On Sufficient Richness for Linear Time-Invariant Systems

TL;DR

This paper establishes necessary and sufficient conditions for persistent excitation in multi-input LTI systems, ensuring parameter convergence in adaptive schemes, and characterizes the set of sufficiently rich input signals.

Contribution

It provides simple PE conditions for discrete and continuous-time LTI systems and characterizes the shape of SR input signal sets, with numerical validation of tightness.

Findings

Derived PE conditions are tight and optimal.

Characterized the set of SR input signals for multi-input LTI systems.

Numerical example confirms the conditions cannot be improved.

Abstract

Persistent excitation (PE) is a necessary and sufficient condition for uniform exponential parameter convergence in several adaptive, identification, and learning schemes. In this article, we consider, in the context of multi-input linear time-invariant (LTI) systems, the problem of guaranteeing PE of commonly-used regressors by applying a sufficiently rich (SR) input signal. Exploiting the analogies between time shifts and time derivatives, we state simple necessary and sufficient PE conditions for the discrete- and continuous-time frameworks. Moreover, we characterize the shape of the set of SR input signals for both single-input and multi-input systems. Finally, we show with a numerical example that the derived conditions are tight and cannot be improved without including additional knowledge of the considered LTI system.