Signal Processing and Linear Systems-B.P.Lathi copy

63 7 if h 8 6 3 9 a nd t he i nput j t is a lout r

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Unformatted text preview: c anonical realization o f t he t ransfer function H (s) = S2 s2 6 .7-1 Y ( s) + 5s + 2 + 4 s + 13 D etermine t he rise t ime t r , t he s ettling t ime ts, t he P O, a nd t he s teady-state errors es, e r , a nd e p for each o f t he fOllOWing systems, whose transfer functions are: 95 (c) s2 + lOs + 100 Y (s) (c) F ig. P 6.6-7 Hint: Normalize t he highest-power coefficient in t he d enominator t o unity. 6 .6-4 6 .7-2 H (s) = (s s is + l )(s + 2) + 5)(s + 6)(s + 8) H (s) = (s Hint: H ere m 6 .6-5 F ig. P 6.7-2 R epeat P roblem 6.6-1 if For a position control system depicted in Fig. P6.7-2, t he u nit s tep response shows t he p eak t ime t p = 1r / 4, t he P O = 9%, a nd t he s teady-state value of t he o utput for t he u nit s tep i nput is Yss = 2. D etermine K I, K2 a nd a . + 1)2(s + 2)(s + 3) = n = 3. R epeat P roblem 6.6-1 if Y (s) 6 .6-6 R epeat P roblem 6.6-1 if H (s) 6 .6-7 = (s + 1)(s2 + 4s + 13) . of com lex conjugate poles may be realized using I n t his p roblem we show how a p au f ~ Show t hat t he t ransfer functions of a cascade o f two first-order transfer unctIOns. t he b lock diagrams in Figs. P 6.6-7a a nd b are ( a) H (s) = 1 (s + a)2 + b2 = 1 S2 + 2as + (a2 + b2) s +a s+a ( b) H (s) = ( s+a)2+b2 = s 2+2as+(a 2 +b 2) Hence, show t hat t he t ransfer function of t he block diagram A s+B A s+B ( c) H (s) = (s + a)2 + b2 = s2 + 2as + (a2 + b2) . 6 .6-8 F ig. P 6.7-3 6.7-3 Show op amp re al· t 'IOn 0 f t he following transfer functIOns: Iza 10 10 . .. ) s + 2 ( ii) - (m + 5 ( .) ~ I s +5 s +5 s .. I II . Fig. P6.6-7c IS 6.7-4 For a position control system illustrated in Fig. P6.7-3 t he following specifications are imposed: t r ::; 0.3, ts ::; 1, P O ::; 30% a nd e . = O. W hich of these specifications c annot b e met by t he s ystem for any value of K ? W hich speCifications c an b e met by simple a djustment o f K ? O pen loop transfer functions of four closed-loop systems are given below. In each case, give a rough sketch of t he r oot locus. ( a) K (s+ 1) s is + 3 )(s + 5) K (s+ 1) ( b) s (s + 3 )(s + 5)(8 + 7) ( e) K (S+5) s is + 3) K (s+ 1) ( d) s is + 4)(8 2 + 2s + 2) 6 Continuous-Time System Analysis Using the Laplace Transform 470 Y (s) ~~I'\' F ig. P 6.7-5 . F ' P 6 7 5 W e a re r equired t o m eet t he For a u nity feedback s ystem s hown I II Ig. . - 0' d < 0 06 Is it possible 0/( < 0 2 t < 0 5 e = a n er - . . specifications P O ~ 1 6.", t r. - b' , s d'- f' , I f n ot s uggest a s uitable form of t o m eet t hese specIficatIOns J ust y a J US I llg. , c ompensator a nd find t he r esulting P O, t s, t r , es a nd e r· .. . f h ( b'l t eral) L aplace t ransform o f t he 6 .8-1 F ind t he r egion of convergence, If It eXIsts, 0 t e l a f ollowing signals: 6 .7-5 K? ( a) 6 .8-2 6 .8-3 e' u (') ( b) e-' u (,) ( c) 1 +t2 1 ( d) l +e' ( e) e - k • 2 dh o nding r egion o f convergence F ind t he ( bilateral) Laplace t ransform a n t e c orresp for t he following signals: ( e) e 'u(t) + e 2tu (_t) ( b) e - 1tl cos t ( a) e - I' I (...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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