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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.19124 |
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| _version_ | 1866915304116846592 |
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| author | Liu, Xingrui Ke, Jieming Zhao, Yanlong |
| author_facet | Liu, Xingrui Ke, Jieming Zhao, Yanlong |
| contents | This paper investigates the optimality analysis of the recursive least-squares (RLS) algorithm for autoregressive systems with exogenous inputs (ARX systems). A key challenge in analyzing is managing the potential unboundedness of the parameter estimates, which may diverge to infinity. Previous approaches addressed this issue by assuming that both the true parameter and the RLS estimates remain confined within a known compact set, thereby ensuring uniform boundedness throughout the analysis. In contrast, we propose a new analytical framework that eliminates the need for such a boundness assumption. Specifically, we establish a quantitative relationship between the bounded moment conditions of quasi-stationary input/output signals and the convergence rate of the tail probability of the RLS estimation error. Based on this technique, we prove that when system inputs/outputs have bounded twentieth-order moments, the RLS algorithm achieves asymptotic normality and the covariance matrix of the RLS algorithm converges to the Cramér-Rao lower bound (CRLB), confirming its asymptotic efficiency. These results demonstrate that the RLS algorithm is an asymptotically optimal identification algorithm for ARX systems, even without the projection operators to ensure that parameter estimates reside within a prior known compact set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19124 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Efficiency Analysis of the Recursive Least-Squares Algorithm for ARX Systems Without Projection Liu, Xingrui Ke, Jieming Zhao, Yanlong Optimization and Control This paper investigates the optimality analysis of the recursive least-squares (RLS) algorithm for autoregressive systems with exogenous inputs (ARX systems). A key challenge in analyzing is managing the potential unboundedness of the parameter estimates, which may diverge to infinity. Previous approaches addressed this issue by assuming that both the true parameter and the RLS estimates remain confined within a known compact set, thereby ensuring uniform boundedness throughout the analysis. In contrast, we propose a new analytical framework that eliminates the need for such a boundness assumption. Specifically, we establish a quantitative relationship between the bounded moment conditions of quasi-stationary input/output signals and the convergence rate of the tail probability of the RLS estimation error. Based on this technique, we prove that when system inputs/outputs have bounded twentieth-order moments, the RLS algorithm achieves asymptotic normality and the covariance matrix of the RLS algorithm converges to the Cramér-Rao lower bound (CRLB), confirming its asymptotic efficiency. These results demonstrate that the RLS algorithm is an asymptotically optimal identification algorithm for ARX systems, even without the projection operators to ensure that parameter estimates reside within a prior known compact set. |
| title | Asymptotic Efficiency Analysis of the Recursive Least-Squares Algorithm for ARX Systems Without Projection |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2505.19124 |