  Since most industrial control applications use PID controllers, PID tuning
and anti-windup measures are significant problems. This paper investigates
tuning the feedback gains of a PID controller via back-calculation and
automatic differentiation tools. In particular, we episodically use a cost
function to generate gradients and perform gradient descent to improve
controller performance. We provide a theoretical framework for analyzing this
non-convex optimization and establish a relationship between back-calculation
and disturbance feedback policies. We include numerical experiments on linear
systems with actuator saturation to show the efficacy of this approach.
