# Exponential Stabilization of a Class of Time-Varying Delay Systems with Nonlinear Perturbations

Authors: Yazhou Tian; Yuanli Cai; Yuangong Sun; Tongxing Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper addresses the problem of exponential stabilization of a class of time-varying delay systems with nonlinear perturbations. These perturbations are related not only with current statext and the delayed state xt−ht but also with βt, where βt is a continuous function defined on [0,+∞). With the delay interval divided into two equidistant subintervals, a novel Lyapunov functional is introduced, and several new exponential stabilization criteria are derived in terms of linear matrix inequalities (LMIs) by employing reciprocally convex approach. Two examples are given to illustrate the effectiveness of the main results.


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## Body

## 1. Introduction

Time delay is commonly encountered in various physical and engineering systems such as aircraft, biological systems, and networked control systems. Since the existence of time delays causes poor performance, oscillation, or even instability, it is very important to investigate stability analysis for systems with time delays before designing control systems. On the other hand, the systems almost present some uncertainties because it is not easy to obtain an exact mathematical model due to environmental noise, uncertain or slowly varying parameters, and so forth. Therefore, considerable amounts of efforts have been done to the stability and stabilization of time-delay systems and time-delay systems with nonlinear perturbations; see, for example, [1–25] and the references cited therein.Recently, Zhang et al. [12] considered interval time-varying delay systems and obtained some delay-dependent conditions by employing Finsler’s lemma. Combining the descriptor model transformation and the integral inequality method, Han [3] investigated the robust stability of linear systems with time-varying delay and nonlinear perturbations and obtained several improved stability conditions. On the basis of free weighting matrices technique, robust stabilization criteria for neutral systems with nonlinear perturbations were reported in [9]. Wang et al. [6] introduced a new parameter in the Lyapunov functional for the time-varying delay systems with nonlinear perturbations and obtained less conservative results, whereas the range of the time delays considered in the paper was assumed from zero to an upper bound. Note that the stability investigated in the above-mentioned papers was primarily focused on asymptotic stability. Using delay decomposition method and Finsler’s lemma, Liu et al. [24] studied the exponential stability of neutral systems with interval time-varying delays and nonlinear perturbations. So far, there are few articles concerning the problem of exponential stabilization of time-varying delay systems with nonlinear perturbations. Thuan et al. [16] provided a detailed analysis for the problem of designing state feedback controllers to exponential stabilization of time-delay systems with nonlinear perturbations by using the integral inequality method and constructing a Lyapunov functional containing the triple integral terms. However, there still exists a gap for reducing both the conservatism and the number of decision variables.In this paper, we study the exponential stabilization of a class of time-delay systems with nonlinear perturbations. The main contributions of this paper can be summarized as follows: (i) a novel Lyapunov functional containing the center point of time-delay interval is constructed; (ii) compared with the systems studied in [3, 8, 16], the nonlinear perturbations of (1) are related not only with the current state x(t) and the delayed state x(t-h(t)) but also with β(t), where β(t) is a continuous function satisfying ∫0+∞β2(s)e2αsds<+∞ and α is a positive constant; new sufficient conditions are obtained that ensure the stability of a closed-loop system, which extend and improve the main results of [16]. Furthermore, the stabilization conditions are shown to be less conservative than those reported in Zhang et al. [12] when there are no nonlinear perturbations in the system. Finally, two numerical examples are presented to demonstrate the effectiveness and advantages of the main results.Notation. Throughout the paper, Rn denotes the n-dimensional Euclidean space with vector norm ·, and Rn×m is the set of all n×m-dimensional real matrices. I denotes the identity matrix of appropriate dimensions, and the superscript “T” stands for matrix transposition. The notation P>0  (≥0) means that P is symmetric and positive (semipositive) definite. λmin(A) and λmax(A) denote the minimum and maximum eigenvalues of A, respectively. In addition, in symmetric block matrices or long matrix expressions, we use an asterisk (∗) to represent a term that is induced by symmetry.


## 2. Problem Description and Preliminaries

Consider the following system with a nonlinear perturbation:(1)x˙t=Axt+Dxt-ht+ft,xt,xt-ht+But,t≥0,xt=ϕt,t∈-h2,0,where x(t)∈Rn is the state vector, u(t)∈Rm is the control input vector, A,D∈Rn×n, and B∈Rn×m, ϕ(t)∈C([-h2,0],Rn) with ϕ=supt∈[-h2,0]{ϕ(t),ϕ˙(t)}, where C([-h2,0],Rn) is the Banach space of continuous functions. The delay h(t) is time-varying and satisfies (2)0≤h1≤ht≤h2,h˙t≤μ,where μ is a positive constant and h1 and h2 are constants representing the lower and upper bounds of the delay, respectively. f(t,x(t),x(t-h(t))) is a nonlinear perturbation satisfying (3)ft,xt,xt-ht≤axt+bxt-ht+βt,where a and b are positive scalars and β(t) satisfies ∫0+∞β2(s)e2αsds<+∞.Remark 1.
In [3, 8, 16], the authors assumed that the nonlinear terms satisfy (4)fTt,xtft,xt≤a2xTtFTFxt,gTt,xt-htgt,xt-ht≤b2xTt-htGTGxt-ht,where F and G are constant matrices and a and b are positive scalars. It is obvious that the assumptions on the nonlinear terms given in (2) and (3) are more general.The following definitions and lemmas will be used for providing the main results in the sequel.Definition 2.
Given a scalarα>0: system (1) with u(t)=0 is α-stable if there exists a positive number γ>0 such that every solution x(t,ϕ) of the system satisfies (5)xt,ϕ≤γe-αt,t≥0.Definition 3.
Given a scalarα>0: system (1) is α-stabilizable if there exists a state feedback control u(t)=Kx(t) such that the closed-loop system (6)x˙t=A+BKxt+Dxt-ht+ft,xt,xt-ht,t≥0,xt=ϕt,t∈-h2,0is α-stable.Lemma 4 (see [25]).
For anyx,y∈Rn and a positive symmetric definite matrix N∈Rn×n, (7)±2yTx≤xTN-1x+yTNy.Lemma 5 (lower bound lemma for reciprocal convexity; see [20]).
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 (8)minαi∣αi>0,∑iαi=1⁡∑i1αifit=∑ifit+maxgi,jt⁡∑i≠jgi,jtsubject to (9)gi,j:Rm⟶R,gj,it≜gi,jt,fitgi,jtgi,jtfjt≥0.


## 3. Main Results

We use the following notation for the convenience:(10)h¯=h1+h22,δ=h2-h12,λ=λminP-1,Γ=λmaxP-1+h1λmaxP-1Q1P-1+h2-h1λmaxP-1Q2P-1+h2-h1λmaxP-1R11P-1P-1R12P-1∗P-1R22P-1+3h1+h2h2-h1216λmaxP-1S1P-1+h1+3h2h2-h1216λmaxP-1S2P-1+h132λmaxP-1S3P-1,Λ=Γϕ2+6∫0+∞β2se2αsds,ξTt=xTtxTt-htxTt-h1xTt-h¯xTt-h2x˙Tt.The following theorem presents an exponential stabilization condition for (1).Theorem 6.
Letα>0 and assume that conditions (2) and (3) are satisfied. If there exist matrices P>0, Q1>0, Q2>0, Si>0(i=1,2,3), R11R12∗R22>0, M1, M2, and Y such that the following LMIs hold (11)Σ1=Σ11DPΣ1300PAT+YTBTa2P0∗Σ22Σ23Σ240PDT0b2P∗∗Σ33Σ340000∗∗∗Σ44Σ45000∗∗∗∗Σ55000∗∗∗∗∗Σ6600∗∗∗∗∗∗-16a2I0∗∗∗∗∗∗0-16b2I<0,(12)Σ2=Σ11DPΣ1300PAT+YTBTa2P0∗Σ¯220Σ¯24Σ¯25PDT0b2P∗∗Σ33Σ¯340000∗∗∗Σ44Σ¯45000∗∗∗∗Σ55000∗∗∗∗∗Σ6600∗∗∗∗∗∗-16a2I0∗∗∗∗∗∗0-16b2I<0,(13)SiMi∗Si≥0i=1,2,where (14)Σ11=AP+PAT+BY+YTBT+2αP+I+Q1-e-2αh1S3,Σ22=-1-μe-2αh2Q2+e-2αh¯M1+e-2αh¯M1T-2e-2αh¯S1,Σ33=e-2αh1Q2+e-2αh1R11-e-2αh1Q1-e-2αh¯S1-e-2αh1S3,Σ44=e-2αh1R22-e-2αh¯R11-e-2αh¯S1-e-2αh2S2,Σ55=-e-2αh¯R22-e-2αh2S2,Σ66=h12S3+δ2S2+δ2S1+I-2P,Σ13=e-2αh1S3,Σ23=e-2αh¯S1-M1,Σ24=e-2αh¯S1-M1T,Σ34=e-2αh1R12+e-2αh¯M1T,Σ45=-e-2αh¯R12+e-2αh2S2,Σ¯22=-1-μe-2αh2Q2+e-2αh2M2+e-2αh2M2T-2e-2αh2S2,Σ¯24=e-2αh2S2-M2,Σ¯25=e-2αh2S2-M2T,Σ¯34=e-2αh1R12+e-2αh¯S1,Σ¯45=-e-2αh¯R12+e-2αh2M2T,then system (1) is robustly α-stabilizable, the state feedback control u(t)=YP-1x(t), and the solution x(t,ϕ) of the closed-loop system satisfies (15)xt,ϕ≤Λλe-αt,t≥0.Proof.
Let us denote(16)Q¯i=P-1QiP-1,M¯i=P-1MiP-1,i=1,2;S¯i=P-1SiP-1,i=1,2,3;R¯11=P-1R11P-1,R¯12=P-1R12P-1,R¯22=P-1R22P-1.Define a Lyapunov functional by (17)Vxt=V1xt+V2xt+V3xt,where (18)V1xt=xTtP-1xt,V2xt=∫t-h1te2αs-txTsQ¯1xsds+∫t-htt-h1e2αs-txTsQ¯2xsds+∫t-h¯t-h1e2αs-txsxs-δT·R¯11R¯12∗R¯22xsxs-δds,V3xt=δ∫-h¯-h1∫t+ste2ατ-tx˙TτS¯1x˙τdτds+δ∫-h2-h¯∫t+ste2ατ-tx˙TτS¯2x˙τdτds+h1∫-h10∫t+ste2ατ-tx˙TτS¯3x˙τdτds.Calculating the time derivative of V(xt) along the trajectories of (6), we conclude that (19)V˙1xt=xTtP-1A+BK+A+BKTP-1xt+2xTtP-1Dxt-ht+2xTtP-1ft,xt,xt-ht,(20)V˙2xt≤-2αV2xt+xTtQ¯1xt-1-μe-2αh2xTt-htQ¯2xt-ht+e-2αh1xTt-h1Q¯2-Q¯1xt-h1+e-2αh1xt-h1xt-h¯TR¯11R¯12∗R¯22xt-h1xt-h¯-e-2αh¯xt-h¯xt-h2TR¯11R¯12∗R¯22xt-h¯xt-h2,(21)V˙3xt≤-2αV3xt+x˙Tth12S¯3+δ2S¯2+δ2S¯1x˙t-δe-2αh2∫t-h2t-h¯x˙TsS¯2x˙sds-δe-2αh¯∫t-h¯t-h1x˙TsS¯1x˙sds-h1e-2αh1∫t-h1tx˙TsS¯3x˙sds.By virtue of Lemma 4, we have (22)2xTtP-1ft,xt,xt-ht≤xTtP-1P-1xt+fTt,xt,xt-htft,xt,xt-ht=xTtP-1P-1xt+ft,xt,xt-ht2.Using (3) and the inequality (H+J+L)2≤3H2+3J2+3L2, where H, J, and L are constants, we obtain (23)ft,xt,xt-ht2≤axt+bxt-ht+βt2≤3a2xt2+3b2xt-ht2+3β2t.It follows from (22) and (23) that (24)2xTtP-1ft,xt,xt-ht≤xTtP-1P-1+3a2Ixt+3b2xTt-htxt-ht+3β2t.Combining (19) and (24), we get (25)V˙1xt≤-2αV1xt+xTtP-1A+BK+A+BKTP-1+2αP-1+P-1P-1+3a2Ixt+2xTtP-1Dxt-ht+3b2xTt-htxt-ht+3β2t.Now, we estimate the upper bounds of the last three integral terms in inequality (21) as follows.
(i) Assume first thath1≤h(t)≤h¯. From Jensen’s inequality [19] and Lemma 5, (26)-δ∫t-h¯t-h1x˙TsS¯1x˙sds=-δ∫t-h¯t-htx˙TsS¯1x˙sds-δ∫t-htt-h1x˙TsS¯1x˙sds≤-δh¯-ht∫t-h¯t-htx˙sdsT·S¯1∫t-h¯t-htx˙sds-δht-h1∫t-htt-h1x˙sdsT·S¯1∫t-htt-h1x˙sds≤-xt-ht-xt-h¯xt-h1-xt-htT·S¯1M¯1∗S¯1xt-ht-xt-h¯xt-h1-xt-ht,and so (27)-δe-2αh¯∫t-h¯t-h1x˙TsS¯1x˙sds≤-xt-ht-xt-h¯xt-h1-xt-htT·e-2αh¯S¯1e-2αh¯M¯1∗e-2αh¯S¯1xt-ht-xt-h¯xt-h1-xt-ht.On the other hand, if h(t)=h¯ or h(t)=h1, then ∫t-h¯t-h(t)x˙(s)ds=0 or ∫t-h(t)t-h1x˙(s)ds=0, respectively. Hence, inequality (27) holds.
Using Jensen’s inequality [19], it is not difficult to arrive at the inequalities (28)-h1e-2αh1∫t-h1tx˙TsS¯3x˙sds≤-xt-xt-h1T·e-2αh1S¯3xt-xt-h1,(29)-δe-2αh2∫t-h2t-h¯x˙TsS¯2x˙sds≤-xt-h¯-xt-h2T·e-2αh2S¯2xt-h¯-xt-h2.It follows now from (6) that (30)2x˙TtP-1A+BKxt+Dxt-ht+ft,xt,xt-ht-x˙t=0.On the other hand, (31)2x˙TtP-1ft,xt,xt-ht≤x˙TtP-1P-1x˙t+fTt,xt,xt-htft,xt,xt-ht=x˙TtP-1P-1x˙t+ft,xt,xt-ht2≤x˙TtP-1P-1x˙t+3a2xTtxt+3b2xTt-htxt-ht+3β2t.Therefore, formulas (19)–(31) imply that (32)V˙xt+2αVxt≤ξTtΦ1ξt+6β2t,where (33)Φ1=Φ11P-1DΦ1300A+BKTP-1∗Φ22Φ23Φ240DTP-1∗∗Φ33Φ3400∗∗∗Φ44Φ450∗∗∗∗Φ550∗∗∗∗∗Φ66,Φ11=P-1A+BK+A+BKTP-1+2αP-1+P-1P-1+Q¯1-e-2αh1S¯3+6a2I,Φ22=-1-μe-2αh2Q¯2+e-2αh¯M¯1+e-2αh¯M¯1T-2e-2αh¯S¯1+6b2I,Φ33=e-2αh1Q¯2+e-2αh1R¯11-e-2αh1Q¯1-e-2αh¯S¯1-e-2αh1S¯3,Φ44=e-2αh1R¯22-e-2αh¯R¯11-e-2αh¯S¯1-e-2αh2S¯2,Φ55=-e-2αh¯R¯22-e-2αh2S¯2,Φ66=h12S¯3+δ2S¯2+δ2S¯1+P-1P-1-2P-1,Φ13=e-2αh1S¯3,Φ23=e-2αh¯S¯1-M¯1,Φ24=e-2αh¯S¯1-M¯1T,Φ34=e-2αh1R¯12+e-2αh¯M¯1T,Φ45=-e-2αh¯R¯12+e-2αh2S¯2.If we pre- and postmultiply Φ1 by diag{P,P,P,P,P,P} and let (34)K=YP-1,then the condition Φ1<0 is equivalent to condition (11) by using Schur Complement Lemma.
(ii) Assume now thath¯≤h(t)≤h2. Applications of Jensen’s inequality [19] and Lemma 5 yield (35)-δ∫t-h2t-h¯x˙TsS¯2x˙sds=-δ∫t-h2t-htx˙TsS¯2x˙sds+∫t-htt-h¯x˙TsS¯2x˙sds≤-δh2-ht∫t-h2t-htx˙sdsT·S¯2∫t-h2t-htx˙sds-δht-h¯∫t-htt-h¯x˙sdsTS¯2∫t-htt-h¯x˙sds≤-xt-ht-xt-h2xt-h¯-xt-htT·S¯2M¯2∗S¯2xt-ht-xt-h2xt-h¯-xt-ht,and hence (36)-δe-2αh2∫t-h2t-h¯x˙TsS¯2x˙sds≤-xt-ht-xt-h2xt-h¯-xt-htT·e-2αh2S¯2e-2αh2M¯2∗e-2αh2S¯2xt-ht-xt-h2xt-h¯-xt-ht.Note that when h(t)=h¯ or h(t)=h2, one can obtain ∫t-h¯t-h(t)x˙(s)ds=0 or ∫t-h(t)t-h2x˙(s)ds=0, respectively. Therefore, inequality (36) is satisfied.
Using Jensen’s inequality [19], we deduce that (37)-δe-2αh¯∫t-h¯t-h1x˙TsS¯1x˙sds≤-xt-h1-xt-h¯T·e-2αh¯S¯1xt-h1-xt-h¯.Combining (19)–(25), (28), and (30)–(37), we obtain (38)V˙xt+2αVxt≤ξTtΦ2ξt+6β2t,where (39)Φ2=Φ11P-1DΦ1300A+BKTP-1∗Φ¯220Φ¯24Φ¯25DTP-1∗∗Φ33Φ¯3400∗∗∗Φ44Φ¯450∗∗∗∗Φ550∗∗∗∗∗Φ66,Φ¯22=-1-μe-2αh2Q¯2+e-2αh2M¯2+e-2αh2M¯2T-2e-2αh2S¯2+6b2I,Φ¯24=e-2αh2S¯2-M¯2,Φ¯25=e-2αh2S¯2-M¯2T,Φ¯34=e-2αh1R¯12+e-2αh¯S¯1,Φ¯45=-e-2αh¯R¯12+e-2αh2M¯2T.Pre- and postmultiplying Φ2 by diag{P,P,P,P,P,P} and letting K=YP-1, the condition Φ2<0 is equivalent to condition (12) by using Schur Complement Lemma.
From the above discussion, if conditions (11)–(13) are satisfied, then (40)V˙xt+2αVxt≤6β2t,t≥0.By virtue of (40) and the definition of V(x(t)), (41)λxt,ϕ2≤Vxt≤e-2αtVx0+6∫0tβ2se∫0s2αdτds≤e-2αtΓϕ2+6∫0+∞β2se2αsds=Λe-2αt.Hence, we have (42)xt,ϕ≤Λλe-αt,t≥0,which implies that the closed-loop system is α-stable. The proof is completed.If there is no perturbation in system (1), that is, f(t,x(t),x(t-h(t)))=0, then system (1) reduces to (43)x˙t=Axt+Dxt-ht+But,t≥0,xt=ϕt,t∈-h2,0.Application of Theorem 6 yields the following result.Corollary 7.
Assume thatα>0 and condition (2) is satisfied. If there exist matrices P>0, Q1>0, Q2>0, Si>0(i=1,2,3), R11R12∗R22>0, M1, M2, and Y such that (44)Ξ1=Ξ11DPΞ1300PAT+YTBT∗Ξ22Ξ23Ξ240PDT∗∗Ξ33Ξ3400∗∗∗Ξ44Ξ450∗∗∗∗Ξ550∗∗∗∗∗Ξ66<0,Ξ2=Ξ11DPΞ1300PAT+YTBT∗Ξ¯220Ξ¯24Ξ¯25PDT∗∗Ξ33Ξ¯3400∗∗∗Ξ44Ξ¯450∗∗∗∗Ξ550∗∗∗∗∗Ξ66<0,SiMi∗Si≥0i=1,2,where (45)Ξ11=AP+PAT+BY+YTBT+2αP+Q1-e-2αh1S3,Ξ22=-1-μe-2αh2Q2+e-2αh¯M1+e-2αh¯M1T-2e-2αh¯S1,Ξ33=e-2αh1Q2+e-2αh1R11-e-2αh1Q1-e-2αh¯S1-e-2αh1S3,Ξ44=e-2αh1R22-e-2αh¯R11-e-2αh¯S1-e-2αh2S2,Ξ55=-e-2αh¯R22-e-2αh2S2,Ξ66=h12S3+δ2S2+δ2S1-2P,Ξ13=e-2αh1S3,Ξ23=e-2αh¯S1-M1,Ξ24=e-2αh¯S1-M1T,Ξ34=e-2αh1R12+e-2αh¯M1T,Ξ45=-e-2αh¯R12+e-2αh2S2,Ξ¯22=-1-μe-2αh2Q2+e-2αh2M2+e-2αh2M2T-2e-2αh2S2,Ξ¯24=e-2αh2S2-M2,Ξ¯25=e-2αh2S2-M2T,Ξ¯34=e-2αh1R12+e-2αh¯S1,Ξ¯45=-e-2αh¯R12+e-2αh2M2T,then system (43) is robustly α-stabilizable, the state feedback control u(t)=YP-1x(t), and the solution x(t,ϕ) of the closed-loop system satisfies (46)xt,ϕ≤Λλe-αt=Γλe-αtϕ,t≥0.


## 4. Numerical Examples

In this section, two numerical examples are given to illustrate the effectiveness of the results obtained in this paper.Example 1.
Consider the system with a nonlinear perturbation(47)x˙t=Axt+Dxt-ht+ft,xt,xt-ht+But,t≥0,xt=11T,t∈-0.4,0,where (48)A=100-1,B=01,D=-2-0.101.1,h(t)=0.1+0.3sin2(5t/3), and the nonlinear perturbation satisfies (49)ft,xt,xt-ht≤0.1xt+0.1xt-ht+e-0.66t.Note that h1=0.1, h2=0.4, μ=0.5, and β(t)=e-0.66t. It is not difficult to check that A and A+D are Hurwitz unstable.
Letα=0.16. Then ∫0+∞β2(s)e2αsds=∫0+∞e-sds=1<+∞. Using the LMI Toolbox in MATLAB, LMIs (11)–(13) in Theorem 6 are satisfied with (50)P=4.4365-0.1306-0.13064.9389,Q1=0.15100.40680.40683.4481,Q2=0.01150.00660.00660.2019,S1=62.2011-5.1166-5.116652.7981,S2=36.2972-1.8408-1.840831.9872,S3=71.6879-4.4063-4.406340.8855,R11=12.6433-1.6878-1.687821.3645,R12=-12.67692.21442.2144-18.8952,R22=13.2723-1.3613-1.361321.8765,M1=11.3018-6.7305-6.730520.9430,M2=-14.5051-2.3510-2.35100.3503,Y=0.2410-6.8850.Furthermore, the solution x(t,ϕ) of the system satisfies (51)xt,ϕ≤6.1258e-0.16t,t≥0,and the stabilizing feedback control (52)ut=0.0133-1.3937xt,t≥0.Observe that the results reported in [3, 8, 16] cannot be applied to (47) since the nonlinear perturbation is related with the term β(t)=e-0.66t.Example 2.
Consider a linear system with an interval time-varying delay(53)x˙t=Axt+Dxt-ht+But,t≥0,xt=ϕt,∀t∈-h2,0,where (54)A=0001,B=01,D=-1-10-0.9,and h(t)=0.5+1.28sin2(25t/64). Note that h1=0.5,h2=1.78, and μ=0.5. It is easy to check that A and A+D are Hurwitz unstable. Given α=0.01, using the LMI Toolbox in MATLAB, LMIs (44) in Corollary 7 are satisfied with (55)P=84.8721-32.8554-32.855418.7095,Q1=0.0864-0.0424-0.042411.7837,Q2=0.20990.20350.203511.4665,S1=109.3732-42.3701-42.370116.5117,S2=102.7380-39.7555-39.755515.5155,S3=124.7698-48.3825-48.382518.8240,R11=32.4350-12.8325-12.83255.1838,R12=-2.31960.93310.9331-0.3969,R22=54.1560-21.0105-21.01058.2532,M1=-22.97339.48819.4881-3.9352,M2=-53.687720.876420.8764-8.2092,Y=23.8997-27.0789.Moreover, the solution x(t,ϕ) of the system satisfies (56)xt,ϕ≤9.3399e-0.01tϕ,t≥0,and the stabilizing feedback control (57)ut=-0.8704-2.9758xt,t≥0.Figure1 shows the trajectories of x1(t) and x2(t) of the open-loop system with the initial condition ϕ(t)=[3010]T, t∈[-1.78,0]. Figure 2 shows the trajectories of x1(t) and x2(t) of the closed-loop system with the state feedback ut=[-0.8704-2.9758]x(t) and the initial condition ϕ(t)=[3010]T, t∈[-1.78,0].Figure 1
Open-loop trajectories ofx1(t) and x2(t).Figure 2
Closed-loop trajectories ofx1(t) and x2(t).Letting the lower and upper bounds of the time delay be the same as in Zhang et al. [12], our results also ensure exponential stability with an α-convergence rate as given in Table 1. Note that Zhang et al. [12] discussed asymptotic stability, whereas the controller derived in this paper provides exponential stability for the closed-loop system. Furthermore, the maximum bound for α is better than Thuan et al. [16] by letting h1 and h2 be the same as in [16]. For selected h1 and μ=0.5, using Corollary 7, one can easily observe that the maximum allowable delay bounds for h2 are better than those reported in the papers by Zhang et al. [12] and Thuan et al. [16].Table 1
Admissible upper boundh2 and K for given h1 with μ=0.5.
Method
h
1
h
2
K
Zhang et al. [12]
0.5
0.967
-
0.5290
-
2.7166
Thuan et al. [16] (α=0.314)
0.5
0.967
-
0.0967
-
1.8338
Corollary7 (α=0.353)
0.5
0.967
-
0.1662
-
2.0605
Thuan et al. [16] (α=0.01)
0.5
1.66
-
0.8765
-
2.9873
Corollary7 (α=0.01)
0.5
1.78
-
0.8704
-
2.9758
Zhang et al. [12]
1.0
1.114
-
0.3096
-
2.1894
Thuan et al. [16] (α=0.225)
1.0
1.114
-
0.6344
-
2.9059
Corollary7 (α=0.236)
1.0
1.114
-
0.6714
-
2.9317
Thuan et al. [16] (α=0.01)
1.0
1.672
-
0.9023
-
2.9983
Corollary7 (α=0.01)
1.0
1.765
-
0.8701
-
2.9712


## 5. Conclusions

In this paper, exponential stabilization of a class of time-varying delay systems with nonlinear perturbations has been investigated. By using the delay decomposition approach and constructing a novel Lyapunov functional, some new delay-dependent stabilization criteria are obtained in order to ensure closed-loop stability of the system with any prescribedα-convergence rate. Numerical examples are given to illustrate that the results obtained are much less conservative than some existing results in the literature. Exponential stabilization of impulsive switched delay systems with nonlinear perturbations will be further investigated in the future.



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