  In this paper, a method is presented to synthesize neural network controllers
such that the feedback system of plant and controller is dissipative,
certifying performance requirements such as L2 gain bounds. The class of plants
considered is that of linear time-invariant (LTI) systems interconnected with
an uncertainty, including nonlinearities treated as an uncertainty for
convenience of analysis. The uncertainty of the plant and the nonlinearities of
the neural network are both described using integral quadratic constraints
(IQCs). First, a dissipativity condition is derived for uncertain LTI systems.
Second, this condition is used to construct a linear matrix inequality (LMI)
which can be used to synthesize neural network controllers. Finally, this
convex condition is used in a projection-based training method to synthesize
neural network controllers with dissipativity guarantees. Numerical examples on
an inverted pendulum and a flexible rod on a cart are provided to demonstrate
the effectiveness of this approach.
