Minimax and Robust Control of Hybrid Stochastic Systems with Jumps of State Vector

Abstract

A class of hybrid in state systems, which modelled as a finite family of differential equations is considered. Each equation of this family describes the individual regime of the system. The transitions between these regimes are discontinuous and are modelled by a homogeneous Markov chain. At the moment of regime change (jump of Markov chain) the state vector can be changed discontinuously too, but its value after jump is uniquelly depend on the same value before one. Two problems are studied in this paper. First a state feedback control law is obtained, which guarantees 2p-stablity of closed-loop hybrid system and at the same time minimizes a 2p-order cost functional with the worst behavior of the input disturbance. A game theoretic approach is used for the solution of this control problem. Further a state feedback control is obtained, which guarantees mean square stability of closed loop system for both all transition probabilities from the given set and independently on transition probabilities. These controls are defined as robust stabilizing control and perfect robust stabilizing control correspondingly.