## Stability of switched systems: a Lie-algebraic condition.(English)Zbl 0948.93048

Summary: We present a sufficient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems.

### MSC:

 93D20 Asymptotic stability in control theory 93D15 Stabilization of systems by feedback 93D30 Lyapunov and storage functions
Full Text:

### References:

 [1] Barabanov, N. E., Absolute characteristic exponent of a class of linear nonstationary systems of differential equations, Siberian Math. J., 29, 521-530 (1988) · Zbl 0688.93051 [3] Gurvits, L., Stability of discrete linear inclusion, Lin. Algebra Appl., 231, 47-85 (1995) · Zbl 0845.68067 [7] Molchanov, A. P.; Pyatnitskiy, Y. S., Criteria of absolute stability of differential and difference inclusions encountered in control theory, System Control Lett., 13, 59-64 (1989) · Zbl 0684.93065 [10] Narendra, K. S.; Balakrishnan, J., A common Lyapunov function for stable LTI systems with commuting $$A$$-matrices, IEEE Trans. Automat. Control, 39, 2469-2471 (1994) · Zbl 0825.93668 [11] Ooba, T.; Funahashi, Y., On a common quadratic Lyapunov function for widely distant systems, IEEE Trans. Automat. Control, 42, 1697-1699 (1997) · Zbl 0899.93035
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.