(363t) Data-Gathering Lyapunov-Based Economic Model Predictive Control: Considering Interpretability and Physics-Based Model Selection

An important consideration for the future of model-building for the process industries is whether models can be developed from process data that are physics-based. The ability to develop such models, automatically and online, could have significant implications, including potentially enabling the development of digital twins with the ability to explain process behavior without engineering time and development. However, a significant question to ask for such cases is what defines a physics-based model. Models in general are not unique, as demonstrated by the large degree of current research in the process systems engineering community in data-driven models which are able to adequately capture process behavior. This raises the question of how to get models that are physics-based, and what their properties might be. Some attempts have been made to develop potentially physics-based models from data using, for example, [1] develops physics-based models by assuming many potential terms that might be in the model and then performing a sparse regression to attempt to locate those which might have a physics meaning. [2] develops a computational technique for attempting to locate physics-based models using partial derivatives and symbolic functions. These, however, primarily view a match to physics as meaning that an equation is simplified.

In this talk, we will provide perspectives on building physics-based process models. We will describe attempts in our recent works to use control designs to gather data which is determined to be desirable to gather [3,4] due to it aiding in discriminating between rival models in a set based on either a prediction error metric or the the steady-state which the closed-loop system is driven to under various control actions. These two methodologies have guaranteed safety properties and are developed in the context of Lyapunov-based economic model predictive control (LEMPC) [5]. They both use nested stability regions for a set of model candidates used in designing the control law (where it is assumed that a representative model of the system dynamics is in this set) for the stability guarantees, but one of the methods repeatedly changes the steady-state around which the stability regions are designed. Simulation examples of a nonideal reactor and level control in a tank demonstrate that there are conditions under which these methods could be valuable for discriminating between rival models being considered for a system, but sensor noise and plant/model mismatch (which would be expected to be inevitable in a real plant) can make this control-assisted discrimination task challenging. Furthermore, these methods do not provide guidance on how to get the different rival models under consideration beyond to postulate a large number of potential models and hope that a sufficiently representative model is in the set. Give the requirements of the stability region nesting, this could be a hard consideration to work with.

This motivates the need for further analysis of when a model is physics-based and how to tell if it has potential to be a priori, before including it, for example, in one of these control-assisted algorithms for gathering data for model discrimination. We hypothesize that an interpretability property of a system could showcase whether it has potential to be physics-based or not, but defining interpretability of a model can be challenging. Interpretability has been given many definitions in the context of, for example, neural networks [6]. We will also describe an attempt to investigate interpretability of a neural network model of a continuous stirred tank reactor (CSTR) inspired by [7] by exploring how weights in the network change when re-training the same model structure after changing a parameter of the CSTR model (e.g., changing a parameter of the reaction rate law). These attempts to investigate interpretability of a neural network for a physical system aid to showcase that a definition of interpretability will require a more rigorous and systematic definition to be useful for building physics-based models in an automated fashion.

We conclude with some discussion of relationships between the data-gathering control designs and cybersecurity-related studies performed in our group (e.g., [8]), where one of the methods has a similar form to what is done in the data-gathering strategies that involve changing steady-states around which the stability regions are designed with time. We discuss to what extent insights from these two different domains provide guidance for the other. We close with a beginning extension of these topics to distributed parameter systems through a computational fluid dynamics model of a flow system where we perform a cyberattack and a data-gathering maneuver to begin to examine how the prior works might extend to systems described by partial differential equations.

References:

1. Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932-3937.

2. Schmidt, M., & Lipson, H. (2009). Distilling free-form natural laws from experimental data. science, 324(5923), 81-85.

3. Oyama, Henrique, and Helen Durand. "Lyapunov-Based Economic Model Predictive Control for Online Model Discrimination." Computers & Chemical Engineering (2022): 107769.

4. Oyama, H., Leonard, A. F., Rahman, M., Gjonaj, G., Williamson, M., and Durand, H., "On-line Process Physics Tests via Lyapunov-based Economic Model Predictive Control and Simulation-Based Testing of Image-Based Process Control, " American Control Conference, paper 1187, 2022.

5. Heidarinejad, M., Liu, J., & Christofides, P. D. (2012). Economic model predictive control of nonlinear process systems using Lyapunov techniques. AIChE Journal, 58(3), 855-870.

6. Chakraborty, S., Tomsett, R., Raghavendra, R., Harborne, D., Alzantot, M., Cerutti, F., Srivastava, M., Preece, A., Julier, S., Rao, R. M., Kelley, T. D., Braines, D. , Sensoy, M., Willis, C. J., & Gurram, P. (2017, August). Interpretability of deep learning models: A survey of results. In 2017 IEEE Smartworld, Ubiquitous Intelligence & Computing, Advanced & Trusted Computed, Scalable Computing & Communications, Cloud & Big Data Computing, Internet of People and Smart City Innovation (SmartWorld/SCALCOM/UIC/ATC/CBDcom/IOP/SCI) (pp. 1-6). IEEE.

[7] Wu, Z., & Christofides, P. D. (2019). Economic machine-learning-based predictive control of nonlinear systems. Mathematics, 7(6), 494.

[8] Oyama, H., & Durand, H. (2020). Integrated cyberattack detection and resilient control strategies using Lyapunov‐based economic model predictive control. AIChE Journal, 66(12), e17084.