lecture25 - 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 9 Lecture 25 Last time: ( ) 1 2 T T K P H HP H R = + If k k z x v = + , ( ) 1 2 H I K P P R = = + Then if P R ± , 1 2 K P R Alternatively, if R P ± , 2 K I The effect is that ( ) ˆ ˆ ˆ x x K z Hx + = + The quantity 2 K is known as the Kalman gain . It is the optimum gain in the mean squared error sense. Substitute it into the expression for P + . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 T T T T T T T T T T T T T T T T T T T T T T T T P I KH P I KH KRK I KH P I KH P H K KRK I KH P P H K KHP H K KRK I KH P P H K K HP H R K I KH P P H K P H HP H R HP H R K I KH P P H K P H K I KH P + = + = + = + + = + + = + + + = + = The form at the top is true for any choice of K . The last form is true only for the Kalman gain . The first form is better behaved numerically if you process a measurement which is very accurate relative to your prior information. So the measurement update step is:
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 9 ( ) ( ) ( ) 1 ˆ ˆ ˆ T T K P H HP H R x x K z Hx P I KH P + + = + = + = The filter operates by alternating time propagation and measurement update steps. The results were derived here on the basis of preserving zero mean errors and minimizing the error variances. If all errors and noises are assumed normally distributed , so the probability density functions can be manipulated, one can derive the same results using the conditional mean approach : define x at every stage to be the mean of the distribution of x conditioned on all the measurements available up to that stage. Example: Conversion of continuous dynamics to discrete time form N 1 2 2 10 2 0 10 0 0 0 2 B A x x x n x x n = = ⎡ ⎤ = + ⎢ ⎥ ⎣ ⎦ ² ² ² ³´µ´¶ We could do time propagation by integration ( ) 1 N = : ˆ ˆ T T x Ax P AP PA BB = = + + ² ² ( ) 2 2 1 ...
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