无线MIMO衰落信道下线性系的在线最优状态反馈控制
Online Optimal State Feedback Control of Linear Systems over Wireless MIMO Fading Channels
摘要：
考虑无线MIMO衰落信道下线性系的最优控制问题，其中MIMO无线衰落和遥控器的随机接入可能导致闭环控制系统的间歇可控或不可控。将随机接入MIMO衰落信道下的最优控制设计表示为一个无限时域平均代价马尔可夫决策过程（MDP），并提出了一种新的状态约简技术，将最优性条件转化为约简状态的时不变Bellman最优性方程.通过对约化状态Bellman最优性方程的分析，给出了最优控制解的存在唯一性的封闭刻画.具体地说，在闭环系统几乎必然可控的情况下，我们证明了最优控制解总是存在的，并且是唯一的。在MIMO衰落信道和遥控器的随机接入破坏闭环可控性的情况下，提出了一种新的可控和不可控半正定（PSD）锥分解方法，该方法通过对MIMO衰落信道污染的控制输入矩阵进行奇异值分解（SVD）得到PSD锥分解.基于分解的细粒度约化状态Bellman最优性方程，进一步给出了最优控制解存在唯一的封闭形式的充分条件.该闭环充分条件揭示了即使闭环系统具有间歇可控性或几乎必然不可控性，最优控制作用仍然存在的事实。
英文摘要：
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.
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