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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 18–1 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 ) δ ( t 1 − t 2 ) ⇒ w ( t ) ∼ N (0 ,R ww ) E [ v ( t 1 ) v ( t 2 ) T ] = R vv ( t 1 ) δ ( 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 18–2 • 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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