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lecture20 - 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 7 Lecture 20 Last time: Completed solution to the optimum linear filter in real-time operation Semi-free configuration: () 0 0 ()( ) 2( ) ( ) ( ) ( ) ( ) j s tp t Li s L L ii L R ii R j p DpF p S p Hs d t e d p e j F sF sSs FpSp π −∞ ⎡⎤ ⎣⎦ 1− = ∫∫ 144424443 Special case: ( ) p is rational: In this solution formula we can carry out the indicated integrations in literal form in the case in which ( ) p is rational. In our work, we deal in a practical way only with rational F , is S , and ii S , so this function will be rational if Dp is rational. This will be true of every desired operation except a predictor. Thus except in the case of prediction, the above function which will be symbolized as [ ] can be expanded into () () () LR p pp =+ ⎡⎤⎡⎤ ⎣⎦⎣⎦ where [ ] L has poles only in LHP and [ ] R has poles only in RHP. The zeroes may be anywhere. For rational [ ] , this expansion is made by expanding into partial fractions, then adding together the terms defining LHP poles to form [ ] L and adding together the terms defining RHP poles to form [ ] R . Actually, only [ ] L will be required. () () {} 00 0 1 2 j st pt st st j dte dp p p e f t e dt f t e dt j ∞∞ −− += + where 1 0 , 0 2 1 0 0 2 j pt L L j j pt R R j ft p ed p t j f e d p t j == < > Note that R f t is the inverse transform of a function which is analytic in LHP; thus () 0 R = for 0 t > and
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 7 0 () 0 st R f te d t = Also L f t is the inverse transform of a function which is analytic in RHP; thus () 0 L ft = for 0 t < . Thus 0 st st LL L t t s ∞∞ −− −∞ == ⎣⎦ ∫∫ Thus finally, 0 () ( ) ( ) Li j Ri i R L i i L DsF s S s Fs S s Hs Fs F s S s ⎡⎤ ⎢⎥ = In the usual case, Fs
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lecture20 - 16.322 Stochastic Estimation and Control, Fall...

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