# Stability Analysis of Systems with Interval Time-Varying Delays via a New Integral Inequality

Authors: Zerong Ren; Jun-kang Tian
Journal: Complexity
 (2020)

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## Abstract

This paper focuses on delay-dependent stability analysis for systems with interval time-varying delays. Based on a new integral inequality and a generalized reciprocally convex combination matrix inequality, a new delay-dependent stability criterion is obtained in terms of a linear matrix inequality (LMI). Finally, the merits of the proposed criterion are shown by two numerical examples.


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## Body

## 1. Introduction

Consider the following systems with interval time-varying delays:(1)x˙t=Axt+Bxt−ht,(2)xt=ϕt,t∈−h2,0,where xt∈ℛn is the state vector and A,B,∈ℛn×n are constant matrices. The time-varying delay ht is continuous and satisfies(3)0≤h1≤ht≤h2,h12≜h2−h1.Over the past decades, providing less conservative stability conditions for linear systems with time-varying delays has attracted considerable attention. The difficulty relies on the handling of the integral terms arising in the derivative of the LKF. The free-weighting-matrix approach [1, 2] was applied to handle the integral terms in the early literature. In recent years, estimating integral terms directly via integral inequalities gradually becomes more popular. Various integral inequalities have been proposed, such as Jensen inequality [3–5], Wirtinger-based inequality [6–8], auxiliary function based inequalities [9], free-matrix-based inequalities [10], and relaxed integral inequalities [11]. Very recently, Bessel–Legendre inequality [12] is proposed to handle the stability of system (1), and some less conservative criteria are obtained. However, the relationship between ∫abs−a+b/2kxsds and ∫abxsds, ∫ab∫u1bxsdsdu1, …, ∫ab∫u1b…∫ukbxsdsduk…du1 was not considered in [12], which may yield conservative results. Then, a new integral inequality was proposed in [13] to consider the relationship fully. But the integral inequality was only used to handle the constant time delay. A new integral inequality for dealing with delays is introduced in [14]. A less conservative stability criterion for linear systems with a time-varying delay is proposed by using the new integral inequality. However, there are two aspects which need to be improved. (1) When estimating the derivative of Vxt, the term ∫abx˙TsRx˙sds is only estimated as ∫abx˙TsRx˙sds≥1/b−aΩ1TRΩ1+3/b−aΩ2TRΩ2+5/b−aΩ3TRΩ3, which may yield conservative results. (2) The assumption on the derivative of time-varying delay ht is included in the stability criterion [14]. Thus, there is still some room for further investigation.In this paper, a new delay-dependent stability criterion for linear systems with interval time-varying delays is developed by using a new integral inequality and a generalized reciprocally convex combination matrix inequality. The highlights of our paper are as follows. (1) The features of the new integral inequality in [13] are fully integrated into the construction of the LKF. (2) A less conservative stability criterion is proposed in terms of an LMI without the assumption on the derivative of time-varying delay ht. (3) Upper bound of h2 in our paper is quite close to the analytical bound. The advantage of the proposed criterion has been illustrated via two numerical examples.Throughout the paper, the setS+n denotes the set of n×n symmetric positive definite matrices and the set Sn denotes the set of n×n symmetric matrices. For any square matrix A, define HeA=A+AT.


## 2. Main Result

Based on the following lemmas, a less conservative stability criterion for systems with interval time-varying delays is established.Lemma 1.
(see [8]). For any matrices Q∈ S+n, N1,N2∈ℛm×n, Γ∈ℛ2n×m, ∀α∈0,1, the following inequality holds:(4)−ΓT1αQ0011−αQΓ≤−ΓTℛαΓ−HeΓT1−αN1TαN2T+αN1Q−1N1T+1−αN2Q−1N2T,where(5)ℛα=2−αQ001+αQ.Lemma 2.
(see [8]). Consider a parameter dependent matrix Φα∈Sm, such that the convex inequality(6)Φα≤1−αΦ0+αΦ1,holds for all α∈0,1. If there exist a matrix R∈S+n and two matrices N1,N2∈ℛm×n, such that the inequality(7)Ψα=Φα−ΓTℛαΓ−HeΓT1−αN1TαN2T∗αN1T+1−αN2T−R<0,holds for α=0,1, then the following inequality holds:(8)Φα−ΓT1αR0011−αRΓ<0,∀α∈0,1.Lemma 3.
(see [13]). For a matrix R∈S+n and any continuously differentiable function x:a,b⟶ℛn, the following inequality holds:(9)∫abx˙TsRx˙sds≥1b−aΩ1TRΩ1+3b−aΩ2TRΩ2+5b−aΩ3TRΩ3+7b−aΩ4TRΩ4,where(10)Ω1=xb−xa,Ω2=xb+xa−2b−a∫abxsds,Ω3=xb−xa+6b−a∫abxsds−12b−a2∫ab∫ubxsdsdu,Ω4=xb+xa−12b−a∫abxsds+60b−a2∫ab∫ubxsdsdu−120b−a3∫ab∫ub∫vbxsdsdvdu.Theorem 1.
For given scalarsh1andh2, system (1) with time-varying delays satisfying (2) is asymptotically stable if there exist matrices P∈S+5n, Q1,Q2,Q3,Q4∈S+n, N1,N2∈ℛ13n×4n, such that the LMI (11)Ψα=Φα−ΓTℛαΓ−HeΓT1−αN1TαN2T∗αN1T+1−αN2T−Q<0,holds for α=0,1, where(12)Φα=HeΣ1TPΣ2+ε1TQ1ε1−ε2TQ1ε2+ε2TQ2ε2−ε4TQ2ε4+h12ε0TQ3ε0+h122ε0TQ4ε0−Σ3TQ3Σ3−3Σ4TQ3Σ4−5Σ5TQ3Σ5−7Σ6TQ3Σ6,Σ1=ε1Th1ε5Tαh12ε6T+1−αh12ε7Th12ε8Th13ε11TT,Σ2=ε0Tε1T−ε2Tε2T−ε4Th1ε1T−h1ε5Th122ε1T−h12ε8TT,Σ3=ε1−ε2,Σ4=ε1+ε2−2ε5,Σ5=ε1−ε2+6ε5−12ε8,Σ6=ε1+ε2−12ε5+60ε8−120ε11,Σ7=ε2−ε3,Σ8=ε2+ε3−2ε6,Σ9=ε2−ε3+6ε6−12ε9,Σ10=ε2+ε3−12ε6+60ε9−120ε12,Σ11=ε3−ε4,Σ12=ε3+ε4−2ε7,Σ13=ε3−ε4+6ε7−12ε10,Σ14=ε3+ε4−12ε7+60ε10−120ε13,ε0=Aε1+Bε3,Γ=Σ7TΣ8TΣ9TΣ10TΣ11TΣ12TΣ13TΣ14TT,Q=diagQ4,3Q4,5Q4,7Q4,and εi∈ℛn×13n is defined as εi=0n×i−1nIn0n×13−in for i=1,2,…,13.Proof.
Consider a LKF candidate given by(13)Vxt=ηTtPηt+∫t−h1txTsQ1xsds+∫t−h2t−h1xTsQ2xsds+h1∫t−h1t∫utx˙TsQ3x˙sdsdu+h12∫t−h2t−h1∫utx˙TsQ4x˙sdsdu,where(14)ηt=xTt∫t−h2txTsds∫t−h2t−h1xTsds∫t−h1t∫utxTsdsdu∫t−h1t∫ut∫stxTrdrdsduT.
Calculate the derivative ofVxt along the solution of system (1) as follows:(15)V˙xt=2ηTtPη˙t+xTtQ1xt−xTt−h1Q1xt−h1+xTt−h1Q2xt−h1−xTt−h2Q2xt−h2+h12x˙TtQ3x˙t+h122x˙TtQ4x˙t−h1∫t−h1tx˙TsQ3x˙sds−h12∫t−h2t−h1x˙TsQ4x˙sds.
Then, it can be rewritten as(16)V˙xt=ξTtHeΣ1TPΣ2+ε1TQ1ε1−ε2TQ1ε2+ε2TQ2ε2−ε4TQ2ε4+h12ε0TQ3ε0+h122ε0TQ4ε0ξt−h1∫t−h1tx˙TsQ3x˙sds−h12∫t−h2t−h1x˙TsQ4x˙sds,where(17)ξt=xTtxTt−h1xTt−htxTt−h2φ1Ttφ2Ttφ3TtT,φ1t=1h1∫t−h1txTsds1ht−h1∫t−htt−h1xTsds1h2−ht∫t−h2t−htxTsdsT,φ2t=1h12∫t−h1t∫utxTsdsdu1ht−h12∫t−htt−h1∫ut−h1xTsdsdu1h2−ht2∫t−h2t−ht∫ut−htxTsdsT,φ3t=1h13∫t−h1t∫ut∫vtxTsdsdvdu1ht−h13∫t−htt−h1∫ut−h1∫vt−h1xTsdsdvdu1h2−ht3∫t−h2t−ht∫ut−ht∫vt−htxTsdsdvduT.
Letα=ht−h1/h12, and applying Lemma 3 yields (18)−h1∫t−h1tx˙TsQ3x˙sds≤ξTt−Σ3TQ3Σ3−3Σ4TQ3Σ4−5Σ5TQ3Σ5−7Σ6TQ3Σ6ξt,(19)−h12∫t−h2t−h1x˙TsQ4x˙sds=−h12∫t−htt−h1x˙TsQ4x˙sds−h12∫t−h2t−htx˙TsQ4x˙sds≤−h12ht−h1ξTtΣ7TQ4Σ7+3Σ8TQ4Σ8+5Σ9TQ4Σ9+7Σ10TQ4Σ10ξt−h12h2−htξTtΣ11TQ4Σ11+3Σ12TQ4Σ12+5Σ13TQ4Σ13+7Σ14TQ4Σ14ξt=−1αξTtΣ7TQ4Σ7+3Σ8TQ4Σ8+5Σ9TQ4Σ9+7Σ10TQ4Σ10ξt−11−αξTtΣ11TQ4Σ11+3Σ12TQ4Σ12+5Σ13TQ4Σ13+7Σ14TQ4Σ14ξt=−ξTtΓT1αQ0011−αQΓξt.
Letα=ht−h1/h12, then 1−α=h2−ht/h12, for any matrices N1,N2∈ℛ13n×4n, and applying Lemma 1 yields(20)−ΓT1αQ0011−αQΓ≤−ΓTℛαΓ−HeΓT1−αN1TαN2T+αN1Q−1N1T+1−αN2Q−1N2T=Ξα,where(21)ℛα=2−αQ001+αQ.
From (16)–(20), the V˙xt can be estimated as(22)V˙xt≤ξTtΦα+Ξαξt.
According to Lemma2, if LMI (11) is verified for α=0,1, then the inequality Φα+Ξα<0 holds for all α∈0,1. This completes the proof.Remark 1.
The integral inequality in Lemma3 is proposed to derive less conservative results because of the term of a triple integral being included. In order to fully consider the features of the integral inequality in Lemma 3, the LKF of our paper includes the terms of a triple integral ∫t−h1t∫ut∫stxTrdrdsdu, which may yield less conservative results.


## 3. Numerical Examples

In this section, two numerical examples are introduced to illustrate the advantage of the proposed criterion.Example 1.
Consider system (1) with(23)A=−200−0.9,B=−10−1−1.
Table1 presents the maximum upper bounds of h2 with respect to different h1 calculated by Theorem 1 and several criteria from [4, 6, 9, 10, 12]. Table 1 shows that our method is more effective than those in [4, 6, 9, 10, 12]. It should be pointed out that our result is quite close to the analytical bound. For h2=3.45, initial state −0.02,0.02T, the simulation of the state trajectories of system (1) is given in Figure 1.Table 1
Upper bound ofh2 for different h1.
h
1
0.0
0.4
0.7
1.0
h
1
=
h
2
Theorem 2 [4]
1.86
1.88
1.95
2.06
4.47
Theorem 7 [6]
2.11
2.17
2.23
2.31
6.05
Theorem 1 [9]
2.14
2.19
2.24
2.31
6.1641
Theorem 1 [10]
2.18
2.21
2.25
2.32
6.05
Theorem 2 [12]
2.26
2.29
2.34
2.40
6.1664
Theorem 1
2.92
3.27
3.38
3.45
6.1686
Analytical bound
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6.1725Figure 1
The state trajectories of system (1) for Example 1.Example 2.
Consider system (1) with(24)A=0.01.0−10.0−1.0,B=0.00.10.10.2.
Table2 presents the maximum upper bounds of h2 with respect to different h1 calculated by Theorem 1 and several criteria from [6, 9, 10, 12]. Table 2 shows that our method is more effective than those in [6, 9, 10, 12]. For h2=4.16, initial state −0.01,0.01T, the simulation of the state trajectories of system (1) is given in Figure 2.Table 2
Upper bound ofh2 for different h1.
h
1
0.0
0.3
0.7
1.0
2.0
Theorem 7 [6]
1.59
2.01
2.41
2.62
3.59
Theorem 1 [9]
1.64
2.13
2.70
2.96
3.63
Theorem 1 [10]
1.80
2.19
2.58
2.79
3.68
Theorem 2 [12]
2.39
2.76
3.15
3.41
4.04
Theorem 1
2.76
3.13
3.48
3.53
4.16Figure 2
The state trajectories of system (1) for Example 2.


## 4. Conclusions

An improved stability criterion of systems with interval time-varying delays has been proposed in our paper. The features of the new integral inequality are fully integrated into the construction of the LKF. Finally, the merits of the proposed criterion are shown by two numerical examples.



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