Learning Koopman Operator under Dissipativity Constraints

TL;DR

This paper proposes a method to learn the Koopman operator for nonlinear systems while ensuring specified dissipativity properties, using convex optimization techniques for efficient computation.

Contribution

It introduces a novel approach to incorporate dissipativity constraints into Koopman operator learning via convex approximation, improving modeling accuracy.

Findings

Achieves high modeling accuracy with dissipativity constraints

Reduces non-convex problem to sequential convex optimization

Demonstrates effectiveness through numerical simulation

Abstract

This paper addresses a learning problem for nonlinear dynamical systems with incorporating any specified dissipativity property. The nonlinear systems are described by the Koopman operator, which is a linear operator defined on the infinite-dimensional lifted state space. The problem of learning the Koopman operator under specified quadratic dissipativity constraints is formulated and addressed. The learning problem is in a class of the non-convex optimization problem due to nonlinear constraints and is numerically intractable. By applying the change of variable technique and the convex overbounding approximation, the problem is reduced to sequential convex optimization and is solved in a numerically efficient manner. Finally, a numerical simulation is given, where high modeling accuracy achieved by the proposed approach including the specified dissipativity is demonstrated.