# Synthesizing Control Laws from Data using Sum-of-Squares Optimization

arXiv: https://arxiv.org/abs/2307.01089 · HTML: https://mlsanity.com/paper/2307.01089/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jul 2023.

## Built on by (most cited first, 2 of 2)

- [SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learning](https://mlsanity.com/paper/2403.09110/) (2024)
- [Neural Sum-of-Squares: Certifying the Nonnegativity of Polynomials with Transformers](https://mlsanity.com/paper/2510.13444/) (2025)

## Abstract (arXiv)

The control Lyapunov function (CLF) approach to nonlinear control design is well established. Moreover, when the plant is control affine and polynomial, sum-of-squares (SOS) optimization can be used to find a polynomial controller as a solution to a semidefinite program. This letter considers the use of data-driven methods to design a polynomial controller by leveraging Koopman operator theory, CLFs, and SOS optimization. First, Extended Dynamic Mode Decomposition (EDMD) is used to approximate the Lie derivative of a given CLF candidate with polynomial lifting functions. Then, the polynomial Koopman model of the Lie derivative is used to synthesize a polynomial controller via SOS optimization. The result is a flexible data-driven method that skips the intermediary process of system identification and can be applied widely to control problems. The proposed approach is used to successfully synthesize a controller to stabilize an inverted pendulum on a cart.