Two-Level Lattice Neural Network Architectures for Control of Nonlinear Systems

In this paper, we consider the problem of automatically designing a Rectified\nLinear Unit (ReLU) Neural Network (NN) architecture (number of layers and\nnumber of neurons per layer) with the guarantee that it is sufficiently\nparametrized to control a nonlinear system. Whereas current state-of-the-art\ntechniques are based on hand-picked architectures or heuristic based search to\nfind such NN architectures, our approach exploits the given model of the system\nto design an architecture; as a result, we provide a guarantee that the\nresulting NN architecture is sufficient to implement a controller that\nsatisfies an achievable specification. Our approach exploits two basic ideas.\nFirst, assuming that the system can be controlled by an unknown\nLipschitz-continuous state-feedback controller with some Lipschitz constant\nupper-bounded by $K_\\text{cont}$, we bound the number of affine functions\nneeded to construct a Continuous Piecewise Affine (CPWA) function that can\napproximate the unknown Lipschitz-continuous controller. Second, we utilize the\nauthors' recent results on a novel NN architecture named as the Two-Level\nLattice (TLL) NN architecture, which was shown to be capable of implementing\nany CPWA function just from the knowledge of the number of affine functions\nthat compromises this CPWA function.\n

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