Robust stability analysis with cycling-based LPTV scaling. Part II: its properties under the use of FIR separators

ABSTRACT

This paper is concerned with robust stability analysis of discrete-time linear periodically time-varying (LPTV) systems using the cycling-based LPTV scaling approach. It consists of applying the separator-type robust stability theorem through the cycling-based treatment of such systems, where this paper aims at revealing fundamental properties of this approach when we confine ourselves to what we call finite impulse response (FIR) separators as a theoretically and numerically very tractable class of separators. Specifically, we clarify such properties of the cycling-based LPTV scaling approach using FIR separators that cannot readily be seen under the treatment of general class of separators. This is accomplished by comparing it with another approach, called lifting-based LPTV scaling using FIR separators, through the framework of representing the associated robust stability conditions with infinite matrices. More precisely, this leads us to clarifying the fundamental relationships between the cycling-based and lifting-based approaches under the use of FIR separators. We also provide a numerical example demonstrating the fundamental relationships clarified in this paper.

KEYWORDS:

• Robust stability analysis
• discrete-time LPTV systems
• cycling
• lifting
• infinite matrices

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. These matrix inequalities are linear if the variable corresponding to (the inverse of) $\bar{\delta }$ is fixed; hence we can solve them with a bisection method (although the details are omitted). The sizes of the (linear) matrix inequalities to be solved are νNp(K_lf + 1) + n and νNp(K_cy + 1) + νNn in the lifting-based and cycling-based cases, respectively.

2. For reference, we state that this was not the case in this example when cycling-based LPTV scaling is applied. However, answering the question of whether this is general will be left for our future work.