Koopman Operator Approximation under Negative Imaginary Constraints

TL;DR

This paper introduces a data-driven approach to approximate nonlinear Negative Imaginary systems with a lifted linear Koopman operator, enabling better control without linearization.

Contribution

It develops a convex optimization framework embedding NI constraints into Koopman operator learning, improving accuracy and control performance for nonlinear NI systems.

Findings

Accurately captures nonlinear dynamics

Achieves better control performance

Provides a tractable solution for NI systems

Abstract

Nonlinear Negative Imaginary (NI) systems arise in various engineering applications, such as controlling flexible structures and air vehicles. However, unlike linear NI systems, their theory is not well-developed. In this paper, we propose a data-driven method for learning a lifted linear NI dynamics that approximates a nonlinear dynamical system using the Koopman theory, which is an operator that captures the evolution of nonlinear systems in a lifted high-dimensional space. The linear matrix inequality that characterizes the NI property is embedded in the Koopman framework, which results in a non-convex optimization problem. To overcome the numerical challenges of solving a non-convex optimization problem with nonlinear constraints, the optimization variables are reformatted in order to convert the optimization problem into a convex one with the new variables. We compare our method with local linearization techniques and show that our method can accurately capture the nonlinear dynamics and achieve better control performance. Our method provides a numerically tractable solution for learning the Koopman operator under NI constraints for nonlinear NI systems and opens up new possibilities for applying linear control techniques to nonlinear NI systems without linearization approximations