Abstract
AbstractThe Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein–Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.
Funder
Carl-Zeiss-Stiftung
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Engineering,Modeling and Simulation
Reference41 articles.
1. Bakry, D., Gentil, I., Ledoux, M.: Analysis and geometry of Markov diffusion operators, vol. 348. Springer Science & Business Media, Berlin (2013)
2. Beck, A., Schwartz, J.T.: A vector-valued random ergodic theorem. Proc. Am. Math. Soc. 8(6), 1049–1059 (1957)
3. Boccia, A., Grüne, L., Worthmann, K.: Stability and feasibility of state constrained MPC without stabilizing terminal constraints. Syst. Control Lett. 72(8), 14–21 (2014)
4. Bruder, D., Fu, X., Vasudevan, R.: Advantages of bilinear Koopman realizations for the modeling and control of systems with unknown dynamics. IEEE Robot. Autom. Lett. 6(3), 4369–4376 (2021)
5. Brunton, S.L., Budišić, M., Kaiser, E., Kutz, J.N.: Modern Koopman theory for dynamical systems. SIAM Rev. 64(2), 229–340 (2022)
Cited by
8 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献