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

Jason J. Bramburger, Steven Dahdah, James Richard Forbes

TL;DR

This work addresses stabilizing nonlinear, control-affine systems without full system identification by integrating Koopman-based data-driven modeling with Control Lyapunov Functions (CLFs) and Sum-of-Squares (SOS) optimization. It introduces a two-step method: (1) estimate the Lie derivative of a CLF from data using EDMD with polynomial lifting functions, and (2) synthesize a polynomial controller via an SOS program that enforces a negative Lie derivative, ensuring asymptotic stability of the equilibrium. The approach is demonstrated on an inverted pendulum on a cart, yielding a state-dependent controller that stabilizes the upright position while highlighting practical considerations like domain choice and sparsity. The framework offers a flexible, data-driven pathway to design stabilizing controllers for a broad class of nonlinear systems without explicit dynamics identification, with potential extensions to discrete-time and stochastic settings.

Abstract

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.

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

TL;DR

This work addresses stabilizing nonlinear, control-affine systems without full system identification by integrating Koopman-based data-driven modeling with Control Lyapunov Functions (CLFs) and Sum-of-Squares (SOS) optimization. It introduces a two-step method: (1) estimate the Lie derivative of a CLF from data using EDMD with polynomial lifting functions, and (2) synthesize a polynomial controller via an SOS program that enforces a negative Lie derivative, ensuring asymptotic stability of the equilibrium. The approach is demonstrated on an inverted pendulum on a cart, yielding a state-dependent controller that stabilizes the upright position while highlighting practical considerations like domain choice and sparsity. The framework offers a flexible, data-driven pathway to design stabilizing controllers for a broad class of nonlinear systems without explicit dynamics identification, with potential extensions to discrete-time and stochastic settings.

Abstract

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.
Paper Structure (10 sections, 29 equations, 1 figure)

This paper contains 10 sections, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Lyapunov functions
  4. Synthesizing controllers with semidefinite programming
  5. The Koopman operator
  6. Main Result
  7. Estimating Lie derivatives from data
  8. Synthesizing control laws from data
  9. Application to the inverted pendulum
  10. Conclusion

Figures (1)

  • Figure 1: Controlled solutions of the inverted pendulum on a cart model \ref{['PendCart']}. Top: The controlled arm angle $\theta$ and its derivative $\dot\theta$, forced to converge to the unstable upright position at $(\theta,\dot\theta) = (0,0)$. Bottom: The monotonic approach inside the Lyapunov function \ref{['PendLyap']} (scaled by $1/10$ for interpretability) according to the controller, $u(\theta,\dot\theta)$, given by \ref{['PendController']}.