Modified fast-sample/fast-hold approximation and γ-independent H∞-discretisation for general sampled-data systems by fast-lifting

This paper is concerned with the fast-lifting approach to H∞ analysis and design of sampleddata systems, and extends our preceding study on modified fast-sample/fast-hold (FSFH) approximation, in which the direct feedthrough matrix D11 from the disturbance w to the controlled output z was assumed to be zero. More precisely, this paper removes this assumption and shows that a γ-independent H∞ discretization is still possible in a nontrivial fashion by applying what we call quasi-finite-rank approximation of an infinite-rank operator and then the loop-shifting technique. As in the case of D11 = 0, the modified FSFH approach retains the feature that both the upper and lower bounds of the H∞-norm or the frequency response gain can be computed, where the gap between the upper and lower bounds can be bounded with the approximation parameter N and is independent of the discrete-time controller. This feature is significant in applying the new method especially to control system design, and this study indeed has a very close relationship to the recent progress in the study of control system analysis/design via noncausal linear periodically time-varying scaling. The significance of a key lemma pertinent to the fast-lifting approach is suggested in connection with such a relationship, and also with its application to time-delay systems.