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Stochastic stabilization and induced $$\ell_2$$-gain for discrete-time Markov jump Lur’e systems with control saturation. (English) Zbl 1297.93175
Summary: This paper addresses the problem of control synthesis for locally stabilizing and minimizing the finite $$\ell_2$$ gain for discrete-time Markov jump Lur’e systems with control saturation and exogenous $$\ell_2$$-type disturbance. We consider that the jump parameter defining the active mode for both the linear and the cone-bounded nonlinearity is governed by a finite state homogeneous Markov chain. The local stochastic stability is established by exploiting the invariant probability measure of the Markov chain to define the level set of the expected value of the stochastic Lur’e type Lyapunov function. Optimization problems for maximizing our estimate of the domain of stochastic stability and minimizing the induced $$\ell_2$$ gain with respect to exogenous disturbances are presented subject to LMI constraints. The paper is concluded with an academic example.

##### MSC:
 93E15 Stochastic stability in control theory 93E03 Stochastic systems in control theory (general) 93C55 Discrete-time control/observation systems 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 93B50 Synthesis problems 93C10 Nonlinear systems in control theory
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