On the sample complexity of observers for unknown linear systems with biased dynamics estimations

Abstract

AbstractObservers have broad applications in power systems, whereas observer models are hard to obtain when the system is unknown. This article considers the observer design problems for unknown noisy linear time‐invariant systems based on biased dynamics estimations. Unlike the unbiased methods, biased methods have stronger generalization ability, which benefits obtaining stable estimations for noisy systems. However, the biased estimation's influence on observer design still needs to be investigated. To analyze the influence, we exploit the impact of estimation bias‐variance trade‐off to observer design. Specifically, we propose a support vector regression (SVR) based estimator to provide biased estimations for the system identification of unknown linear systems. The sample complexity results of SVR with bias‐variance trade‐offs are analyzed and used for observer design and performance analysis. Then, a stable observer gain design algorithm is developed based on biased estimation. The observation performance is evaluated by the mean square observation error, which is shown to be adjustable by tuning the trade‐off between bias and variance, thus achieving higher scalability than the unbiased methods. Finally, observing performance analysis demonstrates the influence of the bias‐variance trade‐off for the observer. Extensive simulation validations are conducted to verify the computed estimation error and performance optimality with different bias‐variance trade‐offs and noise settings.