(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]
 Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX 77845 USA
 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  and chemical vapor decomposition . 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 . 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 .
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 . 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 . In this approach, an improved control policy is derived after starting with suboptimal control policies, while circumventing the âcurse-of-dimensionalityâ 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 âcost-to-goâ 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 âcost-to-goâ 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 , and then applying dynamic mode decomposition (DMD) technique  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.
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