# On Imitation Learning of Linear Control Policies: Enforcing Stability and Robustness Constraints via LMI Conditions

arXiv: https://arxiv.org/abs/2103.12945 · HTML: https://mlsanity.com/paper/2103.12945/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Placed in Imitation learning methods, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 12 of its 25 references that are in the map.
- First posted on arXiv 24 Mar 2021.

## Placed in

- Offline RL and imitation learning from data / [Imitation learning methods](https://mlsanity.com/offlinerl/imitation-learning-methods/)

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

- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [An Algorithmic Perspective on Imitation Learning](https://mlsanity.com/paper/1811.06711/) (2018)
- [Policy Optimization for H2 Linear Control with H_∞ Robustness Guarantee: Implicit Regularization and Global Convergence](https://mlsanity.com/paper/1910.09496/) (2019)
- [Enforcing robust control guarantees within neural network policies](https://mlsanity.com/paper/2011.08105/) (2020)
- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [Derivative-Free Policy Optimization for Linear Risk-Sensitive and Robust Control Design: Implicit Regularization and Sample Complexity](https://mlsanity.com/paper/2101.01041/) (2021)
- [LMI Properties and Applications in Systems, Stability, and Control Theory](https://mlsanity.com/paper/1903.08599/) (2019)
- [Fitting a Linear Control Policy to Demonstrations with a Kalman Constraint](https://mlsanity.com/paper/2001.07572/) (2020)
- [H_∞ Model-free Reinforcement Learning with Robust Stability Guarantee](https://mlsanity.com/paper/1911.02875/) (2019)
- [ADMM for Multiaffine Constrained Optimization](https://mlsanity.com/paper/1802.09592/) (2018)

## Built on by (most cited first, 10 of 10)

- [Provable Guarantees for Generative Behavior Cloning: Bridging Low-Level Stability and High-Level Behavior](https://mlsanity.com/paper/2307.14619/) (2023)
- [Butterfly Effects of SGD Noise: Error Amplification in Behavior Cloning and Autoregression](https://mlsanity.com/paper/2310.11428/) (2023)
- [CCIL: Context-conditioned imitation learning for urban driving](https://mlsanity.com/paper/2305.02649/) (2023)
- [Risk Verification of Stochastic Systems with Neural Network Controllers](https://mlsanity.com/paper/2209.09881/) (2022)
- [Towards a Theoretical Foundation of Policy Optimization for Learning Control Policies](https://mlsanity.com/paper/2210.04810/) (2022)
- [Imitation Learning of Stabilizing Policies for Nonlinear Systems](https://mlsanity.com/paper/2109.10854/) (2021)
- [Fail-Safe Adversarial Generative Imitation Learning](https://mlsanity.com/paper/2203.01696/) (2022)
- [Dissipative Imitation Learning for Robust Dynamic Output Feedback](https://mlsanity.com/paper/2210.00979/) (2022)
- [Learning to control from expert demonstrations](https://mlsanity.com/paper/2203.05012/) (2022)
- [Dissipative Imitation Learning for Discrete Dynamic Output Feedback Control with Sparse Data Sets](https://mlsanity.com/paper/2309.06658/) (2023)

## Abstract (arXiv)

When applying imitation learning techniques to fit a policy from expert demonstrations, one can take advantage of prior stability/robustness assumptions on the expert's policy and incorporate such control-theoretic prior knowledge explicitly into the learning process. In this paper, we formulate the imitation learning of linear policies as a constrained optimization problem, and present efficient methods which can be used to enforce stability and robustness constraints during the learning processes. Specifically, we show that one can guarantee the closed-loop stability and robustness by posing linear matrix inequality (LMI) constraints on the fitted policy. Then both the projected gradient descent method and the alternating direction method of multipliers (ADMM) method can be applied to solve the resulting constrained policy fitting problem. Finally, we provide numerical results to demonstrate the effectiveness of our methods in producing linear polices with various stability and robustness guarantees.