(59ab) Generalization Error Bounds for Neural Networks Modeling Two-Time-Scale System Dynamics with Application to Model Predictive Control of Nonlinear Processes

Two-time scale processes are types of systems that exhibit behavior at two different (slow and fast) time scales, where the slow states evolve on a slower time-scale than the fast states [1]. This type of time-scale multiplicity can be found in many real-life processes, such as catalytic reactors, distillation columns, and biological systems. Two-time scale systems are challenging to analyze due to the interactions between the states evolving at different time scales. However, several techniques have been developed in the literature that have proven to be effective when analyzing two-time scale systems. One technique is the singular perturbation method, where the two-time scale system is decomposed into reduced-order subsystems associated with the fast and slow time-scales [1]. Specifically, to further analyze the system, reduced-order modeling is used to separate the singular perturbation system into two subsystems where one subsystem represents the slow dynamics, and the other subsystem represents the fast dynamics [1]. Despite the extensive work on analysis and control of two-time-scale processes for about half a century, there is very limited work on the use of data-based techniques in the modeling of two-time-scale processes.

To approximate the dynamics of both slow and fast subsystems using data from the two-time-scale process, we plan to design two machine learning models—specifically, neural network models. Assuming stability of the fast dynamics, we will construct a Recurrent Neural Network model to predict the evolution of the slow states of the two-times-scale system and a Feedforward Neural Network to predict the values of the fast states on the slow manifold from the predicted slow states. The generalization error bounds for both networks will be investigated by applying the appropriate theory of statistical machine learning. The generalization error bounds of the neural network models can be used to assess the accuracy and robustness of the models. In addition, since we are dealing with two-time scale systems, the effect of the two different time scales will be accounted for in the generalization error bounds. Moreover, by assuming that the fast dynamics are stable, we will show that, to achieve closed-loop stability for the full two-time-scale system dynamics, it is sufficient to design a model predictive controller (i.e., MPC) based on the RNN model that approximates the slow dynamics to achieve stability, rather than the full two-time-scale system. Finally, we conduct closed-loop simulations using a classical two-time-scale process example to study the factors that affect the generalization error bounds and the closed-loop stability when using a machine learning-based predictive controller.

References:

[1] Kokotović, P., Khalil, H. K., & O'reilly, J. (1999). Singular perturbation methods in control: analysis and design. Society for Industrial and Applied Mathematics.