16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde ={a+b 2 0. otherwise e-a(+rdT dt a+b a-aT (S+a) H0(s) a+b (s+a)S,(s+b) S,(a+b)s+b A here b Note the bandwidth of this filter and the gain which is the correlation between S(O and S(t+T) Example: Semi-free problen with oni-minimumn phase F Optimum compensator K s(d+ S(s)= The s n are uncorrelated Servo example where wed like the output to track the input, so the desired operator, D(s)=1 Page 6 of 716.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 7 ( ) ( ) 1 , 2 0, otherwise j bt T pt T j A A e tT a b e dp a b π j bp ∞ + + − ∞ ⎧ ⎪ < − + = ⎨ + − ⎪ ⎩ ∫ () () 0 0 1 st a t T aT s a t aT A A e e dT e e dt ab ab A e ab sa ∞ ∞ − − + − −+ − = + + = + + ∫ ∫ 0 () 1 ( ) ( )( ) ( ) aT aT n n A s a Ae Hs e a b s aS s b S a b s b − − + = = + + + ++ where 2 n A b a S = + . Note the bandwidth of this filter and the gain ~ aT e− which is the correlation between S t( ) and St T ( ) + Example: Semi-free problem with non-minimum phase F . Optimum compensator ( ) ( ) 2 2 ( ) ( ) ( ) ss nn n c s Fs K sd s A S s a s Ss S − = + = − = The s n, are uncorrelated. Servo example where we’d like the output to track the input, so the desired operator, D s() 1 =