Generalization of Piecewise Constant Approximation in the L∞/L2 Optimal Control of Sampled-Data Systems

This paper is concerned with the
$L_{\infty}/L_{2}$
-induced norm of linear time-invariant (LTI) sampled-data systems, i.e., the induced norm from
$L_{2}$
to
$L_{\infty}$
in LTI sampled-data systems. In our preceding studies, the idea of piecewise constant approximation has been developed for analyzing and minimizing the
$L_{\infty}/L_{2}$
-induced norm in LTI sampled-data systems via the fast-lifting technique together with the Taylor expansion of relevant functions, and it is shown that the piecewise constant approximation has the convergence rate of 1/
$N$
for both the analysis and minimization problems with respect to the fast-lifting parameter N. Along this line, this paper develops a generalized scheme to the piecewise constant approximation by taking advantage of the freedom in the point around which associated functions are expanded to Taylor series. It is further shown in this paper that even though the piecewise constant approximation has the convergence rate of 1/
$N$
regardless of the point at which the Taylor expansion is applied, we can obtain a new discretization method for LTI sampled-data systems with quantitatively improved accuracies than those in the preceding studies for both the analysis and minimization problems through the generalization scheme.