A Study on the Optimal Controller Synthesis for Minimizing the L-2 Norm of the Response to the Worst-Timing Impulse Disturbance in LTI Sampled-Data Systems

Abstract

This paper is concerned with the problem of minimizing the L-2 norm of the response to the worst-timing impulse input in linear time-invariant (LTI) sampled-data systems. Such a measure is recently introduced by the authors as an alternative to the two existing definitions of H-2 norm for LTI sampled-data systems, and it is also called the third H-2 norm of the systems. Taking into account of the linear periodically time-varying (LPTV) nature of LTI sampled-data systems, the third H-2 norm was defined as the supremum of the L-2 norms of all the tau-dependent outputs for the impulse inputs occurring at the instant tau on the sampling interval [0, h). A discretization approach to the continuous-time generalized plant has been developed through the lifting technique together with a gridding approximation method, by which the analysis problem of the third H-2 norm can be approximately reduced to that of the discrete-time H-2 norm. This approach allowed us to compute an upper bound and a lower bound of the third H-2 norm in an asymptotically exact fashion as the gridding parameter N tends to infinity. However, it is unclear whether the discretization approach is valid also in the associated controller synthesis problem since the gap between the upper and lower bounds given for a fixed (finite) parameter N is dependent on the discrete-time controller. In this sense, this paper aims at establishing a theoretical validity of the discretization approach in the optimal controller synthesis problem by deriving an important inequality independent of the discrete-time controller. This inequality verifies that designing an optimal controller based on the discretization approach and letting the gridding approximation parameter N sufficiently large lead to a method of the optimal controller synthesis for minimizing the third H-2 norm of LTI sampled-data systems in the convergence rate of 1/N.