Robust stability and control of complex stochastic systems with jumps of state vector

The paper considers a class of complex control systems which modelled as a finite family of randomly coupled individual stochastic differential equations. Each individual equation describes an individual mode of complex system. Between these modes the discontinuous transitions are seen. This jumping process is modelled by homogeneous Markov chain with discrete set of states. In accordance with these special features the full state vector of the system under study is hybrid and contains both component taking values in continuous space and component taking values in a finite discrete set. At the moment of the jump of Markov chain the continuous state component can be changed by jump too, but its value after jump is uniquely dependent on the value before one. Based on power transformation technique the conditions of robust stability in the sense that the complex system is 2p-stable independently on the transition probabilities of discrete state component (Markov chain above) are obtained in both linear case and in the presence of Lur´e-type nonlinearities. The state feedback control law, which guarantees the robust 2p-stability of the closed-loop complex system in linear case, is also given. All the results are formulated in terms of linear matrix inequalities