## Stability results for discrete-time linear systems with Markovian jumping parameters.(English)Zbl 0790.93108

Summary: Stability results for discrete-time linear systems subject to random jumps in the parameters are obtained. First, necessary and sufficient conditions for mean square stability (MSS), including the case in which the system is driven by an independent wide-sense stationary random sequence, are derived. It is shown that MSS is equivalent to the spectral radius of an augmented matrix being less than one or to the existence of a solution of a certain Lyapunov equation. In addition it is shown that the Lyapunov equation can be derived in four equivalent forms and, as a by-product, each one gives rise to an easier-to-check sufficient condition. These results give, inter alia, a unified and rather complete picture of MSS of Markovian jump linear systems. Next we derive sufficient conditions for almost sure stability of the noiseless case. Finally, we conclude by presenting an illustrative application to adaptive filtering where a mild condition for almost sure convergence is provided.

### MSC:

 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93C55 Discrete-time control/observation systems 93C05 Linear systems in control theory
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