Fuzzy Parameter and State Estimation

Abstract

The field of parameter estimation goes back to Gauss (1795) who has presented the classical least squares method in his book: “Theoria Motus Coelectium”. Least squares wasthen studied by Legendre (1806) in his book: “Nouvekkes Methodes pour la Determination des Orbits des Cametes”. In Gauss’ own words: “The most probable value of the unknown quantities will be that in which the sum of the squares of the differences between the actually observed and the computed values multiplied by numbers that measure the degree of precusion is a minimum”

Since then, an enormous effort was made by scientists and engineers to develop further the methofd and apply it to sophisticated situations (least squares in functions spaces, least squares filtering, etc.) and applications of industrial and real-life nature. Regarding the fuzzification of the least squares problem, the authors have picked-up the work of the Hungarian mathematician Celmins xc1. He has developed an elegant but quite involved, computatioally, method along with a software package (called COLSACC) that implements it. The authors present an alternative (less general) least-squares method where use is made of the concept and the algebraic properties of the so calles L-R fuzzy numbers xc2. Minimizing their distance with respect to the unknown parameters is a classical minimization problem and has a small computational demad. The chapter continuous by formulating and solving the fuzzy state estimation of a particular discrete time state-space model with fuzzy disturbances and initial conditions xc3–7. Similar problems have been considered by several authors under various mathematical formulations.

For example, it is worthwhile to mention the work of Lee xc8 where a state estimation problem is solved for a class of distributed-parameter systems with uncertain parameters. The parameters are assumed to be arbitrary time functions known to be in a closed and bounded region. The resulting estimation algorithm provides an assured accuracy and a “guaranteed” error estimator which gives an upper bound of the estimation error for any allowed variation of uncertain parameters xc9. Other works on fuzzy estimation include xc10–20.