# Improving Delay-Range-Dependent Stability Condition for Systems with Interval Time-Varying Delay

Authors: Wei Qian; Tao Li; Juan Liu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper discusses the delay-range-dependent stability for systems with interval time-varying delay. Through defining the new Lyapunov-Krasovskii functional and estimating the derivative of the LKF by introducing new vectors, using free matrices and reciprocally convex approach, the new delay-range-dependent stability conditions are obtained. Two well-known examples are given to illustrate the less conservatism of the proposed theoretical results.


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## Body

## 1. Introduction

It is well known that time-varying delays are frequently encountered in many practical control systems, and they are usually regarded as a source of instability and poor performance. So the stability issue of time delay systems has received considerable attention [1–16]. In the last years, in order to further reduce conservatism of the stability results, some methods were developed, such as the delay-fraction approach in [1, 2], free weighting matrices method in [3, 4], the convex analysis method in [5, 6], the reciprocally convex approach in [7], LKF constructing method with matrices that depend on the time delays [8], LKF constructing method with triple-integral terms in [9], LKF constructing method with quadruple-integral terms in [10], and simple LKF having quadratic terms multiplied by a higher degree scalar function [11, 12]. These methods reduced the conservatism of the stability results. But when delay is interval time-varying, the information of the delay derivative is not full used, which causes the conservatism of the stability results.Motivated by recent methods, in this paper, we further discuss the stability of linear systems with interval time-varying delay. Firstly, a novel LKF is introduced. Then, by introducing new vectors, using free matrices and reciprocally convex approach, the derivative of LKF is estimated less conservatively, and as a result, the stability criterion is obtained in terms of LMI. Finally, two examples are given to illustrate the effectiveness of the proposed method.Throughout the note, the used notations are standard.Rn denotes the n-dimensional Euclidean space, Rn×m is a set of n×m real matrix, AT is the transpose of A, P>0(P<0) means symmetric positive (negative) definite matrix, and * in the matrix denotes the symmetric element; I is the identity matrix of appropriate dimensions, xt=x(t+θ),θ∈[-h,0]. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Formulations

Consider the following time-delay system:(1)x˙(t)=Ax(t)+A1x(t-h(t)),x(t)=ϕ(t),t∈[-h,0],
where x(t)∈Rn is the state vector, the initial condition ϕ(t) is a continuously differentiable vector-valued function, A,A1∈Rn×n are known real constant matrices, and h(t) is the time-varying delay satisfying
(2)h1≤h(t)≤h2,h˙(t)≤μ≤∞,
where h1, h2, and μ are constants.To obtain the main results, the following lemmas are needed.Lemma 1 (see [9]).
For any symmetric matrixZ=ZT≥0, scalars h2>h1>0 such that the integration is well defined; then
(3)-(h2-h1)∫-h2-h1‍∫t+θt‍xT(s)Zx(s)dsdθ≤-∫-h2-h1‍∫t+θt‍xT(s)dsdθZ∫-h2-h1‍∫t+θt‍x(s)dsdθ.Lemma 2 (see [7]).
Letf1,f2,…,fN:Rm→R have positive values in an open subset D of  Rm. Then, the reciprocally convex combination of fi over D satisfies
(4)min{αi∣αi>0,∑‍αi=1}∑i‍1αifi(t)=∑i‍fi(t)+maxgij(t)∑i≠j‍gij(t)subjecttogi,j:Rm⟶R,gij(t)=gji(t),ppppppppppppp[fi(t)gij(t)gij(t)fj(t)]≥0.Lemma 3 (see [16]).
The symmetric appropriately dimensional matricesΘ>0,Ξ,Υ, if  Ξ-ΥΘΥT<0 hold; then there exists a matrix of appropriate dimension Π such that the following LMI holds:
(5)[Ξ+ΥΠT+ΠΥTΠ*-Θ]<0.


## 3. Main Results

In this section, the stability of system (1) is investigated. Through constructing a novel LKF and estimating the derivative of it, new stability condition is provided.Firstly, define the following vector as(6)ζT(t)=[xT(t)∫t-h1t‍xT(s)ds],ξT(t)=[x˙T(t)xT(t)],
and then construct the L-K functional as follows:
(7)V(xt)=∑i=14‍Vi(xt),
where
(8)V1(xt)=ζT(t)Pζ(t),V2(xt)=∫t-h1t‍xT(s)Q1x(s)ds+∫t-h(t)t-h1‍xT(s)Q2x(s)ds+∫t-h2t-h(t)‍xT(s)Q3x(s)ds,V3(xt)=h1∫-h10‍∫t+θt‍ξT(s)[R10*R2]ξ(s)dsdθ+(h2-h1)∫-h2-h1‍∫t+θt‍ξT(s)[X10*X2]ξ(s)dsdθ,V4(xt)=h122∫-h10‍∫θ0‍∫t+λt‍x˙T(s)U1x˙(s)dsdλdθ+h22-h122∫-h1-h2‍∫θ0‍∫t+λt‍x˙T(s)U2x˙(s)dsdλdθ+12∫t-h2t-h1‍∫θt-h1‍∫λt‍x˙T(s)U3x˙(s)dsdλdθ,
and matrices Q1,Q2,Q3,R1,R2,X1,X2,U1,U2,U3∈Rn×n>0, P=[Pij](1≤i≤j≤2)∈R2n×2n>0.In order to make the proof process clear, the following notations are used:(9)α(t)=h(t)-h1,β(t)=h2-h(t),φ(t)=1α(t)∫t-h(t)t-h1‍xT(s)ds,ϕ(t)=1β(t)∫t-h2t-h(t)‍xT(s)ds,χT(t)=[xT(t)xT(t-h(t))xT(t-h2)∫t-h1t‍xT(s)φT(t)ϕT(t)x˙T(t)]and block entry matrices ei(i=1,2,…,5) (e.g., e2T=[0I000000]).Then taking the time derivatives ofVi(xt) along the trajectory of system (1) yields(10)V˙1(xt)=2ζT(t)Pζ˙(t)=2χT(t)[e1e5]P[e8e1-e3]Tχ(t),V˙2(xt)=xT(t)Q1x(t)-xT(t-h1)(Q1-Q2)x(t-h1)-(1-h˙(t))xT(t-h(t))(Q2-Q3)x(t-h(t))-xT(t-h2)Q3x(t-h2)≤χT(t)[e1Q1e1T-e3(Q1-Q2)e3Tpppppl-(1-μ)e2(Q2-Q3)e2T-e4Q3e4T]χ(t),
where Q2-Q3>0.To reduce the conservatism of the main result, by using the Jesen inequality and Lemma2 with introducing appropriate matrices Y1,Y2∈Rn×n, we can calculate V˙3(xt) as
(11)V˙3(xt)=h12ξT(t)[R10*R2]ξ(t)+(h2-h1)2ξT(t)[X10*X2]ξ(t)-h1∫t-h1t‍ξT(s)[R10*R2]ξ(s)ds-(h2-h1)∫t-h2t-h1‍ξT(s)[X10*X2]ξ(s)ds≤χT(t){e8[h12R1+(h2-h1)2X1]e8Tpppppi+e1[h12R2+(h2-h1)2X2]e1Tpppppi-(e1-e3)R1(e1-e3)T-e5R2e5T}χ(t)-χT(t)[e2-e3e4-e2]×[X1Y1*X1][e2-e3e4-e2]Tχ(t)-Γ1,
where by using Lemma 2 with matrix Y1, one can get
(12)-(h2-h1)∫t-h2t-h1‍x˙T(s)X1x˙(s)ds=-(h2-h1)∫t-h(t)t-h1‍x˙T(s)X1x˙(s)ds-(h2-h1)∫t-h2t-h(t)‍x˙T(s)X1x˙(s)ds≤-[x(t-h(t))-x(t-h1)x(t-h2)-x(t-h(t))]T×[X1Y1*X1][x(t-h(t))-x(t-h1)x(t-h2)-x(t-h(t))]=-χT(t)[e2-e3e4-e2][X1Y1*X1]×[e2-e3e4-e2]Tχ(t),
and by using Lemma 2 with matrix Y2, one can get
(13)-(h2-h1)∫t-h2t-h1‍xT(s)X2x(s)ds=-(h2-h1)∫t-h(t)t-h1‍xT(s)X2x(s)ds-(h2-h1)∫t-h2t-h(t)‍xT(s)X2x(s)ds≤-[∫t-h(t)t-h1‍x(s)ds∫t-h2t-h(t)‍x(s)ds]T×[X2Y2*X2][∫t-h(t)t-h1‍x(s)ds∫t-h2t-h(t)‍x(s)ds]=Γ1.To the time-derivative ofV4(xt), it can be calculated as
(14)V˙4(xt)=h144x˙T(t)U1x˙(t)+(h22-h12)24x˙T(t)U2x˙(t)+(h2-h1)24x˙T(t)U3x˙(t)-h122∫-h10‍∫t+θt‍x˙T(s)U1x˙(s)dsdθ-h22-h122∫-h2-h1‍∫t+θt‍x˙T(s)U2x˙(s)dsdθ-12∫t-h2t-h1‍∫θt-h1‍x˙T(s)U3x˙(s)dsdθ=χT(t)e8[h122U1+(h22-h12)24U2ppppppp+(h2-h1)24U3]e8Tχ(t)-χT(t)(h1e1-e5)U1(h1e1-e5)Tχ(t)-χT(t)[(e3-e6)TU3(e3-e6)ppppppp-(e2-e7)TU3(e2-e7)]χ(t)-Γ2,
where by using Lemma 1, we can get
(15)-h122∫-h10‍∫t+θt‍x˙T(s)U1x˙(s)dsdθ≤-[h1x(t)-∫t-h1t‍x(s)ds]T×U1[h1x(t)-∫t-h1t‍x(s)ds]=-χT(t)(h1e1-e5)U1(h1e1-e5)Tχ(t),-12∫t-h2t-h1‍∫θt-h1‍x˙T(s)U3x˙(s)dsdθ≤-12∫t-h(t)t-h1‍∫θt-h1‍x˙T(s)U3x˙(s)dsdθ-12∫t-h2t-h(t)‍∫θt-h(t)‍x˙T(s)U3x˙(s)dsdθ≤-[(h(t)-h1)x(t-h1)-∫t-h(t)t-h1‍x(s)ds]T×U3α2(t)[(h(t)-h1)x(t-h1)-∫t-h(t)t-h1‍x(s)ds]-[(h2-h(t))x(t-h(t))-∫t-h2t-h(t)‍x(s)ds]T×U3β2(t)[(h2-h(t))x(t-h(t))-∫t-h2t-h(t)‍x(s)ds]=-χT(t)[(e3-e6)TU3(e3-e6)ppppppppppl-(e2-e7)TU3(e2-e7)]χ(t).Similar to the method in (12)-(13), by introducing matrix Y3∈Rn×n and using Lemma 2, we can estimate the following:
(16)-h22-h122∫-h2-h1‍∫t+θt‍x˙T(s)U2x˙(s)dsdθ=-h22-h122∫-h(t)-h1‍∫t+θt‍x˙T(s)U2x˙(s)dsdθ-h22-h122∫-h2-h(t)‍∫t+θt‍x˙T(s)U2x˙(s)dsdθ≤-[(h(t)-h1)x(t)-∫t-h(t)t-h1‍x(s)ds(h2-h(t))x(t)-∫t-h2t-h(t)‍x(s)ds]T×[U2Y3*U2][(h(t)-h1)x(t)-∫t-h(t)t-h1‍x(s)ds(h2-h(t))x(t)-∫t-h2t-h(t)‍x(s)ds]=Γ2.In the last, in order to obtain the stability result base on LMI, we can deal withΓ1,Γ2 as follows:
(17)Γ1+Γ2=-χT(t)[00000α(t)00000000β(t)0]T×[X2Y2*X2][00000α(t)00000000β(t)0]χ(t)-χT(t)[α(t)0000-α(t)00β(t)00000-β(t)0]T×[U2Y3*U2]×[α(t)0000-α(t)00β(t)00000-β(t)0]χ(t)=-χT(t)Υ(t)ΘΥT(t)χ(t),
where
(18)ΥT(t)=[α(t)0000-α(t)00β(t)00000-β(t)000000α(t)00000000β(t)0],Θ=[U2Y300*U20000X2Y200*X2].
Also, for appropriate matrices N1,N2, one can have
(19)2[xT(t)N1+x˙T(t)N2]×[Ax(t)+A1x(t-h(t))-x˙(t)]=0.Therefore, combining (10)–(19), we can obtain
(20)V˙(xt)=∑i=14‍V˙i(xt)≤χT(t)[Ξ-Υ(t)ΘΥT(t)]χ(t),
where
(21)Ξ=[e1e5]P[e8e1-e3]T+[e8e1-e3]P[e1e5]T-e3(Q1-Q2)e3T-(1-μ)e2(Q2-Q3)e2T-e4Q3e4T+e8[h12R1+(h2-h1)2X1+h122U1ppllll+(h22-h12)24U2+(h2-h1)24U3]e8T+e1[Q1+h12R2+(h2-h1)2X2]e1T-(e1-e3)R1(e1-e3)T-e5R2e5T-[e2-e3e4-e2][X1Y1*X1][e2-e3e4-e2]T-(h1e1-e5)U1(h1e1-e5)T-(e3-e6)TU3(e3-e6)-(e2-e7)TU3(e2-e7)+(e1N1+e8N2)(e1TA+e2TA1-e8)T+(e1TA+e2TA1-e8)(e1N1+e8N2)T.
As we know, if Ξ-Υ(t)ΘΥT(t)<0 holds, then V˙(xt)<0, which means that system (1) is asymptotically stable. So, by Lemma 3, there exists a matrix of appropriate dimension Π such that the following LMI holds:
(22)[Ξ+ΥΠT+ΠΥTΠ*-Θ]<0.So, we give the main theorem of this paper as follows.Theorem 4.
The time-delayed system (1) having contrains (2) is asymptotically stable, if there exist matrices Q1,Q2,Q3,R1,R2,X1,X2,U1,U2,U3∈Rn×n>0, matrix P∈R2n×2n>0, and appropriate matrices Y1,Y2,Y3,N1,N2,Π such that the following LMIs hold:
(23)Q2-Q3>0,[Ξ+Υ1ΠT+ΠΥ1TΠ*-Θ]<0,[Ξ+Υ2ΠT+ΠΥ2TΠ*-Θ]<0,
where Ξ, Υ, Π, and Θ are defined in (18) and (21).Proof.
From above, one can see thatΞ+ΥΠT+ΠΥT is a convex combination of matrix Π as
(24)Ξ+ΥΠT+ΠΥT=Ξ+(h(t)-h1)(Υ1ΠT+ΠΥ1T)+(h2-h(t))(Υ2ΠT+ΠΥ2T)<0.
So (22) can be handled nonconservatively by the two corresponding boundary LMIs:
(25)[Ξ+Υ1ΠT+ΠΥ1TΠ*-Θ]<0,[Ξ+Υ2ΠT+ΠΥ2TΠ*-Θ]<0,
where Υ1=Υ(h(t)-h1=0), Υ2=Υ(h2-h(t)=0). This completes the proof.Remark 5.
From (17) and (22), it can be seen that, in order to obtain the less conservative stability condition in terms of LMI, α(t), β(t) are used, but free matrices Π=[Πij](1≤i≤j≤8) are also introduced, which increases the computational burden; this is a disadvantage of the proposed method.Remark 6.
Whenh1≤h(t)≤h2, μ1≤h˙(t)≤μ2, let Ξ1=Ξ(μ=μ1), Ξ2=Ξ(μ=μ2), and using the convex combination again with [Ξ1+Υ1ΠT+ΠΥ1TΠ*-Θ]<0, [Ξ1+Υ2ΠT+ΠΥ2TΠ*-Θ]<0, [Ξ2+Υ1ΠT+ΠΥ1TΠ*-Θ]<0, [Ξ2+Υ2ΠT+ΠΥ2TΠ*-Θ]<0, the corresponding stability criterion can also be obtained.


## 4. Numerical Examples

In this section, the effectiveness of the obtained results in this paper is shown by the two well-known numerical examples.Example 1.
Consider the linear system (1) with
(26)A=[-200-0.9],A1=[-10-1-1].
The purpose is to compare the admissible upper boundsh2 given lower bound h1 and μ.
Forh1=0, μ=0.5,0.9, in [14], the result is h2=2.04,1.37, in [17], the result is h2=2.33,1.87, and our results are h2=2.33,1.87.
Forh1=1, μ=0.5,0.9, in [14], the result is h2=2.07,1.74, in [17], the result is h2=2.33,2.07, and our results are h2=2.43,2.08.
Forh1=2, μ=0.5,0.9, in [14], the result is h2=2.43,2.43, in [17], the result is h2=2.61,2.61, and our results are h2=2.64,2.64.
For varioush1,μ, the maximum upper bounds on delay h2 by different methods are also listed in Table 1. It can be seen that the result obtained in this paper is less conservative.Table 1
Admissible upper boundh2 for various h1 and μ in Example 1.
h
1
Methods
μ
=
0.3
μ
=
0.5
μ
=
0.9
2
Shao [6]
2.69
2.50
2.50
Sun et al. [9]
2.91
2.50
2.50
Theorem4
3.00
2.64
2.64
3
Shao [6]
3.25
3.25
3.25
Sun et al. [9]
3.34
3.34
3.34
Theorem4
3.35
3.35
3.35Example 2.
Consider the linear system (1) with
(27)A=[01-1-2],A1=[00-11].
The purpose is to compare the admissible upper boundsh2 which guarantee the asymptotic stability of the above system for given lower bound h1 and μ. This example is used in many recent papers, such as [6, 9], For μ=0.3, Table 2 lists the comparison of our results with some recent ones, and it is easy to see that the results obtained in this paper are less conservative.Table 2
Admissible upper boundh2 for various h1 and μ=0.3 in Example 1.
Methods
h
1
0.3
0.5
0.8
1
Shao [6]
h
2
2.22
2.22
2.23
2.24
Sun et al. [9]
h
2
2.26
2.28
2.30
2.30
Theorem4
h
2
2.39
2.41
2.42
2.43


## 5. Conclusions

In this note, the stability of interval time-varying delay systems has been discussed. Through constructing a novel LKF and using some new analysis methods, the delay-range-dependent stability criteria were derived. Compared with some previous stability conditions, the obtained main results in this paper have less conservatism. In the end, numerical examples were given to show the superiority of the obtained criteria and their improvements over the existing results.



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