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

From t he a symptotes t he corner frequencies are

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Unformatted text preview: n given in terms of gain a nd p hase margins. 7 .3-2 G H plane ~i!" ----t--t---~>!'_~._s:"':: R e- (c) F ig. 7 .10 Gain and Phase margins of a system with open-loop transfer function .(.+~}~.+4)· 493 Transient Performance in Terms o f Frequency Response For a second-order system in Eq. (6.81), we saw t he dependence of t he t ransient response ( PO, t T , t d a nd t s ) o n t he d ominant pole location. Using this knowledge, we developed in Sec. 6.7 a procedure for designing a control system for a specified transient performance. In order to develop such a procedure from the knowiedge of system's frequency response ( rather t han its transfer function), we m ust know t he r elationship between t he frequency response a nd t he t ransient response of t he s ystem in Eq. (6.81). Figure 7.11 shows t he frequency response of a second-order system in Eq. (6.81). T he p eak frequency response Mp ( the maximum value of the amplitude response), which occurs a t frequency wp , indicates relative stability of t he system. Higher peak response generally indicates smaller ( (see Fig. 7.6a), which implies poles closer t o t he i maginary axis, a nd less relative stability. Higher M p also means higher P O ( the s tep response overshoot). Generally acceptable values of Mp in practice range from 1.1 t o 1.5. T he 3-dB bandwidth W b of t he frequency response indicates t he speed of the system. We can show t hat W b a nd tT tThe Nyquist criterion states as follows: A closed curve C . in the s plane enclosing m zeros and n poles of an open-loop transfer function W (s) maps into a closed curve C w in the W plane encircling the origin of the W plane m - n times, in the same direction as that of C s . I f n - m is negative, then the encirclement is in the opposite direction. 494 7 F requency Response a nd A nalog F ilters 7.3 C ontrol S ystem D esign U sing F requency R esponse 4! Mm dB OdB~----------~~==~",-,~~~~----------------~ 0 0_ lOb -3 dB ... _..............................................................!f!1!.~.5~!~:::,..,..... ' - ........, .... 3.0 2.0 1.0 F ig. 7 .11 Frequency response of a second-order system. are inversely proportional. Hence, higher Wb i ndicates smaller For t he s econd-oder system in Eq. (6.81), we have T (jw) = T o find M p, W (jw)2 tr ( faster response). 0 x_ 2 n + 2j(wnw + w~ - 1.0 we let d IT(jw)l/dw = O. F rom t he s olution of t his e quation, we find - 2.0 1 M ---=== p - 2(v'1=(2 ( :::; 0.707 J I=(2 ( :::; 0.707 wp = Wn Wb = Wn [(1 - 2(2) + J4(4 - 4(2 + 2r /2 - 3.0 (7.27) T hese e quations show t hat we c an d etermine ( a nd W n from M p a nd w p' Knowledge of ( a nd W n allow us t o d etermine t he t ransient p arameters, such as, PO, t r a nd t s as seen from Eqs. (6.83), (6.84) a nd (6.85). Conversely, if we a re given c ertain t ransient s pecifications PO, t r , a nd t ., we c an determine t he r equired M p a nd w p' T hus, t he p roblem now reduces t o designing a system, which h as a c ertain M p a nd W p f or t he closed-loop frequency response. In practice, we know t he o penloop...
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