# H

Authors: C. Emharuethai; P. Niamsup
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

H
∞ control problem for nonlinear system
with time-varying delay is considered by using a set of improved
Lyapunov-Krasovskii functionals including some integral terms, and a matrix-based on quadratic convex, combined with Wirtinger's inequalities and some useful integral inequality. H∞ controller is designed via memoryless state feedback control and new
sufficient conditions for the existence of the H∞ state feedback for the system are given in terms of linear matrix
inequalities (LMIs). Numerical examples are given to illustrate the
effectiveness of the obtained
result.


---

## Body

## 1. Introduction

The phenomena of time delays are often encountered in many practical systems such as process control systems, manufacturing systems, networked control systems, and economic systems. The existence of these delays may be the source of instability and serious deterioration in the performance of the closed-loop systems. In real world systems especially, the delay should be assumed to be time-varying satisfying0≤τ1≤τ(t)≤τ2 and τ1 is not necessarily restricted to be 0, namely, interval time-varying delay. Stability analysis of time-delay system has been investigated extensively in the past decades [1–25].As of time delays, it is well known that the nonlinear perturbations can also cause instability and poor performance of practical systems. Therefore, the stability problem of time-delay systems with nonlinear perturbations has received increasing attention; see [9, 13, 16] and the references cited therein.H
∞ control problem has been widely used to minimize the effects of the external disturbances. The purpose of the problem is to design an H∞ controller to robustly stabilize the systems while guaranteeing a prescribed level of disturbance attenuation γ in the H∞ sense for the systems with external disturbances. A delay-dependent H∞ controller ensures asymptotic stability and a prescribed H∞ performance level of the closed-loop systems. The H∞ performance indexes and the upper bound of the delay are usually two performance indexes to be used to evaluate the conservatism of the derived condition. The conservatism of the delay-dependent H∞ control is measured by the allowable delay size or H∞ performance level bound obtained.Recently, an improved robust stability andH∞ performance analysis criterion has been reported [3–5, 7, 10–12, 15, 17, 18]. In [17], H∞ control problem for uncertain linear system with state delay and parameter uncertainties has been studied, but the time-varying delay is only bounded above by a constant. H∞ control problem for system with interval time-varying delay has been considered in [7], by employing free weighting matrices approach. However, some useful terms in estimating the derivative of Lyapunov-Krasovskii functional are ignored which might lead to some conservatism. In [15], H∞ control problem for nonlinear systems with interval time-varying delay has been studied by using Jensen’s inequality to estimate some integral terms of Lyapunov-Krasovskii functional and deriving delay-dependent sufficient condition for the existence of H∞ control by using reciprocally convex combination technique.In the study of time-delay system, several approaches have been proposed in order to reduce conservatism. For example, it is well known that choosing appropriate Lyapunov-Krasovskii functional and using improved bounding techniques to estimate time-derivative of Lyapunov-Krasovskii functional lead to improvement of stability region. Furthermore, free weighting matrices approach; delay decomposition approach; and convex optimization and reciprocally convex optimization techniques have been widely used to reduce conservatism of stability criterion; see [3, 7, 8, 10, 11, 13, 14, 16]. Recently, the so-called matrix-based quadratic convex approach has been introduced to derive stability criterion for time-delay system which was shown to reduce conservatism; for example, it gives better maximum allowable upper bound for time-varying delay than some other existing approaches; see [20–22].Motivated by the above discussions, in this paper, matrix-based quadratic convex approach will be used to studyH∞ control problem for system with interval time-varying delay and nonlinear perturbations. To the best of our knowledge, this is one of the first reports of such investigation. By introducing new augmented Lyapunov-Krasovskii functional which has not been considered yet in stability analysis of H∞ control problem, a delay-dependent stability criterion and H∞ performance analysis are derived in terms of linear matrix inequalities (LMIs). This new Lyapunov-Krasovskii functional consists of integral terms of the form ∫t-ht‍(h-t-s)jx˙T(s)Rjx˙(s)ds  (j=1,2) which allows us to use the matrix-based quadratic convex approach introduced in [20–22]. With the use of this new Lyapunov-Krasovskii functional, matrix-based quadratic convex approach combined with some improved bounding techniques for integral terms such as Wirtinger-based integral inequality [14, 20], some new cross terms will be introduced which enhance the feasible stability criterion. Through two numerical examples, it is shown that the obtained stability criterion may give a larger maximum allowable upper bound of time-varying delay than some existing results.


## 2. Mathematical Model and Preliminaries

The following notations will be used in this paper:R+ denotes the set of all nonnegative real numbers; Rn denotes the n-dimensional space with the Euclidean norm ·; Mn×r denotes the space of all matrices of (n×r)-dimensions.A
T denotes the transpose of matrix A; A is symmetric if A=AT; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λmax⁡(A)=max⁡⁡{Reλ;λ∈λ(A)}.x
t
:
=
x
t
+
s
:
s
∈
[
-
h
,
t
0
], xt=sups∈[-h,t0]x(t+s); C([t0,t],Rn) denotes the set of all Rn-valued continuous functions on [t0,t]; L2([t0,t],Rm) denotes the set of all the Rm-valued square integrable functions on [t0,t].MatrixA is called semipositive definite (A≥0) if 〈Ax,x〉≥0, for all x∈Rn; A is positive definite (A>0) if 〈Ax,x〉>0 for all x≠0; A>B means A-B>0. The symmetric term in a matrix is denoted by ∗.Consider the following system with time-varying delays and control input:(1)x˙t=Axt+Dxt-τt+But+Cwt+ft,xt,xt-τt,ut,wt,zt=Ext+Gxt-τt+Fut+gt,xt,xt-τt,ut,(2)xt0+θ=ϕθ,θ∈-τ2,0,t0,ϕ∈R+×C-τ2,t0,Rn,where x(t)∈Rn is the state; u(t)∈Rm is the control input; w(t)∈L2([0,∞],Rr) is a disturbance input; and z(t)∈Rs is the observation output. The delay τ(t) is time-varying continuous function which satisfies (3)0≤τ1≤τt≤τ2,μ1≤τ˙t≤μ2.Let xτ=x(t-τ(t)). The nonlinear functions f(t,x,xτ,u,w):R+×Rn×Rn×Rm×Rr→Rn, g(t,x,xτ,u):R+×Rn×Rn×Rm→Rs satisfy the following growth condition:(4)∃a,b,c,d>0:ft,x,xτ,u,w≤ax+bxτ+cu+dw,xxxxxxxxxxxxxxx∀x,xτ,u,w,∃a1,b1,c1>0:gt,x,xτ,u2≤a1x2+b1xτ2+c1u2,xxxxxxxxxxxxxxx∀x,xτ,u.Definition 1.
Givenγ>0, the H∞ control problem for system (1) is to seek if there exists a memoryless state feedback controller u(t)=Kx(t) such that we have the following.(i)
The zero solution of the closed-loop system, wherew(t)=0,(5)x˙=A+BKxt+Dxt-τt+ft,x,xτ,u,0,is asymptotically stable.(ii)
TheH∞ performance (6)z(t)2<γw(t)2of the closed-loop system (5) is guaranteed for all nonzero w(t)∈L2([0,∞],Rr) and a prescribed γ>0 under the condition x(t)=0,∀t∈[-τ2,t0]. In this case, we say that the feedback control u(t)=Kx(t) asymptotically stabilizes the system.We introduce the following technical lemmas, which will be used in the proof of our results.Lemma 2 (see [14]).
For a given matrixR>0, the following inequality holds for any continuously differentiable function w:[a,b]→Rn:(7)∫ab‍ω˙TuRω˙udu≥1b-aΓ1TRΓ1+3Γ2TRΓ2,where (8)Γ1≔ωb-ωaΓ2:=ωb+ωa-2b-a∫ab‍ωudu.Remark 3.
It is clear to see that the inequality in Lemma2 provides a tighter lower bound for ∫ab‍ω˙T(u)Rω˙(u)du than Jensen’s inequality since 3Γ2TRΓ2>0 for Γ2≠0. Thus, inequality (7) is an improvement over Jensen’s inequality.Before we introduce some useful integral inequalities, we denote(9)ν1t:=1τ2-τt∫t-τ2t-τt‍ysds,ν2t:=1τt-τ1∫t-τtt-τ1‍ysds,ν3t:=1τ1∫t-τ1t‍ysds.Lemma 4 (see [20]).
For a given scalarτ1≥0 and any n×n real matrices Y1>0 and Y2>0 and a vector y˙:[-τ1,0]→Rn such that the integration concerned below is well defined, the following inequality holds for any vector-valued function π1(t):[0,∞)→Rk and matrices M1∈Rk×k and N1∈Rk×n satisfying M1N1N1TY1≥0, (10)℘1:=∫t-τ1t‍τ1-t+sy˙TsY1y˙sds≥-τ122π1TtM1π1t-2τ1π1TN1yt-ν3t,℘2:=∫t-τ1t‍τ1-t+s2y˙TsY2y˙sds≥τ1yt-ν3tTY2yt-ν3t,where ν3(t) is defined in (9).Lemma 5 (see [21]).
Letτ(t) be a continuous function satisfying 0≤τ1≤τ(t)≤τ2. For any n×n real matrix R2>0 and a vector y˙:[-τ2,0]→Rn such that the integration concerned below is well defined, the following inequality holds for any ϕi1∈Rq and real matrices Zi∈Rq×q, Fi∈Rq×n satisfying ZiFiFiTR2≥0, (i=1,2),(11)-∫t-τ2t-τ1‍τ2-t+sy˙TsR2y˙sds≤12τ2-τt2ϕ11TZ1ϕ11+2τ2-τtϕ11TF1ϕ12+12τ2-τ12-τ2-τt2ϕ21TZ2ϕ21+2ϕ21TF2τ2-τtϕ22+τt-τ1ϕ23,where(12)ϕ12:=yt-τt-ν1t,ϕ22:=yt-τ1-xt-τt,ϕ23:=yt-τ1-ν2t,with νi(t) (i=1,2) being defined in (9).Lemma 6 (see [20]).
Letτ(t) be a continuous function satisfying 0≤τ1≤τ(t)≤τ2. For any n×n real matrix R1>0 and a vector y˙:[-τ2,0]→Rn such that the integration concerned below is well defined, the following inequality holds for any 2n×2n real matrix S1 satisfying R~1S1S1TR~1≥0,(13)-τ2-τ1∫t-τ2t-τ1‍y˙TsR1y˙sds≤2ψ11TS1ψ21-ψ11TR~1ψ11-ψ21TR~1ψ21,where R~1:=diagR1,3R1 and(14)ψ11:=yt-τt-yt-τ2yt-τt+yt-τ2-2ν1t,ψ21:=yt-τ1-yt-τty(t-τ1)+yt-τt-2ν2t,with νi(t) (i=1,2) being defined in (9).Remark 7.
In Lemma6, when τ1=0, the inequality reduces to similar one in [14]. It contains slack matrix variable S1 with dimension 2n×2n comparing to slack matric variable with dimension 2n×5n introduced in [8].Lemma 8 (see [20]).
Letχ0, χ1, and χ2 be m×m real symmetric matrices and a continuous function τ satisfy τ1≤τ≤τ2, where τ1 and τ2 are constants satisfying 0≤τ1≤τ2. If χ0≥0, then(15)τ2χ0+τχ1+χ2<0≤0,∀τ∈τ1,τ2,⟺τi2χ0+τiχ1+χ2<0≤0,i=1,2,or(16)τ2χ0+τχ1+χ2>0≥0,∀τ∈τ1,τ2⟺τi2χ0+τiχ1+χ2>0≥0,i=1,2.


## 3. Main Results

In this section, we give a design of memorylessH∞ feedback control for system (1). First, we present delay-dependent asymptotical stabilizability analysis conditions for the nonlinear system with time-varying delay (1). Now, we operate the matrix-based quadratic convex approach with the integral inequalities in [20] to formulate a new stability criterion for system (1). For our goal, we choose the following Lyapunov-Krasovskii functional: (17)Vt,xt,x˙t=V1t+V2t+V3t,where xt denotes the function x(t) defined on the interval [t-τ2,t]. Set P1=P-1, y(t)=P1x(t), τ21:=τ2-τ1, and(18)V1t:=yTtPyt+∫t-τ1t‍y˙TsQ0y˙sds,V2t≔∫t-τ1t‍yTtyTsQ1yTtyTsTds+∫t-τ(t)t-τ1‍yTtyTsQ2yTtyTsTds+∫t-τ2t-τ(t)‍yTtyTsQ3yTtyTsTds,V3t≔∫t-τ1t‍τ1τ1-t+sy˙TsW1y˙sxxxx+τ1-t+s2y˙TsW2y˙sds+∫t-τ2t-τ1τ21τ2-t+sy˙TsR1y˙sxxxxxx+τ2-t+s2y˙TsR2y˙sds,where Qj>0, (j=0,1,2,3), W1>0, W2>0, R1>0, R2>0, and P are real matrices to be determined. Before introducing the main theorem, for simplicity, we set (19)ɛ=a+b+c+4d2γ.Theorem 9.
Givenγ>0, then system (1) is asymptotically stabilizable and satisfies z(t)2<γw(t)2 for all nonzero w∈L2[0,∞) if there exist positive definite matrices P, Qj>0, (j=0,1,2,3), W1, W2, R1, R2, S1, Z1, Z2, Z3, N1, N2, N3, and Y such that the following LMIs hold:(20)R~1S1S1TR~1≥0,ZiNiNiTR2≥0,i=1,2,Z3N3N3TW2≥0,Z1≥Z2,(21)Ξ2τ1,μ1+Ξ3τ1+Ξ^4<0,Ξ2τ1,μ2+Ξ3τ1+Ξ^4<0,Ξ2τ2,μ1+Ξ3τ2+Ξ^4<0,Ξ2τ2,μ2+Ξ3τ2+Ξ^4<0,where R~1=diag{R1,3R1} and(22)Ξ2τt,τ˙t:=Ξ20+τt-τ1Ξ21+τ2-τtΞ22,(23)Ξ3τt:=φ~1TS1φ~2+φ~2TS1Tφ~1-φ~1TR~1Tφ~1+τ2-τt2Z1-Z2+τ2-τtΞ31+τt-τ1Ξ32+τ212Z2-φ~2TR~1φ~2,(24)Ξ^4≔τ12Z3-φ~3TW~1φ~3+e9Tτ12W1+τ12W2e9+2τ1N3e1-e7+e8Tτ21TR1+τ21TR2e8+2τ1e1-e7TN3T+e1TAP+PAT+BY+YTBT+4γCCT+ɛIe1+e1TPDe2+e2TDTPe1+e1TAP+YBTe9+e9TPAT+YTBe1+e2TDPe9+e9TPDTe2+e8TQ0e8+e9T-2P+4γCCT+ɛI+Q0e9+e1TPe10+e10TPe1+e1TYTe11+e11TYe1+e1TPETe12+e12TEPe1+e2TPGTe13+e13TGPe2+e2TPe14+e14TPe2+e10T-12a+4a1Ie10+e11T-12+2c+4c1Ie11+e12T-I3e12+e13T-I3e13+e14T(-12b+4b1I)e14,with ei∈Rn×14n (i=1,2,…,14) denoting the ith row-block vector of the 14n×14n identity matrix W~1=diag{W1,3W1} and(25)Ξ20:=e1Te3TQ2-Q1e1Te3TT+τ1e9T0Q1e1Te7TT+τ1e1Te7TQ1e9T0T-1-τ˙te1Te2TQ2-Q3e1Te2TT-e1Te4TQ3e1Te4TT+e1Te1TQ1e1Te1TT,Ξ21:=e1Te6TQ2e9T0T+e9T0Q2e1Te6TT,Ξ22:=e1Te5TQ3e9T0T+e9T0Q3e1Te5TT,Ξ31:=2N1e2-e5+2N2e3-e2+2e3-e2TN2T+2e2-e5TN1T,Ξ32:=2N1e3-e6+2e3-e6TN1T,φ~1≔cole2-e4,e2+e4-2e5,φ~2≔cole3-e2,e3+e2-2e6,φ~3:=cole1-e3,e1+e3-2e7.Moreover, the feedback control is given by(26)ut=YP-1xt,t≥0.Proof.
Taking the derivative ofV along the solution of system (1), we obtain(27)V˙1=2yTtPy˙t+y˙TtQ0y˙t-y˙Tt-τ1Q0y˙t-τ1,(28)V˙2=yTtyTtQ1yTtyTtT-yTtyTt-τ1Q1yTtyTt-τ1T+∫t-τ1t‍2yTtyTsQ1y˙tT0Tds+yTtyTt-τ1Q2·yTtyTt-τ1T-1-τ˙tyTtyTt-τtQ2·yTtyTt-τtT+2∫t-τ(t)t-τ1yTtyTsQ2y˙Tt0Tds-yTtyTt-τ2Q3yTtyTt-τ2T+1-τ˙tyTtyTt-τtQ3·yTtyTt-τtT+∫t-τ2t-τ(t)‍2yTtyTsQ2y˙Tt0Tds,V˙3=τ1y˙Ttτ1W1y˙t+τ12y˙TtW2y˙t-∫t-τ1t‍y˙Tsτ1W1y˙sds-2∫t-τ1t‍τ1-t+sy˙TsW2y˙sds+τ21y˙Tt-τ1τ21R1y˙t-τ1+τ212y˙Tt-τ1R2y˙t-τ1-τ21∫t-τ2t-τ1‍y˙TsR1y˙sds-2∫t-τ2t-τ1‍τ2-t+sy˙TsR2y˙sds.From (4) and Cauchy inequality, we get the following inequalities:(29)2xTP1ft,x,xh,u,w≤2P1xft,x,xh,u,w≤2P1xax+bxh+cu+dw≤aP1x2+ax2+bP1x2+bP1xh2+cP1x2+cu2+4d2γP1x2+γ4w2=ax2+bxh2+cu2+γ4w2+ɛP1x2.Similarly, (30)2x˙TP1ft,x,xτ,u,w≤ax2+bxτ2+cu2+γ4w2+ɛP1x˙2,2xtTP1Cw≤γ4w2+4γxtTP1CCTP1xt,2x˙tTP1Cw≤γ4w2+4γx˙tTP1CCTP1x˙t.By using the following identity relation: (31)-2x˙TtP1x˙t-Axt-Dxt-τtf·-But-Cwt-f·=0,we obtain the following:(32)0=-2x˙TtP1x˙t-Axt-Dxt-τt-But-Cwt-f·≤-2x˙TtP1x˙t-Axt-Dxt-τt222222-BYP1xt+2x˙TtP1Cwt+2x˙TtP1f·≤-2y˙TtPy˙t+2y˙TtAPyt+2y˙TtDPyt-τt+2y˙TtBYyt+4γy˙tTCCTy˙t+aytTP2yt+byTt-τtP2yt-τt+cyTtYTYyt+γ2wtTwt+ɛy˙Tty˙t.From (27) and (29), (32), V˙1 is estimated as(33)V˙1≤yTtAP+PAT+BY+YTBT4γ+4γCCT+aP2+cYTY+ɛyt+γ2wt2+2yTtDPyTt-τt+byTt-τtP2y·t-τt-2y˙TtPy˙t+2y˙TtAPyt+2y˙TtDPyt-τt+2y˙TtBYyt+4γy˙tTCCTy˙t+aytTP2yt+byTt-τtP2yt-τt+cyTtYTYyt+γ2wtTwt+ɛy˙Tty˙t+y˙TtQ0y˙t-y˙Tt-τ1Q0y˙t-τ1=ξTtΞ1ξt,where(34)ξt:=colyt,yt-τt,yt-τ1,yt-τ2,t-τtν1t,ν2t,ν3t,y˙t-τ1,y˙t,Ξ1≔e1T4γAP+PAT+BY+YTBT+4γCCT+2aP2+2cYTY+ɛIe1+e1TPDe2+e2TDTPe1+e1TAP+YBTe9+e9TPAT+YTBe1+e2TDPe9+e9TPDTe2+e9T-2P+4γCCT+ɛI+Q0e9+e2T2bP2e2+e8T-Q0e8.With the consideration of the three terms of V˙2(t), we obtained the following inequality: (35)∫t-τ1t‍2yTtyTsQ1y˙tT0Tds≤2∫t-τ1t‍yTtds∫t-τ1t‍yTsdsQ1y˙Tt0T=2τ1yTtν3TQ1y˙Tt0T,∫t-τ(t)t-τ1‍2yTtyTsQ2y˙tT0Tds≤2∫t-τtt-τ1‍yTtds∫t-τtt-τ1‍yTsdsQ2y˙Tt0T=2τt-τ1yTtν2TQ2y˙Tt0T,∫t-τ2t-τ(t)‍2yTtyTsQ3y˙tT0Tds≤2∫t-τ2t-τt‍yTtds∫t-τ2t-dt‍yTsdsQ3y˙Tt0T=2τ2-τtyTtν1TQ3y˙Tt0.Therefore, the estimation of V˙2(t) is as follows:(36)V˙2t≤Ξ20+τt-τ1Ξ21+τ2-τtΞ22=ξTtΞ2τt,τ˙tξt,where Ξ2 is defined in (22). Similarly, V˙3(t) is estimated as (37)V˙3t=ξTtΞ30ξt+δ1t+δ2t,where(38)Ξ30:=e9Tτ12W1+τ12W2e9+e8Tτ212R1+τ212R2e8,δ1t=-τ21∫t-τ2t-τ1‍y˙TsR1y˙sds-2∫t-τ2t-τ1‍τ2-t+sy˙TsR2y˙sds,δ2t=-τ1∫t-τ1t‍y˙TsW1y˙sds-2∫t-τ1t‍τ1-t+sy˙TsW2y˙sds.By Lemmas 5 and 6, we obtain the following: (39)-τ2-τ1∫t-τ2t-τ1‍y˙TsR1y˙sds≤2ψ11TS1ψ21-ψ11TR~1ψ11-ψ21TR~1ψ21,where R~1:=diag{R1,3R1} and (40)ψ11:=yt-τt-yt-τ2yt-τt+yt-τ2-2ν1t,ψ21:=yt-τ1-yt-τtyt-τ1+yt-τt-2ν2t,-2∫t-τ2t-τ1‍τ2-t+sy˙TsR2y˙sds≤-212τ2-τt2ξTtZ1ξt+2τ2-τtξTtN1yt-τt-ν1+12τ212τ2-τt2ξTtZ2ξt+2ξTtN2·τ2-τtyt-τ1-yt-τt12+τt-τ1yt-τ1-ν2.Thus,(41)δ1t≤2ψ11TS1ψ21-ψ11TR~1ψ11-ψ21TR~1ψ21-212τ2-τt2ξTtZ1ξt+2τ2-τtξTtN1yt-τt-ν1+12τ212τ2-τt2ξTtZ2ξt+2ξTtN2·τ2-τtyt-τ1-yt-τt12+τt-τ1yt-τ1-ν2.=ξTtΞ3τtξt,where Ξ3(τ(t)) is given in (23). From Lemmas 2 and 4, we obtain (42)-τ1∫t-τ1t‍y˙TsW1y˙sds≤yt-yt-τ1TW1yt-yt-τ1+3Ω~1TW1Ω~1,-2∫t-τ1t‍τ1-t+sy˙TsW2y˙sds≤-τ12ξTtZ3ξt-2τ1ξTtN3yt-ν3,from which it follows that(43)δ2t≤yt-yt-τ1TW1yt-yt-τ1+3Ω~1TW1Ω~1-τ12ξTtZ3ξt-2τ1ξTtN3yt-ν3,=ξTtΞ33ξt,where (44)Ω~1=yt+yt-τ1-2ν3,Ω~2=yt-τt+yt+τ2-ν1,Ω~3=yt-τ1+yt-τt-ν2,Ξ33:=-φ~3TdiagW1,3W1φ~3+τ12Z3+2τ1N3e1-e7+2τ1e1-e7TN3T.Hence, from (41) and (43), we obtain(45)V˙3≤ξTtΞ3τt+Ξ4ξt,where Ξ4:=Ξ30+Ξ33. From (33), (36), and (45), we obtain V˙(t,yt,y˙t) along the solution of system (1) as (46)V˙t,xt≤ξTΔτt,τ˙tξt,where (47)Δτt,τ˙t=Ξ2τt,τ˙t+Ξ3τt+Ξ1+Ξ4.Therefore, we have(48)V˙t,xt≤ξTtΔτt,τ˙tξt+γwt2+yTt3PETEP+4a1P2+2+4c1YTYyt-yTt3PETEP+4a1P2+2+4c1YTYyt+yTt-τt3PGTGP+4b1P2yt-τt-yTt-τt3PGTGP+4b1P2yt-τt=ξTtΔ^τt,τ˙tξt+γwt2-yTt3PETEP+4a1P2+2+4c1YTYyt-yTt-τt3PGTGP+4b1P2yt-τt,where (49)Δ^τt,τ˙t=Ξ2τt,τ˙t+Ξ3τt+Ξ^1+Ξ4=Ξ2τt,τ˙t+Ξ3τt+Ξ^4,Ξ^1=Ξ1+e1T2+4c1YTY+3PETEP+4a1e1+e2T3PETEP+4b1P2e2,and Ξ^4=Ξ^1+Ξ4 is defined in (24). Observe that Δ^(τ(t),τ˙(t)) may be rewritten as(50)Δ^τt,τ˙t=τ2tΔ0+τtΔ1+Δ2,where Δ0=Z1-Z2 and Δ1, Δ2 are τ(t)-independent real matrices. By Lemma 8, if Z1-Z2≥0 and the inequalities in (21) hold, then Δ^(τ(t),τ˙(t))<0, ∀τ(t)∈[τ1,τ2], ∀τ˙(t)∈[μ1,μ2]. Moreover, Δ^(τ(t),τ˙(t)) may be rewritten as a convex combination of τ˙(t) as follows:(51)Δ^τt,τ˙t=1-τ˙t□0+τ˙t□1+□2,where □0=Q2-Q3 and □1, □2 are τ˙(t)-independent real matrices. By utilizing the Schur complement lemma, it follows from (21), (50), and (51) that Δ^(τ(t),τ˙(t))<0 holds, from which it follows from inequality (48) that(52)V˙t,xt≤γwtTwt-yTt3PETEP+4a1P2+2+4c1YTYyt-yTt-τt3PGTGP+4b1P2yt-τt.Letting w(t)=0 and from(53)-yTt3PETEP+4a1P2+2+4c1YTYyt≤0,-yTt-τt3PGTGP+4b1P2yt-τt≤0,there exists a scalar ε3>0 such that (54)V˙(t,xt)≤-ε3x(t)2<0,∀t≥0.Therefore, system (1) with w(t)≡0 is asymptotically stable. To complete the proof of theorem, next we consider the H∞ performance z2<γw2. By assuming that x(t)=0, t∈[-τ2,t0], it follows from definition of z(t) that(55)z2≤Ex2+Gxt-τt2+utT+2xTtETGxt-τt+2xTtETg·+2xTt-τtETg·+2uTtFTg·+g·2≤3Ext2+3Gxt-τt2+2ut2+4g·2≤xTt3ETE+4a1xt+xTt-τt·3GTG+4b1xt-τt+2+4c1ut2=yTt3PETEP+4a1P2+2+4c1YTYyt+yTt-τt3PGTGP+4b1P2yt-τt.From (52), we obtain(56)V˙t,xt≤γwtTwt-yTt3PETEP+4a1P2+2+4c1YTYyt-yTt-τt3PGTGP+4b1P2yt-τt.From estimations of V˙(t,xt) and z(t)2 in (48) and (55), we obtain (57)V˙t,xt+zTtzt-γ2wTtwt<0.Integrating both sides of the above equation from t0 to t, we get (58)∫t0t‍V˙t,xt+zTtzt-γ2wTtwtdt<0.It follows that(59)∫t0t‍zTtzt-γ2wTtwtdt≤Vt0,xt0-Vt,xt≤0.Therefore, under zero initial condition x(t)=0, t∈[-τ2,t0], by letting t→+∞ in (59), we get(60)∫t0∞‍zTtztdt<γ2∫t0∞‍wTtwtdt,which gives z2<γw2. This completes the proof.Whenτ1=0, B=0, and C=0, the nonlinear function is f(t,x,xτ), and (1) reduces to(61)x˙t=Axt+Dxt-τt+ft,xt,xt-τt,where 0≤τ(t)≤τ2, μ1≤τ˙(t)≤μ2. In the following, we present a stability criterion for the case when τ1=0. We consider the following Lyapunov-Krasovskii functional candidate:(62)V^t,xt,x˙t=xTtPxt+∫t-τ(t)t‍xTtxTsQ2xTtxTsTds+∫t-τ2t-τ(t)‍xTtxTsQ3xTtxTsTds+∫t-τ2t‍τ2τ2-t+sx˙TsR1x˙s+τ2-t+s2x˙TsR2x˙sds.Corollary 10.
For given scalarsτ2, μ1, μ2, a, and b, (61) is asymptotically stable if there exist symmetric positive definite matrices P, R1, R2, N1, N2, Z1, Z2, Q2, Q3, and S1, such that the following LMIs hold:(63)τ2Υ1+1-μ1Υ3+Υ4<0,τ2Υ1+1-μ2Υ3+Υ4<0,τ2Υ2+1-μ1Υ3+Υ4<0,τ2Υ2+1-μ2Υ3+Υ4<0.R~1S1S1TR~1≥0,Z1≥Z2,ZiNiNiTR2≥0,i=1,2,Qj=Qj1Qj2∗Qj3≥0,i=2,3,where R~1≜diag{R1,3R1}(64)Υ1:=e~1Te~4TQ3e~6T0T+e~6T0Q3e~1Te~4TT+τ2Z1-Z2+2N1e~2-e~4+2e~2-e~4TN1T+2N2e~1-e~2+2e~1-e~2TN2T,Υ2≔e~1Te~5TQ2e~6T0T+e~6T0Q2e~1Te~5TT+2N2e~1-e~5+2e~1-e~5TN2T,Υ3:=e~1e~2Q3-Q2e~1e~2T,Υ4:=e~1TATP+PAe~1+e~1Ta+bIe~1,+e~1TPDe~2+e~2TDTPe~1+e~1TPAe~6+e~6TATPe~1+e~1TPTe~7+e~7TPe~1+e~2TPDe~6+e~6TDTPe~2+e~2TPTe~8+e~8TPe~2+e~6T-2Pe~6+e~6Ta+bIe~6+e~7T-12aIe~7+e~8T-12bIe~8+e~1Te~1TQ2e~1Te~1TT-e~1Te~3TQ3e~1Te~3TT+τ22e~6TR1+R2e^6+Θ1TS1Θ2+Θ2TS1TΘ1-Θ1TR~1Θ1-Θ2TR~1Θ2+τ22Z2,Θ1:=cole~2-e~3,e~2+e~3-2e~4,Θ2:=cole~1-e~2,e~1+e~2-2e~5with e~1=I0000000,…,e~8=0000000I.Proof.
The proof is the same as in Theorem9 by using Lyapunov-Krasovskii functional (62). The proof is omitted.


## 4. Numerical Examples

In this section, we provide numerical examples to show the effectiveness of theoretical results.Example 1.
Consider the nonlinear system with interval time-varying delays (1) which was considered in [15], where(65)A=-1.30.30.50.1,D=-0.010.020.03-0.04,B=0.200.30,C=-0.020.010.02-0.03,E=G=0.06-0.06-0.080.08,F=0.800.60,f·=g·=0.01x12t+x22t-τtx22t+x12t-τt,a=b=c=d=a1=b1=c1=0.01, τ1=0.3, τ2=0.5, μ1=-0.1, μ2=0.1, and γ=4. By using LMI Toolbox in MATLAB, the LMIs in Theorem 9 are feasible. From (26) the H∞ controller feedback gain can be computed as(66)K=YP-1=-3.4638-6.8069-4.3243-4.1846.For simulation, we choose τ(t)=0.4+0.1cos⁡(t), ϕ(t)=[-5cos⁡(t),3cos⁡(t)], ∀t∈[0,10], and w(t) is the Gaussian noise which is the set of random numbers with fluctuation range between −1 and 1. Figure 1 shows the trajectories of solutions x1(t) and x2(t) of system (1) without feedback control (u(t)=0) and Figure 2 shows the trajectories of solutions x1(t) and x2(t) of the system with feedback control u(t). Moreover, in Tables 1 and 2, by using Theorem 9, we give the minimum allowable value γ with μ1=-0.1, μ2=0.1 and with μ1=0.05, μ2=0.1, for some given τ1 and τ2, respectively.Table 1
The minimum allowable value of disturbance attenuationγ with μ1=-0.1 and μ2=0.1.
a
=
b
=
c
=
d
=
a
1
=
b
1
=
c
1
=
0.01
Method
τ
1
τ
2
min
⁡
γ
Theorem9
0.1
0.3
0.2377
0.1
0.5
0.2474
a
=
b
=
c
=
d
=
a
1
=
b
1
=
c
1
=
0.05
Method
τ
1
τ
2
min
⁡
γ
Theorem9
0.1
0.3
0.8991
0.1
0.5
0.9643Table 2
The minimum allowable value of disturbance attenuationγ with μ1=0.05 and μ2=0.1.
a
=
b
=
c
=
d
=
a
1
=
b
1
=
c
1
=
0.01
Method
τ
1
τ
2
min
⁡
γ
Theorem9
0.1
0.5
0.2487Figure 1
The trajectories ofx1(t) and x2(t) of (1) in Example 1 without feedback control.Figure 2
The trajectories ofx1(t) and x2(t) of (1) in Example 1 with feedback control.Example 2.
Consider the following nonlinear system with interval time-varying delays which was considered in [18]: (67)x˙t=Axt+Dxt-τt+ft,xt,xt-τt,where(68)A=-200-1,D=-10-1-1,a=b=0.1, and 0≤τ(t)≤τ2, μ1≤τ˙(t)≤μ2. By using LMI Toolbox in MATLAB, the LMIs in Corollary 10 are feasible. Table 3 shows the maximum allowable upper bound τ2 with different values of μ1 and μ2.Table 3
The maximum allowable upper boundτ2 in Example 2 with τ1=0.
a
=
0.1
b
=
0.1
Method
μ
1
μ
2
max
⁡
τ
2
By Corollary10
−0.1
0.1
2.1443
−0.3
0.3
2.1126
−0.5
0.5
2.0833
−0.8
0.8
2.0473
−1
1
2.0332Example 3.
Consider the following nonlinear system with interval time-varying delays which was considered in [13]: (69)x˙t=Axt+Dxt-τt+ft,xt,xt-τt,where (70)A=-1.20.1-0.1-1,D=-0.60.7-1-0.8,a=b=0.1, and 0≤τ(t)≤τ2, μ1≤τ˙(t)≤μ2. By using LMI Toolbox in MATLAB, the LMIs in Corollary 10 are feasible. For comparison with other existing results, we now calculate the admissible maximum allowable upper bounds of τ2 by setting μ=μ2=-μ1 for various values of μ. Table 4 shows that the maximum allowable upper bounds of τ2 with given τ1=0 obtained from Corollary 10 are greater than the upper bounds obtained in [13, 23, 25].Table 4
Comparison of maximum allowable upper boundτ2 in Example 3 with τ1=0.
a
=
0.1
b
=
0.1
Method
μ
=
0.5
μ
=
0.9
[25]
1.009
0.714
[23]
1.284
1.209
[13]
1.287
1.279
Corollary10
1.7706
1.7355


## 5. Conclusions

In this paper, we have investigated theH∞ control problem for a class of nonlinear systems with interval time-varying delay. A new Lyapunov-Krasovskii functional is constructed to obtain new delay-dependent sufficient condition for the H∞ control and asymptotic stability condition in terms of LMIs. Numerical examples are given to illustrate the effectiveness of the theoretical results.



---

*Source: 473165-2015-05-11.xml*