Robust stability analysis with cycling-based LPTV scaling: part I. Fundamental results on its relationship with lifting-based LPTV scaling

Cycling-Based Synthesis of Robust Output Estimators for Uncertain LPTV Systems

“…This theorem, together with the equivalence relation about robust stability shown in [7], guarantees that the problem of designing a robust output estimator E for the LPTV system shown in Fig. 1 can equivalently reduce to that of designing the formal cycled counterpartĚ for the LTI system shown in Fig.…”

“…A very closely related issue has been studied in [7], where robust stability of the formal cycled representation was shown to be equivalent to that of the original (cycling-free) system. In addition, we can show that a similar assertion also holds in our present situation about robust performance and obtain the following theorem, which constitutes a theoretical contribution in this paper.…”

“…The matrixΘ(ζ) in the above theorem is called a cyclingbased separator. By searching forΘ(ζ) satisfying (17) and 18, we can analyze robust l 2 performance of G Δ ; we call such a robustness analysis approach cycling-based LPTV scaling [7], [8].…”

“…Hence, one of the simplest ways of designing robust output estimators for uncertain LPTV systems is to exploit the result in [1] for using the separator-type robust stability theorem in the synthesis through the lifting-based treatment of systems. Nevertheless, this paper considers dealing with LPTV systems through the use of the technique called cycling [5], [6], instead of lifting; we call the cycling-based robustness analysis approach using the separator-type theorem the cycling-based LPTV scaling [7], [8]. The reason why we consider using not lifting but cycling in the synthesis is closely related to the following motivation of this study.…”

“…Since we evaluate the performance of the estimators by using the l 2 -induced norm, we first show in this paper the equivalence between the l 2 -induced norm of the cycled system with the structured signals and that of the system obtained by viewing the cycled system as the usual time-invariant system (i.e., without the structural constraints on signals). Based on such equivalence and the ideas in [1], [7], [8], we develop a method of designing robust output estimators, which allow us to predetermine the estimator period regardless of the system period. Note that the same topic has been already dealt with in our conference paper [9].…”