A Generalized Framework for the L1$$ {L}_1 $$ Optimal Controller Synthesis in Sampled‐Data Systems

ABSTRACT This article introduces a generalized framework for the optimal control of sampled‐data systems, in which an optimal discrete‐time controller is designed to minimize the continuous‐time ‐induced norm of the feedback system for a given continuous‐time plant. This article aims at deriving a discrete‐time plant (approximately) equivalent to the original continuous‐time plant in the sense of the optimal control, by which the hybrid continuous/discrete‐time optimal control problem is reducible to the pure discrete‐time optimal control problem. To this end, we first develop a method for approximating the input function and the kernel function of the input operator together with the hold function of the output operator in the lifted description of sampled‐data systems by piecewise linear functions. More precisely, this method is based on dividing the sampling interval by subintervals with equal width and taking the freedom in the point on each subinterval around which associated functions are expanded to the Taylor series. The piecewise linear approximations lead to discretizing the continuous‐time plant, and a discrete‐time optimal controller for the resulting discrete‐time system is shown to approximately minimize the ‐induced norm of the original sampled‐data system in the relevant order of . We further show that the Taylor expansion taken around the central point of each of the subintervals might achieve the most quantitatively improved discretization in terms of the optimal control. Finally, the overall arguments developed in this article are demonstrated through some numerical studies.