Certified Learning of Incremental ISS Controllers for Unknown Nonlinear Polynomial Dynamics

TL;DR

This paper presents a data-driven method to design incremental input-to-state stability controllers for unknown nonlinear polynomial systems, ensuring robustness without requiring explicit system models.

Contribution

It introduces a novel approach to synthesize delta-ISS Lyapunov functions and controllers using minimal data and sum-of-squares optimization, bypassing the need for system knowledge.

Findings

Successfully designed controllers for a physical case study.

Achieved delta-ISS property without explicit system models.

Utilized only two sets of input-state trajectories.

Abstract

Incremental input-to-state stability (delta-ISS) offers a robust framework to ensure that small input variations result in proportionally minor deviations in the state of a nonlinear system. This property is essential in practical applications where input precision cannot be guaranteed. However, analyzing delta-ISS demands precise knowledge of system dynamics to assess the state's incremental response to input changes, posing a challenge in real-world scenarios where mathematical models are unknown. In this work, we develop a data-driven approach to design delta-ISS Lyapunov functions together with their corresponding delta-ISS controllers for continuous-time input-affine nonlinear systems with polynomial dynamics, ensuring the delta-ISS property is achieved without requiring knowledge of the system dynamics. In our data-driven scheme, we collect only two sets of input-state trajectories from sufficiently excited dynamics. By fulfilling a specific rank condition, we design delta-ISS controllers using the collected samples through formulating a sum-of-squares optimization program. The effectiveness of our data-driven approach is evidenced by its application to a physical case study.