Nonlinear model reduction

Owing to the presence of the small parameter , the model in Equation (7.1) is stiff and can potentially exhibit a dynamic behavior with multiple time scales. Proceeding in a manner similar to the one adopted in Chapter 6, we define the fast time scale r = t/e and rewrite (7.1) as 

We continue to refer to the analysis in Chapter 6 and consider the limit e — 0, making the important distinction that in this case the limit corresponds to an 

The observations above indicate that, upon defining uf as a function of the state variables 0 (e.g., via feedback control laws), the Jacobian matrix 

On substituting the solution 0 into the model, we obtain a description of the dynamics of the process after the fast temperature boundary layer  

Remark 7.1. The energy flow rates Hi. .. 77/v-i C cu1 are typically functions of the flow rates of (internal) material streams and cannot be set independently. In order to preserve the simplicity of the presentation, this dependence has not been explicitly accounted for the results presented thus far are, however, independent of this consideration. It is also intuitive that the following statements apply. 

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

Hahn, J., Edgar, T. F., An improved method for nonlinear model reduction using balancing of empirical gramians, Computers Chem. Eng. 2002, 26 1379— 1397. 

Finally, nonlinear wave can also be used for nonlinear model reduction for applications in advanced, nonlinear model-based control. Successful applications were reported for nonreactive distillation processes with moderately nonideal mixtures [21]. For this class of mixtures the column dynamics is entirely governed by constant pattern waves, as explained above. The approach is based on a wave function which can be used for the approximation of the concentration profiles inside the column. The wave function can be derived from analytical solutions of the corresponding wave equations for some simple limiting cases. It is given by... 

A. Kienle, Nonlinear model reduction for nonreactive and reactive distillation processes using nonlinear wave propagation theory. 

Time-scale decomposition and nonlinear model reduction... 

Kumar, A. and Daoutidis, P. (2003). Nonlinear model reduction and control for high-purity distillation columns. Ind. Eng. Chem. Res., 42, 4495-4505. 

Vora, N.P. (2000). Nonlinear Model Reduction and Control of Multiple Time Scale Chemical Processes Chemical Reaction Systems and Reactive Distillation Columns. PhD thesis, University of Minnesota - Twin Cities. 

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. 

Vora and Daoutidis (2001) developed a nonlinear model reduction method for non-isothermal reaction systems that exhibit dynamics on two different timescales. The method identifies the independent algebraic constraints (possibly of QSSA origin) that define the low-dimensional state space where the slow dynamics of the reaction system are constrained to evolve. 

Chu, Y., Serpas, M., Hahn, J. State-preserving nonlinear model reduction procedme. Chem. Eng. Sd. 66, 3907-3913 (2011)... 

Vora, N., Daoutidis, P. Nonlinear model reduction of chemical reaction systems. AIChE J. 47, 2320-2332 (2001)... 

Simulations show negligible differences in the transient temperature and concentration profiles as a result of this quasi-steady-state approximation. The major advantage of this assumption should be apparent in control system design, where a reduction in the size of the state vector is computationally beneficial or in the time-consuming simulations of the full nonlinear model. 

Many methods have been developed for model analysis for instance, bifurcation and stability analysis [88, 89], parameter sensitivity analysis [90], metabolic control analysis [16, 17, 91] and biochemical systems analysis [18]. One highly important method for model analysis and especially for large models, such as many silicon cell models, is model reduction. Model reduction has a long history in the analysis of biochemical reaction networks and in the analysis of nonlinear dynamics (slow and fast manifolds) [92-104]. In all cases, the aim of model reduction is to derive a simplified model from a larger ancestral model that satisfies a number of criteria. In the following sections we describe a relatively new form of model reduction for biochemical reaction networks, such as metabolic, signaling, or genetic networks. 

Unit Model reduction technique Full nonlinear model Reduced model... 

The variables (wavelengths) associated with the IR emission spectra were highly correlated. Principal components analysis (PCA), linear and nonlinear PLS showed that at least 86% of the total variance could be explained by the two primary latent dimensions. The forward and reverse modelling results showed that dimensional reduction with a linear model (PLS) produced better models than a nonlinear model (multilayer perceptron neural network trained with the back propagation algorithm) without dimensional reduction. 

Reutlinger M, Schneider G. Nonlinear dimensionality reduction and mapping of compound libraries for drug discovery. J Mol Graphics Model 2012 34 108—117. 

This section provides an example to illustrate the mechanics of the ECI-based model reduction algorithm and emphasize its advantages, namely its applicability to nonlinear systems, ability to achieve graph-level reduction, and ability to reduce the order and structure of the model, while taking into account the scenario of interest and preserving the realization of the model. 

Model reduction of nonlinear state-space models Non-minimal input-affine state-space models can be transformed to a minimal realization form by applying a suitable nonlinear state transformation followed by state elimination. Such a model reduction is based on finding quantities which are constant along any trajectory in the state-space. These can be termed hidden conserved quantities (Szederkdnyi et al. 2002). 

The first step of the model reduction is to carry out nonlinear controllability and/or observability analysis to check if the model to be reduced is jointly controllable and observable (and thus minimal). If this is not the case, then the hidden conserved quantities can be determined from the result of this analysis by solving sets of nonlinear partial differential equations (see Isidori, 1995). 

Model reduction The fermentor model in equations (E1)-(E4) has been analyzed by nonlinear techniques and found not to be reachable (Szederkenyi et al. 2002). The constant (hidden conserved) quantity generating the model reduction was found to be ... 

To overcome this limitation, a sequence of model reduction steps is commonly employed. Perhaps the best-known of these is Step 2 in Fig. 2, the linearization of the nonlinear fundamental model Mpto obtain a linear approximation Ml- Note that the process characterization cube may be applied to both the process V and all of the approximating models considered here. Since the fundamental model is intended as a detailed description of the process V, we can... 

The use of complex models is hindered by two obstacles. First, the models contain large numbers of unknown kinetic parameters regression to determine the parameters of complex nonlinear models is both difficult and unreliable. Secondly, because of their sheer size and the presence of multiple time-scales, these models are difficult to solve. For these reasons, model simplification and order reduction are central problems for complex reaction systems. Ideally, a model order reduction algorithm would have broad applicability, permit analysis at several levels of detail, and provide an assessment of the modeling error. 

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