Computing the L∞-induced norm of linear time-invariant systems via Kernel approximation and its comparison with input approximation

Abstract

This study deals with the L₁ analysis of stable finite-dimensional linear time-invariant (LTI) systems, by which the authors mean the computation of the L∞-induced norm of these systems. To compute this norm, they need to integrate the absolute value of the impulse response of the given system, which corresponds to the kernel function in the convolution formula for the input/output relation. However, it is very difficult to compute this integral exactly or even approximately with an explicit upper bound and lower bound. They first review an approach named input approximation, in which the input of the LTI system is approximated by a staircase or piecewise linear function and computation methods for an upper bound and lower bound of the L∞-induced norm are given. They further develop another approach using an idea of kernel approximation, in which the kernel function in the convolution is approximated by a staircase or piecewise linear function. These approaches are introduced through fast-lifting, by which the interval [0, h) with a sufficiently large h is divided into M subintervals with an equal width. It is then shown that the approximation errors in staircase or piecewise linear approximation are ensured to be reciprocally proportional to M or M², respectively. The effectiveness of the proposed methods is demonstrated through numerical examples.