Lyapunov Spectrum Local Assignability of Linear Discrete Time-Varying Systems by Static Output Feedback

Adam Czornik,Vasilii Zaitsev,Svetlana Popova,Evgenii Makarov,Michal Niezabitowski

doi:10.1109/access.2021.3114767

Adam Czornik, Vasilii Zaitsev + Show 3 more

Open Access

https://doi.org/10.1109/access.2021.3114767

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Abstract

We consider a linear discrete time-varying input-output system. Our goal is to study the problem of local assignability of the Lyapunov spectrum by static output feedback control. To this end we introduce the notion of uniform consistency for discrete-time linear systems which is the extension of the notion of uniform complete controllability to input-output systems. The property of uniform consistency is investigated, some necessary and sufficient conditions for this property are obtained. The notion of uniform local attainability is introduced for the closed-loop system. We prove that uniform consistency implies uniform local attainability of the closed-loop system. The property of local Lyapunov reducibility is introduced for the closed-loop system. We prove that uniform local attainability implies local Lyapunov reducibility. We prove that, for a locally Lyapunov reducible system, the Lyapunov spectrum is locally assignable, if the free system is diagonalizable or regular (in the Lyapunov sence) or has the stable Lyapunov spectrum.

Highlights

It is well known that a well-designed feedback controller is expected to produce the required output, and to ensure the satisfactory quality for the transition process, e.g., provide the required overall decay rate of the solutions or some appropriate oscillatory properties
The property of uniform consistency have been developed in detail, the necessary and sufficient conditions for this property have been obtained
The notions of uniform local attainability and local Lyapunov reducibility have been introduced, which were previously introduced for continuous-time systems

Summary

INTRODUCTION

It is well known that a well-designed feedback controller is expected to produce the required output, and to ensure the satisfactory quality for the transition process, e.g., provide the required overall decay rate of the solutions or some appropriate oscillatory properties. A. Czornik et al.: Lyapunov Spectrum Local Assignability of Linear Discrete Time-Varying Systems by Static Output Feedback. Necessary and sufficient conditions for assignability of the dichotomy spectrum for one-sided discrete time-varying linear systems are obtained in [29]. A series of papers [30]–[33] discussed investigations of the problem of Lyapunov exponents placement for discrete-time systems In these works, sufficient conditions are obtained for the solvability of the problem of assigning the Lyapunov spectrum of discrete non-stationary systems in various formulations. The definition of uniform consistency was given in [34] for continuous-time systems, and in [35] for discrete-time systems This new notion allows us to obtain sufficient condition for the above formulated problem.

DEFINITIONS

PROPERTIES OF UNIFORMLY CONSISTENT SYSTEMS

UNIFORM LOCAL ATTAINABILITY

INTERCONNECTION BETWEEN UNIFORM

CONCLUSION