16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde System dynamics are continuous in time Measurements are processed at discrete points in time discrete measurement X(t) D P(t1) k-1 continuous time propagation Where a discrete time measurement is processed there is a discontinuity in the estimate, the estimate error and the error covariance matrix. We use the superscript"-to indicate values before incorporating the measurement at a measurement time and the superscript"+to indicate values after incorporating he measurement at a measurement time The filter operates in a sequence of two-part steps 1. Propagate i(n) and P() in time starting from i'(-)and P*(k-)to i(4) andPˉ 2. Incorporate the measurement at t to produce i'() Time Propagation After processing the measurement at Ik- we have the estimate i'('k-) and the estimation error covariance matrix P*(tk-D) Then the filter operates open loop until the next measurement point at tr. Note that we are not considering any effects of computation delays here In the interval from tk- to tt the system dynamics are described as A(Ox+G(u+b(on The estimate i'(tk-1)is the mean of the distribu measurements processed up to that point. So the estimate error is zero We want to preserve the error unbiased until the next measurement, and we can achieve this driving the estimate with the mean or expectation of the system Page 3 of 716.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 7 • System dynamics are continuous in time • Measurements are processed at discrete points in time Where a discrete time measurement is processed, there is a discontinuity in the estimate, the estimate error, and the error covariance matrix. We use the superscript “-“ to indicate values before incorporating the measurement at a measurement time and the superscript “+” to indicate values after incorporating the measurement at a measurement time. The filter operates in a sequence of two-part steps 1. Propagate xˆ( )t and P t( ) in time starting from xˆ (tk 1 ) + − and P t( ) k 1 + − to xˆ (tk ) − and ( ) P tk − 2. Incorporate the measurement at kt to produce xˆ (tk ) + and P t( ) k + Time Propagation After processing the measurement at k 1 t − we have the estimate ( ) 1 ˆ k x t + − and the estimation error covariance matrix ( ) P tk 1 + − . Then the filter operates open loop until the next measurement point at kt . Note that we are not considering any effects of computation delays here. In the interval from k 1 t − to kt the system dynamics are described as x& =++ At x Gt u Bt n () () () The estimate ( ) 1 ˆ k x t + − is the mean of the distribution of x conditioned on all the measurements processed up to that point. So the estimate error is zero mean. We want to preserve the error unbiased until the next measurement, and we can achieve this driving the estimate with the mean or expectation of the system dynamics