Reliably-Stabilizing Piecewise-Affine Neural Network Controllers

Filippo Fabiani and Paul J. Goulart

@article{FG:2023,
  author = {Fabiani, Filippo and Goulart, Paul J.},
  title = {Reliably-Stabilizing Piecewise-Affine Neural Network Controllers},
  journal = {IEEE Transactions on Automatic Control},
  year = {2023},
  volume = {68},
  number = {9},
  pages = {5201-5215},
  url = {https://ieeexplore.ieee.org/document/9928332},
  doi = {10.1109/TAC.2022.3216978}
}

A common problem affecting neural network (NN) approximations of model predictive control (MPC) policies is the lack of analytical tools to assess the stability of the closed-loop system under the action of the NN-based controller. We present a general procedure to quantify the performance of such a controller, or to design minimum complexity NNs with rectified linear units (ReLUs) that preserve the desirable properties of a given MPC scheme. By quantifying the approximation error between NN-based and MPC-based state-to-input mappings, we first establish suitable conditions involving two key quantities, the worst-case error and the Lipschitz constant, guaranteeing the stability of the closed-loop system. We then develop an offline, mixed-integer optimization-based method to compute those quantities exactly. Together these techniques provide conditions sufficient to certify the stability and performance of a ReLU-based approximation of an MPC control law.