Output Regulation of Linear Stochastic Systems With Unstable Linear Exogenous Systems

This article investigates the output regulation problem of multi-input-multi-output (MIMO) linear stochastic systems and unstable linear exogenous systems. First, stochastic output regulator equations are constructed to characterize the solvability of the output regulation problem. Then, a necessary and sufficient condition on the solvability of the output regulator equation is established. Under this condition, not only the relationship between the output regulation of linear stochastic systems and that of linear deterministic systems is established, but also the solvability in some special cases is discussed. In the case where the control input is not invoked in the tracking error, there almost always exists a bounded and deterministic solution if the stochastic output regulator equation has a bounded solution, which provides a new approach without estimating the Brownian motion. This approach can deal with the crucial issue that the stochastic output regulator equation may not be solved in a practical sense, since the solution depends on the Brownian motion which may not be measured in most real scenes. Several numerical examples are presented to illustrate the main results.