Research On Recursive Bayesian Estimation Using Random-Fuzzy Variables

With the rapid development of the modern science and technology,measurement system and measured object become more and more complex,which leads to increasingly difficulty for the evaluation and expression of measurement data and its uncertainty.Especially in many cases the measurement data is not only influenced by random noise,but also affected by non-random error,such as system error or unknown error.The uncertainties due to random effects and the non-random effects have to be evaluated according to different theory,thus a more accurate and more reasonable measured results and its uncertainty can be obtained.RFVs(random-fuzzy variables)are fuzzy variables of type-2,which are comprised of two PDs(possibility distributions): one is random PD,representing the random contributions to uncertainty;and the other is internal PD,representing the non-random contributions to uncertainty.RFVs approach is reasonable in the theoretical sense,and easy to implement,and it's an effective and more general way to measurement uncertainty evaluation and expression than the one defined by the GUM(the Guide to the Expression of Uncertainty in Measurement).Aim to reasonably evaluate and express measurement data,study and development of the analyzing and treating approach to measurement data based on RFVs is achieved in this thesis.Especially,theoretical and applied research of the extension to the RBE(recursive Bayesian estimation)method is significantly carried out.The main work of this paper is:(1)The RFVs method and its theory are improved.The RFVs method is based on the evidence theory,which is a general theory to deal with uncertainty information,including two special branches of probability and possibility theories.Therefore,the RFVs method is more general than GUM method to uncertainty evaluation.The relationship and the difference among possibility,probability and evidence theories are especially analyzed,which leads to the conclusion that the RFVs method,combining probability with possibility theory,is the generalization of the GUM method.The concept and connotation of RFVs are deeply discussed;the methods to construct the internal and the random PDs of RFVs according to the available metrological information are separately proposed.In addition,the way to propagate uncertainty in RFVs is improved.(2)An innovative method of combining random PDs is proposed.The RFVs uses t-norms to obtain joint PDs,which is a key step to propagate random contributions to uncertainty.Aiming at the problem that the error would accumulate and become bigger when Frank t-norm is applied to combine multiple random contributions,a new solution,which applies GDO(general Dombi operator)for the combination of random contributions is proposed in this paper.Different optimal parameters of GDO are obtained for the combination of different PDs by a designed method of choosing optimal parameters,and simulated analysis presents an important improvement for reducing approximation error in the combination of the random contributions.The GDO is applied to the case of evaluation of measurement uncertainty for active power and compared with the results by Frank t-norm,traditional GUM and experiment data.The confidence intervals issued from GDO is thus a good approximation of those provided by traditional GUM and experiment data.The GDO is a very flexible operator due to two parameters,and it can present a satisfied solution to the propagation of multiple random contributions to uncertainty.(3)A generalization method of the RBE based on possibility theory is proposed.The classical RBE,based on probability theory,deals only with the cases that system state quantity and their observations are modeled as random variables.In addition,although the RBE provides optimal theoretical solution,from the practical point of view,the RBE can only provide closed-form solution for a few cases.Starting from the key concept of conditional PDs,a realization method of the RBE in the possibility domain is put forward in this paper.A system for measuring rotation speed of DC motor is built up and the extended RBE is applied to obtain the optimal estimation of rotation speed.The application example shows that the extended RBE allows one to estimate the system state variables with any distribution and to take into account all random and systematic contributions to the uncertainty.Furthermore,the extended RBE can always reasonably provide a closed-form solution.(4)The extended KF(Kalman filter)algorithm based on RFVs is proposed.This paper presents the extended KF algorithm based on RFVs,according to RFVs mathematics and the concepts of mean value and covariance for PDs.The extended KF can correctly consider both random and systematic contributions to uncertainty,without the restriction on the distribution type.It's easy and effective to implement KF equations by RFVs mathematics.In addition,the characteristic and performance of the extended KF,the extended RBE and the Matía fuzzy KF are analyzed and compared by an application example of estimating angular position of rotating motor.Both the extended KF and the extended RBE use RFVs to present the estimated results and can consider separately random and non-random contributions to uncertainty.The former has less computational burden and can provide good approximation results,while the latter has more rigorous theory and provide more conservative results.The work mentioned above,as the underlying technology of measurement uncertainty evaluation and expression,and optimal estimation of system states,can be widely used in the fields of metrology support,automatic test system,global positioning system,inertial navigation and image detection and so on.