5, pp , X.-G. Yan and S. K. Spurgeon, Christopher edwards, global stabilisation for a class of nonlinear time-delay systems based on dynamical output feedback sliding mode control, International Journal of Control, vol.82, no.12, pp , H. H. Choi, Adaptive controller design for uncertain fuzzy systems using variable structure control approach, Automatica, vol.45, pp , J. Lian, J. Zhao and G. M. Dimirovski, Model reference adaptive integral sliding mode control for switched delay systems, International Journal of Innovative Computing, Information and Control, vol.4, no.8, pp , H. Xing, D. Li and X. Zhong, Sliding mode control for a class of parabolic uncertain distributed parameter systems with time-varying delays, International Journal of Innovative Computing, Information and Control, vol.5, no.9, pp , Y. Yao, K. Zheng, K. Ma and T. Shen, Sliding mode control design with a class of piecewise smooth switching surfaces, International Journal of Modelling, Identification and Control, vol.7, no.3, pp , L. Dugard and E. I. Vettiest, Stability and Control of Time-Delay Systems, Springer, London, 1998.