(644h) Offset-Free Koopman-Based Model Predictive Control of a Batch Pulp Digester

Kraft pulping is a dominant pulp manufacturing process where lignin, a highly branched polymer that links cellulose microfibrils in wood chips, is removed by an alkaline solution known as white liquor at elevated temperature. Since the amount of residual lignin in the pulp significantly affects the physical and mechanical properties of final products, previous studies have been carried out to describe the degree of delignification, which is indicated as Kappa number, by developing the first principle and empirical models [1,2]. Subsequently, model-based controllers have been developed to efficiently attain the desired Kappa number during the pulping process [3,4].

Meanwhile, recent studies have revealed that microscopic characteristics of fibers such as porosity, cell wall thickness (CWT), and fiber length considerably affect the properties of paper products. Hence, a multiscale modeling framework, which integrates the extended Purdue model and the kinetic Monte Carlo (kMC) model, has been suggested to capture the evolution of both macroscopic and microscopic characteristics during the pulping process [5]. Based on this framework, futher studies have been carried out to consider the influence of fiber collapse and breakage phenomena, and fiber-to-fiber heterogeneity on delignification rate and fiber attributes [6-8]. However, due to the high computational cost, it is challenging to directly implement these high-fidelity multiscale models to the model-based controller to regulate desired pulp properties.

To this end, a reduced-order model was derived via extended dynamic mode decomposition (EDMD), which is one of the most commonly used Koopman operator-based data-driven model identification methods for controlled systems [9,10] due to its capability of providing a reliable linear representation for nonlinear systems that enables the utilization of established control methods for linear systems. However, the inherent plant-model mismatch in EDMD due to the finite-dimensional approximation of the infinite-dimensional Koopman operator degrades the closed-loop performance of model-based control systems. To overcome such limitation, an offset-free control framework that compensates for the influence of plant-model mismatch and disturbances using the disturbance model was designed [11,12]. Subsequently, the designed offset-free Koopman-based model predictive control system was implemented to a batch pulp digester to obtain the desired Kappa number and the CWT value. Consequently, the closed-loop numerical experiment demonstrated that the developed control system is able to drive the Kappa number and CWT to the set-points under plant-model mismatch and additional disturbance.

REFERENCES

[1] Wisnewski, P. A., Doyle III, F. J., & Kayihan, F. (1997). Fundamental continuous‐pulp‐digester model for simulation and control. AIChE journal, 43(12), 3175-3192.

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[9] Williams, M. O., Kevrekidis, I. G., & Rowley, C. W. (2015). A data–driven approximation of the koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 25(6), 1307-1346.

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[11] Pannocchia, G., & Bemporad, A. (2007). Combined design of disturbance model and observer for offset-free model predictive control. IEEE Transactions on Automatic Control, 52(6), 1048-1053.

[12] Maeder, U., Borrelli, F., & Morari, M. (2009). Linear offset-free model predictive control. Automatica, 45(10), 2214-2222.