Model reduction

In this chapter several model reduction techniques will be discussed. The first method is based on firequency response matching, other methods make use oficonversion ofi the model structure to a state space model and subsequently truncating the states that have a minimum impact on the input-output relationship. The main indicator used fior this purpose is the so-called Hankel singular value. In addition, the model structure is converted to a balanced realization, afiter which the reduction techniques can be applied. Several examples are given on how to apply the dififierent methods. 

Example 1. Let us consider an example that exemplifies step 3 in the core-box modeling framework. The system to be studied consists of one substance, A, with concentration x = [A]. There are two types of interaction that affect the concentration negatively degradation and diffusion. Both processes are assumed to be irreversible and to follow simple mass action kinetics with rate constants p and P2, respectively. Further, there is a synthesis of A, which increases its concentration. This synthesis is assumed to be independent of x, and its rate is described by the constant parameter p3. Finally, it is possible to measure x, and the measurement noise is denoted d. The system is thus given in state space form by the following equations  

We assume that in-vitro experiments have resulted in estimates p = 3 2 and P3 = 0.02 0.01, but that there is no estimate for P2- We further assume that there is one experimental time-series, the one shown in Fig. 5.3. The time-series has been generated by simulations using the parameters 

This example illustrates the problem of both structural and practical identifiabil-ity. The structural identifiability is present because p and P2 may never be distinguished from each other with only the present type of measurement possibilities. This is because the first two reactions may be written as — (p + P2)x, where one 

Example 3.1 highlights the fact that the presence of (and need for) material recycle streams with significant flow rates is entirely a steady-state design feature of a process. In what follows, we will focus on investigating the profound impact of this feature on the dynamics and control of the processes under consideration. 

To this end, let us rewrite the model in Equation (3.4) in a more general form  

According to the developments in Section 2.3, the model of Equation (3.10) is in a nonstandard singularly perturbed form. We thus expect its dynamics (and, consequently, the dynamics of integrated process systems with large material recycle) to feature two distinct time scales. However, the analysis of the system dynamics is complicated by the presence of the term u1, which, as we will see below, precludes the direct application of the methods presented in Chapter 2 for deriving representations of the slow and fast components of the system dynamics. 

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Model Reduction Through Parameter Estimation in the s-Domain... 

Quite often we are face with the task of reducing the order of a transfer function without losing essential dynamic behavior of the system. Many methods have been proposed for model reduction, however quite often with unsatisfactory results. A reliable method has been suggested by Luus (1980) where the deviations between the reduced model and the original one in the Nyquist plot are minimized. 

It is important to understand and be able to identify dominant poles if they exist. This is a skill that is used later in what we call model reduction. This is a point that we first observed in Example 2.6. Consider the two terms such that 0 < a < a2 (Fig. 2.4),... 

Edwards, K. V. Manousiouthakis and T. F. Edgar. Kinetic Model Reduction Using Genetic Algorithms. Comput Chem Eng 22 239-246 (1998). 

Which approach to model reduction is the most important Population is not the ultimate judge, and popularity is not a scientific criterion, but "Vox populi, vox Dei", especially in the epoch of citation indexes, impact factors and bibliometrics. Let us ask Google. It gave on 31st December 2006 ... 

This basic approach is really divided into several distinct categories. Two of these, Davison s method and Marshall s method, provide suitable modal reduction for the state-space representation of the methanation reactor to a 12th-order model. Comparisons of the models and discussion of additional model reduction are presented in the next section. 

Figure 28 shows comparisons of the transient gas and solid axial temperature profiles for a step-input change with the full model and the reduced models. The figure shows negligible differences between the profiles at times as short as 10 sec. Concentration results (not shown) show even smaller discrepancies between the profiles. Additional simulations are not shown since all showed minimal differences between the solutions using the different linear models. Thus for the methanation system, Marshall s model reduction provides an accurate 2Nth-order reduced state-space representation of the original 5/Vth-order linear model. 

Practical identifiability is not the only problem that can be adressed by principal component analysis of the sensitivity matrix. In (refs. 29-30) several examples of model reduction based on this technique are discussed. 

L. Petzold and W. Zhu. Model Reduction for Chemical Kinetics An Optimization Approach. AIChEJ., 45 869-886,1999. 

Reprint H is a late paper written as a festschrift paper in honor of Davidson s retirement. It combines the continuous mixture techniques with the model reduction method given in Chapter 2, the section entitled Scaling and Partial Solution in Linear Systems. A closely similar paper was requested... 

Beliczynski et al., 1992] Beliczynski, B., Kale, I., and Cain, G. D. (1992). Approximation of FIR by HR digital filters An algorithm based on balanced model reduction. IEEE Trans. Acoustics, Speech, Signal Processing, 40(3) 532-542. 

Vol. 536 R. Brtiggemann, Model Reduction Methods for Vector Autoregressive Processes. X, 218 pages. 2004. 

S. Skogestad, Simple analytic rules for model reduction and PID controller tuning, J. Process Control 13 (2003) 291-309. 

Gorban, A. N., Kazantzis, N., Kevrekidis, I. G., Ottinger, H. C. and Theodoropoulos, K., "Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena". Springer, New York (2006). 

The model reduction procedure must be adapted to the use of the simplified models and to the availability of experimental data needed to evaluate the unknown parameters, as discussed in Chap. 3. In general, more complex models are used for the design of the reactor and for the simulation of the entire process, whereas more simplified models are best fit for feedback control. In the following chapters it is shown that fairly accurate results are obtained when a strongly simplified kinetic model is used for control and fault diagnosis purposes. 

K. Edwards, T.F. Edgar, and V.I. Manousiouthakis. Kinetic model reduction using genetic algorithms. Computers and Chemical Engineering, 22 239-246, 1998. 

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

P.F. Tupper. Adaptive model reduction for chemical kinetics. BIT Numerical Mathematics, 42 447-465, 2002. 

As with atherosclerosis itself, recruitment of inflammatory cells is now recognized as an essential step in the pathogenesis of neointima formation in humans (I 1,12). In various animal models, reduction of leukocyte recruitment by selective blockade of adhesion molecules significantly reduced neointima formation and restenosis (13-16). Recent studies also concluded a role of pre-existing inflammation within the... 

Rorije, E., Langenberg, J.H., Richter, J., and Peijnenburg, W.J.G.M., Modeling reductive dehalogenation with quantum-chemically derived descriptors, SAP QSAR Environ, Res., 4, 237-252, 1995. 

Model reduction involves the identification and elimination of such parts of a model that are unrelated to some specific features of a model. The nature of such features might vary from situation to situation. In the core-box modeling framework, the feature in focus is identifiability (and agreement) with respect to the available data. We first review the state-of-the-art methods for identifiability, and then those for model reduction. 

Example 3. In Example 1, the core model was obtained using model reduction, and the mapping is thus already given... 

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

Biosimulation has a dominant role to play in systems biology. In this chapter, we briefly outline two approaches to systems biology and the role that mathematical models has to play in them. Our focus is on kinetic models, and silicon cell models in particular. Silicon cell models are kinetic models that are firmly based on experiment. They allow for a test of our knowledge and identify gaps and the discovery of unanticipated behavior of molecular mechanisms. These models are very complicated to analyze because of the high level of molecular-mechanistic detail included in them. To facilitate their analysis and understanding of their behavior, model reduction is an important tool for the analysis of silicon cell models. We present balanced truncation as one method to perform model reduction and apply it to a silicon cell model of glycolysis in Saccharomyces cerevisiae. 

Model reduction aims at simplifying without losing the essence of the dynamic behavior of a model. Reduction of silicon cells should thereby facilitate the understanding of real cells. Strategies for model reduction, pinpointing molecular organizational properties that are essential for network behavior, are essential to make silicon cell models understandable. 

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