Exponential Estimates and Stabilization of Discrete-Time Singular Time-Delay Systems Subject to Actuator Saturation

Abstract

This paper is concerned with exponential estimates and stabilization of a class of discrete-time singular systems with time-varying state delays and saturating actuators. By constructing a decay-rate-dependent Lyapunov-Krasovskii function and utilizing the slow-fast decomposition technique, an exponential admissibility condition, which not only guarantees the regularity, causality, and exponential stability of the unforced system but also gives the corresponding estimates of decay rate and decay coefficient, is derived in terms of linear matrix inequalities (LMIs). Under the proposed condition, the exponential stabilization problem of discrete-time singular time-delay systems subject actuator saturation is solved by designing a stabilizing state feedback controller and determining an associated set of safe initial conditions, for which the local exponential stability of the saturated closed-loop system is guaranteed. Two numerical examples are provided to illustrate the effectiveness of the proposed results.

1. Introduction

Singular time-delay systems (STDSs) arise naturally in many engineering fields such as electric networks, chemical processes, lossless transmission lines, and so forth [1]. A STDS is a mixture of delay differential equations and delay difference equations; such a complex nature of STDS leads to abundant dynamics, for example, non-strictly proper transcendental equations, irregularity, impulses or non-causality. Therefore, the study of such systems is much more complicated than that for normal state-space time-delay systems. In the past two decades, a great number of stability results on STDSs have been reported in the literature; see, for example, [2–8] and the references therein.

It is noted that many stability results for STDSs are concerned with asymptotic stability. Practically, however, exponential stability is more important because the transient process of a system can be described more clearly once the decay rate is determined [9]. Therefore, in recent years, the study of exponential estimates problem of STDSs has received increasing attention, and a few approaches have been proposed. For example, in [10, 11], the STDS was decomposed into slow (differential) and fast (algebraic) subsystems and the exponential stability of the slow subsystem was proved by using the Lyapunov method. Subsequently, the solutions of the fast subsystem was bounded by an exponential term using a function inequality. However, this approach cannot give an estimate of the convergence rate of the system. To overcome this difficulty, Shu and Lam [12] and Lin et al. [13] adopted the Lyapunov-Krasovskii function method [14, 15] and some improvements have been obtained. In [16, 17], an exponential estimates approach for SSTDs was presented by employing the graph theory to establish an explicit expression of the state variables of fast subsystem in terms of those of slow subsystem and the initial conditions, which allows to prove the exponential stability of the fast subsystem. However, all of the above results are related to continuous-time STDSs. To the best of the authors' knowledge, the problem of exponential estimates of discrete-time STDSs has not been investigated yet. One possible reason is the difficulty in obtaining the estimates for solutions of the corresponding fast subsystem. Therefore, the first aim is to develop effective approach to give the exponential estimates of discrete-time STDSs.

On the other hand, actuator saturation is also an important phenomenon arising in engineering. Saturation nonlinearity not only deteriorates the performance of the closed-loop systems but also is the source of instability. Stabilization of normal state-space systems subject to actuator saturation has therefore attracted much attention from many researchers; see, for example, [18–22], and the references cited therein. Recently, some results for normal state-space systems have been generalized to singular systems. For example, semiglobal stabilization and output regulation of continuous-time singular system subject to input saturation were addressed in [23] by assuming that the open-loop system is semistable and impulse free which allows a state transformation such that the singular system is transformed into a normal system. Also, an algebraic Riccati equation approach to semiglobal stabilization of discrete-time singular linear systems with input saturation was proposed in [24] without any transformation of the original singular system. The invariant set approach developed for state-space system in [18] was extended to general, not necessarily semistable, continuous-time singular system in [25]. This approach was further extended to the analysis of the ℒ₂ gain and ℒ_∞ performance for continuous-time singular systems under actuator saturation [26] and the analysis and design of discrete-time singular systems under actuator saturation [27, 28], respectively. In [17], estimation of domain of attraction for continuous-time STDSs with actuator saturation and the design of static output feedback controller that maximize it were proposed. However, so far, few work exists to address the stabilization problem for discrete-time STDSs subject to actuator saturation, which forms the second object of this paper.

In this paper, we investigate the problems of exponential estimates and stabilization for a class of discrete-time singular systems with time-varying delays and saturating actuators. The main contributions of the paper are twofold: (1)In terms of linear matrix inequalities (LMIs), an exponential admissibility condition, which not only guarantees the regularity, causality and exponential stability of the unforced system but also gives the corresponding estimates of decay rate and decay coefficient, is derived by constructing a decay-rate-dependent Lyapunov-Krasovskii function and using the slow-fast decomposition. (2)The exponential stabilization problem of STDSs with saturating actuators is solved by designing a stabilizing state feedback controller and determining an associated set of safe initial conditions, for which the local exponential stability of the saturated closed-loop system is guaranteed. The existence criterion of the desired controller is formulated, and an LMI optimization approach is proposed to enlarge the domain of safe initial conditions.

The paper is organized as follows. Problem statement and the preliminaries are given in Section 2. In Section 4, we present the exponential estimates for the STDSs and the solutions to the stabilization problem for the system with saturating actuators. Numerical examples will be given in Section 4 to illustrate the effectiveness of the proposed method. The paper will be concluded in Section 5.

Notation 1. For real symmetric matrices 𝑃, 𝑃>0(𝑃≥0) means that matrix 𝑃 is positive definite (semipositive definite). 𝜆_max(𝑃)(𝜆_min(𝑃)) denotes the largest (smallest) eigenvalue of the positive definite matrix 𝑃. 𝐑^𝑛 is the 𝑛-dimensional real Euclidean space and 𝐑^𝑚×𝑛 is the set of all real 𝑚×𝑛 matrices. 𝐙⁺ represents the sets of all non-negative integers. The superscript “𝑇” represents matrix transposition, and “∗” in a matrix is used to represent the term which is induced by symmetry. diag{⋯} stands for a block-diagonal matrix. Sym{𝐴} is the shorthand notation for 𝐴+𝐴^𝑇. For two integers 𝑛₁ and 𝑛₂ with 𝑛₁≤𝑛₂, we use 𝐈[𝑛₁,𝑛₂] to denote the integer set {𝑛₁,𝑛₁+1,…,𝑛₂}. Let 𝒞_𝑛,𝑑={𝜙∶𝐈[−𝑑,0]→𝐑^𝑛} denote the Banach space of family continuous vector valued functions mapping the interval 𝐈[−𝑑,0] to 𝐑^𝑛 with the topology of uniform convergence. Denote 𝑥_𝑘(𝑠)=𝑥(𝑘+𝑠), ∀𝑠∈𝐈[−𝑑,0]. ‖⋅‖ refers to either the Euclidean vector norm or the induced matrix two-norm. For a function 𝜙∈𝒞_𝑛,𝑑, its norm is defined as ‖𝜙‖_𝑐=sup_{𝑠∈𝐈[−𝑑,0]}‖𝜙(𝑠)‖.

2. Problem Statement and Preliminaries

2.1. Problem Statement

Consider a class of discrete-time singular system subject to time-varying delay and actuator saturation as follows: 𝐸𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴_𝑑𝑥((𝑘(𝑢(𝑘,𝑥(𝑠)(𝑠)−𝑘−𝑑))+𝐵sat))=𝜙,𝑠∈𝐈𝑑,0, (2.1) where 𝑥(𝑘)∈𝐑^𝑛 is the system state, 𝑢(𝑘)∈𝐑^𝑚 is the control input, and 𝜙(𝑠)∈𝒞_𝑛,𝑑 is a compatible vector valued initial function. 𝑑(𝑘) is a time-varying delay satisfying 0<𝑑≤𝑑(𝑘)≤𝑑, where 𝑑 and 𝑑 are constant positive scalars representing the minimum and maximum delays, respectively. The matrix 𝐸 is singular and rank𝐸=𝑟<𝑛. 𝐴, 𝐴_𝑑 and 𝐵 are known constant matrices. The function sat∶𝐑^𝑚→𝐑^𝑚 is the standard saturation function defined as sat(𝑢(𝑘))=[sat(𝑢₁(𝑘))sat(𝑢₂(𝑘))⋯sat(𝑢_𝑚(𝑘))]^𝑇, where sat(𝑢_𝑖(𝑘))=Sign(𝑢_𝑖(𝑘))min{1,|𝑢_𝑖(𝑘)|}. Note that the notation of sat(⋅) is slightly abused to denote scalar values and vector valued saturation functions. Also note that it is without loss of generality to assume unity saturation level [20].

In this paper, we consider the design of a linear state feedback control law of the following form [27]: 𝑢(𝑘)=𝐹𝑥(𝑘),𝐹=𝐹𝐸, (2.2) where 𝐹∈𝐑^𝑚×𝑛 and 𝐹∈𝐑^𝑚×𝑛. The closed-loop system under this feedback is given by 𝐸𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴_𝑑𝑥(𝑘−𝑑(𝑘))+𝐵sat𝐹𝐸𝑥(𝑘). (2.3)

Remark 2.1. The state feedback of the form (2.2) is used to guarantee the uniqueness of the solution of system (2.1). For that, since rank𝐸=𝑟<𝑛, there exist two nonsingular matrices 𝐺,𝐻∈𝐑^𝑛×𝑛 such that 𝐼𝐺𝐸𝐻=^𝑟0𝐴00,𝐺𝐴𝐻=¹¹𝐴₁₂𝐴₂₁𝐴₂₂,𝐺𝐴_𝑑𝐴𝐻=^𝑑11𝐴_𝑑12𝐴_𝑑21𝐴_𝑑22,𝐵𝐺𝐵=¹𝐵₂,𝐹𝐻=𝐹₁𝐹₂,𝐹𝐺⁻¹=𝐹₁𝐹₂,𝑥(𝑘)𝑥=𝐻¹(𝑘)𝑥₂(𝑘),𝑥₁(𝑘)∈𝐑^𝑟,𝑥₂(𝑘)∈𝐑^𝑛−𝑟. (2.4) If the state feedback is taken as the general form, that is, 𝑢(𝑘)=𝐹𝑥(𝑘), by using (2.4), then system (2.3) is restricted system equivalent (r.s.e.) to the following one: 𝑥₁(𝑘+1)=𝐴₁₁𝑥₁(𝑘)+𝐴₁₂𝑥₂(𝑘)+𝐴_𝑑11𝑥₁(𝑘−𝑑(𝑘))+𝐴_𝑑12𝑥₂(𝑘−𝑑(𝑘))+𝐵₁sat𝐹₁𝑥₁(𝑘)+𝐹₂𝑥₂(𝑘),0=𝐴₂₁𝑥₁(𝑘)+𝐴₂₂𝑥₂(𝑘)+𝐴_𝑑21𝑥₁(𝑘−𝑑(𝑘))+𝐴_𝑑22𝑥₂(𝑘−𝑑(𝑘))+𝐵₂sat𝐹₁𝑥₁(𝑘)+𝐹₂𝑥₂(𝑘). (2.5) From the second equation of (2.5), it can be seen that 𝑥₂(𝑘) is in the function sat(⋅). Hence, if 𝐵₂≠0, for given 𝑥₁(0), the solution of 𝑥₂(0) is not unique even 𝐴₂₂ is nonsingular. However, if 𝑢(𝑘)=𝐹𝐸𝑥(𝑘), then system (2.3) is r.s.e. to 𝑥₁(𝑘+1)=𝐴₁₁𝑥₁(𝑘)+𝐴₁₂𝑥₂(𝑘)+𝐴_𝑑11𝑥₁(𝑘−𝑑(𝑘))+𝐴_𝑑12𝑥₂(𝑘−𝑑(𝑘))+𝐵₁sat𝐹₁𝑥₁(𝑘),0=𝐴₂₁𝑥₁(𝑘)+𝐴₂₂𝑥₂(𝑘)+𝐴_𝑑21𝑥₁(𝑘−𝑑(𝑘))+𝐴_𝑑22𝑥₂(𝑘−𝑑(𝑘))+𝐵₂sat𝐹₁𝑥₁(𝑘), (2.6) which implies that, for given 𝑥₁(𝑘), the unique solution of 𝑥₂(𝑘) can be obtained when 𝐴₂₂ is nonsingular. Nevertheless, it should be pointed that the state feedback (2.2) may be conservative due to its special structure.

To describe the main objective of this paper more precisely, we introduce the following definitions.

Definition 2.2 (see [29]). System 𝐸𝑥(𝑘+1)=𝐴𝑥(𝑘) (or the pair (𝐸,𝐴)) is said to be regular if det(𝑧𝐸−𝐴) is not identically zero, and if deg(det(𝑧𝐸−𝐴))=rank𝐸, then it is further said to be causal.

Definition 2.3 (see [6]). System (2.1) with 𝑢(𝑘)=0 is said to be regular and causal, if the pair (𝐸,𝐴) is regular and causal.

Note that regularity and causality of system (2.1) with 𝑢(𝑘)=0 ensure that the solution to this system exists and is unique for any given compatible initial value 𝜙(𝑠).

Definition 2.4. System (2.1) under feedback law (2.2) is said to be exponentially stable with decay rate 𝜆 (𝜆>1) if, for any compatible initial conditions 𝑥_𝑘0(𝑠)=𝑥(𝑘₀+𝑠), 𝑠∈𝐈[−𝑑,0], its solution 𝑥(𝑘,𝑥_𝑘0) satisfies ‖𝑥(𝑘,𝑥_𝑘0)‖≤𝜖𝜆^{−(𝑘−𝑘0)}‖𝑥_𝑘0‖_𝑐 for all 𝑘≥𝑘₀, where ‖𝑥_𝑘0‖_𝑐=sup_{𝑠∈𝐈[−𝑑,0]}‖𝑥(𝑘₀+𝑠)‖, 𝑘₀ is the initial time step, and 𝜖>0 is the decay coefficient.

We are interested in the exponential estimates and design for system (2.3). For any compatible initial condition 𝑥₀=𝜙∈𝒞_𝑛,𝑑, denote the state trajectory of system (2.1) as 𝑥(𝑘,𝑥₀); then the domain of attraction of the origin is 𝒮=𝜙∈𝒞_𝑛,𝑑∶lim_𝑘→∞𝑥𝑘,𝑥₀=0. (2.7) In general, for a given stabilizing state feedback gain 𝐹, it is impossible to determine exactly the domain of attraction of the origin with respect to system (2.3). Therefore, the purpose of this paper is to design a state feedback gain 𝐹 and determine a suitable set of initial condition 𝒟={𝜙∈𝐶_𝑛,𝑑∶‖𝜙‖²_𝑐≤𝛿}∈𝒮 from which the regularity, causality, and exponential stability of the closed-loop system (2.3) is ensured. Also, we are interested in maximizing the size of this set, that is, obtaining the maximal value of 𝛿.

2.2. Preliminary Results

Define 𝜁(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝑥^𝑇(𝑘−𝑑)𝑥^𝑇(𝑘−𝑑)²,𝑦(𝑘)=𝑥(𝑘+1)−𝑥(𝑘). (2.8)

Lemma 2.5. For any appropriately dimensioned matrices 𝑅>0 and 𝑁, two positive time-varying integer 𝑑(𝑘₁) and 𝑑(𝑘₂) satisfying 𝑑(𝑘₁)+1≤𝑑(𝑘₂)≤𝑑, and a scalar 𝜆>0, the following equality holds −^{𝑘−𝑑(𝑘1)−1}_{𝑘𝑙=𝑘−𝑑2}𝑦^𝑇(𝑙)𝐸^𝑇𝜆^𝑘−𝑙𝑅𝐸𝑦(𝑙)=𝑐𝜁^𝑇(𝑘)𝑁𝑅⁻¹𝑁^𝑇𝜁(𝑘)+2𝜁^𝑇(𝑘)𝑁^{𝑘−𝑑(𝑘1)−1}_{𝑙=𝑘−𝑑(𝑘2)}(𝑙)−𝐸𝑦_{𝑘−𝑑(𝑘1)−1𝑘𝑙=𝑘−𝑑2}𝜁^𝑇(𝑘)𝑁+𝜆^𝑘−𝑙𝑦^𝑇(𝑙)𝐸^𝑇𝑅𝜆^𝑘−𝑙𝑅⁻¹𝑁^𝑇𝜁_𝑘+𝜆^𝑘−𝑙𝑅𝐸𝑦(𝑙), (2.9) where 𝑐=(𝜆^{−𝑑(𝑘2)}−𝜆^{−𝑑(𝑘1)})/(1−𝜆).

Proof. See the Appendix.

Lemma 2.6 (see [30]). Given a matrix 𝐷, let a positive-definite matrix 𝑆 and a positive scalar 𝜂∈(0,1) exist such that 𝐷^𝑇𝑆𝐷−𝜂²𝑆<0 (2.10) then the matrix 𝐷 satisfies the bound 𝐷^𝑖≤𝜒𝑒^−𝜆𝑖,𝑖=0,1,…, (2.11) where 𝜒=𝜆_max(𝑆)/𝜆_min(𝑆) and 𝜆=−ln(𝜂).

Lemma 2.7. Let 0<𝑑≤𝑑(𝑘)≤𝑑. Consider the following system: 𝑥(𝑘)+𝐷𝑥(𝑘−𝑑(𝑘))=𝑓(𝑘),𝑘≥0, (2.12) where ‖𝐷^𝑖‖≤𝜒𝑒^−𝜆𝑖,  𝜆>0,  𝑖=0,1,…,  and  ‖𝑓(𝑘)‖≤𝜅𝑒^−𝛽𝑘, 𝑘≥0. If 𝛽𝑑−𝜆<0. (2.13) Then, for any compatible initial function 𝜙∈𝒞_𝑛,𝑑, the solution 𝑥(𝑘,𝜙) of (2.10) satisfies 𝑥𝑘,𝜙≤𝜒𝜙_𝑐1+𝜒𝜅1−𝑒^{𝛽𝑑−𝜆}+𝜅𝑒^−𝑟𝑘,𝑘≥0, (2.14) where 𝑟=min{𝜆/𝑑,𝛽}.

Proof. See the Appendix.

For a matrix 𝐹∈𝐑^𝑚×𝑛, denote the 𝑗th row of 𝐹 as 𝑓_𝑗 and define ℒ(𝐹)=𝑥(𝑘)∈𝐑^𝑛∶𝑓_𝑗𝑥(𝑘)≤1,𝑗∈𝐈[1,𝑚]. (2.15) Let 𝑃∈𝐑^𝑛×𝑛 be a positive-definite matrix and 𝐸^𝑇𝑃𝐸≥0, the set Ω(𝐸^𝑇𝑃𝐸,1) is defined by Ω𝐸^𝑇𝑃𝐸,1=𝑥(𝑘)∈𝐑^𝑛∶𝑥^𝑇(𝑘)𝐸^𝑇𝑃𝐸𝑥(𝑘)≤1. (2.16) Also, let 𝒱 be the set of 𝑚×𝑚 diagonal matrices whose diagonal elements are either 1 or 0. There are 2^𝑚 elements in 𝒱. Suppose that each element of 𝒱 is labeled as 𝐷_𝑖,𝑖∈𝐈[1,2^𝑚] and denote 𝐷⁻_𝑖=𝐼−𝐷_𝑖. Clearly, 𝐷_𝑖 is also an element of 𝒱 if 𝐷_𝑖∈𝒱.

Lemma 2.8 (see [18]). Let 𝐹,𝐻∈𝐑^𝑚×𝑛 be given. If 𝑥(𝑘)∈ℒ(𝐻), then sat(𝐹𝑥(𝑘) can be expressed as sat(𝐹𝑥(𝑘))=^2𝑚_𝑖=1𝛼_𝑖(𝑘)𝐷_𝑖𝐹+𝐷⁻_𝑖𝐻𝑥(𝑘), (2.17) where 𝛼_𝑖(𝑘) for 𝑖∈𝐈[1,2^𝑚] are some variables satisfying 𝛼_𝑖(𝑘)≥0 and ∑^2𝑚_𝑖=1𝛼_𝑖(𝑘)=1.

Lemma 2.9 (see [31]). Given matrices 𝑋, 𝑌, and 𝑍 with appropriate dimensions, and 𝑌 is symmetric. Then there exists a scalar 𝜌>0, such that 𝜌𝐼+𝑌>0 and −Sym{𝑋^𝑇𝑍}−𝑍^𝑇𝑌𝑍≤𝑋^𝑇(𝜌𝐼+𝑌)⁻¹𝑋+𝜌𝑍^𝑇𝑍.

3. Main Results

In this section, we will first present a delay-dependent LMI condition which guarantees the regularity, causality and exponential stability of the unforced system (2.1) (i.e., with 𝑢(𝑘)=0) with a predefined decay rate.

Theorem 3.1. Given constants 0<𝛼<1 and 0<𝑑<𝑑. If there exist symmetric matrices 𝑋>0, 𝑄_𝑙>0, 𝑙=1,2,3, 𝑍_𝑣>0, 𝑣=1,2 and 𝑆, and matrices 𝑀=[𝑀^𝑇₁𝑀^𝑇₂𝑀^𝑇₃𝑀^𝑇₄]^𝑇, 𝑁=[𝑁^𝑇₁𝑁^𝑇₂𝑁^𝑇₃𝑁^𝑇₄]^𝑇 and 𝑇=[𝑇^𝑇₁𝑇^𝑇₂𝑇^𝑇₃𝑇^𝑇₄]^𝑇 such that the following inequality holds ΦΨ∗Γ<0, (3.1) where Φ=diag−(1−𝛼)𝐸^𝑇𝑋𝐸+1+𝑑𝑄₁+𝑄₂+𝑄₃,−(1−𝛼)^𝑑𝑄₁,−(1−𝛼)^𝑑𝑄₂,−(1−𝛼)^𝑑𝑄₃+Sym𝑀Λ₁+𝑁Λ₂+𝑇Λ₃+𝒜^𝑇₁𝑋𝒜₁−𝒜^𝑇₁𝑅^𝑇𝑆𝑅𝒜₁+𝒜^𝑇₂𝑈𝒜₂,Ψ=𝜚₁𝑀𝜚₂𝑁𝜚₂𝑇,Γ=diag−𝜚₁𝑍₁,−𝜚₂𝑍₁+𝑍₂,−𝜚₂𝑍₂,𝒜₁=𝐴𝐴_𝑑00,𝒜₂=𝐴−𝐸𝐴_𝑑,Λ00₁=𝐸−𝐸00,Λ₂=,Λ0𝐸0−𝐸₃=0𝐸0−𝐸,𝑑=𝑑−𝑑,𝑈=𝑑𝑍₁+𝑑𝑍₂,𝜚₁=(1−𝛼)^−𝑑−1𝛼,𝜚₂=(1−𝛼)^−𝑑−(1−𝛼)^−𝑑𝛼, (3.2) and 𝑅∈𝐑^𝑛×𝑛 is any constant matrix satisfying 𝑅𝐸=0 and rank(𝑅)=𝑛−𝑟, then the unforced system (2.1) is regular, causal, and exponentially stable with 𝜆=1/√1−𝛼.

Proof. The proof is divided into three parts: (i) To show the regularity and causality; (ii) to show the exponential stability of the difference subsystem; (iii) to show the exponential stability of the algebraic subsystem.
Part (i): Regularity and causality. Since 𝐸 is regular and rank(𝐸)=𝑟, there exist two nonsingular matrices 𝐺₁ and 𝐻₁ such that 𝐺₁𝐸𝐻₁=𝐼_𝑟000,𝐺₁𝐴𝐻₁=𝐴₁₁𝐴₁₂𝐴₂₁𝐴₂₂,𝐺₁^−𝑇𝑋𝐺₁⁻¹=𝑋₁₁𝑋₁₂𝑋^𝑇₁₂𝑋₂₂,𝐻^𝑇₁𝑀₁𝐺₁⁻¹=𝑀₁₁𝑀₁₂𝑀₂₁𝑀₂₂,𝑅𝐺₁⁻¹=𝑅₁𝑅₂. (3.3) Note that 𝑅𝐸=0 and rank(𝑅)=𝑛−𝑟, it can be verified that 𝑅₁=0, rank(𝑅₂)=𝑛−𝑟 and 𝑅₂∈𝐑^{𝑛×(𝑛−𝑟)}, that is, 𝑅𝐺₁⁻¹=0𝑅₂. (3.4) From (3.1), it is easy to obtain Φ₁₁<0. In view of 𝑋>0,  𝑄_𝑙>0,  𝑙=1,2,3,  𝑍_𝑙>0,  𝑍₂>0,  𝑑>0,  and 𝑑>0, it can be further obtained that −𝐴^𝑇𝑅^𝑇𝑆𝑅𝐴−(1−𝛼)𝐸^𝑇𝑋𝐸+Sym𝑀₁𝐸<0. (3.5) Pre- and postmultiplying (3.5) by 𝐻^𝑇₁ and 𝐻₁, respectively, and using (3.3) and (3.4), it is obtained that 𝐴⋆⋆⋆−^𝑇₂₂𝑅^𝑇₂𝑆𝑅₂𝐴₂₂<0, (3.6) where ⋆ represents matrices that are not relevant in the following discussion. Thus, −𝐴^𝑇₂₂𝑅^𝑇₂𝑆𝑅₂𝐴₂₂<0. (3.7) Now, we assume that the matrix 𝐴₂₂ is singular, then, there exists a vector 𝜂∈𝐑^𝑛−𝑟 and 𝜂≠0 such that 𝐴₂₂𝜂=0. Pre- and postmultiplying (3.7) by 𝜂^𝑇 and 𝜂, respectively, result in 𝜂^𝑇𝐴^𝑇₂₂𝑅^𝑇₂𝑆𝑅₂𝐴₂₂𝜂=0. Then, it is easy to see that (3.7) is a contradiction since 𝜂^𝑇𝐴^𝑇₂₂𝑅^𝑇₂𝑆𝑅₂𝐴₂₂𝜂>0. Thus, 𝐴₂₂ is nonsingular, which implies that the unforced system (2.1) is regular and causal by Definition 2 and Theorem 1 in [4].
Part (ii): Exponential stability of the difference subsystem. From [29], the regularity and causality of the unforced system (2.1) imply that there exist two nonsingular matrices 𝐺₂ and 𝐻₂ such that 𝐸=𝐺₂𝐸𝐻₂=𝐼_𝑟0,00𝐴=𝐺₂𝐴𝐻₂=𝐴₁₁00𝐼_𝑛−𝑟. (3.8) According to (3.8), define 𝐴_𝑑=𝐺₂𝐴_𝑑𝐻₂=𝐴_𝑑11𝐴_𝑑12𝐴_𝑑21𝐴_𝑑22,𝑋=𝐺₂^−𝑇𝑋𝐺₂⁻¹=𝑋₁₁𝑋₁₂𝑋^𝑇₁₂𝑋₂₂,𝑄_𝑙=𝐻^𝑇₂𝑄_𝑙𝐻₂=𝑄_𝑙11𝑄_𝑙12𝑄^𝑇_𝑙12𝑄_𝑙22,𝑍_𝑣=𝐺₂^−𝑇𝑍_𝑣𝐺₂⁻¹=𝑍_𝑣11𝑍_𝑣12𝑍^𝑇_𝑣12𝑍_𝑣22,𝑀_𝑣=𝐻^𝑇₂𝑀_𝑣𝐺₂⁻¹=𝑀_𝑣11𝑀_𝑣12𝑀_𝑣21𝑀_𝑣22,𝑁_𝑣=𝐻^𝑇₂𝑁_𝑣𝐺₂⁻¹=𝑁_𝑣11𝑁_𝑣12𝑁_𝑣21𝑁_𝑣22,𝑇_𝑣=𝐻^𝑇₂𝑇_𝑣𝐺₂⁻¹=𝑇_𝑣11𝑇_𝑣12𝑇_𝑣21𝑇_𝑣22,𝑅=𝑅𝐺₂⁻¹=0𝑅₂,𝑙=1,2,3,𝑣=1,2. (3.9) By using Schur complement on (3.1), we get Φ^′₁₁Φ^′₁₂∗Φ^′₂₂<0, (3.10) where Φ^′₁₁=−A^𝑇𝑅^𝑇𝑆𝑅𝐴−(1−𝛼)𝐸^𝑇𝑃𝐸+1+𝑑𝑄₁+𝑄₂+𝑄₃+Sym𝑀₁𝐸,Φ^′₁₂=−𝐴^𝑇𝑅^𝑇𝑆𝑅𝐴_𝑑−𝑀₁𝐸+𝑁₁𝐸−𝑇₁𝐸+𝐸^𝑇𝑀^𝑇₂,Φ^′₂₂=−𝐴^𝑇_𝑑𝑅^𝑇𝑆𝑅𝐴_𝑑−(1−𝛼)^𝑑𝑄₁+Sym−𝑀₂𝐸+𝑁₂𝐸−𝑇₂𝐸. (3.11) Substituting (3.8) and (3.9) into the above inequality, pre- and postmultiplying by diag{𝐻^𝑇₂,𝐻^𝑇₂} and diag{𝐻₂,𝐻₂}, respectively, and using Schur complement yields −𝑅^𝑇₂𝑆𝑅₂+1+𝑑𝑄₁₂₂+𝑄₂₂₂+𝑄₃₂₂−𝑅^𝑇₂𝑆𝑅₂𝐴_𝑑22∗−(1−𝛼)^𝑑𝑄₁₂₂−𝐴^𝑇_𝑑22𝑅^𝑇₂𝑆𝑅₂𝐴_𝑑22<0. (3.12) Pre- and postmultiplying this inequality by [−𝐴^𝑇_𝑑22𝐼] and its transpose, respectively, and noting 𝐴^𝑇_𝑑22(𝑄₂₂₂+𝑄₃₂₂)𝐴_𝑑22≥0, we have 𝐴^𝑇_𝑑22𝑄₁₂₂𝐴_𝑑22−(1−𝛼)^𝑑1+𝑑𝑄₁₂₂<0. (3.13) Therefore, according to Lemma 2.6, there exist constants 𝜒=𝜆_max(𝑄₁₂₂)/𝜆_min(𝑄₁₂₂) and 𝜆=−ln(1−𝛼)^𝑑/2+ln(1+𝑑)^1/2 such that 𝐴^𝑖_𝑑22≤𝜒𝑒^−𝜆𝑖,𝑖=1,2,… (3.14)
Let 𝜉(𝑘)=𝐻₂⁻¹𝑥(𝑘)=[𝜉^𝑇₁(𝑘)𝜉^𝑇₂(𝑘)]^𝑇, where 𝜉₁(𝑘)∈𝐑^𝑟 and 𝜉₂(𝑘)∈𝐑^𝑛−𝑟. Then, the unforced system (2.1) is r.s.e. to the following one: 𝜉₁(𝑘+1)=𝐴₁₁𝜉₁(𝑘)+𝐴_𝑑11𝜉₁(𝑘−𝑑(𝑘))+𝐴_𝑑12𝜉₂(𝑘−𝑑(𝑘)),0=𝜉₂(𝑘)+𝐴_𝑑21𝜉₁(𝑘−𝑑(𝑘))+𝐴_𝑑22𝜉₂(𝑘−𝑑(𝑘)). (3.15) Now, choose the following Lyapunov-Krasovskii function: 𝑉𝜉_𝑘=⁴_𝑠=1𝑉_𝑠𝜉_𝑘, (3.16) where 𝑉₁𝜉_𝑘=𝜉^𝑇₁(𝑘)𝑋₁₁𝜉₁(𝑘)=𝜉^𝑇(𝑘)𝐸^𝑇𝑋(𝑘),𝑉𝐸𝜉₂𝜉_𝑘=^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝜉^𝑇(𝑙)(1−𝛼)^{𝑘−1−𝑙}𝑄₁𝜉(𝑙)+^−𝑑_{𝜃=−𝑑+1}^𝑘−1_{𝑙=𝑘+𝜃}𝜉^𝑇(𝑙)(1−𝛼)^{𝑘−1−𝑙}𝑄₁𝜉(𝑙),𝑉₃𝜉_𝑘=^𝑘−1_{𝑙=𝑘−𝑑}𝜉^𝑇(𝑙)(1−𝛼)^{𝑘−1−𝑙}𝑄₂𝜉(𝑙)+^𝑘−1_{𝑙=𝑘−𝑑}𝜉^𝑇(𝑙)(1−𝛼)^{𝑘−1−𝑙}𝑄₃𝜉(𝑙),𝑉₄𝜉_𝑘=⁻¹_𝜃=−𝑑^𝑘−1_{𝑙=𝑘+𝜃}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^{𝑘−1−𝑙}𝑍₁𝐸𝜂(𝑙)+^−𝑑−1_𝜃=−𝑑^𝑘−1_{𝑙=𝑘+𝜃}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^{𝑘−1−𝑙}𝑍₂𝐸𝜂(𝑙), (3.17) with 𝜉_𝑘(𝑠)=𝜉(𝑘+𝑠), ∀𝑠∈𝐈[−𝑑,0] and 𝜂(𝑘)=𝜉(𝑘+1)−𝜉(𝑘). Define ▵𝑉𝜉_𝑘=⁴_𝑠=1▵𝑉_𝑠𝜉_𝑘=⁴_𝑠=1𝑉_𝑠𝜉_𝑘+1−(1−𝛼)𝑉_𝑠𝜉_𝑘. (3.18) Then, it follows from (2.1) that ▵𝑉₁𝜉_𝑘=𝜉^𝑇(𝑘+1)𝐸^𝑇𝑋𝐸𝜉(𝑘+1)−(1−𝛼)𝜉^𝑇(𝑘)𝐸^𝑇𝑋(𝑘)=𝐸𝜉𝐴𝜉(𝑘)+𝐴_𝑑𝜉(𝑘−𝑑(𝑘))^𝑇𝑋𝐴𝜉(𝑘)+𝐴_𝑑𝜉(𝑘−𝑑(𝑘))−(1−𝛼)𝜉^𝑇(𝑘)𝐸^𝑇𝑋𝐸𝜉(𝑘),▵𝑉₂𝜉_𝑘≤1+𝑑𝜉^𝑇(𝑘)𝑄₁𝜉(𝑘)−𝜉^𝑇(𝑘−𝑑(𝑘))(1−𝛼)^𝑑(𝑘)Q₁𝜉(𝑘−𝑑(𝑘)),▵𝑉₃𝜉_𝑘=𝜉^𝑇(𝑘)𝑄₂+𝑄₃𝜉(𝑘)−𝜉^𝑇𝑘−𝑑(1−𝛼)^𝑑𝑄₂𝜉𝑘−𝑑−𝜉^𝑇𝑘−𝑑(1−𝛼)^𝑑𝑄₃𝜉𝑘−𝑑,▵𝑉₄𝜉_𝑘=𝜂^𝑇(𝑘)𝐸^𝑇𝑑𝑍₁+𝑑𝑍₂𝐸𝜂(𝑘)−^𝑘−1_{𝑙=𝑘−𝑑}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^𝑘−𝑙𝑍₁(𝑙)−𝐸𝜂_{𝑘−𝑑−1𝑙=𝑘−𝑑}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^𝑘−𝑙𝑍₂𝐸𝜂(𝑙)=𝜂^𝑇(𝑘)𝐸^𝑇𝑑𝑍₁+𝑑𝑍₂𝐸𝜂(𝑘)−^𝑘−1_{𝑙=𝑘−𝑑(𝑘)}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^𝑘−𝑙𝑍₁(𝑙)−𝐸𝜂_{𝑘−𝑑(𝑘)−1𝑙=𝑘−𝑑}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^𝑘−𝑙𝑍₁+𝑍₂𝐸𝜂(𝑙)−^{𝑘−𝑑−1}_{(𝑘)𝑙=𝑘−𝑑}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^𝑘−𝑙𝑍₂𝐸𝜂(𝑙). (3.19) Let 𝜓(𝑘)=𝜉^𝑇(𝑘)𝜉^𝑇(𝑘−𝑑(𝑘))𝜉^𝑇(𝑘−𝑑)𝜉^𝑇(𝑘−𝑑)^𝑇. (3.20) Using Lemma 2.5 for the last three terms of ▵𝑉₄(𝜉_𝑘), respectively, and noting 𝑑≤𝑑(𝑘)≤𝑑, we have −^𝑘−1_{𝑙=𝑘−𝑑(𝑘)}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^𝑘−𝑙𝑍₁𝐸𝜂(𝑙)≤𝜚₁𝜓^𝑇(𝑘)𝑀𝑍₁⁻¹𝑀^𝑇𝜓(𝑘)+2𝜓^𝑇(𝑘)𝑀^𝑘−1_{𝑙=𝑘−𝑑(𝑘)}(𝑙)−𝐸𝜂_{𝑘−1(𝑘)𝑙=𝑘−𝑑}𝜓^𝑇(𝑘)𝑀+(1−𝛼)^𝑘−𝑙𝜂^𝑇(𝑙)𝐸^𝑇𝑍₁(1−𝛼)^𝑘−𝑙𝑍₁⁻¹×𝑀^𝑇𝜓(𝑘)+(1−𝛼)^𝑘−𝑙𝑍₁(𝑙−𝐸𝜂)_{𝑘−𝑑(𝑘)−1𝑙=𝑘−𝑑}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^𝑘−𝑙𝑍₁+𝑍₂𝐸𝜂(𝑙)≤𝜚₂𝜓^𝑇(𝑘)𝑁𝑍₁+𝑍₂⁻¹𝑁^𝑇𝜓(𝑘)+2𝜓^𝑇(𝑘)𝑁^{𝑘−𝑑(𝑘)−1}_{𝑙=𝑘−𝑑}(𝑙)−𝐸𝜂_{𝑘−𝑑(𝑘)−1𝑙=𝑘−𝑑}𝜓^𝑇(𝑘)𝑁+(1−𝛼)^𝑘−𝑙𝜂^𝑇(𝑙)𝐸^𝑇𝑍₁+𝑍₂(1−𝛼)^𝑘−𝑙𝑍₁+𝑍₂⁻¹⋅𝑁^𝑇𝜓(𝑘)+(1−𝛼)^𝑘−𝑙𝑍₁+𝑍₂(𝑙−𝐸𝜂)_{𝑘−𝑑−1𝑙=𝑘−𝑑(𝑘)}𝜂^𝑇(𝑙)𝐸^𝑇(1−𝛼)^𝑘−𝑙𝑍₂𝐸𝜂(𝑙)≤𝜚₂𝜓^𝑇(𝑘)𝑇𝑍₂⁻¹𝑇^𝑇𝜓(𝑘)+2𝜓^𝑇(𝑘)𝑇^{𝑘−𝑑−1}_{𝑙=𝑘−𝑑(𝑘)}(𝑙)−𝐸𝜂_{𝑘−𝑑−1(𝑘)𝑙=𝑘−𝑑}𝜓^𝑇(𝑘)𝑇+(1−𝛼)^𝑘−𝑙𝜂^𝑇(𝑙)𝐸^𝑇𝑍₂(1−𝛼)^𝑘−𝑙𝑍₂⁻¹×𝑇^𝑇𝜓(𝑘)+(1−𝛼)^𝑘−𝑙𝑍₂𝐸𝜂(𝑙). (3.21) Note that 𝜂(𝑙)=𝜉(𝑙+1)−𝜉(𝑙) provides 𝐸𝜂(^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝑙)=𝐸−𝐸00_{𝑘−𝑑(𝑘)−1𝑙=𝑘−𝑑}𝜓(𝑘),𝐸𝜂(𝑙)=0𝐸0−𝐸𝜓(𝑘),_{𝑘−𝑑−1(𝑘)𝑙=𝑘−𝑑}𝐸𝜂(𝑙)=0−𝐸𝐸0𝜓(𝑘). (3.22)(3.23)(3.24) Also, it follows from 𝑅𝐸=0 that 0=−𝜉^𝑇(𝑘+1)𝐸^𝑇𝑅^𝑇𝑆𝑅𝐸𝜉^𝑇(𝑘+1)=−𝐴𝜉(𝑘)+𝐴_𝑑𝜉(𝑘−𝑑(𝑘))^𝑇𝑅^𝑇𝑆𝑅𝐴𝜉(𝑘)+𝐴_𝑑𝜉(𝑘−𝑑(𝑘)). (3.25) Then, substituting (3.19)–(3.24) into (3.18), using (3.25), and noting 1−𝛼>0, 𝑍₁>0 and 𝑍₂>0, we get ▵𝑉𝜉_𝑘≤𝜓^𝑇(𝑘)Φ+𝜚₁𝑀𝑍₁⁻¹𝑀^𝑇+𝜚₂𝑁𝑍₁+𝑍₂⁻¹𝑁^𝑇+𝜚₂𝑇𝑍₂⁻¹𝑇^𝑇𝜓(𝑘)−^𝑘−1_{(𝑘)𝑙=𝑘−𝑑}𝜓^𝑇(𝑘)𝑀+(1−𝛼)^𝑘−𝑙𝜂^𝑇(𝑙)𝐸^𝑇𝑍₁(1−𝛼)^𝑘−𝑙𝑍₁⁻¹×𝑀^𝑇𝜓(𝑘)+(1−𝛼)^𝑘−𝑙𝑍₁(𝑙−𝐸𝜂)_{𝑘−𝑑(𝑘)−1𝑙=𝑘−𝑑}𝜓^𝑇(𝑘)𝑁+(1−𝛼)^𝑘−𝑙𝜂^𝑇(𝑙)𝐸^𝑇𝑍₁+𝑍₂(1−𝛼)^𝑘−𝑙𝑍₁+𝑍₂⁻¹⋅𝑁^𝑇𝜓(𝑘)+(1−𝛼)^𝑘−𝑙𝑍₁+𝑍₂(𝑙−𝐸𝜂)_{𝑘−𝑑−1(𝑘)𝑙=𝑘−𝑑}𝜓^𝑇(𝑘)𝑇+(1−𝛼)^𝑘−𝑙𝜂^𝑇(𝑙)𝐸^𝑇𝑍₂(1−𝛼)^𝑘−𝑙𝑍₂⁻¹×𝑇^𝑇𝜓(𝑘)+(1−𝛼)^𝑘−𝑙𝑍₂𝐸𝜂(𝑙)≤𝜓^𝑇(𝑘)Φ+𝜚₁𝑀𝑍₁⁻¹𝑀^𝑇+𝜚₂𝑁𝑍₁+𝑍₂⁻¹𝑁^𝑇+𝜚₂𝑇𝑍₂⁻¹𝑇^𝑇𝜓(𝑘), (3.26) where Φ follows the same definition as Φ defined in (3.1) with 𝐴, 𝐴_𝑑, 𝑋, 𝑄_𝑙, 𝑍_𝑣, 𝑅, 𝑀, 𝑁, and 𝑇 instead of 𝐴, 𝐴_𝑑, 𝑋, 𝑄_𝑙, 𝑍_𝑣, 𝑅, 𝑀, 𝑁, and 𝑇. Performing a congruence transformation on (3.1) by diag{𝐻^𝑇₂,𝐻^𝑇₂,𝐻^𝑇₂,𝐻^𝑇₂,𝐻^𝑇₂,𝐻^𝑇₂,𝐻^𝑇₂}, and then using the Schur complement implies Φ+𝜚₁𝑀𝑍₁⁻¹𝑀^𝑇+𝜚₂𝑁(𝑍₁+𝑍₂)⁻¹𝑁^𝑇+𝜚₂𝑇𝑍₂⁻¹𝑇^𝑇<0. Thus, it follows from (3.18) and (3.26) that ▵𝑉(𝜉_𝑘)=𝑉(𝜉_𝑘+1)−(1−𝛼)𝑉(𝜉_𝑘)≤0, which leads to 𝑉𝜉_𝑘+1≤(1−𝛼)𝑉𝜉_𝑘. (3.27) By iterative substitutions, inequality (3.27) yields 𝑉𝜉_𝑘≤(1−𝛼)^𝑘𝑉𝜉₀. (3.28) On the other hand, it follows from the Lyapunov functional (3.16) that 𝛽₁𝜉₁(𝑘)²≤𝑉𝜉_𝑘,𝑉𝜉₀≤𝛽₂𝜙²_𝑐, (3.29) where 𝛽₁=𝜆_min𝑋₁₁,𝛽₂=𝜆_max𝑋₁₁+𝑑+𝑑𝑑−1𝜆_max𝑄₁+𝑑𝜆_max𝑄₂+𝑑𝜆_max𝑄₃+𝑑²𝜆_max𝑍₁+𝑍₂, (3.30)
Then, combining (3.28) and (3.29) leads to 𝜉₁(𝑘)≤𝛽₂𝛽₁(1−𝛼)^𝑘/2𝜙_𝑐=𝛽₂𝛽₁𝑒^{−(ln(1−𝛼)−1/2)𝑘}𝜙_𝑐. (3.31) Therefore, the difference subsystem of (3.15) is exponentially stable with a decay rate which is not less than 𝑒^{ln(1−𝛼)−1/2}. The remaining task is to show the exponential stability of the algebraic subsystem.
Part (iii): Exponential stability of the algebraic subsystem. Set 𝑓(𝑘)=−𝐴_𝑑21𝜉₁(𝑘−𝑑(𝑘)); then, it follows from (3.31) that 𝑓(𝑘)≤(1−𝛼)^−𝑑/2𝐴_𝑑21𝛽₂𝛽₁𝑒^{−(ln(1−𝛼)−1/2)𝑘}𝜙_𝑐. (3.32) Using the second equation in (3.15), (3.14) and Lemma 2.7, one gets 𝜉₂(𝑘)≤𝛽₃𝑒^{−(ln(1−𝛼)−1/2)𝑘}𝜙_𝑐, (3.33) where 𝛽₃=𝜆_max𝑄₁₂₂𝜆_min𝑄₁₂₂𝜙_𝑐+𝜆_max𝑄₁₂₂𝜆_min𝑄₁₂₂11−𝑒^{−(1/2)ln(1+𝑑)}+1𝛽₂𝛽₁(1−𝛼)^−𝑑/2𝐴_𝑑21. (3.34) Combining (3.31) and (3.33) yields that ‖𝑥(𝑘)‖²=𝜉^𝑇(𝑘)𝐻^𝑇₂𝐻₂𝜉(𝑘)≤𝐻₂²𝜉₁(𝑘)²+𝜉₂(𝑘)²≤𝛽₂𝛽₁+𝛽²₃𝑒^{−(ln(1−𝛼)−1)𝑘}𝜙²_𝑐. (3.35) Thus, we have ‖𝑥(𝑘)‖≤𝛽₂𝛽₁+𝛽²₃𝑒^{−(ln(1−𝛼)−1/2)𝑘}𝜙_𝑐=𝛽₂𝛽₁+𝛽²₃1√1−𝛼^−𝑘𝜙_𝑐 (3.36) The proof is completed.

Remark 3.2. Theorem 3.1 is obtained by applying a Lyapunov-Krasovskii function method to both the difference and algebraic subsystems of a discrete-time singular system with time-varying delay. Such a method in dealing with the algebraic subsystem of the discrete-time singular delay system has not been reported in the literature.

Remark 3.3. When 𝐸=𝐼 in system (2.1), then 𝑅=0, and we get the exponential stability condition for the standard delay systems 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴_𝑑𝑥(𝑘−𝑑(𝑘)) from (3.1).
Based on the result of Theorem 3.1, we now present the existence conditions of a stabilizing state feedback controller for system (2.1) and the corresponding set of initial condition.

Theorem 3.4. Given constants 0<𝛼<1, 0<𝑑<𝑑, 𝜌>0, 𝜀₁>0, 𝜀₂>0, 𝜀₃>0, 𝜀₄, and 𝜀₅. If there exist symmetric matrices 𝑋>0, 𝑄_𝑙>0, 𝑙=1,2,3, 𝑍_𝑣>0, 𝑣=1,2, and 𝑆, and matrices 𝑀=[𝑀^𝑇₁𝑀^𝑇₂𝑀^𝑇₃𝑀^𝑇₄]^𝑇, 𝑁=[𝑁^𝑇₁𝑁^𝑇₂𝑁^𝑇₃𝑁^𝑇₄]^𝑇, 𝑇=[𝑇^𝑇₁𝑇^𝑇₂𝑇^𝑇₃𝑇^𝑇₄]^𝑇, 𝐹 and 𝐻 such that the following inequalities hold: Φ_𝑗ΨΥ_𝑗𝑑Π_𝑗𝑑Π_𝑗ΞΥ_𝑗𝑅^𝑇∗Γ00000∗∗−2𝜀₁𝐼+𝜀²₁𝑋0000∗∗∗−𝑑2𝜀₂𝐼−𝜀²₂𝑍₁𝑑000∗∗∗∗−2𝜀₃𝐼−𝜀²₃𝑍₂00∗∗∗∗∗−𝜌𝐼−𝑆0∗∗∗∗∗∗−𝜌⁻¹𝐼<0,𝑗∈𝐈1,2^𝑚,1ℎ_𝑙∗𝐸^𝑇[],𝑋𝐸≥0,𝑙∈𝐈1,𝑚 (3.37)(3.38) where ℎ_𝑙 denotes the 𝑙th row of 𝐻, Φ_𝑗=diag−(1−𝛼)𝐸^𝑇𝑋𝐸+1+𝑑𝑄₁+𝑄₂+𝑄₃,−(1−𝛼)^𝑑𝑄₁,−(1−𝛼)^𝑑𝑄₂,−(1−𝛼)^𝑑𝑄₃+SymΞ𝑅Υ^𝑇_𝑗+𝑀Λ₁+𝑁Λ₂+𝑇Λ₃,Υ_𝑗=𝐴_𝑗𝐴_𝑑00^𝑇,Π_𝑗=𝐴_𝑗−𝐸𝐴_𝑑00^𝑇Ξ=𝜀₄𝐼𝜀₅𝐼00^𝑇,𝐴_𝑗=𝐴+𝐵𝐷_𝑗𝐹𝐸+𝐷⁻_𝑗𝐻. (3.39)Λ₁, Λ₂, Λ₃, Ψ, Γ are defined in (3.1), and 𝑅∈𝐑^𝑛×𝑛 is any constant matrix satisfying 𝑅𝐸=0 with rank(𝑅)=𝑛−𝑟, then system (2.3) is regular, causal, and locally exponentially stable with 𝜆=1/√1−𝛼 for any compatible initial condition in the ball ℬ(𝛿)=𝜙∈𝐶_𝑛,𝑑∶𝜙²_𝑐≤𝛿, (3.40) where 1𝛿=𝜆_max𝐸^𝑇𝑋𝐸+𝑑+𝑑𝑑−1𝜆_max𝑄₁+𝑑𝜆_max𝑄₂+𝑑𝜆_max𝑄₃+𝑑²𝜆_max𝑍₁+𝑍₂. (3.41)

Proof. If (3.38) holds, then the ellipsoid Ω(𝐸^𝑇𝑋𝐸,1) is included in ℒ(𝐻) [27] (For more details about the ellipsoids and ellipsoid algorithm, we refer the readers to [32–34]). Suppose that 𝑥(𝑘)∈ℒ(𝐻), ∀𝑘>0 (will be proved later). Hence, by Lemma 2.8, sat(𝐹𝐸𝑥(𝑘)) can be expressed as sat𝐹𝐸𝑥(𝑘)=^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝐷_𝑗𝐹𝐸+𝐷⁻_𝑖𝐻𝑥(𝑘) (3.42) and it follows that 𝐸𝑥(𝑘+1)=^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝐴_𝑗𝑥(𝑘)+𝐴_𝑑𝑥(𝑘−𝑑(𝑘)). (3.43) Choose a Lyapunov function as in (3.16), and then, by Theorem 3.1, system (3.43) is regular, causal and locally exponentially stable if there exist symmetric matrices 𝑋>0, 𝑄_𝑙>0, 𝑙=1,2,3, 𝑍_𝑣>0, 𝑣=1,2, and 𝑆, and matrices 𝑀, 𝑁, and 𝑇 defined in (3.1) such that ΦΨ∗Γ<0, (3.44) where Φ=diag−(1−𝛼)𝐸^𝑇𝑋𝐸+1+𝑑𝑄₁+𝑄₂+𝑄₃,−(1−𝛼)^𝑑𝑄₁,−(1−𝛼)^𝑑𝑄₂,−(1−𝛼)^𝑑𝑄₃+𝒜^𝑇_1𝑗𝑋𝒜_1𝑗−𝒜^𝑇_1𝑗𝑅^𝑇𝑆𝑅𝒜_1𝑗+𝒜^𝑇_2𝑗𝑈𝒜_2𝑗+Sym𝑀Λ₁+𝑁Λ₂+𝑇Λ₃,𝒜_1𝑗=∑^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝐴_𝑗𝐴_𝑑00,𝒜_2𝑗=∑^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝐴_𝑗−𝐸𝐴_𝑑00. (3.45)(3.46) Now, provided that inequalities (3.37) hold, then by using Schur complement and Lemma 2.9, one has Φ_𝑗+Ξ𝜌𝐼+𝑆⁻¹Ξ^𝑇ΨΥ_𝑗𝑑Π_𝑗𝑑Π_𝑗Υ_𝑗𝑅^𝑇∗Γ0000∗∗−𝑋⁻¹000∗∗∗−𝑑𝑍₁⁻¹00∗∗∗∗−𝑑𝑍₂⁻¹0∗∗∗∗∗−𝜌⁻¹𝐼<0,𝑗∈𝐈1,2^𝑚. (3.47) This, together with 𝛼_𝑗(𝑘)≥0 and ∑^2𝑚_𝑗=1𝛼_𝑗(𝑘)=1, implies that ^2𝑚_𝑗=1𝛼_𝑗(𝑘)Φ_𝑗+Ξ𝜌𝐼+𝑆⁻¹Ξ^𝑇ΨΥ_𝑗𝑑Π_𝑗𝑑Π_𝑗Υ_𝑗𝑅^𝑇∗Γ0000∗∗−𝑋⁻¹000∗∗∗−𝑑𝑍₁⁻¹00∗∗∗∗−𝑑𝑍₂⁻¹0∗∗∗∗∗−𝜌⁻¹𝐼<0, (3.48) that is, ^2𝑚∑_𝑗=1𝛼_𝑗(𝑘)Φ_𝑗+Ξ𝜌𝐼+𝑆⁻¹Ξ^𝑇Ψ∑^2𝑚_𝑗=1𝛼_𝑗(𝑘)Υ∑_𝑗^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑑Π∑_𝑗^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑑Π∑_𝑗^2𝑚_𝑗=1𝛼_𝑗(𝑘)Υ_𝑗𝑅^𝑇∗Γ0000∗∗−𝑋⁻¹000∗∗∗−𝑑𝑍₁⁻¹00∗∗∗∗−𝑑𝑍₂⁻¹0∗∗∗∗∗−𝜌⁻¹𝐼<0. (3.49) Applying Schur complement to (3.49) leads to Λ_𝑗Ψ∑^2𝑚_𝑗=1𝛼_𝑗(𝑘)Υ∑_𝑗^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑑Π∑_𝑗^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑑Π_𝑗∗Γ000∗∗−𝑋⁻¹00∗∗∗−𝑑𝑍₁⁻¹0∗∗∗∗−𝑑𝑍₂⁻¹<0, (3.50) where Λ_𝑗=^2𝑚_𝑗=1𝛼_𝑗(𝑘)Φ_𝑗+Ξ𝜌𝐼+𝑆⁻¹Ξ^𝑇+𝜌^2𝑚_𝑗=1𝛼_𝑗(𝑘)Υ_𝑗𝑅^𝑇2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑅Υ^𝑇_𝑗. (3.51) Rewrite ^2𝑚_𝑗=1𝛼_𝑗(𝑘)Φ_𝑗=^2𝑚_𝑗=1𝛼_𝑗(𝑘)SymΥ_𝑗𝑅^𝑇Ξ^𝑇+Φ, (3.52) where Φ=diag−(1−𝛼)𝐸^𝑇𝑋𝐸+1+𝑑𝑄₁+𝑄₂+𝑄₃,−(1−𝛼)^𝑑𝑄₁,−(1−𝛼)^𝑑𝑄₂,−(1−𝛼)^𝑑𝑄₃+Sym𝑀Λ₁+𝑁Λ₂+𝑇Λ₃. (3.53) From (3.37), it follows that 𝜌𝐼+𝑆>0, and then the following inequality holds from Lemma 2.9−^2𝑚_𝑗=1𝛼_𝑗(𝑘)Υ_𝑗𝑅^𝑇𝑆^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑅Υ^𝑇_𝑗≤^2𝑚_𝑗=1𝛼_𝑗(𝑘)SymΥ_𝑗𝑅^𝑇Ξ^𝑇+Ξ𝜌𝐼+𝑆⁻¹Ξ^𝑇+𝜌^2𝑚_𝑗=1𝛼_𝑗(𝑘)Υ_𝑗𝑅^𝑇2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑅Υ^𝑇_𝑗, (3.54) which, together with (3.50) and (3.52), implies that −∑^2𝑚_𝑗=1𝛼_𝑗(𝑘)Υ_𝑗𝑅^𝑇𝑆∑^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑅Υ^𝑇_𝑗+ΦΨ∑^2𝑚_𝑗=1𝛼_𝑗(𝑘)Υ∑_𝑗^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑑Π∑_𝑗^2𝑚_𝑗=1𝛼_𝑗(𝑘)𝑑Π_𝑗∗Γ000∗∗−𝑋⁻¹00∗∗∗−𝑑𝑍₁⁻¹0∗∗∗∗−𝑑𝑍₂⁻¹<0. (3.55) Applying Schur complement, it results that inequality (3.55) is equivalent to inequality (3.44). Therefore, if inequalities (3.37) hold, then inequality (3.44) holds, that is, system (3.43) is regular, causal, and exponentially stable, provided that 𝑥(𝑘)∈ℒ(𝐻), ∀𝑘≥0.
On the other hand, by the proof in Theorem 3.1, inequality (3.44) holding implies 𝑉(𝜉_𝑘+1)<(1−𝛼)𝑉(𝜉_𝑘)<𝑉(𝜉_𝑘). Assume the initial condition 𝑥(𝑠)=𝜙(𝑠) is included in the ball ℬ(𝛿), ∀𝑠∈𝐈[−𝑑,0], and inequalities (3.37) and (3.38) hold. Then, from the definition of 𝑉(𝜉_𝑘), we get 𝑥^𝑇(0)𝐸^𝑇𝑋𝐸𝑥(0)=𝜉^𝑇(0)𝐸^𝑇𝑋𝐸𝜉(0)≤𝑉(𝜉₀)≤𝛿⁻¹‖𝜙‖²_𝑐≤1, that is, 𝑥(0)∈Ω(𝐸^𝑇𝑋𝐸,1). Since inequalities (3.38) are satisfied, it follows that 𝑥(0)∈ℒ(𝐻). Furthermore, because 𝑉(𝜉_𝑘+1)<𝑉(𝜉_𝑘), we can conclude that 𝑥^𝑇(𝑘)𝐸^𝑇𝑋𝐸𝑥(𝑘)=𝜉^𝑇(𝑘)𝐸^𝑇𝑋𝐸𝜉(𝑘)≤𝑉(𝜉_𝑘)≤𝑉(𝜉₀)≤𝛿⁻¹‖𝜙‖²_𝑐≤1, which means that 𝑥(𝑘)∈ℒ(𝐻), ∀𝑘>0. This completes the proof.

Remark 3.5. In practice, we may be interested to obtain a ball ℬ(𝛿) of initial condition as large as possible. To maximize the ball of initial condition, the following approximate optimization procedure can be used: min_{𝑋>0,𝑄1>0,𝑄2>0,𝑄3>0,𝑍1>0,𝑍2>0,𝐹,𝐻}𝑟s.t.(3.44),(3.45)and𝐸^𝑇𝑋𝐸≤𝑤₁,𝑄₁≤𝑤₂,𝑄₂≤𝑤₃,𝑄₃≤𝑤₄,𝑍₁≤𝑤₅,𝑍₂≤𝑤₆, (P1) where 𝑟=𝑤₁+(𝑑+𝑑(𝑑−1))𝑤₂+𝑑𝑤₃+𝑑𝑤₄+𝑑²𝑤₅+𝑑²𝑤₆.

In the special case that 𝐸=𝐼, the following corollary directly from Remark 3.3 and Theorem 3.4.

Corollary 3.6. Given constants 0<𝛼<1, 0<𝑑<𝑑, 𝜌>0, 𝜀₁>0, 𝜀₂>0, and 𝜀₃>0. If there exist matrices 𝑋>0, 𝑄_𝑙>0, 𝑙=1,2,3, 𝑍_𝑣>0, 𝑣=1,2, 𝑀=[𝑀^𝑇₁𝑀^𝑇₂𝑀^𝑇₃𝑀^𝑇₄]^𝑇, 𝑁=[𝑁^𝑇₁𝑁^𝑇₂𝑁^𝑇₃𝑁^𝑇₄]^𝑇, 𝑇=[𝑇^𝑇₁𝑇^𝑇₂𝑇^𝑇₃𝑇^𝑇₄]^𝑇, 𝐹 and 𝐻 such that the following hold: Φ_𝑗ΨΥ_𝑗𝑑Π_𝑗𝑑Π_𝑗∗Γ000∗∗−2𝜀₁𝐼+𝜀²₁𝑋00∗∗∗−𝑑2𝜀₂𝐼−𝜀²₂𝑍₁0∗∗∗∗−𝑑2𝜀₃𝐼−𝜀²₃𝑍₂<0,𝑗∈𝐈1,2^𝑚,1ℎ_𝑙[],∗𝑋≥0,𝑙∈𝐈1,𝑚 (3.56) where ℎ_𝑙 denotes the 𝑙th row of 𝐻, Φ_𝑗=diag−(1−𝛼)𝑋+1+𝑑𝑄₁+𝑄₂+𝑄₃,−(1−𝛼)^𝑑𝑄₁,−(1−𝛼)^𝑑𝑄₂,−(1−𝛼)^𝑑𝑄₃+Sym𝑀Λ₁+𝑁Λ₂+𝑇Λ₃,Υ_𝑗=𝐴_𝑗𝐴_𝑑00^𝑇,Π_𝑗=𝐴_𝑗−𝐼𝐴_𝑑00^𝑇,Λ₁=𝐼−𝐼00,Λ₂=,Λ0𝐼0−𝐼₃=0−𝐼𝐼0,𝐴_𝑗=𝐴+𝐵𝐷_𝑗𝐹+𝐷⁻_𝑗𝐻, (3.57) and Ψ and Γ are defined in (3.1), then, under the state feedback 𝑢(𝑘)=𝐹𝑥(𝑘), system (2.3) with 𝐸=𝐼 is locally exponentially stable with 𝜆=1/√1−𝛼 for any compatible initial condition in the ball ℬ(𝛿)=𝜙∈𝐶_𝑛,𝑑∶𝜙²_𝑐≤𝛿, (3.58) where 1𝛿=𝜆_max(𝑋)+𝑑+𝑑𝑑−1𝜆_max𝑄₁+𝑑𝜆_max𝑄₂+𝑑𝜆_max𝑄₃+𝑑²𝜆_max𝑍₁+𝑍₂. (3.59)

Also, as Remark 3.5, the corresponding approximate optimization problem of Corollary 3.6 becomes lim_{𝑋>0,𝑄1>0,𝑄2>0,𝑄3>0,𝑍1>0,𝑍2>0,𝐹,𝐻}𝑟s.t.(3.56),(3.58)and𝑋≤𝑤₁,𝑄₁≤𝑤₂,𝑄₂≤𝑤₃,𝑄₃≤𝑤₄,𝑍₁≤𝑤₅,𝑍₂≤𝑤₆, (P2) where 𝑟=𝑤₁+(𝑑+𝑑(𝑑−1))𝑤₂+𝑑𝑤₃+𝑑𝑤₄+𝑑²𝑤₅+𝑑²𝑤₆.

Remark 3.7. Scalars 𝜌, 𝜀_𝑓, 𝑓=1,2,…,5, in Theorem 3.4 are tuning parameters which need to be given first. In fact, (3.37) and (3.38), for fixed 𝛼>0 and 0<𝑑<𝑑, are bilinear matrix inequalities (BMIs) regarding to these tuning parameters. If one can accept more computational burden, the optimal values of these parameters can be obtained by applying some global optimization algorithms [35] to solve the BMIs.

4. Numerical Examples

In this section, two examples are given to show the effectiveness of the proposed method.

Example 4.1. Consider the discrete-time singular time-delay system (2.1) without input and with the following parameters: 𝐸=1000,𝐴=0.500.350.35,𝐴_𝑑=.0.1300.20.1 (4.1) Let 𝑅=⁰⁰₀₁. For various delay upper bound 𝑑 and by using Theorem 3.1 in this paper, the maximum allowable values of decay rate 𝜆, that guarantees the regularity, causality and exponential stability of the system for given delay lower bound 𝑑, are listed in Table 1. It can be seen from Table 1 that the delay upper bound 𝑑 is related to the decay rate 𝜆, and a smaller 𝜆 allows for a larger 𝑑. Figure 1 depicts the simulations results of 𝑥₁(𝑘) and 𝑥₂(𝑘) as compared to 1.25^−𝑘 when 𝑑(𝑘)=2+sin(2.5𝜋𝑘) and initial function is 𝜙_𝑘=[−1,1]^𝑇, 𝑘=−3,−2,…,0. It is clearly observed from Figure 1 that the states 𝑥₁(𝑘) and 𝑥₂(𝑘) exponentially converge to zero with a decay rate greater than 1.25.

Table 1 

Maximum allowable decay rates 𝜆 for different 𝑑 with d   =1.

Figure 1 

Simulation results of and as compared 1.25−𝑘.

Example 4.2. Consider the discrete-time singular time-delay system (2.1) with the following parameters: 𝐸=2.9100,𝐴=1.20.21.22.2,𝐴_𝑑=.0.1−0.10.3−0.2,𝐵=−0.1−0.4 (4.2) The time-delay is 1≤𝑑(𝑘)≤3. Our purpose is to design the stabilizing controllers for different decay rate 𝜆=1/√1−𝛼 and estimate the domain of attraction for the above system.
Let 𝜌=0.001, 𝜀₁=2.6, 𝜀₂=67, 𝜀₃=71, 𝜀₄=−0.08, 𝜀₅=0.01, and 𝑅=⁰⁰₀₅. For various 𝛼, solving (P1) by using the mincx problem in LMI Toolbox [36], the maximal values of 𝛿, denoted 𝛿_max, and the corresponding controller gains 𝐹 are listed in Table 2. Especially, when 𝛼=0.3, the feasible solutions to (P1) are 𝑋=0.01510.01180.01180.0288,𝑄₁=0.01200.00260.00260.0011,𝑄₂=,𝑄0.01800.00260.00260.0016₃=0.02100.00150.00150.0016,𝑍₁=0.00910.00190.00190.0043,𝑍₂=,,0.00970.00230.00230.0013𝐻=1.62870.5616𝐹=1.21240, (4.3) Furthermore, applying the obtained controller and giving a possible time-varying delay 𝑑(𝑘)=2+sin(2.5𝜋𝑘), we obtain the state response of the resulting closed-loop system and the corresponding actuator output as shown in Figures 2 and 3 for the given initial function 𝜙_𝑘=[−1.6,1]^𝑇, 𝑘=−3,−2,…,0.

Table 2 

𝛿_max and 𝐹 for different values of 𝛼.

Figure 2 

State response of the closed-loop system under saturated control.

Figure 3 

The actuator output under saturated control.

For clarity, we give some comments to explain the above simulation results. Firstly, it is easily seen from Table 2 that the smaller 𝛼, that is, the smaller decay rate 𝜆, then the larger is the domain 𝐵(𝛿) for which we guarantee the exponential stability of the saturated system. It shows the tradeoff between the size of the ball of the admissible initial condition and the desired decay rate. Secondly, from the curves in Figures 2 and 3, one can see that despite the partly unknown transition probabilities, either the designed state-feedback or the designed output-feedback controllers are feasible and effective ensuring that the resulting closed-loop systems are stable.

It should be pointed out that the results in [27] cannot be applied to this system, because the system considered here contains state delay.

5. Conclusions

The problems of exponential estimates and stabilization for a class of discrete-time singular systems with time-varying state delays and saturating actuators have been investigated in this paper. An exponential admissibility condition has been developed for the system. Also, a saturated state feedback controller and a domain of safe initial conditions have been determined by using LMI optimization-based approach. The effectiveness of the results has been illustrated by numerical examples. Note that in the present work, a quadratic Lyapunov-Krasovskii function was used for the saturated closed-loop system modeled by differential inclusion, which may lead to conservatism. Applying the saturation-dependent Lyapunov function approach proposed in [20] to capture the real-time information on the severity of saturation and thus leading to less conservative results will be interesting topics for further research.

Appendix

Proof of Lemma 2.5. For given 𝜁(𝑘) and 𝑦(𝑘) defined in (2.8), the following equation holds −^{𝑘−𝑑(𝑘1)−1}_{𝑘𝑙=𝑘−𝑑2}𝜁^𝑇(𝑘)𝑁+𝜆^𝑘−𝑙𝑦^𝑇(𝑙)𝐸^𝑇𝑅𝜆^𝑘−𝑙𝑅⁻¹𝑁^𝑇𝜁(𝑘)+𝜆^𝑘−𝑙𝑅𝐸𝑦(𝑙)=−𝜁^𝑇(𝑘)𝑁𝑅⁻¹𝑁^𝑇𝜉(𝑘)^{𝑘−𝑑(𝑘1)−1}_{𝑘𝑙=𝑘−𝑑2}𝜆^𝑙−𝑘−2𝜁^𝑇(𝑘)𝑁^{𝑘−𝑑(𝑘1)−1}_{𝑘𝑙=𝑘−𝑑2}𝐸𝑦(𝑙)−^{𝑘−𝑑(𝑘1)−1}_{𝑘𝑙=𝑘−𝑑2}𝑦^𝑇(𝑙)𝐸^𝑇𝜆^𝑘−𝑙𝑅𝐸𝑦(𝑙). (A.1) Rearranging the above equation yields (2.9). This completes the proof.

Proof of Lemma 2.7. Case  1 (𝑘≤𝑑(𝑘)). Clearly, 𝑘−𝑑(𝑘)∈[−𝑑,0], and then ‖𝑥(𝑘,𝜙)‖≤‖𝐷‖‖𝜙‖_𝑐+‖𝑓(𝑘)‖≤𝜒𝑒^−𝜆‖𝜙‖_𝑐+𝜅𝑒^−𝛽𝑘≤𝜒𝑒^{−𝜆(𝑘/𝑑)}‖𝜙‖_𝑐+𝜅𝑒^−𝛽𝑘≤(𝜒‖𝜙‖_𝑐+𝜅)𝑒^−𝑟𝑘.
Case  2 (𝑘>𝑑(𝑘)). Inspired by the idea in [16], to model the dependency of 𝑥(𝑘) on past instances, define 𝑘₀=𝑘,𝑘_𝑖=𝑘_𝑖−1−𝑑𝑘_𝑖−1,𝑖=1,2,… (A.2) Note that 𝑘_𝑖<𝑘_𝑖−1 then, by iterative substitutions of (2.10), one gets 𝑥𝑘,𝜙=(−𝐷)^𝑘(𝑡)𝑥𝑘_𝑘(𝑡)+^{𝑘(𝑡)−1}_𝑖=1(−𝐷)^𝑖𝑓𝑘_𝑖+𝑓𝑘₀,𝑘≥0, (A.3) where 𝑘(𝑡) is a positive integer.
(i) There exists a finite positive integer 𝑘(𝑡) such that 0≤𝑘_{𝑘(𝑡)−1} and −𝑑≤𝑘_𝑘(𝑡)<0. Then it follows from (A.2) that 𝑘≤𝑘(𝑡)𝑑,𝑘_𝑖=𝑘₀−^𝑖−1_𝑗=0𝑑𝑘_𝑗≥𝑘₀−𝑖𝑑. (A.4) By (A.3), (A.4), (2.11) and 𝑟=min{𝜆/𝑑,𝛽}, the following is true: 𝑥𝑘,𝜙≤𝐷^𝑘(𝑡)𝜙_𝑐+^{𝑘(𝑡)−1}_𝑖=1𝐷^𝑖𝑓𝑘_𝑖+𝑓𝑘₀≤𝜒𝑒^{−𝜆𝑘(𝑡)}𝜙_𝑐+^{𝑘(𝑡)−1}_𝑖=1𝜒𝑒^−𝜆𝑖𝜅𝑒^{−𝛽𝑘𝑖}+𝜅𝑒^−𝛽𝑘0≤𝜒𝑒^{−𝜆(𝑘/𝑑)}𝜙_𝑐+𝜒𝜅^{𝑘(𝑡)−1}_𝑖=1𝑒^−𝜆𝑖𝑒^{−𝛽(𝑘0−∑𝑖−1𝑗=0𝑑(𝑘𝑗))}+𝜅𝑒^−𝛽𝑘0≤𝜒𝑒^{−𝜆(𝑘/𝑑)}𝜙_𝑐+𝜒𝜅^{𝑘(𝑡)−1}_𝑖=1𝑒^−𝜆𝑖𝑒^𝛽𝑖𝑑𝑒^−𝛽𝑘+𝜅𝑒^−𝛽𝑘≤𝜒𝜙_𝑐+𝜒𝜅^{𝑘(𝑡)−1}_𝑖=1𝑒^{(𝛽𝑑−𝜆)𝑖}+𝜅𝑒^−𝑟𝑘≤𝜒𝜙_𝑐1+𝜒𝜅1−𝑒^{𝛽𝑑−𝜆}+𝜅𝑒^−𝑟𝑘. (A.5)
(ii)  𝑘₀−∑^𝑖−1_𝑗=0𝑑(𝑘_𝑗)≥0 for all positive integer 𝑖 with 𝑖≥1. Since 𝑑(𝑘)≥0, the sequence {𝑘₀−∑^𝑖−1_𝑗=0𝑑(𝑘_𝑗)}^+∞_𝑖=1 is monotonically decreased and has a greatest lower bound 𝜖 with 𝜖≥0; that is lim_𝑖→+∞𝑘₀−∑^𝑖−1_𝑗=0𝑑(𝑘_𝑗)=𝜖. Noting 𝜆>0, we have lim_𝑖→+∞𝑒^{−𝜆𝑘𝑖}‖𝑥_𝑘𝑖‖=0 [8]. Then, similar to the process to derive (A.5), the results in the lemma holds.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant 60904020, Open Fund of Key Laboratory of Measurement and Control of CSE (no. MCCSE2012A06), Ministry of Education, China, Southeast University, and the Scientific Research Foundation of Nanjing University of Posts and Telecommunications (NY210080).