# Synthesis of Stabilizing Recurrent Equilibrium Network Controllers

arXiv: https://arxiv.org/abs/2204.00122 · HTML: https://mlsanity.com/paper/2204.00122/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 23 references that are in the map.
- First posted on arXiv 31 Mar 2022.

## Builds on (most cited first, 12 of 12)

- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Continuous-time system identification with neural networks: Model structures and fitting criteria](https://mlsanity.com/paper/2006.02915/) (2020)
- [Recurrent Neural Network Controllers Synthesis with Stability Guarantees for Partially Observed Systems](https://mlsanity.com/paper/2109.03861/) (2021)
- [Learning over All Stabilizing Nonlinear Controllers for a Partially-Observed Linear System](https://mlsanity.com/paper/2112.04219/) (2021)
- [Neural network training under semidefinite constraints](https://mlsanity.com/paper/2201.00632/) (2022)
- [Youla-REN: Learning Nonlinear Feedback Policies with Robust Stability Guarantees](https://mlsanity.com/paper/2112.01253/) (2021)
- [Robust Control Barrier Functions with Sector-Bounded Uncertainties](https://mlsanity.com/paper/2109.02537/) (2021)
- [System Identification Through Lipschitz Regularized Deep Neural Networks](https://mlsanity.com/paper/2009.03288/) (2020)
- [Backward Reachability for Polynomial Systems on A Finite Horizon](https://mlsanity.com/paper/1907.03225/) (2019)

## Built on by (most cited first, 8 of 8)

- [TorchDEQ: A Library for Deep Equilibrium Models](https://mlsanity.com/paper/2310.18605/) (2023)
- [Learning Over Contracting and Lipschitz Closed-Loops for Partially-Observed Nonlinear Systems (Extended Version)](https://mlsanity.com/paper/2304.06193/) (2023)
- [Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guarantees](https://mlsanity.com/paper/2404.07373/) (2024)
- [Quantization Robustness of Monotone Operator Equilibrium Networks](https://mlsanity.com/paper/2603.10562/) (2026)
- [RobustNeuralNetworks.jl: a Package for Machine Learning and Data-Driven Control with Certified Robustness](https://mlsanity.com/paper/2306.12612/) (2023)
- [Robust Recurrent Neural Network to Identify Ship Motion in Open Water with Performance Guarantees -- Technical Report](https://mlsanity.com/paper/2212.05781/) (2022)
- [Rational Neural Network Controllers](https://mlsanity.com/paper/2307.06287/) (2023)
- [Convex neural network synthesis for robustness in the 1-norm](https://mlsanity.com/paper/2405.19029/) (2024)

## Abstract (arXiv)

We propose a parameterization of a nonlinear dynamic controller based on the recurrent equilibrium network, a generalization of the recurrent neural network. We derive constraints on the parameterization under which the controller guarantees exponential stability of a partially observed dynamical system with sector bounded nonlinearities. Finally, we present a method to synthesize this controller using projected policy gradient methods to maximize a reward function with arbitrary structure. The projection step involves the solution of convex optimization problems. We demonstrate the proposed method with simulated examples of controlling nonlinear plants, including plants modeled with neural networks.