Discrete-Time Noncausal Linear Periodically Time-Varying Scaling for Robustness Analysis and Controller Synthesis

Since modeling of real plants inevitably gives rise to modeling errors regarded as uncertainties, considering robustness for the uncertainties is important in actual control problems. For tackling issues of analyzing robust stability of closed-loop systems in a less conservative fashion, the μ-analysis method is known to be effective. As an alternative approach to robust stability analysis, on the other hand, discrete-time noncausal linear periodically time-varying (LPTV) scaling has been proposed recently. This approach can be naturally introduced through the lifting-based treatment of systems, and the associated conservativeness can be reduced by increasing the period of lifting. This thesis is concerned with this lifting-based scaling approach. In this thesis, we first review the definition and properties of noncausal LPTV scaling. This scaling approach is a generalization of the conventional causal linear time-invariant (LTI) scaling, and coincides with the latter scaling when we take the lifting period N = 1 (i.e., without lifting-based treatment). We call such a framework for robust stability analysis without lifting-based treatment the lifting-free framework, and that with liftingbased treatment the lifting-based framework. It is known that noncausal LPTV scaling induces dynamic causal LTI scaling in the lifting-free framework even if it is confined to static in the lifting-based framework. We show two theorems associated with this promising property, and confirm them numerically. After such reviews, we consider applying static noncausal LPTV scaling to robust controller synthesis. If we take account of only robust stability in the synthesis, however, responses of the resulting control systems may become oscillatory at an unacceptable level. To avoid this problem, we develop a controller synthesis method taking account of not only robust stability but also robust H∞ performance. Effectiveness of the developed method for robust control is discussed theoretically and numerically, and also demonstrated by experiments with a cart inverted pendulum whose length can be set from three choices (we regard the difference as the uncertainty in the experiments). The obtained results will indicate that the developed synthesis framework is indeed effective and practical in actual control problems.