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Delay-dependent robust stability and \(H_\infty\) control of jump linear systems with time-delay. (English) Zbl 1015.93069

The authors consider stability properties of a stochastic control system \(\Sigma_1\) that is linear in the state with random coefficients and delay \(\tau\), given by the dynamics \[ \dot{x}_t = A(r_t)x_t + A_1(r_t)x_{t-\tau} + B(r_t)u_t + B_w(r_t)w_t, \] the initial condition \(x_t = \Phi(t)\), and \(r_t = r_0\) for \(t \in [-\tau, 0], (\tau>0)\) and the output \[ z_t = C(r_t)x_t + D(r_t)u_t + E(r_t)w_t, \] where \(u_t\) is a control, \(w_t\) is a noise process and \(r_t\) a finite state Markov process with values in \(S = \{1,2,\dots ,N\}\) and \(A, A_1,\dots ,E\) are matrix-valued functions of polytype \({\mathcal M}(i),\) meaning that they belong to a family of matrix-valued functions \(M\) on \(S\) with the property that there is a number \(\nu \in {\mathbb N}\) and non-negative coefficients \(t_j\) \((j=1,\dots ,\nu)\) with \(\sum_{j=1}^{\nu}t_{j} = 1\) that are independent of \(M\) such that any value \(M(i)\) can be represented as the convex combination \(M(i) = \sum_{j=1}^{\nu}t_jM_{ij}\) of suitable matrices \(M_{ij}\).
Sufficient conditions are given in terms of algebraic inequalities for the involved coefficient matrices and the delay \(\tau\) (known as linear matrix inequalities) for (i) the uncontrolled system \(\Sigma_1\) with \(u_t=0\) and (ii) the closed-loop feedback system \(\Sigma_1\) with \(u_t=K(r_t)x_t\) to be robustly stochastically stable or, respetively, robustly stable with noise attenuation level \(\gamma\) (\(\gamma>0\) given) that is satisfying \( \int_{0}^{\infty}{\mathbb E}{\mathbf [} \|x_t\|^2 \mid r_0,\Phi(\cdot){\mathbf ]} dt \leq M_0(r_0, \Phi(\cdot), \|w\|_2) \) or \( {\mathbb E} \{ \int_{0}^{\infty} \|z_t\|^2 dt \}^{\frac{1}{2}} \leq \gamma [M(r_0, \Phi(\cdot)) + \|w\|_2] \), respectively, with suitable constants \(M_0\) and \(M\) (depending only on the initial conditions [and noise \(w\)]) satisfying \(M_0(r_0,0,0) = M(r_0,0) = 0 \).
If the system \(\Sigma_1\) is modified to a system \(\Sigma_3\) with delay also in the control, an algorithm is given for constructing a feedback control such that \(\Sigma_3\) with that control is robustly stable for all delays \(\tau\) that vary over a given interval.

MSC:

93E15 Stochastic stability in control theory
93D09 Robust stability
93C23 Control/observation systems governed by functional-differential equations
93B36 \(H^\infty\)-control
93E20 Optimal stochastic control
34K50 Stochastic functional-differential equations
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