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This note deals with the problem of robust H∞ control for stochastic system with interval time delay. By using both, Itô’s differential formula and Lyapunov function, a new design method is proposed. Meanwhile, design a feedback controller, such that solve the problem of robust stochastic stabilization, the expressions of feedback controlle ris given. Finally, an acdemic example is include to illustrate the effectiveness of result. Introduction During the past years, people’s attention have been devoted to the study of time delay due to the fact that time delay is often the main cause for instability and poor performance of a control system[1, 2]. For example, delay-dependent stability criterion for systems with uncertain time-invariant delays was discussed in [3]. The H∞control problem for stochastic systems was studied, and a stochastic bounded real lemma was derived in [4] by using the Moons inequalities approach, [5] investigated the problem of delay dependent robust stochastic stabilization and H∞ control for stochastic systems with norm-bounded uncertainties and state delay. For the case of uncertain linear systems, i.e. for proving the robust stability, the problem has been partially solved, either by using Lyapunov functionals[6], or robustness tools(quadratic separation [7]). All resulting stability conditions are based on Linear Matrix Inequality (LMI) and allow to conclude on stability region with respect to the delay. However, when revisiting this problem, we find that the lower bound of delay range for the most present research is limited to zero, the selected Lyapunov function mostly does not contain the information of the lower bound of delay. Although much attention has been focused on the robustH∞ control for stochastic systems, the method therein still leaves much room for improvement. This paper aims at design H∞ controller for stochastic system with interval time delay, which use proper Lyapunov function, Itô’s formula and introducing free weighting matrix. Finally, an example illustrating the proposed approach is provided. Problem Formulation Consider the following stochastic system with state delay and input delay: dxx(t)=[Axx(t)+AAττxx(t-τ(t))+Bu(t)+BBττu(t-τ(t))+BBVVv(t)]dt+[Exx(t)+EEττxx(t-τ(t))+EEVVv(t)]dωω(t)(1) zz(tt) = CCxx(tt) + CCττxxtt − ττ(tt) + DDDD(tt) + DDττDD(tt − ττ(tt))(2) xx(tt) = φφ(tt),DD(tt) = φφ(tt),∀tt ∈ [−hMM, 0](3) wherexx(tt) ∈ RRnn is the state;DD(tt) ∈ RRmm is the control input; vv(tt) ∈ RRpp denotes for the disturbance which belongs to LL2[0,∞); z(t) ∈ Rqq stands for the controlled output; v(t) is a one-dimensional Brownian motion satisfying εε{ddωω(tt)} = 0 and εε{ddωω(tt)2} = ddtt ,A, AAττ ,B, BBττ,BBvv,E,EEττ,EEvv,C,CCττ,D,DDττare known real constant matrices matrices, and ττ (tt)is the time-varying delay satisfying hmm < ττ(tt) < hMM, ?̇?τ ≤ DD (4) where hmm andhMare real constant scalars. Lemma 1. For any constant matrix M > 0, any scalars a and b with a < b , and a vector function International Conference on Education, Sports, Arts and Management Engineering (ICESAME 2016) © 2016. The authors Published by Atlantis Press 1366 xx(tt) ∶ [aa, bb] → RRnn such that the integrals concerned are well defined the following holds: ∫ xx(ss)ddss bb aa  TT MM ∫ xx(ss)ddss bb aa  ≤ (bb − aa)∫ xx TT(ss)MMxx bb aa (ss)ddss (5) HH∞ Performance Analysis We design a feedback gain matrix u(t) = Kx(t), the closed-loop stochastic system consisting of (1) and (2) can be rewritten in the following form: xx(tt) = ?̅?AKKxx(tt) + ?̅?AττKKxx(tt − ττ(tt)) + BBvvvv(tt)ddtt + EExx(tt) + EEττxxtt − ττ(tt) + EEvvvv(tt)ddωω(tt) (6) zz(tt) = CCKKxx(tt) + DDττKKDD(tt − ττ (tt)) (7) where ?̅?AKK = AA + BBBB ?̅?AττKK = AA ττ + BBττBB EE = EE EEττ = EEττ C KK = C + DK DττKK = Cττ + DττK In this section, a delay-dependent approach is proposed to solve the problem of robust stochastic stabilization with disturbance attenuation level. We first assume that the feedback gain matrix K is known. For robust HH∞ performance analysis of the system (6-7), we have the following result. Theorem 1. Consider the closed-loop system (6)-(7). For given scalars γ > 0, hmm > 0,hMM > 0, μ > 0, and feedback gain K, the stochastic system (6) and (7) is robustly stochastically stable with disturbance attenuationsγγ for anyττ(tt)satisfyinghmm < ττ(tt) < hMM, ?̇?τ(tt) ≤ DD , if there exist matrices PP > 0,QQqq > 0,ZZ1 > 0,ZZ2 > 0,NNpp,GGpp,MMqq , SSqq, (pp 1, 2, qq = 1, 2, 3) , satisfying the following LMI ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ φφ11 φφ12 HH1 −GG1 φφ15 φφ16 −NN1 −GG1 −HH1 φφ1,10 φφ1,11 ∗ φφ22 HH2 −GG2 φφ25 φφ26 −NN2 −GG2 −HH2 φφ2,10 φφ2,11 ∗ ∗ −QQ2 0 0 0 0 0 0 0 0 ∗ ∗ ∗ −QQ1 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ φφ55 0 0 0 0 φφ5,10 0 ∗ ∗ ∗ ∗ ∗ φφ66 0 0 0 φφ6,10 0 ∗ ∗ ∗ ∗ ∗ ∗ − ZZ1 hMM 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ − ZZ1 + ZZ2 hMM − hmm 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ − ZZ2 hMM − hmm 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γγ2II 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −II ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ φφ11 = QQ1 + QQ2 + QQ3 + NN1 + NN1 + MM1(AA + BBBB) + (AA + BBBB)MM1 + SS1EE + EESS1 φφ12 = −NN1 + NN2 + GG1 − HH1 + MM1(AAττ + BBττBB) + (AA + BBBB)MM2 + SS1EEττ + EEττSS1 φφ22 = −(1 − DD)QQ3 − NN2 − NN2 + GG2 − HH2 − HH2 + MM2(AAττ + BBττBB) + (AAττ + BBττBB) MM2 + SS2EEττ + EEττSS2 φφ15 = PP −MM1 + (AA + BBBB)MM3 φφ16 = −SS1 + EESS3 φφ1,10 = MM1BBVV + SS1EEVV φφ1,11 = CCTT + BBTTDDTT φφ25 = −MM2 + (AAττ + BBττBB)MM3 φφ26 = −SS2 + EEττSS3 φφ2,10 = MM2BBvv + SS2EEvv φφ2,11 = CCττ + BBDDττ φφ55 = hMMZZ1 + (hMM − hmm)ZZ2 − MM3 −MM3 φφ66 = PP − SS3 − SS3

A delay-dependent approach to robust H control for stochastic systemwith interval time delay