Models nonlinear process

The scope of this book deals primarily with the parameter estimation problem. Our focus will be on the estimation of adjustable parameters in nonlinear models described by algebraic or ordinary differential equations. The models describe processes and thus explain the behavior of the observed data. It is assumed that the structure of the model is known. The best parameters are estimated in order to be used in the model for predictive purposes at other conditions where the model is called to describe process behavior. 

The targets for the MPC calculations are generated by solving a steady-state optimization problem (LP or QP) based on a linear process model, which also finds the best path to achieve the new targets (Backx et al., 2000). These calculations may be performed as often as the MPC calculations. The targets and constraints for the LP or QP optimization can be generated from a nonlinear process model using a nonlinear optimization technique. If the optimum occurs at a vertex of constraints and the objective function is convex, successive updates of a linearized model will find the same optimum as the nonlinear model. These calculations tend to be performed less frequently (e.g., every 1-24 h) due to the complexity of the calculations and the process models. 

NN applications, perhaps more important, is process control. Processes that are poorly understood or ill defined can hardly be simulated by empirical methods. The problem of particular importance for this review is the use of NN in chemical engineering to model nonlinear steady-state solvent extraction processes in extraction columns [112] or in batteries of counter-current mixer-settlers [113]. It has been shown on the example of zirconium/ hafnium separation that the knowledge acquired by the network in the learning process may be used for accurate prediction of the response of dependent process variables to a change of the independent variables in the extraction plant. If implemented in the real process, the NN would alert the operator to deviations from the nominal values and would predict the expected value if no corrective action was taken. As a processing time of a trained NN is short, less than a second, the NN can be used as a real-time sensor [113]. 

Patwardhan, A. A., Rawlings, J. B., and Edgar, T. F., Model predictive control of nonlinear processes in the presence of constraints, presented at annual AIChE Meeting, Washington, D.C. (1988). 

For continuous process systems, empirical models are used most often for control system development and implementation. Model predictive control strategies often make use of linear input-output models, developed through empirical identification steps conducted on the actual plant. Linear input-output models are obtained from a fit to input-output data from this plant. For batch processes such as autoclave curing, however, the time-dependent nature of these processes—and the extreme state variations that occur during them—prevent use of these models. Hence, one must use a nonlinear process model, obtained through a nonlinear regression technique for fitting data from many batch runs. 

Ansari, R.M. arid M.O. Tade Nonlinear Model-Bated Process Control Applications in Petroleum Refining, Springer-Vedag, Inc-, New York. NY, 2000. 

The implementation of the on-line optimization strategy requires the knowledge of current states and/or parameters in nonlinear process models in order to modify a new optimal profile defined as a set point for a controller. It is known that some measurements i.e. concentration are available at low sampling rate with significant time delay. To overcome this difficulty, state and parameter estimation is incorporated into the proposed on-line optimization algorithm. 

For the sake of simplicity, simple monophasic pharmacokinetics (one compartment and one half-life) was assumed in the above example and in many other examples in this report. In real life, most chemicals express biphasic or polyphasic pharmacokinetics (several compartments and several half-lives). Squeezing a polyphasic pharmacokinetic behavior into a one-compartment model by assuming a single half-life may lead to negligible errors for some chemicals and serious misinterpretation of biomarker concentrations for others. The same can be said about nonlinear processes, such as metabolic induction, inhibition, and saturation. A good way to check the accuracy of a simple pharmacokinetic model is to verify its performance by comparing with a physiologically based pharmacokinetic (PBPK) model that may encompass the mentioned factors. 

Particularly strong and complex interactions prevail among reaction and separation systems that are generally not at all or not fully exploited as a result of the application of the available synthesis methods for reactor networks and separation systems in isolation. The lack of generality in the synthesis methods is a tribute to the nonlinear process models required to capture the reaction and separation phenomena as well as to the vast number of feasible process design candidates. These complexities even make it difficult to synthesize the decomposed subsystems, which are typically reactor networks, separation systems, reactor-separator-recycle systems, and reactive separation systems. The development of reliable synthesis tools for these sub-systems is still an active research area. 

Simulation of Dynamic Models Linear dynamic models are particularly useful for analyzing control system behavior. The insight gained through linear analysis is invaluable. However, accurate dynamic process models can involve large sets of nonlinear equations. Analytical solution of these models is not possible. Thus, in these cases, one must turn to simulation approaches to study process dynamics and the effect of process control. Equation (8-3) will be used to illustrate the simulation of nonlinear processes. If dcjdt on the left-hand side of Eq. (8-3) is replaced with its finite difference approximation, one gets... 

The structural submodel describes the central tendency of the time course of the antibody concentrations as a function of the estimated typical pharmacokinetic parameters and independent variables such as the dosing regimen and time. As described in Section 3.9.3, mAbs exhibit several parallel elimination pathways. A population structural submodel to mechanistically cover these aspects is depicted schematically in Fig. 3.14. The principal element in this more sophisticated model is the incorporation of a second elimination pathway as a nonlinear process (Michaelis-Menten kinetics) into the structural model with the additional parameters Vmax, the maximum elimination rate, and km, the concentration at which the elimination rate is 50% of the maximum value. The addition of this second nonlinear elimination process from the peripheral compartment to the linear clearance process usually significantly improves the fit of the model to the data. Total clearance is the sum of both clearance parts. The dependence of total clearance on mAb concentrations is illustrated in Fig. 3.15, using population estimates of the linear (CLl) and nonlinear clearance (CLnl) components. At low concentra-... 

J.B. Balchen, B. Lie, and I. Solberg. Internal decoupling in nonlinear process control. Modeling Identification and Control, 9 137-148, 1988. 

Model-free adaptive (MFA) control does not require process models. It is most widely used on nonlinear applications because they are difficult to control, as there could be many variations in the nonlinear behavior of the process. Therefore, it is difficult to develop a single controller to deal with the various nonlinear processes. Traditionally, a nonlinear process has to be linearized first before an automatic controller can be effectively applied. This is typically achieved by adding a reverse nonlinear function to compensate for the nonlinear behavior so that the overall process input-output relationship becomes somewhat linear. It is usually a tedious job to match the nonlinear curve, and process uncertainties can easily ruin the effort. 

Kravaris, C., Niemiec, M., Berber, R., and Brosilow, C.B. (1998). Nonlinear model-based control of nonminimum-phase processes. In R. Berber and C. Kravaris, eds., Nonlinear Model Based Process Control, pp. 115-143. Dordrecht Kluwer Academic Publishers. 

Kumar, A., Christofides, P. D., and Daoutidis, P. (1998). Singular perturbation modeling of nonlinear processes with non-explicit time-scale separation. Chem. Eng. Sci., 53, 1491-1504. 

Liu, J., Munoz de la Pena, D., and Christofides, P. D. (2009). Distributed model predictive control of nonlinear process systems. AIChE J., 55, 1171-1184. 

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