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arXiv · 2502.04062

On Sufficient Richness for Linear Time-Invariant Systems

Marco Borghesi·Simone Baroncini·Guido Carnevale·Alessandro Bosso·Giuseppe Notarstefano
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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.

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Marco Borghesi, Simone Baroncini, Guido Carnevale, Alessandro Bosso, Giuseppe Notarstefano. 2025-02-06. On Sufficient Richness for Linear Time-Invariant Systems. https://arxiv.org/abs/2502.04062

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