Mathematical models of real life processes pose challenges when used in numerical simulations, due to high dimensionality and complexity. Model order reduction aims to lower the complexity of such problems, for example, in simulations of large-scale dynamical and control systems. By a reduction in dimension of the model’s associated state space, parameter space or degrees of freedom, an approximation to the original model is computed. This reduced order model can then be solved to approximate the high-order solution but in significantly less time. However, in many chemical and industrial processes these reduced order models must be customized to capture the local dynamics to yield a better approximation accuracy, especially in cases of systems with moving boundaries. To achieve this, the concepts of domain partitioning and projection based reduced order modeling are combined to provide a systematic framework for model reduction of nonlinear complex distributed parameter systems.

In the first step of the framework, the partitioning of the temporal (or spatial) domain is formulated as a Mixed Integer nonlinear programming problem and solved to obtain an optimum configuration as well as an optimal number of subdomains. Next, reduced order models are developed within each subdomain which are subsequently used in dynamic optimization or design of feedback control strategies. This idea of dimensionality reduction can be further extended to the parameter space and combined with a data assimilation technique for the estimation of unknown spatially varying parameters in complex dynamical processes. Parameter estimation combined with model reduction provides an overall methodology for the design of feedback control schemes for nonlinear dynamical systems. Specifically, the proposed reduction scheme is applied to a hydraulic fracturing process to describe the fracture propagation and achieve specified control objectives such as the desired fracture geometry and uniform proppant concentration. Further research in this area involves developing online based parameter estimation and model reduction schemes for process control.

##### Related Publications

A. Narasingam, P. Siddhamshetty, J. S. Kwon, “Handling of spatial heterogeneity using POD-based EnKF in model-based feedback control of hydraulic fracturing,” *Ind. & Eng. Chem. Res.*, 2018, 57, 3977–3989. DOI: 10.1021/acs.iecr.7b04927

A. Narasingam, J. S. Kwon,”Development of local dynamic mode decomposition with control: Application to model predictive control of hydraulic fracturing,” *Comp. & Chem. Eng.*, 2017, 106, 501-511. DOI: 10.1016/j.compchemeng.2017.07.002

A. Narasingam, P. Siddhamshetty, J. S. Kwon, “Temporal clustering for order reduction of nonlinear parabolic PDE systems with time-dependent spatial domains: Application to a hydraulic fracturing process,” *AIChE J.*, 2017, 63, 3818-3831. DOI: 10.1002/aic.15733