Assumed density filter with application to homing missile guidance

Abstract

A maximum likelihood estimation method is developed for a class of problems where the dynamics are linear and the measurement function is nonlinear. In this method, called the assumed density filter(ADF), the form of the conditional probability density function(CPDF) is selected to be a function of a finite number of quantities. These quantities which describe the approximate shape of the CPDF around the mode are propagated through each measurement interval. At the measurement the CPDF is updated using Bayes theorem and its mode, computed numerically, is defined to be the best estimate of the state. The posteriori CPDF is then approximated by a Taylor series expansion about its mode to preserve the assumed functional form. The numerical results for a target-intercept problem indicate that the ADF is superior to the extended Kalman filter. Howe:er:fhe ADF has a negative range bias. It is analytically proved,with some approximations, that the maximum likelihood range estimates are smallerthan the mean range estimates. I . INTRODUCTIONTactical weapon systems require accurate tracking of maneuverable vehicles such as submarines and airplanes. During the last several years, there has been an active interest in the development of sophisticated filtering algorithms for tracking with bearings-only as the observations. Mathematically,this problem can be described in an inertial rectangular coordinate frame by a linear dynamical model and a nonlinear discrete observation model or in an inertial polar coordinate frame by a nonlinear dynamical model and a linear discrete observation model. Satisfactory results for this class of problems have been difficult to obtain using current mechanizable filters because of the nonlinearity and the passive nature of the observations. As a result, considerable research has been going on to improve the existing methods in order to obtain better estimates. APPROXIMATIONS IN NONLINEAR FILTERING THEORYThe target tracking problem is stochastic in nature. Analyses of stochastic problems are possible through statistical interpretations. In order to obtain mathematical expressions for the statistics, assumed to represent the best estimates of the states associated with a problem, knowledge of the underlying probability density function (PDF) is essential.If the system dynamics andlor the measurement function are nonlinear,a finite set of statistics sufficient to describe the conditional probability density function (CPDF) is not available (1,2,3). Even if the initial states and the process noise are assumed Gaussian,the nonlinear dynamical system results in a non-Gaussian CPDF. Second,the propagation equation for the conditional mean consists of expectations of nonlinear functions that are very difficult . -to evaluate. Third,since the CPDF is not Gaussian, the system of equations to describe the conditional moments and,hence,the filter as a whole becomes coupled and infinite-dimensional.To circumvent these difficulties,approximations have been attempted to realize estimation methods consisting o...