Feedback stabilization of discrete-time homogeneous semi-linear systems

Lars Grüne,Fabian Wirth

doi:10.1016/s0167-6911(98)00110-8

Lars Grüne, Fabian Wirth

Open Access

https://doi.org/10.1016/s0167-6911(98)00110-8

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Abstract

For discrete-time semi-linear systems satisfying an accessibility condition asymptotic null controllability is equivalent to exponential feedback stabilizability using a piecewise constant feedback. A constructive procedure that yields such a feedback is presented.

Highlights

We consider the problem of feedback stabilizability of homogeneous semi-linear discrete time systems, that is systems linear in the state where the entries of the transition matrix are functions of the control
This class is a generalization of the frequently studied bilinear systems. Systems of this form occur as linearizations with respect to the state only of nonlinear systems at singular points
In the more general semi-linear case it has been recently shown in Grune 10] that null controllability is equivalent to feedback stabilizability by discretized feedbacks and a numeric procedure for the calculation of stabilizing feedbacks has been presented

Summary

Introduction

We consider the problem of feedback stabilizability of homogeneous semi-linear discrete time systems, that is systems linear in the state where the entries of the transition matrix are functions of the control This class is a generalization of the frequently studied bilinear systems. In the more general semi-linear case it has been recently shown in Grune 10] that null controllability is equivalent to feedback stabilizability by discretized feedbacks and a numeric procedure for the calculation of stabilizing feedbacks has been presented. For general nonlinear systems in continuous time it has been shown in 6] that asymptotic controllability is equivalent to feedback stabilizability by means of a sampled feedback This approach, does not lead to exponential stabilization and is only constructive up to the fact that the knowledge of a control Lyapunov function is required.

Problem Formulation

Main result

Construction of the Feedback

A Numerical Construction of the Feedback