(711f) Local Dynamic Mode Decomposition with Control: It’s Application to Model Predictive Control of Hydraulic Fracturing

More often than not, a chemical process can be accurately described using a mathematical model which is generally comprised of nonlinear partial differential equations (PDE). In order to accurately capture the dynamics of systems described by PDEs, a large number of state snapshots are required and this makes it computationally expensive to design online control strategies. Dynamic Mode decomposition with control (DMDc) is an effective model reduction technique that has been developed by the fluids community to analyze observational data arising from high-dimensional dynamic systems with external inputs applied to the system [1]. It is a data driven, equation free framework that approximates the underlying dynamics from snapshot measurements alone. Computationally, DMDc assumes a linear model that best represents the underlying dynamics, even if those dynamics stem from a nonlinear process. Although it might seem equivocal describing a nonlinear system by superposition of modes whose dynamics are governed by the corresponding eigenvalues, DMDc can be considered a numerical approximation to Koopman spectral analysis providing theoretical justification for characterizing nonlinear systems [2].

However, for a highly nonlinear system, the assumption of linear relation might not work well especially, only if limited number of measurements are available to characterize the data. Additionally, it fails to capture the local dynamics when the process parameters change with space and time. Intuitively, for a system with nonlinear dynamics, this assumption of linear relation can still be applicable when we consider shorter periods of time (i.e., time-clustered data). Based on these observations, in order to capture the local dynamics of a complex nonlinear system more effectively, the embedded coherent structures must be tailored to the local behavior of every portion of the solution trajectory [3].

In this work, we extend the concepts of DMDc to better capture the local dynamics associated with highly nonlinear processes and develop local reduced-order models that accurately describe the fully-resolved data. In this context, we first partition the data into clusters using a Mixed Integer Nonlinear Programming based optimization algorithm, the Global Optimum Search, which incorporates an added feature of predicting the optimal number of clusters [4]. Next, we compute the reduced-order models that are tailored to capture local system dynamics by applying DMDc within each cluster. The developed models can be subsequently used to compute approximate solutions to the original high-dimensional system and to design feedback control systems. We successfully applied the proposed local model reduction method to design a model predictive controller (MPC) of a hydraulic fracturing process, which is characterized by a system of nonlinear highly-coupled PDEs with time-dependent spatial domains, to achieve uniform proppant concentration at the end of pumping [5]. We also compared the performance of MPC based on the reduced-order models developed by the proposed local DMDc method and the Multivariable Output Error State Space (MOESP) identification method (of same order). The results showed that the proposed method outperforms the MOESP by approximating the nonlinear system more accurately.

Literature cited:

[1] Proctor, J.L., Brunton, S.L., Kutz, J.N. Dynamic mode decomposition with control. SIAM J. Appl. Dyn. Syst. 2016;15:142-161.

[2] Rowley, C.W., Mezic, I., Bagheri, S., Schlatter, P., Henningson, D.S. Spectral analysis of nonlinear flows. J. Fluid Mech. 2009;641:115-127.

[3] Amsallem D, J. Zahr M., Farhat C. Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Methods Eng. 2012;92:891-916.

[4] Tan M. P., Broach J. R., Floudas C. A. A novel clustering approach and prediction of optimal

number of clusters: global optimum search with enhanced positioning. J. Glob. Optim. 2007;39:323-346.

[5] Economides, M.J., Nolte, K.G., 2000. Reservoir stimulation. Chichester, Wiley.