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Publication# Koopman-based Data-driven Robust Control of Nonlinear Systems Using Integral Quadratic Constraints

Abstract

This paper introduces a novel method for data-driven robust control of nonlinear systems based on the Koopman operator, utilizing Integral Quadratic Constraints (IQCs). The Koopman operator theory facilitates the linear representation of nonlinear system dynamics in a higher-dimensional space. Data-driven Koopman-based representations inherently yield only approximate models due to various factors. In addressing this, we focus on effective characterization of the modeling error, which is crucial for ensuring closed-loop guarantees. We identify non-parametric IQC multipliers to characterize the modeling error in a data-driven fashion through the solution of frequency domain (FD) linear matrix inequalities (LMIs), treating it as additive uncertainty for robust control design. These multipliers provide a convex set representation of stabilising robust controllers. We then obtain the optimal controller within this set by solving a different set of FD LMIs. Lastly, we propose an iterative approach alternating between IQC multiplier identification and robust controller synthesis, ensuring monotonic convergence of a robust performance index.

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Nonlinear system

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Control theory

Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality. To do this, a controller with the requisite corrective behavior is required.

Optimal control

Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure.

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