(644b) Development of Offset-Free MPC Framework for Hydraulic Fracturing Process Using Sindy

Data-fueled modeling and control of complex systems are currently undergoing a revolution, driven by the confluence of big data, advanced algorithms in machine learning, and modern computational hardware [1]. Model-based control strategies, such as model predictive control (MPC), rely on accurate and efficient models that capture the relevant dynamics for a given system. In this regard, technical advancement in data-driven modeling provides a tremendous opportunity to extend the reach of model predictive control (MPC) [2]. Moreover, since plant-model mismatch and un-measured disturbance always exist in real systems, MPC usually cannot achieve optimal closed-loop performance. Although these data-driven models show promise and are successful, many leading modern techniques such as neural networks rely on access to large amounts of data, may not be interpretable, and do not incorporate known constraints. Hence, in this work, we use the sparse identification of nonlinear dynamics (SINDy) modeling procedure and demonstrate its ability to identify the fewest terms needed to explain the plant-model mismatch, making the model interpretable and generalizable [3]. Specifically, SINDy is combined with MPC for enhanced data-driven control of a hydraulic fracturing process, improving the prediction accuracy and control performance.

Hydraulic fracturing is a highly complicated process where many vital characteristics are not considered when developing a mathematical model [4]. In the proposed framework, we use SINDy to identify a model that captures the offset between plant measurements and first-principles model predictions of a fracturing process, namely fracture width, width at the wellbore, and fracture length. The resulting sparse model, which describes the offset, is considered as a disturbance term in the offset-free MPC formulation. Thus, the disturbance model in the framework represents the nonlinear dynamics of the offset which in this case represents the structural uncertainty. Structural uncertainty is a term in the offset-free MPC that describes the model bias which comes from the underlying knowledge of the process. This disturbance model is then augmented to the state-space model, and the states are driven to their set points using the modeled structural uncertainty. These state and disturbance estimates are used to initialize the MPC problem [5]. Generally speaking, offset free control is guaranteed when there are no constraints or when the disturbance is either a constant or linear in nature [6,7]. Here, we apply the identified plant-model mismatch to the shrinking horizon optimal control problem to improve the prediction accuracy and closed-loop performance of the MPC [8]. The disturbance estimates from the SINDy model strengthen the prediction accuracy in the shrinking horizon problem as the offset-free framework captures the system's nonlinear dynamics well.

The proposed SINDy-based offset-free MPC framework has high reference tracking performance, requires significantly lesser data to cope with rapid system changes by representing the offset's nonlinear nature. The framework is computationally more efficient and robust to noise than the nominal MPC models that use either neural networks for plant model mismatch or complex high-fidelity models. Furthermore, the closed-loop simulation results demonstrate that the proposed framework can efficiently model the intrinsic plant-model mismatch that should be handled for the performance of offset-free MPC. Thus, the proposed method is effective in enhancing the performance of offset-free MPC for real-time applications.

References:

1. Kutz, J. Nathan, et al. Dynamic mode decomposition: data-driven modeling of complex systems. Society for Industrial and Applied Mathematics, 2016.
2. Kaiser, Eurika, J. Nathan Kutz, and Steven L. Brunton. "Sparse identification of nonlinear dynamics for model predictive control in the low-data limit." Proceedings of the Royal Society A 474.2219 (2018): 20180335.
3. Brunton, Steven L., Joshua L. Proctor, and J. Nathan Kutz. "Discovering governing equations from data by sparse identification of nonlinear dynamical systems." Proceedings of the national academy of sciences 113.15 (2016): 3932-3937.
4. Siddhamshetty, Prashanth, Seeyub Yang, and Joseph Sang-Il Kwon. "Modeling of hydraulic fracturing and designing of online pumping schedules to achieve uniform proppant concentration in conventional oil reservoirs." Computers & Chemical Engineering 114 (2018): 306-317.
5. Maeder, Urban, Francesco Borrelli, and Manfred Morari. "Linear offset-free model predictive control." Automatica 45.10 (2009): 2214-2222.
6. Pannocchia, Gabriele, and James B. Rawlings. "Disturbance models for offset‐free model‐predictive control." AIChE journal 49.2 (2003): 426-437.
7. Badgwell, Thomas A., and Kenneth R. Muske. "Disturbance model design for linear model predictive control." Proceedings of the 2002 American Control Conference (IEEE Cat. No. CH37301). Vol. 2. IEEE, 2002.
8. Das, Buddhadeva, and Prashant Mhaskar. "Lyapunov‐based offset‐free model predictive control of nonlinear process systems." The Canadian Journal of Chemical Engineering 93.3 (2015): 471-478.