General nonlinear models

The balance equations analyzed in Sections 8.2 and 8.3 are examples of nonlinear models (sets of constraints). We limit ourselves to steady-state models, in the sense explained in Section 4.2, first paragraph. Generally, such model reads 

Introducing the vector p makes sense when the dependence of the solutions on the a priori parameters is analyzed. Among the components of p, we can also have certain thermodynamic data such as specific heats, standard enthalpies, and the like. Some of the components of p can be certain constants of the model specific to the given system, but not known a priori then the problem reads finding the values of these constants based on certain measured data. We shall not deal with problems of this kind. Hence the model will read more simply 

A meaningful model is clearly solvable. But generally, the set laf can have rather strange properties and the analysis is difficult. In the case of balance 

The situation can change when certain components of vector z have been fixed (for example as the measured ones). See the examples in Section 8.1. The analysis is then considerably more difficult. To some extent, it is facilitated when the feasible set is a manifold, as will be shown in the next sections. 

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If a model is nonlinear with respect to the model parameters, then nonlinear least squares rather than linear least squares has to be used to estimate the model coefficients. For example, suppose that experimental data is to be fit by a reaction rate expression of the form rA = kCA. Here rA is the reaction rate of component A, CA is the reactant concentration, and k and n are model parameters. This model is linear with respect to rate constant k but is nonlinear with respect to reaction order n. A general nonlinear model can be written as... 

For a general nonlinear model f(xt, [1 where x is the vector of the independent model variables and P is the vector of parameters,... 

With the coding 4, subroutine ADVAN 6, which implements a general nonlinear model user-dehned with differential equations describing the process, was used. The model was parameterized in clearance and volume. The distribution volume of M8 was hxed to 1L. The differential equations used are presented below ... 

Lindsey, J.K., Byrom, W.D., Wang, J., Jarvis, P., and Jones, B. Generalized nonlinear models for pharmacokinetic data. Biometrics 2000 56 81-88. 

The same idea is applied to more general cases of balance constraints. The (generally) nonlinear model is linearized. We then admit that certain variables are already measured in a given plant, and some other measuring points... 

M-vector function of A -vector variable z introduced by the general nonlinear model (set of constraints) g(z) = 0 (8.4.5), later specified (possibly with different notation) in different particular cases... 

This model is linear with respect to rate constant k but is nonlinear with respect to reaction order n. A general nonlinear model can be written as... 

Although the posterior density above appears to be of a simple functional form, for a general nonlinear model, S(0) (8.58) may depend in a very comphcated manner upon 6. 

Thus the world of nonlinear models and equations is very important to the field of engineering. An advantage of approaching everything directly from the start as a nonlinear model is that cases of linear models become simply special cases of general nonlinear models. A brief discussion of how the analysis approach for nonlinear models differs from that for linear models is thus appropriate. A vast array of books and analysis tools exist for linear models and linear equations. No such array of material exists for nonlinear models. Even proofs that solutions exist for nonlinear models are in the vast majority of cases not available. So what... 

From this brief discussion one can see that the solution of nonlinear models is closely tied to a toolbox of methods for solving linear models. A linear set of model equations requires only one loop through the L I iterative loop and thus can be considered a special case of solving nonlinear models. Thus of necessity considerable discussion will be presented in subsequent chapters of solution methods for linear models. However, this will be considered only as an intermediate step in embedding such linear models within an iterative loop for the solution of more general nonlinear models. Considerable effort will be given to the linearization of various models, on the testing of solution accuracy and on the final accuracy of various solution methods for nonlinear models. 

The usual practice in these appHcations is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be appHed only to problems in which there is a weU-defined, clear association between the independent and dependent variables. The generalization of statistics to the associated confidence intervals for nonlinear coefficients is not well developed. 

The methods concerned with differential equation parameter estimation are, of course, the ones of most concern in this book. Generally reactor models are non-linear in their parameters and therefore we are concerned mostly with nonlinear systems. 

As seen in Chapter 2 a suitable measure of the discrepancy between a model and a set of data is the objective function, S(k), and hence, the parameter values are obtained by minimizing this function. Therefore, the estimation of the parameters can be viewed as an optimization problem whereby any of the available general purpose optimization methods can be utilized. In particular, it was found that the Gauss-Newton method is the most efficient method for estimating parameters in nonlinear models (Bard. 1970). As we strongly believe that this is indeed the best method to use for nonlinear regression problems, the Gauss-Newton method is presented in detail in this chapter. It is assumed that the parameters are free to take any values. 

In principle, any type of process model can be used to predict future values of the controlled outputs. For example, one can use a physical model based on first principles (e.g., mass and energy balances), a linear model (e.g., transfer function, step response model, or state space-model), or a nonlinear model (e.g., neural nets). Because most industrial applications of MPC have relied on linear dynamic models, later on we derive the MPC equations for a single-input/single-output (SISO) model. The SISO model, however, can be easily generalized to the MIMO models that are used in industrial applications (Lee et al., 1994). One model that can be used in MPC is called the step response model, which relates a single controlled variable y with a single manipulated variable u (based on previous changes in u) as follows ... 

In general, the model of a plant operating under steady-state conditions is made up of a system of nonlinear algebraic equations of the form... 

The estimation of model parameters is an important activity in the design, evaluation, optimization, and control of a process. As discussed in previous chapters, process data do not satisfy process constraints exactly and they need to be rectified. The reconciled data are then used for estimating parameters in process models involving, in general, nonlinear differential and algebraic equations (Tjoa and Biegler, 1992). 

If the nonlinear estimation procedure is carefully applied, a minimum in the sums-of-squares surface can usually be achieved. However, because of the fitting flexibility generally obtainable with these nonlinear models, it is seldom advantageous to fit a large number of models to a set of data and to try to eliminate inadequate models on the basis of lack of fit (see Section IV). For example, thirty models were fitted to the alcohol dehydration data just discussed (K2). As is evident from the residual mean squares of Table II, approximately two-thirds of the models exhibit an acceptable fit of the data... 

The general approach used with nonlinear models, such as Eq. (40) is to linearize by a Taylor expansion [Eq. (41)] and apply the linear theory of Section III,C,1. 

A matrix of independent variables, defined for a linear model in Eq. (29) and for nonlinear models in Eq. (43) Transpose of matrix X Inverse of the matrix X Conversion in reactor A generalized independent variable... 

J. Paul, C. Christopoulos, and D. W. P. Thomas, "Generalized Material Models in TLM -Part 3 Materials With Nonlinear Properties," IEEE Trans. Antennas Propagat. 50, 997-1004 (2002). 

The convective terms also introduce wavelike characteristics into the flow equations. A model for these generally nonlinear, coupled terms is Berger s equation,... 

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