lecture19

Lecture19 - 16.322 Stochastic Estimation and Control Fall 2004 Prof Vander Velde Lecture 19 Last time w0 3)Rii 1 3)d 3 w D 3)Ris 1 3)d 3 = 0 for 1

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 6 Lecture 19 Last time: 03 1 3 3 3 1 3 3 1 ( ) () ( ) ()0 f o r 0 ii D is wR d w R d ττ τ ∞∞ −∞ −∞ −− −= ∫∫ Solution in the Free Configuration, Non-Real-Time Filter Case The applicable condition in this important case is: 1 3 3 3 1 3 3 0 ii D is d w R d −∞ −∞ for all 1 . [] 1 1 0 s ed −∞ = Since this function of 1 must be zero for all 1 , its transform must also be zero. 13 3 3 0 3 1 3 133 1 3 0 ss ii D is d d w R ee d d w R τττ −∞ −∞ −∞ −∞ 0 0 () () () () 0 ii is is ii HsSs D sSs DsS s Hs Ss = with Ds = desired signal transfer is ss ns ii ss sn ns nn Ss Ss S s Ss S s S s S s S s =+ =+++ This is the optimum filter “transfer function.” Since it is non-real-time, it is usually implemented as a weighting sequence in a digital computer. The continuous weighting function is 00 1 ( ) 2 j st j wt Hsed s j π −∞ = I will give an integral form which is useful in evaluating this integral. 0 typically has poles in both left and right hand planes. This does not imply instability in the case of a two-sided transform.
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 6 () 0 ( ) ot i w t d τ ττ −∞ =− In terms of s , the integrals are written: 1. If the real part of a is positive, 1 1 0 1 1! 2 00 na t j st n j te t e n ds j sa t π −− −∞ > = + < 2. If the real part of a is negative, 1 1 0 1 2 t j st n j t e n ds j t < = + > In writing power density spectra, which are rational functions of 2 ω - if rational at all – we note that 2 is replaced by 2 s . Note that the inverse transform of 0 Hs can be done by expanding 0 in a partial fraction expansion and integrating each term separately. This expended problem is more general than the case treated in the text for two reasons: 1. A more general expression for the desired output is permitted 2. Non minimum phase fixed parts of the system, Fs , may be handled without stability problems due to cancellation of unstable modes.
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This note was uploaded on 11/07/2011 for the course AERO 16.322 taught by Professor Wallacevandervelde during the Fall '04 term at MIT.

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Lecture19 - 16.322 Stochastic Estimation and Control Fall 2004 Prof Vander Velde Lecture 19 Last time w0 3)Rii 1 3)d 3 w D 3)Ris 1 3)d 3 = 0 for 1

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