Physical Model Parameter Estimation of a Nonlinear Process

Abstract

Abstract-A physical model of a nonlinear subprocess in a continuous paper pulp digester is discussed and simplified. Model approximation is carried out in order to produce a simple linear model to be used for unknown parameter estimation of the physical model. The Taylor series expansion and the orthogonal collocation method are applied for the model linearization and model lumping, respectively. The reduced model is expressed as a standard state space form. The model parameters are estimated in the least squares sense, and the parameters retain their own physical meanings. The results of the parameter estimation are discussed and the model is verified using validation data. I. INTRODUCTIONProcess models play a more and more important role in the development of control technology. Modelling of a continuous paper pulp digester is a troublesome task due to a complex mixture of chemical reactions and transport phenomena within the process. System identification is a good alternative to model a complex process, for example, a digester. Comparing with black-box models, the parameters in a physical model bear certain physical meanings, which can be very useful in the diagnostic procedure or the controller design. The current study aims to deliver a simple linear model for a distributed parameter nonlinear process. For this process a complex nonlinear physical model is already available, and there exist unknown parameters in the physical model. The unknown parameters can be identified based on the simple linear model that is an approximation of the complex physical model, and the parameter estimates retain their own physical meanings. A subprocess in a continuous paper pulp digester is selected as an application example. In the next section, a physical digester model is simplified for the selected subprocess. Then the model is linearized and lumped. In Section 3, a software tool developed for the parameter estimation is applied and the parameter estimation results are analyzed. Model validation is discussed in Section 4. Section 5 gives conclusions.

Dynamic modeling and model predictive control of a continuous pulp digester

“…For simplicity, the degradation of carbohydrates is assumed to follow the lignin reduction linearly, and the concentrations of dissolved solids in the entrapped liquor and free liquor can be omitted. [27][28][29] There is a linear approximation relationship between the lignin and carbohydrate F I G U R E 3 Cook zone and wash zone in a digester in the solid phase, ρ carb ¼ 0:36 þ 0:84ρ lig , and, as a result, the lignin and carbohydrate need not necessarily be regarded as being two states of the system. 27 In addition, according to Gustafson et al, 9 the carbohydrate reaction rate in the bulk delignification period is proportional to the lignin reaction rate: R carb ¼ 0:47R lig .…”

“…The cross‐sectional area is assumed to be constant in the digester. For simplicity, the degradation of carbohydrates is assumed to follow the lignin reduction linearly, and the concentrations of dissolved solids in the entrapped liquor and free liquor can be omitted 27–29 . There is a linear approximation relationship between the lignin and carbohydrate in the solid phase, , and, as a result, the lignin and carbohydrate need not necessarily be regarded as being two states of the system 27 .…”

Sliding Mode Control with Adaptive Approaching Law for Bioreactor

“…System identification is used to linearize the model of the bioreactor [8][9]. For the bioreactor mathematical model shown in equation ( 1), the linearized state space equation can be described as…”