Fundamental Entropic Laws and $\mathcal{L}_p$ Limitations of Feedback Systems: Implications for Machine-Learning-in-the-Loop Control

-- In this paper, we study the fundamental performance limitations for generic feedback systems in which both the controller and the plant may be arbitrarily causal while the disturbance can be with any distributions. We also examine the implications of the generic bounds for machine-learning-in-the-loop control; in other words, fundamental limits in general exist to what machine learning elements in feedback loops can achieve. Machine learning techniques are becoming more and more prevalent nowadays in the feedback control of dynamical systems, where system dynamics that are determined by physical laws will play an indispensable role. In this trend, it is becoming more and more critical to be fully aware of the performance limits of the machine learning algorithms that are to be embedded in the feedback loop, especially in scenarios where performance guarantees are required and must be strictly imposed. In conventional performance limitation analysis [1] of feedback systems such as the Bode integral [2], however, specific restrictions on the classes of the controller that can be implemented must be imposed in general. These restrictions would normally render the analysis invalid if machine learning elements such as deep learning or reinforcement learning are to be placed at the position of the controller, as a result of the complexity of the learning algorithms.