zbMATH — the first resource for mathematics

Robust $$H_\infty$$ stabilisation with definite attenuance of an uncertain impulsive switched system. (English) Zbl 1123.93073
The 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.

MSC:
 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93D09 Robust stability
Keywords:
robust stability; Lyapunov stability
Full Text:
References:
 [1] Lakshmikantham, Theory of impulsive differential equations (1989) · Zbl 0718.34011 [2] DOI: 10.1109/9.50357 · Zbl 0707.93060 [3] Boyd, Linear matrix inequalities in system and control theory (1994) · Zbl 0816.93004 [4] DOI: 10.1109/9.763234 · Zbl 0954.49022 [5] DOI: 10.1109/9.376091 · Zbl 0827.93021 [6] Lakshmikantham, Stability analysis in terms of two measures (1994) [7] DOI: 10.1016/0167-6911(87)90102-2 · Zbl 0618.93056 [8] DOI: 10.1142/S0218127402005029 · Zbl 1051.93511 [9] DOI: 10.1216/rmjm/1181072290 · Zbl 0832.34039 [10] DOI: 10.1016/S0005-1098(97)00082-4 [11] DOI: 10.1016/0167-6911(90)90088-C · Zbl 0698.93054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.