L1 analysis of LTI systems via piecewise higher-order approximation

This paper deals with the L₁ analysis of linear time-invariant (LTI) systems, by which we mean the L_∞-induced norm analysis of LTI systems. It is well known that this induced norm corresponds to the L₁ norm of the impulse response of the given system, i.e., integral of the absolute value of the kernel function in the convolution formula for the input/output relation. However, because it is very hard to compute this integral exactly or even approximately with explicit upper and lower bounds, the ideas of piecewise constant and piecewise linear approximations have been developed to compute upper and lower bounds of the L_∞-induced norm in our preceding study. These ideas 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 piecewise constant or piecewise linear approximation converge to 0 at the rate of 1/M or 1/M², respectively. Motivated by the success of the L_∞-induced norm analysis in that study, this paper aims at developing extended schemes named piecewise quadratic and piecewise cubic approximations. These approximations are also developed through fast-lifting, and it is shown that the piecewise quadratic and piecewise cubic approximation leads to approximation errors converging to 0 at the rate of 1/M³ and 1/M⁴, respectively.