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

TL;DR

This paper develops a novel framework for optimal control of linear systems over wireless MIMO fading channels, addressing challenges of intermittent controllability and providing conditions for the existence and uniqueness of optimal solutions.

Contribution

It introduces a state reduction technique and a Bellman optimality equation for control over MIMO fading channels, including a PSD cone decomposition for uncontrollable cases.

Findings

Optimal control solutions exist and are unique when the system is almost surely controllable.

A novel PSD cone decomposition helps analyze uncontrollable scenarios.

Closed-form conditions ensure the existence of optimal control even under intermittent controllability.

Abstract

We consider the optimal control of linear systems over wireless MIMO fading channels, where the MIMO wireless fading and random access of the remote controller may cause intermittent controllability or uncontrollability of the closed-loop control system. We formulate the optimal control design over random access MIMO fading channels as an infinite horizon average cost Markov decision process (MDP), and we propose a novel state reduction technique such that the optimality condition is transformed into a time-invariant reduced-state Bellman optimality equation. We provide the closed-form characterizations on the existence and uniqueness of the optimal control solution via analyzing the reduced-state Bellman optimality equation. Specifically, in the case that the closed-loop system is almost surely controllable, we show that 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 (PSD) cone decomposition induced by the singular value decomposition (SVD) of the MIMO fading channel contaminated control input matrix. Based on the decomposed fine-grained reduced-state Bellman optimality equation, we further propose a closed-form 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.