6 ENGINEERING BACKGROUN D If B is stable as in (l.S), then K = 0 andb is in //^(R.H.P.) with sup^ |6(io )| = \\B\\. For a system with n inputs and m outputs everything goes through as above with b an m x n matrix-valued function and i(s) a Cn-valued function. Also all b which occur in engineering are real on the real axis. 1 Practically al l function s b in engineerin g ar e meromorphi c o n al l o f C , an d most b are actually rational. Consequentl y th e theorem gives a more conservativ e impression tha n necessary . IDEA O F PROO F FO R m = n 1. I f 1 were i n H 2 , the n w e woul d tak e b = Bl. Thi s i s because Be~as = e- as {Bl){s) = e -asb{s) for R e a 0 and s o [Bp](s) p(s)b(s) fo r an y p whic h i s a linea r combinatio n of the e~ as . Thes e ar e dense i n H 2 . No w 1 is not i n H 2 , bu t i f / i s any functio n in H 2 formall y b = Bf/f. Thi s ca n b e use d t o giv e a proof . ALTERNATIVE PROOF . Se t b = B6 0 wher e 6 a denote s th e 6 functio n sup - ported a t th e poin t a. Us e superposition . Thi s ha s th e physica l interpretatio n that w e send a sharp puls e 6o into th e box . Measur e th e outpu t BSQ, called th e impulse response function, an d Fourie r transfor m i t t o ge t b. The functio n b is calle d th e transfer function o r frequency response function (F.R.F.) o f the system whose input-output operato r i s B. I t has a simple physical interpretation befittin g it s name . Fo r m = n = 1 and a strictl y stabl e system , think o f an inpu t signa l s'm ujt going into the system . T o be more precise nothin g is goin g in—w e flip a switc h t o sen d i n sinut. Afte r a whil e th e effec t o f th e sudden transition die s out and what w e observe must be of the form A sin(ujt+(/). This i s alway s tru e fo r a strictl y stabl e linea r tim e invarian t system . Algebrai c manipulation i s more convenient i n complex notation namely , think (fictionally ) of sendin g a signa l e luJt int o th e system—the n Ae^e ljjt come s out . Fo r a stabl e system the frequency response function b could be defined at frequency UJ to be the complex number Ae^. To "prove " thi s intuitivel y recal l tha t th e Fourie r transfor m o f e tut i s 6^. Since th e syste m i s stable, b y Theore m 1. 1 b is defined o n th e iuj axis an d Beiu,t = b(s)6i„ = b(iuj)6 luJ = b(iu)e luJt . This i s no t exactl y correc t sinc e her e w e us e e luJt fo r al l oc t oc . Thu s our formul a ignore s the effec t o f the transitio n fro m 0 to e luJt whic h th e Laplac e transform metho d account s for . A more complete "proof " i s also instructive (bu t can be skipped). Th e Laplac e transform o f an inpu t e luJt i s poo j ^ LetuJt = / e lwt -ts dt = i e ^ - ^ b 0 0 = ^— fo r R e s 0 Jo IUJ s s iu B alway s maps real valued function s t o real valued functions .
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