Strong Consistency and Rate of Convergence of Switched Least Squares System Identification for Autonomous Markov Jump Linear Systems

• Published in: IEEE transactions on automatic control 2024-06, Vol.69 (6), p.3952-3959
• Main Authors: Sayedana, Borna, Afshari, Mohammad, Caines, Peter E., Mahajan, Aditya
• Format: Article
• Language: English
• Subjects: Asymptotic stability; Autonomous systems; Convergence; Least mean squares methods; Least squares method; Linear systems; Numerical stability; parameter estimation; Stability; Stability analysis; statistical learning; Switches; switching systems; System identification
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2024.3351806

In this article, we investigate the problem of system identification for autonomous Markov jump linear systems (MJS) with complete state observations. We propose switched least squares method for identification of MJS, show that this method is strongly consistent, and derive data-dependent and data-independent rates of convergence. In particular, our data-independent rate of convergence shows that, almost surely, the system identification error is \mathcal {O}(\sqrt{\log (T)/T}) where T is the time horizon. These results show that the switched least squares method for MJS has the same rate of convergence as the least squares method for autonomous linear systems. We derive our results by imposing a general stability assumption on the model called stability in the average sense. We show that stability in the average sense is a weaker form of stability compared with the stability assumptions commonly imposed in the literature. We present numerical examples to illustrate the performance of the proposed method.