16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Continuous time filter As noted in the beginning of this section, we rarely process measurements ontinuously in a real system. However, there may be cases in which the measurements are processed so rapidly that one can approximate the process as tegration of a continuous time differential equati Also, since integration of a single differential equation for the estimate and one for the covariance matrix is logically simpler than alternating time propagation and measurement update steps, we sometimes approximate the hybrid continuous-discrete filter with a nearly equivalent continuous filter for analysis The relationship between continuous and discrete measurement processing can be seen in the following. Suppose we had the continuous measurement 三(1)=H(1)x(1)+y(1 where v(t) is an unbiased white noise process. Define a nearly equivalent discrete measurement to be the average of the continuous measurement over an interval Er(y()'=R(S(t-T tk (1)d [H(1)x()+y(1)d (4)x()△+Jy Hx()+」y(16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 9 Continuous time filter As noted in the beginning of this section, we rarely process measurements continuously in a real system. However, there may be cases in which the measurements are processed so rapidly that one can approximate the process as integration of a continuous time differential equation. Also, since integration of a single differential equation for the estimate and one for the covariance matrix is logically simpler than alternating time propagation and measurement update steps, we sometimes approximate the hybrid continuous-discrete filter with a nearly equivalent continuous filter for analysis purposes. The relationship between continuous and discrete measurement processing can be seen in the following. Suppose we had the continuous measurement zt Htxt vt () () () () = + where v t( ) is an unbiased white noise process. Define a nearly equivalent discrete measurement to be the average of the continuous measurement over an interval. () ( ) () ( ) T E vtv Rt t ⎡ ⎤ τ = − δ τ ⎣ ⎦ [ ] 1 1 1 1 1 ( ) 1 () () () 1 1 ( ) ( ) () 1 ( ) () k k k k k k k k t k t t t t k k t t k k t z ztd t H t x t v t dt t H t x t t v t dt t t H x t v t dt t τ − − − − = ∆ = + ∆ ≈ ∆+ ∆ ∆ = + ∆ ∫ ∫ ∫ ∫