This article focuses on the adaptive linear quadratic Gaussian control problem, where both the state matrix A and the control gain matrix B are unknown. We only assume that (A, B) is stabilizable and (A, Q(1/2)) is detectable, where Q is the weighting matrix of the state in the quadratic cost function. This significantly weakens the classic assumptions used in the literature. To design an optimal adaptive control, a weighted least squares algorithm is modified by using random regularization method, which can ensure uniform stabilizability and uniform detectability of the family of estimated models. At the same time, a diminishing excitation is incorporated into the design of the proposed adaptive control to guarantee strong consistency of the desired components of the estimates. Finally, although some components of the estimates may not converge to the true values, it is still demonstrated that a certainty equivalence control with diminishing excitation remains optimal for an ergodic quadratic cost function.

Publication:

IEEE TRANSACTIONS ON AUTOMATIC CONTROL

http://dx.doi.org/10.1109/TAC.2025.3568032

Author:

Nian Liu and Shaolin Tan

Zhongguancun Laboratory, Beijing 100094, China

e-mail: liunian@amss.ac.cn; shaolintan@hnu.edu.cn

Cheng Zhao

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100190, China

e-mail:zhaocheng@amss.ac.cn

Jinhu Lü

Zhongguancun Laboratory, Beijing 100094, China,and also with the Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

e-mail: jhlu@iss.ac.cn