16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 23 ast time d(t,r)=A(1)(t,r d(r,z)=1 So the covariance matrix for the state at time t is x(1)=x(1)-x(1)x(t)-x(r) =x(1)x(1) =Ea(4)x4)+j()B(7厘(xM1y(24)+J(yB()()dt =Φ(14)x0)x()d(t,) +∫a()x(6)(z)B(z)(z)d2 +∫a(.x)B()x)7a(4)dr +∫d可Jdr(t.x)B(x)(工(可)B(2)(x)y The two middle terms are zero For t>to, n(r) and x(to) are uncorrelated because n(r) is white(impulse correlation function For T=to, n(t) has a finite effect on x(to) because n(r) is white. But the integral of a finite quantity over one point is zero X()=(4)X()(5)+ dr dt2(x)B()N()5(z2-z)B(z2yo(x2 =(4)X((4)+」o(t:)B)N()(ry(r)yar This is an integral expression for the state covariance matrix. But we would prefer to have a differential equation. So take the derivative with respect to time Page 1 of 316.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 3 Lecture 23 Last time: (, ) () (, ) (,) d t At t dt I τ τ τ τ Φ =Φ Φ = So the covariance matrix for the state at time t is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 1 1 11 0 2 2 2 2 0 0 002 2 2 2 () () () () () () () , ( ) , ( ) ( ) () , ( ) ( ) , , () () , , ()( ) ( ) , T T t t T TT T T t t T T t T T T t Xt xt xt xt xt xtxt E tt xt t B n d xt tt n B t d tt xtxt tt tt xt n B t d t τ τ ττ τ τ τ τ τ τ ττ =− − ⎡ ⎤⎡ ⎤ ⎣ ⎦⎣ ⎦ = ⎡ ⎤⎡ ⎤ = Φ +Φ Φ + Φ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ =Φ Φ +Φ Φ + Φ ∫ ∫ ∫ % % % %% % % % % % ( ) ( ) ( ) 0 0 0 1 1 10 0 1 12 1 11 2 2 2 , ( )( )() , , ( ) ( ) ( ) ( ) (, ) t T T t t t TT T t t B n xt tt d dd t Bn n B t τ ττ τ ττ τ τττ τ τ Φ +Φ Φ ∫ ∫ ∫ % % % % The two middle terms are zero: - For 0 τ > t , n%( ) τ and 0 x%( ) t are uncorrelated because n%( ) τ is white (impulse correlation function) - For 0 τ = t , n%( ) τ has a finite effect on 0 x%( ) t because n%( ) τ is white. But the integral of a finite quantity over one point is zero. () () ( ) ( ) ( ) ( ) () 0 0 0 0 0 0 1 2 1 1 1 21 2 2 00 0 () , ( ) , , ( ) ( ) ( ) (, ) , ( ) , , ( ) ( ) ( ) (, ) t t T T T t t t T T T t Xt tt Xt tt d d t B N B t tt Xt tt t B N B t d τ τ τ τ τ δτ τ τ τ τ τ ττ ττ =Φ Φ + Φ − Φ =Φ Φ + Φ Φ ∫ ∫ ∫ This is an integral expression for the state covariance matrix. But we would prefer to have a differential equation. So take the derivative with respect to time