On identification of nonlinear systems

The development of accurate models is very important for analyzing problems concerning simulation, prediction, control, etc. Therefore it is not astonishing that many studies in applied science are about the modeling of these processes. In this thesis we will focus on the building of models that are used to describe some nonlinear processes in hydrology and meteorology; the first process is the movement of water in porous media and the second process is the large-scale atmospheric circulations.

The process of model development can be divided in three essential subprocesses: selection of a model structure, determination of a "best fit" criterion and experimental design. In literature, their are several examples of "case-studies" known, where the specific combination of model structure, criterion and experimental design did not lead to unique estimates of the unknown parameters of the model. This situation is designated by the term: "the model is not identifiable".

A model may not be identifiable (given a certain choice of the experimental design) because the chosen object function is insensitive to some linear combinations of the parameters. In this case the identification problem will not have a unique solution. On the other hand, due to noise in the system, the optimization problem may have many local optima. One can then easily be misled because an optimization algorithm may converge to such a local optimum. It will be studied how such a situation can be recognized. Furthermore, it will be studied how the identifiability can be improved by an appropriate choise of the experimental design.

There are also other situations where the chosen combination of model structure, "best fit" criterion and experimental design will not lead to a unique solution. Such a case occurs when we are dealing with chaotic systems. For chaotic systems the optimization problem, using the output-error criterion as "best- fit" criterion, is ill-posed, because the model's solution depends sensitively on its initial state. The observed values and the model values will then diverge due to the limited accuracy of the initial state. Several criteria are analyzed on their capability for detecting small perturbations in the system and for estimating unknown parameters in the system.

In chapter 2 of this thesis the ONE-STEP method is described. This method is developed to identify the parameters in a model for the movement of water in the unsaturated soils. The motivation to analyze the identifiability of this model comes from the statement made by several authors that not all model parameters can be estimated uniquely. In this chapter we will analyze first some numerical schemes to solve the mathematical model, because the efficiency and the accuracy- of a numerical scheme are very important for applicability of the ONE- STEP method.

In chapter 3 the concept of "structural identifiability" is further developted. The term "numerical identifiable" is introduced, so that we can take into account the accuracy of the sensitivity matrix. The identifiability analysis of the ONE- STEP method shows that not all parameters can be estimated uniquely. In the best case, where the pressure in the pressure cell is increased during the experiment at certain time instants, only 5 of the 6 model parameters can be estimated uniquely. Analyzing the structure of the model, we can derive that the object function depends on 5 independent parameters only, which explains the identifiability problem. Only by adding some other measurements, for example the pressure head at a certain position in the soil core, one may, expect better results of this method.

As already mentioned above, the output-error criterion in combination with chaotic systems, leads to ill-posed problems. In chapter 4 it is analyzed whether a criterion, based on a modified sentinel function, can be used to detect an external perturbation in a chaotic system. We found that fast varying perturbations are often "stealthy" for this function. Therefore this criterion can only be used to detect slowly varying perturbations.

The sentinel function can also be used to estimate uncertain parameters that are used to describe such a small perturbation term. We have compared the performance of the sentinel approach with an adaptive extended Kalman filter in a test-case. In the example that is presented, the size of a perturbation in the equator-pole temperature gradient is estimated. The equator-pole temperature gradient characterizes the driving force in a low-order spectral model of the atmospheric circulation and therefore a change in the equator-pole temperature gradient may be important in studing the greenhouse effect. In this test-case the performance of the adaptive extended Kalman filter was better then the performance of the sentinel approach. The less accurate results of the sentinel method are caused by the relative slow sampling frequency. The effect of neglecting higher order terms in the Taylor expansion and the influence of observation errors is then felt.

A disadvantage of extended Kalman filtering is that the filter easily diverges. In chapter 5 this problem is studied for chaotic systems. A reasonable approach to solve the divergence problem is to add an artificial noise term to the state equations. This noise term is used to control the accuracy of the state estimates and so preventing that the filter learns the wrong state too well. With the extended Kalman filter one can easily obtain an approximation of the value of the loglikelihood function. For this problem we have developed an optimization procedure that can be used together with the extended Kalman filter to estimate the unknown parameters in the model description as well as the parameters that are used to describe the covariance matrix of the artificial noise term. This method is successfully applied to determine the optimal extended Kalman filter for a T11-spectral model of the atmospheric circulation.