Stabilization and $$H^{\infty}$$ control of symmetric systems: An explicit solution.(English)Zbl 0986.93028

Summary: We address the $$H^{\infty}$$ control analysis, the output feedback stabilization, and the output feedback $$H^{\infty}$$ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system, we obtain an explicit expression to compute the $$H^{\infty}$$ norm of the system. For the output feedback stabilization problem, we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the $$H^{\infty}$$ control synthesis problem, we derive an explicit expression for the optimally achievable closed-loop $$H^{\infty}$$ norm and the optimal control gains. Extensions to robust and positive real control of such systems are also examined. These results are obtained from linear matrix inequality formulations of the stabilization and the $$H^{\infty}$$ control synthesis problems using simple matrix algebraic tools.

MSC:

 93B36 $$H^\infty$$-control 93D15 Stabilization of systems by feedback 15A39 Linear inequalities of matrices
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References:

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