# Robust stabilization of polytopic systems via fast and reliable neural network-based approximations

arXiv: https://arxiv.org/abs/2204.13209 · HTML: https://mlsanity.com/paper/2204.13209/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 57 references that are in the map.
- First posted on arXiv 27 Apr 2022.

## Builds on (most cited first, 14 of 14)

- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Deep Reinforcement Learning in Parameterized Action Space](https://mlsanity.com/paper/1511.04143/) (2015)
- [Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums](https://mlsanity.com/paper/2104.08135/) (2021)
- [Real-time Power System State Estimation and Forecasting via Deep Neural Networks](https://mlsanity.com/paper/1811.06146/) (2018)
- [Near-Optimal Rapid MPC using Neural Networks: A Primal-Dual Policy Learning Framework](https://mlsanity.com/paper/1912.04744/) (2019)
- [Reliably-stabilizing piecewise-affine neural network controllers](https://mlsanity.com/paper/2111.07183/) (2021)
- [Stability and feasibility of neural network-based controllers via output range analysis](https://mlsanity.com/paper/2004.00521/) (2020)
- [Polytopic Input Constraints in Learning-Based Optimal Control Using Neural Networks](https://mlsanity.com/paper/2105.03376/) (2021)
- [Stabilization of affine systems with polytopic control value sets](https://mlsanity.com/paper/2112.02451/) (2021)

## Built on by (most cited first, 1 of 1)

- [A neural network-based approach to hybrid systems identification for control](https://mlsanity.com/paper/2404.01814/) (2024)

## Abstract (arXiv)

We consider the design of fast and reliable neural network (NN)-based approximations of traditional stabilizing controllers for linear systems with polytopic uncertainty, including control laws with variable structure and those based on a (minimal) selection policy. Building upon recent approaches for the design of reliable control surrogates with guaranteed structural properties, we develop a systematic procedure to certify the closed-loop stability and performance of a linear uncertain system when a trained rectified linear unit (ReLU)-based approximation replaces such traditional controllers. First, we provide a sufficient condition, which involves the worst-case approximation error between ReLU-based and traditional controller-based state-to-input mappings, ensuring that the system is ultimately bounded within a set with adjustable size and convergence rate. Then, we develop an offline, mixed-integer optimization-based method that allows us to compute that quantity exactly.