Periodically time-varying memory state-feedback for Robust H₂ control of uncertain discrete-time linear systems

This study is concerned with the synthesis of periodically time-varying memory state-feedback controllers (PTVMSFCs) for discrete-time linear systems. In our preceding studies, we have already established a solid theoretical basis for linear matrix inequality (LMI)-based (robust) H _∞-PTVMSFCs synthesis, and the goal of this paper is to extend those results to the H ₂ performance criterion. In the H ₂ case, the main difficulty stems from the fact that we have to ensure the existence of common auxiliary variables for multiple LMI conditions that are related to the Lyapunov inequality and the inequalities for bounding traces that characterize the H ₂ norm. We can overcome this difficulty and derive a necessary and sufficient LMI condition for the optimal H ₂-PTVMSFC synthesis. On the basis of this result, we also consider robust H ₂-PTVMSFC synthesis for LTI systems with parametric uncertainties.