F This leads to an unstable mode which is not observable at the output thus cannot be controlled by feeding back.
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1 Lecure 8 Las ime: Semi-free configuraion design This is equivalen o: Noe ns, ener he sysem a he same place. is fixed. We design C (and perhaps B. We mus sabilize if i is given as unsable. Cs ( H( s = + CssBs ( ( ( so ha having he opimum H, we deermine C from H( s Cs ( = H ( s ( s B ( s We do no collec H and ogeher because if is non-minimum phase, we would no wish o define H by ( H H = op This leads o an unsable mode which is no observable a he oupu hus canno be conrolled by feeding back. Associae weighing funcions wih he given ransfer funcions. H( s wh ( s ( w ( Ds ( w( D Page of 5
2 If s ( is unsable, pu a sabilizing feedback around i, laer associae i wih he res of he sysem. Error Analysis We require he mean squared error. c ( = w ( ( i d H o ( = w ( ( c d = d w ( d w ( i( H d ( = w( ( s d D e ( = o ( d ( e ( = o ( od ( ( + d ( o( = d w( dwh( ( i d4w( 4 d3wh( 3( i 3 4 = d w ( d w ( d w ( d w ( i( i( H 3 H = d w ( d w ( d w ( d4w( 4 Rii( H 3 H 3 od ( ( = d w( dwh( ( i d3wd( 3( s 3 = d w ( d w ( d w ( i( s( H 3 D 3 3 = d w ( d w ( d w ( R ( + H 3 D 3 is 3 We shall no require d ( in inegral form. Page of 5
3 The problem now is o choose wh ( so as o minimize his use variaional calculus. Le: w ( = w ( + δ w( H e (, for which we where w ( is he opimum weighing funcion (o be deermined and δ w ( is an arbirary variaion arbirary excep ha i mus be physically realizable. Calculae he opimum e and is firs and second variaions. e = e + δe + δ e e = o( + o( d( + d( The opimum e ( e for δ w ( = : = ii dw( dw( d3wd( 3 Ris( d( e( d w ( d w ( d w ( d w ( R ( The firs variaion in e ( is ( = ( ( 3 ( 3 4 ( 4 ii( δ e dδw d w d w d w R + d w ( d w ( d δw( d w ( R ( ii 3 4 dδw( d w ( d w ( R ( + 3 D 3 is 3 In he second erm, le: = 3 = 4 3 = 4 = and inerchange he order of inegraion. nd erm = d δw( d w( d 3w ( 3 d 4w( 4 Rii( 3+ 4 Page 3 of 5
4 bu since Rii( 3+ 4 = Rii( we see ha he second erm is exacly equal o he firs erm. Collecing hese erms and separaing ou he common inegral wih respec o gives δ e( = dδ w( dw( d3w( 3 d4w( 4 Rii( dw( d3wd( 3 Ris( + 3 The second variaion of e ( is δ e = dδ w dw dδ 3 w 3 d4w 4 Rii ( ( ( ( ( ( By comparison wih he expression for oupu of he sysem o (, his is seen o be he mean squared ( δ e oupu = ( >, non-zero inpu This second variaion mus be greaer han zero, so he saionary poin defined by he vanishing of he firs variaion is shown o be a minimum. In he expression for he firs variaion, δ w( = for < by he requiremen ha he variaion be physically realizable. Bu δ w( is arbirary for, so we can be assured of he vanishing of δ e ( only if he { } erm vanishes almos everywhere for. The condiion which defines he minimum in e ( is hen d w ( d w ( d w ( R ( ii 3 4 d w ( d w ( R ( + = 3 D 3 is 3 for all, non-real-ime. Using his condiion in he expression for for < gives he resul e ( and remembering ha w ( = Page 4 of 5
5 e ( = d ( o ( which is convenien for he calculaion of e (. Also since o ( = d ( e (, his says he opimum mean squared oupu is always less han he mean squared desired oupu. Auocorrelaion uncions We have arrived a an exended form of he Wiener-Kopf equaion which defines he opimum linear sysem under he ground rules saed before. Recall ha: Rii( = Rss ( + Rsn( + Rns ( + Rnn ( Ris( = Rss ( + Rns ( since i = s+ n. The free configuraion problem is a specializaion of he semi-free configuraion. In his expression we would ake s= (, or w ( = δ (. In ha case we have d d( d w ( d δ( R ( ii 3 4 dδ ( dw ( R ( + = 3 D 3 is 3 w ( R ( d w ( R ( d = for 3 ii 3 3 D 3 is 3 3 Page 5 of 5
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