Periodically time-varying controller synthesis for multiobjective H ₂ H_∞ control of discrete-time systems and analysis of achievable performance

In this paper, we propose a linear periodically time-varying (LPTV) controller synthesis approach for the multiobjective H₂H _∞ control problem of discrete-time linear time-invariant (LTI) systems. By artificially regarding the LTI plant as N-periodic and applying the discrete-time system lifting, we first derive a sequence of semidefinite programmings (SDPs) indexed by N for the synthesis of suboptimal multiobjective LPTV controllers. Furthermore, we show that we can reduce the conservatism and improve the control performance gradually by simply increasing the controller period N. On the other hand, in the latter part of the paper, we propose another sequence of SDPs for the computation of the lower bound of the multiobjective control performance that is achievable via LPTV controllers of any period and order. Similarly to the LPTV controller synthesis, the SDP is derived based on the lifting-based treatment of the LTI plant. Again, it is shown that we can improve the lower bound gradually by increasing the fictitious period N. We validate all of these theoretical results through illustrative examples.