(182h) Learning Linear Representations of Nonlinear Dynamics Using Deep Learning

The control and optimization of nonlinear systems is an important task as most systems of practical interest exhibit nonlinear behaviour. However, at the same time, this task is a formidable challenge with no general and scalable solution. This is especially relevant for large-scale systems, the likes of which are often encountered in process systems engineering, which encompasses multiscale spatiotemporal phenomena which can be difficult to accurately model. Moreover, even if such a model could be identified, the resulting model structure may be too complex for the tractable control and optimization of the system of interest [1].

In contrast, the study of linear systems is well developed with the scalable design, analysis, control, and optimization of linear systems thoroughly detailed within the literature [2]. To take advantage of these developments, one can obtain linear approximations of nonlinear systems by linearizing around an operating point. While this may prove to be an effective strategy for some nonlinear systems, this may not be generally applicable for systems exhibiting strong nonlinearities. Instead, for such systems, a more general approach is required. In such instances, it may be possible to consider a coordinate transformation or a change of variables such that the original nonlinear system is transformed into an equivalent linear system. For example, such ideas are leveraged in feedback linearization to control nonlinear systems [3].

Along these lines, there has been renewed interest in the seminal work of Bernard Koopman on Koopman operator theory [4]. In essence, Koopman operator theory allows for a nonlinear system to be represented as an infinite dimensional linear system. This is achieved by considering transformations of the original system’s variables such that the resulting system is linear in the new variables. It could be argued that Koopman operator theory does not solve the original problem as nonlinearity is traded for infinite dimensionality. However, with the advent of more powerful data-driven methods in machine learning and system identification, finite dimensional approximations can be obtained of the infinite dimensional linear representation. In particular, Dynamic Mode Decomposition (DMD) was proposed to compute these finite dimensional approximations [5]. Consequently, DMD, and by extension, Koopman operator theory, has been applied with great success to a broad range of fields, ranging from fluid mechanics [6] to neuroscience [7], all of which are underpinned by nonlinear dynamics.

Despite this success, DMD is restricted to linear transformations of the original system variables which renders the approach restrictive for many nonlinear systems of interest [8]. Consequently, to allow for a richer set of transformations, many alternative approaches have been proposed from the use of kernel functions to neural networks. Additionally, Koopman operator theory, in its original form, does not consider the effects of inputs and control for the nonlinear system. However, in recent years, many advances have been made to generalise Koopman operator theory to circumvent this challenge. These extensions could be used to represent complex nonlinear systems with finite dimensional linear approximations which could subsequently be used for the tractable control of the system.

For the above reasons, we propose a deep learning framework to discover a transformation of a nonlinear system to an equivalent higher dimensional linear system. In the proposed framework, the neural network serves to parameterize the function space over which the transformation is searched for, allowing for a tractable optimization problem to be solved. Additionally, we leverage both forward and backpropagation to relate the original dynamics to the learned transformed dynamics by using the chain rule to approximate the Jacobian of the transformation. Moreover, we demonstrate that the use of neural networks, through the collection of activation functions, allows for a broader range of transformations of the original system’s variables. Consequently, through simulation results we can accurately capture the original system dynamics using the learned linear representation for a wider range of operating conditions than would be possible with standard linearization. This is depicted in Fig. 1 which shows the simulation results of the reactor temperature in a CSTR. The black line corresponds to the full nonlinear CSTR model while the red line depicts the simulation results from a model obtained by linearizing around a fixed point. The discrepancy between these results is clear. On the other hand, the green dashed line shows the results of the model learned by the proposed deep learning framework which matches the results from the full nonlinear model. In addition to this, we demonstrate that the resulting learned linear representations can be used to successfully devise model-based control strategies to control the original nonlinear system.

References

[1] Antoulas, A., 2005. An overview of approximation methods for large-scale dynamical systems. Annual Reviews in Control, 29(2), pp.181-190.

[2] Katsuhiko Ogata, 2010. Modern Control Engineering. Uppern Saddle River, N.J.: Prentice Hall.

[3] Rubio, J., 2018. Robust feedback linearization for nonlinear processes control. ISA Transactions, 74, pp.155-164.

[4] Koopman, B., 1931. Hamiltonian Systems and Transformation in Hilbert Space. Proceedings of the National Academy of Sciences, 17(5), pp.315-318.

[5] Schmid, P., 2010. Dynamic mode decomposition of numerical and experimental data. Journal of Fluid Mechanics, 656, pp.5-28.

[6] Schmid, P., Li, L., Juniper, M. and Pust, O., 2010. Applications of the dynamic mode decomposition. Theoretical and Computational Fluid Dynamics, 25(1-4), pp.249-259.

[7] Brunton, B., Johnson, L., Ojemann, J. and Kutz, J., 2016. Extracting spatial–temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition. Journal of Neuroscience Methods, 258, pp.1-15.

[8] Kaiser, E., Kutz, J. and Brunton, S., 2021. Data-driven discovery of Koopman eigenfunctions for control. Machine Learning: Science and Technology, 2(3), p.035023.