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Unformatted text preview: 4) w hen K = 24. T he s ame information is p resented in polar form in
t he c orresponding Nyquist plot in Fig. 7.lOc. For example, a t w = 1, IH(jw)1 = 2.6
a nd L.H (jw) =  130.6°. We plot a point a t a d istance 2.6 u nits a nd a t a n angle
 130.6° from t he h orizontal axis. This point is labeled as w = 1 for identification
(see Fig. 7.lOc). We p lot such points for several frequencies in t he r ange from w = 0
t o 0 0 a nd d raw a smooth curve through them t o o btain t he N yquist plot. T he same
information is presented in Cartesian form in Nichols plot. For instance, a t w = 1,
t he log magnitude is 20 log 2.6 = 8.3 dB, a nd t he p hase a t w = 1 is  130.6°. We
p lot a point a t c oordinates x = 8.3, Y =  130.6° a nd label this point as w = 1 for
identification. We do this for several values of w from 0 t o 0 0 a nd d raw a curve
through these points t o o btain Nichols plot. Using Bode or Nyquist (or Nichols)
plots of t he openloop transfer function, we c an readily investigate stability aspect
of t he c orresponding closedloop system as discussed below. 7.31 Relative Stability: Gain and Phase margins For t he s ystem in Fig. 7.10a, t he c haracteristic equation is 1 + K G(8)H (8) = 0
a nd t he c haracteristic roots are t he r oots of K G (8) H (8) =  1. T he s ystem becomes
unstable when t he r oot loci cross over t o RHP. T he crossing occurs on t he i maginary
axis where 8 = jw (see Fig. 6.43). Hence, a t t he verge of instability (marginal
stability) K G(jw)H(jw) T he t imedomain method of control system design discussed in Sec. 6.7 works only
when t he t ransfer function of t he p lant (the system t o be controlled) is known and is
a r ational function (ratio of two polynomials in 8). T he i nputoutput description of
practical s ystems is often unknown a nd is more likely to b e nonrational. A system
containing a n ideal time delay (dead time) is a n example of a nonrational system.
In such cases, we c an determine t he frequency response of t he openloop system
empirically a nd use this d ata t o design t he (closedloop) system. In this section
we shall discuss feedback system design procedure based on frequency response
description o f a system. However, t he frequency response design method is n ot as
convenient a s t he t imedomain design method from the viewpoint of t he t ransient
a nd t he s teadystate error specifications. Consequently, t he t imedomain method in =  1 = 1 e±j1f T hus, a t t he verge of instability, t he m agnitude a nd angle of t he o pen loop gain
K G(jw)H(jw) a re I KG(jw)H(jw)1 = 1,
K G(8)
T(8) = 1 + K G(8)H(8) 491 a nd L.G(jw)H(jw) = ±7r T hus, on t he verge of instability, t he openloop transfer function has unity gain
a nd p hase of ±7r. I n order t o u nderstand t he significance of these conditions, let us
consider t he s ystem in Fig. 7.lOa, with openloop transfer function K / 8 (s+2)(8+4).
T he Bode plot for this transfer function (for K = 24) is d epicted in Fig. 7.lOb. T he
r oot locus for this system is illustrated in Fig....
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 Spring '13
 Bayliss
 Signal Processing, The Land

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