# Novel Robust Stability Criteria of Uncertain Systems with Interval Time-Varying Delay Based on Time-Delay Segmentation Method and Multiple Integrals Functional

Authors: Xing He; Li-Jun Song; Yu-Bin Wu; Zi-Yu Zhou
Journal: Complexity
 (2020)

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## Abstract

Interval time-varying delay is common in control process, e.g., automatic robot control system, and its stability analysis is of great significance to ensure the reliable control of industrial processes. In order to improve the conservation of the existing robust stability analysis method, this paper considers a class of linear systems with norm-bounded uncertainty and interval time-varying delay as the research object. Less conservative robust stability criterion is put forward based on augmented Lyapunov-Krasovskii (L-K) functional method and reciprocally convex combination. Firstly, the delay interval is partitioned into multiple equidistant subintervals, and a new Lyapunov-Krasovskii functional comprising quadruple-integral term is introduced for each subinterval. Secondly, a novel delay-dependent stability criterion in terms of linear matrix inequalities (LMIs) is given by less conservative Wirtinger-based integral inequality approach. Three numerical comparative examples are given to verify the superiority of the proposed approach in reducing the conservation of conclusion. For the first example about closed-loop control systems with interval time-varying delays, the proposed robust stability criterion could get MADB (Maximum Allowable Delay Bound) about 0.3 more than the best results in the previous literature; and, for two other uncertain systems with interval time-varying delays, the MADB results obtained by the proposed method are better than those in the previous literature by about 0.045 and 0.054, respectively. All the example results obtained in this paper clearly show that our approach is better than other existing methods.


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## Body

## 1. Introduction

Many dynamic model systems in the real world contain very significant time delays in the transmission of data and materials, in automatic robot control system, the acquisition and transmission of sensor signals, and the calculation of controller and the drive of brake may lead to time delay. In many kinds of time-delay types, the interval time-varying delay is more representative. The lower bound of its time delay is not necessarily zero, and the time delay is within a changing interval. It is common in practical application of engineering, especially in chemical reactors, internal combustion engines, and network control [1, 2]. Consequently, the stability analysis about interval time-delay systems has attracted wide attention in these years.Generally, aiming to analyze the stability of time-delay system, the most common method is to construct an appropriate LK functional (Lyapunov–Krasovskii functional, LKF) in time domain and combine it with linear matrix inequalities (LMIs). In general, the free weight matrix method, the time-delay segmentation method, the integral inequality method, the interactive convex combination method, and so forth are used to analyze its stability. Augmented functional method [3–5] can make full use of the system’s time-delay information to reduce the conservativeness of conclusion, but the introduction of matrix variables inevitably burdens the theoretical analysis and engineering calculation. Zhang et al. and Shen et al. [6, 7] obtain a conservative less stable stability criterion for linear systems with time-varying delays by constructing LKF with triple integral functional terms and optimize the stability conditions of time-delay systems. The integral inequality method has the characteristics of simple form and few matrix variables, which can promote the stability analysis of time-delay systems. Gu [8] first introduced Jensen’s inequality into the stability analysis of time-delay systems, and then Ramakrishnan [9, 10], Zhang [11], and Gouaisbaut [12] further promoted Jensen’s inequality, resulting in different and novel forms. In various forms, we have obtained effective conclusions of different conservation. As an innovative method, the interactive convex combination method [13, 14] can solve the stability problems of systems with interactive convex combination. Wu et al. [15] studied the issue of robust stability analysis for a sort of uncertain neutral system with mixed time-varying delays, and a novel discrete and neutral delay-dependent stability criterion based on linear matrix inequalities was given, which could greatly reduce the complexity of theoretical derivation and computation. Li et al. [16] deal with a set of positive functions combined with inverse convex weighting parameters by interactive convex combination technique instead of directly ignoring the term and deduce the stability criterion for uncertain neutral systems with mixed time delay. This criterion reduces the number of relevant decision variables while ensuring the conservativeness and avoids the complexity of numerical calculation.Farnam et al. [17] studied the robust stability problem for a class of linear systems with time-varying delays. By constructing LKF with more time-delay information, the stability condition of LMIs is obtained by means of interactive convex combination definition technique. Finally, the numbers are used to demonstrate that the given stability conditions are less conservative in computational efficiency. Ding et al. [18] construct an augmented functional with specific time-delay information based on the idea of time-delayed partitioning. The free-weight matrix inequality is used to define the cross terms generated by the functional derivatives, and a lower-conservative stability criterion is obtained. Senthilraj et al. [19] introduced a novel method to study the robust stability problem of an interval-delayed neural network system by using a nonuniform time-delay segmentation method and the integral inequality definition technique. Delay-dependent stability conditions for ensuring the stability of the system are obtained. Cheng et al. [20] studied a time-delay-related state feedback control problem for a class of time-varying delay continuous systems via improved interactive convex combination techniques and Wirtinger-based integral inequalities, and new stability conditions and state feedback control are obtained. Zhang et al. [21] proposed a robust stability criterion for a class of linear systems with time-varying delays by using the Wirtinger-based integral inequality and the interactive convex combination lemma to effectively define the cross terms emerging in the LKF derivatives. The conclusions obtained are superior in terms of stability analysis. Chang et al. studied the control problem with time-varying norm bounded uncertainties and discrete-time nonlinear systems with parametric uncertainties; LMI are used to obtain the sufficient conditions for robust stabilization [22, 23].On the basis of the above research results, for the sake of further revealing the relationship between the asymptotical stability of uncertain systems with interval time-varying delay and the constructed LKF and then to lower the conservatism caused by dealing with the functional derivatives, this paper attempts to study the robust stability problem of uncertain systems with interval time-varying delay by constructing a novel LKF and realizing less conservative integral inequalities. The main contributions of this paper include the following:In this paper, the robust stability criterion is proposed based on the time-delay segmentation method. Specifically, the time-delay interval is divided intoN equal parts. Then, a new LKF with quadruple integral term is constructed for different subintervals.The constructed LKF is augmented with single integral terms and multiple integrals terms, which can make more connections among different vectors and then eliminate the redundant conservatism arising from estimating the interval time-varying delay. Moreover, in addition to the single integral, the double integral, and the triple integral, the quadruple integral is used as a term to construct the integral functional, which would make full use of more information about the upper and lower bounds of the time delay existing in the systems.The Wirtinger-based integral inequality and interactive convex combination technique are used to give conclusion in the form of LMIs without any extra parameters.ℝndenotes n-dimensional Euclidean space. ℝn×m denotes the set of all n×m real matrices. ∗ denotes symmetric terms in symmetric matrices. I denotes the identity matrix with proper dimensions. M=MT>0 denotes that Mis symmetric matrix. ei denotes block input matrix with proper dimensions; for instance, e6T=0000010000.


## 2. Problem Description

The uncertain linear systems with interval time-varying delay are as follows:(1)x˙t=A+ΔAtxt+B+ΔBtxt−ht,xt=φt,t∈−hM,0,where xt∈ℝn is the state vector of the system, A and B are system matrices with appropriate dimensions, h(t) is time-varying delay satisfying 0≤hm≤ht≤hM, and ΔA(t) and ΔB(t) are unknown matrices with time-varying structure uncertainty. When ΔA(t) and ΔB(t) have norm bounded uncertainty, they can be described as follows:(2)ΔAtΔBt=DFtEaEb,where D, Ea, and Eb are known matrices with appropriate dimensions, while F(t) is an uncertain matrix with measurable elements satisfying FtTFt≤I,∀t, in which I represents the unit matrix of the appropriate dimension. When Ft=0, system (1) becomes a nominal system.In this paper, assuming thatN is a positive integer greater than zero, hii=1,2,…,N+1 are scalars, and the time-delay interval [hm, hM] can be averaged as follows:(3)hm=h1<h2<h3<⋯<hN<hN+1=hM,where hm=h1; hM=hN+1; then hΔ represents the length of subinterval [hi, hi+1]; namely, hΔ=hi+1−hi=hM−hm/N.To facilitate the proof of stability criteria, the following lemmas are summarized as follows:Lemma 1 (see [12]).
Assuming any positive definite matrixM=MT>0, scalar h>0, and continuous vector functions xt:0,h⟶ℝn, the following inequality is established:(4)−h∫t−htx˙TsMx˙sds≤−xt−xt−hTMxt−xt−h−3ΘTMΘ,where Θ=xt+xt−h−2/h∫t−htxsds.Lemma 2 (see [17]).
Assuming any positive definite matrixM=MT>0, scalar h>0, and continuous vector functions xt:0,h⟶ℝn, the following inequality is established:(5)−h∫t−htxTsMxsds≤−∫t−htxTsdsM∫t−htxsds−h2/2∫−h0∫t+βtxTsMxsdsdβ≤−∫−h0∫t+βtxTsdsdβM∫−h0∫t+βtxsdsdβ,−h3/6∫−h0∫β0∫t+λtxTsMxsdsdλdβ≤−∫−h0∫β0∫t+λtxTsdsdλdβM∫−h0∫β0∫t+λtxsdsdλdβ.Lemma 3 (see [17]).
Assuming any positive definite matrixM=MT>0, scalars 0≤α,ε≤1, α=ht−hi/hi+1−hi, ε=ht2−hi2/hi+12−hi2, hi≤ht≤hi+1, and vector functions xt:0,h⟶ℝn, the following inequality is established:(6)−hi+1−hi∫t−hi+1t−hixTsMxsds≤−ζTte7Me7T+e6Me6Tζt−αζTte7Me7Tζt−1−αζTte6Me6Tζt,−hi+12−hi2/2∫−hi+1−hi∫t+βtxTsMxsdsdβ≤−ζTte10Me10T+e9Me9Tζt−εζTte10Me10Tζt−1−εζTte9Me9Tζt,where(7)ζTt=xtxt−htxt−hixt−hi+1∫t−hitxsds∫t−htt−hixsds∫t−hi+1t−htxsds∫−hi0∫t+βtxsdsdβ∫−ht−hi∫t+βtxsdsdβ∫−hi+1−ht∫t+βtxsdsdβ.


## 3. Main Results

In this section, the stability of the system is discussed in two steps. First, the stability criterion of the nominal system is given, and then the stability of the uncertain system is analyzed. The nominal system of system (1) is as follows:(8)x˙t=Axt+Bxt−ht,xt=φt,t∈−hM,0.For nominal systems (8), a new quadruple integral term L-K functional containing more time-delay information is constructed in each subinterval. The following conclusions are obtained by combining Lemmas 1–3.Theorem 1.
For given scalarshm, hM , and λ1, λ2 (λ1>λ2), it is asymptotically stable for the nominal system (8), if there exist positive definite symmetric matrices Pii=1,2,3,4,5,Q1,Q2,U1,U2,Xj,Rjj=1,2,3,4, such that the following linear matrix inequalities (LMIs) hold:(9)Φ=Φi,j10×10<0,where(10)Φ11=P1A+ATP1+Q1+hi2X1+hi2ATX2A−X2+hΔ2X3+hΔ2ATX4A+hi44R1−hi2R2+hi44ATR2A+hi+12−hi224R3−2hΔ2R4+hi+12−hi224ATR4A−hi44U1+hi636ATU1A+hi+12−hi224U2−hi+13−hi3236ATU2A,Φ12=P1B+hi2ATX2B+hΔ2ATX4B+hi44ATR2B+hi+12−hi224ATR4B+hi636ATU1B+hi+13−hi3236ATU2B,Φ13=X,Φ14=0,Φ15=P2+hiR2,Φ16=Φ17=hΔR4,Φ18=hiP4+hi2/2U1,Φ19=Φ110=hΔP5+hi+12−hi22U2,Φ22=hi2BTX2B−2X4+hΔ2BTX4B+hi44BTR2B+hi636BTU1B+hi+12−hi224BTR4B+hi+13−hi3236BTU2B,Φ23=Φ24=X4,Φ25=Φ26=Φ27=Φ28=Φ29=Φ210=0,Φ33=−Q1+Q2−X2−X4,Φ34=0,Φ35=−P2,Φ36=Φ37=P3,Φ38=Φ39=Φ310=0,Φ44=−Q2−X4,Φ45=0,Φ46=Φ47=−P3,Φ48=Φ49=Φ410=0,Φ55=−X1−R2,Φ56=Φ57=0,Φ58=−P4,Φ59=Φ510=0,Φ66=−X3−R4,Φ67=Φ68=0,Φ69=Φ610=−P5,Φn=−X3−R4,Φ78=0,Φ79=Φ710=−P5,Φ88=−R1−U1,Φ89=Φ810=0,Φ99=−R3−U2,Φ910=U2,Φ1010=−R3−U2,hΔ=hi+1=hi=hM−hmN,hi=h1+i−1hM−hmN.Proof.
For the sake of simplicity, Theorem1 holds when ht∈h2,h3 first; and then Theorem 1 is generalized to be established when ht∈hi+hi+1i=1,3,…,N.
Whenht∈h2,h3, the L-K functional is constructed as follows:(11)V2xt=V21xt+V22xt+V23xt+V24xt+V25xt,where(12)V21xt=xTtP1xt+∫t−h2txTsdsP2∫t−h2txsds+∫t−h3t−h2xTsdsP3∫t−h3t−h2xsds+∫−h20∫t+βtxTsdsdβP4∫−h20∫t+βtxsdsdβ+∫−h3−h2∫t+βtxTsdsdβP5∫−h3−h2∫t+βtxsdsdβ,V22xt=∫t−h2txTsQ1xsds+∫t−h3t−h2xTsQ2xsds,V23xt=h2∫−h20∫t+βtxTsX1xsdsdβ+h2∫−h20∫t+βtx˙TsX2x˙sdsdβ+h3−h2∫−h3−h2∫t+βtxTsX3xsdsdβ+h3−h2∫−h3−h2∫t+βtx˙TsX4x˙sdsdβ,V24xt=h222∫−h20∫β0∫t+λtxTsR1xsdsdλdβ+h222∫−h20∫β0∫t+λtx˙TsR2x˙sdsdλdβ+h32−h222∫−h3−h2∫β0∫t+λtxTsR3xsdsdλdβ+h32−h222∫−h3−h2∫β0∫t+λtx˙TsR4x˙sdsdλdβ,V25xt=h236∫−h20∫β0∫λ0∫t+φtx˙TsU1x˙sdsdφdλdβ+h33−h236∫−h3−h2∫β0∫λ0∫t+φtx˙TsU2x˙sdsdφdλdβ.
The derivative of L-K functionalVt along the nominal system (8) is calculated as follows:(13)V˙2t=V˙21t+V˙22t+V˙23t+V˙24t+V˙25t,where(14)V˙21t=2xTtATP1xt+2xTt−htBTP1xt+2xTtP2∫t−h2txsds−2xTt−h2P2∫t−h2txsds+2xTt−h2P3∫t−h3t−h2xsds−2xTt−h3P3∫t−h3t−h2xsds−2∫t−h2txTsdsP4∫−h20∫t+βtxsdsdβ+2h2xTtP4∫−h20∫t+βtxsdsdβ+2h3−h2xTtP5∫−h3−h2∫t+βtxsdsdβ−2∫t−h3t−h2xTsdsP5∫−h3−h2∫t+βtxsdsdβ,V˙22t=xTtQ1xt−xTt−h2Q1xt−h2+xTt−h2Q2xt−h2−xTt−h3Q2xt−h3,V˙23t=h22xTtX1xt−h2∫t−h2txTsX1xsds−h2∫t−h2tx˙TsX2x˙sds+h22x˙TtX2x˙t+h3−h22xTtX3xt+h3−h22x˙TtX4x˙t−h3−h2∫t−h3t−h2xTsX3xsds−h3−h2∫t−h3t−h2x˙TsX4x˙sds,V˙24t=h244xTtR1xt−h222∫−h20∫t+βtxTsR1xsdsdβ+h244x˙TtR2x˙t−h222∫−h20∫t+βtx˙TsR2x˙sdsdβ+h32−h2224xTtR3xt−h32−h222∫−h3−h2∫t+βtxTsR3xsdsdβ+h32−h2224x˙TtR4x˙t−h32−h222∫−h3−h2∫t+βtx˙TsR4x˙sdsdβ,V˙25t=h2636x˙TtU1x˙t+h33−h23236x˙TtU2x˙t−h236∫−h20∫β0∫t+λtx˙TsU1x˙sdsdλdβ−h33−h236∫−h3−h2∫β0∫t+λtx˙TsU2x˙sdsdλdβ.
From Lemmas1 and 2, we can obtain the following:(15)−h2∫t−h2txTsX1xsds≤−ζTte5X1e5Tζt,(16)−h2∫t−h2tx˙TsX2x˙sds≤−ζTte1−e3X2e1T−e3Tζt−3ζTte1+e3−2h2e5X2e1T+e3T−2h2e5Tζt,where ζt is consistent with i=2 in Lemma 3.
From Lemma3, we can obtain the following:(17)−h3−h2∫t−h3t−h2xTsX3xsds≤−ζTte7X3e7Tζt−ζTte6X3e6Tζt−αζTte7X3e7Tζt−1−αζTte6X3e6Tζt.
Similarly, according to Lemma3, we can obtain the following:(18)−h3−h2∫t−h3t−h2x˙TsX4x˙sds≤−ζTte2−e4X4e2T−e4Tζt−ζTte3−e2X4e3T−e2Tζt−αζTte2−e4X4e2T−e4Tζt−1−αζTte3−e2X4e3T−e2Tζt,(19)−h222∫−h20∫t+βtxTsR1xsdsdβ≤−ζTte8R1e8Tζt,(20)−h222∫−h20∫t+βtx˙TsR2x˙sdsdβ≤−ζTth2e1−e5R2h2e1T−e5Tζt,(21)−h32−h222∫−h3−h2∫t+βtxTsR3xsdsdβ≤−ζTte10R3e10Tζt−ζTte9R3e9Tζt−εζTte10R3e10Tζt−1−εζTte9R3e9Tζt,(22)−h32−h222∫−h3−h2∫t+βtx˙TsR4x˙sdsdβ≤−ζTth3−h2e1−e7R4h3−h2e1T−e7Tζt−ζTth3−h2e1−e6R4h3−h2e1T−e6Tζt−εζTth3−h2e1−e7R4h3−h2e1T−e7Tζt−1−εζTth3−h2e1−e6R4h3−h2e1T−e6Tζt,(23)−h236∫−h20∫β0∫t+λtx˙TsU1x˙sdsdλdβ≤−ζTth222e1−e8U1h222e1T−e8Tζt,(24)−h33−h236∫−h3−h2∫β0∫t+λtx˙TsU2x˙sdsdλdβ≤−ζTth32−h222e1−e9−e10U2h32−h222e1T−e9T−e10Tζt.
Substituting (15)～(24) into (13), V˙2xt can be expressed as follows:(25)V˙2xt≤ζTtαΓ1+1−αΓ2+εΓ3+1−εΓ2ζt,where(26)Γi1=Φ2−e7X3e7T−e2−e4X4e2T−e4T,Γi2=Φ2−e6X3e6T−e3−e2X4e3T−e2T,Γi3=Φ2−e10R3e10T−hi+1−hie1−e7R4hi+1−hie1T−e7T,Γi4=Φ2−e9R3e9T−hi+1−hie1−e6R4hi+1−hie1T−e6T.
For0≤α,ε≤1, according to convex combination technique, the following inequality is established:(27)αΓ1+λ1I+1−αΓ2+λ1I<0,εΓ3+λ2I+1−εΓ4+λ2I<0.
Namely,(28)αΓ1+1−αΓ2<−λ1I,(29)εΓ3+1−εΓ4<λ2I.
As a result ofλ1>λ2, combining (28) and (29), the following formula is available:(30)αΓ1+1−αΓ2+εΓ3+1−εΓ4<λ2−λ1I<0.
IfαΓ1+1−αΓ2+εΓ3+1−εΓ4<0, according to L-K stability theorem, there exists a sufficient small positive number δ2 for V˙2t<−δ2xt2 to hold, and then the nominal system (8) is asymptotically stable.
Without losing generality, whenht∈hi,hi+1i=1,3,…,N, the L-K function is constructed as follows:(31)Vixt=Vi1xt+Vi2xt+Vi3xt+Vi4xt+Vi5xt,where(32)Vi1xt=xTtP1xt+∫t−hitxTsdsP2∫t−hitxsds+∫t−hi+1t−hixTsdsP3∫t−hi+1t−hixsds+∫−hi0∫t+βtxTsdsdβP4∫−hi0∫t+βtxsdsdβ+∫−hi+1−hi∫t+βtxTsdsdβP5∫−hi+1−hi∫t+βtxsdsdβ,Vi2xt=∫t+βtxTsQ1xsds+∫t−hi+1hixTsQ2xsds,Vi3xt=hi∫−hi0∫t+βtxTsX1xsdsdβ+hi∫−hi0∫t+βtx˙TsX2x˙sdsdβ+hi+1−hi∫−hi+1−hi∫t+βtxTsX3xsdsdβ+hi+1−hi∫−hi+1−hi∫t+βtx˙TsX4x˙sdsdβ,Vi4xt=hi22∫−hi0∫β0∫t+λtxTsR1xsdsdλdβ+hi22∫−hi0∫β0∫t+λtx˙TsR2x˙sdsdλdβ+hi+12−hi22∫−hi+1−hi∫β0∫t+λtxTsR3xsdsdλdβ+hi+12−hi22∫−hi+1−hi∫β0∫t+λtx˙TsR4x˙sdsdλdβ,Vi5xt=hi36∫−hi0∫β0∫λ0∫t+φtx˙TsU1x˙sdsdφdλdβ+hi+13−hi36∫−hi+1−hi∫β0∫λ0∫t+φtx˙TsU2x˙sdsdφdλdβ,where the definition of ζt is the same as that in Lemma 3. Pii=1,2,3,4,5, Q1, Q2, Q3, U1, U2, Xj, and Rjj=1,2,3,4 are the matrices defined in the same formula (9). The same method is available. The following conclusions can be reached by the same method:(33)V˙ixt≤ζTtαΓi1+1−αΓi2+εΓi3+1−εΓi4ζt,where(34)Γi1=Φ2−e7X3e7T−e2−e4X4e2T−e4T,Γi2=Φ2−e6X3e6T−e3−e2X4e3T−e2T,Γi3=Φ2−e10R3e10T−hi+1−hie1−e7R4hi+1−hie1T−e7T,Γi4=Φ2−e9R3e9T−hi+1−hie1−e6R4hi+1−hie1T−e6T.
In the same way, it is known that there exists a sufficient small positive numberδi to make V˙tt<−δxt2 hold, and then the nominal system (8) is asymptotically stable.
The combination of (25) and (33) is equivalent to (9). This fulfills the proof.Remark 1.
Firstly, different from [15], in which the delay range was divided into two equidistant subintervals, new LKF comprising quadruple-integral term and quadratic forms of double-integral term was constructed. In this paper, for each subinterval, the time-delay interval is divided into N equal parts by using the method of time-delay partitioning. A new LKF with four integral terms is designed for each partitioned interval, and the quadratic form of double integral is introduced, such as ∬xTsdsdβM∬xsdsdβ. Although the double integral functional term ∫−h0∫t+βtxsdsdβ is also used in [1, 10], it is not introduced into the definition of augmented vector. Secondly, the triple integral functional term integrand used in the new LKF contains the state vector x, and the lower bound information of the delay interval is introduced. Thanks to the coexistence of the quadratic integral functional term and the quadratic term ∫xTsdsdβ, the conservativeness of the stability conclusion is significantly reduced.Remark 2.
In formula (9), the new stability criterion does not involve redundant free-weight matrices but skillfully uses Wirtinger-based integral inequality to define the cross terms generated by LKF derivatives and uses a few free matrices to represent the relationship between the relevant terms. Therefore, the complexity of theoretical derivation and computation is reduced, and the conservatism of conclusions is reduced.Remark 3.
For a given scalarμ and time-delay rate h˙t satisfying 0<h˙t≤μ, substituting the functional term ∫t−httxTsQ3xsds into the LKF constructed, the stability criterion containing the time-delay rate μ can be obtained according to the proof process of Theorem 1. The form is shown in Theorem 2.Theorem 2.
For the scalarshm,hM, and μ, λ1,λ2λ1>λ2, it is asymptotically stable for the nominal system (8), if there exist positive definite symmetric matrices Pii=1,2,3,4,5, Q1, Q2, Q3, U1, U2, Xj, and Rjj=1,2,3,4, such that the following LMIs hold:(35)Φ˜=Φ˜i,j10×10<0,where Φ˜11=Φ11+Q3 and Φ˜22=Φ22−μQ3; other items in Φ˜ are defined the same as in Φ Theorem 1.
Next, the robust stability of uncertain systems with interval time-varying delays (1) is considered.Theorem 3.
For the scalars0<hm<hM and μ, λ1,λ2λ1>λ2, it is asymptotically stable for the uncertain system (1), if there exist positive definite symmetric matrices Pii=1,2,3,4,5, Q1, Q2, Q3, U1, U2, Xj, and Rjj=1,2,3,4, the scalar δ>0, and the free matrices with suitable dimension T1,T2, such that the following LMIs hold:(36)Φ˜Γ1DδΓ2T∗−δI0∗∗−δI<0,where(37)Γ1=T1T00000T2T,Γ2=Ea0Eb0000.Proof.
For the uncertain system (1), A and B in equation (9) are replaced by A + ΔA and B + ΔB, respectively. According to the proof of Theorem 1, the asymptotic stability of system (1) is obtained. This fulfills the proof.


## 4. Numerical Examples

The following three numerical examples are used to compare the results of the existing literature with the method proposed in this paper. MADB (Maximum Allowable Delay Bound) is defined as the upper bound of the maximum allowable delay to ensure the stability of the system, and it is the most common criterion to compare the conservativeness of the stability conclusions of time-delay systems.Example 1.
First consider the following closed-loop control systems with interval time-varying delays:(38)x˙t=01−1−2xt+00−11xt−ht.
For givenhm, according to (35) in Theorem 2 and (9) in Theorem 1, Tables 1 and 2 give corresponding MADB from two aspects, μ = 0.3 and μ = any, respectively. It can be clearly seen from Tables 1 and 2 that the method proposed in this paper is obviously better than the conclusion in the existing literature.
To verify the validity of the results, givenμ=0.3, hm=1, and hM=3.0796 and given initial condition xt=2−2T, the state response curve of xt is shown in Figure 1. It can be seen that the state trajectory of the above-mentioned system can quickly reach a stable state under the action of the obtained MADB, which further verifies the correctness of the proposed stability criterion.Table 1
In Example1, MADB is simulated to be obtained for different hm and different methods.
μMethodhm=1hm=2hm=30.3Literature [24]2.40422.58703.4766Literature [18]2.43282.6322——Literature [11] (N = 2)2.52783.07443.9136Literature [11] (N = 3)2.73683.48364.2857Literature [25]3.163.504.32Theorem23.07963.90644.4152Table 2
In Example1, when μ = any, MADB is simulated to be obtained for different hm and different methods.
Methodhm=0.3hm=0.5hm=0.8Literature [13] (N = 2)1.16771.30781.5333Literature [13] (N = 4)1.20431.34291.5633Literature [18]1.35311.46631.6592Literature [26]1.43471.53361.7140Literature [14]1.68371.81202.0209Literature [27]1.781.811.90Theorem11.92362.13842.2473Figure 1
State response curve ofxt when hM=3.0796.Example 2.
Uncertain systems with interval time-varying delays are considered:(39)x˙t=−2+λ100−1+λ1xt+−1+λ30−1−1+λ4xt−ht,where λ1, λ2, λ3, and λ4 are unknown parameters satisfying λ1≤1.6,λ2≤0.05,λ3≤0.1,λ4≤0.3.
For givenhm, according to (36) in Theorem 3, Table 3 gives corresponding MADB in the simulation. From the comparison results, it can be seen that, for this example, this method improves the conclusions of the existing literature.
Given the initial conditionxt=0.10.2T, the state response curve of xt is shown in Figure 2 when the constant of time-varying delay ht is 1.4723. When ht takes variable 1.51+0.51sint, the state response curve of xt is shown in Figure 3. It can be seen that xt can quickly reach a stable state under the action of the nonlinear disturbance and the obtained MADB, thus verifying the correctness of the proposed stability criterion.Table 3
In Example2, MADB is simulated to be obtained for different hm and different methods.
Methodhm=0.2hm=0.4hm=0.6Literature [10] (N = 2)1.13371.17031.2123Literature [28] (N = 2)1.17831.21231.2527Literature [28] (N = 4)1.18711.22461.2686Literature [11] (N = 2)1.33691.35711.3817Literature [11] (N = 3)1.38091.40031.4216Theorem31.42411.44131.4723Figure 2
State response curve ofxt when ht=1.4723.Figure 3
State response curve ofxt when ht=1.51+0.51sint.Example 3.
Consider another uncertain system with interval time-varying delays. The system parameters are as follows:(40)A=−0.400−1,B=−0.90−1−0.7,D=1001,Ea==Eb=0.2000.2.
Similarly, according to (36) in Theorem 3, for given hm and μ=any, Table 4 gives corresponding MADB in the simulation. As can be seen from Table 4, the robust stability theorem proposed in this paper enlarges the upper bound of the maximum allowable delay to guarantee the stability of the system. It has lower conservatism.Table 4
In Example3, when μ = any, MADB is simulated to be obtained for different hm and different methods.
Methodhm=0hm=0.4hm=0.8Literature [9]1.05711.13851.2392Literature [10] (N = 2)1.10301.17031.2594Literature [11] (N = 2)1.32131.35711.4102Literature [11] (N = 3)1.36341.40031.4445Theorem31.41271.45941.4987


## 5. Conclusion

In this paper, we study the robust stability of a class of uncertain systems with interval time-varying delays. A new stability criterion based on LMI is proposed by constructing a new LKF containing a generalized term of quadruple integral. In order to improve the computational efficiency and simplify the conclusion, the criterion avoids the use of model transformation and free weight matrix definition techniques. Instead, Wirtinger-based integral inequalities and interactive convex combination techniques with tighter definition techniques are adopted, which make full use of the lower bound information of the delay and obtain a lower conservative conclusion. Finally, numerical simulations show that the proposed criterion enlarges the upper bound of the maximum delay allowed to guarantee the stability of the system and is more competitive than the existing methods.However, the new stability criterion proposed in this paper mainly focuses on a class of linear systems with norm-bounded uncertainty and interval time-varying delay. How to get the similar conclusion for nonlinear system is another interesting topic and the next work for us; and some related researches are hopeful to supply reference to us [29, 30].



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