Non-Linear Systems Parameter Estimation

The methods concerned with differential equation parameter estimation are, of course, the ones of most concern in this book. Generally reactor models are non-linear in their parameters and therefore we are concerned mostly with nonlinear systems. 

A model described by this differential equation is linear in the parameters ki. .. k , if 

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in non-linear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. A simple but flexible and useful error model is used in SIMUSOLV (Steiner et al., 1986 Burt, 1989). 

If basic assumptions concerning the error structure are incorrect (e.g., non-Gaussian distribution) or cannot be specified, more robust estimation techniques may be necessary. In addition to the above considerations, it is often important to introduce constraints on the estimated parameters (e.g., the parameters can only be positive). Such constraints are included in the simulation and parameter estimation package SIMUSOLV. Beeause of numerical inaccuracy, scaling of parameters and data may be necessary if the numerical values are of greatly differing order. Plots of the residuals, difference between model and measurement value, are very useful in identifying systematic or model errors. 

Non-linear parameter estimation is far from a trivial task, even though it is greatly simplified by the availability of user-friendly program packages such as a) SIMUSOLV (Steiner et al., 1986), b) ESL, c) a set of BASIC programs (supplied with the book of Nash and Walker-Smith, 1987) or d) by mathematical software (MATLAB). ISIM itself does not supply these advanced features, but ISIM programs can easily be translated into other more powerful languages. 

---

Jang, S. S., Josepth, B and Mukai, H. (1986). Comparison of two approaches to on-line parameter and state estimation problem of non-linear systems. Ind. Eng. Chem. Process Des. Dev. 25, 809-814. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory. Academic Press, New York. Liebman, M. J., Edgar, T. F., and Lasdon, L. S. (1992). Efficient data reconciliation and estimation for dynamic process using non-linear programming techniques. Comput. Chem. Eng. 16, 963-986. McBrayer, K. F., and Edgar, T. F. (1995). Bias detection and estimation on dynamic data reconciliation. J Proc. Control 15, 285-289. 

Billings, S.A. and Voon, W.S.F. 1984. Least-squares parameter estimation algorithms for non-linear systems. Int. J. Syst. Sci. 15 601. 

III. Parameter Estimation for Linear Models of Non-Linear Systems. 

As seen previously for some specific applications such as wastewater treatment plants, software sensors can be envisaged to provide on-line estimation of non-measurable variables, model parameters or to overcome measurement delays [81-83]. Software sensors have been developed mainly for monitoring bioprocesses because the control system design of bioreactors is not straightforward due to [84] significant model uncertainty, lack of reliable on-line sensors, the non-linear and time-varying nature of the system or slow response of the process. 

In order to develop a suitable kinetic model of the full NH3-N0-N02/02 SCR reacting system, first the active reactions depending on N0/N02 feed ratio and temperature were identified then a dedicated study was performed aimed at clarifying the catalytic mechanism of the fast SCR reaction on the basis of such a reaction chemistry a detailed kinetic model was eventually derived, whose intrinsic rate parameters were estimated from global non-linear regression of a large set of experimental transient runs. 

There is an increasing interest in technologies that maximize the production of various essential enzymes and therapeutic proteins based on E. coli cultivation. The costs of developing mathematical models for bioprocesses improvements are often too high and the benefits are too low. The main reason for this is related to the intrinsic complexity and non-linearity of biological systems. The important part of model building is the choice of a certain optimization procedure for parameter estimation. The estimation of model parameters with high parameter accuracy is essential for successful model development. 

Parameter estimation problem of the presented non-linear dynamic system is stated as the minimization of the distance measure J between the experimental and the model predicted values of the considered state variables ... 

The term a gives the slope of the left-hand ascending side of the curve and (a - b) that of the right-hand descending side. The non-linear parameter jS, which must be estimated by a stepwise iteration procedure, relates to the volume ratio of the aqueous and lipid phases in the system. Setting jS = 1 and b 2a produces the original McFarland model. Kubiny s bilinear model can be derived from kinetically controlled model systems as well as from equilibrium models, indicating that it is valid under diffusion control as well as under equilibrium or pseudo-equilibrium conditions. For many data sets, the bilinear function aptly fits the experimental observations. Difficulties in calculations may arise from unbalanced data sets, which often occur in environ-... 

Since the variation of thermophysical parameters with respect to temperature and to solvent composition for real interacting systems is usually non-linear, estimated values of V, p and q for liquid mixtures corresponding to the experimental data gaps are subjected to substantial interpolation errors, especially when the experimentally determined data for a given solvent system are sparse. 

Compartmental modeling involves the specification of a structural mathematical model (commonly using either explicit or ordinary differential equations) and system parameters are estimated from fitting the model to pharmacokinetic data via non linear regression analysis or population mixed effects modeling. One popular structural model is the open two-compartment model shown in Figure 6.10. 

For updating with the smaller-amplitude data set, Pec, estimates of inter-story stiffnesses are fairly well-constrained, and close to the actual values. However, since there is relatively little non-linear behavior, there are larger uncertainties associated with the strength and elastic-to-plastic transition parameters, and especially the breaking ductility ratio in model class Mi, since there is very little deterioration in the actual system. Figure 4 shows the posterior samples obtained by updating model class Mi... 

---