lecture23

lecture23 - 16.322 Stochastic Estimation and Control, Fall...

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 3 Lecture 23 Last time: (, ) () (, ) (,) d tA t t dt I τ ττ Φ= Φ So the covariance matrix for the state at time t is () 0 0 0 00 1 11 1 0 2 2 2 2 002 2 2 2 () () , , ()() ( ) , , ,( ) ( ) , ) ( ) ( ) , T T tt TT T T T t T t Xt xt xtxt Et t x t t B n dx t t t n B t d tt xtxt tt xt n B t d t ⎡⎤ =− ⎣⎦ = + Φ Φ + Φ ⎢⎥ Φ Φ ∫∫ %% % % 0 1 0 0 1 12 1 11 2 2 2 , ( )( )() , ) ( ) ( ) ( ) ( , ) t T T t T Bnx t t td dd t Bnn B t τττ Φ Φ The two middle terms are zero: - For 0 t > , n % and 0 x t % are uncorrelated because n % is white (impulse correlation function) - For 0 t = , n % has a finite effect on 0 x t % because n % is white. But the integral of a finite quantity over one point is zero. ( ) 0 0 1 2 1112 2 0 , ( ) , , ( ) ( ) ) , , ( ) ( ) ( ) ( , ) T t T t tt Xt d d t B N B t t B N B t d δ ττττ 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.
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 3 () 0 0 00 0 0 , ( ) , ,( ) , ( ) , ( ) ( ) ( ) , ) ( ) ( ) , ( ) () () () () () T T T
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lecture23 - 16.322 Stochastic Estimation and Control, Fall...

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