Input Redundancy under Input and State Constraints (Extended version of the submission accepted to Automatica)

TL;DR

This paper investigates how input redundancy in dynamical systems is affected by input and state constraints, providing conditions under which redundancy is preserved or destroyed, especially in linear constraint scenarios.

Contribution

It extends the concept of input redundancy to constrained systems and offers a general condition for its preservation, including linear constraint cases.

Findings

Redundancy can be destroyed when trajectories lie on constraint borders.

A sufficient condition for redundancy preservation under arbitrary constraints.

Specialized results for linear input and state constraints.

Abstract

For a given unconstrained dynamical system, input redundancy has been recently redefined as the existence of distinct inputs producing identical output for the same initial state. By directly referring to signals, this definition readily applies to any input-to-output mapping. As an illustration of this potentiality, this paper tackles the case where input and state constraints are imposed on the system. This context is indeed of foremost importance since input redundancy has been historically regarded as a way to deal with input saturations. An example illustrating how constraints can challenge redundancy is offered right at the outset. A more complex phenomenology is highlighted. This motivates the enrichment of the existing framework on redundancy. Then, a sufficient condition for redundancy to be preserved when imposing constraints is offered in the most general context of arbitrary constraints. It is shown that redundancy can be destroyed only when input and state trajectories lie on the border of the set of constraints almost all the time. Finally, those results are specialized and expanded under the assumption that input and state constraints are linear.