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Robust stabilization of nonlinear systems: The LMI approach. (English) Zbl 0968.93075
In the first section the authors consider a system \[ \dot x= Ax+ h(t,x),\tag{1} \] where the matrix \(A\) is Hurwitz and \(h(t,x)\) is a possibly discontinuous function which represents nonlinear uncertainties. The admissible functions \(h(t,x)\) belong to a set \(H_\alpha\) defined by the inequality \[ h(t,x)^T h(t,x)\leq \alpha^2 x^T H^T Hx, \] where \(H\) is a given matrix. The parameter \(\alpha\) is thought of as a measure of the size of \(H_\alpha\).
The maximal value of \(\alpha\) is identified by solving an optimization problem, with constraints expressed in the form of a linear matrix inequality.
In the second section the author considers the case where \(A\) is not Hurwitz. Here, (1) is replaced by \[ \dot x= Ax+ Bu+ h(t,x)\tag{2} \] and a solution is sought in feedback form. In the following sections the author considers systems which satisfy the matching condition. Finally, the results are applied to decentralized control and interconnected systems.

MSC:
93D21 Adaptive or robust stabilization
93D09 Robust stability
93A14 Decentralized systems
93C73 Perturbations in control/observation systems
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