(299c) Deep Learning Based Koopman System Identification of Nonlinear Controlled Systems

Koopman operator theory, developed in 1931 [1], provides a linear, albeit infinite-dimensional, embedding of a dynamical system even when the underlying system is nonlinear. Representing nonlinear dynamics in a linear framework is particularly appealing because it enables the analysis and control of nonlinear systems using linear systems theory. Advances in numerical techniques such as extended dynamic mode decomposition (EDMD) [2], combined with accessibility to huge amounts of data, has sparked renewed interest in Koopman analysis of nonlinear dynamical systems. In EDMD, the idea is to lift the set of system observables from state space to a higher dimensional space, usually through a nonlinear transformation, where their trajectories then evolve according to the linear Koopman operator. Although proven successful for some systems, a clear challenge of EDMD is the requirement to select a priori a suitable dictionary of basis functions for the nonlinear transformation which is usually left to the discretion of the user. To deal with this challenge, several works utilized the power of machine learning to train the dictionary employed in EDMD [3]-[6]. However, all these works consider unforced systems and for controlled dynamical systems, the predictive capability of the Koopman operator can be significantly impacted unless the role of actuation (i.e., the manipulated inputs) is appropriately accounted.

The challenge of including actuation is that since the input affects the Koopman operator and its eigenfunctions in a nonlinear way, it would be necessary to include many nonlinear functions of the input in addition to the state variables within the dictionary of EDMD. For this reason, the function space may quickly become prohibitively large to cover a sufficiently large range of the dynamics. In contrast, here, we seek to leverage the power of deep learning to discover parsimonious representations of the nonlinear dictionary. Specifically, the proposed architecture is made up of a deep autoencoder neural network and has three high-level requirements: (1) the dictionary function values are given as the outputs of an encoder network, i.e., the states and inputs are lifted using the encoder; (2) a linear time evolution of these nonlinear functions is given by the finite dimensional approximation of the Koopman operator, parameterized by the state and inputs; (3) a nonlinear decoder reconstructs the original state values from the dictionary values. The three requirements are associated with different loss functions namely, the reconstruction error of the encoder/decoder, the error associated with enforcing linear dynamics with the Koopman operator, and the future state prediction error with respect to a fixed horizon. A regularization term is also added for generality and to avoid overfitting as is the standard practice. Simultaneous learning of the operator with the encoder and decoder networks in this fashion has the capability to extract more information about the original states from fewer features provided by the encoder. This enables nonlinear reconstruction of the states from descriptive dictionary functions while still keeping the dictionary size to be extremely small. This is actually the same principle that has led to the incredible success of autoencoders for feature extraction, manifold learning, and dimensionality reduction in many applications. Based on numerical simulations, we show that the proposed deep learning based Koopman system identification performs favorably in terms of prediction accuracy for a controlled system compared to several standard techniques.

Literature cited:

[1] Koopman, B.O. Hamiltonian systems and transformation in Hilbert space. Proceedings of National Academy of Sciences USA, 17(5):315, 1931.

[2] Williams, M. O., Kevrekidis, I. G. & Rowley, C. W. A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. Journal of Nonlinear Science, 6: 1307–1346, 2015.

[3] Li, Q., Dietrich, F., Bollt, E. M. & Kevrekidis, I. G. Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the koopman operator. Chaos, 27:103111, 2017.

[4] Takeishi, N., Kawahara, Y. and Yairi, T., Learning Koopman invariant subspaces for dynamic mode decomposition. In Advances in Neural Information Processing Systems, pp. 1130-1140, 2017.

[5] Yeung, E., Kundu, S. and Hodas, N, Learning deep neural network representations for Koopman operators of nonlinear dynamical systems. In 2019 American Control Conference (ACC), Philadelphia, USA, July 10-12, pp. 4832-4839.

[6] Lusch, B., Kutz, J.N. and Brunton, S.L., Deep learning for universal linear embeddings of nonlinear dynamics. Nature communications, 9(1):1-10, 2018.