Optimal Observer Design Using Reinforcement Learning and Quadratic Neural Networks

Abstract: This paper introduces an innovative approach based on policy iteration (PI), a reinforcement learning (RL) algorithm, to obtain an optimal observer with a quadratic cost function. This observer is designed for systems with a given linearized model and a stabilizing Luenberger observer gain. We utilize two-layer quadratic neural networks (QNN) for policy evaluation and derive a linear correction term using the input and output data. This correction term effectively rectifies inaccuracies introduced by the linearized model employed within the observer design. A unique feature of the proposed methodology is that the QNN is trained through convex optimization. The main advantage is that the QNN's input-output mapping has an analytical expression as a quadratic form, which can then be used to obtain a linear correction term policy. This is in stark contrast to the available techniques in the literature that must train a second neural network to obtain policy improvement. It is proven that the obtained linear correction term is optimal for linear systems, as both the value function and the QNN's input-output mapping are quadratic. The proposed method is applied to a simple pendulum, demonstrating an enhanced correction term policy compared to relying solely on the linearized model. This shows its promise for addressing nonlinear systems.