[Paper Review] Memoryless output nullification and canonical forms, for time varying systems

Gera Weiss|ArXiv.org|Mar 15, 2005

Stability and Control of Uncertain Systems11 references3 citations

TL;DR

This paper presents a memoryless output feedback control algorithm that uniformly nullifies finite-dimensional linear time-varying single-input single-output systems within a bounded number of steps, provided the system is completely controllable, completely observable, and satisfies the condition $ c_k \operatorname{adj}(A_k)b_k \neq 0 $ for all $ k $. The method leverages a time-varying controller canonical form and guarantees nullification in at most $ 2(n^4 + n^3 + n^2) $ steps, with a constructive feedback law derived from system invariants.

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.

Motivation & Objective

• To establish sufficient conditions under which time-varying linear systems can be nullified using memoryless output feedback.
• To develop an explicit algorithm for computing feedback gains that achieve state nullification within finite time.
• To derive an upper bound on the nullification time that depends only on system dimension.
• To analyze the preservation of controllability under zero-hold sampling for continuous-time systems with analytic coefficients.

Proposed method

• Represent the system in time-varying controller canonical form via algebraic equivalence transformations.
• Use the invariant $ c_k \operatorname{adj}(A_k)b_k \neq 0 $ as a key criterion for nullifiability under memoryless feedback.
• Construct a feedback law $ u_k = F_k y_k $ such that the closed-loop system reaches the origin in finite steps.
• Apply a recursive nullification procedure that successively drives components of the state to zero over $ n $ cycles.
• Leverage the adjugate matrix and determinant invariance under similarity transformations to preserve the nullification condition across canonical forms.
• Analyze sampled-data systems by showing that controllability is preserved under zero-hold sampling at almost any rate for analytic-coefficient systems.

Experimental results

Research questions

• RQ1 Under what conditions can a time-varying linear system be nullified using memoryless output feedback?
• RQ2 Can a finite upper bound on the nullification time be derived that depends only on system dimension?
• RQ3 How does the controller canonical form facilitate the design of memoryless feedback for nullification?
• RQ4 Does zero-hold sampling preserve controllability in time-varying continuous-time systems with analytic coefficients?
• RQ5 Is the condition $ c_k \operatorname{adj}(A_k)b_k \neq 0 $ for all $ k $ both necessary and sufficient for nullification in time-varying systems?

Key findings

• Any completely controllable and completely observable time-varying discrete-time system satisfying $ c_k \operatorname{adj}(A_k)b_k \neq 0 $ for all $ k $ is uniformly nullifiable via memoryless output feedback.
• The nullification time is bounded above by $ 2(n^4 + n^3 + n^2) $ steps, where $ n $ is the system dimension.
• The controller canonical form representation ensures that the nullification condition reduces to checking the first coordinate of the output vector $ c_k $.
• The condition $ c_k \operatorname{adj}(A_k)b_k \neq 0 $ is invariant under algebraic equivalence transformations, enabling canonical form analysis.
• For continuous-time systems with analytic coefficients, controllability is preserved under zero-hold sampling at almost any sampling rate.
• The algorithm constructs feedback coefficients $ F_k $ explicitly using the controller canonical form, enabling practical implementation.