# Reliably-stabilizing piecewise-affine neural network controllers

arXiv: https://arxiv.org/abs/2111.07183 · HTML: https://mlsanity.com/paper/2111.07183/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 72 references that are in the map.
- First posted on arXiv 13 Nov 2021.

## Builds on (most cited first, 20 of 20)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Fast inference of deep neural networks in FPGAs for particle physics](https://mlsanity.com/paper/1804.06913/) (2018)
- [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)
- [Safe, Efficient, and Comfortable Velocity Control based on Reinforcement Learning for Autonomous Driving](https://mlsanity.com/paper/1902.00089/) (2019)
- [Efficient representation and approximation of model predictive control laws via deep learning](https://mlsanity.com/paper/1806.10644/) (2018)
- [Learning an Approximate Model Predictive Controller with Guarantees](https://mlsanity.com/paper/1806.04167/) (2018)
- [Learning Deep Control Policies for Autonomous Aerial Vehicles with MPC-Guided Policy Search](https://mlsanity.com/paper/1509.06791/) (2015)
- [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)
- [Reachable Set Estimation and Safety Verification for Piecewise Linear Systems with Neural Network Controllers](https://mlsanity.com/paper/1802.06981/) (2018)
- [A Neural Network Architecture to Learn Explicit MPC Controllers from Data](https://mlsanity.com/paper/1911.10789/) (2019)
- [Piecewise Linear Regression via a Difference of Convex Functions](https://mlsanity.com/paper/2007.02422/) (2020)
- [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)
- [On feasibility, stability and performance in distributed model predictive control](https://mlsanity.com/paper/1302.1974/) (2013)

## Built on by (most cited first, 13 of 13)

- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Stability Verification of Neural Network Controllers using Mixed-Integer Programming](https://mlsanity.com/paper/2206.13374/) (2022)
- [Approximate non-linear model predictive control with safety-augmented neural networks](https://mlsanity.com/paper/2304.09575/) (2023)
- [Self-Supervised Learning of Iterative Solvers for Constrained Optimization](https://mlsanity.com/paper/2409.08066/) (2024)
- [Improved Scalable Lipschitz Bounds for Deep Neural Networks](https://mlsanity.com/paper/2503.14297/) (2025)
- [YANNs: Y-wise Affine Neural Networks for Exact and Efficient Representations of Piecewise Linear Functions](https://mlsanity.com/paper/2505.07054/) (2025)
- [Robust stabilization of polytopic systems via fast and reliable neural network-based approximations](https://mlsanity.com/paper/2204.13209/) (2022)
- [Mapping back and forth between model predictive control and neural networks](https://mlsanity.com/paper/2404.12030/) (2024)
- [An active learning method for solving competitive multi-agent decision-making and control problems](https://mlsanity.com/paper/2212.12561/) (2022)
- [A neural network-based approach to hybrid systems identification for control](https://mlsanity.com/paper/2404.01814/) (2024)
- [Geometric Fault-Tolerant Neural Network Tracking Control of Unknown Systems on Matrix Lie Groups](https://mlsanity.com/paper/2505.04725/) (2025)
- [ReLU Networks for Model Predictive Control: Network Complexity and Performance Guarantees](https://mlsanity.com/paper/2601.16764/) (2026)
- [Generalization as a robust performance property of learning-enabled dynamical systems](https://mlsanity.com/paper/2608.30431/) (2026)

## Abstract (arXiv)

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.