  In this paper, we present a method to analyze local and global stability in
offset-free setpoint tracking using neural network controllers and we provide
ellipsoidal inner approximations of the corresponding region of attraction. We
consider a feedback interconnection of a linear plant in connection with a
neural network controller and an integrator, which allows for offset-free
tracking of a desired piecewise constant reference that enters the controller
as an external input. Exploiting the fact that activation functions used in
neural networks are slope-restricted, we derive linear matrix inequalities to
verify stability using Lyapunov theory. After stating a global stability
result, we present less conservative local stability conditions (i) for a given
reference and (ii) for any reference from a certain set. The latter result even
enables guaranteed tracking under setpoint changes using a reference governor
which can lead to a significant increase of the region of attraction. Finally,
we demonstrate the applicability of our analysis by verifying stability and
offset-free tracking of a neural network controller that was trained to
stabilize a linearized inverted pendulum.
