Computing the L∞-induced norm of LTI systems

This paper studies computing the L_∞-induced norm of stable finite-dimensional linear time-invariant (LTI) systems. To compute this norm, we 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. We first review an approach named input approximation, in which the inputs of the LTI system are 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. We 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, and it is shown that the approximation errors in staircase or piecewise linear approximation are ensured to be reciprocally proportional to M or M², respectively.