Guaranteed level-\(\gamma\) \(\mathcal{H}_ \infty\) control in uncertain linear systems via linear matrix inequalities. (English) Zbl 0870.93013

The paper deals with a guaranteed level \( H_\infty \) control problem for uncertain linear systems. For time-varying systems, where the finite horizon cost function is considered, the authors derive a time-varying nonlinear differential equation to determine a guaranteed level \( H_\infty \) control law. For time-invariant systems, linear matrix inequalities (LMIs) are presented which allow to use convex optimization techniques to solve the optimal guaranteed level \( H_\infty \) problem as well as the guaranteed level-\(\gamma \) \( H_\infty \) problem with the maximal stability margin. The proposed approach allows to handle additional constraints, parameter uncertainties and to overcome other difficulties that arise when \( H_\infty \) problems are reduced to those of solving algebraic Riccati-type equations.


93B36 \(H^\infty\)-control
15A39 Linear inequalities of matrices
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