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Deep neural networks have revolutionized many fields, including imageprocessing, inverse problems, text mining and more recently, give verypromising results in systems and control. Neural networks with hidden layershave a strong potential as an approximation framework of predictive controllaws as they usually yield better approximation quality and smaller memoryrequirements than existing explicit (multi-parametric) approaches. In thispaper, we first show that neural networks with HardTanh activation functionscan exactly represent predictive control laws of linear time-invariant systems.We derive theoretical bounds on the minimum number of hidden layers and neuronsthat a HardTanh neural network should have to exactly represent a givenpredictive control law. The choice of HardTanh deep neural networks isparticularly suited for linear predictive control laws as they usually requireless hidden layers and neurons than deep neural networks with ReLU units forrepresenting exactly continuous piecewise affine (or equivalently min-max)maps. In the second part of the paper we bring the physics of the model andstandard optimization techniques into the architecture design, in order toeliminate the disadvantages of the black-box HardTanh learning. Morespecifically, we design trainable unfolded HardTanh deep architectures forlearning linear predictive control laws based on two standard iterativeoptimization algorithms, i.e., projected gradient descent and acceleratedprojected gradient descent. We also study the performance of the proposedHardTanh type deep neural networks on a linear model predictive controlapplication.

Exact representation and efficient approximations of linear model predictive control laws via HardTanh type deep neural networks