(183k) Dynamic Mode Decomposition Based Model Reduction for Control of Moving Boundary Problems via Approximate Dynamic Programming

Mohammed Saad Faizan Bangi^{[1, 2]}, Harwinder Singh Sidhu^{[1, 2]}, Prashanth Siddhamshetty ^{[1, 2]},

Joseph Sang-Il Kwon ^{[1, 2]}

[1] Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX 77845 USA

[2] Texas A&M Energy Institute, Texas A&M University, College Station, TX 77845 USA

There are a large number of industrial control problems which can be characterized by highly nonlinear distributed parameter systems (DPS) with moving boundaries, such as catalytic reactors [1] and chemical vapor decomposition [2]. In general, mathematical models of these processes are derived from first-principles and consist of nonlinear parabolic partial differential equations (PDEs) with time-varying spatial domains. The literature over the last three decades is affluent with the development in the modeling and nonlinear control of parabolic PDEs and has mainly focused on systems with time-invariant spatial domains [3]. Despite the progress on nonlinear control of parabolic PDE systems with time-invariant spatial domains, few investigations are available on control and estimation of parabolic PDE systems with time-varying spatial domains [4].

In the past 15 years, owing to its ability to handle a large-scale multivariable system with both input and state constraints, significant efforts have been made to apply model predictive control (MPC) to systems modeled by nonlinear PDEs [5]. Despite the popularity of this approach, there are two drawbacks of the MPC approach. First, the online computational load to calculate the optimal control moves at each sampling time can be significant and demand the use of long prediction/control horizons to ensure satisfactory performance. The second is its insufficiency to handle the uncertainty, which is an inherent feature while solving the real-time optimization problems in chemical plants.

Alternatively, both the issues of MPC formulation can be addressed by the approximate dynamic programming (ADP) approach [6]. In this approach, an improved control policy is derived after starting with suboptimal control policies, while circumventing the ‘curse-of-dimensionality’ of the traditional dynamic programming (DP) approach. Function approximators such as k-nearest neighbor or artificial neural networks are used to obtain the map between the ‘cost-to-go’ values and system states, and then Bellman equation is solved offline with the approximator in an iterative manner. However, it is not straightforward to apply ADP to systems characterized by nonlinear PDEs with time-varying spatial domains because of the high computational efforts required to derive the ‘cost-to-go’ function. Therefore, it is necessary to develop reduced-order models (ROMs) for the application of ADP to such systems.

The primary objective of this work is two-fold. First, we develop a temporally local ROM that reproduces the essential features of the underlying parabolic PDE system with time-varying spatial domains. This is accomplished by partitioning the temporal domain into multiple temporal subdomains using global optimum search (GOS) framework [7], and then applying dynamic mode decomposition (DMD) technique [8] within each cluster. Second, we employ the developed temporally local ROM in ADP based controller. In application, we will apply the proposed ADP based controller to several parabolic PDE systems with moving boundaries.

[1] Bremer, J., Goyal, P., Feng, L., Benner, P. and Sundmacher, K., 2017. POD-DEIM for efficient reduction of a dynamic 2D catalytic reactor model. Computers & Chemical Engineering, 106, pp.777-784.

[2] Baker, J. and Christofides, P.D., 1999. Output feedback control of parabolic PDE systems with nonlinear spatial differential operators. Industrial & Engineering Chemistry Research, 38(11), pp.4372-4380.

[3] Christofides, P.D., 2012. Nonlinear and robust control of PDE systems: Methods and applications to transport-reaction processes. Springer Science & Business Media.

[4] Armaou, A. and Christofides, P.D., 2001. Robust control of parabolic PDE systems with time-dependent spatial domains. Automatica, 37(1), pp.61-69.

[5] Dufour, P., Touré, Y., Blanc, D. and Laurent, P., 2003. On nonlinear distributed parameter model predictive control strategy: on-line calculation time reduction and application to an experimental drying process. Computers & Chemical engineering, 27(11), pp.1533-1542.

[6] Lee, J.H. and Lee, J.M., 2006. Approximate dynamic programming based approach to process control and scheduling. Computers & Chemical engineering, 30(10-12), pp.1603-1618.

[7] Tan, M.P., Broach, J.R. and Floudas, C.A., 2007. A novel clustering approach and prediction of optimal number of clusters: global optimum search with enhanced positioning. Journal of Global Optimization, 39(3), pp.323-346.

[8] Schmid, P.J., 2010. Dynamic mode decomposition of numerical and experimental data. Journal of fluid mechanics, 656, pp.5-28.