Non-linear Process Modeling

Many papers about fuzzy modeling use optimization of model parameters. Often a least squares method is used. Optimization of eonsequence parameters may give a perfect global fit, but can also lead to bad loeal representation of the system. In this ease a minor change in the premise parameters may give a major ehange in the consequenee parameters. 

Optimization of premise parameters can lead to crisp boundaries of rules instead of fuzzy boundaries. The fuzzy clustering algorithm is already an optimization algorithm, for these reasons it is suggested not to perform an additional optimization step. 

In this section three examples fuzzy models will be discussed. 

The first model that is used for illustration of the proposed identilleation teehiuque is a nonlinear namic process described by Zhao et al. (1994)  

Let the input signal u for identification be a uniformly distributed noise in the interval [-2, 2], the number of sample dataiV= 1001, and assume the number of fuzzy rules is not known 

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This problem is extensively studied for refinery operations planning leading to non-linear process models, see Zhang et al. (2001), Li et al. (2005), or Alhajri et al. (2008). 

In this chapter, discrete linear-state space models will be discussed and their similarity to ARX models will be shown. In addition Wiener models are introduced. They are suitable for non-linear process modeling and consist of a linear time variant model and a non-linear static model. Several examples show how to develop both types of models. 

Khandalekar, P.D. and Riggs, J.B. (1995) Non-linear process model-based control and optimization of a model IV FCC unit. Computers and Chemical Engineering, 19 (11), 1153-68. 

This approach is of course limited by the availability of reliable data and the resolution of the data. An inherent problem in the np-scaling process is the interaction between variance in input parameters and non-linearity in models. This may prodnce chaotic behaviour, van Bodegom et al. (2002) discuss this in relation to CH4 emission from rice. The point at which inpnt data are averaged before making model runs may also be limited by the available computing... 

Viscoelasticity has already been introduced in Chapter 1, based on linear viscoelasticity. However, in polymer processing large deformations are imposed on the material, requiring the use of non-linear viscoelastic models. There are two types of general non-linear viscoelastic flow models the differential type and the integral type. 

A non-linear mathematical model, which is a set of ordinary differential equations, for the process in the SPBER was developed.19 The model accounts for the heterogeneous electrochemical reaction and homogeneous reaction in the bulk solution. The lateral distributions of potential, current density and concentration in the reactor were studied. The potential distribution in the lateral dimension, x, of the packed bed was described by a one dimensional Poisson equation as ... 

Sorption of organic contaminants onto aquifer solids is frequently described as a partitioning process, where the hydrophobic organic compound partitions into natural organic material associated with the aquifer solids [8]. Sorption can be characterized as either an equilibrium or rate-limited phenomenon. Equilibrium sorption can be modeled as either a linear or non-linear process. Equilibrium sorption may be assumed when the flow of groundwater and other processes affecting contaminant transport are slow compared to the rate of sorption. In this event the sorption of the contaminant can be considered instantaneous. If we assume equilibrium sorption, the relationship between sorbed and aqueous contaminant concentrations may be described by a sorption isotherm. 

The paper is organized as follows. The problem formulation - parameter identification of the non-linear dynamic model of an E. coli cultivation process - is given in Sect. 2. In Sect. 3 the hybrid schemes between GA and FA are presented. The numerical results and discussion are presented in Sect. 4. Conclusion remarks are done in Sect. 5. 

Linear filter Throughout we have assiuned that the system operates as a time invariant linear (LTI) filter of the type described in Chapter 10. While it is well known that there are many non-linear processes present in vocal tract soimd propagation, in general the linear model provides a very good approximation to these. 

Thus, we have derived the non-linear mathematic model of the electrochemical process with low-conductive surface film that can decompose with an exponential temperature-dependent rate. Essentially, this is an example of a thermokinetic model. 

As mentioned above, one purpose of this paper is to show how rigorous models can be used offline to quantify key non-linear process relationships, including those related to manufacturing clean fuels. 

Rantanen J, Rasanen E, Antikainen O, Mannermaa J-P, Yliruusi J. In-line moisture measurement during granulation with a four-wavelength near-infrared sensor an evaluation of process-related variables and a development of non-linear calibration model. Chemom Intell Lab Syst 2001 56 51-58. 

Sometimes, linear techniques can be used to describe non-linear process behavior. An example is a fuzzy model, discussed in chapter 28, which is a combination of local linear models in distinct operating areas. Developing a non-linear model requires much insight and understanding of the developer as to what mechanism underlies the observed data. Application of empirical techniques for modeling non-linear process behavior has therefore become very popular, such as the application of neural networks, described in chapter 27. 

The non-linear and linear process models are simulated in Matlab Simulink and stored in file F0606.mdl. The design of the simulation will be explained in chapter 8. 

One could choose as file name, for example out.mat and as variable name out. The level would then be stored as follows row one in outmat would contain values of the time, row two would contain the level from the non-hnear process simulation and row three the values of the level from the linearized process model. If one would like to plot the results, one could type ... 

As already discussed in the previous chapters, process behavior is usually non-Unear. Whether or not the empirical model to be developed should also be non-linear depends on the operating range in which the model will be used. If the process is controlled and the operating range is small, a linear process model may be an adequate approximation of reality. The application of the model will determine whether the model needs to be dynamic or static. For control and prediction type applications, models are usually dynamic. 

If the process conditions vary over a wide range, there may be a need for a non-linear empirical model. In case of a dynamic non-linear model there are a few possibilities for developing such a model, for example a dynamic neural network or a dynamic fuzzy model. One could also develop a Wiener model, in which the process dynamics are represented by a hnear model, such as a state space model. The static characteristics of the process are then modeled by a polynomial, able to represent the non-linearity. 

This section describes two examples of PLS modeling. The first one describes the model for the reactor already introduced in the previous chapter the second example describes the model for a non-linear process. 

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