  In this paper, we consider the problem of automatically designing a Rectified
Linear Unit (ReLU) Neural Network (NN) architecture (number of layers and
number of neurons per layer) with the guarantee that it is sufficiently
parametrized to control a nonlinear system. Whereas current state-of-the-art
techniques are based on hand-picked architectures or heuristic based search to
find such NN architectures, our approach exploits the given model of the system
to design an architecture; as a result, we provide a guarantee that the
resulting NN architecture is sufficient to implement a controller that
satisfies an achievable specification. Our approach exploits two basic ideas.
First, assuming that the system can be controlled by an unknown
Lipschitz-continuous state-feedback controller with some Lipschitz constant
upper-bounded by $K_\text{cont}$, we bound the number of affine functions
needed to construct a Continuous Piecewise Affine (CPWA) function that can
approximate the unknown Lipschitz-continuous controller. Second, we utilize the
authors' recent results on a novel NN architecture named as the Two-Level
Lattice (TLL) NN architecture, which was shown to be capable of implementing
any CPWA function just from the knowledge of the number of affine functions
that compromises this CPWA function.
