Computation of the Induced Norm from L-2 to L-infinity in SISO Sampled-Data Systems: Discretization Approach with Convergence Rate Analysis

Abstract

This paper provides a discretization method for computing the induced norm from L-2 to L-infinity in single-input/single-output (SISO) linear time-invariant (LTI) sampled-data systems. We first follow the lifting-based treatment for the induced norm from L-2 to L-infinity of SISO LTI sampled-data systems, but further apply the key idea of fast-lifting, by which the sampling interval [0; h) is divided into M subintervals with an equal width. Such an idea allows us to develop two methods for computing the induced norm with gridding and piecewise constant approximations. These methods leads to approximately equivalent discretization methods of the generalized plant that can be used for readily computing upper and lower bounds of the induced norm together with the derivation of the associated convergence rates. More precisely, it is shown that the approximation error converges to 0 at the rate of 1/root M and 1/M in the gridding and piecewise constant approximation methods, respectively.