Robust L2–L∞L2–L∞ control of uncertain differential linear repetitive processes

For two-dimensional (2-D) systems, information propagates in two independent directions. 2-D systems are known to have both system-theoretical and applications interest, and the so-called linear repetitive processes (LRPs) are a distinct class of 2-D discrete linear systems. This paper is concerned with the problem of L2–L∞L2–L∞ (energy to peak) control for uncertain differential LRPs, where the parameter uncertainties are assumed to be norm-bounded. For an unstable LRP, our attention is focused on the design of an L2–L∞L2–L∞ static state feedback controller and an L2–L∞L2–L∞ dynamic output feedback controller, both of which guarantee the corresponding closed-loop LRPs to be stable along the pass and have a prescribed L2L2–L∞L∞ performance. Sufficient conditions for the existence of such L2L2–L∞L∞ controllers are proposed in terms of linear matrix inequalities (LMIs). The desired L2–L∞L2–L∞ dynamic output feedback controller can be found by solving a convex optimization problem. A numerical example is provided to demonstrate the effectiveness of the proposed controller design procedures.