  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.
