Michael Hanss

The transformation method for the simulation and analysis of systems with uncertain parameters

In this paper, the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters. Based on the concept of α-cuts, the method represents a special implementation of fuzzy arithmetic that avoids the well-known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation. Systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results. As a by-product of the implementation scheme, the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output. By this, a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic. Finally, to show the efficiency of the transformation method, the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection.

A new uncertainty analysis for the transformation method

In this paper, a new uncertainty analysis for the transformation method (TM) is proposed. As a practical implementation of fuzzy arithmetic, the TM is a convenient tool for the simulation and analysis of systems with uncertain parameters that are expressed by fuzzy numbers. The proposed uncertainty analysis and the sensitivity analysis of the TM complete each other in providing some quantification of the relationship between the uncertainties of the system input and the system output. The computation of gain factors is proposed, which allows the estimation of the absolute and relative measures of uncertainty. These measures allow the quantification of the influence of the uncertainty of the input on the uncertainty of the output.

On the reliability of the influence measure in the transformation method of fuzzy arithmetic

The transformation method has been proposed as a practical tool for the simulation and the analysis of systems with uncertain parameters using fuzzy arithmetic. A major advantage of this method, among others, is the fact that in the analysis part of the method the degree of influence for each uncertain parameter can be computed with a rather simple formula. However, a thorough examination of how accurate this measure of the degree of influence really is has not yet been performed. In the paper, the influence measure provided by the transformation method is compared to a rather classical, analytical approach based on differential calculus. Furthermore, an additional measure is introduced which can serve as an indicator for the effectiveness and the reliability of the influence measure calculated by the transformation method.

On the implementation of fuzzy arithmetical operations for engineering problems

Fuzzy arithmetic is a successful tool to solve engineering problems with uncertain parameters. The generalized mathematical operations for fuzzy numbers can theoretically be defined making use of Zadehs extension principle. Practical real world applications of fuzzy arithmetic, however require an appropriate form of implementation for the fuzzy numbers and the fuzzy arithmetical operations. For this reason, the often applied concept of triangular fuzzy numbers and the more promising approach of discretized fuzzy numbers are presented and rated with respect to their practical application to solve engineering problems with uncertain parameters. As an example, a rather simple but typical problem of mechanical engineering is considered, consisting of determining the displacements in a two-component massless rod under tensile load with uncertain elasticity parameters.

Identification of a bolted-joint model with fuzzy parameters loaded normal to the contact interface

Abstract Modeling and identification of a joint with load normal to the contact interface of two connected rods is discussed in this paper. An experimental setup for the analysis of the joint is proposed and measurement results are presented. The perception that both the damping behavior and the stiffness of the joint are influenced by a large number of effects that can hardly be modeled motivates the use of a rather simple model, but with fuzzy-valued model parameters, instead of crisp ones. In this concept, the uncertainty and variability of the model parameters can be taken into account by representing the parameters as fuzzy numbers that can be identified on the basis of the measured data. The identification of the fuzzy parameters proves to be a non-trivial problem which can be solved by applying the transformation method as a special implementation of fuzzy arithmetic.

A nearly strict fuzzy arithmetic for solving problems with uncertainties

Fuzzy arithmetic is a powerful tool to solve engineering problems with uncertain parameters. In doing so, the uncertain parameters in the model equations are expressed by fuzzy numbers, and the problem is solved by using fuzzy arithmetic to carry out the mathematical operations in a generalized form. The practical use of standard fuzzy arithmetic, however turns out to be very problematic, basically because of the overestimation effect which is responsible for a more or less large discrepancy between the proper arithmetical solution of the problem and the calculated one. In this paper a new implementation of fuzzy arithmetic is presented by which those discrepancies in general can be reduced to a slight remainder and in many cases can even be totally avoided. The effectiveness of the method is illustrated by some typical examples.

Simulation of the human glucose metabolism using fuzzy arithmetic

With the object of future improvement of the medical treatment of diabetes, human glucose metabolism is simulated using fuzzy arithmetic to take into account the uncertainties in the physiological model. For this purpose, the uncertain model parameters and model inputs are represented by fuzzy numbers with their shape derived from experimental data or expert knowledge. To avoid the disadvantageous consequences of standard fuzzy arithmetic, a new implementation of fuzzy arithmetic is used which guarantees a proper solution of the problem. Finally, as a byproduct of the new implementation of fuzzy arithmetic, a measure of influence is introduced which allows a quantification of the percentile influence of each varying factor in the physiological model on the variation of the overall model output. By doing this, one can detect to what extent the model output is influenced by the different model parameter which is of extreme importance for the practical optimization of medical therapy.

Identification of enhanced fuzzy models with special membership functions and fuzzy rule bases

Abstract A special fuzzy modeling method for developing multi-variable fuzzy models on the basis of measured input and output data is presented. Forming the crucial point in fuzzy modeling, the fuzzy model identification procedure is carried out by applying a special clustering method, the fuzzy c -elliptotypes method, to provide the parameters of the fuzzy model. To enhance the efficiency of the fuzzy model, the rather simple membership functions defined at first for the input fuzzy sets are replaced by a special class of functions. Additionally, the conventional rule base expressing the main links between the inputs and the outputs of the fuzzy model is replaced by a fuzzy rule base with fuzzy assignments, leading to further improvements of the fuzzy model.

An interdependency index for the outputs of uncertain systems

The study of mechanical systems with uncertain parameters is gaining increasing interest in the field of system analysis to provide an expedient model for the prediction of the system behavior. Making use of the Transformation Method, the uncertain parameters of the system are modeled by fuzzy numbers in contrast to random numbers used in stochastic approaches. As a result of this analysis, a quantification of the overall uncertainty of the system outputs, including a worst-case scenario, is obtained. The inputs of the resulting fuzzy-valued model are a priori uncorrelated but after the uncertainties are propagated through the model, interdependency (or interaction) between the outputs may arise. If such interdependency is neglected, a misinterpretation of the results may occur. For example, in the case of applying uncertainty analysis in the early design phase of a product to determine the relevant design-parameter space, the interdependency between the design variables may reduce significantly the available part of the design space. This paper proposes a measure of interdependency between the uncertain system outputs. The interdependency index can be derived by a postprocessing of the data gained by the analysis with the Transformation Method. Such information can be obtained by a negligible amount of extra computation time.

On applying fuzzy arithmetic to finite element problems

Fuzzy arithmetic, based on Zadehs (1973) extension principle, is applied to solve finite element problems with uncertain parameters. As an example, a rather simple, one-dimensional static problem consisting of a two-component massless rod under tensile load is considered. Application of fuzzy arithmetic directly to the traditional techniques for the numerical solution of finite elements, i.e. primarily on the algorithms for solving systems of linear equations, however turns out to be impracticable in all circumstances. In contrast to the use of exclusively crisp numbers, the results for the calculations including fuzzy numbers usually differ to a large extent depending on the solution technique applied. The uncertainties expressed in the different calculation results are then basically twofold. On one hand, uncertainty is caused by the presence of parameters with fuzzy value, whilst on the other, an additional undesirable uncertainty is artificially created by the solution technique itself. For this reason, an overview of the most common techniques for solving finite element problems is offered, rating them with respect to minimizing the occurence of artificial uncertainties. Moreover a special technique is outlined which leads to modified solution procedures with reduced artificial uncertainties.