On improved delay-dependent robust H control for systems with interval time-varying delay

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Abstract

This paper considers the design problem of delay-dependent robust H control for interval time-delay systems (ITDS). Based on the Lyapunov–Krasovskii functional approach and the convexity of the matrix equations, a delay-dependent bounded real lemma (BRL) is firstly proposed, which is shown in terms of linear matrix inequalities. Then by applying the BRL, sufficient conditions can be obtained for the asymptotic stabilizability of the uncertain ITDS with a prescribed H performance index. Finally, numerical examples are provided to illustrate the less conservativeness of the proposed methods than some existing results.

Introduction

As we know, the lower bound of the time delay may not be zero, which can be viewed as the case with interval time delay. A typical example for this case is networked control system [18], [17]. In the past few years, the stability analysis and stabilization of the interval time-delay systems (ITDS) have been investigated in [5], [7], [13], [18], [17]. Computation results in these references have shown that when the lower bound of the delay is larger than 0, the upper bound of the time delay can be improved.

Recently, the problem of delay-dependent robust H control design has received considerable attentions for system with both time-invariant delay [2], [3], [11], [12], [15] and time-varying delay [1], [7], [8], [9], [16]. In these references, one of the main efforts has been paid on effective reduction of the conservation of the time delay. For this purpose, new bounding technique for cross terms [10], [11], descriptor system method [3] and free weighting matrix method [6], [16] have been proposed. However, conservatism still exists and further investigation is needed. For example, when the Lyapunov–Krasovskii functional method is used, the time-varying delay τ(t) often appears in the derivation of the Lyapunov functional or the introduced free weighing matrix equations, such as tτ(t)tx˙T(s)Rx˙(s)ds and τ(t)ζT(t)Xζ(t)(R>0 and X>0) [5], [6], [7], [16], the usual method to treat them is to enlarge them as tτMtx˙T(s)Rx˙(s)ds and τMζT(t)Xζ(t), then the estimation errors tτMtτ(t)x˙T(s)Rx˙(s)ds and (τMτ(t))ζT(t)Xζ(t) are ignored, which will unavoidably lead to some conservativeness.

In this paper, we are concerned with the delay-dependent robust H control design for uncertain systems with interval time-varying delay. By using the convexity of the matrix equations the conservatism caused by enlarging τ(t) to τM can be avoided. Then a less conservative delay-dependent bounded real lemma (BRL) is obtained by using the Lyapunov–Krasovskii functional approach. Based on the derived BRL, delay-dependent conditions for the existence of a state feedback H controller are obtained in terms of linear matrix inequalities. At last, numerical examples are given to show the less conservativeness of the proposed method.

Section snippets

System description

The uncertain time-delay system is described asx˙(t)=(A+ΔA(t))x(t)+(Ad+ΔAd)x(tτ(t))+(B+ΔB(t))u(t)+Bωω(t)z(t)=Cx(t)+Cdx(tτ(t))+Du(t)x(t)=ϕ(t),t[τM,0]where x(t)Rn, u(t)Rm and z(t)Rr are the state, input and controlled output, respectively; ω(t)L2[0,) is the exogenous disturbance signal, A,Ad,B,Bω, C,Cd and D are known parameter matrices of appropriate dimensions, ΔA(t),ΔAd(t) and ΔB(t) are unknown matrices of appropriate dimensions and satisfying [ΔA(t)ΔAdΔB(t)]=HF(t)[E1E2E3]where H and E

Main results

Before giving the main results, we establish a new version of delay-dependent bounded real lemma for the system (1), (2), (3) with u(t)0, i.e.,x˙(t)=(A+ΔA(t))x(t)+(Ad+ΔAd)x(tτ(t))+Bωω(t)z(t)=Cx(t)+Cdx(tτ(t))x(t)=ϕ(t),t[τM,0]

Lemma 2

For given scalars γ>0, τm and τM, the system (11), (12), (13) is asymptotically stable and satisfies z(t)2<γω(t)2 for any non-zero ω(t)L2[0,) if there exist matrices P>0, Qi>0, Ri>0(i=1,2), Nj, Mj, Vj(j=1,2,3,4,5), Sk(k=1,2,3) and a scalar ɛ>0 such that the

Numerical examples

Example 1

Consider the time-delay system (11), (12), (13) with ΔA=ΔAd=ΔB=0 and A=0.62381.01322.01160.2106,Ad=0.50110.78710.30020.5231,Bω=0.43260.12531.66560.2877C=0.21340.01910.11190.1665,Cd=0.08160.12900.07120.0669,D=0000To compare our results with those in Fridman and Shaked [3], Gao and Wang [4], Lee et al. [11] and Xu et al. [16], we use Corollary 1 with h=0, Table 1 gives the comparison results on the upper bound of the time delay for given γ among [3], [4], [11], [16] and Corollary 1.

Conclusion

This paper has investigated the problem of delay-dependent robust H control design for interval time-varying delay system (ITDS). Based on the convexity of the matrix equations and the Lyapunov–Krasovskii functional approach, sufficient conditions have been obtained as a set of linear matrix inequalities. Numerical examples have been provided to show the less conservativeness of the proposed method.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant nos. 60904013, 60904061, 51075215, 61074024 and 61074025) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant no. 09KJB510004).

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