# Further Stability Analysis of Time-Delay Systems with Nonlinear Perturbations

Authors: Jie Sun; Jing Zhang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

In this study, we focus on stability analysis for systems with time-varying delay and nonlinear perturbations. In order to cut down the conservatism of the existing stability criteria, we utilize the triple integral forms of Lyapunov-Krasovskii functional (LKF). In addition, by using single and double integral forms of Wirtinger-based inequality, we overcome some conservatism which come from Jensen’s inequality. Three well-known numerical examples are given at the end. Compared with some existing results, our results have less conservatism.


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## Body

## 1. Introduction

Time-delay has attracted a lot of interests because it is widely encountered in communication systems, neural networks, economic systems, biological systems, and networked control systems [1–6]. Since time-delay will cause serious degradation of system performance, a lot of researchers are making considerable effort for stability analysis of time-delay systems in the last few decades. The LKF approach and LMI method are the efficient instrument to get the delay-dependent stability criterion. However, only a sufficient condition we can get by utilizing these methods. Therefore, we focus on cutting down the conservatism of stability criterion. As we know, the maximum allowable delay bound (MADB) of the time-delay can measure the conservatism of stability criterion. We can get the larger MADB of time-varying delay according to the stability criterion with less conservatism.In a lot of existing results, Jensen’s inequality has played an important role. However, it will induce some conservatism hard to overcome. To cut down the conservatism, Wirtinger-based integral inequality [7], which can be used to obtain much tighter lower bound of single integral terms, was proposed. Very recently, based on Wirtinger-based integral inequality, a Wirtinger-based double integral inequality was proposed to get much tighter lower bound of double integral terms [8].In a lot of recent literatures [9–11], researchers only utilize single and double integral forms of LKF to derive delay-dependent stability criterion. We believe that triple integral form of LKF is helpful for improving the performance of former criteria.Motivated by the previous discussions, we are concerned about the stability analysis for the time-varying delay systems with nonlinear perturbations. We introduce the triple integral forms of LKF to cut down the conservatism. Taking the time derivative of∫t-ht‍∫st‍∫ut‍x˙TvRix˙vdvduds, we obtain -∫t-ht‍∫st‍x˙T(u)Rix˙(u)duds. Instead of Jensen’s inequality, Wirtinger-based double integral inequality was utilized to find much tighter upper bound of -∫ab‍∫ub‍x˙T(u)Rix˙(u)duds. Compared with some existing results, our results have less conservatism.The organization of this paper is as follows. The system is presented in Section2, and some useful lemmas are given. Then the new stability criterion and proof are given in Section 3. In Section 4, the effectiveness of our results can be illustrated by three examples. Section 5 is the conclusion of our investigation.Notations. In this paper, the superscript T means the transpose of a matrix; col{·} denotes the column vector.


## 2. Problem Formulation and Preliminaries

The time-varying delay systems with nonlinear perturbations are considered as follows:(1)x˙t=Axt+Bxt-τt+ϑt,ϑt=hxt,t+pxt-τt,t,xt=ϕt,t∈-dM,0with x(t)∈Rn as the state variate; A,B∈Rn∗n are predefined constant matrices; d(t)≥0 represent a time-varying delay satisfying(2)dt∈dm,dM,d˙t≤dd.The nonlinear perturbations h(x(t),t)∈Rn and p(x(t-dt),t)∈Rn can be abbreviated as h and p, assumed as follows:(3)hTh≤α2xTtCTCxt,pTp≤β2xTt-dtDTDxt-dt,where α and β are positive scalars and C,D∈Rn∗n are constant matrices.In order to derive improved stability criterion, the following lemmas will be used.Lemma 1 (see [7]).
Z is a symmetric positive definite matrix, for differentiable function x∈[d1,d2]→Rn, and we can obtain (4)∫d1d2x˙TsZx˙sds≥1d2-d1ε1ε2TZ0∗Zε1ε2,where(5)ε1=xd1-xd2,ε2=3xd1+3xd2-23d2-d1∫d1d2xsds.Lemma 2 (see [28]).
For positive definitedv:Rn→R, dv∈G⊆Rn(v=1⋯N), reciprocally convex combination of dv can be written as(6)minρm∣ρm>0,∑mρm=1∑m1ρmdmt=∑mdmt+maxlm,wt∑m≠wlm,wtsubjecttolm,w:Rn⟶R,lw,mt≜lm,wt,dmtlm,wtlm,wtdwt≥0.Lemma 3 (see [8]).
R is a symmetric positive definite matrix, for differentiable function x∈[ha,hb]→Rn, and we have(7)hb-ha22∫hahb∫shbx˙TuRx˙ududs≥ω1ω2TR0∗Rω1ω2,where(8)ω1=hb-haxhb-∫hahbxsds,ω2=2hb-ha2xhb+2∫hahbxsds-32hb-ha∫hahb∫shbxududs.


## 3. Stability Criterion

Theorem 4.
System (1) satisfying (2)-(3) will be asymptotically stable if there exist given scalars dm,dM, and dd and ϵ1>0 and ϵ2>0 and positive symmetric matrices P∈R5n∗5n, Ws∈Rn∗n(s=1,2,3), and Zq∈Rn∗n(q=1,…,4) and appropriate dimensions matrices Sm, satisfying the following LMIs:(9)ΦΔTN∗-N<0,Z2Sm∗Z2≥0,m=1,2,where(10)Φ=G1TPG2+G2TPG1+Q+M1+M2-dm2∑i=12ΓiTZ1Γi-∑i=36ΓiTZ2Γi-Γ3TS1Γ5-Γ5TS1TΓ3-Γ4TS2Γ6-Γ6TS2TΓ4-dm2∑i=78ΓiTZ3Γi-dMm2∑i=910ΓiTZ4Γi,G1=cole1,e5,e6,e7,e8,G2=colAe1+Be3+e11+e12,e1-e2,e2-e4,dme1-e5,dMme2-e6,W=diagW1+W2+W3-W2-1-ddW1-W300000000,M1=diagϵ1α2CTC000000000-ϵ1I0,M2=diag00ϵ2β2DTD00000000-ϵ2I,Γ1=e1-e2,Γ2=3e1+3e2-23dme5,Γ3=e2-e3,Γ4=3e2+3e3-23e9,Γ5=e3-e4,Γ6=3e3+3e4-23e10,Γ7=dme1-e5,Γ8=22dme1+2e5-32dme7,Γ9=dMme2-e6,Γ10=2dMm2e2+2e6-32dMme8,Δ=Ae1+Be3+e11+e12,dMm=dM-dm,ei=0n×k-1nI0n×12-kn,k=1,2,…,12N=N1+N2=dm4Z1+dMm2Z2+dm64Z3+dMm64Z4.Proof.
Construct the following LKF:(11)Vt=∑i=14Vit,where(12)V1t=νTtPνt,V2t=∫t-dttxTsW1xsds+∫t-dmtxTsW2xsds+∫t-dMtxTsW3xsds,V3t=dm3∫-dm0∫t+stx˙TuZ1x˙ududs+dMm∫-dM-dm∫t+stx˙TuZ2x˙ududs,V4t=dm42∫t-dmt∫st∫utx˙TvZ3x˙vdvduds+dMm42∫t-dMt-dm∫st-dm∫utx˙TvZ4x˙vdvduds,u1t=∫t-dmtxTsds,u2t=∫t-dMt-dmxTsds,u3t=∫t-dmt∫stxTududs,u4t=∫t-dMt-dm∫st-dmxTududs,νTt=xTtu1tu2tu3tu4t.The time derivative of V(t) is as follows:(13)V˙t=∑i=14V˙it,where(14)V˙1t=2νTtPν˙t=ξTtG1TPG2+G2TPG1ξt,V˙2t=∑i=13xTtWixt-1-d˙txTt-dt∗W1xt-dt-xTt-dmW2xt-dm-xTt-dMW3xt-dM≤ξTtQξt,V˙3t=x˙Ttdm4Z1+dMm2Z2x˙t+∑i=13ζi=ξTtΔTN1Δξt+∑i=13ζi,V˙4t=dm42∫t-dmt∫stx˙TtZ3x˙t-x˙TuZ3∗x˙ududs+dMm42∫t-dMt-dm∫st-dmx˙Tt∗Z4x˙t-x˙TuZ4x˙ududs=ξTtΔTN2Δξt+∑i=45ζi,where(15)N1=dm4Z1+dMm2Z2,N2=dm64Z3+dMm64Z4,ζ1t=-dm3∫t-dmtx˙TsZ1x˙sds,ζ2t=-dMm∫t-dtt-dmx˙TsZ2x˙sds,ζ3t=-dMm∫t-dMt-dtx˙TsZ2x˙sds,ζ4t=-dm42∫t-dmt∫stx˙TuZ3x˙ududs,ζ5t=-dMm42∫t-dMt-dm∫st-dmx˙TuZ4x˙ududs,u5t=1dt-dm∫t-dtt-dmxTsds,u6t=1dM-dt∫t-dMt-dtxTsds,ξTt=xTtxTt-dmxTt-dtxTt-dMu1t⋯u6thTpT.
According to Lemma1, we obtain(16)ζ1t≤-dm2ε1tε2tTZ10∗Z1ε1tε2t,(17)ζ2t≤-1ρ1ε3tε4tTZ20∗Z2ε3tε4t,(18)ζ3t≤-1ρ2ε5tε6tTZ20∗Z2ε5tε6t,where(19)ρ1=dt-dmdMm,ρ2=dM-dtdMm,ε1t=xt-xt-dm,ε2t=3xt+3xt-dm-23dmu1t,ε3t=xt-dm-xt-dt,ε4t=3xt-dm+3xt-dt-23u5t,ε5t=xt-dt-xt-dM,ε6t=3xt-dt+3xt-dM-23u6tand inequality (16) can be denoted as(20)ζ1t≤-ξTtdm2Γ1TZ1Γ1+Γ2TZ1Γ2ξt.It is clear that the real numbers ρ1 and ρ2 satisfy ρ1>0, ρ2>0, and ρ1+ρ2=1. Then introduce appropriate dimensions matrices S1 and S2, such that(21)Z2S2∗Z2≥0,Z2S1∗Z2≥0.Applying Lemma 2 to (17) and (18)(22)ζ2t+ζ3t≤-1ρ1ε3TtZ2ε3t+1ρ2ε5TtZ2ε5t-1ρ1ε4TtZ2ε4t+1ρ2ε6TtZ2ε6t≤-ε3tε5tTZ2S1∗Z2ε3tε5t-ε4tε6tTZ2S2∗Z2ε4tε6t=-ξTt∑i=36ΓiTZ2Γi+Γ3TS1Γ5+Γ5TS1TΓ3+Γ4TS2Γ6+Γ6TS2TΓ4ξt.Using Lemma 3 can lead to (23)ζ4t≤-dm2ω1tω2tTZ30∗Z3ω1tω2t=-ξTtdm2∑i=78ΓiTZ3Γiξt,ζ5t≤-dMm2ω3tω4tTZ40∗Z4ω3tω4t=-ξTtdMm2∑i=910ΓiTZ4Γiξt,where(24)ω1t=dmxt-u1t,ω2t=2dm2xt+2u1t-32dmu3t,ω3t=dMmxt-dm-u2t,ω4t=2dMm2xt-dm+2u2t-32dMmu4t.Form (3), we have(25)ϵ1hTh-α2xTtCTCxt≤0,ϵ2pTp-β2xTt-dtDTDxt-dt≤0.Combining (13), (14), (20), (22), (23), and (25), we obtain(26)V˙t≤ξTtG1TPG2+G2TPG1+W+M1+M2+ΔTNΔ-dm2∑i=12ΓiTZ1Γi-∑i=36ΓiTZ2Γi-Γ3TS1Γ5-Γ5TS1TΓ3-Γ4TS2Γ6-Γ6TS2TΓ4-dm2∑i=78ΓiTZ3Γi-dMm2∑i=910ΓiTZ4Γiξt.Using Schur complement, (26) can be transform to the first LMI of (9), which completed the proof of Theorem 4.Whenh=p=0, that means system without perturbation. We can consider the following system:(27)x˙t=Axt+Bxt-dt,xt=ϕt,t∈-dM,0.Stability analysis of system (26) is studied in a lot of literatures. From Theorem 4, the stability criterion of system (26) can be obtained easily.Corollary 5.
System (26) under condition (2) will be asymptotically stable if there exist given scalars dm,dM, and dd and positive symmetric matrices P∈R5n∗5n, Ws∈Rn∗n(s=1,2,3), and Zq∈Rn∗n(q=1,…,4) and appropriate dimensions matrices Sm, satisfying the following LMIs:(28)ΦΔTN∗-N<0,Z2Sm∗Z2≥0,m=1,2,where(29)Φ=G1TPG2+G2TPG1+W-dm2∑i=12ΓiTZ1Γi-∑i=36ΓiTZ2Γi-Γ3TS1Γ5-Γ5TS1TΓ3-Γ4TS2Γ6-Γ6TS2TΓ4-dm2∑i=78ΓiTZ3Γi-dMm2∑i=910ΓiTZ4Γi,G1=cole1,e5,e6,e7,e8,G2=colAe1+Be3,e1-e2,e2-e4,dme1-e5,dMme2-e6,W=diagW1+W2+W3-W2-1-ddW1-W3000000,Δ=Ae1+Be3,ei=0n×k-1nI0n×10-kn,k=1,2,…,10,Γ1⋯Γ10 and N are defined in Theorem 4. If the information of delay is unavailable, by setting Q1=0, we obtainCorollary 6.
If there exist given scalarsdm,dM, and dd and positive symmetric matrices P∈R5n∗5n, Ws∈Rn∗n(s=1,2,3), and Zq∈Rn∗n(q=1,…,4) and appropriate dimensions matrices Sm, such that (27) with W1=0 are feasible, system (26) under condition (2) will be asymptotically stable.Remark 7.
From the viewpoint of control theory, the changed trend of system in reality is not only decided by the current states but also related to its past states. This is why the study of time-delay system is significant. This paper is interested in the stability analysis of the system with time-delay which has strong background in reality. The stability criterion we proposed can be used to design effective control strategy in a specific engineering file, such as synchronization of coronary artery. In order to compare with exiting results, a lot of papers related to stability analysis are concerned about the MADB which can be calculated by stability criterion. We can see the same particular examples in [10–30] as Examples 1–3 in our manuscript. In these examples structure of the matrices is prescribed. The purpose of these examples is to measure the conservatism of proposed criterion. The larger MADB, the less conservatism.


## 4. Numerical Examples

The advantages of our results can be illustrated by the following examples.Example 1.
Nonlinear system (1) is subject to (2) and (3) with(30)A=-1.20.1-0.1-1,B=-0.60.7-1-0.8,C=D=1001.
For givendm, α, β, and dd, utilizing Theorem 4, we calculate the MADB dM which can guarantee the stability of system (1). The result is listed in Table 1. We can see that the stability criterion we proposed has less conservatism than others. It is worth mentioning that when α=0, Theorem 4 can get much better results. Then we consider system (1) with constant time-delay. When dd=0, the MADB dM under different α and β is listed in Table 2. We can see that Theorem 4 provides much less conservative results than others.
Letf(x(t))=0.1x(t)∗sin⁡(x(t)), g(x(t-d(t)))=0.1x(t-d(t))∗cos⁡(x(t-d(t))), and x(0)=(0.2,0.3). Figure 1(a) shows the trajectories of variable x(t) with α=0.1, β=0.1, dd=0.5, dm=0.5, and d(t)=1.0922+0.5922∗sin⁡(1.5t). Figure 1(b) shows the trajectories of variable x(t) with α=0.1, β=0.1, dd=0, dm=0, and d(t)=2.0497.Table 1
MaximaldM for given dm=0.5.
Methods
α, β
α
=
0, β=0.1
α
=
0.1, β=0.1
d
d
=
0.5
d
d
=
0.9
d
d
=
1.1
d
d
=
0.5
d
d
=
0.9
d
d
=
1.1
[12]
1.4420
1.3380
1.3380
1.2840
1.2450
1.2450
[10] (N=2)
1.5500
1.5500
1.5500
1.3690
1.3690
1.3690
[13]
1.5580
1.5580
1.5580
1.3840
1.3840
1.3840
[14]
1.5636
1.5636
1.5636
1.3858
1.3858
1.3858
[10] (N=4)
1.8240
1.8240
1.8240
1.5240
1.5240
1.5240
[15]
1.8599
1.8599
1.8599
1.6622
1.6622
1.6622
Theorem4
2.2561
2.2561
2.2561
1.6844
1.6844
1.6844Table 2
MaximaldM with given dm=0 for dd=0.
Methods
α
,
β
α
=
0, β=0.1
α
=
0.1, β=0.1
[16]
0.6811
0.6129
[17]
1.3279
1.2503
[18]
2.7422
1.8753
[19]
2.7423
1.8753
[20]
2.7757
1.8959
[21]
2.7758
1.8959
[22]
2.9816
1.9805
Theorem4
3.0715
2.0497Figure 1
Trajectories ofx(t).
(a)
d
(
t
)
=
1.0922
+
0.5922
∗
sin
⁡
⁡
1.5
t
(b)
d
(
t
)
=
2.0497Example 2.
Consider system (26) with(31)A=-200-0.9,B=-10-1-1.
For givendm=3,4,5 and different dd=0.1,0.3,0.5,0.9, we calculate the MADB dM which is listed in Table 3. Table 3 illustrates the methods presented in Corollary 5 providing less conservative results than others. It is worth noting that the effect is very obvious when dd≤0.5.
Letd(t)=5.1964+0.1964∗sin⁡(4.58∗t), and trajectories of variable x(t) can be showed in Figure 2.Table 3
MaximaldM with given dm for different dd.
dm
Methods
d
d
=
0.1
d
d
=
0.3
d
d
=
0.5
d
d
=
0.9
3
[23]
4.3979
3.3408
3.3408
3.3408
[24]
4.4506
3.4186
3.4186
3.4186
[11]
4.8247
3.6616
3.6616
3.6616
Corollary5
5.1213
4.2496
4.2208
3.5974
4
[23]
4.1978
4.1690
4.1690
4.1690
[24]
4.2367
4.2097
4.2097
4.2097
[11]
4.5762
4.3788
4.3788
4.3788
Corollary5
5.1916
4.9391
4.9338
4.4987
5
[23]
5.0275
5.0275
5.0275
5.0275
[24]
5.0440
5.0440
5.0440
5.0440
[11]
5.1453
5.1453
5.1453
5.1453
Corollary5
5.7065
5.7043
5.6965
5.3928Figure 2
Trajectories ofx(t) with d(t)=5.1964+0.1964∗sin⁡4.58∗t.Example 3.
Consider another system (26) with(32)A=01-1-2,B=00-11.
Sometimes the information ofd(t) is unavailable or d(t) is undifferentiable. We calculate the MADB dM under different dm for unknown dd. From Table 4, it is observed that the results obtained by Corollary 6 are much less conservative than others. Finally, for given dm=0 and different dd=0.1,0.3, using Corollary 5 we get the MADB dM listed in Table 5. It is clear to see that our results have much less conservatism when the lower bound is dm=0.
Letd(t)=1.54685+1.54685∗sin⁡(0.194∗t), and trajectories of variable x(t) can be showed in Figure 3.Table 4
MaximaldM with given dm for unknown dd.
Methods
dm=0.3
dm=0.5
dm=0.8
[25]
1.0715
1.2191
1.4539
[23] (N=2)
1.0716
1.2196
1.4552
[26]
1.2043
1.3429
1.5663
[27]
1.2246
1.3619
1.5838
[28]
1.2400
1.3800
1.6000
[29]
1.2700
1.3900
1.6100
[30]
1.3500
1.4700
1.6800
Corollary6
1.5607
1.6426
1.7101Table 5
MaximaldM with given dm=0 for dd=0.1,0.3.
Methods
d
d
=
0.1
d
d
=
0.3
[25]
5.4630
2.2160
[23] (N=2)
5.4764
2.2160
[26]
5.4780
2.2850
[27]
5.4940
2.3070
Corollary5
7.6301
3.0937Figure 3
Trajectories ofx(t) with d(t)=1.54685+1.54685∗sin⁡0.194∗t.


## 5. Conclusion

We have researched the stability analysis problem for the time-varying delay systems. We utilize the new augmented LKF which is constructed based on single and double integral forms of Wirtinger-based inequality. Combining reciprocally convex method, we get an improved delay-dependent stability criterion. Finally, examples illustrate that the stability criterion we obtained is less conservative than some existing results.



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