{"title":"Data-driven predictor of control-affine nonlinear dynamics: Finite discrete-time bilinear approximation of koopman operator","authors":"Sara Iman, Mohammad-Reza Jahed-Motlagh","doi":"10.1016/j.amc.2024.129068","DOIUrl":null,"url":null,"abstract":"<div><div>This paper introduces a novel and efficient data-driven approach for approximating a finite discrete-time bilinear model of control affine nonlinear dynamical systems with a tunable parameter that balances model dimension and prediction accuracy. An approximation of the Koopman operator based on the evolutions of the nonlinear system measurements used to lift a control-affine nonlinear system to a higher dimensional model. However, higher dimensional spaces can result in a long learning time and the curse of dimensionality in control analysis. The proposed approach addresses these challenges by introducing a convex optimization which identifies informative observable functions. This technique allows for the adjustment of a parameter to strike a balance between model dimension and accuracy in prediction. The main contribution of this study is to introduce a reduced dimensional bilinear model for a nonlinear complex system. This achievement is made possible by implementing convex sparse optimization, enabling the exploration of informative estimated Koopman eigenfunctions while minimizing the number of system measurements required. The optimization problem is solved using the alternating direction method of multipliers. The effectiveness of the proposed method is evaluated on three different nonlinear systems: a numerical nonlinear system, a Van der Pol oscillator, and a Duffing oscillator. In the last simulation, an estimation of the Koopman linear model is considered as a special case, and the policy iteration algorithm is employed to evaluate optimal control designed for different reduced-dimensional models.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"487 ","pages":"Article 129068"},"PeriodicalIF":3.5000,"publicationDate":"2024-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0096300324005290","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper introduces a novel and efficient data-driven approach for approximating a finite discrete-time bilinear model of control affine nonlinear dynamical systems with a tunable parameter that balances model dimension and prediction accuracy. An approximation of the Koopman operator based on the evolutions of the nonlinear system measurements used to lift a control-affine nonlinear system to a higher dimensional model. However, higher dimensional spaces can result in a long learning time and the curse of dimensionality in control analysis. The proposed approach addresses these challenges by introducing a convex optimization which identifies informative observable functions. This technique allows for the adjustment of a parameter to strike a balance between model dimension and accuracy in prediction. The main contribution of this study is to introduce a reduced dimensional bilinear model for a nonlinear complex system. This achievement is made possible by implementing convex sparse optimization, enabling the exploration of informative estimated Koopman eigenfunctions while minimizing the number of system measurements required. The optimization problem is solved using the alternating direction method of multipliers. The effectiveness of the proposed method is evaluated on three different nonlinear systems: a numerical nonlinear system, a Van der Pol oscillator, and a Duffing oscillator. In the last simulation, an estimation of the Koopman linear model is considered as a special case, and the policy iteration algorithm is employed to evaluate optimal control designed for different reduced-dimensional models.
期刊介绍:
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.