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$$H_\infty$$ control of linear singularly perturbed systems with small state delay. (English) Zbl 0965.93049
An infinite horizon $$H_\infty$$ state-feedback control problem for singularly perturbed linear systems with a small state delay is considered. The system under consideration is: \begin{aligned} \dot x & = A_1x(t)+ A_2y(t)+ H_1x(t- \varepsilon h)+ H_2y(t- \varepsilon h)+ B_1u(t)+ F_1 w(t),\\ \varepsilon\dot y(t) & = A_3x(t)+ A_4y(t)+ H_3 x(t-\varepsilon h)+ H_4y(t- \varepsilon h)+ B_2u(t)+ F_2 w(t),\\ V(t) & = \text{col}\{C_1x(t)+ C_2y(t), u(t)\},\quad t>0,\end{aligned} where $$x$$, $$y$$ are state variables, $$u$$ is a control, $$w$$ is an exogenous disturbance, $$v$$ is a control output, $$\varepsilon> 0$$ is a small parameter.
An asymptotic solution of the hybrid system of Riccati type algebraic, ordinary differential, and partial differential equations with deviating arguments, associated with this $$H_\infty$$ control problem, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original $$H_\infty$$ problem, independent of the singular perturbation parameter, are derived. A simplified controller with parameter-independent gain matrices, solving the original problem for all sufficiently small values of this parameter, is obtained. An illustrative example is presented.

##### MSC:
 93B36 $$H^\infty$$-control 93C23 Control/observation systems governed by functional-differential equations 93C70 Time-scale analysis and singular perturbations in control/observation systems
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