AReN: Assured ReLU NN Architecture for Model Predictive Control of LTI Systems

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.