Online Optimal State Feedback Control of Linear Systems Over Wireless MIMO Fading Channels

In this article, we consider the optimal control of linear systems over wireless multiple-input multiple-output (MIMO) fading channels, where the wireless MIMO fading and random access of the remote controller may cause intermittent controllability or uncontrollability of the closed-loop control system. We consider the finite-time horizon optimal control first and then check the conditions for the existence and uniqueness of optimal control for the infinite-time horizon case. Specifically, for the finite-time horizon case, we show that the optimal control gain matrix can be computed offline based on the distribution of the wireless MIMO fading and random access of the controller. For the infinite-time horizon case, we show that, when the closed-loop system is almost surely controllable, the optimal control solution always exists and is unique. In the case that MIMO fading channels and the random access of the remote controller destroy the closed-loop controllability, we propose a novel controllable and uncontrollable positive semidefinite cone decomposition induced by the singular value decomposition of the MIMO fading channel contaminated control input matrix. This enables a closed-form characterization of the sufficient condition for both the existence and the uniqueness of the optimal control solution. The closed-form sufficient condition reveals the fact that the optimal control action may still exist even if the closed-loop system suffers from intermittent controllability or almost sure uncontrollability. We further propose a novel stochastic approximation (SA)-based online learning algorithm that can learn the optimal control action on the fly based on the plant state observations. We finally extend the technical results to the case in which the channel state information (CSI) is unavailable at the controller. We show that the sufficient condition for the existence and uniqueness of the optimal control solution is more difficult to satisfy due to the penalty caused by the unknown CSI. The proposed scheme is also compared with various baselines, and we show that significant performance gains can be achieved.