Yohei Hosoe

Properties of discrete-time noncausal linear periodically time-varying scaling and their relationship with shift-invariance in lifting-timing

This article is concerned with the technique called discrete-time noncausal linear periodically time-varying (LPTV) scaling for robust stability analysis. Noncausal LPTV scaling has already been shown to be effective for reducing the conservativeness of robustness analysis in theoretical and numerical ways. However, there still remain some issues to be resolved for further understanding and exploiting noncausal LPTV scaling, e.g. its relationship with the conventional analysis approach of causal linear time-invariant scaling. In this article, by introducing the key idea of shift-invariance in lifting-timing, we discuss the difference and corresponding relationship between the conventional approach and noncausal LPTV scaling.

Relationship between Noncausal Linear Periodically Time-Varying Scaling and Causal Linear Time-Invariant Scaling for Discrete-Time Systems

Abstract In this paper, we discuss the relationship between noncausal linear periodically time-varying (LPTV) scaling and causal linear time-invariant (LTI) scaling against discrete-time LTI closed-loop system. Noncausal LPTV scaling is naturally introduced via lifting technique, and can induce some frequency-dependent scaling in the lifting-free (i.e., usual) framework. However, it has not been clear what classes of noncausal LPTV scaling and causal LTI scaling have equivalent abilities in their respective frameworks. It is an important issue for sophisticating the theoretical base of noncausal LPTV scaling, and this paper studies such a relationship.

State feedback synthesis for robust stabilization of discrete-time linear systems characterized by stochastic polytopes

This paper discusses robustly stabilizing state feedback synthesis of discrete-time stochastic plants whose dynamics are characterized by convex polytopes (called stochastic polytopes) consisting of random matrices (i.e., matrices involving random variables). The stochastic polytopes enable us to describe the uncertainties in the probability distributions underlying the stochastic systems. Hence, we can study robust stability (in the stochastic sense) of the systems with respect to the uncertainties in the distributions, through dealing with stochastic polytopes. This paper gives a synthesis-oriented sufficient condition for robust closed-loop stability, and states a numerical design method exploiting the condition. The effectiveness of the method is also demonstrated with a numerical example.

Robust stability analysis of discrete-time linear systems characterized by stochastic polytopes

This paper discusses robust stability analysis of discrete-time stochastic systems whose system matrices belong to convex polytopes (called stochastic polytopes) consisting of random matrices (i.e., matrices involving random variables). The stochastic polytopes enable us to describe the uncertainty in the probability distribution of the system matrix. Hence, we can tackle the problem of deciding whether the system is robustly stable (in the stochastic sense) with respect to the uncertainty in the distribution, through dealing with the stochastic polytopes. This paper gives sufficient conditions for analyzing such robust stability, and provides a numerical example showing the effectiveness of the developed analysis framework.

Unified treatment of robust stability conditions for discrete-time systems through an infinite matrix framework☆

Abstract This paper is motivated by the study on clarifying further relationship between the conventional lifting-free causal linear time-invariant (LTI) scaling and lifting-based noncausal linear periodically time-varying (LPTV) scaling approaches to robust stability analysis. To facilitate such a study, this paper gives the infinite matrix representation counterparts of the robust stability conditions in the separator-type robust stability theorems for these approaches. These counterparts lead to the idea of infinite-dimensional separators and provide us with a unified framework for studying the mutual relationship between these two approaches. This paper takes causal LTI and noncausal LPTV separators characterized by finite impulse response, which are respectively defined in the lifting-free and lifting-based frameworks, and compares them with respect to conservativeness of robust stability analysis by means of the unified framework. Through such a discussion, it is demonstrated that the unified framework can lead to a very comprehensible and intuitive study on the mutual relationship between causal LTI and noncausal LPTV scaling approaches.

Robust stability analysis based on discrete-time FIR scaling

This paper is concerned with robust stability analysis of discrete-time linear time-invariant (LTI) systems via the separator-type robust stability theorem. It develops a framework for dealing with finite impulse response (FIR) scaling, as a special class of dynamic causal LTI scaling. FIR separators to be searched for in FIR scaling are dynamic in general, and they are much more difficult to directly deal with than static LTI separators. This paper resolves such a difficulty by exploiting some relevant results on the technique called noncausal linear periodically time-varying (LPTV) scaling and the well-known KYP lemma. The FIR scaling applied in this way enables us to analyze robust stability of closed-loop systems in a less conservative fashion than conventional static LTI scaling, and is shown to be more effective than μ-analysis through a numerical example.

Synthesis of Robust Performance Controllers Based on Discrete-Time Noncausal Linear Periodically Time-Varying Scaling

Abstract This paper is concerned with the discrete-time noncausal linear periodically time-varying (LPTV) scaling technique. It is defined through the discrete-time lifting, and it has been shown that even static noncausal LPTV scaling induces some frequency-dependent scaling when it is interpreted in the context of lifting-free treatment. It has been also shown that this feature together with the fact that the uncertainties come to be dealt with in their lifted forms leads to an effective method for robust stability analysis. This paper shows that noncausal LPTV scaling is effective also in controller design. More precisely, we study the periodic robust performance controller synthesis problem, and discuss how the properties of noncausal LPTV scaling lead to the performance improvement when the period of the controller is increased.

Extension of the concept of random polytopes and robust stabilization synthesis

In this paper, we introduce families of what we call random polytopes, which are defined through random matrices. Then, we discuss robustly stabilizing state feedback synthesis for discrete-time plants whose dynamics are characterized by the families. Random polytopes enable us to deal with uncertainties in the distributions underlying stochastic systems. Through extending the concept of the polytopes (from the viewpoint of probability measures), we develop a framework of robust stabilization synthesis that can tackle larger classes of uncertainties in the distributions.

Robust stability analysis based on noncausal LPTV FIR scaling: Explicit procedure and relationship with causal LTI FIR scaling

This paper develops a framework of a robust stability analysis approach called discrete-time noncausal linear periodically time-varying (LPTV) finite impulse response (FIR) scaling. This approach is an extension of (frequency-dependent) causal linear time-invariant FIR scaling, obtained by further introducing time dependence through lifting treatment. This paper first provides a theoretical result showing how such newly introduced time dependence contributes to reduction of conservativeness of robust stability analysis. We further provide linear matrix inequality conditions for robust stability and develop an explicit and feasible numerical method by exploiting noncausal LPTV FIR scaling. The effectiveness of this scaling approach is demonstrated and the above theoretical result is confirmed through numerical examples.

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

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. To study the properties of this approach in comparison with the lifting-based LPTV scaling approach, we consider exploiting the framework of representing the associated robust stability conditions with infinite matrices. Since it serves as a common framework for comparing the two different LPTV scaling approaches, it provides us with new insights into the relationship between the cycling-based and lifting-based scaling approaches. In particular, we derive fundamental results that enable us to reduce the comparison, with respect to conservativeness in robust stability analysis, of the two scaling approaches with restricted and tractable classes of separators to a modified comparison of the associated classes of what we call infinite-dimensional separators arising in the above infinite matrix framework.