16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Lecture 20 ast time: Completed solution to the optimum linear filter in real-time operation emi-free configuration: H0(s) dp D()F(p),s 2TJF(S),F(S)LS(S)L N F(PRS(P) [(p)] Special ca 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, Sis, and S, so this function will be rational if D(p)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 [(p)]=[(p)]+(p) where[l, has poles only in LHP and 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[I, and adding together the terms defining RHP poles to form[ Actually, only [l, will be required where f(1) [(p)]e"4=0.1<0 (0=n1.()]hb=0.1>0 ote that fr( is the inverse transform of a function which is analytic in LHP, thus fR(=0 for t>0 and Page 1 of 716.322 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 st pt L is L L ii L R ii R j p DpF p S p H s dte dp e π jF s F s S s F p S p ∞ ∞ − − ∞ ⎡ ⎤ ⎣ ⎦ 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 D p( ) 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 () () () L R ⎡ ⎤⎡ ⎤ ⎡ ⎤ p = + p p ⎣ ⎦⎣ ⎦ ⎣ ⎦ 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. { } () () 0 00 1 () () 2 j st pt st st L R L R j dte dp p p e f t e dt f t e dt π j ∞ ∞∞ ∞ − −− − ∞ ⎡⎤⎡⎤ += + ∫∫ ∫ ∫ ⎣⎦⎣⎦ where ( ) ( ) 1 ( ) 0, 0 2 1 ( ) 0, 0 2 j pt L L j j pt R R j f t p e dp t j f t p e dp t j π π ∞ − ∞ ∞ − ∞ = =< ⎡ ⎤ ⎣ ⎦ = => ⎡ ⎤ ⎣ ⎦ ∫ ∫ Note that ( ) Rf t is the inverse transform of a function which is analytic in LHP; thus () 0 Rf t = for t > 0 and