  In this paper, we consider the problem of automatically designing a Rectified
Linear Unit (ReLU) Neural Network (NN) architecture that is sufficient to
implement the optimal Model Predictive Control (MPC) strategy for an LTI system
with quadratic cost. Specifically, we propose AReN, an algorithm to generate
Assured ReLU Architectures. AReN takes as input an LTI system with quadratic
cost specification, and outputs a ReLU NN architecture with the assurance that
there exist network weights that exactly implement the associated MPC
controller. AReN thus offers new insight into the design of ReLU NN
architectures for the control of LTI systems: instead of training a
heuristically chosen NN architecture on data -- or iterating over many
architectures until a suitable one is found -- AReN can suggest an adequate NN
architecture before training begins. While several previous works were inspired
by the fact that both ReLU NN controllers and optimal MPC controller are both
Continuous, Piecewise-Linear (CPWL) functions, exploiting this similarity to
design NN architectures with correctness guarantees has remained elusive. AReN
achieves this using two novel features. First, we reinterpret a recent result
about the implementation of CPWL functions via ReLU NNs to show that a CPWL
function may be implemented by a ReLU architecture that is determined by the
number of distinct affine regions in the function. Second, we show that we can
efficiently over-approximate the number of affine regions in the optimal MPC
controller without solving the MPC problem exactly. Together, these results
connect the MPC problem to a ReLU NN implementation without explicitly solving
the MPC and directly translates this feature to a ReLU NN architecture that
comes with the assurance that it can implement the MPC controller. We show
through numerical results the effectiveness of AReN in designing an NN
architecture.
