16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde =A(1)+G(1) ( The error dynamics are then (1)=x(1)-X() A(De+b(o)n The mean of the error then satisfies since n(t)=0, and since the initial condition is (t-)=0 we have e(0=0, (+-p4) The form of the error dynamics are a linear system driven by white noise, which is the form of dynamics we just analyzed. The covariance matrix for the estimation error therefore satisfies the differential equation P=AP+ Pa+BNB with the initial condition P(-)=P() So the time propagation step can be taken by integrating the differentia x= ax+gu (-)=x(-) P=AP+PA+ BNB P(k-1)=P*(4-1) Both of these relations can be expressed in the form of discrete time transitions but especially if A and/ or B and/or G are time-varying, this may not be convenient The transition matrix would be the solution to the differential equation d Φ(t1)=AO)(,1-) with the initial condition d Then the estimate at the next measurement point would be given by Page 4 of 716.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 7 ( ) 1 1 ( ) ˆ ˆ () () ˆ ˆ k k x At x Gt u xt x t + − − = + = & The error dynamics are then () () () ˆ ˆ () () et xt xt e xx At e Bt n = − = − = + & & & The mean of the error then satisfies e At e = ( ) & since n t() 0 = , and since the initial condition is ( ) 1 0 k e t − = we have e t() 0 = , ( ) 1, k k t t + − − The form of the error dynamics are a linear system driven by white noise, which is the form of dynamics we just analyzed. The covariance matrix for the estimation error therefore satisfies the differential equation T T P AP PA BNB & =+ + with the initial condition () () Pt P t k k 1 1 + − − = So the time propagation step can be taken by integrating the differential equations ( ) ( ) () () 1 1 1 1 ˆ ˆ ˆ ˆ k k T T k k x Ax Gu xt x t P AP PA BNB Pt P t + − − + − − = + = =+ + = & & Both of these relations can be expressed in the form of discrete time transitions, but especially if A and/or B and/or G are time-varying, this may not be convenient. The transition matrix would be the solution to the differential equation () () 1 1 , () , k k d tt At tt dt Φ =Φ − − with the initial condition ( ) 1 1 , k k tt I Φ = − − Then the estimate at the next measurement point would be given by