16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde ∫f(e-"t=0 Also fi(t) is the inverse transform of a function which is analytic in RHP; thus f(t=0 for t<0. Thus ∫f(e-"t=f(e"=[(s) Thus finally D(SF(SS(S) F(SRS(S) H0(s) F(s)2F(-s)2S2(s) In the usual case, F(s)is a stable, minimum phase function. In that case, F(S)=F(S), F(S =l; that is, all the poles and zeroes of F(s) are in the LHP. Similarly, F(S)=1. Then D(sS(s) S(s) H()=F(s)S2(S) Thus in this case the optimum transfer function from input to output D(S)S(s) F(s)H0(s)= S(s)2 S2(s) nd the optimum function to be cascaded with the fixed part is obtained from nis by division by F(s), so that the fixed part is compensated out by cancellation Free configuration problem: D(sS,(s) S,(S) H0(s) S(S) Optimum free configuration filter H(S) F(s) Page 2 of 716.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 7 0 () 0 st Rf t e dt ∞ − = ∫ Also ( ) Lf t is the inverse transform of a function which is analytic in RHP; thus () 0 Lf t = for t < 0 . Thus ( ) 0 () () st st L L L f t e dt f t e dt s ∞ ∞ − − −∞ = =⎡ ⎤ ∫ ∫ ⎣ ⎦ Thus finally, 0 () ( ) () () () ( ) () ( ) () L ij R ii R L L L ii L DsF s S s Fs S s H s Fs F s S s ⎡ ⎤ − ⎢ ⎥ ⎣ ⎦ = − In the usual case, F s( ) is a stable, minimum phase function. In that case, () () Fs Fs L = , () 1 F s R = ; that is, all the poles and zeroes of F s( ) are in the LHP. Similarly, () 1 F s − = L . Then 0 () () ( ) ( ) () () ij ii R L ii L DsS s S s H s FsS s ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ = Thus in this case the optimum transfer function from input to output is 0 () () ( ) () () ( ) ij ii R L ii L DsS s S s FsH s S s ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ = and the optimum function to be cascaded with the fixed part is obtained from this by division by F s( ) , so that the fixed part is compensated out by cancellation. Free configuration problem: 0 () () ( ) ( ) ( ) ij ii R L ii L DsS s S s H s S s ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ = Optimum free configuration filter: