# Neural System Level Synthesis: Learning over All Stabilizing Policies for Nonlinear Systems

arXiv: https://arxiv.org/abs/2203.11812 · HTML: https://mlsanity.com/paper/2203.11812/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 32 references that are in the map.
- First posted on arXiv 22 Mar 2022.

## Builds on (most cited first, 15 of 15)

- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Analysis of the Optimization Landscape of Linear Quadratic Gaussian (LQG) Control](https://mlsanity.com/paper/2102.04393/) (2021)
- [Recurrent Neural Network Controllers Synthesis with Stability Guarantees for Partially Observed Systems](https://mlsanity.com/paper/2109.03861/) (2021)
- [A Neural Network Approach Applied to Multi-Agent Optimal Control](https://mlsanity.com/paper/2011.04757/) (2020)
- [Learning over All Stabilizing Nonlinear Controllers for a Partially-Observed Linear System](https://mlsanity.com/paper/2112.04219/) (2021)
- [Distributed neural network control with dependability guarantees: a compositional port-Hamiltonian approach](https://mlsanity.com/paper/2112.09046/) (2021)
- [Offset-free setpoint tracking using neural network controllers](https://mlsanity.com/paper/2011.14006/) (2020)
- [A System Level Approach to Controller Synthesis](https://mlsanity.com/paper/1610.04815/) (2016)
- [Sparsity Invariance for Convex Design of Distributed Controllers](https://mlsanity.com/paper/1906.06777/) (2019)
- [Graph Neural Networks for Distributed Linear-Quadratic Control](https://mlsanity.com/paper/2011.05360/) (2020)
- [First Order Methods For Globally Optimal Distributed Controllers Beyond Quadratic Invariance](https://mlsanity.com/paper/1909.10999/) (2019)
- [A System Level Approach to Discrete-Time Nonlinear Systems](https://mlsanity.com/paper/2004.08004/) (2020)
- [Incremental Dissipativity based Control of Discrete-Time Nonlinear Systems via the LPV Framework](https://mlsanity.com/paper/2110.00290/) (2021)

## Built on by (most cited first, 11 of 11)

- [Safety Certification for Stochastic Systems via Neural Barrier Functions](https://mlsanity.com/paper/2206.01463/) (2022)
- [Learning to optimize with convergence guarantees using nonlinear system theory](https://mlsanity.com/paper/2403.09389/) (2024)
- [Learning to Boost the Performance of Stable Nonlinear Systems](https://mlsanity.com/paper/2405.00871/) (2024)
- [Safety Beyond the Training Data: Robust Out-of-Distribution MPC via Conformalized System Level Synthesis](https://mlsanity.com/paper/2602.12047/) (2026)
- [MAD: A Magnitude And Direction Policy Parametrization for Stability Constrained Reinforcement Learning](https://mlsanity.com/paper/2504.02565/) (2025)
- [React to Surprises: Stable-by-Design Neural Feedback Control and the Youla-REN](https://mlsanity.com/paper/2506.01226/) (2025)
- [Controller Design for Structured State-space Models via Contraction Theory](https://mlsanity.com/paper/2604.07069/) (2026)
- [Convex neural network synthesis for robustness in the 1-norm](https://mlsanity.com/paper/2405.19029/) (2024)
- [Distributed Control of Network Systems in the Space of Stabilizing Graph Neural Network Policies](https://mlsanity.com/paper/2512.18540/) (2025)
- [Stability-Preserving Online Adaptation of Neural Closed-loop Maps](https://mlsanity.com/paper/2603.22469/) (2026)
- [Global Convergence of Policy Gradient Methods for ReLU Controllers in Linear Quadratic Regulation](https://mlsanity.com/paper/2604.22138/) (2026)

## Abstract (arXiv)

We address the problem of designing stabilizing control policies for nonlinear systems in discrete-time, while minimizing an arbitrary cost function. When the system is linear and the cost is convex, the System Level Synthesis (SLS) approach offers an effective solution based on convex programming. Beyond this case, a globally optimal solution cannot be found in a tractable way, in general. In this paper, we develop a parametrization of all and only the control policies stabilizing a given time-varying nonlinear system in terms of the combined effect of 1) a strongly stabilizing base controller and 2) a stable SLS operator to be freely designed. Based on this result, we propose a Neural SLS (Neur-SLS) approach guaranteeing closed-loop stability during and after parameter optimization, without requiring any constraints to be satisfied. We exploit recent Deep Neural Network (DNN) models based on Recurrent Equilibrium Networks (RENs) to learn over a rich class of nonlinear stable operators, and demonstrate the effectiveness of the proposed approach in numerical examples.