Modeling Nonlinear Control Systems via Koopman Control Family: Universal Forms and Subspace Invariance Proximity

2023/07/28 by Haseli, Masih, Cortés, Jorge · 9 citations
#FOS: Electrical engineering #FOS: Mathematics #Optimization and Control (math.OC) #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · doi:10.48550/arxiv.2307.15368

Abstract

This paper introduces the Koopman Control Family (KCF), a mathematical framework for modeling general (not necessarily control-affine) discrete-time nonlinear control systems with the aim of providing a solid theoretical foundation for the use of Koopman-based methods in systems with inputs. We demonstrate that the concept of KCF captures the behavior of nonlinear control systems on a (potentially infinite-dimensional) function space. By employing a generalized notion of subspace invariance under the KCF, we establish a universal form for finite-dimensional models, which encompasses the commonly used linear, bilinear, and linear switched models as specific instances. In cases where the subspace is not invariant under the KCF, we propose a method for approximating models in general form and characterize the model's accuracy using the concept of invariance proximity. We end by discussing how the proposed framework naturally lends itself to data-driven modeling of control systems.

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• Control-Oriented System Identification: Classical, Learning, and Physics-Informed Approaches
• On the Exponential Stability of Koopman Model Predictive Control
• Two Roads to Koopman Operator Theory for Control: Infinite Input Sequences and Operator Families
• Regularization in Data-driven Predictive Control: A Convex Relaxation Perspective
• An overview of Koopman-based control: From error bounds to closed-loop guarantees
• From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma
• Gramians for a New Class of Nonlinear Control Systems Using Koopman and a Novel Generalized SVD
• Data-Enabled Predictive Control for Nonlinear Systems Based on a Koopman Bilinear Realization
• Controller Design for Bilinear Neural Feedback Loops

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