On Sufficient Richness for Linear Time-Invariant Systems
Fuente:
arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912222096130048 |
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| author | Borghesi, Marco Baroncini, Simone Carnevale, Guido Bosso, Alessandro Notarstefano, Giuseppe |
| author_facet | Borghesi, Marco Baroncini, Simone Carnevale, Guido Bosso, Alessandro Notarstefano, Giuseppe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04062 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Sufficient Richness for Linear Time-Invariant Systems Borghesi, Marco Baroncini, Simone Carnevale, Guido Bosso, Alessandro Notarstefano, Giuseppe Systems and Control 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. |
| title | On Sufficient Richness for Linear Time-Invariant Systems |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2502.04062 |