# A New Stability Criterion for Systems with Distributed Time-Varying Delays via Mixed Inequalities Method

Authors: Jun-kang Tian; Yan-min Liu
Journal: Complexity
 (2020)

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## Abstract

This paper is concerned with the delay-dependent stability of systems with distributed time-varying delays. The novelty relies on the use of some new inequalities which are less conservative than some existing inequalities. A less conservative stability criterion is obtained by constructing some new augmented Lyapunov–Krasovskii functionals, which are given in terms of linear matrix inequalities. The effectiveness of the presented criterion is demonstrated by two numerical examples.


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## Body

## 1. Introduction

Consider the systems with distributed time-varying delays:(1)x˙t=Axt+Bxt−ht+C∫t−httxsds,(2)xt=ϕt,t∈−h,0,where xt∈Rn is the system state, A,B,C∈Rn×n are constant matrices, and ht is the time-varying delay satisfying(3)0≤ht≤h,−u≤h˙t≤u<1.Since time delays occur in many dynamic systems, stability analysis of the time delay system [1–5] has become a hot topic in the past few decades. Due to the representation of linear systems with time-varying delays, the delay-dependent stability analysis via the LKF method has attracted much attention. The conservatism of the LKF method comes from two aspects: the construction of the LKF and the bound on its derivative. Selecting the LKF is crucial to derive less conservative criteria. An augmented LKF [6] is proposed to reduce the conservatism in the early literature. Recently, a new augmented LKF [7] is introduced by employing the information of a second-order Bessel–Legendre inequality. It is necessary to take the derivative of the LKF to derive a stability criterion. The difficulty lies in the bounds of the integrals that arise in the derivative of the LKF. There are two main methods for dealing with such integrals: the free-weighting matrix method [8] and the integral inequality method. The integral inequality method includes various integral inequalities, such as Jensen inequality [9–11], Wirtinger-based inequality [12–15], free matrix-based inequality [16, 17], auxiliary function-based inequality [18], relaxed integral inequality [19], and Bessel–Legendre inequality [20]. Very recently, the improved inequality-based functions approach [21] is proposed to derive less conservative results for systems with time-varying delays. However, when estimating V˙xt, ∫abx˙TsRx˙sds is only estimated as ∫abx˙TsRx˙sds≥1/b−aΩ1TRΩ1+3/b−aΩ2TRΩ2+5/b−aΩ3TRΩ3. Ω1,Ω2,andΩ3 are the same as in Lemma 2. Then, a new integral inequality was proposed in [22] to further reduce the conservatism. But the integral inequality can only deal with the constant time delay. On the other hand, stability analysis for systems with distributed delays is of both practical and theoretical importance. Then, it is desirable to extend the system model to include distributed delays. In recent years, the stability analysis of systems with distributed delays has been received considerable attention [23–27]. But only the authors in [25, 26] consider the systems with distributed time-varying delays.This paper is concerned with the delay-dependent stability of systems with distributed time-varying delays. Based on some new inequalities and some new augmented LKFs, a less conservative stability criterion is obtained in terms of LMIs. Our paper has two characteristics: (1) the integral∫abx˙TsRx˙sds is estimated as ∫abx˙TsRx˙sds≥1/b−aΩ1TRΩ1+3/b−aΩ2TRΩ2+5/b−aΩ3TRΩ3+7/b−aΩ4TRΩ4, which includes those in [9, 13, 20] as special cases. Ωi,i=1,2,3,4, is the same as in Lemma 2. (2) An augmented LKF which contains more information about ht is proposed to reduce the conservatism. The effectiveness of the presented criterion is demonstrated by two numerical examples.Throughout this paper, the setSn denotes the set of symmetric matrices and the set S+n denotes the set of symmetric positive definite matrices. For any square matrix P, we define SymP=P+PT.


## 2. Main Results

In this section, the following lemmas are introduced to derive the main results.Lemma 1 (see [20]).
For any matricesΘ∈S+n, M1,M2∈Rm×n, Υ∈R2n×m, and ∀α∈0,1, the inequality(4)−ΥT1αΘ0011−αΘΥ≤−ΥTΣαΥ−SymΥT1−αM1TαM2T+αM1Θ−1M1T+1−αM2Θ−1M2T,holds, where(5)Σα=2−αΘ001+αΘ.Lemma 2 (see [22]).
For a matrixR∈S+n and any continuously differentiable function x:a,b⟶Rn, the inequality(6)∫abx˙TsRx˙sds≥1b−aΩ1TRΩ1+3b−aΩ2TRΩ2+5b−aΩ3TRΩ3+7b−aΩ4TRΩ4,holds, where(7)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.Lemma 3 (see [28]).
Suppose thatΩ,Ωiji,j=1,2 are the constant matrices of appropriate dimensions, α∈0,1,β∈−u,u, 0≤u<1, then(8)Ω+αΩ11+1−αΩ12+βΩ21+1−βΩ22<0,holds if and only if the following inequalities hold:(9)Ω+Ω11−uΩ21+1+uΩ22<0,Ω+Ω12−uΩ21+1+uΩ22<0,Ω+Ω11+uΩ21+1−uΩ22<0,Ω+Ω12+uΩ21+1−uΩ22<0.Based on Lemmas1–3, a novel stability criterion is derived for system (1) with distributed time-varying delays.Theorem 1.
For given scalarsh>0,u>0, if there exist matrices P,Q1,Q2,∈S+2n, Q3∈S+n, M1,M2∈R10n×4n, such that the LMI(10)Φα,β=ϕα,β−ΥTΣαΥ−SymΥT1−αM1TαM2T∗αM1T+1−αM2T−Θ<0,holds for α=0,1,h˙t=β=−u,u, i.e.,(11)Φ0,−u<0,(12)Φ0,u<0,(13)Φ1,−u<0,(14)Φ1,u<0,then, system (1) is asymptotically stable, where(15)ϕα,β=SymΠ1TPΠ2+Π3TQ1Π3−1−βΠ4TQ1Π4+SymΠ5TQ2Π6+h2ε0TQ3ε0,Π1=ε1Tαhε5T+1−αhε6TT,Π2=Aε1T+Bε2T+Cαhε5Tε1T−ε3TT,Π3=ε1TAε1T+Bε2T+Cαhε5TT,Π4=ε2Tε4TT,Π5=ε1Tε2TT,Π6=Aε1T+Bε2T+Cαhε5T1−βε4TT,Π7=ε1−ε2,Π8=ε1+ε2−2ε5,Π9=ε1−ε2+6ε5−12ε7,Π10=ε1+ε2−12ε5+60ε7−120ε9,Π11=ε2−ε3,Π12=ε2+ε3−2ε6,Π13=ε2−ε3+6ε6−12ε8,Π14=ε2+ε3−12ε6+60ε8−120ε10,ε0=Aε1+Bε2+Cαhε5,Υ=Π7TΠ8TΠ9TΠ10TΠ11TΠ12TΠ13TΠ14TT,Θ=diagQ3,3Q3,5Q3,7Q3,and εi∈ℛn×10n is defined as εi=0n×i−1nIn0n×10−in for i=1,2,…,10.Proof.
Introduce an LKF candidate as(16)Vxt=V1xt+V2xt+V3xt+V4xt,where(17)V1xt=xt∫t−htxsdsTPxt∫t−htxsds,V2xt=∫t−httxsx˙sTQ1xsx˙sds,V3xt=xtxt−htTQ2xtxt−ht,V4xt=h∫t−ht∫utx˙TsQ3x˙sdsdu.
Calculate the derivative ofVxt along the solution of system (1) as follows:(18)V˙1xt=2xt∫t−htxsdsTPx˙txt−xt−h=2xt∫t−htxsdsTPAxt+Bxt−ht+C∫t−httxsdsxt−xt−h,(19)V˙2xt=xtx˙tTQ1xtx˙t−1−h˙txt−htx˙t−htTQ1xt−htx˙t−ht=xtAxt+Bxt−ht+C∫t−httxsdsTQ1xtAxt+Bxt−ht+C∫t−httxsds−1−h˙txt−htx˙t−htTQ1xt−htx˙t−ht,(20)V˙3xt=2xtxt−htTQ2x˙t1−h˙tx˙t−ht,(21)=2xtxt−htTQ2Axt+Bxt−ht+C∫t−httxsds1−βx˙t−ht,(22)V˙4xt=h2x˙TtQ3x˙t−h∫t−htx˙TsQ3x˙sds.
According to (18)–(22), we can obtain(23)V˙xt=ζTtsymΠ1TPΠ2+Π3TQ1Π3−1−βΠ4TQ1Π4+SymΠ5TQ2Π6+h2ε0TQ3ε0ζt−h∫t−htx˙TsQ3x˙sds,where(24)ζt=txTtxTt−htxTt−hx˙Tt−htπ1Ttπ2Ttπ3TtT,π1t=1ht∫t−httxTsds1h−ht∫t−ht−htxTsdsT,π2t=1ht2∫t−htt∫utxTsdsdu1h−ht2∫t−ht−ht∫ut−htxTsdsT,π3t=1ht3∫t−htt∫ut∫vtxTsdsdvdu1h−ht3∫t−ht−ht∫ut−ht∫vt−htxTsdsdvduT.
Letα=ht/h, then 1−α=h−ht/h, applying Lemma 2, and we have(25)−h∫t−htx˙TsQ3x˙s=−h∫t−httx˙TsQ3x˙sds−h∫t−ht−htx˙TsQ3x˙sds≤−hhtζTtΠ7TQ3Π7+3Π8TQ3Π8+5Π9TQ3Π9+7Π10TQ3Π10ζt−hh−htζTtΠ11TQ3Π11+3Π12TQ3Π12+5Π13TQ3Π13+7Π14TQ3Π14ζt=−1αζTtΠ7TQ3Π7+3Π8TQ3Π8+5Π9TQ3Π9+7Π10TQ3Π10ζt−11−αζTtΠ11TQ3Π11+3Π12TQ3Π12+5Π13TQ3Π13+7Π14TQ3Π14ζt=−ζTtΥT1αΘ0011−αΘΥζt.
For any matricesM1,M2∈R10n×4n and applying Lemma 1, we can obtain(26)−ΥT1αΘ0011−αΘΥ≤−ΥTΣαΥ−SymΥT1−αM1TαM2T+αM1Θ−1M1T+1−αM2Θ−1M2T=ηα.
From (23)–(26), we get(27)V˙xt≤ζtϕα,β+ηαζt.
By Lemma2 [20], if the LMI (10) is true for α=0,1, β=−u,u, then ϕα,β+ηα<0 holds for all α∈0,1 and β∈−u,u. By Lemma 3, LMI (10) holds if and only if LMIs (11)–(14) hold. This completes the proof.Remark 1.
The integral∫abx˙TsRx˙sds in [9, 13, 20] is estimated as ∫abx˙TsRx˙sds≥1/b−aΩ1TRΩ1, ∫abx˙TsRx˙sds≥1/b−aΩ1TRΩ1+3/b−aΩ2TRΩ2, ∫abx˙TsRx˙sds≥1/b−aΩ1TRΩ1+3/b−aΩ2TRΩ2+5/b−aΩ3TRΩ3, respectively. In this paper, the integral ∫abx˙TsRx˙sds is estimated as ∫abx˙TsRx˙sds≥1/b−aΩ1TRΩ1+3/b−aΩ2TRΩ2+5/b−aΩ3TRΩ3+7/b−aΩ4TRΩ4, which includes those in [9, 13, 20] as special cases. So our method can yield less conservative results.Remark 2.
An augmented LKF which contains more information about time-varying delayht which was proposed to reduce the conservatism. x˙t−ht is added as a state vector, which may yield less conservative criteria.


## 3. Numerical Examples

Two numerical examples are given to demonstrate advantages of the proposed criterion.Example 1.
Consider system (1) with(28)A=0.01.0−1.0−2.0,B=0.00.0−1.01.0,C=0000.
For differentu, Table 1 presents the allowable upper bound of ht, which guarantees the stability of system (1). Table 1 shows that our method produces the larger upper bound h than those in [7, 12, 13, 16, 17, 21]. In this sense, our stability criterion is less conservative than those in [7, 12, 13, 16, 17, 21].Table 1
Upper bound ofh for Example 1 with different u.
u
0.1
0.2
0.5
0.8
[13]
6.590
3.672
1.411
1.275
[12]
7.125
4.413
2.243
1.662
[16]
7.148
4.466
2.352
1.768
[17]
7.167
4.517
2.415
1.838
[7]
7.230
4.556
2.509
1.940
[21]
7.297
4.625
2.264
2.038
Theorem1
10.095
6.808
3.676
2.615Example 2.
Consider system (1) with(29)A=−200−0.9,B=−1.00.0−1.0−1.0,C=0000.
For differentu, Table 2 presents the allowable upper bound of ht, which guarantees the stability of system (1). Table 2 shows that our method produces the larger upper bound h than those in [7, 11, 13, 15, 16, 21]. In this sense, our stability criterion is less conservative than those in [7, 11, 13, 15, 16, 21].Table 2
Upper bound ofh for Example 2 with different u.
u
0.1
0.2
0.5
0.8
[13]
4.703
3.834
2.420
2.137
[11]
4.753
—
2.429
2.183
[16]
4.788
4.060
3.055
2.615
[15]
4.93
4.22
3.09
2.66
[7]
4.910
—
3.233
2.789
[21]
4.996
4.308
3.251
2.867
Theorem1
5.650
4.913
3.793
3.251


## 4. Conclusions

This paper focus on delay-dependent stability analysis for systems with distributed time-varying delays. The novelty relies on the use of some new inequalities which are less conservative than some existing inequalities. A less conservative stability criterion is obtained by constructing some new augmented LKFs. The effectiveness of the presented criterion is demonstrated by two numerical examples. In addition, the proposed method can be applied to stability analysis of other dynamic systems such as fuzzy systems with time-varying delay and neutral systems with time-varying delay.



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