Control Under Stochastic Multiplicative Uncertainties: Part II, Optimal Design for Performance
This paper studies the optimal control design problem for linear discrete-time systems with stochastic multiplicative uncertainties. These uncertainties are assumed to be present in the control inputs and modeled as independent and identically distributed (i.i.d.) random processes. The optimal performance under study is defined in the mean-square sense, referred to as the mean-square optimal <inline-formula><tex-math notation="LaTeX">$\mathcal{H}_{2}$</tex-math></inline-formula> performance. It is shown that the mean-square optimal <inline-formula><tex-math notation="LaTeX">$\mathcal{H}_{2}$</tex-math></inline-formula> control problem via state feedback can be solved using a mean-square stabilizing solution to a modified algebraic Riccati equation (MARE). A necessary and sufficient condition for the existence of this solution is presented, which constitutes a generalization of the solution to the classic optimal <inline-formula><tex-math notation="LaTeX">$\mathcal{H}_{2}$</tex-math></inline-formula> state feedback design problem, whereas the latter can be obtained by solving an algebraic Riccati equation (ARE). It is also proven that the optimal control design problem can be cast as an eigenvalue problem (EVP). For the output feedback case with possible input delays, we show that the mean-square optimal <inline-formula><tex-math notation="LaTeX">$\mathcal{H}_{2}$</tex-math></inline-formula> control problem also amounts to solving an MARE, when the plant has no nonminimum phase zeros from the inputs to the measurement outputs. That is, the global optimal solution is obtained by solving an MARE incorporating the delays. The implication is that in this case a separation principle still holds.