Optimization nonlinear models

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. 

The nature of the optimization problem can mm out to be linear or nonlinear depending on the mass transfer model chosen14. If a model based on a fixed outlet concentration is chosen, the model turns out to be a linear model (assuming linear cost models are adopted). If the outlet concentration is allowed to vary, as in Figure 26.35a and Figure 26.35b, then the optimization turns out to be a nonlinear optimization with all the problems of local optima associated with such problems. The optimization is in fact not so difficult in practice as regards the nonlinearity, because it is possible to provide a good initialization to the nonlinear model. If the outlet concentrations from each operation are initially assumed to go to their maximum outlet concentrations, then this can then be solved by a linear optimization. This usually... 

Zhang, X., 1995. Algorithms for optimal scheduling using nonlinear models. PhD thesis, University of London, London. 

Constraints (4.1), (4.2), (4.3), (4.4), (4.5) and (4.6) constitute a nonconvex nonlinear model due to constraints (4.3) and (4.4), which involve bilinear terms. Nonconvexity, and not necessarily nonlinearity, is a disadvantageous feature in any model, since global optimality cannot be guaranteed. Therefore, if can be avoided, it should. This is achieved by either linearizing the model or using convexification techniques where applicable. In this instance, the first option was proven possible as shown below. 

There is a need for methods that can determine global optima for arbitrary nonlinear functions, and that can handle extremely large nonlinear models for real-time optimization (on the order of millions of variables). 

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. 

Like ANNs, SVMs can be useful in cases where the x-y relationships are highly nonlinear and poorly nnderstood. There are several optimization parameters that need to be optimized, including the severity of the cost penalty , the threshold fit error, and the nature of the nonlinear kernel. However, if one takes care to optimize these parameters by cross-validation (Section 12.4.3) or similar methods, the susceptibility to overfitting is not as great as for ANNs. Furthermore, the deployment of SVMs is relatively simpler than for other nonlinear modeling alternatives (such as local regression, ANNs, nonlinear variants of PLS) because the model can be expressed completely in terms of a relatively low number of support vectors. More details regarding SVMs can be obtained from several references [70-74]. 

Electronic descriptors were calculated for the ab initio optimized (RHG/STO-3G) structures. In addition, logP as a measure of hydrophobicity and different topological indices were also calculated as additional descriptors. A nonlinear model was constructed using ANN with back propagation. Genetic algorithm (GA) was used as a feature selection method. The best ANN model was utilized to predict the log BB of 23 external molecules. The RMSE of the test set was only... 

The method proposed for improving the batch operation can be divided into two phases on-line modification of the reactor temperature trajectory and on-line tracking of the desired temperature trajectory. The first phase involves determining an optimal temperature set point profile by solving the on-line dynamic optimization problem and will be described in this section. The other phase involves designing a nonlinear model-based controller to track the obtained temperature set point and will be presented in the next section. 

It is obvious that the simultaneous inclusion of all possible reactor-reactive separation-separator design options into an automated design framework quickly leads to combinatorial explosion. In combination with the nonlinear models used to describe the reaction, mass, and heat transfer phenomena that occur in the processing units, the resulting synthesis problem is beyond the scope of existing optimization technology, even for relatively small problems. This has led to the... 

For models in which the dependent variables are linear functions of the parameters, the solution to the above-mentioned optimization problems can be obtained in closed form when the least squares objective functions (3.22) and (3.24) are considered. However, in chemical kinetics, linear problems are encountered only in very simple cases, so that optimization techniques for nonlinear models must be considered. 

The main methods for both linear and nonlinear optimization are presented in the following, with reference to the objective functions C/Ls and Hwls, since they allow for a more straightforward analysis. Hence, in the following, U = C/Ls or U = i/wi.s nevertheless, the analysis of nonlinear models, which is discussed in Sect. 3.5, can be extended with a little computational effort to the more rigorous maximum likelihood objective function (3.21). 

W. Marquardt. Nonlinear model reduction for optimization based control of transient chemical processes. AIChE Symposium Series 326, 98 12-42, 2001. 

It can be easily argued that the choice of the process model is crucial to determine the nature and the complexity of the optimization problem. Several models have been proposed in the literature, ranging from simple state-space linear models to complex nonlinear mappings. In the case where a linear model is adopted, the objective function to be minimized is quadratic in the input and output variables thus, the optimization problem (5.2), (5.4) admits analytical solutions. On the other hand, when nonlinear models are used, the optimization problem is not trivial, and thus, in general, only suboptimal solutions can be found moreover, the analysis of the closed-loop main properties (e.g., stability and robustness) becomes more challenging. 

Wave models were successfully used for the design of a supervisory control system for automatic start-up of the coupled column system shown in Fig. 5.15 [19] and for model-based measurement and online optimization of distillation columns using nonlinear model predictive control [15], The approach was also extended to reactive distillation processes by using transformed concentration variables [22], However, in reactive - as in nonreactive - distillation, the approach applies only to processes with constant pattern waves, which must be checked first. 

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

A second proposition relies on the idea that the on-line optimization problem is unconstrained after a certain time step in the finite moving horizon. Where in the finite horizon that happens is determined by examining whether the state has entered a certain invariant set (Mayne, 1997). Once that happens, then closed-form expressions can be used for the objective function from that time point the end of the optimization horizon, p. The idea is particularly useful for MFC with nonlinear models, for which the computational load of the on-line optimization is substantial. A related idea was presented by Rawlings and Muske (1993), where the on-line optimization problem has a finite control horizon length, m, and infinite prediction horizon length, p, but the objective function is truncated, because the result of the optimization is known after a certain time point. 

The model predictive control used includes all features of Quadratic Dynamic Matrix Control [19], furthermore it is able to take into account soft output constraints as a non linear optimization. The programs are written in C++ with Fortran libraries. The manipulated inputs (shown in cm Vs) calculated by predictive control are imposed to the full nonlinear model of the SMB. The control simulations were made to study the tracking of both purities and the influence of disturbances of feed flow rate or feed composition. Only partial results are shown. 

Marquard, W, Nonlinear Model Reduction for Optimization Based Control of Transient Chemical Processes, Proceedings of the 6 International Conference of Chemical Process Control, AlChe Symp. Ser. 326, Vol. 98 (12), 2002. 

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