Unformatted text preview: n given in terms of gain a nd
p hase margins.
7 .32 G H plane ~i!" tt~>!'_~._s:"':: R e (c) F ig. 7 .10 Gain and Phase margins of a system with openloop transfer function .(.+~}~.+4)· 493 Transient Performance in Terms o f Frequency Response For a secondorder 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 secondorder
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 3dB 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 openloop 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 secondorder system. are inversely proportional. Hence, higher Wb i ndicates smaller
For t he s econdoder 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 closedloop frequency response. In practice, we know t he o penloop...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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