Further Results Concerning Delay-Dependent 𝐻_∞ Control for Uncertain Discrete-Time Systems with Time-Varying Delay

Abstract

This paper addresses the problem of 𝐻_∞ control for uncertain discrete-time systems with time-varying delays. The system under consideration is subject to time-varying norm-bounded parameter uncertainties in both the state and controlled output. Attention is focused on the design of a memoryless state feedback controller, which guarantees that the resulting closed-loop system is asymptotically stable and reduces the effect of the disturbance input on the controlled output to a prescribed level irrespective of all the admissible uncertainties. By introducing some slack matrix variables, new delay-dependent conditions are presented in terms of linear matrix inequalities (LMIs). Numerical examples are provided to show the reduced conservatism and lower computational burden than the previous results.

1. Introduction

During the past decades, considerable attention has been paid to the problems of stability analysis and control synthesis of time-delay systems. Many methodologies have been proposed and a large number of results have been established (see, e.g., [1–4] and the references therein). All these results can be generally divided into two categories: delay-independent stability conditions [5, 6] and delay-dependent stability conditions [7–12]. The delay-independent stability condition does not take the delay size into consideration, and thus is often conservative especially for systems with small delays, while the delay-dependent stability condition makes fully use of the delay information and is usually less conservative than the delay-independent one.

Up to now, the most important approach to deal with delay in the states of the systems is the use of Lyapunov-Krasovskii functionals, which has been largely employed to obtain convex conditions mainly for continuous-time systems subjected to retarded states. However, discrete-time systems with state delay have received little attention. This mainly because that for precisely known discrete-time systems with constant delay, it is always possible to derive a delay-free system by state augmentation [10, 11]. Although, such an approach is valid for the system with constant delays, it fails to deal with time-varying delay case, which is more frequently encountered than the constant case in practice. Recent results on discrete time-delay systems can be found in [13] where delay-dependent stability criteria were considered using a sum inequality. In [14], stability conditions for discrete time-delay systems were presented, while less conservative results were given in [15] by using a more general Lyapunov-Krasovskii functional than that in [14]. In [16], the authors summarized the recent results concerning robust stabilization of discrete-time systems with state delay. Sufficient LMI conditions were presented checking the robust stability for a class of linear discrete-time systems with time-varying delay and polytopic uncertainties; robust state feedback gains with memory were also designed. These results were mainly with the stability analysis and state feedback controller design. Very few people have investigated the delay-dependent 𝐻_∞ control problem of discrete time-delay systems. In [17], the authors proposed an exponential output feedback 𝐻_∞ controller. Delay-dependent robust 𝐻_∞ control conditions for uncertain linear systems with lumped delays were given in [18], which were proved to be less conservative than some previous results. Also delay-dependent results were derived in [19] by combining a descriptor model transformation approach with Moon's bounding technique [9]. Very recently, in order to reduce the conservatism of the result in [19], a finite sum inequality approach was proposed in [20] and some less conservative 𝐻_∞ control condition was derived. Although the result in [20] is superior to that in [19], it is still a sufficient condition and has conservatism to some extent, which leaves open room for further improvement.

Naturally, one may say that whether we can employ the similar Lyapunov functional, fewer variables, and reduced complexity of the algorithm to obtain less conservatism than the existing results. In this paper, we will further study the robust 𝐻_∞ control problem for uncertain discrete-time system with time-varying delays. By introducing some slack matrix variables, new delay-dependent conditions for 𝐻_∞ control problem are proposed in terms of LMI form, while no model transformation and bounding technique are employed. It is also shown that the complexity of the algorithm is considerably reduced and the result in this paper is less conservative than that in [18–20]. Numerical examples are finally provided to demonstrate the effectiveness of the main results.

Notations
Throughout this paper, ℝ^𝑛 represents the 𝑛-dimensional Euclidean space; ℝ^𝑚×𝑛 is the set of all 𝑚×𝑛 real matrices. For real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite). The superscript “𝑇’’ denotes the transpose. 𝐼 is an identity matrix with appropriate dimension. ℤ⁺ denotes the set of {0,1,2,…}. ℒ₂ refers to the space of square summable infinite vector sequences. In symmetric block matrices, we use an asterisk “∗’’ to represent a term that is induced by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Problem Formulation

Consider the following uncertain discrete-time systems with time-varying delay [20]: (Σ)∶𝑥(𝑘+1)=(𝐴₀+Δ𝐴₀(𝑘))𝑥(𝑘)+(𝐴₁+Δ𝐴₁(𝑘))𝑥(𝑘−𝑑(𝑘))+(𝐵₁+Δ𝐵₁(𝑘))𝜔(𝑘)+(𝐵₂+Δ𝐵₂(𝑘))𝑢(𝑘),𝑧(𝑘)=(𝐶₀+Δ𝐶₀(𝑘))𝑥(𝑘)+(𝐶₁+Δ𝐶₁(𝑘))𝑥(𝑘−𝑑(𝑘))+(𝐷₁₁+Δ𝐷₁₁(𝑘))𝜔(𝑘)+(𝐷₁₂+Δ𝐷₁₂(𝑘))𝑢(𝑘),𝑥(𝑘)=𝜙(𝑘),∀𝑘∈[−ℎ,0], (2.1) where 𝑥(𝑘)∈ℝ^𝑛, 𝑢(𝑘)∈ℝ^𝑚, and 𝑧(𝑘)∈ℝ^𝑝 are the state, control input, and controlled output, respectively; 𝜔(𝑘)∈ℝ^𝑞 is the exogenous disturbance input, which belongs to ℒ₂. 𝜙(𝑘) is the initial condition; 𝐴₀, 𝐴₁, 𝐵₁, 𝐵₂, 𝐶₀, 𝐶₁, 𝐷₁₁, and 𝐷₁₂ are known real constant matrices. The time-varying parameter uncertainties are norm-bounded and meet with Δ𝐴₀(𝑘)Δ𝐴₁(𝑘)Δ𝐵₁(𝑘)Δ𝐵₂(𝑘)Δ𝐶₀(𝑘)Δ𝐶₁(𝑘)Δ𝐷₁₁(𝑘)Δ𝐷₁₂=𝒟(𝑘)𝒟¹₂𝐹(𝑘)𝐸₁𝐸₂𝐸₃𝐸₄, (2.2) where 𝐹(𝑘) is an unknown real time-varying matrix and satisfies the following bound condition: 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼. (2.3)𝒟₁,𝒟₂, and 𝐸_ℓ(ℓ=1,2,3,4) are known constant matrices of appropriate dimensions describing how the uncertainty 𝐹(𝑘) enters the nominal matrices of system (Σ). 𝑑(𝑘) denotes the time-varying delay satisfying ℎ≤𝑑(𝑘)≤ℎ,∀𝑘∈ℤ⁺, (2.4) where ℎ and ℎ are positive integer numbers.

Remark 2.1. The parameter uncertain structure in (2.2) and (2.3) has been widely used in the issues of robust control and filtering for uncertain systems; see, for example, [10, 21]. It comprises the “matching conditions’’ and many physical systems can be either exactly modeled in this manner or overbounded by (2.3).

Now, consider the following memoryless state feedback controller: 𝑢(𝑘)=𝐾𝑥(𝑘). (2.5) Applying this controller to system (Σ) results in the following closed-loop system:

(Σ_cl)∶𝑥(𝑘+1)=(𝐴₀+𝒟₁𝐹(𝑘)(𝐸₁+𝐸₄𝐾))𝑥(𝑘)+(𝐴₁+Δ𝐴₁(𝑘))𝑥(𝑘−𝑑(𝑘))+(𝐵₁+Δ𝐵₁(𝑘))𝜔(𝑘),𝑧(𝑘)=(𝐶₀+𝒟₂𝐹(𝑘)(𝐸₁+𝐸₄𝐾))𝑥(𝑘)+(𝐶₁+Δ𝐶₁(𝑘))𝑥(𝑘−𝑑(𝑘))+(𝐷₁₁+Δ𝐷₁₁(𝑘))𝜔(𝑘),𝑥(𝑘)=𝜙(𝑘),∀𝑘∈[−ℎ,0], (2.6) where 𝐴₀=𝐴₀+𝐵₂𝐾,𝐶₀=𝐶₀+𝐷₁₂𝐾.

The robust 𝐻_∞ control problem to be addressed in this paper can be formulated as developing a state feedback controller in the form of (2.5) such that

(1)the closed-loop system (Σ_cl) is robustly asymptotically stable when 𝜔(𝑘)=0, for all 𝑘≥0; (2)the 𝐻_∞ performance ‖𝑧‖₂<𝛾‖𝜔‖₂ is guaranteed for all nonzero 𝜔(𝑘)∈ℒ₂ and a prescribed 𝛾>0 under the zero-initial condition, for all admissible uncertainties and time-varying delays satisfying (2.2)–(2.4).

At the end of this section, let us introduce some important lemmas which will be used in the sequel.

Lemma 2.2 (Schur complement [22]). Given constant matrices 𝑀, 𝐿, 𝑄 of appropriate dimensions, where 𝑀 and 𝑄 are symmetric, then 𝑄>0 and 𝑀+𝐿^𝑇𝑄⁻¹𝐿<0 if and only if 𝑀𝐿^𝑇𝐿−𝑄<0, (2.7) or equivalently 𝐿−𝑄𝐿_𝑇𝑀<0. (2.8)

Lemma 2.3 (see [10]). Let 𝒟, ℰ, and ℱ be matrices with appropriate dimensions. Suppose ℱ^𝑇ℱ≤𝐼, then for any scalar 𝜇>0, there holds 𝒟ℱℰ+ℰ^𝑇ℱ^𝑇𝒟^𝑇≤𝜇𝒟𝒟^𝑇+𝜇⁻¹ℰ^𝑇ℰ. (2.9)

3. Main Results

In this section, some delay-dependent LMI-based conditions will be developed to solve the robust 𝐻_∞ control problem formulated in the previous section. First, we will consider the nominal system of system (Σ_cl) with 𝐹(𝑘)=0, for all 𝑘>0, that is, (Σ_ncl)∶𝑥(𝑘+1)=𝐴₀𝑥(𝑘)+𝐴₁𝑥(𝑘−𝑑(𝑘))+𝐵₁𝜔(𝑘),𝑧(𝑘)=𝐶₀𝑥(𝑘)+𝐶₁𝑥(𝑘−𝑑(𝑘))+𝐷₁₁𝜔(𝑘),𝑥(𝑘)=𝜙(𝑘),∀𝑘∈[−ℎ,0], (3.1) where 𝐴₀=𝐴₀+𝐵₂𝐾,𝐶₀=𝐶₀+𝐷₁₂𝐾.

Theorem 3.1. System (Σ_ncl) is asymptotically stable with a prescribed 𝐻_∞ disturbance attenuation level 𝛾>0, if there exist matrices 𝑃>0, 𝑄>0, 𝑅>0, 𝑊, and 𝑌 of appropriate dimensions such that Θ₁₁−𝑌+𝑊^𝑇−ℎ𝑌0𝐴^𝑇₀𝑃(𝐴₀−𝐼)^𝑇ℎ𝑅𝐶^𝑇₀∗−𝑊−𝑊^𝑇−𝑄−ℎ𝑊0𝐴^𝑇₁𝑃𝐴^𝑇₁ℎ𝑅𝐶^𝑇₁∗∗−ℎ𝑅0000∗∗∗−𝛾²𝐼𝐵^𝑇₁𝑃𝐵^𝑇₁ℎ𝑅𝐷^𝑇₁₁∗∗∗∗−𝑃00∗∗∗∗∗−ℎ𝑅0∗∗∗∗∗∗−𝐼<0, (3.2) where Θ₁₁=−𝑃+𝑌+𝑌^𝑇+(ℎ−ℎ+1)𝑄.

Proof. Let 𝑦(ℓ)=𝑥(ℓ+1)−𝑥(ℓ). (3.3) Then, it is easy to see that 𝑥(𝑘−𝑑(𝑘))=𝑥(𝑘)−^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ). (3.4) Now, choose a Lyapunov-Krasovskii functional candidate for the time-delay system (Σ_ncl) as 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘), (3.5) where 𝑉₁(𝑘)=𝑥^𝑇𝑉(𝑘)𝑃𝑥(𝑘),₂(𝑘)=^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑥^𝑇(ℓ)𝑄𝑥(ℓ)+^−ℎ+1_{𝑗=−ℎ+2}^𝑘−1_{ℓ=𝑘+𝑗−1}𝑥^𝑇𝑉(ℓ)𝑄𝑥(ℓ),₃(𝑘)=⁰_{𝑗=−ℎ+1}^𝑘−1_{ℓ=𝑘−1+𝑗}𝑦^𝑇(ℓ)𝑅𝑦(ℓ). (3.6) Taking the forward difference, we have Δ𝑉₁(𝑘)=𝑉₁(𝑘+1)−𝑉₁(𝑘)=[𝐴₀𝑥(𝑘)+𝐴₁𝑥(𝑘−𝑑(𝑘))+𝐵₁𝜔(𝑘)]^𝑇𝑃[𝐴₀𝑥(𝑘)+𝐴₁𝑥(𝑘−𝑑(𝑘))+𝐵₁𝜔(𝑘)]−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)[𝐴^𝑇₀𝑃𝐴₀−𝑃]𝑥(𝑘)+2𝑥^𝑇(𝑘)𝐴^𝑇₀𝑃𝐴₁𝑥(𝑘−𝑑(𝑘))+2𝑥^𝑇(𝑘)𝐴^𝑇₀𝑃𝐵₁𝜔(𝑘)+𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁𝑃𝐴₁𝑥(𝑘−𝑑(𝑘))+2𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁𝑃𝐵₁𝜔(𝑘)+𝜔^𝑇(𝑘)𝐵^𝑇₁𝑃𝐵₁𝜔(𝑘). (3.7) For any two matrices of appropriate dimensions 𝑌 and 𝑊, there holds 0=2𝑥^𝑇(𝑘)𝑌^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)+2𝑥^𝑇(𝑘−𝑑(𝑘))𝑊^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)−[2𝑥^𝑇(𝑘)𝑌^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)+2𝑥^𝑇(𝑘−𝑑(𝑘))𝑊^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)]. (3.8) Substituting (3.4) and the previous equality into (3.7) gives Δ𝑉₁(𝑘)=𝑥^𝑇(𝑘)[𝐴^𝑇₀𝑃𝐴₀−𝑃]𝑥(𝑘)+2𝑥^𝑇(𝑘)𝐴^𝑇₀𝑃𝐴₁[𝑥(𝑘)−^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)]+2𝑥^𝑇(𝑘)𝐴^𝑇₀𝑃𝐵₁𝜔(𝑘)+𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁𝑃𝐴₁𝑥(𝑘−𝑑(𝑘))+2𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁𝑃𝐵₁𝜔(𝑘)+𝜔^𝑇(𝑘)𝐵^𝑇₁𝑃𝐵₁𝜔(𝑘)=𝑥^𝑇(𝑘)[𝐴^𝑇₀𝑃𝐴₀−𝑃+𝐴^𝑇₀𝑃𝐴₁+𝐴^𝑇₁𝑃𝐴₀]𝑥(𝑘)+2𝑥^𝑇(𝑘)[𝑌−𝐴^𝑇₀𝑃𝐴₁]×^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)+2𝑥^𝑇(𝑘−𝑑(𝑘))𝑊^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)−[2𝑥^𝑇(𝑘)𝑌^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)+2𝑥^𝑇(𝑘−𝑑(𝑘))𝑊^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)]+2𝑥^𝑇(𝑘)𝐴^𝑇₀𝑃𝐵₁𝜔(𝑘)+𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁𝑃𝐴₁𝑥(𝑘−𝑑(𝑘))+2𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁𝑃𝐵₁𝜔(𝑘)+𝜔^𝑇(𝑘)𝐵^𝑇₁𝑃𝐵₁𝜔(𝑘)=𝑥^𝑇(𝑘)[𝐴^𝑇₀𝑃𝐴₀−𝑃+𝐴^𝑇₀𝑃𝐴₁+𝐴^𝑇₁𝑃𝐴₀]𝑥(𝑘)+2𝑥^𝑇(𝑘)[𝑌−𝐴^𝑇₀𝑃𝐴₁]×[𝑥(𝑘)−𝑥(𝑘−𝑑(𝑘))]+2𝑥^𝑇(𝑘−𝑑(𝑘))𝑊[𝑥(𝑘)−𝑥(𝑘−𝑑(𝑘))]−[2𝑥^𝑇(𝑘)𝑌^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)+2𝑥^𝑇(𝑘−𝑑(𝑘))𝑊^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦(ℓ)]+2𝑥^𝑇(𝑘)𝐴^𝑇₀𝑃𝐵₁𝜔(𝑘)+𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁𝑃𝐴₁𝑥(𝑘−𝑑(𝑘))+2𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁𝑃𝐵₁𝜔(𝑘)+𝜔^𝑇(𝑘)𝐵^𝑇₁𝑃𝐵₁=1𝜔(𝑘)𝑑(𝑘)^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}{𝑥^𝑇(𝑘)[𝐴^𝑇₀𝑃𝐴₀−𝑃+𝑌+𝑌^𝑇]𝑥(𝑘)+2𝑥^𝑇(𝑘)[𝐴^𝑇₀𝑃𝐴₁−𝑌+𝑊^𝑇]𝑥(𝑘−𝑑(𝑘))+2𝑥^𝑇(𝑘)𝐴^𝑇₀𝑃𝐵₁𝜔(𝑘)+2𝑥^𝑇(𝑘)[−𝑑(𝑘)𝑌]𝑦(ℓ)+𝑥^𝑇(𝑘−𝑑(𝑘))[𝐴^𝑇₁𝑃𝐴₁−𝑊−𝑊^𝑇]𝑥(𝑘−𝑑(𝑘))+2𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁𝑃𝐵₁𝜔(𝑘)+2𝑥^𝑇(𝑘−𝑑(𝑘))[−𝑑(𝑘)𝑊]𝑦(ℓ)+𝜔^𝑇(𝑘)𝐵^𝑇₁𝑃𝐵₁𝜔(𝑘)}. (3.9) Similar to [17], we have Δ𝑉₂(𝑘)=𝑉₂(𝑘+1)−𝑉₂(𝑘)≤(ℎ−ℎ+1)𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−𝑥^𝑇=1(𝑘−𝑑(𝑘))𝑄𝑥(𝑘−𝑑(𝑘))𝑑(𝑘)^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}{(ℎ−ℎ+1)𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑(𝑘))𝑄𝑥(𝑘−𝑑(𝑘))}. (3.10) After some manipulations, we obtain Δ𝑉₃(𝑘)=𝑉₃(𝑘+1)−𝑉₃(𝑘)=𝑦^𝑇(𝑘)ℎ𝑅𝑦(𝑘)−^𝑘−1_{ℓ=𝑘−ℎ}𝑦^𝑇(ℓ)𝑅𝑦(ℓ). (3.11) Observe that −^𝑘−1_{ℓ=𝑘−ℎ}𝑦^𝑇(ℓ)𝑅𝑦(ℓ)≤−^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦^𝑇(ℓ)𝑅𝑦(ℓ),𝑦(𝑘)=𝑥(𝑘+1)−𝑥(𝑘)=(𝐴₀−𝐼)𝑥(𝑘)+𝐴₁𝑥(𝑘−𝑑(𝑘))+𝐵₁𝜔(𝑘). (3.12) This together with (3.11) gives Δ𝑉₃(𝑘)≤𝑦^𝑇(𝑘)ℎ𝑅𝑦(𝑘)−^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦^𝑇(ℓ)𝑅𝑦(ℓ)=[(𝐴₀−𝐼)𝑥(𝑘)+𝐴₁𝑥(𝑘−𝑑(𝑘))+𝐵₁𝜔(𝑘)]^𝑇ℎ𝑅×[(𝐴₀−𝐼)𝑥(𝑘)+𝐴₁𝑥(𝑘−𝑑(𝑘))+𝐵₁𝜔(𝑘)]−^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦^𝑇=1(ℓ)𝑅𝑦(ℓ)𝑑(𝑘)^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}{𝑥^𝑇(𝑘)(𝐴₀−𝐼)^𝑇ℎ𝑅(𝐴₀−𝐼)𝑥(𝑘)−𝑦^𝑇(ℓ)𝑑(𝑘)𝑅𝑦(ℓ)+2𝑥^𝑇(𝑘)(𝐴₀−𝐼)^𝑇ℎ𝑅𝐴₁𝑥(𝑘−𝑑(𝑘))+2𝑥^𝑇(𝑘)(𝐴₀−𝐼)^𝑇ℎ𝑅𝐵₁𝜔(𝑘)+𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁ℎ𝑅𝐴₁𝑥(𝑘−𝑑(𝑘))+2𝑥^𝑇(𝑘−𝑑(𝑘))𝐴^𝑇₁ℎ𝑅𝐵₁𝜔(𝑘)+𝜔^𝑇(𝑘)𝐵^𝑇₁ℎ𝑅𝐵₁𝜔(𝑘)}. (3.13) Then, from (3.7)–(3.13), we have 1Δ𝑉(𝑘)=𝑉(𝑘+1)−𝑉(𝑘)=𝑑(𝑘)^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝜂^𝑇(𝑘,ℓ)Ψ₁(𝑑(𝑘))𝜂(𝑘,ℓ), (3.14) where 𝜂(𝑘,ℓ)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝑦^𝑇(ℓ)𝜔^𝑇(𝑘)^𝑇,Ψ₁Ψ(𝑑(𝑘))=¹(1,1)Ψ₁(1,2)−𝑑(𝑘)𝑌Ψ₁(1,4)∗Ψ₁(2,2)−𝑑(𝑘)𝑊Ψ₁(2,4)∗∗−𝑑(𝑘)𝑅0∗∗∗Ψ₁,(4,4) (3.15) with Ψ₁(1,1)=−𝑃+𝑌+𝑌^𝑇+(ℎ−ℎ+1)𝑄+𝐴^𝑇₀𝑃𝐴₀+(𝐴₀−𝐼)^𝑇ℎ𝑅(𝐴₀Ψ−𝐼),₁(1,2)=−𝑌+𝑊^𝑇+𝐴^𝑇₀𝑃𝐴₁+(𝐴₀−𝐼)^𝑇ℎ𝑅𝐴₁,Ψ₁(1,4)=𝐴^𝑇₀𝑃𝐵₁+(𝐴₀−𝐼)^𝑇ℎ𝑅𝐵₁,Ψ₁(2,2)=−𝑊−𝑊^𝑇−𝑄+𝐴^𝑇₁𝑃𝐴₁+𝐴^𝑇₁ℎ𝑅𝐴₁,Ψ₁(2,4)=𝐴^𝑇₁𝑃𝐵₁+𝐴^𝑇₁ℎ𝑅𝐵₁,Ψ₁(4,4)=𝐵^𝑇₁𝑃𝐵₁+𝐵^𝑇₁ℎ𝑅𝐵₁. (3.16) In the next, we will prove the conclusion from two aspects. First, we establish the asymptotic stability of system (Σ_ncl) with 𝜔(𝑘)=0 if (3.2) is satisfied. For this situation, (3.14) becomes 1Δ𝑉(𝑘)≤𝑑(𝑘)^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝜓^𝑇(𝑘,ℓ)Ψ₂(𝑑(𝑘))𝜓(𝑘,ℓ), (3.17) where 𝜓(𝑘,ℓ)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝑦^𝑇(ℓ)^𝑇,Ψ₂Ψ(𝑑(𝑘))=¹(1,1)Ψ₁(1,2)−𝑑(𝑘)𝑌∗Ψ₁(2,2)−𝑑(𝑘)𝑊.∗∗−𝑑(𝑘)𝑅 (3.18) By Lemma 2.2, it can be verified that Δ𝑉(𝑘)<0 if (3.2) is true. Therefore, system (Σ_ncl) with 𝜔(𝑘)=0 is asymptotically stable according to the Lyapunov-Krasovskii stability theorem.
Second, we show that subject to the zero initial condition, the discrete time-delay system (Σ_ncl) has a prescribed 𝐻_∞ disturbance attenuation level 𝛾>0, that is, ‖𝑧‖₂<𝛾‖𝜔‖₂ for all nonzero 𝜔∈ℒ₂. To this end, we introduce the following performance index: 𝐽_𝑁=^𝑁_𝑘=0[𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝜔^𝑇(𝑘)𝜔(𝑘)], (3.19) where the scalar 𝑁∈ℕ. Noting the zero initial condition and (3.14), one can verify that 𝐽_𝑁=^𝑁_𝑘=0[𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝜔^𝑇≤(𝑘)𝜔(𝑘)+Δ𝑉(𝑘)]−𝑉(𝑁+1)_∞𝑘=0[𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝜔^𝑇=(𝑘)𝜔(𝑘)+Δ𝑉(𝑘)]_{∞𝑘=0𝑘−1ℓ=𝑘−𝑑(𝑘)}1𝜂𝑑(𝑘)^𝑇(𝑘,ℓ)Ψ₃(𝑑(𝑘))𝜂(𝑘,ℓ), (3.20) where Ψ₃Ψ(𝑑(𝑘))=³(1,1)Ψ₃(1,2)−𝑑(𝑘)𝑌Ψ₃(1,4)∗Ψ₃(2,2)−𝑑(𝑘)𝑊Ψ₃(2,4)∗∗−𝑑(𝑘)𝑅0∗∗∗Ψ₃,(4,4) (3.21) with Ψ₃(1,1)=Ψ₁(1,1)+𝐶^𝑇₀𝐶₀,Ψ₃(1,2)=Ψ₁(1,2)+𝐶^𝑇₀𝐶₁,Ψ₃(1,4)=Ψ₁(1,4)+𝐶^𝑇₀𝐷₁₁,Ψ₃(2,2)=Ψ₁(2,2)+𝐶^𝑇₁𝐶₁,Ψ₃(2,4)=Ψ₁(2,4)+𝐶^𝑇₁𝐷₁₁,Ψ₃(4,4)=Ψ₁(4,4)+𝐷^𝑇₁₁𝐷₁₁−𝛾²𝐼. (3.22) Now, by Lemma 2.2, it follows from (3.2) that Ψ₃(𝑑(𝑘))<0, which together with (3.20) ensures that 𝐽_𝑁<0. This further implies that ‖𝑧‖₂<𝛾‖𝜔‖₂ holds under the zero initial condition. This complements the proof.

Remark 3.2. In Theorem 3.1, two slack variables 𝑌 and 𝑊 are introduced to reduce some conservatism in the existing delay-dependent conditions for the 𝐻_∞ control problem, while no bounding techniques for cross terms are involved. By doing so, we have provided a more flexible condition in (3.2). The advantage of these introduced variables can be seen from the numerical example later.

Remark 3.3. In [19], based on a descriptor system transformation method, a delay-dependent condition on the 𝐻_∞ control issue for system (Σ_ncl) was proposed. However, there is an additional constraint on the matrix 𝐴₁, that is, 𝐴₁ should be nonsingular. While, Theorem 3.1 in this paper gets rid of this constraint.

Very recently, for discrete time-delay system (Σ_ncl), a less conservative delay-dependent 𝐻_∞ condition was proposed in [20]. The rationale behind the method lies in providing a finite sum inequality as follows.

Lemma 3.4 (finite sum inequality [20, Lemma  1]). For any matrices 𝑀₁, 𝑀₂, 𝑍₁₁, 𝑍₁₂, 𝑍₂₂, 𝑅∈ℝ^𝑛×𝑛, where 𝑅=𝑅^𝑇≥0, and 𝑍₁₃, 𝑍₂₃, 𝑀₃∈ℝ^𝑛×𝑞, 𝑍₃₃∈ℝ^𝑞×𝑞, the following inequality holds: −^𝑘−1_{ℓ=𝑘−𝑑(𝑘)}𝑦^𝑇(ℓ)𝑅𝑦(ℓ)≤𝜉^𝑇𝑣(𝑘)¹¹𝑣₁₂𝑀₃+ℎ𝑍₁₃∗𝑣₂₂−𝑀₃+ℎ𝑍₂₃∗∗ℎ𝑍₃₃𝜉(𝑘), (3.23) where 𝜉(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝜔^𝑇(𝑘)^𝑇,𝑅𝑀₁𝑀₂𝑀₃∗𝑍₁₁𝑍₁₂𝑍₁₃∗∗𝑍₂₂𝑍₂₃∗∗∗𝑍₃₃≥0, (3.24) with 𝑣₁₁=𝑀^𝑇₁+𝑀₁+ℎ𝑍₁₁,𝑣₁₂=−𝑀^𝑇₁+𝑀₂+ℎ𝑍₁₂,𝑣₂₂=−𝑀^𝑇₂−𝑀₂+ℎ𝑍₂₂. (3.25)

By Theorem 3.1, we can obtain the following delay-dependent 𝐻_∞ disturbance attenuation condition, which has been reported in [20] recently.

Corollary 3.5 (see [20, Proposition  1]). For a given 𝛾>0, system (Σ_ncl) is asymptotically stable with a prescribed 𝐻_∞ disturbance attenuation level 𝛾 for any time-varying delay satisfying (2.4) if there exist matrices 𝑃>0,𝑄>0,𝑅>0,𝑀_𝑖,𝑍_𝑖𝑗(𝑖,𝑗=1,2,3) with appropriate dimensions such that (3.24) and the following inequality hold: Ω₁Φ∶=¹¹Φ₁₂Φ₁₃(𝐴₀−𝐼)^𝑇ℎ(𝐴₀−𝐼)^𝑇𝐶^𝑇₀∗Φ₂₂Φ₂₃𝐴^𝑇₁ℎ𝐴^𝑇₁𝐶^𝑇₁∗∗Φ₃₃𝐵^𝑇₁ℎ𝐵^𝑇₁𝐷^𝑇₁₁∗∗∗−𝑃⁻¹00∗∗∗∗−ℎ𝑅⁻¹0∗∗∗∗∗−𝐼<0, (3.26) where Φ₁₁=𝑃(𝐴₀−𝐼)+(𝐴₀−𝐼)^𝑇𝑃+𝑀^𝑇₁+𝑀₁+(ℎ−ℎ+1)𝑄+ℎ𝑍₁₁,Φ₁₂=𝑃𝐴₁−𝑀^𝑇₁+𝑀₂+ℎ𝑍₁₂,Φ₁₃=𝑃𝐵₁+𝑀₃+ℎ𝑍₁₃,Φ₂₂=−𝑀^𝑇₂−𝑀₂−𝑄+ℎ𝑍₂₂,Φ₂₃=−𝑀₃+ℎ𝑍₂₃,Φ₃₃=−𝛾²𝐼+ℎ𝑍₃₃. (3.27)

In the following, we will show that the result in [20] can be deduced from Theorem 3.1.

Proof. It is easy to see that (3.26) is equivalent to Φ₁₁Φ₁₂Φ₁₃(𝐴₀−𝐼)^𝑇𝑃(𝐴₀−𝐼)^𝑇ℎ𝑅𝐶^𝑇₀∗Φ₂₂Φ₂₃𝐴^𝑇₁𝑃𝐴^𝑇₁ℎ𝑅𝐶^𝑇₁∗∗Φ₃₃𝐵^𝑇₁𝑃𝐵^𝑇₁ℎ𝑅𝐷^𝑇₁₁∗∗∗−𝑃00∗∗∗∗−ℎ𝑅0∗∗∗∗∗−𝐼<0. (3.28) By (3.24) and using the Schur complement formula, we have 𝑍₁₁𝑍₁₂𝑍₁₃∗𝑍₂₂𝑍₂₃∗∗𝑍₃₃−𝑀^𝑇₁𝑀^𝑇₂𝑀^𝑇₃𝑅⁻¹𝑀₁𝑀₂𝑀₃≥0. (3.29) This together with (3.28) implies Φ₁₁Φ₁₂Φ₁₃(𝐴₀−𝐼)^𝑇𝑃(𝐴₀−𝐼)^𝑇ℎ𝑅𝐶^𝑇₀ℎ𝑀^𝑇₁∗Φ₂₂Φ₂₃𝐴^𝑇₁𝑃𝐴^𝑇₁ℎ𝑅𝐶^𝑇₁ℎ𝑀^𝑇₂∗∗−𝛾²𝐼𝐵^𝑇₁𝑃𝐵^𝑇₁ℎ𝑅𝐷^𝑇₁₁ℎ𝑀^𝑇₃∗∗∗−𝑃000∗∗∗∗−ℎ𝑅00∗∗∗∗∗−𝐼0∗∗∗∗∗0−ℎ𝑅<0, (3.30) where Φ₁₁=𝑃(𝐴₀−𝐼)+(𝐴₀−𝐼)^𝑇𝑃+𝑀^𝑇₁+𝑀₁+(ℎ−ℎ+1)𝑄,Φ₁₂=𝑃𝐴₁−𝑀^𝑇₁+𝑀₂,Φ₁₃=𝑃𝐵₁+𝑀₃,Φ₂₂=−𝑀^𝑇₂−𝑀₂−𝑄,Φ₂₃=−𝑀₃. (3.31) Denote 𝐿₁=.𝐼00−𝐼0000𝐼0000000𝐼0000000𝐼0000000𝐼0000000𝐼0000000𝐼 (3.32) Pre- and postmultiplying (3.30) by 𝐿₁ and 𝐿^𝑇₁, respectively, yields ϝ−𝑀^𝑇₁+𝑀₂𝑀₃𝐴^𝑇₀𝑃(𝐴₀−𝐼)^𝑇ℎ𝑅𝐶^𝑇₀ℎ𝑀^𝑇₁∗−𝑀^𝑇₂−𝑀₂−𝑄−𝑀₃𝐴^𝑇₁𝑃𝐴^𝑇₁ℎ𝑅𝐶^𝑇₁ℎ𝑀^𝑇₂∗∗−𝛾²𝐼𝐵^𝑇₁𝑃𝐵^𝑇₁ℎ𝑅𝐷^𝑇₁₁ℎ𝑀^𝑇₃∗∗∗−𝑃000∗∗∗∗−ℎ𝑅00∗∗∗∗∗−𝐼0∗∗∗∗∗0−ℎ𝑅<0, (3.33) where ϝ=−𝑃+𝑀^𝑇₁+𝑀₁+(ℎ−ℎ+1)𝑄.
Letting 𝑀₁=𝑌^𝑇,𝑀₂=𝑊^𝑇,𝑀₃=0, we have Θ₁₁−𝑌+𝑊^𝑇0𝐴^𝑇₀𝑃(𝐴₀−𝐼)^𝑇ℎ𝑅𝐶^𝑇₀ℎ𝑌∗−𝑊−𝑊^𝑇−𝑄0𝐴^𝑇₁𝑃𝐴^𝑇₁ℎ𝑅𝐶^𝑇₁ℎ𝑊∗∗−𝛾²𝐼𝐵^𝑇₁𝑃𝐵^𝑇₁ℎ𝑅𝐷^𝑇₁₁0∗∗∗−𝑃000∗∗∗∗−ℎ𝑅00∗∗∗∗∗−𝐼0∗∗∗∗∗0−ℎ𝑅<0, (3.34) where Θ₁₁ is defined in Theorem 3.1.
Denote 𝐿₂=.𝐼0000000𝐼00000000000𝐼00𝐼0000000𝐼0000000𝐼0000000𝐼0 (3.35) Pre- and postmultiplying (3.34) by 𝐿₂ and its transpose, respectively, yields (3.2). This completes the proof.

Remark 3.6. From Corollary 3.5, it is noted that Theorem 3.1 in this paper is less conservative than Corollary 1 which was reported in [20]. It should be pointed out that neither model transformation (e.g., [23]) nor bounding technique (e.g., [19]) is employed here. Although it is proved that the finite sum inequality approach in [20] is better than other reported ones when dealing with delay-dependent stability analysis problem for discrete time-delay systems, it still gives relatively conservative results.

Remark 3.7. Compared with the delay-dependent 𝐻_∞ disturbance attenuation condition in [20], it is worth noting that one of the advantages in our paper is that the inequality in (3.2) involves significantly fewer variables than those in [20]. Specifically, in the case when 𝑥(𝑘)∈ℝ^𝑛, the number of the variables to be solved in (3.2) is (𝑛(7𝑛+3))/2, while in [20] the number of variables is (13𝑛²+2𝑞²+6𝑛𝑞+3𝑛)/2. When 𝑞=𝑛, that is, 𝜔(𝑘)∈ℝ^𝑛, the number of variables in [20] becomes (21𝑛(𝑛+3))/2, which is around 3 times more than those in Theorem 3.1. Therefore, from mathematical and practical points of view, our condition is more desirable than that in [20].

Now, we are in a position to solve the controller gain 𝐾 from (3.2).

Define Π₁=diag{𝐼,𝐼,𝐼,𝐼,𝐼,𝑅⁻¹,𝐼}. Multiplying (3.2) by Π^𝑇₁ and Π₁ on the left-hand side and the right-hand side, respectively, yields Θ₁₁−𝑌+𝑊^𝑇−ℎ𝑌0𝐴^𝑇₀𝑃ℎ(𝐴₀−𝐼)^𝑇𝐶^𝑇₀∗−𝑊−𝑊^𝑇−𝑄−ℎ𝑊0𝐴^𝑇₁𝑃ℎ𝐴^𝑇₁𝐶^𝑇₁∗∗−ℎ𝑅0000∗∗∗−𝛾²𝐼𝐵^𝑇₁𝑃ℎ𝐵^𝑇₁𝐷^𝑇₁₁∗∗∗∗−𝑃00∗∗∗∗∗−ℎ𝑅⁻¹0∗∗∗∗∗∗−𝐼<0. (3.36) Let 𝑃⁻¹=𝑋. Defining Π₂=diag{𝑋,𝑋,𝑋,𝐼,𝑋,𝐼,𝐼}, then after performing congruence transformations on (3.36) by Π₂, we have 𝑋Θ₁₁𝑋𝑋(−𝑌+𝑊^𝑇)𝑋−ℎ𝑋𝑌𝑋0𝑋𝐴^𝑇₀ℎ𝑋(𝐴₀−𝐼)^𝑇𝑋𝐶^𝑇₀∗𝑋(−𝑊−𝑊^𝑇−𝑄)𝑋−ℎ𝑋𝑊𝑋0𝑋𝐴^𝑇₁ℎ𝑋𝐴^𝑇₁𝑋𝐶^𝑇₁∗∗−ℎ𝑋𝑅𝑋0000∗∗∗−𝛾²𝐼𝐵^𝑇₁ℎ𝐵^𝑇₁𝐷^𝑇₁₁∗∗∗∗−𝑋00∗∗∗∗∗−ℎ𝑅⁻¹0∗∗∗∗∗∗−𝐼<0. (3.37) Setting 𝑌=𝑋𝑌𝑋, 𝑄=𝑋𝑄𝑋, 𝑊=𝑋𝑊𝑋, 𝐾=𝐾𝑋, 𝑅=𝑅⁻¹, we obtain Θ₁₁−𝑌+𝑊^𝑇−ℎ𝑌0Θ₁₂Θ₁₃Θ₁₄∗−𝑊−𝑊^𝑇−𝑄−ℎ𝑊0𝑋𝐴^𝑇₁ℎ𝑋𝐴^𝑇₁𝑋𝐶^𝑇₁∗∗−ℎ𝑋𝑅⁻¹𝑋0000∗∗∗−𝛾²𝐼𝐵^𝑇₁ℎ𝐵^𝑇₁𝐷^𝑇₁₁∗∗∗∗−𝑋00∗∗∗∗∗−ℎ𝑅0∗∗∗∗∗∗−𝐼<0, (3.38) where Θ₁₁=−𝑋+𝑌+𝑌^𝑇+(ℎ−ℎ+1)𝑄,Θ₁₂=𝑋𝐴^𝑇₀+𝐾^𝑇𝐵^𝑇₂,Θ₁₃=ℎ𝑋𝐴^𝑇₀+ℎ𝐾^𝑇𝐵^𝑇₂−ℎ𝑋,Θ₁₄=𝑋𝐶^𝑇₀+𝐾^𝑇𝐷^𝑇₁₂. (3.39)

It is clear that (3.38) is a nonlinear matrix inequality in the matrix variables 𝑋, 𝑄, 𝑅, 𝑌, 𝑊, and 𝐾, due to the existence of the nonlinear term −ℎ𝑋𝑅⁻¹𝑋. In order to solve the desired controller 𝐾, we will propose three methods in the sequel.

Let 𝑅=𝑃, that is, take a particular Lyapunov-Krasovskii functional in (3.5). Then, the following result holds naturally.

Theorem 3.8. System (Σ_ncl) is asymptotically stable with a prescribed 𝐻_∞ disturbance attenuation level 𝛾>0, if there exist matrices 𝑋>0, 𝑄>0, 𝑌, 𝑊, and 𝐾 of appropriate dimensions such that the following LMI holds: Ξ₁=Θ₁₁−𝑌+𝑊^𝑇−ℎ𝑌0Θ₁₂Θ₁₃Θ₁₄∗−𝑊−𝑊^𝑇−𝑄−ℎ𝑊0𝑋𝐴^𝑇₁ℎ𝑋𝐴^𝑇₁𝑋𝐶^𝑇₁∗∗−ℎ𝑋0000∗∗∗−𝛾²𝐼𝐵^𝑇₁ℎ𝐵^𝑇₁𝐷^𝑇₁₁∗∗∗∗−𝑋00∗∗∗∗∗−ℎ𝑋0∗∗∗∗∗∗−𝐼<0. (3.40) Moreover, a robustly stabilizing state feedback controller is given by (2.5) with 𝐾=𝐾𝑋⁻¹.

Remark 3.9. Theorem 3.8 provides a simple method in solving the controller gain 𝐾 by introducing a special Lyapunov-Krasovskii functional. Although it has some good merits, it may bring some conservatism due to the restriction of 𝑅=𝑃.

Note that (𝑅−𝑋)^𝑇𝑅⁻¹(𝑅−𝑋)≥0, (3.41) which implies −ℎ𝑋𝑅⁻¹𝑋≤ℎ𝑅−2ℎ𝑋. (3.42) From (3.38) and (3.42), the following theorem follows immediately.

Theorem 3.10. System (Σ_ncl) is asymptotically stable with a prescribed 𝐻_∞ disturbance attenuation level 𝛾>0, if there exist matrices 𝑋>0, 𝑄>0, 𝑅>0, 𝑌, 𝑊, and 𝐾 of appropriate dimensions such that the following LMI holds: Ξ₂=Θ₁₁−𝑌+𝑊^𝑇−ℎ𝑌0Θ₁₂Θ₁₃Θ₁₄∗−𝑊−𝑊^𝑇−𝑄−ℎ𝑊0𝑋𝐴^𝑇₁ℎ𝑋𝐴^𝑇₁𝑋𝐶^𝑇₁∗∗ℎ𝑅−2ℎ𝑋0000∗∗∗−𝛾²𝐼𝐵^𝑇₁ℎ𝐵^𝑇₁𝐷^𝑇₁₁∗∗∗∗−𝑋00∗∗∗∗∗−ℎ𝑅0∗∗∗∗∗∗−𝐼<0. (3.43) Moreover, a robustly stabilizing state feedback controller is given by (2.5) with 𝐾=𝐾𝑋⁻¹.

Remark 3.11. It is clear that there also exists conservatism because of the replacement −ℎ𝑋𝑅⁻¹𝑋 with ℎ𝑅−2ℎ𝑋.

In the sequel, we will resort to the cone complementary linearization method [24] to further reduce the conservatism. Introduce a new matrix variable 𝑆>0, which satisfies 𝑋𝑅⁻¹𝑋≥𝑆. (3.44) It is easily seen that inequality (3.44) is more general than that in (3.42). Note that (3.44) is equivalent to 𝑆⁻¹𝑋⁻¹∗𝑅⁻¹≥0. (3.45) Letting 𝒮=𝑆⁻¹, 𝒳=𝑋⁻¹, ℛ=𝑅⁻¹, we obtain the following theorem.

Theorem 3.12. System (Σ_ncl) is asymptotically stable with a prescribed 𝐻_∞ disturbance attenuation level 𝛾>0, if there exist matrices 𝑋>0, 𝑄>0, 𝑅>0, ℛ>0, 𝑆>0, 𝒮>0, 𝒳>0, 𝑌, 𝑊, and 𝐾 of appropriate dimensions such that Ξ₃=Θ₁₁−𝑌+𝑊^𝑇−ℎ𝑌0Θ₁₂Θ₁₃Θ₁₄∗−𝑊−𝑊^𝑇−𝑄−ℎ𝑊0𝑋𝐴^𝑇₁ℎ𝑋𝐴^𝑇₁𝑋𝐶^𝑇₁∗∗−ℎ𝑆0000∗∗∗−𝛾²𝐼𝐵^𝑇₁ℎ𝐵^𝑇₁𝐷^𝑇₁₁∗∗∗∗−𝑋00∗∗∗∗∗−ℎ𝑅0∗∗∗∗∗∗−𝐼<0,𝒮𝒳∗ℛ≥0,𝑆𝒮=𝐼,𝑋𝒳=𝐼,ℛ𝑅=𝐼. (3.46)(3.47)(3.48) Moreover, a robustly stabilizing state feedback controller is given by (2.5) with 𝐾=𝐾𝑋⁻¹.

Remark 3.13. As one can see that the inequality conditions in Theorem 3.12 are not strict LMI conditions due to the equation constraints in (3.48). However, by resorting to the cone complementary linearization method in [24] and the optimization solver in [25], the nonconvex feasibility problem formulated by (3.46), (3.47), and (3.48) can be transformed into the following nonlinear minimization problem subject to LMIs: minimizeTr(𝑆𝒮+𝑋𝒳+ℛ𝑅)subjectto(3.46)and𝐼𝑅(3.47),𝒮𝐼𝐼𝑆≥0,𝒳𝐼𝐼𝑋≥0,ℛ𝐼≥0. (3.49) According to the cone complementarity problem (CCP) in [24], if the solution of the above minimization problem is 6𝑛, we can say from Theorem 3.12 that system (Σ_ncl) is asymptotically stable with a prescribed 𝐻_∞ disturbance attenuation level 𝛾>0 via the controller (2.5) with 𝐾=𝐾𝑋⁻¹. Although it is very difficult to always find the global optimal solution, the proposed nonlinear minimizatiion problem is easier to solve than the original nonconvex feasibility problem. Based on the linearization method in [24], we can solve the above nonlinear minimization problem using an iterative algorithm presented in the following.

Algorithm 3.14. We have the following steps.Step 1. Choose a sufficiently initial 𝛾_ini>0 such that (3.46), (3.47), and (3.49) are feasible. Set 𝛾_𝑠0=𝛾_ini.Step 2. Find a feasible set (𝑋⁰,𝒳⁰,𝑆⁰,𝒮⁰,ℛ⁰,𝑅⁰,𝑌⁰,𝑊⁰,𝐾⁰,𝑄⁰,𝜀⁰) satisfying (3.46), (3.47), and (3.49). Set 𝑘=0.Step 3. Solve the following LMI problem: minimizeTr(𝑆^𝑘𝒮+𝑋^𝑘𝒳+ℛ^𝑘𝑅+𝑆𝒮^𝑘+𝑋𝒳^𝑘+ℛ𝑅^𝑘)subjectto(3.46),(3.47),and(3.49). (3.50) Set 𝑆^𝑘+1=𝑆, 𝑋^𝑘+1=𝑋, ℛ^𝑘+1=ℛ, 𝒮^𝑘+1=𝒮, 𝒳^𝑘+1=𝒳, 𝑅^𝑘+1=𝑅.Step 4. If matrix (3.46) is satisfied and |Tr(𝑆^𝑘𝒮+𝑋^𝑘𝒳+ℛ^𝑘𝑅+𝑆𝒮^𝑘+𝑋𝒳^𝑘+ℛ𝑅^𝑘)−6𝑛|<𝛿 (3.51) for some sufficient small scalar 𝛿>0, then decrease 𝛾_ini to some extent and set 𝛾_𝑠0=𝛾_ini and go to Step 2. If one of the conditions in (3.47) and (3.51) is not satisfied within a specified number of iterations, then exit. Otherwise, set 𝑘=𝑘+1 and go to Step 3.

Now, we are in a position to present the delay-dependent robust conditions concerning 𝐻_∞ control of system (Σ) with uncertainties based on Theorems 3.8, 3.10, and 3.12, respectively. By Lemma 2.3, we can easily have the following results.

Theorem 3.15. System (Σ) is asymptotically stable with a prescribed 𝐻_∞ disturbance attenuation level 𝛾>0, if there exist a scalar 𝜀>0, matrices 𝑋>0, 𝑄>0, 𝑌, 𝑊, and 𝐾 of appropriate dimensions such that the following LMI holds: Ξ₁𝜀𝒟ℰ^𝑇∗−𝜀𝐼0∗∗−𝜀𝐼<0, (3.52) where Ξ₁ is defined in (3.40), and 𝒟=0000𝒟^𝑇₁ℎ𝒟^𝑇₁𝒟^𝑇₂^𝑇,ℰ=𝐸₁𝑋+𝐸₄𝐾𝐸₂𝑋0𝐸₃000. (3.53)(3.54) Moreover, a robustly stabilizing state feedback controller is given by (2.5) with 𝐾=𝐾𝑋⁻¹.

Theorem 3.16. System (Σ) is asymptotically stable with a prescribed 𝐻_∞ disturbance attenuation level 𝛾>0, if there exist a scalar 𝜀>0, matrices 𝑋>0, 𝑄>0, 𝑅>0, 𝑌, 𝑊, and 𝐾 of appropriate dimensions such that the following LMI holds: Ξ₂𝜀𝒟ℰ^𝑇∗−𝜀𝐼0∗∗−𝜀𝐼<0, (3.55) where Ξ₂, 𝒟, ℰ are defined in (3.43), (3.53), and (3.54), respectively. A robustly stabilizing state feedback controller is given by (2.5) with 𝐾=𝐾𝑋⁻¹.

Theorem 3.17. System (Σ) is asymptotically stable with a prescribed 𝐻_∞ disturbance attenuation level 𝛾>0, if there exist a scalar 𝜀>0, matrices 𝑋>0, 𝑄>0, 𝑅>0, ℛ>0, 𝑆>0, 𝒮>0, 𝒳>0, 𝑌, 𝑊, and 𝐾 of appropriate dimensions such that (3.47), (3.49) and the following LMI holds: Ξ₃𝜀𝒟ℰ^𝑇∗−𝜀𝐼0∗∗−𝜀𝐼<0, (3.56) where Ξ₃, 𝒟, ℰ are defined in (3.46), (3.53), and (3.54), respectively. 𝐾=𝐾𝑋⁻¹ is the corresponding controller which is derived.

4. Examples

In this section, two examples are used to demonstrate the effectiveness of the proposed methods.

Example 4.1. Consider the following discrete-time systems with time-varying delay: 𝑥(𝑘+1)=(𝐴₀+𝒟₁𝐹(𝑘)𝐸₁)𝑥(𝑘)+(𝐴₁+𝒟₁𝐹(𝑘)𝐸₂)𝑥(𝑘−𝑑(𝑘))+𝐵₁𝜔(𝑘)+𝐵₂𝑢(𝑘),𝑧(𝑘)=𝐶₀𝑥(𝑘)+𝐷₁₂𝑢(𝑘),𝑥(𝑘)=0,∀𝑘≤0, (4.1) where 𝐴₀=1001.01,𝐴₁=−0.02−0.0050−0.01,𝐵₁=01,𝐵₂=00.01,𝐶₀=10,𝐷₁₂=0.1,𝒟₁𝐸=0.2𝐼,₁=𝐸₂=0.01𝐼,𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼, (4.2) and 𝑑(𝑘) is a delay satisfying (2.4).

In the following, different cases of 𝑑(𝑘) are involved.

Case 1. Delay 𝑑(𝑘) is time invariant.
First, suppose ℎ=ℎ=64. For this situation, we will compare the results in Theorems 3.15, 3.16, and 3.17. For this reason, we calculate the minimum value of 𝛾 for which system (4.1) is robustly stabilizable via state feedback (2.5). The obtained results are listed in Table 1, from which we can see that the conditions in Theorem 3.17 are less conservative than those in Theorems 3.15 and 3.16. Therefore, we will only compare the results in Theorem 3.17 with those in the previous literatures in the sequel.
Second, when ℎ=ℎ=64, Zhang and Han [20] also calculated the achieved minimum 𝐻_∞ performances 𝛾, the corresponding controller gain 𝐾, and the iterations. Here, in order to show much less conservative results (or lower computational burden) of Theorem 3.17 than [20], we give Table 2. Noting from this table, we conclude that in order to achieve the same disturbance attenuation level, Theorem 3.17 needs significantly less iterations and smaller gain. From this table, we also have verified Remark 3.9.
Third, in [19, 20], the authors also calculated the achieved minimum 𝐻_∞ performance for ℎ=64, respectively. However, according to Theorem 3.17, much less 𝐻_∞ performance is obtained, which is listed in Table 3. From this table, one can see that Theorem 3.17 in this paper provides much less 𝐻_∞ performances.
Now, we are in a position to calculate the maximum delay bound ℎ, which can guarantee that system (4.1) is robustly stabilizable via state feedback (2.5). The details are given in Table 4.

Table 1 

The achieved minimum 𝐻_∞ performances 𝛾 in this paper and corresponding controller gain 𝐾 for ℎ=ℎ=64.

Table 2 

The iterations and corresponding controller gain 𝐾 for ℎ=ℎ=64 under different cases of 𝛾.

Table 3 

The achieved minimum 𝐻_∞ performances 𝛾 and corresponding controller gain 𝐾 for ℎ=ℎ=64.

Table 4 

The maximum delay bound ℎ for system (4.1) under Case 1.

Case 2. Delay 𝑑(𝑘) is time varying.
Under this case, Fridman and Shaked [19] concluded that system (4.1) can be stabilized for all ℎ≤43. In [20], it is obtained that system (4.1) is robustly stabilizable for ℎ≤48. However, by Theorem 3.17, we have that system (4.1) is robustly stabilizable for ℎ≤52. The details are shown in Table 5.
In [20], Zhang and Han also gave the minimum 𝐻_∞ performances 𝛾 and corresponding controller gain 𝐾 for a set of ℎ when ℎ=48. Here, we also present a table to demonstrate the lower computational complexity and smaller controller gains than those in [20], which are listed in Table 6.

Table 5 

The maximum delay bound ℎ for system (4.1) under Case 2.

Table 6 

The corresponding controller gain 𝐾 for different ℎ and 𝛾 when ℎ=48.

Example 4.2. Consider the discrete-time systems (4.1) (this example was first presented in [27]) with matrices 𝐴₀=2101,𝐴₁=0.20.100.1,𝐵₁=,𝐵0.10.1₂=11,𝐶₀=11,𝐷₁₂=1,𝒟₁𝐸=0,₁=𝐸₂=0. (4.3)

From Table 7, we can see that the condition in our paper can obtain a smaller 𝐻_∞ performance 𝛾 than [18, 27] for this example.

Table 7 

The achieved minimum 𝐻_∞ performances 𝛾 and corresponding controller gain 𝐾.

5. Conclusions

The problem of 𝐻_∞ control for uncertain discrete-time systems with time-varying delay has been studied. By introducing slack matrix variables, delay-dependent LMI based conditions have been developed to design a stable state feedback controller, which ensures the asymptotic stability of the resulting closed-loop system and guarantees a prescribed disturbance attenuation level irrespective of all the admissible uncertainties. Numerical examples have been provided to demonstrate the effectiveness and applicability of the proposed approach.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (nos. 60674027, 60904022, 60974127, 60875039), Key Projects of Science and Technology of Ministry of Education of China (no. 208074), and the Scientific Research Foundation of Qufu Normal University (no. XJ200845).