Robust \(H_\infty\) stabilisation with definite attenuance of an uncertain impulsive switched system.

*(English)*Zbl 1123.93073The paper uses and combines various standard ingredients for robust and \(H_{\infty}\) control of linear systems to deal with the problem of disturbance rejection for uncertain linear systems subject to polytopic switching. Stability and performance analysis conditions are derived using mainstream Lyapunov theory. These conditions are not formulated as convex linear matrix inequality (LMI) since they involve products of Lyapunov matrices with several scalar decision variables (denoted \(\gamma\), \(\epsilon\) and \(\lambda\) in the main Theorem 3.5). Moreover, the conditions are valid only provided the (possibly unrealistic) knowledge of an explicit \(H_{\infty}\) bound between the disturbance signal and the system state (see Assumption 3.4), and under a restrictive (and certainly difficult to enforce) condition on the spectrum of the Lyapunov matrix (via the variable \(\beta_k\) in Theorem 3.5). For these reasons, one can be skeptical about the usefulness of the proposed analysis conditions.

Reviewer: Didier Henrion (Toulouse)

##### MSC:

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93D09 | Robust stability |

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