Surveys

Stochastic problems in $H_\infty$ and $H_2/H_\infty$ control

M. E. Shaikin

Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

Abstract: We consider stochastic control systems subjected simultaneously to stochastic and determinate perturbations. Stochastic perturbations are assumed to be state-multiplicative stochastic processes, while determinate perturbations can be any processes with finite energy on infinite time interval. The results of the determinate $H_\infty$-theory are compared to their stochastic analogs. The determinate and stochastic theories are linked together by the lemma that establishes the equivalence between the stability and boundness of the $\|L\|_\infty<\gamma$ norm of the perturbation operator $L$, from one side, and the solvability of certain linear matrix inequalities (LMIs), from the other side. As soon as the stochastic version of the lemma is proven, the $\gamma$-controller analysis and design problems are solved, in general, identically in the frame of the united LMI methodology.

PACS: 02.30.Yy

Presented by the member of Editorial Board: B. T. Polyak

Received: 06.06.2008

 English version: Automation and Remote Control, 2009, 70:2, 173–202

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