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All controllers for the general \({\mathcal H}_ \infty\) control problem: LMI existence conditions and state space formulas. (English) Zbl 0806.93017
Summary: This paper presents all controllers for the general \({\mathcal H}_ \infty\) control problem (with no assumptions on the plant matrices). Necessary and sufficient conditions for the existence of an \({\mathcal H}_ \infty\) controller of any order are given in terms of three linear matrix inequalities (LMIs). Our existence conditions are equivalent to Scherer’s results, but with a more elementary derivation. Furthermore, we provide the set of all \({\mathcal H}_ \infty\) controllers explicitly parametrized in the state space using the positive definite solutions to the LMIs. Even under standard assumptions (full rank, etc.), our controller parametrization has an advantage over the \(Q\)-parametrization. The freedom \(Q\) (a real-rational stable transfer matrix with the \({\mathcal H}_ \infty\) norm bounded above by a specified number) is replaced by a constant matrix \(L\) of fixed dimension with a norm bound, and the solutions \((X,Y)\) to the LMIs. The inequality formulation converts the existence conditions to a convex feasibility problem, and also the free matrix \(L\) and the pair (\(X,Y)\) define a finite dimensional design space, as opposed to the infinite dimensional space associated with the \(Q\)- parametrization.

MSC:
93B36 \(H^\infty\)-control
93B50 Synthesis problems
93B25 Algebraic methods
93B51 Design techniques (robust design, computer-aided design, etc.)
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