Reliably-stabilizing piecewise-affine neural network controllers

Filippo Fabiani and Paul J. Goulart

@misc{FG:2021b,
  author = {Filippo Fabiani and Paul J. Goulart},
  title = {Reliably-stabilizing piecewise-affine neural network controllers},
  year = {2021},
  url = {https://arxiv.org/abs/2111.07183}
}

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.

The optimal transport paradigm enables data compression in data-driven robust control

F. Fabiani and P. J. Goulart

@inproceedings{FG:2021,
  author = {F. Fabiani and P. J. Goulart},
  title = {The optimal transport paradigm enables data compression in data-driven robust control},
  booktitle = {American Control Conference},
  year = {2021},
  pages = {3431-3436}
}

A new data-enabled control technique for uncertain linear time-invariant systems, recently conceived by Coulson et al., builds upon the direct optimization of controllers over input/output pairs drawn from a large dataset. We adopt an optimal transport-based method for compressing such large dataset to a smaller synthetic dataset of representative behaviours, aiming to alleviate the computational burden of controllers to be implemented online. Specifically, the synthetic data are determined by minimizing the Wasserstein distance between atomic distributions supported on both the original dataset and the compressed one.We show that a distributionally robust control law computed using the compressed data enjoys the same type of performance guarantees as the original dataset, at the price of enlarging the ambiguity set by an easily computable and well-behaved quantity. Numerical simulations confirm that the control performance with the synthetic data is comparable to the one obtained with the original data, but with significantly less computation required.