Robust stability and robust stabilization of discrete-time linear stochastic systems

The main goal of this chapter is to investigate several aspects of the problem of robust stability and robust stabilization for a class of discrete-time linear stochastic systems subject to sequences of independent random perturbations and Markov jump perturbations. As a measure of the robustness of the stability of an equilibrium of a nominal system a concept of stability radius is introduced. A crucial role in determining the lower bound of the stability radius is played by the norm of a linear bounded operator associated with the given plant. This operator is called the input.output operator and it is introduced in Section 8.2. In Section 8.3 a stochastic version of the so-called bounded real lemma is proved. This result provides an estimation of the norm of the input.output operator in terms of the feasibility of some linear matrix inequalities (LMIs) or in terms of the existence of stabilizing solutions of a discrete-time generalized algebraic Riccati-type equation. In Section 8.4 the stochastic version of the so-called small gain theorem is proved. Then this result is used to derive a lower bound of robustness with respect to linear structured uncertainties. In the second part of this chapter we consider the robust stabilization problem of systems subject to both multiplicative white noise and Markovian jumps with respect to some classes of parametric uncertainty. Based on the bounded real lemma we obtain a set of necessary and sufficient conditions for the existence of a stabilizing feedback gain that ensures a prescribed level of attenuation of the exogenous disturbance. We also show that in the case of full state measurement if the disturbance attenuation problem has a solution in a dynamic controller form then the same problem is solvable via a control in a state feedback form. Finally a problem of H∞ filtering is solved.