Automatica. Sliding mode control in the presence of input delay: A singular perturbation approach. X. Han a,1, E. Fridman b, S.K.

Automatica ( ) Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Sliding mode control in the presence of input delay: A singular perturbation approach X. Han a,1, E. Fridman b, S.K. Spurgeon a a School of Engineering and Digital Arts, Kent University, Canterbury, Kent, CT 7NZ, United Kingdom b School of Electrical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel a r t i c l e i n f o a b s t r a c t Article history: Received 5 February 11 Received in revised form 1 December 11 Accepted March 1 Available online xxxx Keywords: Sliding mode control Input delay Singularly perturbed system Ultimate bound Sliding Mode Control (SMC) in the presence of small, unavoidable input delay as may be present in controller implementation is studied. Linear systems with bounded matched disturbances and uncertain system matrices are considered, where input delay in the SMC will produce oscillations or potentially even unbounded solutions. Without a priori knowledge of the bounds on the state-dependent terms as required by existing methods, the design objective is to achieve ultimate boundedness of the closed-loop system with a bound proportional to the delay and disturbance bounds. This is a non-trivial problem because the relay gain depends on the state bound, whereas the latter bound depends on the relay gain. A controller with linear gain proportional to the scalar 1 is proposed, which for small enough > produces a closed-loop singularly perturbed system and yields the desired ultimate bound. A constructive Linear Matrix Inequality (LMI)-based solution for evaluation of both the design parameters and the ultimate bound is derived. The superiority of the proposed control over existing methodologies that ignore input delay within the design is demonstrated through an example. Crown Copyright 1 Published by Elsevier Ltd. All rights reserved. 1. Introduction Sliding Mode Control is well known for its invariance properties and has received a great deal of attention in the area of robust control. SMC can provide asymptotic stability in the presence of matched uncertainties and disturbances. However, in practical control systems, ideal sliding motion cannot usually be achieved due to model imperfections, time delays etc. The combination of delay phenomenon with relay actuators induces oscillations of finite frequency around the sliding surface and even instability (Fridman, 1997; Fridman, Fridman, & Shustin, 1993; Levaggi & Punta, 6). The degree to which the robustness of this SMC design paradigm can translate into systems with input delay is thus of considerable interest and an open research question. Many results are available which apply various control methods to time-delay systems. For example, adaptive control (Wen, Soh, & Zhang, ; Zhou, Wen, & Wang, 9), finite-time stabilization This work was partially supported by Israel Science Foundation (grant no 754/1). The material in this paper was partially presented at the 8th IFAC Symposium on Nonlinear Control Systems (NOLCOS 1), September, 1 3, 1, Bologna, Italy. This paper was recommended for publication in revised form by Associate Editor Antonio Loria under the direction of Editor Andrew R. Teel. E-mail addresses: hxr87@hotmail.com (X. Han), emilia@eng.tau.ac.il (E. Fridman), S.K.Spurgeon@kent.ac.uk (S.K. Spurgeon). 1 Tel.: 44 774819579; fax: 44 17 45684. (Karafyllis, 6), to name a few. Delays are unavoidable in the implementation of any feedback loop. Sampled-data control can be considered as control with a delayed input (Fridman, Seuret, & Richard, 4). When control engineers approach SMC, the choice of sampling rate is an immediate, and extremely critical design decision (Utkin, 199). The existing work on sampled-data SMC transforms the system to discrete-time. However, this approach becomes complicated for uncertain or state-delay systems. While the study of SMC in the presence of state delay has been ongoing (Chou & Cheng, 3; Shyu, Liu, & Hsu, 5), results on the effect of input delay in SMC are scarce. It was shown in Akian, Bliman, and Sorine () and Fridman et al. (1993); Fridman, Fridman, and Shustin () that even in the simplest one dimensional delayed relay control system, only oscillatory solutions can occur. Despite the oscillatory performance, relay delay control has advantages over a linear delayed controller in keeping an inverted pendulum upright (Sieber, 6). A simple example in Gouaisbaut, Perruquetti, and Richard () pointed out behavioral changes (bifurcations) arising when designing a controller without taking the input delay into consideration. This work motivates the study of specific SMC design methods for systems with input delays. In the existing results (Fridman et al., ; Gouaisbaut et al., ), an a priori constant bound is assumed on the state-dependent terms of the system, which is restrictive. The relay gain is chosen to be greater than this bound. In Fridman, Strygin, and Polyakov (4) for linear 5-198/$ see front matter Crown Copyright 1 Published by Elsevier Ltd. All rights reserved. doi:1.116/j.automatica.1.6.16

X. Han et al. / Automatica ( ) time-invariant systems with bounded nonlinear uncertainties, relay delay control was designed to achieve practical and adaptive stabilization provided the signs of the appropriately transformed states are available for measurement. For the case of known and constant input delay, the predictorbased SMC was designed in Roh and Oh (1999). Stability was achieved without any restriction on the time delay and spectral properties of the open-loop system. However, it was pointed out in Nguang (1) that the method cannot compensate for matched uncertainties. A summary of some contributions to the field of SMC with relay delay was provided in Richard (3). A frequency domain method of analysis was given in Boiko (9). Recently, sampled-data high gain output feedback SMC for linear systems with matched disturbances has been designed via discretization and singular perturbation analysis (Nguyen, Su, & Gajic, 1). Note that discretization of systems with uncertain matrices may lead to complicated conditions. Also, there may be additional difficulties in the presence of additional input/output delay. In the present paper, output-feedback SMC for linear systems with bounded disturbances and polytopic type uncertainties is considered under uncertain time-varying input delays. The design objective is to achieve ultimate boundedness of the closed-loop system with a bound proportional to the size of the delay and the disturbance. For small enough delay such a controller should have advantages over a corresponding linear controller, because the linear control will produce a bound proportional to the disturbance only. The main contribution is a general framework for SMC in the presence of input delay without any a priori knowledge of the bounds on the system states. The following design difficulty arises, which does not appear in the absence of input delay: the relay gain depends on the ultimate bound on the state, whereas the latter bound depends on the relay gain. To overcome this difficulty, a sliding mode controller is designed with a linear gain proportional to the scalar 1, which for small enough > produces a closedloop singularly perturbed system and which allows the desired ultimate bound to be achieved for the closed-loop system. The design process seeks to enlarge avoiding a high gain control. The resulting ultimate bound is proportional to the size of the delay, disturbance and the switching gain. Therefore trade-offs can be made between the linear and discontinuous controller elements in order to minimize delay effects. Preliminary results were presented in Fridman, Han, and Spurgeon (1). Notation. Throughout the paper, the superscript T stands for matrix transposition, R n denotes the n-dimensional Euclidean space with vector norm, R n m is the set of all n m real matrices, and the notation P >, for P R n n means that P is symmetric and positive definite. The symmetric elements of the symmetric matrix are denoted by. The symbol stands for essential supremum.. Problem formulation Consider the following uncertain dynamical system with time varying input delay τ(t) and disturbance w(t) ẋ(t) = Ax(t) B (u(t τ(t)) w(t)), y(t) = Cx(t) (1) where x(t) R n, x(t ) = x, u(t) R m, y(t) R p with m < p < n. It is assumed that u(t) = for t < t. Matrix A may be uncertain with polytopic type uncertainty. The matched disturbance w(t) R m and the uncertain input delay τ(t) are assumed to be bounded w(t), τ(t) [, τ ], where and τ are known bounds and τ is supposed to be sufficiently small. It is assumed that the delay is either fast varying (without any constraints on the delay derivative) or slowly varying, where the delay-derivative satisfies the bound τ d < 1. Assuming B and C are both of full rank, a controller will be designed which for sufficiently large t induces the motion of the closed-loop system in the Mτ -neighborhood (with M > independent of τ ) of the surface S = {x R n : z (t) = FCx(t) = } () for some selected matrix F R m p. The ideal sliding motion z (t) = is only possible with τ =. Remark 1. Since a static output feedback control is designed, the results are applicable to both input delay, τ i, and output delay, τ o, where in the closed-loop system the resulting delay is τ = τ i τ. 3. Sliding manifold design If rank(cb) = m, there exists a change of coordinates x r = T r x, where T r R n n is non-singular, in which the system has the regular form (Edwards & Spurgeon, 1995) A11 A ẋ r (t) = 1 x A r (t) (u(t τ(t)) w(t)) I m A 1 y(t) = T x r (t) (3) where x r (t) = col{x 1 (t), x (t)}, T R p p is invertible, A 11 R (n m) (n m). Given K R m (p m), let F = K I m T 1. As a result F T = KC 1 I m, C1 = (p m) (n p) I (p m). Defining the sliding manifold as z (t) = Fy(t) = x (t) KC 1 x 1 (t) (4) the reduced-order dynamics is governed by the system ẋ 1 (t) = (A 11 A 1 KC 1 )x 1 (t) A 1 z (t) (5) with input z. The system triple A 11, A 1, C 1 is assumed to be stabilizable. In the presence of input delay, z in (5) will not vanish in finite time. Therefore, a K is sought which not only stabilizes (5) (as in the case without delay), but also produces input-to-state stability (with the smallest gain possible). Sufficient conditions for the input-to-state stability of (5) are given by the following lemma: Lemma 1. Given tuning parameters α >, ε, ε 1, b, and M R (p m) (n p), if there exists an (n m) (n m) matrix P >, and matrices Q R (p m) (p m), Q 11 R (n p) (n p), Q 1 R (n p) (p m), Y R m (p m) so that LMI Θ = θ 1,1 θ 1, A 1 εq εq T εa 1 < bi m θ 1,1 = A 11 Q A 1 [YM ε 1 Y] αp Q T AT 11 [YM ε 1Y] T A T 1, θ 1, = P Q εq T AT 11 ε[ym ε 1Y] T A T 1 (6) Q11 Q holds, where Q = 1 Q M ε, then the solution of (5) with K 1 Q = YQ 1 and with the initial condition x 1(t ) at initial time t is bounded by x T 1 (t)ˆpx1 (t) < e α(t t ) x T 1 (t )ˆPx1 (t ) b α z [t,t] where ˆP = Q T PQ 1 (Han, Fridman, & Spurgeon, 1). Remark. To minimize the ultimate bound on x 1, the following procedure is adopted from Fridman and Dambrine (9). The ζ R is minimized subject to LMI (6) and P Q T ζ I n m which leads to lim sup x 1 (t) < ζ b α lim sup z (t). <, A delayed sliding mode controller is designed which ensures the closed-loop system is ultimately bounded with bound proportional to the delay, disturbance and switching gain.

X. Han et al. / Automatica ( ) 3 4. Controller design: a singular perturbation approach that Defining z = In m KC 1 I m x r = col{z 1 (t), z (t)} in (3) it follows switching gain to be chosen. Denoting w(t) = w(t) (1 δ) sign z 1 (t ξ(t)) sign z m (t ξ(t)) T (13) ż 1 (t) = Ā 11 z 1 (t) Ā 1 z (t) ż (t) = Ā 1 z 1 (t) Ā z (t) u (t τ(t)) w(t) where Ā 11 = A 11 A 1 KC 1, Ā 1 = A 1, Ā 1 = KC 1 Ā 11 A KC 1 A 1, A = KC 1 Ā 1 A. For i = 1,..., m, denote the i-th component z by z i. A control law of the form u(t) = F y(t) (1 δ) sign z 1 (t) sign z m (t) T will be designed for (7), where > and δ > are tuning parameters. The design objective is to achieve ultimate boundedness of the closed-loop system with a bound proportional to the delay and the disturbance bounds. The closed-loop system (7) and (8) has the form ż 1 (t) = Ā 11 z 1 (t) Ā 1 z (t) (9) ż (t) = Ā 1 z 1 (t) Ā z (t) z (t ξ(t)) [w(t) (1 δ) [sign z 1 (t ξ(t)) sign z m (t ξ(t))] T ] (1) with the initial condition z(t ) = z, z(t) =, t < t (11) where ξ(t) = τ(t), ξ(t) h and h = τ. For small > (7) and (8) is a singularly perturbed system. The delay is scaled by in order to guarantee robust (input to state) stability with respect to small enough delay (Fridman, ). Remark 3. For ξ, a conventional SMC is designed as follows Edwards and Spurgeon (1995): find > so that the linear controller u l (t) = F y(t) asymptotically stabilizes (7) with w. For all δ >, (8) asymptotically stabilizes (7) with non-zero w. The bound [Ā 1i Ā i ]z(t) < δ is valid for big enough t, which implies finite time convergence of the closed-loop system to z =. The Lyapunov-based proofs of stability and finite time convergence use the relation z i (t) sign z i (t). For non-zero ξ(t), the product z i (t) sign z i (t ξ(t)) may change sign and the closed-loop system (7) and (8) is not asymptotically stable. Given h >, the main problem is the choice of > and of δ > (if any) that ensures the bound lim sup [Ā 1 Ā ]z(t) < δ, ξ(t) [, h] (1) holds for solutions of (9), (1). In the following, matrix inequalities are derived for finding and δ via a singular perturbation approach, which guarantees the feasibility of these matrix inequalities for small enough. The design process seeks to enlarge avoiding a high gain control. Finally, it will be proved that the closed-loop system is ultimately bounded with a bound proportional to τ. For recent results on stability of singularly perturbed systems with small delay, refer to Chen, Yang, Lu, and Shen (1) and Glizer (9). 4.1. Input-to-state stability of the time-delay system Considering the switching component of the SMC as a perturbation, this allows a bound on the state and an appropriate (7) (8) the closed-loop system (9), (1) can be presented as ż 1 (t) = Ā 11 z 1 (t) Ā 1 z (t) (14) ż (t) = Ā 1 z 1 (t) Ā z (t) z (t ξ(t)) w(t) where w(t) [1 (1 δ) m]. Let P R n n be positive definite with structure (Kokotovic, Khalil, & O Reilly, 1986) P P = 1 P T > (15) P 3 where P 1 R n m. For (14), choose the Lyapunov Krasovskii functional of the form V (t) = eᾱ(s t) z T (s)sz (s)ds t ξ(t) z (s)ds h h eᾱ(s t) z T (s)g t h eᾱ(s t) ż T (s)rż (s)dsdθ z T (t)p z(t) (16) where G, R and S R m are positive matrices. The inequality W(t) = d dt V (t) ᾱv (t) b w T (t) w(t) < (17) along the trajectories of (9), (1) for z w [t,t] > guarantees (19) (Fridman & Dambrine, 9). The following lemma can be stated (for proof see the Appendix) as follows: Lemma. Given positive tuning scalars, h, ᾱ and b, let there exist P > in (15) with (n m) (n m) matrix P 1 >, m (n m)- matrix P and m m positive matrices P 3, G, R, S such that the following LMI with its entries θ Θ = 1,1 θ 1,6 < (18) θ 6,6 θ 1,1 = P 1 Ā 11 Ā T 11 P 1 P T Ā1 Ā T 1 P ᾱp 1, θ 1, = P 1 Ā 1 Ā T 1 P 3 Ā T 11 P T P T Ā ᾱp T, θ 1,4 = P T, θ 1,5 = P T, θ 1,6 = hā T 1 R, θ, = P Ā 1 Ā T 1 P T P 3Ā Ā T P 3 ᾱp 3 G e ᾱh R S, θ,4 = P 3 e ᾱh R, θ,5 = P 3, θ,6 = hā T R, θ 3,3 = e ᾱh G e ᾱh R, θ 3,4 = e ᾱh R, θ 4,4 = e ᾱh R (1 d)se ᾱh, θ 4,6 = hr, θ 5,5 = bim, θ 5,6 = hr, θ 6,6 = R is feasible. Then solutions of (9) (11) satisfy the bound z T (t)p z(t) < e ᾱ(t t ) z T (t )P z(t ) b ᾱ w [t,t] (19) for all ξ(t) [, h] with ξ(t) d < 1 (and thus (9) (1) is input-to-state stable). Moreover, solutions of (9) (11) satisfy (19) for all fast-varying delays ξ(t) [, h] if LMI (18) is feasible with S =.

4 X. Han et al. / Automatica ( ) 4.. LMIs for the controller design Conditions will now be derived that guarantee the bound (1) for the solutions of (9), (1). Taking into account (19) and, thus, lim sup z T (t)p z(t) < b ᾱ [1(1δ) m], it may be concluded that (1) holds if the inequality z T (t)[ā 1 Ā ] T [Ā 1 Ā ] z(t) < ᾱzt (t)pz(t)δ b[1(1δ) is satisfied for t. Hence, the inequality m] ᾱδ b[1 (1 δ) m] P ᾱδ 1 b[1 (1 δ) m] P T Ā T 1 ᾱδ b[1 (1 δ) m] P 3 Ā T I m < () guarantees that the solutions of (9), (1) satisfy the bound (1). By Schur complements, () is feasible if the following matrix inequality is feasible ᾱδ b[1 (1 δ) m] P 1 P T P 3 ĀT 1 Ā1 Ā T Ā <. (1) Matrix inequalities (15), (18) and () have been derived for finding the parameters and δ of the controller (8). It will now be shown that if the -independent LMI Θ = Θ = < () is feasible, then for all δ > inequalities (15), (18) and () are feasible for all small enough. Let P 1, P, P 3 satisfy Θ <. Then for small enough >, (15) and (18) are feasible for the same - independent matrices P 1, P, P 3. Hence, given δ >, (1) is feasible for small enough >. It is easily seen that Θ < guarantees exponential stability with decay rate ᾱ/ of the slow subsystem ż s (t) = Ā 11 z s (t), z s (t) R n m and asymptotic stability of the fast subsystem of (14) ż f (t) = z f (t ξ(t)), ξ(t) [, h], z f (t) R m. Since Ā 11 is Hurwitz, there exists P 1 > satisfying P 1 Ā 11 Ā T 11 P 1 ᾱp 1 < for small enough ᾱ >. Choose next P =, G = S = and R = P 3 = p 3 I m. By using Schur complements, it can be shown that Θ < holds for big enough p 3 >, b and small enough h. The sufficient conditions below for the feasibility of (15), (18) and () have been proved: Proposition 1. (i) Given positive tuning scalars h, ᾱ and b, let there exist < P 1 R (n m) (n m), P R m (n m) and positive m m- matrices P 3, G, R, S such that LMI () is feasible. Then, for all δ > there exists (δ) > such that for all (, (δ)] LMIs (15), (18) and () are feasible and, thus, solutions of (9) (1) satisfy the bound (1). (ii) LMI () is feasible for small enough h, ᾱ and big enough b. 4.3. Main result Let φ(t, t, ) be the fundamental solution of the equation ζ (t) = ζ (t ξ(t)), ζ (t) R with φ(t, t, ) = 1 and φ(t, t, ) = for t < t. By using the arguments of Lemma and choosing V = ψ hr e α(s t) ζ (s)ds q t ξ(t) h e α(s t) ζ (s)ds t h e α (s t) ζ (s)dsdθ ρζ (t) with positive scalars ρ, q, r, ψ, it can be shown that the feasibility of the -independent LMI ψ q r ρ r q r r (1 d)ψ r hr < (3) r yields the following bound φ(t, t, ) e α (t t ) (4) for small enough α > and >, ξ(t) h, ξ d < 1. Note that (3) is feasible for h 1.414 if d = and for h 1. if d is unknown (i.e. for fast varying delay). The main result is now formulated (for proof see the Appendix) as follows: Theorem 1. Let the conditions of Lemma 1 hold. Given positive tuning scalars, h, ᾱ, b and δ let there exist < P1 R (n m) (n m), P R m (n m), positive m m-matrices P 3, G, R, S and positive scalars ρ, q, r, ψ such that LMIs (15), (18), () and (3) are feasible. Then for all ξ [, h], ξ d < 1 the solutions z(t) of the closed-loop system (9) (1) satisfy the following bounds: lim sup z i (t) M h, M = (1 δ)(1 m) (5) where i = 1,..., m denotes the i-th component of z, and lim sup z T 1 (t)ˆpz1 (t) 4 b α mm h. (6) Moreover, the solutions of (9)- (1) satisfy (5) and (6) for all fast varying delays ξ(t) [, h] if the above LMIs are feasible with S = and ψ =. Remark 4. The singular perturbation approach allows the choice of tuning parameters to occur in two stages: (i) Given b >, h = find the tuning parameter ᾱ > that b ᾱ minimizes by solving the -independent LMI () (which corresponds to the slow and the fast subsystems). Increase h arriving to some maximum achievable h 1. which preserves the feasibility of (). (ii) With ᾱ, b and h as found in (1) search for the remaining tuning parameters > and δ > such that the -dependent LMIs (15), (18), () and (1) (which correspond to the full-order system) are feasible. Start with small and big δ for which the above LMIs are feasible (as guaranteed by Proposition 1). Then increase (to avoid the high-gain control and to treat bigger delays) by decreasing h and ᾱ such that h is maximized and δ is minimized. The latter leads to a smaller ultimate bound. Note increase in leads to increase of the switching parameter δ. Therefore, a trade-off exists between bound minimization and the acceptable control magnitude. Remark 5. Consider now (1) with the linear controller u l (t) = F y(t). Then the closed-loop system has the form (14) with w(t) = w(t). Under the conditions of Lemma, the solutions of the resulting closed-loop system satisfy z T (t)p z(t) < e ᾱ(t t ) z T P z b ᾱ. Given and δ satisfying the conditions of Theorem 1, the ultimate bounds under the proposed SMC are given by (5) and (6) and these bounds vanish for h, i.e. the performance under the proposed SMC recovers the performance under the ideal SMC without input delay. Given > satisfying Lemma, the ultimate bounds under the linear controller u l (t) = F y(t) are proportional to the disturbance bound only and do not vanish for h. Therefore, the linear controller leads to vanishing bounds only for, i.e. by using very high gain (even with no input delay).

X. Han et al. / Automatica ( ) 5 5. Extension to input and state delay The following uncertain dynamical system is considered with state and input time varying delay r(t) and τ(t), respectively, and with matched disturbance w(t) ẋ(t) = Ax(t) A d x(t r(t)) B (u(t τ(t)) w(t)), y(t) = Cx(t) where x(t) R n, u(t) R m, w(t) R m and y(t) R p with m < p < n. The delays and disturbance are bounded by: r(t) [, r ], τ(t) [, τ ] and w(t). The delays may be either slowly varying with ṙ(t) d 1 < 1, τ(t) d < 1 or fast varying (with no constraint on the delay derivatives). The input and output matrices B and C are both of full rank. The sliding manifold can be defined by (). 5.1. Sliding manifold design In regular form, the system (5) becomes A11 A ẋ r (t) = 1 Ad11 A x A 1 A r (t) d1 x A d1 A r (t r(t)) d (u(t τ(t)) w(t)), I m y(t) = T x r (t). (7) Defining the sliding manifold as in (4), the reduced-order system with inputs z (t) and z (t r(t)) is ẋ 1 (t) = (A 11 A 1 KC 1 )x 1 (t) (A d11 A d1 KC 1 )x 1 (t r(t)) A 1 z (t) A d1 z (t r(t)) (8) where (A 11 A d11, A 1 A d1, C 1 ) is assumed stabilizable. Lemma 3 (Han et al. (1)). Given tuning scalars α >, ε, ε 1, b 1, b > and a matrix M R (p m) (n p), let there exist (n m) (n m) matrices P >, G, S, R and matrices Q R (p m) (p m), Q 11 R (n p) (n p), Q 1 R (n p) (p m), Y R m (p m), K = YQ 1 such that LMI ˆθ ˆΘ = 1,1 ˆθ 1,6 < (9) ˆθ 6,6 where ˆθ 11 = A 11 Q A 1 [Y ε 1 Y] Q T AT 11 αp [YM εy] T A T 1 G S Re αr, ˆθ 15 = A 1, ˆθ 34 = Re αr ˆθ 1 = P Q εq T AT 11 ε[ym ε 1Y] T A T 1, ˆθ 66 = b I m ˆθ 14 = A d11 Q A d1 [YM ε 1 Y] Re αr, ˆθ 16 = A d1 ˆθ = εq εq T r R, ˆθ 33 = (G R)e αr, ˆθ 4 = εa d11 Q εa d1 [YM ε 1 Y], ˆθ 5 = εa 1, ˆθ 6 = εa d1 ˆθ 44 = e αr R (1 d 1 )Se αr, ˆθ 55 = b 1 I m, holds, then for all ultimately bounded z, solutions of (8) satisfy the inequality lim sup x T 1 (t)ˆpx1 (t) < b 1b lim sup α z (t), where ˆP and Q are in the form given in Lemma 1. 5.. Controller design and the resulting ultimate bound By similar change of coordinates as in (7) and (7) becomes ż 1 (t) = Ā 11 z 1 (t) Ā d11 z 1 (t r(t)) Ā 1 z (t) Ā d1 z (t r(t)) ż (t) = Ā 1 z 1 (t) Ā d1 z 1 (t r(t)) Ā z (t) Ā d z (t r(t)) u(t τ(t)) w(t). With the controller given in (8) the closed-loop system is ż 1 (t) = Ā 11 z 1 (t) Ā d11 z 1 (t r(t)) Ā 1 z (t) Ā d1 z (t r(t)) ż (t) = Ā 1 z 1 (t) Ā d1 z 1 (t r(t)) Ā z (t) Ā d z (t r(t)) z (t ξ(t)) w(t) (3) where w(t) is given by (13) with w(t) [1 (1 δ) m], ξ(t) = τ(t), ξ(t) h, z(t) = col{z 1 (t), z (t)}. Let P be of the same structure as (15), then input-to-state stability can be derived using the Lyapunov Krasovskii functional V (t) = z T (t)p z(t) t r eᾱ(s t) z T 1 (s)g 1z 1 (s)ds eᾱ(s t) z T 1 (s)s 1z 1 (s)ds t r(t) eᾱ(s t) z T (s)g z (s)ds t h eᾱ(s t) z T (s)s z (s)ds t ξ(t) eᾱ(s t) z T (s)s 3z (s)ds t r(t) eᾱ(s t) z T t r (s)s 4z (s)ds r eᾱ(s t) ż T r 1 (s)r 1 ż 1 (s)dsdθ h eᾱ(s t) ż T (s)r ż (s)dsdθ h r eᾱ(s t) ż T r (s)r 3ż (s)dsdθ with positive matrices G 1, G, S 1, S, S 3, S 4, R 1, R and R 3. Lemma 4. Given positive tuning scalars r,, h, ᾱ and b1, let there exist P > in (15) with (n m) (n m) matrix P 1 >, m (n m)-matrix P, m m positive matrix P 3, (n m) (n m) positive matrices G 1, S 1, R 1, m m positive matrices G, S, S 3, S 4, R and R 3 such that the LMI θ Θ = 1,1 θ 1,1 < (31) θ 1,1 with entries θ 1,1 = ᾱp 1 P 1 Ā 11 Ā T 11 P 1 P T Ā1 Ā 1 P G 1 S 1 R 1 e ᾱr, θ 1, = P 1 Ā d11 R 1 e ᾱr P T Ād1, θ 1,4 = P 1 Ā 1 Ā T 1 P 3 Ā T 11 P T P T Ā αp T, θ 1,5 = P 1 Ā d1 P T Ād, θ 1,7 = P T, θ 1,9 = P T, θ 1,1 = r Ā T 11 R 1, θ 1,11 = hā T 1 R, θ 1,1 = r Ā T 1 R 3,

6 X. Han et al. / Automatica ( ) θ, = R 1 e ᾱr (1 d 1 )S 1 e ᾱr, θ,3 = R 1 e ᾱr, θ,4 = Ā T d1 P 3 Ā T d11 P T, θ,1 = r Ā T d11 R 1, θ,11 = hā T d1 R, θ,1 = r Ā T d1 R 3, θ 5,1 = r Ā T d1 R 1, θ 3,3 = e ᾱr (R 1 G 1 ), θ 4,7 = P 3 R e ᾱh, θ 4,4 = P 3 Ā Ā T P 3 P Ā 1 Ā T 1 P T ᾱp 3 R e ᾱh e ᾱr R 3 G S S 3 S 4, θ 4,5 = P 3 Ā d P Ā d1 e ᾱr R 3, θ 4,9 = P 3, θ 4,1 = r Ā T 1 R 1, θ 4,11 = hā T R, θ 5,5 = (1 d 1 )S 3 e ᾱr e ᾱr R 3, θ 5,6 = e ᾱr R 3, θ 5,11 = hā T d R, θ 5,1 = r Ā T d R 3, θ 4,1 = r Ā T R 3, θ 6,6 = e ᾱr ( R 3 S 4 ), θ 9,9 = b1 I m, θ 1,1 = R 3 θ 7,7 = R e ᾱh (1 d )S e ᾱh, θ 7,8 = R e ᾱh, θ 7,11 = hr, θ 7,1 = r R 3, θ 8,8 = (R G )e ᾱh, θ 9,11 = hr, θ 9,1 = r R 3, θ 1,1 = R 1, θ 11,11 = R. Then solutions of (3) satisfy the bound lim sup z T (t)p z(t) < b1 1 (1 δ) m (3) ᾱ for all r(t) [, r ] and ξ(t) [, h] with ṙ(t) d 1 < 1 and ξ d < 1. Moreover, solutions of (3) satisfy (3) for all fast-varying delays r(t) [, r ] or ξ(t) [, h] if LMI (31) is feasible with S 1 = S 3 = or S = respectively. Conditions will be derived that guarantee lim sup [Ā 1 Ā ]z(t) < κ 1 δ, lim sup [Ā d1 Ā d ]z(t r(t)) < κ δ for solutions of (3), where κ 1 κ 1. Given (3) and lim sup z T (t r(t))p z(t r(t)) < b1 1 (1 δ) m are true, (33) holds if the following inequalities are satisfied ᾱκ z T (t)[ā 1 Ā ] T 1 [Ā 1 Ā ]z(t) < δ z T (t)p z(t) b 1 (1 (1 δ) m) z T (t r(t))[ā d1 Ā d ] T [Ā d1 Ā d ]z(t r(t)) < ᾱκ δ z T (t r(t))p z(t r(t)) b 1 (1 (1 δ) m) for t. Hence, the inequalities κ ϖ 1 P 1 κ ϖ 1 P T Ā T 1 κ ϖ 1 P 3 Ā T <, I m κ ϖ P 1 κ ϖ P T Ā T d1 κ ϖ P 3 Ā T < d I m ᾱ (33) (34) ᾱδ where ϖ = b, guarantee that the solutions of (3) 1(1(1δ) m) satisfy the bound (33). Proposition. Given positive tuning scalars r,, h, ᾱ, b1, κ 1, κ, δ let there exist < P 1 R (n m) (n m), P R m (n m) and positive (n m) (n m) matrices G 1, S 1, R 1, positive m m matrices P 3, G, S, S 3, S 4, R and R 3 such that LMI Θ < is feasible, where Θ is given by (31) with =. Then, for positive scalars κ 1, κ, where κ 1 κ 1 and all δ >, there exists (δ) > such that for all (, (δ)] LMIs (15), (31) and (34) are feasible and, thus, solutions of (3) satisfy the bound (33). Theorem. Let the conditions of Lemma 4 hold. Given positive tuning scalars r,, h 1., ᾱ, b1, κ 1, κ, δ let there exist < P 1 R (n m) (n m), P R m (n m) and positive (n m) (n m) matrices G 1, S 1, R 1, positive m m matrices P 3, G, S, S 3, S 4, R and R 3 such that LMIs (15), (31) and (34) are feasible. Then for all ξ [, h], ξ d 1 < 1, r(t) [, r ], ṙ(t) d < 1, the solutions of the closed-loop system (3) satisfy (5) and (6). Moreover, the solutions of (3) satisfy (5) and (6) for all fast varying delays r(t) [, r ] or ξ(t) [, h] if the above LMIs are feasible with S 1 = S 3 = or S = respectively. Remark 6. Since LMIs (15), (18), () and (3), as well as (9), (31) and (34), are affine in the system matrices, the results are applicable where these matrices have polytopic type uncertainties. 6. Example The following model of combustion in a liquid monopropellant rocket motor has been considered in Zheng, Cheng, and Gao (1995), where the system is given by (5) with A =.ρ(t) 1 1 1 1, 1 1 1.ρ(t) 1 A d = 1 C =, B = 1 T. (35) 1 Here ρ(t) = sin(t) and the exogenous disturbance w(t) satisfy w(t) = 5. Time-varying delays in the states and input are due to pressure force propagation in the combustion chamber and gas injector respectively. LMI solutions for the controller design incorporate matrices A and A d with two vertices corresponding to ρ = ±1. The controller is designed for fast varying state delay r(t). s and fast varying input delay τ(t).5 s. Setting r =., α =.9, b 1 =., b =.1, ε = 1.5, ε 1 = 3.5, M = [4.4] in LMI (9) and ζ = 147 in the LMI in Remark 4, the reduced order system (8) is ultimately bounded with K = 1.15. Choosing LMI tuning parameters according to the algorithm in Remark 4, it is obtained by solving the -independent LMI () (with = ), that ᾱ =.44, b1 =.5 and maximum h =.7. Substituting the above ᾱ, b1 and h into the -dependent LMI (31), we find that LMIs (31) and (34) are feasible for =.17, h =.5 s with a smaller ᾱ =.8, h =.9 and κ 1 =.9999, κ =.1, δ = 5.3. Therefore, the system is ultimately bounded under the linear controller u(t) = F y(t) with F = [1 1.15]. The controller (8) has been fully synthesized to guarantee the bound z (t) 6.3 according to (5) for all fast varying state delays r(t). s

X. Han et al. / Automatica ( ) 7 1 1 5 1 15 5 3 (a) Conventional SMC with state delay only. 1 1 5 1 15 5 3 (b) Conventional SMC with state and input delays. 1 1 5 1 15 5 3 (c) Proposed SMC with both state and input delays. 1 1 5 1 15 5 3 (d) Proposed SMC with both state and input delays. Fig. 1. Conventional and proposed SMC under slowly varying state delay r(t). s and fast varying input delay τ(t).5 s. 1 1 5 1 15 5 3 (a) Using linear controller. 1 1 5 1 15 5 3 (b) Using proposed SMC. Fig.. Comparison of proposed SMC and the linear controller of the SMC under fast varying delays r(t). s and τ(t).1 s. and input delays τ(t).5 s. The control input does not produce high gain (here F/ 8.4 since =.17 has been chosen large enough). The designed controller is simulated, where the disturbance is w(t) = 5 sin 3t. The advantages of the designed SMC over the conventional SMC from Han et al. (1) (which ignores the input delay) u(t) = [15.8 43]y(t) 67 Fy(t), F = [1 3.1] (36) Fy(t) are first demonstrated. Note that (36) has been designed for (35) with the slowly varying state delay r(t). s, ṙ.5. For simulation, the slowly varying state delay is chosen as r(t) =.1 sin(5t).1. s and the fast varying input delay is τ(t) =.5 sin(4t).5.5 s. The conventional SMC (36) achieves asymptotic stability in the presence of the state delay (see Fig. 1(a), where τ ). In the presence of the input delay the controller leads to the outputs bounded by y t 3 s 4 (see Fig. 1(b)). In comparison the proposed SMC leads to a smaller output bounds with y(t) t 3 s 1.6 as shown in Fig. 1(c). The resulting switching variable is bounded by z t 3 s 1.7 as shown in Fig. 1(d), which is in line with the theoretical estimation. In the simulations under both SMC methods, chattering of high frequency is observed due to the delayed switching component sign u(t τ(t)). In the real implementations, many methods are introduced for reducing the chattering. A comprehensive review is given in Young, Utkin, and Özgüner (1999). For sufficiently small input delay the proposed SMC should have advantage over its linear control component leading to smaller bounds on the system outputs. Consider next the fast varying state delay r(t) =.1 sin(15t).1. s and the fast varying input delay τ(t) =.5 sin(t).5.1 s, where the disturbance was kept the same. The outputs of the system under the linear controller u(t) = F y(t)( =.17) and under the proposed SMC are shown in Fig. (a) and (b) respectively. The bound on the outputs produced by the linear controller is y(t) t 3 s.85, whereas the bound obtained by the proposed SMC is smaller with y(t) t 3 s.35. Note that the linear controller leads to the same bound y(t) t 3 s.35 as SMC by more than twice higher gain with =.7. Remark 7. Without input delay, the new SMC design method has advantages over existing methods (Edwards & Spurgeon, 1995; Han et al., 1). The matrix P for the analysis of the closedloop system is full and not diagonal as in existing methods. The conservativeness of the diagonally structured P was verified by setting P = in (15) for the above example while keeping all the other tuning parameters in the LMIs unchanged. The bound on the feasible input delay in this case was found to be h =. s, which is smaller than h =.5 s obtained using the full P. 7. Conclusion Sliding mode control for systems with matched bounded disturbances in the presence of input time-varying delay has been studied using a singular perturbation approach. Unlike existing results on relay control with input delay (Fridman et al., ; Gouaisbaut et al., ) a priori knowledge of the bounds on the system states is not needed. Ultimately bounded solutions of the delayed system are found based on LMI formulations and various Lyapunov-based methods. The ultimate bound is proportional to the delay, the disturbances and the switching gain. The proposed SMC brings the input delay analysis into the design phase which is shown in the example to have key advantages when compared with an existing SMC that ignores the input delay and with a linear control (for sufficiently small input delay). The method is applicable to linear systems with polytopic uncertainties in all blocks of the system matrices. In the extension to state delays, for the first time a static output feedback, a SMC is designed via the Krasovskii method for systems with fast varying delays. Appendix A. Proof of Lemma Differentiating V of the structure (15) and (16) along (14) it follows from (17) that W(t) z T 1 (t)p 1[Ā 11 Ā 1 ]z(t) z T (t)p [Ā 11 Ā 1 ]z(t) z T 1 (t)p T [Ā1 Ā ]z(t) z (t ξ(t)) w(t) z T (t)p 3 [Ā1 Ā ]z(t) z (t ξ(t)) w(t) b w T (t) w(t) ᾱz T 1 (t)p 1z 1 (t) ᾱz T (t)p z 1 (t)

8 X. Han et al. / Automatica ( ) ᾱz T 1 (t)p T z (t) ᾱz T (t)p 3z (t) h ż T (t)rż (t) h t h e ᾱh ż T (s)rż (s)ds z T (t)gz (t) e ᾱh z T (t h)gz (t h) z T (t)sz (t) (1 d)e ᾱξ(t) z T (t ξ(t))sz (t ξ(t)). Using the identity ξ(t) e ᾱh ż T (s)rż (s)ds = e ᾱh ż T (s)rż (s)ds t h t h e ᾱh ż T (s)rż (s)ds t ξ(t) apply Jensen s inequality ξ(t) h e ᾱh ż T (s)rż (s)ds t h e ᾱh [z T (t ξ(t)) zt (t h)]r[z (t ξ(t)) z (t h)] h e ᾱh ż T (s)rż (s)ds t ξ(t) e ᾱh [z T (t) zt (t ξ(t))]r[z (t) z (t ξ(t))]. Then, setting ζ (t) = col{z 1 (t), z (t), z (t h), z (t ξ), w(t)} and applying Schur complements to the term h ż T (t) Rż (t), where ż (t) is substituted by the right-hand side of (14), it is established that W(t) < if Θ <. Appendix B. Proof of Theorem 1 The i-th component of differential equation (1) with the initial condition (11) can be represented in the form of an integral equation (Kolmanovskii & Myshkis, 199) z i (t) = φ(t, t, )z i (t ) φ(t, s, ) [Ā 1i Ā i ]z(s) t w i (s) (1 δ) sign z i (s ξ(s)) ds. (B.1) The feasibility of () implies the bound (1), then the following inequality holds for t : [Ā 1i Ā i ]z(s) w i (s) (1 δ) sign z i (s ξ(s)) < M. (B.) Taking into account (4) and (B.), it is established from (B.1) that for t z i (t θ) z i (t) φ(t, s, ) [Ā 1i Ā i ]z(s) w i (s) (1 δ) sign z i (s ξ(s)) ds < M e α (t s) ds < M 1 e α h α M h where θ [ h, ]. Therefore, z i (t) M h < z i (t θ) < z i (t) M h for t and the following implication holds z i (t) M h sign z i (t θ) = sign z i (t) (B.3) for large enough t. Thus, from (1), (B.) and (B.3) for sufficiently large t the following implication follows: z i (t) M h z T i (t)[[ā 1i Ā i ]z(t θ) w i (t θ) (1 δ) sign z i (t θ)] < z i (t) ( [Ā 1i Ā i ]z(t θ) ) (1 δ) z i (t) <. (B.4) It will be shown next that the z i -component of the solutions to (1) exponentially converges to the ball (5). Moreover, for sufficiently large t, whenever z i (t) achieves the ball (5), it will never leave it. Taking into account (B.4), for sufficiently large t it follows that z i (t) M h d dt z i (t) = z i (t)ż i (t) = z i (t)[ z i (t ξ(t)) ([Ā 1i Ā i ]z(t) w i (t) (1 δ) sgn z i (t))] z i (t) z i (t) = z i (t) z i (t) t ξ(t) ż i (s)ds t ξ(t) z i (s ξ(s)) [Ā 1i Ā i ]z(s) w i (s) (1 δ) sgn z i (s) ds z i (t) z i (t) t ξ(t) z i (s ξ(t))ds. Therefore, given (B.3) holds for large enough t, it follows that t ξ(t) z i (t)z i (s ξ(t))ds. Hence z i (t) M h d dt z i (t) z i (t). (B.5) Assume now that for large enough t 1 the z i component of the solution to (1) is outside the ball (5). Then from (B.5) it follows that for all t t 1 such that z i (t) M h the inequality holds z i (t) e (t t 1) z i (t 1 ), i.e. z i exponentially converges to the ball (5). Let t > t 1 be the time when z i (t ) = M h. Then due to (B.5) z i (t ) < z i (t ). Therefore, whenever z i (t) attains the ball (5), it will never leave it. Then (6) follows from Lemma 1 and (5). References Akian, M., Bliman, P. A., & Sorine, M. (). Control of delay systems with relay. IMA Journal of Mathematical Control and Information, 19, 133 155. Boiko, I. (9). Oscillations and transfer properties of relay feedback systems with time delay linear plants. Automatica, 45, 897 9. Chen, W. H., Yang, S. T., Lu, X., & Shen, Y. (1). Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time-varying delay. International Journal of Robust and Nonlinear Control,, 1 44. Chou, C. H., & Cheng, C. C. (3). A decentralized model reference adaptive variable structure controller for large-sclae time-varying delay systems. IEEE Transaction on Automatic Control, 48, 113 117. Edwards, C., & Spurgeon, S. K. (1995). Sliding mode stabilization of uncertain systems using only output information. International Journal of Control, 6, 119 1144. Fridman, L. (1997). The separation of motions in different speeds discontinuous control systems with delay. Automation and Remote Control, 58, 163 175. Fridman, E. (). Effects of small delays on stability of singularly perturbed systems. Automatica, 38, 897 9. Fridman, E., & Dambrine, M. (9). Control under quantization, saturation and delay: an LMI approach. Automatica, 45, 58 64. Fridman, L., Fridman, E., & Shustin, E. (1993). Steady modes in a discontinuous control system with time delay. Pure Mathmatics and Applications, 4, 59 73.

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Adaptive control of linear systems with unknown time delay. In Adaptive control of nonsmooth dynamic systems. UK: Springer Verlag. Young, K. D., Utkin, V. I., & Özgüner, Ü. (1999). A control engineer s guide to sliding mode control. IEEE Transaction on Automatic Control, 7, 38 34. Zheng, F., Cheng, M., & Gao, W. (1995). Variable structure control of time delay systems with a simulation study on stabilizing combustion in liquid propellant rocket motors. Automatica, 31, 131 137. Zhou, J., Wen, C., & Wang, W. (9). Adaptive backstepping control of uncertain systems with unknown input time delay. Automatica, 45, 1415 14. X. Han received an M.Sc. in electrical and electronic engineering from the University of Leicester, UK in 6 and a Ph.D. in control engineering from the University of Kent, Canterbury, UK in 11. His research interests include sliding mode control and time-delay systems. E. Fridman received the M.Sc. degree from Kuibyshev State University, USSR, in 1981 and the Ph.D. degree from Voroneg State University, USSR, in 1986, all in mathematics. From 1986 to 199 she was an Assistant and Associate Professor in the Department of Mathematics at Kuibyshev Institute of Railway Engineers, USSR. Since 1993 she has been at Tel Aviv University, where she is currently Professor of Electrical Engineering-Systems. She has held visiting positions at the Weierstrass Institute for Applied Analysis and Stochastics in Berlin (Germany), INRIA in Rocquencourt (France), Ecole Centrale de Lille (France), Valenciennes University (France), Leicester University (UK), Kent University (UK), CINVESTAV (Mexico), Zhejiang University (China), St. Petersburg Institute of Mechanics (Russia). Her research interests include time-delay systems, distributed parameter systems, robust control, singular perturbations, nonlinear control and asymptotic methods. She has published about 9 articles in international scientific journals. Currently she serves as Associate Editor in Automatica, SIAM Journal of Control and Optimization and in IMA Journal of Mathematical Control & Information. S.K. Spurgeon received the B.Sc. and D.Phil. degrees from the University of York, York, UK, in 1985 and 1988, respectively. She has held academic positions at the University of Loughborough and the University of Leicester in the UK and is currently Professor of Control Engineering and Head of the School of Engineering and Digital Arts at the University of Kent. She is a member of the Editorial Board of the International Journal of Systems Science, a member of the Editorial Board of the IET Proceedings D, a Subject Editor for the International Journal of Robust and Nonlinear Control and an Editor of the IMA Journal of Mathematical Control and Information. Her research interests are in the area of robust nonlinear control and estimation, particularly via sliding mode techniques in which area she has published in excess of 7 refereed papers. Professor Spurgeon received the IEEE Millennium Medal in and was awarded the 1 Honeywell International Medal in recognition of her outstanding contribution to control theory and systems engineering. She is a Fellow of the IET, a Fellow of the IMA, a Fellow of the InstMC, was elected a Fellow of the Royal Academy of Engineering in 8 and is currently an IEEE Distinguished Lecturer for the Control Systems Society.