(432a) Learning-Based Estimation for Distributed Parameter Systems

Many important chemical engineering plants are of distributed nature in space and time and thus can be described by distributed parameter systems (DPS), e.g., chemical reactor plants, heat exchangers, fluid flow processes, etc. Model discovery and estimation of distributed parameter systems have been active research topics in science and engineering [1-3], which is fundamental to realize advanced modelling, control and monitoring. Recently, one learning-based approach is proposed by using sparse identification of nonlinear dynamical systems (i.e., SINDy algorithm [4]) and its variants [5-6]. There are several limitations of the proposed SINDy algorithms. First, it is designed as a batch estimation approach, implying that all measurements (collected from initial time instance to the current time instance) are simultaneously used for calculation. The complexity of the estimation problem increases at least linearly as the increase of the size of measurements. This would cause the estimation problem intractable as the time instance goes to infinity. Moreover, the parameter distributions and their uncertainties could not be inferred from the measurements by using the SINDy algorithms. Furthermore, it is difficult to properly handle the measurement noises/disturbances as well as practical measurements with delay, multi-rate, and/or missing data, based on the SINDy algorithms.

Variational Bayesian as a cornerstone parametric statistical estimation technique has been widely used for model identification, parameter estimation, and soft sensing of lumped parameter systems [7-8]. In this work, a variational Bayesian (VB) learning-based approach is proposed for model identification and parameter estimation of distributed parameter systems. The proposed method has the following advantages. Instead of point estimation, the proposed method enables us to estimate the probability distributions of the parameters of DPSs accounting for corresponding uncertainties and providing better probabilistic interpretability, compared to SINDy algorithms. In addition, the proposed method converts the batch estimation in SINDy algorithms into a recursive estimation framework which is amenable for online estimation. The measurement noises/disturbances can be naturally addressed under the VB learning scheme. Moreover, the delay and/or multi-rate issues in the measurements can be readily addressed under the presented VB learning scheme [9-10]. The proposed design can be applied to model identification and parameter estimation of linear and nonlinear distributed parameter systems modelled by partial differential equations (PDE) and partial differential integral equations (PIDE). Specifically, the effectiveness of the proposed design is validated through simulation examples on heat exchangers (described by two first-order hyperbolic PDEs) and transport processes with accumulative terms (described by a first-order hyperbolic PIDE) that are widely utilized in chemical engineering practice.

References

[1]. J. Bongard, and H. Lipson, 2007. Automated reverse engineering of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 104(24), pp.9943-9948.

[2]. M. Schmidt, and H. Lipson, 2009. Distilling free-form natural laws from experimental data. Science, 324(5923), pp.81-85.

[3]. M. A. Demetriou, and I. G. Rosen, 1993. On the persistence of excitation in the adaptive estimation of distributed parameter systems. In 1993 American Control Conference, pp. 454-458. IEEE.

[4]. S. L. Brunton, J. L. Proctor, and J. N. Kutz, 2016. Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proceedings of the national academy of sciences, vol. 113, no. 15, pp. 3932-3937.

[5]. S. H. Rudy, S. L. Brunton, J. L. Proctor, and J. N. Kutz, 2017. Data-driven discovery of partial differential equations, Science Advances, vol. 3, no. 4, p. e1602614.

[6]. S. H. Rudy, A. Alla, S. L. Brunton, and J. N. Kutz, 2019. Data-driven identification of parametric partial differential equations, SIAM Journal on Applied Dynamical Systems, vol. 18, no. 2, pp. 643-660.

[7]. S. Ludger, and D. Ostwald, 2017. Variational Bayesian parameter estimation techniques for the general linear model." Frontiers in neuroscience 11: 504.

[8]. J. Xie, B. Huang, and S. Dubljevic, 2021. Transfer Learning for Dynamic Feature Extraction Using Variational Bayesian Inference. IEEE Transactions on Knowledge and Data Engineering.

[9]. R. Han, Y. Salehi, B. Huang, and V. Prasad, 2021. Parameter estimation for nonlinear systems with multirate measurements and random delays. AIChE Journal 67, no. 9 (2021): e17327.

[10]. Y. Ma, S. Kwak, L. Fan, and B. Huang, 2018. A variational Bayesian approach to modelling with random time-varying time delays." In 2018 Annual American Control Conference (ACC), pp. 5914-5919. IEEE.