Discrete-Time Sliding-Mode Control of Uncertain Systems with Time-Varying Delays via Descriptor Approach

Abstract

This paper considers the problem of robust discrete-time sliding-mode control (DT-SMC) design for a class of uncertain linear systems with time-varying delays. By applying a descriptor model transformation and Moon's inequality for bounding cross terms, a delay-dependent sufficient condition for the existence of stable sliding surface is given in terms of linear matrix inequalities (LMIs). Based on this existence condition, the synthesized sliding mode controller can guarantee the sliding-mode reaching condition of the specified discrete-time sliding surface for all admissible uncertainties and time-varying delays. An illustrative example verifies the effectiveness of the proposed method.

1. Introduction

Time delay often appears in many engineering systems, such as networked control systems, telecommunication systems, nuclear reactors, chemical engineering systems, and so on. Since time delay is often an important source of poor performance and instability, considerable attention has been dedicated to stability analysis and control synthesis for time-delay systems. Many research results have been reported and well documented in [1, 2], and the references therein. A linear matrix inequalities (LMIs-) based approach to delay-dependent stability for continuous-time uncertain systems with time-varying delays is proposed by using the Leibniz-Newton formula in [3]. In [4], a new technique is studied by incorporating both the time-varying-delayed state and the delay-upper-bounded state to make full use of all available information in the design. It is worth noting that the wide use of digital computers in practical control systems requires the stability analysis and controller synthesis for discrete-time time-delay systems. In [5], delay-dependent sufficient conditions for the existence of the guaranteed cost controller in terms of LMIs are derived for uncertain discrete-time systems with constant delays. A necessary and sufficient condition is established for stabilizing a networked control system by modeling both the sensor-to-controller and controller-to-actuator delays as Markov chains in [6]. In [7], Gao et al. propose an output-feedback stabilization method for discrete-time systems with time-varying delays. This method is dependent on the lower and upper delay bounds, and is further extended to discrete-time systems with norm-bounded mismatched uncertainties. Further, a less conservative result on stability of discrete-time systems with time-varying delays, which also incorporates both the time-varying-delayed state and the delay-upper-bounded state, is obtained by defining new Lyapunov functions in [8].

It is well known that the sliding-mode control (SMC) is an effective method to achieve robustness and invariance to matched uncertainties and disturbances on the sliding surface [9]. In recent years, increasing attention has been paid to the design of SMC for uncertain systems with time delay. One approach based on lumped sliding surface has been applied to design an SMC law for state-delay systems in [10]. In [11], the SMC method is proposed for systems with mismatched parametric uncertainties and time delay, and a delay-independent sufficient condition for the existence of linear sliding surface is derived in terms of LMIs; the synthesized sliding-mode controller can guarantee the reaching condition of the specified sliding surface. In [12], an observer-based sliding mode control is studied for continuous-time state-delayed systems with unmeasurable states and nonlinear uncertainties, and a sufficient condition of asymptotic stability is derived for the overall closed-loop systems.

Recently, a new descriptor model transformation is introduced for stability analysis [13], controller design [5, 14], and filtering design [15] of time-delay systems. This approach can significantly reduce the overdesign entailed in the existing methods because of the model equivalence and less conservative bound on cross terms [14]. In [16], a Lyapunov-Krasovskii techniques-based delay-dependent descriptor approach to stability and control of linear systems with time-varying delays is proposed, and is further combined with the sliding mode control result to deal with the mismatched uncertainties and unknown nonlinear functions. However, to the best of authors' knowledge, the descriptor approach has not been fully investigated for research on delay-dependent DT-SMC design of linear systems with mismatched uncertainties and time-varying delays. This motivates our research work in this paper.

In this paper, we first propose a delay-dependent sufficient condition for the existence of stable sliding surface via the descriptor approach. Then, based on this existence condition, the synthesized discrete-time sliding-mode controller is designed to guarantee the sliding mode reaching condition of the specified discrete-time sliding surface. The rest of the paper is organized as follows. Section 2 describes the problem formulation and some necessary preliminary results. In Section 3, sufficient conditions for the existence of stable sliding surface are presented in terms of LMIs and a robust DT-SMC law is presented. This DT-SMC law can guarantee the sliding-mode reaching condition of the specified discrete-time sliding surface for all admissible uncertainties and time-varying delays. An illustrative example is used to demonstrate the validity and effectiveness of the proposed method in Section 4. Finally, Section 5 offers some concluding remarks.

Throughout this paper, superscript “T” stands for matrix transposition, ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space, ℝ^𝑛×𝑚 is the set of all real matrices of dimension 𝑛×𝑚,𝑃>0 means that 𝑃 is real symmetric and positive-definite matrix, 𝐼 and 0 represent identity matrix and zero matrix with appropriate dimensions. In symmetric block matrices or long matrix expressions, we use an asterisk (∗) to represent a term that is induced by symmetry. diag{⋅} stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation

Consider the following discrete-time uncertain system with time-varying delays in the regular form:𝑥₁(𝑘+1)=+(𝐴(𝐴₁₁+₁₂Δ𝐴₁₁)𝑥₁(𝑘)+(𝐴_𝑑11+Δ𝐴_𝑑11)𝑥₁(𝑘−𝑑(𝑘))𝑥+Δ𝐴₁₂)𝑥₂(𝑘),₂(𝑘+1)=²_𝑖=1𝐴_2𝑖𝑥(𝑘)=𝜑𝑥(_𝑖𝑘()𝑘,)+𝑘𝐴=−_𝑑2𝑖𝑥𝑑_𝑖𝑀(𝑘−𝑑(𝑘))+𝐵𝑢(𝑘)+𝑓(𝑘,𝑥(𝑘)),,−𝑑_𝑀+1,…,0,(1)where 𝑥₁∈ℝ^𝑛−𝑚, 𝑥₂∈ℝ^𝑚, 𝑥(𝑘)=(𝑥₁,𝑥₂)^T is the system state vector, 𝑢(𝑘)∈ℝ^𝑚 is the control input, 𝑓(𝑘,𝑥(𝑘))∈ℝ^𝑚 is an unknown nonlinear function representing the unmodeled dynamics and external disturbances, 𝑑(𝑘) is the unknown time-varying delays, 𝐴₁₁,𝐴_𝑑11,𝐴₁₂,𝐴₂₁,𝐴₂₂,𝐴_𝑑21,𝐴_𝑑22, and 𝐵 represents real constant system matrices with appropriate dimensions, Δ𝐴₁₁,Δ𝐴_𝑑11, and Δ𝐴₁₂ are time-varying matrix functions representing parameter uncertainties, 𝜑(𝑘)∈ℝ^𝑛 represents initial values of 𝑥(𝑘). It is assumed that the system uncertainties are norm-bounded with the following form:[Δ𝐴₁₁Δ𝐴₁₂]=𝐸₁𝐹₁(𝑘)[𝐻₁𝐻₃],Δ𝐴_𝑑11=𝐸₂𝐹₂(𝑘)𝐻₂,(2)where 𝐸₁, 𝐸₂, and 𝐻_𝑖(𝑖=1,2,3) are known constant matrices with compatible dimensions, and the properly dimensioned matrix 𝐹_𝑖(𝑘) is an unknown and time-varying matrix of uncertain parameters, but norm bounded as 𝐹^T_𝑖(𝑘)𝐹_𝑖(𝑘)≤𝐼,𝑖=1,2. Associated with system (2), we make the following assumptions.

Assumption 1. The time-varying delays 𝑑(𝑘) is assumed to satisfy 𝑑_𝑚≤𝑑(𝑘)≤𝑑_𝑀, where 𝑑_𝑚 and 𝑑_𝑀 are constant positive scalars representing the lower and upper bounds on the time delay, respectively.

Assumption 2. The pair (𝐴₁₁,𝐴₁₂) in the nominal system of (2) is controllable.

Assumption 3. Nonlinear function 𝑓(𝑘,𝑥(𝑘)) is unknown but bounded in the sense of the Euclidean norm.

The design procedure of DT-SMC consists of two steps: design of the stable sliding surface and the reaching motion control law. For discrete-time uncertain systems with time-varying delays, the linear sliding surface is chosen as𝑆(𝑘)=𝐶𝑥(𝑘)=𝐶𝐼𝑥(𝑘)=𝐶𝑥₁(𝑘)+𝑥₂(𝑘)=0(3)with 𝐶∈ℝ^𝑚×𝑛 and 𝐶∈ℝ^{𝑚×(𝑛−𝑚)} is a real matrix to be designed. Once sliding surface satisfies the discrete-time sliding mode reaching condition, substituting 𝑥₂(𝑘)=−𝐶𝑥₁(𝑘) into the first subsystem of (2), we obtain the following sliding motion:𝑥₁(𝑘+1)=+(𝐴(𝐴₁₁−_𝑑11𝐴₁₂𝐶+Δ𝐴₁₁−Δ𝐴₁₂𝐶)𝑥₁(𝑘)𝑥+Δ𝐴_𝑑11)𝑥₁(𝑘−𝑑(𝑘)),₁(𝑘)=𝜑₁(𝑘),𝑘=−𝑑_𝑀,−𝑑_𝑀+1,…,0.(4)Here, 𝜑₁(𝑘)∈ℝ^𝑛−𝑚 stands for initial values of 𝑥₁(𝑘). In order to derive our main results, the following two lemmas are necessary.

Lemma 1 (See [17], Moon's Inequality). Assume that 𝑎∈ℝ^𝑛𝑎, 𝑏∈ℝ^𝑛𝑏, and 𝑁∈ℝ^{𝑛𝑎×𝑛𝑏}. Then for any matrices 𝑋∈ℝ^{𝑛𝑎×𝑛𝑎}, 𝑌∈ℝ^{𝑛𝑎×𝑛𝑏}, and 𝑍∈ℝ^{𝑛𝑏×𝑛𝑏}, the following holds: −2𝑎^T𝑁𝑏≤𝑎𝑏^T𝑌−𝑋𝑁^T𝑌−𝑍𝑁^T𝑎𝑏,(5)where 𝑋𝑌𝑍𝑌≥0.(6)

Lemma 2 (See [18]). For any matrices 𝐷∈ℝ^𝑛×𝑝, 𝐸∈ℝ^𝑝×𝑛, 𝐹∈ℝ^𝑝×𝑝 with 𝐹^T(𝑘)𝐹(𝑘)≤𝐼 and scalars 𝜀>0, 𝐷𝐹𝐸+𝐸^T𝐹^T𝐷^T≤𝜀⁻¹𝐷𝐷^T+𝜀𝐸^T𝐸.(7)

The main objective of this paper is to design the sliding surface 𝑆(𝑘) and a DT-SMC law 𝑢(𝑘) such that

(1)the sliding surface is asymptotically stable;(2)the DT-SMC law can guarantee discrete-time sliding mode reaching condition.

3. Main Results

In this section, the results of slidng surface design and robust DT-SMC law will be presented for a class of uncertain linear systems with time-varying delays. Let us first consider the problem of sliding surface design. In order to reduce the overdesign entailed in the stability analysis methods for time-delay systems, we employ the descriptor approach to derive the delay-dependent sufficient conditions for the existence of the sliding surface in terms of LMIs. The first result on the asymptotic stability of designing sliding surface is presented in the following theorem.

Theorem 1. If, for certain prescribed positive number 𝜅, there exist positive scalars 𝛿₁>0,𝛿₂>0,(𝑛−𝑚)×(𝑛−𝑚), matrices 𝐿₁>0,𝐿₂,𝐿₃,𝑈₁>0,𝑈₂>0,𝑊₁,𝑊₂,𝑊₃, and 𝑚×(𝑛−𝑚) matrix 𝐾 such that the following LMIs hold: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Θ¹Θ₂00𝐿₁𝜗𝐿^T₂𝐿^T₂∗Θ₃𝜍000𝐿^T₃𝐿^T₃∗∗∗∗−∗𝑈₁𝑈−₁𝛿𝐻^T₂₂0000∗∗∗∗∗∗∗∗𝐼𝜒∗0−𝛿00₁0000∗∗∗∗∗∗𝐼−0𝐿₁0∗∗∗∗∗∗∗−𝑑0_𝑀−1𝑈₂⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(8)⎡⎢⎢⎢⎣𝑊∗₁𝑊₂0𝑊₃𝜅𝐴_𝑑𝑈₁∗∗𝑈₁⎤⎥⎥⎥⎦≥0,(9)where 𝜗 denotes 𝐿₁𝐻^T₁−𝐾^T𝐻^T₃,𝜍 denotes (1−𝜅)𝐴_𝑑𝑈₁, and 𝜒 denotes −(𝑑_𝑀−𝑑_𝑚+1)⁻¹𝑈₁, ΘΘ₁₂=𝐿₂+𝐿^T₂+𝑑_𝑀𝑊₁,Θ₃=𝐿₃+𝐿₁(𝐴^T+𝜅𝐴^T_𝑑−𝐼)−𝐾^T𝐴^T₁₂−𝐿^T₂+𝑑_𝑀𝑊₂,=−𝐿₃−𝐿^T₃+𝑑_𝑀𝑊₃+²_𝑖=1𝛿_𝑖𝐸_𝑖𝐸^T_𝑖.(10)Then, the system (4) with Assumption 1 is asymptotically stable, and the sliding surface of (3) is given by 𝑆(𝑘)=𝐾𝐿¹⁻¹𝑥₁(𝑘)+𝑥₂(𝑘)=0.(11)

Proof. Let 𝜂(𝑘)=𝑥₁(𝑘+1)−𝑥₁(𝑘), the system (4) can be rewritten in an equivalent descriptor form:0=𝑥₁(𝑘+1)=𝑥₁(𝑘)+𝜂(𝑘),𝐴₁₁(𝑘)+𝐴_𝑑11(𝑘)−𝐼𝑥₁(𝑘)−𝜂(𝑘)−𝐴_𝑑11(𝑘)∑^𝑘−1_{𝑖=𝑘−𝑑(𝑘)}𝜂(𝑖),(12)where𝐴₁₁(𝑘)=𝐴₁₁−𝐴₁₂𝐶+Δ𝐴₁₁−Δ𝐴₁₂𝐶,𝐴_𝑑11(𝑘)=𝐴_𝑑11+Δ𝐴_𝑑11.(13)Choose the Lyapunov-Krasovskii functional candidate as𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘).(14)Here,𝑉𝑉₁(𝑘)=𝑥^T₁(𝑘)𝑃₁𝑥₁(𝑘),₂(𝑘)=^𝑘−1_𝑖=𝑘_{−𝑑(𝑘)}𝑉𝑥^T₁(𝑖)𝑄𝑥₁(𝑖),₃(𝑘)=^{−𝑑𝑚+1}_{𝑗=−𝑑𝑀+2𝑘−1𝑖=𝑘+𝑗−1}𝑉𝑥^T₁(𝑖)𝑄𝑥₁(𝑖),₄(𝑘)=⁻¹_{𝑗=𝑑𝑀}^𝑘−1_{𝑖=𝑘+𝑗}𝜂^T(𝑖)𝐺𝜂(𝑖),(15)where 𝑃₁,𝑄,𝐺∈ℝ^{(𝑛−𝑚)×(𝑛−𝑚)} are symmetric positive definite matrices to be determined. The Lyapunov-Krasovskii functional candidate 𝑉(𝑘) is positive definite for all 𝑥₁(𝑘)≠0. Now, in order to evaluate the forward difference Δ𝑉(𝑘)=𝑉(𝑘+1)−𝑉(𝑘), in what follows we calculate Δ𝑉₁,Δ𝑉₂,Δ𝑉₃, and Δ𝑉₄, respectively. For Δ𝑉₁(𝑘), by using the descriptor system form (12), we obtain=𝜂_TΔ𝑉₁(𝑘)=𝜂_T(𝑘)𝑃₁𝜂(𝑘)+2𝑥^T₁(𝑘)𝑃₁𝜂(𝑘)(𝑘)𝑃₁𝜂(𝑘)+2𝑥^T₁(𝑘)𝑃^T=𝜂_T𝜂(0𝑘)(𝑘)𝑃₁𝜂(𝑘)+2𝑥₁(𝑘)𝑃^T−2𝐴₁₁(𝑘)+𝐴_𝑑11(𝜂𝑘()𝑘−)𝐼𝑥₁(𝑘)−𝜂(𝑘)𝑥^T₁(𝑘)𝑃∑^T𝑘−1_{𝑖=𝑘−𝑑(𝑘)}𝐾_𝑛−𝑚𝐴_𝑑11(𝑘)𝜂(𝑖),(16)where 𝑥^T₁(𝑘)=[𝑥^T₁(𝑘)𝜂^T(𝑘)],𝐾_𝑛−𝑚=[0𝐼]^T, and 𝑃=[^𝑃10𝑃_2𝑃3]. If identifying 𝑁∶=𝑃^T𝐾_𝑛−𝑚𝐴_𝑑11(𝑘), 𝑎∶=𝑥₁(𝑘), and 𝑏∶=𝜂(𝑖) in (16), then applying Lemma 1 gives rise to the following inequality:≤𝜂_TΔ𝑉₁(𝑘)(𝑘)𝑃₁𝜂(𝑘)+2𝑥^T₁(𝑘)𝑃^T×𝜂(𝑘)+𝐴₁₁(𝑘)+𝐴_𝑑11(𝑘)−𝐼𝑥₁(𝑘)−𝜂(𝑘)_𝑘−1_𝑖=𝑘∑_{−𝑑(𝑘)}𝑥𝜂₁((𝑖𝑘))^T𝑊𝑀−𝑃^T≤𝜂_T∗𝐾𝐺_𝑛−𝑚𝐴_𝑑11(𝑘)𝑥𝜂₁((𝑖𝑘))(𝑘)𝑃₁𝜂(𝑘)+2𝑥^T₁(𝑘)𝑃^T+𝑑_𝑀(𝐴₁₁(𝑘)+𝐴_𝑑11(𝜂𝑘()𝑘−)𝐼)𝑥₁(𝑘)−𝜂(𝑘)+2𝑥^T₁(𝑘)𝑊𝑥₁(𝑘)+∑^𝑘−1_{𝑖=𝑘−𝑑𝑀}𝜂^T(𝑖)𝐺𝜂(𝑖)𝑥^T₁(𝑘)𝑀−𝑃^T𝐾_𝑛−𝑚𝐴_𝑑11(𝑘)𝑥₁(𝑘)−𝑥₁(𝑘−𝑑(𝑘)),(17)where 𝑊, 𝑀∈ℝ^{(𝑛−𝑚)×(𝑛−𝑚)} are constant matrices with appropriate dimensions satisfying𝑊∗𝑀𝐺≥0.(18)For Δ𝑉₂, we haveΔ𝑉₂=^𝑘_{𝑖=𝑘−𝑑}_(𝑘+1)+1𝑥^T₁(𝑖)𝑄𝑥₁(𝑖)−^𝑘−1_𝑖=𝑘_{−𝑑(𝑘)}=𝑥_T1𝑥^T₁(𝑖)𝑄𝑥₁(𝑖)+(𝑘)𝑄𝑥₁(𝑘)−𝑥^T₁(𝑘−𝑑(𝑘))𝑄𝑥₁(𝑘−𝑑(𝑘))_{𝑘−1𝑖=𝑘−𝑑}_(𝑘+1)+1𝑥^T₁(𝑖)𝑄𝑥₁(𝑖)−^𝑘−1_𝑖=𝑘−_𝑑(𝑘)+1≤𝑥_T1𝑥^T₁(𝑖)𝑄𝑥₁(𝑖)+(𝑘)𝑄𝑥₁(𝑘)−𝑥^T₁(𝑘−𝑑(𝑘))𝑄𝑥₁(𝑘−𝑑(𝑘))_{𝑘−𝑑𝑚𝑖=𝑘}_{−𝑑𝑀+1}𝑥^T₁(𝑖)𝑄𝑥₁(𝑖).(19)For Δ𝑉₃, we getΔ𝑉₃=^{−𝑑𝑚+1}_𝑗=−_𝑑𝑀+2𝑥^T₁=(𝑑_𝑀(𝑘)𝑄𝑥₁(𝑘)−𝑥^T₁(𝑘+𝑗−1)𝑄𝑥₁(𝑘+𝑗−1)]−𝑑_𝑚)𝑥^T₁(𝑘)𝑄𝑥₁(𝑘)−^{𝑘−𝑑𝑚}_𝑗=𝑘_{−𝑑𝑀+1}𝑥^T₁(𝑗)𝑄𝑥₁(𝑗).(20)For Δ𝑉₄, we obtainΔ𝑉₄=⁻¹_𝑗=_−𝑑𝑀𝜂^T=𝑑_𝑀(𝑘)𝐺𝜂(𝑘)−𝜂^T(𝑘+𝑗)𝐺𝜂(𝑘+𝑗)𝜂^T(𝑘)𝐺𝜂(𝑘)−^𝑘−1_𝑗=_{𝑘−𝑑𝑀}𝜂^T(𝑗)𝐺𝜂(𝑗).(21)Based on the above results in (17), (19)–(21), and using Lemma 2, we can deriveΔ𝑉(𝑘)=≤Δ𝑉₁(𝑘)+Δ𝑉₂(𝑘)+Δ𝑉₃(𝑘)+Δ𝑉₄(𝑘)×𝑥^T₁(𝑃(𝑘_T)Ψ𝑥₁(𝑘)+2𝑥^T₁(𝑘)+𝑥_T1𝐾_𝑛−𝑚𝐴_𝑑11−𝑀)𝑥₁(𝑘−𝑑(𝑘))=𝑥(𝑘−𝑑(𝑘))(𝛿²⁻¹𝐻^T₂𝐻₂−𝑄)𝑥₁(𝑘−𝑑(𝑘))_T1×⎡⎢⎢⎢⎣Ψ(𝑘)𝑃𝜂^T_T(𝑘)𝑥^T₁(𝑘−𝑑(𝑘))∗𝛿₂₋₁𝐴0_𝑑11−𝑀𝐻^T₂𝐻₂−𝑄⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝑥^T1𝜂(𝑘)_T𝑥(𝑘)_T1(𝑘−𝑑(𝑘))⎤⎥⎥⎥⎦,(22)whereΨ=𝑃^T+0𝐴𝐴_T11−𝐴0₁₂𝐶−𝐼−𝐼₁₁𝐼𝐼−𝐶−^T𝐼𝐴^T₁₂−𝐼𝑃+𝑑_𝑀+𝑀𝑊+𝑀0_T0+𝜔0𝑃₁+0𝑑_𝑀𝐺+²_𝑖=1𝛿_𝑖𝑃^T𝐸0_𝑖0𝐸^T𝑖𝑃(23)where 𝜔 denotes 𝛿¹⁻¹(𝐻^T₁−𝐶^T𝐻^T₃)(𝐻₁−𝐻₃𝐶)+(𝑑_𝑀−𝑑_𝑚+1)𝑄 and 𝛿₁, 𝛿₂ are any positive scalars. Let 𝜉=[𝑥^T₁(𝑘)𝜂^T(𝑘)𝑥^T₁(𝑘−𝑑(𝑘))] andΦ=⎡⎢⎢⎢⎣Ψ∗𝑃𝛿^T₂₋₁𝐴0_𝑑11−𝑀𝐻^T₂𝐻₂−𝑄⎤⎥⎥⎥⎦.(24)ThenΔ𝑉(𝑘)≤𝜉^T(𝑘)Φ𝜉(𝑘).(25)In order to obtain a convenient LMI, we restrict the choice of 𝑀 to be𝑀=𝜅𝑃^T𝐴0_𝑑11(𝑘).(26)Define𝑃⁻¹=𝐿∶=𝐿¹𝐿₂0𝐿₃,𝐾=𝐶𝐿¹𝐺₋₁,𝑄⁻¹=𝑈₁,=𝑈₂,𝑊=(𝑃⁻¹)^T𝑊(𝑃⁻¹)=⎡⎢⎢⎣𝑊₁𝑊₂𝑊^T₂𝑊₃⎤⎥⎥⎦.(27)Multiply (18) by diag[(𝑃⁻¹)^T,𝑈₂] and diag[(𝑃⁻¹),𝑈₂] on the left- and right-hand sides, respectively. Alternatively, multiply (24) by diag[(𝑃⁻¹)^T,𝑈₁] and diag[(𝑃⁻¹),𝑈₁] on the left- and right-hand sides, respectively. By using the Schur complement [19], we obtain that (8) is equivalent to Φ<0, and (18) is equivalent to (9), which yieldsΔ𝑉(𝑘)<0,∀𝜉(𝑘)≠0.(28)It follows from Lyapunov stability theory [20] that the system (4) is asymptotically stable. Moreover, based on the above definition of the sliding surface in (3), it can be obtained with 𝐶=𝐾𝐿¹⁻¹.

Once the sliding surface is appropriately designed according to Theorem 1, the next result on the design of DT-SMC is summarized in the following theorem.

Theorem 2. Consider the system (2) with Assumptions 1–3. If the linear sliding surface is designed as in (11), then the following DT-SMC law 𝑢(𝑘)=−𝐵⁻¹{²_𝑖=1(𝐴_2𝑖+𝜖𝜏+⋅s𝐶g𝐴n(𝑆_1𝑖)(𝑥𝑘_𝑖()𝑘))+−𝛼(₀1−𝑞𝜏)𝑆(𝑘)+(𝜌_𝛼+𝛽₀+𝛾₀+𝜌_𝛽+𝜌_𝛾)⋅sgn(𝑆(𝑘))}(29)can guarantee the discrete-time sliding mode reaching condition. Here, sgn(𝑆𝑆((𝑘𝑘)))==𝑆sg₁n(𝑘(𝑆)₁𝑆₂(𝑘)⋯𝑆_𝑚(𝑘),𝛼₀)sgn(𝑆₂)⋯sgn(𝑆_𝑚),=𝛼^𝑈+𝛼_𝐿2,𝛽⁰=𝛽^𝑈+𝛽_𝐿2,𝛾⁰=𝛾^𝑈+𝛾_𝐿𝜌_𝛼2,=𝛼^𝑈−𝛼_𝐿2,𝜌^𝛽=𝛽^𝑈−𝛽_𝐿2,𝜌^𝛾=𝛾^𝑈−𝛾_𝐿2,(30)𝐶=𝐾𝐿¹⁻¹, 𝛼_𝑈 and 𝛼_𝐿 are the upper bound and lower bound of 𝛼(𝑘)=𝐶Δ𝐴₁₁𝑥₁(𝑘)+𝐶Δ𝐴₁₂𝑥₂(𝑘), 𝛽_𝑈 and 𝛽_𝐿 are the upper bound and lower bound of 𝛽̃(𝑘)=𝐶Δ𝐴_𝑑11𝑥₁(𝑘−𝑑(𝑘))+𝐶𝐴_𝑑11𝑥₁(𝑘−𝑑(𝑘))+∑^2𝑖=1[𝐴_𝑑2𝑖𝑥_𝑖(𝑘−𝑑(𝑘))], 𝛾_𝑈 and 𝛾_𝐿 are the upper bound and lower bound of 𝛾̃(𝑘)=𝑓(𝑘,𝑥_𝑘), 𝑞>0, 𝜀>0, 1−𝑞𝜏>0, and 𝜏>0 is the sampling period.

Proof. The design of DT-SMC law guarantees that the reaching condition must be satisfied when there exist uncertainties and time-varying delays in systems. We consider the reaching condition for the DT-SMC proposed in [21] as follows:ΔΔ𝑆𝑆((𝑘𝑘))≤≥==𝑆𝑆−−((𝑘𝑘𝜖𝜖𝜏𝜏++⋅⋅11ss))gg−−nn((𝑆𝑆𝑆𝑆((((𝑘𝑘𝑘𝑘))))))−−𝑞𝑞𝜏𝜏𝑆𝑆((𝑘𝑘)),,iiff𝑆𝑆((𝑘𝑘))><00,.(31)From the designed sliding surface𝑆(𝑘)=𝐶𝑥(𝑘)=𝐶𝐼𝑥(𝑘),(32)we haveΔ𝑆(𝑘)=+(𝐶(𝐴𝐶𝐴₁₁+₁₂𝐶Δ𝐴₁₁)𝑥₁(𝑘)+(𝐶𝐴_𝑑11+𝐶Δ𝐴_𝑑11)𝑥₁(𝑘−𝑑(𝑘))++𝐶Δ𝐴₁₂)𝑥₂(𝑘)₂_𝑖=1𝐴_2𝑖𝑥_𝑖(𝑘)+𝐴_𝑑2𝑖𝑥_𝑖(𝑘−𝑑(𝑘))+𝐵𝑢(𝑘)+𝑓(𝑘,𝑥_𝑘).(33)Since it is assumed that Δ𝐴₁₁, Δ𝐴_𝑑11, Δ𝐴₁₂, and 𝑓(𝑘,𝑥_𝑘) are bounded, using the discrete-time version of improved Razumikhin theorem in [22], for any solution 𝑥(𝑘−𝑑(𝑘)) of (2), there exists a constant 𝜃 such that the following inequality is satisfied:‖‖𝑥(𝑘−𝑑(𝑘))‖‖<𝜃‖‖𝑥(𝑘)‖‖,∀𝑘,𝑑_𝑚≤𝑑(𝑘)≤𝑑_𝑀.(34)We can also obtain that 𝛼(𝑘)=𝐶Δ𝐴₁₁𝑥₁(𝑘)+𝐶Δ𝐴₁₂𝑥₂(𝑘), 𝛽̃(𝑘)=𝐶(𝐴_𝑑11+Δ𝐴_𝑑11)𝑥₁(𝑘−𝑑(𝑘))+∑^2𝑖=1[𝐴_𝑑2𝑖𝑥_𝑖(𝑘−𝑑(𝑘))], and 𝛾̃(𝑘)=𝑓(𝑘,𝑥_𝑘) will be bounded with upper and lower bounds. Let the bounds be𝛼_𝐿≤𝛼(𝑘)=𝐶Δ𝛽𝐴_11𝐿𝑥₁(𝑘)+𝐶Δ𝐴₁₂𝑥₂(𝑘)≤𝛼_𝑈,=𝐶(𝐴_𝑑11≤𝛽̃(𝑘)+Δ𝐴_𝑑11)𝑥₁(𝑘−𝑑𝛾(𝑘_𝐿))+∑²_𝑖=1𝐴_𝑑2𝑖𝑥_𝑖(𝑘−𝑑(𝑘))≤𝛽_𝑈,≤𝛾̃=𝑓(𝑘,𝑥_𝑘)≤𝛾_𝑈.(35)Here, the inequality𝛼_𝐿=𝛼_𝐿,1𝛼_𝐿,2⋯𝛼_𝐿,𝑚^T≤𝛼(𝑘)=𝛼₁𝛼₂⋯𝛼_𝑚^T≤𝛼_𝑈=𝛼_𝑈,1𝛼_𝑈,2⋯𝛼_𝑈,𝑚^T(36)implies that 𝛼_𝐿,𝑖≤𝛼_𝑖(𝑘)≤𝛼_𝑈,𝑖,𝑖=1,2,…,𝑚; similar notations are used for 𝛽̃(𝑘) and 𝛾̃(𝑘).
Then, substituting (29) into (33), we haveΔ𝑆(𝑘)=−𝛼((𝑘𝛼)₀+𝛽̃(𝑘)+𝛾̃(𝑘)−𝑞𝜏𝑆(𝑘)−𝜖𝜏⋅sgn(𝑆(𝑘))+𝛽₀+𝛾₀)−(𝜌_𝛼+𝜌_𝛽+𝜌_𝛾)⋅sgn(𝑆(𝑘)).(37)According to previous discussion, the following relations hold:𝛼𝛼((𝑘𝑘))≥≤𝛼𝛼₀₀+𝜌_𝛼⋅sgn(𝑆(𝑘))if𝑆(𝑘)>0,𝛽̃(𝑘)≤𝛽₀+𝜌_𝛼⋅sgn(𝑆(𝑘))if𝑆(𝑘)<0,𝛽̃(𝑘)≥𝛽₀+𝜌_𝛽⋅sgn(𝑆(𝑘))if𝑆(𝑘)>0,𝛾̃(𝑘)≤𝛾₀+𝜌_𝛽⋅sgn(𝑆(𝑘))if𝑆(𝑘)<0,𝛾̃(𝑘)≥𝛾₀+𝜌_𝛾⋅sgn(𝑆(𝑘))if𝑆(𝑘)>0,+𝜌_𝛾⋅sgn(𝑆(𝑘))if𝑆(𝑘)<0.(38)Thus, the sign change of Δ𝑆(𝑘) in (37) is opposite to that of 𝑆(𝑘), irrespective of the value of the uncertainties 𝛼(𝑘), 𝛽̃(𝑘), and 𝛾̃(𝑘). Moreover, the closed-loop control system (2) satisfies the reaching condition (31).

Remark 1. The descriptor system approach [13, 14, 16] can greatly reduce the overdesign entailed in the existing methods because the descriptor model leads to a system that is equivalent to the original one (from the point of view of stability) and requires bounding of fewer cross-terms. Therefore, in this work, we apply the descriptor system approach to the SMC design for discrete-time systems with uncertainties and time-varying delays.

Remark 2. SMC approach is an effective method to achieve the robustness and invariant property to matched uncertainties and disturbance on the sliding surface [9, 21]. In this paper, the results of Theorem 1 guarantee that the sliding motion dynamics will be robustly asymptotically stable. Furthermore, the descriptor-system-approach-based DT-SMC in Theorem 2 provides a solution to satisfy the sliding mode reaching condition of the designed discrete-time sliding surface for discrete-time systems with uncertainties and time-varying delays.

4. Numerical Example

In this section, an illustrative example is given to verify the design method proposed in this paper. Consider the discrete-time system (2) with the following description:𝑥₁(𝑘)=𝑥¹¹𝑥(𝑘)₁₂(𝑘),𝐴¹¹𝐴=00..07500.8_𝑑11,𝐴=−00.2−00.1,𝐴₁₂=00.3₂₁,𝐴=−00.5,𝐴₂₂=2.0,_𝑑21Δ𝐴₁₁=00.1,𝐴_𝑑22=0.05,𝐵=1,Δ𝐴₁₂=00..0012csoins((00..0011𝑘𝑘𝜋𝜋))00..0024ssiinn((00..0011𝑘𝑘𝜋𝜋)),Δ𝐴_𝑑11=00..0054,𝑓(𝑘,𝑥(𝑘))==0.004..0s0i11nc(s𝑥oins₁₁((00..0011𝑘𝑘𝜋𝜋))00..0045ssiinn((00..0011𝑘𝑘𝜋𝜋)),(𝑘)).(39)Initial states of the system are 𝑥₁(𝑘)=[1.0−0.5]^T,𝑥₂(𝑘)=0.6, for 𝑘∈[−𝑑_𝑀0]. Using Theorem 1 and choosing 𝜅=0.5, we obtain 𝐶=[0.93050.5168]. Then, the linear sliding surface is 𝑆(𝑘)=[𝐶1]𝑥(𝑘). According to Theorem 2, the robust DT-SMC law is designed as𝑢(𝑘)=−𝐵⁻¹{²_𝑖=1(𝐴_2𝑖+(+𝜖𝜏𝐶+𝐴𝜌_1𝑖𝛼)𝑥_𝑖(𝑘)−(1−𝑞𝜏)𝑆(𝑘)+𝜌_𝛽+𝜌_𝛾)⋅sgn(𝑆(𝑘))},(40)where 𝜌_𝛼=0.0238|𝑥₁₁(𝑘)|+0.0393|𝑥₁₂(𝑘)|+0.0672|𝑥₂(𝑘)|, 𝜌_𝛽=0.5014|𝑥₁₁(𝑘)|+0.2785|𝑥₁₂(𝑘)|+0.125|𝑥₂(𝑘)|, 𝜌_𝛾=0.4|sin𝑥₁₁(𝑘)|, 𝜏=0.1, 𝑞=5, and 𝜖=0.1. The time-varying delay 1≤𝑑(𝑘)≤3 is shown in Figure 1. System state trajectories are illustrated in Figure 2. The resulting sliding surface is in Figure 3. Figure 4 depicts the control input signal.

Figure 1 

Time-varying delays 𝑑(𝑘).

Figure 2 

System states 𝑥11,𝑥12,and𝑥2.

Figure 3 

Sliding surface 𝑆(𝑘).

Figure 4 

Control input 𝑢(𝑘).

It is observed from Figure 2 that the state trajectories of the system all converge to the origin quickly. The system can be stabilized quickly by the proposed method and the reaching motion satisfies the sliding reaching condition in spite of the time-varying delays and uncertainties. Simulation results illustrate that the proposed approach in this paper is feasible and effective for discrete-time uncertain linear systems with time-varying delays.

5. Conclusion

In this paper, the problem of robust DT-SMC for uncertain systems with time-varying delays has been studied. By using the descriptor model transformation and a recent result on bounding cross products of vectors, a delay-dependent sufficient condition for the existence of stable sliding surfaces is constructed for all admissible uncertainties. Based on this existence condition, the corresponding reaching motion controller is developed such that the reaching motion satisfies the discrete-time sliding mode reaching condition for uncertain systems with time-varying delays. An example shows the validity and effectiveness of the proposed DT-SMC design method.

Acknowledgments

The authors wish to thank the associate editor and anonymous reviewers for providing constructive suggestions which have improved the presentation of the paper. This research was supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Foundation of Innovation.