Data-driven H∞ Optimal Controller Design using the Koopman Operator: Case Study

We follow up on the idea of using linear control methods for controlling non-linear systems by representing the non-linear system as an approximate linear system in a lifted state-space using the Koopman framework. H∞ control techniques have been successfully applied to linear multi-variable systems and the main idea in this paper is to use them on a data-driven identified lifted system within the Koopman framework. We present results in the form of a case study on two example systems. We choose the Van der Pol (VdP) oscillator as the first system which is a commonly used example in the Koopman literature. Even though a linear controller designed for the linearized VdP system shows good results for our test scenarios, this system illuminates some pitfalls of using the naive S/KS design structure on the identified Koopman model by showing poor closed-loop performance. The performance can be recovered by using a 4-block design or a 2 degree of freedom control architecture which shows that the well known techniques for improving performance of linear systems can be applied to the identified Koopman model. As a more challenging second example, we choose a Two-Mass-Duffing-Spring (TMDS) system where the linear controller for the linearized system fails to stabilize the non-linear system. We show that controllers with good closed loop performance can be successfully synthesized for this system via standard H∞ optimal control methods.