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An LMI approach to output-feedback-guaranteed cost control for uncertain time-delay systems. (English) Zbl 0951.93032
Linear time-delay systems with uncertainties of the form \[ \dot x(t)= (A + D\Delta(t)E_1)x(t) + (F + D\Delta(t)E_3)x(t-\tau) \] are considered, where the uncertain parameters \(\Delta(t)\) are assumed to satisfy \(\Delta'(t)\Delta(t)\leq I\). Using a standard Lyapunov function of the form \[ V(x, t) = x'(t)Px(t) + \int^t_0 x'(t-s)P_1x(t-s)ds \] it is shown that the system is quadratically stable iff the LMI \[ \left[\begin{matrix} {\l}\quad & {\l}\quad & {\l}\quad & {\l}\\ \varepsilon(A'P+PA+P_1) & \varepsilon PD & \varepsilon PF & E_1'\\ \varepsilon D'P & -I & 0 & 0\\ \varepsilon F'P & 0 & -\varepsilon P_1 & E_3\\ E_1 & 0 & E_3' & -I\end{matrix}\right] < 0 \] is satisfied by \(P, P_1\) for some \(\varepsilon > 0\). It is also shown that by using a full-order dynamic controller of the form \[ \begin{aligned} & \dot\xi(t)=A_c\xi(t)+B_cy(t),\\ & u(t) = C_c\xi(t)\end{aligned} \] a quadratic cost function satisfies a fixed bound for all admissible \(\Delta(t)\).

MSC:
93B51 Design techniques (robust design, computer-aided design, etc.)
15A39 Linear inequalities of matrices
34K35 Control problems for functional-differential equations
Software:
LMI toolbox
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