topic18 - Topic #18 Optimal Estimators Bryson and Ho...

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Unformatted text preview: Topic #18 Optimal Estimators Bryson and Ho Applied Optimal Control (Chapter 12) Gelb Applied Optimal Estimation Crassidis and Junkins Optimal Estimation of Dynamic Systems Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 181 Optimal Estimator Can also develop an optimal estimator for linear systems. First introduce a more general system model that includes various types of random noise in the system on the estimator: 1 x ( t ) = A x ( t ) + B u u ( t ) + B w w ( t ) y ( t ) = C y x ( t ) + v ( t ) w ( t ) : process noise models uncertainty in the system model. v ( t ) : sensor noise models uncertainty in the measure- ments. Key issue then is to balance the effects of these noises on the estimation error. Note that the (relative) level of these noises can be used to indicate which you trust more, the model or the measurements. Typically assume that w ( t ) and v ( t ) are: Zero mean, so that E [ w ( t )] = 0 Uncorrelated Gaussian white random noises no correlation between the noises at one time instant and another, or between each other. E [ w ( t 1 ) w ( t 2 ) T ] = R ww ( t 1 ) ( t 1 t 2 ) w ( t ) N (0 ,R ww ) E [ v ( t 1 ) v ( t 2 ) T ] = R vv ( t 1 ) ( t 1 t 2 ) v ( t ) N (0 ,R vv ) E [ w ( t 1 ) v ( t 2 ) T ] =0 1 If B w not specified for a system, then take B w = B u November 7, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 182 Note that R ww and R vv specify the covariances associated with the Gaussian noise and thus effectively determine the noise levels...
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topic18 - Topic #18 Optimal Estimators Bryson and Ho...

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