# Stability Analysis of Time-Varying Delay Systems with an Improved Type of LKF

Authors: Wenhao Wang; Hong Zhu; Kaibo Shi; Shouming Zhong; Can Zhao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper further investigates the problem of stability for a general linear system with time-varying delays. Firstly an improved type of Lyapunov–Krasovskii functional is introduced with integral and nonintegral terms and time-correlation terms. Referring a few existing papers, some valid inequalities mathematical analysis techniques are used in this paper in order to reduce the conservatism of the system. Finally, two examples are presented to demonstrate the advantages of the proposed tactics in this paper.


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## Body

## 1. Introduction

It is widely acknowledged that, in practical systems such as switched systems [1], T-S fuzzy systems [2], and neural networks [3], time delays inevitably occur and affect the system performance [4, 5]. Therefore, in recent decades, researchers have paid so much attention to analyzing the stability and accomplishing the stabilization of time-delayed systems. The Lyapunov–Krasovskii functional (LKF), recognized as one of the most efficient tools for these questions, offers both the methods as well as the challenges for dealing with such problems. On one hand, like in a previous work [6–9], the way to construct LKF with augmented vector is commonly accepted. Meanwhile, adding more multi-integral seems also will be helpful to reduce the conservatism in some cases [10]. On the other hand, many innovative integral inequalities have been proposed to deal with the derivative of LKF, for example, Jensen inequality [11, 12], Wirtinger-based inequality [13], Bessel–Legendre inequality [14], free-matrix-based integral inequality [15, 16], and many new refined types of these integral inequalities such as shown in [17, 18]. Combining the recent studies of LMIs discussed above [13, 19] and improved quadratic integral inequalities such as in [20, 21], a general type of LKF may be employed for stability analysis of time-delayed systems [22, 23], with both the construction of LKF and the way to cope with the derivative of LKF been considered [24, 25].Motivated by the contents above, an attempt to construct the LKF has been made for the time-varying delay systems in this paper. Consulting the method of LMIs in previous articles with an augmented quadratic term, analysis of a time-varying delay system with less conservatism is revealed.This paper consists of the following parts: Section2 states the description and definition of the system, as well as the lemmas used later. Then, the achievements of this paper are proved in Theorem 1. Two numerical examples are provided to confirm the results at the end of this paper.Throughout this paper,ℝn denotes the n-dimensional Euclidean space. P>0 means that P is a real symmetric and positive definite matrix. The superscripts “−1” and “T” indicate the inverse and the transpose of a matrix, respectively. SymX=X+XT,col…=x1T,x2T,…,xnTT, diag{…} represents the block diagonal matrix.


## 2. Problem Statement

Firstly, we take the following system model with a time-varying delay into account:(1)x˙t=Axt+Adxt−dtxt=ϕtt∈−d,0,where xt∈Rn represents the state vector. ϕt is the initial condition with t∈−d,0,d>0.A, and Ad are system matrices with compatible dimensions. The time delay function dt is continuous and defined as(2)0≤dt≤d,μ1≤d˙t≤μ2.The scalarμ1 and μ2, respectively, represent the lower and upper bounds of the time-varying delay.In order to deduce the results of this paper, some lemmas are introduced before the main results.Lemma 1 (See [26])..
LetN∈ℕ,ζ∈ℝm and x be a continuous and differentiable function: α,β⟶Rn. Matrix Z∈ℝn×n>0,M∈ℝN+1n×m. The following inequality holds:(3)−∫αβx˙TuZx˙udu≤2ζNTΓNTMξ+β−αξTMTZ˜Mξ,with(4)ΓN=πNT0,πNT1…πNTN,Z˜=1Z,13Z…12N+1Z,ζN=xTβxTαifN=0xTβxTα1β−αΘ0T…1β−αΘN−1TifN>0,πNk=I−IifN=0I−1k+1IθNk0I…θNkN−1IifN>0,θNkj=2j+1−1k+j−1ifj≤k0ifj>k,Fku=−1k∑i=0k−1ikik+iiu−αβ−αi,Θk=∫αβFkuxudu.Lemma 2 (See [27])..
Letfdt=a2d2+a1d+a0, where a2,a1 and a0∈ℝ,(5)fd≤0,∀d∈0,d,if the following conditions hold:(i)
f0≤0(ii)
fd≤0(iii)
f0−da2≤0Lemma 3 (See [28])..
For any symmetric positive definite matrixM>0, scalar γ>0, and vector function ω:0,γ⟶ℝ such that the integrations concerned are well defined, the following inequality holds:(6)∫0γωsdsTM∫0γωsds≤γ∫0γωTsMωsds.Lemma 4 (See [29])..
Given positive integers m,n and a scalarβ∈0,1, given matrices R1>0,R2>0 and H1,H2∈ℝn×m. For all vecotrs ξ∈ℝm, the function Θβ,R1,R2 is given by(7)Θβ,R1,R2=1βξTH1TR1H1ξ+11−βξTH2TR2H2ξ.Then, if there exists a matrixX∈ℝn×n such that R1XXTR2>0, the following equality holds:(8)minΘβ,R1,R2︸β∈0,1≥H1ξH2ξTR1XXTR2H1ξH2ξ.


## 3. Main Result

In this section, the stability of the system is analyzed. For brevity, the following nomenclature is used to simplify vector and matrix representations:(9)η1t=colxt,xt−dt,xt−d,∫t−dttxsds,∫t−dt−dtxsds1dt∫−dt0∫t+θtxsdsdθ,1d−dt∫−d−dt∫t+θt−τtxsdsdθ,η2s=colx˙s,xs,xt−dt,∫stx˙θdθ,∫t−dsx˙θdθ,∫t−dtsx˙θdθ,η3t=colxt,x​t−dt,∫t−dttxsds,1dt∫−dt0∫t+θtxsdsdθ,η4t=colxt−dt​,xt−d,∫t−dt−dtxsds,1d−dt∫−d−dt∫t+θt−τtxsdsdθ,ξt=colξ1t,ξ2t,ξ3t,ξ1t=colxt,xt−dt,xt−d,x˙t−dt,x˙t−d,ξ2t=col1dt∫t−dttxsds,1d−dt∫t−dt−dtxsds,ξ3t=col1d2t∫−dt0∫t+θtxsdsdθ,1d−dt2∫−d−dt∫t+θt−dtxsdsdθ,ei=0n×i−1In0n×9−i,i=1,2,…,9.Theorem 1.
In this section, the stability of system (1) is analyzed under the condition of the new LKF. With known scalars μ1,μ2<1 and d>0, system (1) is asymptotically stable if there exist appropriately dimensional symmetrical matrices P>0,Q1,Q2>0,Z1,ℤ=Z2+Z3>0,U1,U2>0, and V1,V2>0 satisfying the following LMIS:(10)ΨdN1∗−Z3˜<0,(11)ΨdN2∗−Z3˜<0,(12)ΨdN2∗−Z3˜−d2Ωd2t<0,(13)Φ1=SymΠ1TPΠ2,Φ2=Π3TQ1Π3+1−d˙tΠ4TQ2−Q1Π4−Π5TQ2Π5T+SymΠ6TQ1Π8+Π7TQ2Π8,Φ3=e1TU1+dZ1e1+e2TU2−U1e2−e3TU2e3+de0TZ2e0−1dΠ13TΥΠ13,Φ4=de0Z3e0+SymN1Π14+N2Π15,Φ5=d˙tΠ9TV1Π9+dtSymΠ9TV1Π10−d˙tΠ11TV2Π11+d−dtSymΠ11TV1Π12,Φ6=d−dtN1Z3˜−1N1T+dtN2Z3˜−1N2T,Ψ=Φ1+Φ2+Φ3+Φ4+Φ5−Φ6,Π1=cole1,e2,e3,dte6,d−dte7,dte8,d−dt9,Π2=coled,1−d˙te4,e5,e1−1−d˙te2,1−d˙te2−e3,e1−1−d˙te6−d˙te8,1−d˙te2−e7+d˙te9,Π3=coled,e1,e2,0,e1−e3,e1−e2,Π4=cole4,e2,e2,e1−e2,e2−e3,0,Π5=cole5,e3,e2,e1−e3,0,e3−e2,Π6=cole1−e2,dte6,dte2,dte1−e6,dte6−e3,dte6−e2,Π7=cole2−e3,d−dte7,d−dte2,​d−dte1−e7,d−dte7−e3,d−dte7−e2,Π8=col0,0,1−d˙te4,ed,−e5,−1−d˙te4,Π9=cole1,e2,dte6,dte8Π9a=cole1,e2,0,0Π9b=col0,0,e6,e8,Π10=coled,1−d˙te4,e1−1−d˙te2,−d˙te8+e1−1−d˙te6,Π11=cole2,e3,d−dte7,d−dte9Π11a=cole2,e3,0,0Π11b=col0,0,e7,e9,Π12=col1−d˙te4,e5,1−d˙te2−e3,d˙te9+1−d˙te2−e7,Π13=coldte6,e1−e2,d−dte7,e2−e3,Π14=cole1−e2,e1+e2−2e6,e1−e2+6e6−12e8,Π15=cole2−e3,e2+e3−2e7,e2−e3+6e7−12e9,ed=Ae1+Ade2,U¯1=Z1U1∗Z2,U¯2=Z1U2∗Z2,H=H1H2∗H3.Proof.
We design the functional below for system(14)Wt=∑i=13Vit+wt,with each part of Wt as(15)V1t=η1TtPη1t,V2t=∫t−dttη2TsQ1η2sds+∫t−dt−dtη2TsQ2η2sds,V3t=∫−d0∫t+θtxTsZ1xsdsdθ+∫−d0∫t+θtx˙Tsℤx˙sdsdθ,wt=dtη3TtV1η3t+d−dtη4TtV2η4t.
The positive ofWt can be proved with matrices P,Q1,Q2,Z1,ℤ,V1,V2>0, respectively. The derivation of each part in Wt is(16)V˙1t=2η1TtPη˙1t=ξTtSymΠ1TPΠ2ξt,(17)V˙2t=η2TtQ1η2t+1−d˙tη2Tt−dtQ2−Q1η2t−dt−η2Tt−dQ2η2t−d+2∫t−dttη2TsQ1η˙2sds+2∫t−dt−dtη2TsQ2η˙2sds,=ξTtΠ3TQ1Π3+1−d˙tΠ4TQ2−Q1Π4−Π5Q2Π5+SymΠ6TQ1Π8+Π7TQ2Π8ξt,(18)w˙t=​d˙tη3TtV1η3t−d˙tη4TtV2η4t+2dtη3TV1η˙3t+2d−dtη4TV2η˙4t,=ξTtd˙tΠ9TV1Π9+dtSymΠ9TV1Π10−d˙tΠ11TV2Π11+d−dtSymΠ11TV1Π12,and V˙3t could be divided into two parts:(19)V˙3at=ξTtde1TZ1e1+de0TZ2e0ξt−∫t−dtxTsZ1xsds−∫t−dtx˙TsZ2x˙sds,(20)V˙3bt=ξTtde0TZ3e0ξt−∫t−dtx˙TsZ3x˙sds.
Based on the formula of integration by parts, the following zero equalities hold for symmetric matricesU1 and U2 with appropriate dimension.(21)0=xTtU1xt−xTt−dtU1xt−dt−2∫t−dttxTsU1x˙sds,=ξTte1TU1e1−e2TU1e2ξt−2∫t−dttxTsU1x˙sds,(22)0=xTt−dtU2xt−dt−xTt−dU2xt−d−2∫t−dt−dtxTsU2x˙sds,=ξTte2TU2e2−e3TU2e3ξt−2∫t−dttxTsU1x˙sds.
Adding equations (21) and (22) to (20), one has(23)V˙3at=ξTte1TU1+dZ1e1+e2TU2−U1e2−e3TU2e3+de0TZ2e0ξt−∫t−dttxsx˙sTU1¯xsx˙s−∫t−dt−dtxsx˙sTU2¯xsx˙s.
Then, through applying Lemmas3 and 4, equation (23) would yield(24)V˙3at≤ξTte1TU1+dZ1e1+e2TU2−U1e2−e3TU2e3+de0TZ2e0ξt−1dtξTtdte6e1−e2TU1¯dte6e1−e2ξt−1d−dtξtTd−dte7e2−e3TU2¯d−dte7e2−e3ξt≤ξtTe1TU1+dZ1e1+e2TU2−U1e2−e3TU2e3+de0TZ2e0−1ddte6e1−e2d−dte7e2−e3TU1¯H∗U2¯dte6e1−e2d−dte7e2−e3ξt≤ξtTe1TU1+dZ1e1+e2TU2−U1e2−e3TU2e3+de0TZ2e0−1dΠ13TΥΠ13.
According to Lemma1, with N=2, the integration in V˙3bt is expressed as(25)−∫t−dtx˙TsZ3x˙sds=−∫t−dt−dtx˙TsZ3x˙sds−∫t−dttx˙TsZ3x˙sds≤−d−dtξtTN1Z3−1N1T+13N1Z3−1N1T+15N1Z3−1N1Tξt+2ξTtN1Π14−dtξTtN2Z3−1N2T+13N2Z3−1N2T+15N2Z3−1N2Tξt+2ξTtN2Π15,=ξTtSymN1Π14+N2Π15−d−dtN1Z3−1N1T+13N1Z3−1N1T+15N1Z3−1N1T+dtN2Z3−1N2T+13N2Z3−1N2T+15N2Z3−1N2T​ξt,=ξTtSymN1Π14+N2Π15−d−dtN1Z3˜−1N1T+dtN2Z3˜−1N2T​ξt.
For equation (19), the expressions of η3t and η4t are divided into two parts, respectively, i.e.,(26)η3t=ξTtcole1,e2,0,0+col0,0,dte6,dte8ξt=ξTtΠ9a+dtΠ9bξt,(27)η4t=ξTtcole2,e3,0,0+col0,0,d−dte7,d−dte9ξt=ξTtΠ11a+d−dtΠ11bξt.
Therefore, all the quadratic terms ofW˙t exist in equations (24), (26), and (27), and the coefficient Ωd2t of the quadratic terms is(28)Ωd2t=d˙tΠ9aTU1Π9a+SymΠ9aU1Π10−d˙tΠ11aTV2Π11a+SymΠ11aV2Π12−1de6TZ1+U1+H1+H2e6−Syme6TZ1+U1+H1+H2e7+e7TZ1+U2+H1+H2e7.
Combining equations (16)–(18) and (24)–(28) and referring Lemma 4 simultaneously, W˙t would be negative if LMIs (10)–(12) hold through utilizing Schur complement. Based on Lyapunov stable analysis of dynamic system, the system discussed in this paper is stable.Remark 1.
To consider both the conservatism and computational complexity of the system, a nonintegral term ofwt is introduced into this with flexibal coefficients dt and d−dt. In addition, the vector ηit is not fixed as well, which makes the LKF more general for some conditions of practical system. If the upper bound is much more high than the lower bound, the part dtξ3TtV1ξ3t could be ignored and so the same is in reverse.Remark 2.
Depending on the definition of Lemma1, it is not difficult to get tighter bounds of the system through adjusting the number of order. With the increase of the order N, the double integral state vectors in ξ1t are required to translate into triple or multiple integrals in order to reduce the conservatism. Thus, these state vectors may satisfy the demand of a complex system.


## 4. Numerical Example

In this section, two numerical examples are provided while comparing other results existed in previous research to demonstrate the proposed methods.We define the system matricesA and Ad as(29)A=−200−0.9,Ad=−10−1−1.The example is aimed at searching the maximum value for the upper bounds of the delay (MVUBD) with the derivative of known delay. We setμ1=−μ2=μ and assign a value to μ, and the results of the MVUBD computed by Theorem 1 are shown in Table 1. Some other results from former research are also listed in Table 1 for the sake of comparison. Figures 1–3 show the state trajectories with different μ and MVUBD. From the simulation results, it is concluded that the achieved improvements in this paper are less conservative than other results.Table 1
The MVUBD under the conditionμ1=−μ2 with a certain μ.
μ0.10.20.8[16]4.7884.0602.615[4]4.8314.1412.713[30]4.9084.1992.735[31]4.9104.2162.789[26]4.9214.2182.792[32]4.9304.2352.807Theorem14.9404.2642.868Figure 1
The state trajectory withμ=0.1 and MVUBD 4.940.Figure 2
The state trajectory withμ=0.2 and MVUBD 4.264.Figure 3
The state trajectory withμ=0.8 and MVUBD 2.868.In this example, the state matricesA and Ad have been set as(30)A=01−1−2,Ad=00−11.In this example, the value ofμ1 is still set as μ2, and the results of the MVUBD which is computed by Theorem 1 are shown in Table 2. Figures 4–6 also show the state trajectory with different μ and MVUBD. The results of MVUBD with different μ in Table 2 are much better, which means the method proposed in not only valid but also advanced. It is worth noticing that the MVUBD reaches almost 18.97% than the data published lately when μ is 0.5.Table 2
The MVUBD under the conditionμ1=−μ2 and a certain μ.
μ0.10.20.5[13]6.5903.6721.411[16]7.1484.4662.352[4]7.1674.5172.415[32]7.1764.5432.496[31]7.2304.5562.509[26]7.3084.6702.664Theorem17.3484.7592.897Figure 4
The state trajectory withμ=0.1 and MVUBD 7.348.Figure 5
The state trajectory withμ=0.2 and MVUBD 4.759.Figure 6
The state trajectory withμ=0.5 and MVUBD 2.897.


## 5. Conclusions

A further investigation of the stability for system with time-varying delays has been figured out in this paper. With different methods of solving inequalities, results of less-conservative stability criteria have been achieved. At the end of the paper, two examples, compared with results from other papers, are represented in order to show the availability of the such methods. Based on the construction of LKF in this paper, some other problem such as sampled data control may also be considered in the same way. In the future work, the improved type of LKF proposed in this paper will be extended to a chaotic Lurie system, T-S fuzzy systems, Markov jump systems, quaternion-valued or memristor-based neural networks, and complex dynamical networks.



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