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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 t A t 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 1 1 0 2 2 2 2 0 0 0 0 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) , ( ) , ( ) ( ) ( ) , ( ) ( ) , , ( ) ( ) , , ( ) ( ) ( ) , T T t t T T T T T t t T T t T T T t X t x t x t x t x t x t x t E t t x t t B n d x t t t n B t d t t x t x t t t t t x t n B t d t τ τ τ τ τ τ τ τ τ τ τ τ ⎤ ⎡ = ⎦ ⎣ = ⎤ ⎡ = Φ + Φ Φ + Φ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ = Φ Φ + Φ Φ + Φ % % % % % % % % % % ( ) ( ) ( ) 0 0 0 1 1 1 0 0 1 1 2 1 1 1 2 2 2 , ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) ( , ) t T T t t t T T T t t B n x t t t d d d t B n n B t τ τ τ τ τ τ τ τ τ τ τ τ Φ + Φ Φ % % % % The two middle terms are zero: - For 0 t τ > , ( ) n τ % and 0 ( ) x t % are uncorrelated because ( ) n τ % is white (impulse correlation function) - For 0 t τ = , ( ) 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 2 1 2 2 0 0 0 ( ) , ( ) , , ( ) ( ) ( ) ( , ) , ( ) , , ( ) ( ) ( ) ( , ) t t T T T t t t T T T t X t t t X t t t d d t B N B t t t X t t t t B N B t d τ τ τ

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