Nonlinear Process Identification

Although in many circumstances linear system identification can provide a sufficiently good model of the system for the intended purpose, it is occasionally necessary to consider nonlinear system identification. 

Nonlinear system identification attempts to fit a nonlinear model to the given data. However, since there is a large number of potential nonlinear models that could be fit, nonlinear identification simplifies the available functions. Instead of choosing any arbitrary function, a basis function, k(x), is selected. The basis function can also be called the generating function or the mother function. Then, the goal becomes to fit the following model to the data 

6 Modelling Dynamic Processes Using System Identification Methods 

Wavelet Function k x) = This is only one example of many different 

It can be shown that for a sufficiently large value of n, for almost any choice of the basis function, except a polynomial basis, any reasonable nonlinear function can be approximated arbitrarily well. 

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Y Cheng, W Karjala, and DM Himmelblan. Resolving problems in closed loop nonlinear process identification nsing IRN. Comput. Chem. Engg., 20(10) 1159-1176, 1996. 

RK Pearson and BA Ogunnaike. Nonlinear Process Control, chapter Nonlinear Process Identification. Prentice-Hall PTR, Upper Saddle River, NJ, 1997. 

Su, H. T., and T. J. McAvoy, Artificial Neural Networks for Nonlinear Process Identification and Control, Chapter 7 in Nonlinear... 

For continuous process systems, empirical models are used most often for control system development and implementation. Model predictive control strategies often make use of linear input-output models, developed through empirical identification steps conducted on the actual plant. Linear input-output models are obtained from a fit to input-output data from this plant. For batch processes such as autoclave curing, however, the time-dependent nature of these processes—and the extreme state variations that occur during them—prevent use of these models. Hence, one must use a nonlinear process model, obtained through a nonlinear regression technique for fitting data from many batch runs. 

Fesch, C., W. Simon, S. B. Haderlein, P. Reichert, and R. P. Schwarzenbach, Nonlinear sorption and nonequilibrium solute transport in aggregated porous media Experiments, process identification and modeling , J. Contam. Hydrol., 31, 373-407 (1998). 

J.B. Balchen, B. Lie, and I. Solberg. Internal decoupling in nonlinear process control. Modeling Identification and Control, 9 137-148, 1988. 

M Pottmann and DE Seborg. Identification of nonlinear processes using reciprocal multiquadric functions. J. Process Control, 2 189-203, 1992. 

Most work on the development of dynamic process models has been empirical this work is usually referred to as process identification. As mentioned earlier, two classes of empirical identification techniques are available one uses deterministic (step, pulse, etc.) functions, the other stochastic (random) identification functions. With either technique, the process is perturbed and the resulting variations of the response are measured. The relationship between the perturbing variable and the response is expressed as a transfer function. This function is the process model. Empirical identification of process models by the deterministic method has been reported by various workers [55-58]. A drawback of this method is the difficulty in obtaining a measurable response while restricting the process to a linear response (small perturbation). If the perturbation is large, the process response will be nonlinear and the representations of the process with a linear process model will be inaccurate. 

Instead of fitting a fully nonlinear model, another approach to nonlinear system identification is to partition the nonUnearities from the linear component A common application of this approach is the Wiener-Hammerstein model. A Wiener-Hammerstein model is a generalisation of the Hammerstein model, where non-linearities are assumed cmly to be in the input and the Wiener model, where nonlinearities are assumed only to be in the output, which allows nonlinearities to be present in both the input and output The process model is assumed to be linear. Thus, the general form of the model can be written as... 

The measure of nonlinearity inherently relies only on input/output data and is, therefore, readily computable. While it may be difficult to obtain perfectly sinusoidal input/output data in order to compute the lower bound (5), the more general optimization based computation can be performed given any set of input/output data. It should be expected that, similar to process identification, if the inputs in question do not significantly excite the process nonlinearity, a artificially low value of Eq. (1) may be obtained. Other measures of nonlinearity computable from input/output data records exist as well (e.g., see Haber [10]). 

J. Wang, T. Chen, L. Wang, A blind approach to identification of Hammerstein-Wiener systems corrupted by nonlinear-process noise, Proceedings of the 7th Asian Control Conference, Hong Kong, China, August 27-29,2009. 

Nonlinear System Identification Particle-Based Methods, Fig. 1 The unscented Kalman filter process for a two-dimensional state... 

P. H. Menold, F. AUgower, and P. K. Peaison, Nonlinear structure identification of chemical processes, Comput. Chem. Eng. 21, S137-S142 (1997). 

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

In this work, the AD model proposed and validated by Bernard et ah, 2001 is used in the development of the robust nonlinear scheme. The use of this model is justified by two facts i) this model has demonstrated to be useful in the monitoring and control of AD processes and, ii) the semi-industrial fixed-bed anaerobic digester located in the LBEhlNRA used in the validation and identification of the model will be also used in the experimental implementation of the robust nonlinear approach here proposed. Therefore, in what follows the model is briefly described. 

System Identification Techniques. In system identification, the (nonlinear) resi pnses of the outputs of a system to the input signals are approximated by a linear model. The parameters in this linear model are determined by minimizing a criterion function that is based on some difference between the input-output data and the responses predictedv by the model. Several model structures can be chosen and depending on this structure, different criteria can be used (l ,IX) System identification is mainly used as a technique to determine models from measured input-output data of processes, but can also be used to determine compact models for complex physical models The input-output data is then obtained from simulations of the physical model. 

Then, the healthy signal is used to feed a bank of /Vp + 1 nonlinear adaptive observers (where /Vp is the number of the possible process/actuator faults). The first observer is in charge of detecting the occurrence of process/actuator faults. The other /Vp observers, each corresponding to a particular type of process/actuator fault, achieve fault isolation and identification by adopting a suitable adaption mechanism. Figure 6.3 shows a block diagram representation of the overall architecture. 

Once a process/actuator fault has been detected, isolation and identification can be achieved via N-p nonlinear adaptive observers. Each observer is designed in such a way to be insensitive to a particular type of fault. In fact, the ith observer (hereafter i =, ..., Np) has the form... 

In all the cases considered, auto catalytic processes must be present, whether linear or nonlinear. To understand the actual mechanism of autocatalysis for the Soai reaction, identification of the process at a molecular level is necessary, but is out of scope of the present review. 

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