Memoryless output nullification and canonical forms, for time varying systems

TL;DR

This paper demonstrates that for generic controllable and observable linear time-varying systems, it is possible to nullify the state using memoryless output feedback within finite time, with algorithms and bounds provided, and extends results to sampled-data systems.

Contribution

It introduces a nullification algorithm for time-varying systems in controller canonical form and proves controllability preservation under zero-hold sampling for systems with analytic coefficients.

Findings

Any state can be driven to zero in finite time using memoryless output feedback.

An explicit nullification algorithm with a dimension-dependent upper bound is provided.

Controllability is preserved under zero-hold sampling at almost any rate for systems with analytic coefficients.

Abstract

We study the possibility of nullifying time-varying systems with memoryless output feedback. The systems we examine are linear single-input single-output finite-dimensional time-varying systems. For generic completely controllable and completely observable discrete-time systems, we show that any state at any time can be steered to the origin within finite time. An algorithm for nullification and an upper bound for nullification time, depending only on the system's dimension, are provided. The algorithm is described using a representation of the system in time-varying controller canonical form. We verify that every completely controllable system has such a representation. The application of the nullification algorithm to sampled-data systems is also analysed: we show that a controllable continuous-time time-varying system with analytic coefficients can be nullified utilising zero-hold sampling of the output and time-varying memoryless linear feedback; for generic observables, almost any sampling period can be used. We also prove that controllability of time-varying systems with analytic coefficients is preserved under zero-hold sampling at almost any rate.