# New Results for Finite-Time Stability of Discrete-Time Linear Systems with Interval Time-Varying Delay

Authors: Sreten B. Stojanovic
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

The problem of finite-time stability for linear discrete time systems with state time-varying delay is considered in this paper. Two finite sum inequalities for estimating weighted norms of delayed states are proposed in order to obtain less conservative stability criteria. By using Lyapunov-Krasovskii-like functional with power function, two sufficient conditions of finite-time stability are proposed and expressed in the form of  linear matrix inequalities (LMIs), which are dependent on the minimum and maximum delay bounds. The numerical example is presented to illustrate the applicability of the developed results. It was shown that the obtained results are less conservative than some existing ones in the literature.


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## Body

## 1. Introduction

Up to now, most of the existing literature related to stability systems is focused on Lyapunov asymptotic stability, which is defined over an infinite time interval. However, often, this stability concept is insufficient for practical purposes, because there are some real systems which require that the system states do not exceed the specified values over a fixed short time. A system could be Lyapunov stable but completely useless because it possesses undesirable transient performances, for instance, in the presence of saturation or controlling the trajectory of a space vehicle from an initial point to a final one in a prescribed time interval. Consequently, it is of particular interest to investigate the trajectories of the system only over a finite time interval. The described stability concept, based on the stability investigation in a limited time frame, is named finite-time stability (FTS). In that sense, the system is stable if the states of the system do not exceed the predefined boundaries on some fixed time interval. This stability concept was introduced in the era of modern control systems [1, 2], and it is still widely used nowadays as well. Initially, the concept had an academic value, and its practical applications were applied later on. With the development of the linear inequalities, the stability conditions that could be used for practical purposes were developed for both continuous [3–5] and discrete-time systems [6–8].It should be noted that, in the existing literature, parallel to the previously defined FTS, there is another stability concept denoted by FTS, but with a different meaning. The second FTS concept is used to describe system whose states approach equilibrium point in a finite time [5]. In our paper, we consider the first concept of FTS.Furthermore, FTS in the presence of exogenous input leads to the concept of finite-time boundedness (FTB) [4, 9, 10] (a system is said to be FTB if, given a bound on the initial condition and characterization of the set of admissible inputs, the state variables remain below the prescribed limit for all inputs in the set).In some practical systems (such as biological systems, mechanical systems, and networked control systems), time delay is inevitable. The existence of time delay might result in the performance deterioration or even instability of the system. Frequently, the time delay is not constant but time varying. A large number of researchers were involved in the study of time-delay systems (see, e.g., [11–13] and the references therein).Similar to regular systems, the FTS concept can be applied to the class of time-delay systems. However, up to date, there are few works dealing with the FTS of time-delay systems. Some early results of FTS for time-delay systems can be found in [14]. The results of these investigations are conservative because they are based on the application of simple algebraic inequalities. Recently, using the theory of linear matrix inequality (LMI), more practical and precise results are obtained for continuous and discrete-time-delay systems.In [15], the finite-time control problem for networked control systems with time-varying delays is investigated and a finite-time controller via state feedback is proposed to guarantee FTB. In [16, 17], the finite-time stability and stabilization for a class of linear continuous-time systems with constant time delay are studied. By using Lyapunov-Krasovskii-like (LKL) functional, some sufficient conditions that ensure finite-time performance of the considered systems are derived in the form of LMIs. The robust FTS is discussed in [17]. The controller design problem is solved by using the LMIs and the cone complementarity linearization iterative algorithm. In order to get less conservative criteria, a new form of LKL functional with power weighting function is introduced in [18]. In the last few years, with the great development of the switched system theory, the FTS and FTB problems of switched systems with time delay have received increasing attention (see, e.g., [19, 20] and the references therein). Recently, there has been rapidly growing interest in the problem of FTS analysis and synthesis of discrete-time systems with time-varying delay. In [21], the sufficient delay-dependent FTS conditions are derived by using new LKL functional and an appropriate model transformation of the original system. A less conservative criterion is proposed in [22] by choosing a new LKL functional with power weighting function. In order to get less conservative criteria in [23], the original time-delay system is firstly transformed into two interconnected subsystems. Then, new sufficient conditions of FTS and stabilization are derived by using a two-term approximation of the time-varying delay.The main purpose of this paper is to obtain less conservative stability criteria for finite-time stability for discrete-time linear system with time-varying delay. Firstly, two finite sum inequalities for estimating weighted norms of delayed states are proposed. By using LKL functional with power function and finite sum inequalities, sufficient conditions of finite-time stability are proposed and expressed in the form of linear matrix inequalities (LMIs), which are dependent on the minimum and maximum delay bounds. Finally, a numerical example is given to illustrate the applicability of the developed results and the comparisons with some previous ones. It was shown that the obtained results are less conservative than some existing ones in the literature.The rest of this paper is organized as follows. Section2 gives the description of the considered system firstly and some necessary lemmas and remarks are listed. In Section 3, on the basis of the proposed finite sum inequalities and LKL functional, sufficient conditions are presented to guarantee the finite-time stability of the considered system. Section 4 gives one numerical example to illustrate the applicability of the developed results. Section 5 shows the summary of this paper.Notations. The matrix transposition was denoted by a superscript “T.” Rn and Rn×m are the n-dimensional Euclidean spaces and the set of all real matrices having dimension n×m, respectively. X>0 denotes a real positive definite matrix, while X>Y implies that the matrix X-Y is a positive definite matrix. λmax(X)(λmin(X)) denotes the maximum (minimum) of eigenvalues of a real symmetric matrix X.


## 2. Preliminaries and Problem Formulation

Consider the following discrete time system with time-varying delay:(1)xk+1=Axk+Adxk-dk,where x(k)∈Rn is the state vector and A∈Rn×n and Ad∈Rn×n are known constant matrices. The time delay d(k) is assumed to be time varying and has lower dm and upper dM bounds such that (2)0<dm≤dk≤dM.The initial condition ϕ(θ) is defined as(3)xθ=ϕθ,θ∈-dM,-dM+1,…,-1,0with(4)supθ∈-dM,-dM+1,…,-1⁡ϕθ+1-ϕθTϕθ+1-ϕθ≤δ.In this paper, the purpose is to derive sufficient conditions which guarantee the finite-time stability of discrete-time system with time-varying delay (1). To study the finite-time stability, the following definition is necessarily introduced.Definition 1.
Discrete-time system with time-varying delay (1) is said to be finite-time stable with respect to (α,β,N), where 0≤α<β, if ∀k∈1,2,…,N(5)supj∈-dM,-dM+1,…,0⁡ϕTjϕj≤α⟹xTkxk<β.The following lemmas will be used in the derivation of the main results.Lemma 2.
For any appropriately dimensioned matricesZ=ZT>0, M and positive integers d1, d2>d1, and γ, the inequality (6)-∑j=k-d2k-d1-1γk-jyTjZyj≤ξTkcMZ-1MTξk+2ξTkMxk-d1-xk-d2holds, where y(k)=x(k+1)-x(k), ξ(k)=xT(k)xT(k-d(k))xT(k-d1)xT(k-d2)T, and(7)c=γ-d1-γ-d2γ-1.Proof.
Using the inequality(8)0≥-∑j=k-d2k-d1-1MTξk+γk-jZyjTγk-jZ-1·MTξk+γk-jZyjwe obtain(9)0≥-∑j=k-d2k-d1-1ξTkMγk-jZ-1MTξk-∑j=k-d2k-d1-12ξTkMγk-jZ-1γk-jZyj-∑j=k-d2k-d1-1γk-jyTjZγk-jZ-1γk-jZyj=-ξTkMZ-1MTξk∑j=k-d2k-d1-1γk-j-2ξTkM∑j=k-d2k-d1-1yj-∑j=k-d2k-d1-1γk-jyTjZyjwhich implies(10)-∑j=k-d2k-d1-1γk-jyTjZyj≤ξTkMZ-1MTξk∑j=k-d2k-d1-1γk-j+2ξTkM∑j=k-d2k-d1-1yj=ξTkcMZ-1MTξk+2ξTkMxk-d1-xk-d2.This completes the proof.Lemma 3.
For any appropriately dimensioned matricesZ=ZT>0, X, Y and positive integers d1, d2>d1, and γ, the inequality (11)-∑j=k-d2k-d1-1γk-jyTjZyj≤ξTkcXξk+2ξTkYxk-d1-xk-d2holds, where y(k)=x(k+1)-x(k), ξ(k)=xT(k)xT(k-d(k))xT(k-d1)xT(k-d2)T, and(12)XY∗Z≥0,(13)c=γ-d1-γ-d2γ-1.Proof.
From (12), we have(14)abTXY∗Zab≥0.Let a=γ(k-j)/2ξ(k), b=γ(k-j)/2y(j). Then(15)-γ-k-j/2ξTkγ-k-j/2Xξk≤γk-j/2yTjZγk-j/2yj+2γ-k-j/2ξTkYγk-j/2yj,-γ-k-jξTkXξk≤γk-jyTjZyj+2ξTkYyj.Summarizing the previous expression per j, from j=k-d2 to j=k-d1-1, we get(16)-∑j=k-d2k-d1-1γ-k-jξTkXξk≤∑j=k-d2k-d1-1γk-jyTjZyj+2ξTkY∑j=k-d2k-d1-1yj=∑j=k-d2k-d1-1γk-jyTjZyj+2ξTkYxk-d1-xk-d2.Rearranging the above equation yields (17)-∑j=k-d2k-d1-1γk-jyTjZyj≤∑j=k-d2k-d1-1γ-k-jξTkXξk+2ξTkYxk-d1-xk-d2=cξTkXξk+2ξTkYxk-d1-xk-d2.This completes the proof.Remark 4.
LetY=M. Then, the right sides of inequalities (6) and (11) become (18)ξTkcMZ-1MTξk+2ξTkMxk-d1-xk-d2,ξTkcXξk+2ξTkMxk-d1-xk-d2,respectively. From (12), for Z>0, the following inequality holds:(19)X>MZ-1MT,which implies(20)ξTkcXξk>ξTkcMZ-1MTξk.Based on (18) and (20), we have (21)ξTkcMZ-1MTξk+2ξTkMxk-d1-xk-d2<ξTkcXξk+2ξTkMxk-d1-xk-d2.Hence, it can be concluded that inequality (6) is less restrictive compared to inequality (11) if Y=M.Remark 5.
If we adoptX=MZ-1MT and Y=M, then the right sides of inequalities (6) and (11) are equal to each other and(22)XY∗Z=MZ-1MTM∗Z=Z-1/2MTZ1/200TZ-1/2MTZ1/200≥0.Then, Lemma 2 can be considered as a particular case of Lemma 3.


## 3. Main Results

In the sequel, we will present two finite-time stability criteria for system (1).Theorem 6.
The discrete-time system (1) with time-varying delay is finite-time stable with respect to α,β,N if there exist a scalar γ>1, positive scalars λ1, λ2, λ3, λ4, λ5, λ6, λ7, and λ8, positive definite symmetric matrices P, Q1, Q2, R1, and R2, and matrices L=L1TL2TL3TL4TT, S=S1TS2TS3TS4TT, and T=T1TT2TT3TT4TT, such that the following inequalities hold:(23)Σ1+Σ2c1Lc1Sc2T∗-c1R100∗∗-c1R10∗∗∗-c2R2<0,(24)λ1I<P<λ2I,λ3I<Q1<λ4I,λ5I<Q2<λ6I,R1<λ7I,R2<λ8I,(25)γNαλ2+μ1λ4+μ2λ6+γNδμ3λ7+μ4λ8-βλ1+μ1λ3+μ2λ5<0,where(26)Σ1=Ω11Ω1200∗Ω2200∗∗-γdmQ20∗∗∗-γdMQ1,Ω11=ATPA-γP+Q1+Q2+dM-dmA-ITR1A-I+dmA-ITR2A-I,Ω12=ATPAd+dM-dmA-ITR1Ad+dmA-ITR2Ad,Ω22=AdTPAd+dM-dmAdTR1Ad+dmAdTR2Ad,Σ2=T1+T1TL1-S1+T2TS1-T1+T3T-L1+T4T∗L2+L2T-S2-S2TL3T+S2-S3T-T2-L2+L4T-S4T∗∗S3+S3T-T3-T3T-L3+S4T-T4T∗∗∗-L4-L4T,c1=γ-dm-γ-dMγ-1,c2=1-γ-dmγ-1,μ1=γdM-1γ-1,μ2=γdm-1γ-1,μ3=γdM+1-γdm+1-γ-1dM-dmγ-12,μ4=γγdm-1-γ-1dmγ-12.Proof.
Let us consider the following discrete LKL functional:(27)Vxk=V1xk+V2xk+V3xk,where(28)V1xk=xTkPxk,V2xk=∑j=k-dMk-1γk-1-jxTjQ1xj+∑j=k-dmk-1γk-1-jxTjQ2xj,V3xk=∑j=-dM-dm-1∑θ=k+jk-1γk-1-θyTθR1yθ+∑j=-dm-1∑θ=k+jk-1γk-1-θyTθR2yθ.Define ΔVx(k)=Vx(k+1)-x(k). Then, along with solution of system (1), we have(29)ΔV1xk=xTk+1Pxk+1-xTkPxk=Axk+Adxk-dkTPAxk+Adxk-dk-xTkPxk+γxTkPxk-γxTkPxk=xTkATPA-γPxk+2xTkATPAdxk-dk+xTk-dk·AdTPAdxk-dk+γ-1V1,ΔV2xk=∑j=k+1-dMkγk-jxTjQ1xj-∑j=k-dMk-1γk-1-jxTjQ1xj+∑j=k+1-dmkγk-jxTj·Q2xj-∑j=k-dmk-1γk-1-jxTjQ2xj=γ∑j=k-dMk-1γk-1-jxTjQ1xj+xTkQ1xk-γdMxTk-dMQ1xk-dM-∑j=k-dMk-1γk-1-jxTjQ1xj+γ∑j=k-dmk-1γk-1-jxTjQ2xj+xTkQ2xk-γdmxTk-dmQ2xk-dm-∑j=k-dmk-1γk-1-jxTjQ2xj=xTkQ1+Q2·xk-γdMxTk-dMQ1xk-dM-γdmxTk-dmQ2xk-dm+γ-1·V2xk,ΔV3xk=∑j=-dMdm-1∑θ=k+1+jkγk-θyTθR1yθ-∑j=-dMdm-1∑θ=k+jk-1γk-1-θyTθR1yθ+∑j=-dm-1∑θ=k+1+jkγk-θyTθR2yθ-∑j=-dm-1∑θ=k+jk-1γk-1-θyTθR2yθ=∑j=-dMdm-1γ∑θ=k+jk-1γk-1-θyTθR1yθ+yTkR1yk-γ-jyTk+jR1yk+j+∑j=-dm-1γ∑θ=k+jk-1γk-1-θyTθR2yθ+yTkR2yk-γ-jyTk+jR2yk+j-∑j=-dMdm-1∑θ=k+jk-1γk-1-θyTθR1yθ-∑j=-dm-1∑θ=k+jk-1γk-1-θyTθR2yθ,ΔV3xk=γ-1∑j=-dMdm-1∑θ=k+jk-1γk-1-θyTθR1yθ+γ-1∑j=-dm-1∑θ=k+jk-1γk-1-θyTθR2yθ+∑j=-dMdm-1yTkR1yk-γ-jyTk+jR1yk+j+∑j=-dm-1yTkR2yk-γ-jyTk+jR2yk+j=dM-dmyTk·R1yk-∑j=-dMdm-1γ-jyTk+jR1yk+j+dmyTkR2yk-∑j=-dm-1γ-jyTk+jR2yk+j+γ-1V3xk=A-Ixk+Adxk-dkTdM-dmR1+dmR2A-Ixk+Adxk-dk-∑j=k-dMk-dm-1γk-jyTjR1yj-∑j=k-dmk-1γk-jyTjR2yj+γ-1V3xk.
Hence, it is concluded that(30)ΔVxk=γ-1Vxk+xTkΩ11xk+2xTkΩ12xk-dk+xk-dkΩ22xk-dk-γdMxTk-dMQ1xk-dM-γdmxTk-dmQ2xk-dm-∑j=k-dMk-dm-1γk-jyTjR1yj-∑j=k-dmk-1γk-jyTjR2yj.According to Lemma 2, we have(31)-∑j=k-dMk-dm-1γk-jyTjR1yj=-∑j=k-dMk-dk-1γk-jyTjR1yj-∑j=k-dkk-dm-1γk-jyTjR1yj≤ξTkc1′LZ-1LTξk+2ξTkLxk-dk-xk-dM+ξTkc1′′SZ-1STξk+2ξTkSxk-dm-xk-dk≤ξTkc1LZ-1LTξk+2ξTk0L0-Lξk+ξTkc1SZ-1STξk+2ξTk0-SS0ξk=ξTkc1LZ-1LTξk+ξTk0L0-L+0L0-LTξk+ξTkc1SZ-1STξk+ξTk0-SS0+0-SS0Tξk,-∑j=k-dmk-1γk-jyTjR2yj≤ξTkc2TZ-1TTξk+2ξTkTxk-xk-dm=ξTkc2TZ-1TTξk+2ξTkT0-T0ξk=ξTkc2TZ-1TTξk+ξTkT0-T0+T0-T0Tξk,where(32)c1′=γ-dk-γ-dMγ-1≤γ-dm-γ-dMγ-1≙c1,c1′′=γ-dm-γ-dkγ-1≤γ-dm-γ-dMγ-1≙c1.Taking (30)-(31) into account, we obtain(33)ΔVxk-γ-1Vxk=ξTk·Σ1+Σ2+c1LZ-1LT+c1SZ-1ST+c2TZ-1TT·ξk.If(34)Σ1+Σ2+c1LZ-1LT+c1SZ-1ST+c2TZ-1TT<0then(35)ΔVxk-γ-1Vxk=Vxk+1-Vxk-γ-1Vxk=Vxk+1-γVxk<0.Applying (35) iteratively, we obtain(36)Vxk<γVxk-1<γ2Vxk-2<⋯<γkVx0,k=1,2,3,….Based on the fact that γ>1, ϕT(j)ϕ(j)≤α, and (4), we get(37)Vx0=xT0Px0+∑j=-dM-1γ-1-jxTjQ1xj+∑j=-dm-1γ-1-jxTjQ2xj+∑j=-dM+1dm-1∑θ=j-1γ-1-θyTθR1yθ+∑j=-dm-1∑θ=j-1γ-1-θyTθR2yθ≤αλmaxP+λmaxQ1∑j=-dM-1γ-1-jxTjxj+λmaxQ2∑j=-dm-1γ-1-jxTjxj+λmaxR1∑j=-dM+1dm-1∑θ=j-1γ-1-θyTθyθ+λmaxR2∑j=-dm-1∑θ=j-1γ-1-θyTθyθ<αλmaxP+αλmaxQ1∑j=-dM-1γ-1-j+αλmaxQ2∑j=-dm-1γ-1-j+δλmaxR1∑j=-dM+1dm-1∑θ=j-1γ-1-θ+δλmaxR2∑j=-dm-1∑θ=j-1γ-1-θ;that is,(38)Vx0<αλmaxP+μ1λmaxQ1+μ2λmaxQ2+δμ3λmaxR1+μ4λmaxR2,where(39)μ1=∑j=-dM-1γ-1-j=γdM-1γ-1,μ2=∑j=-dm-1γ-1-j=γdm-1γ-1,μ3=∑j=-dM+1dm-1∑θ=j-1γ-1-θ=∑θ=-dM-1γ-1-θ+∑θ=-dM+1-1γ-1-θ+⋯+∑θ=-dm-1-1γ-1-θ=γdM-1γ-1+γdM-1-1γ-1+⋯+γdm+1-1γ-1=γdM+1-γdm+1-γ-1dM-dmγ-12,μ4=∑j=-dm-1∑θ=j-1γ-1-θ=∑θ=-dm-1γ-1-θ+∑θ=-dm+1-1γ-1-θ+⋯+∑θ=-1-1γ-1-θ=γdm-1γ-1+γdm-1-1γ-1+⋯+γ1-1γ-1=γγdm-1-γ-1dmγ-12.Note that(40)Vxk>λminPxTkxk+λminQ1∑j=k-dMk-1γk-1-jxTjxj+λminQ2∑j=k-dmk-1γk-1-jxTjxj.If the following condition is valid:(41)γkVx0<βλminP+μ1λminQ1+μ2λminQ2,then(42)λminPxTkxk+λminQ1∑j=k-dMk-1γk-1-jxTjxj+λminQ2∑j=k-dmk-1γk-1-jxTjxj<Vxk<γkVx0<γkαλmaxP+μ1λmaxQ1+μ2λmaxQ2+γkδμ3λmaxR1+μ4λmaxR2<βλminP+μ1λminQ1+μ2λminQ2,which implies xT(k)x(k)<β for k=1,2,…,N. Therefore, system (1) is finite-time stable.
By using inequalities (24) we have(43)γNαλ2+μ1λ4+μ2λ6+γNδμ3λ7+μ4λ8<βλ1+μ1λ3+μ2λ5which is equivalent to (25). This completes the proof.Theorem 7.
The discrete-time system with time-varying delay (1) is finite-time stable with respect to α,β,N if there exist a scalar γ>1, positive scalars λ1, λ2, λ3, λ4, λ5, λ6, λ7, and λ8, positive definite symmetric matrices P, Q1, Q2, R1, and R2, and matrices G, H, M, L=L1TL2TL3TL4TT, S=S1TS2TS3TS4TT, and T=T1TT2TT3TT4TT, such that the following inequalities hold:(44)Σ1+Σ2+c1G+H+c2M<0,(45)GL∗R1≥0,HS∗R1≥0,MT∗R2≥0,(46)λ1I<P<λ2I,λ3I<Q1<λ4I,λ5I<Q2<λ6I,R1<λ7I,R2<λ8I,(47)γNαλ2+μ1λ4+μ2λ6+γNδμ3λ7+μ4λ8-βλ1+μ1λ3+μ2λ5<0,where Σ1, Σ2, c1, c2, μ1, μ2, μ3, and μ4 are defined by (26).Proof.
This proof is based on the proof of Theorem6. Starting from expression (30) and using Lemma 3, we have(48)-∑j=k-dMk-dm-1γk-jyTjR1yj=-∑j=k-dMk-dk-1γk-jyTjR1yj-∑j=k-dkk-dm-1γk-jyTjR1yj≤ξTkc1′Gξk+2ξTkLxk-dk-xk-dM+ξTkc1′′Hξk+2ξTkSxk-dm-xk-dk≤ξTkc1Gξk+2ξTk0L0-Lξk+ξTkc1Hξk+2ξTk0-SS0ξk=ξTkc1Gξk+ξTk0L0-L+0L0-LTξk+ξTkc1Hξk+ξTk0-SS0+0-SS0Tξk,-∑j=k-dmk-1γk-jyTjR2yj≤ξTkc2Mξk+2ξTkTxk-xk-dm=ξTkc2Mξk+2ξTkT0-T0ξk=ξTkc2Mξk+ξTkT0-T0+T0-T0Tξk.In addition, by combining (30) and inequalities (48), we can obtain(49)ΔVxk-γ-1Vxk=ξTkΣ1+Σ2+c1G+H+c2Mξk.Based on inequality (44), we get(50)Vxk+1<γVxk;that is,(51)Vxk<γkVx0,k=1,2,3,….Further procedure is identical to the proof of Theorem 6.Remark 8.
In the finite-time stability theory, the main problem is how to obtain the inequalityVx(k+1)<γVx(k) (see (35) and (50)) using a minimum of approximations. By introducing the power function γk-j in LKL functional, the previous inequality can be obtained using only approximations of the following quadratic form, which was used in (6) and (11):(52)-∑j=k-d2k-d1-1γk-jyTjZyj.Therefore, it is important that the quadratic form should be approximated as precisely as possible, in order to reduce the conservatism of the FTS criterion. In this paper, this problem is solved by using Lemmas 2 and 3.
In most of the existing literature (see, e.g., [21, 23] and the references therein), the power function is not used in the LKL functional. Consequently, various techniques for approximation have been used for obtaining the inequality Vx(k+1)<γVx(k), which lead to increasing the conservatism of FTS criteria. On the other hand, the LKL functional in [22] is proposed with the power function, but the quadratic form with the power function is approximated using the following quadratic form without power function:(53)-∑j=k-d2k-d1-1γk-jyTjZyj≤-γd1+1∑j=k-d2k-d1-1yTjZyj,which introduces additional conservatism in FTS criteria. Thus, it can be expected that introducing the power function in LKL functional and using the less restrictive inequalities (6) and (11) lead to smaller conservatism of FTS criteria.Remark 9.
Inequality (12) is nonstrict LMIs. However, Lemma 3 also remains valid in the case of the following strict LMIs:(54)XY∗Z>0,but its result becomes somewhat restrictive. Accordingly, nonstrict LMIs in (45) can be replaced by the following restrictive strict LMIs:(55)GL∗R1>0,HS∗R1>0,MT∗R2>0,which can be easily calculated using some strict LMI solvers, similar to LMI Toolbox in Matlab. Note that the nonstrict LMIs (55) are used in Numerical Example.Remark 10.
Taking into account Remarks4 and 9, it can be expected that the result of Theorem 7 will be somewhat restrictive compared to Theorem 6. This claim will be demonstrated in Example 1.Remark 11.
In order to compare the results of this paper with those from the existing literature, the smallest eligible value of the parameterβ, βm, and the largest eligible value of time delay d(k), dMm, are adopted as criteria. If the given criterion, compared to other criteria, has a higher (smaller) computed value of βm (dMm), then its conservatism is greater. For the given criterion, the computed value of βm (dMm) is always greater (smaller) than its theoretical value βmt (dMmt), which represents the minimal (maximal) values of βm (dMm) such that the considered system is FTS. The system simulation can be used for the estimation of the parameter βmt (dMmt) (see Examples 1–3).The algorithm for computing the smallest eligible value of the parameterβm is described by the following steps.Algorithm 12.
Consider the following steps.Step 1.
Adopt a value of computational precisionΔ>0 of the smallest eligible value of the parameter β(βm).Step 2.
Setk=1. Choose a sufficiently large value for initial value for β0 such that LMIs (23)–(25) (Theorem 6) or (44)–(47) (Theorem 7) are feasible. Select new value β1∈α,β0.Step 3.
Setk=2. If there is a feasible solution of LMIs (23)–(25) (Theorem 6) or (44)–(47) (Theorem 7) for given βk-1, then select new value βk∈α,βk-1; otherwise, βk∈βk-1,βk-2.Step 4.
Setk=k+1. If there is a feasible solution of LMIs (23)–(25) (Theorem 6) or (44)–(47) (Theorem 7) for given βk-1, then select new value βk∈βk-2,βk-1; otherwise, βk∈βk-1,βk-2.Step 5.
Ifβk-βk1<Δ, then βm=βk and exit. Otherwise, go to Step 4.A similar algorithm can be proposed for computing the largest eligible value of time delaydMm.


## 4. Numerical Example

The effectiveness of the results presented in the previous section is now shown by means of numerical examples. We investigate the finite-time stability by comparing the smallest eligible values of parameterβm or the largest eligible values of time delay dMm.Example 1.
Consider the following discrete-time system with time-varying delay:(56)xk+1=0.20.10.10.20.10.40.210.2xk+0.10.010.10.10.010.10.10.10.01xk-dk,dm=2,dM=5.
The aim is to examine the FTS of system (56) with respect to (3,β,N) for N=5, 10, 20 and find the smallest eligible value of the parameter βm.
Applying Theorem6, for δ=1.1, N=10, and γ=1.14, we get the following feasible solution:(57)βm=βm1=37,P=7.2962-0.0953-0.4689∗7.2793-0.7771∗∗3.6941,Q1=0.7662-0.0008-0.0010∗0.7498-0.0057∗∗0.7463,Q2=0.78110.01240.0816∗0.2411-0.0673∗∗0.2316,R1=2.4430-0.13240.3298∗1.21041.0944∗∗1.4342,R2=0.16910.0056-0.0337∗0.15700.0918∗∗0.0811,λ1=3.4667,λ2=7.4649,λ3=0.7408,λ4=0.7709,λ5=0.1581,λ6=0.8030,λ7=2.6171,λ8=0.2496,L=0.0092-0.00250.0166-0.1793-0.0973-0.0251-0.01840.0006-0.0251-0.3490-0.0166-0.0756-0.0074-0.1930-0.1200-0.0814-0.1272-0.2417-0.09890.0020-0.01850.0033-0.0690-0.0681-0.0254-0.0712-0.0898-0.1159-0.0000-0.0238-0.0073-0.0533-0.0468-0.0210-0.0438-0.0648,S=-0.02410.0046-0.01640.21450.09730.1207-0.0034-0.02200.00010.32160.01870.07180.01030.14150.06550.09040.07540.1862-0.01200.02430.03440.0307-0.1504-0.09650.0179-0.1066-0.16900.06450.00220.01620.00660.02140.02000.02020.01660.0282,T=-0.0261-0.0022-0.00240.05150.02490.0005-0.0478-0.0498-0.02320.0154-0.0258-0.0184-0.0125-0.00210.00370.01120.00490.00100.0222-0.0105-0.01150.00610.03630.0188-0.01160.01180.01570.0133-0.0099-0.00820.00110.00400.00320.00360.00160.0011.
Based on Theorem7, for α=3, δ=1.1, N=10, γ=1.16, and βm=βm2=43, we also get a feasible solution. Therefore, system (56) with time-varying delay is finite-time stable with respect to (3,10,37)  (Theorem 6) and (3,10,43)  (Theorem 7).
Further, the smallest eligible values of the parameterβm, in this paper and the literature [21–23], are shown in Table 1 for N=5, 10, and 20. In this paper, the values of βm are smaller than those in [21–23]. Thus, for this example, the approach in this paper is less conservative than those in [21–23]. Theorem 7 shows a slightly greater conservatism than Theorem 6. The explanation for this is given in Remarks 4, 9, and 10.Table 1
The smallest eligible value of the parameterβ, βm.
N
5
10
20
[21, Theorem 1]
215
2402
273783
[22, Theorem 1]
32
57
140
[23, Theorem 1]
56
125
518
Theorem6
15
37
119
Theorem7
18
43
350In order to verify the stability properties, a computer simulation of the system with the random time-varying delay,2≤d(k)≤5, is performed under the conditions(58)ϕθ∈-5,-4,…,0=ϕ-5⋯ϕ0=111111111111111111T.It is obvious that(59)supθ∈-5,-4,…,0⁡ϕTθϕθ≤α=3,supθ∈-5,-4,…,-1⁡ϕθ+1-ϕθTϕθ+1-ϕθ≤δ=1.1.Figures1–3 show the initial state responses of system (56), time-varying delay as random variable, and the norm of the initial state responses. Obviously, the considered system is not asymptotically stable. From Figure 3(b), the theoretical value of βm, βmt, which is obtained by computer simulation amounts to βmt=6.25. As the conditions βm1=37>βmt and βm2=43>βmt are satisfied, based on Theorems 6 and 7, we can conclude that system (56) is finite-time stable.Figure 1
The initial state responses of system (56).Figure 2
Time-varying delay2≤d(k)≤5.Figure 3
The norm of initial state responses of system (56) for (a) -10≤k≤50 and (b) -5≤k≤15.
(a)
(b)Example 2.
Consider the following discrete-time system with time-varying delay:(60)xk+1=0.600.000.350.70xk+0.100.000.200.10xk-dk,dm=2≤dk≤dM,which has been considered in [21–23]. In addition, we choose α,β,N=(2.1,80,100). By using Theorem 6, for fixed γ=1.000002, the following feasible solution with the largest eligible value of time delay dMm is obtained:(61)dMm=10,P=4961.738.038.0306.8,Q1=715.81-36.91-36.9176.77,Q2=1088.8-41.1-41.127.3,R1=1031.4109.0109.065.0,R2=59.35-52.43-52.4347.07,λ1=306.4,λ2=4963.9,λ3=74.6,λ4=718.1λ5=25.7,λ6=1091.3,λ7=1043.7,λ8=107.2L=-0.04-0.021.190.10-110.05-11.55-11.68-7.82-11.34-1.24-1.72-0.2427.212.9413.357.52,S=-2.31-0.060.68-0.05110.4511.5812.097.868.580.69-16.43-8.0111.401.270.810.15,T=-23.5920.8223.79-21.372.23-1.960.02-0.01-1.421.27-22.2820.020.22-0.190.33-0.29.
To compare our results with those in [21–23], we give Table 2 in the following. From Table 2 it can be seen that, for this example, our method leads to less conservative results than that obtained in [21] and gives the same results as those obtained in [22, 23].Table 2
The largest eligible value of time delayd(k) (dMm).
[21, Theorem 1]
7
[22, Theorem 1]
10
[23, Theorem 1]
10
Theorem6
10In order to verify previous results, we choose the following initial conditions:(62)ψθ∈-10,-9,…,0=ψ-10⋯ψ0=-1-1-1-1-1-1-1-1-1-1-101111111111T,and we compute the initial state responses of system (60). Obviously, it can be seen that the initial values satisfy the following conditions:(63)supθ∈-10,-9,…,0⁡ϕTθϕθ<α=2.1,supθ∈-10,-9,…,-1⁡ϕθ+1-ϕθTϕθ+1-ϕθ≤δ=1.1.Figures4-5 show the initial state responses x(k) and the norm of initial state responses xT(k)x(k) for the random time-varying delay 2≤d(k)≤10. It can be seen that, under the above initial conditions, the state responses of system (60) satisfy the following conditions:(64)xTkxk<β=80,∀k∈1,2,…,100,which means that the system is finite-time stable with respect to (2.1,80,100).Figure 4
Initial state responses of system (60) with time-varying delay 2≤d(k)≤10.Figure 5
The norm of initial state responses of system (60).Example 3.
Consider the following discrete-time-delay system with time-varying delay:(65)xk+1=0.600.000.350.70xk+0.200.250.250.15xk-dk,which has been considered in [21, 22].
Based on Theorem6, for dm=1≤d(k)≤5=dM, α=2.1, N=10, δ=1.1, and γ=1.44, we find a feasible solution of (23)–(25) for the smallest eligible value βm1=299. Further, the condition of Theorem  1 in [22] is only feasible for βm2=3017, while the condition of Theorem  1 in [21] is not feasible for any value of the parameter β. At the same time, Theorem  1 in [21] gives feasible solution for βm3=27183 under the following, more restrictive, conditions: dm=1≤d(k)≤3=dM, α=2.1, N=7, δ=1.1, and γ=2.98. Therefore, from this example, we can conclude that our method leads to less conservative results than those obtained in [21, 22].In order to verify the stability properties, a computer simulation of the system is performed under the conditions(66)ϕθ∈-5,-4,…,0=ϕ-5⋯ϕ0=111111111111T,with the random time-varying delay, 1≤d(k)≤5. It is obvious that (67)supθ∈-5,-4,…,0⁡ϕTθϕθ≤α=2.1,supθ∈-5,-4,…,-1⁡ϕθ+1-ϕθTϕθ+1-ϕθ≤δ=1.1.Figures 6-7 show the initial state responses of the above system and the norm of initial state responses. Obviously, the considered system is not asymptotically stable. From Figure 7, the theoretical smallest eligible value of the parameterβ, βmt, which is obtained by computer simulation amounts to βmt=23.6. As the following conditions are satisfied: βm1=299>βmt, βm2=3017>βmt, and βm3=27183>βmt, we can conclude that system (65) is finite-time stable.Figure 6
Initial state response of system (65) with time-varying delay 1≤d(k)≤5.Figure 7
The norm of initial state response of system (65).Remark 4.
From the previous examples, the following conclusion can be made. In case when the system is not asymptotically stable, the derived results in this paper are less conservative in comparison with those from the existing literature [21–23]. When the system is asymptotically stable, this paper gives the same results as [22, 23] and less conservative results compared to [21].


## 5. Conclusion

In this paper, the problem of finite-time stability of discrete-time systems with time-varying delay has been considered. First, in order to obtain less conservative stability criteria, two finite sum inequalities for estimating weighted norms of delayed states are proposed. Then, by using Lyapunov-Krasovskii-like functionals with power function, new sufficient conditions of finite-time stability are obtained. The stability criteria are expressed in the form of linear matrix inequalities. It was shown that the obtained results are less conservative than some existing ones in the literature.



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