L1 optimal controller synthesis for sampled‐data systems via piecewise linear kernel approximation

Abstract

AbstractThis article provides a new framework for the so‐called L1 optimal control problem of sampled‐data systems, that is, the synthesis problem of a discrete‐time optimal controller minimizing the continuous‐time L∞‐induced norm of the closed‐loop system for a given continuous‐time plant. The main idea is to develop the approximation method called the piecewise linear kernel approximation (PLKA) method, by which the kernel function of the input operator together with the hold function of the output operator in the lifted representation of sampled‐data systems are approximated by piecewise linear functions. By defining adequate preadjoint operators, the PLKA method is shown to have the associated convergence rate of 1/N2 for the deterioration of the attainable L∞‐induced norm performance when the optimal controller synthesis is conducted approximately under the approximation parameter N. Compared with another existing procedure for controller synthesis through different approximation treatment called the piecewise linear input approximation method, we further show that the proposed PLKA method has a quantitatively improved bound on the performance deterioration. Finally, numerical examples are studied to verify the effectiveness of the proposed PLKA method.