Page 39 - Servo Motors and Industrial Control Theory

2.6 Nyquist Plot 31

Fig. 2.9 Atypical negative
feedback control system x + e G(s) y
–

H(s)

the locus when ω goes from 0 to infinity about the real axis. In Fig. 2.18 the locus
shows when ω travels from zero to infinity.
In the preceding discussion GH has been the ratio of two polynomials of s. Thus,
the transport lag e − Ts has been excluded. Note, however, that a similar discussion
applies to systems with transport lag. The proof of this is beyond the scope of this
book. The stability of a system with transport lag can be determined from the open
loop frequency response curves by examining the number of encirclement 0 f −1
point, just as in the case of a system whose open loop transfer function is a ratio of
two polynomials in s.
With the previous discussion now the Nyquist stability criterion can be estab-
lished. For special case where there are no poles or zeros of the GH function on the
imaginary axis, which is a special case, and if the GH transfer function has k poles
( )( )Hs = constant, then the
on the right half of the imaginary axis and lim Gs
s→infinity
closed loop function as s travels from ω = − ∞ to ω = + ∞ is stable if the GH function
encircle the − 1 point k times. Note that when there are poles of the open loop trans-
fer function in the right half of the imaginary axis, the open loop transfer function
is unstable. The above stability criterion states that the closed loop transfer function
becomes stable.
Now special cases of the system with various location of poles and zeros will be
considered. The Nyquist stability criteria as described in the previous section can
be defined as follows, Let n be number of zeros for the function 1 + G( s)H( s) and k
the number clockwise encirclement of the − 1 point and let m be number of poles of
G( s)H( s) in the right half of the s plane. Then the following statement can be made,
n = k + m. If m is not zero for a stable control system the value of n must be zero,
then k = −m, which means that there be m counter clockwise encirclement of the − 1
point. If G( s)H( s) does not have any poles in the right half of s plane then n = k.
Thus, for stability there must be no encirclement of the − 1 point on the real axis.
This is the proof of the Nyquist stability used in the next section.

2.6 Nyquist Plot

A control system with forward path transfer function G( s) and feedback path trans-
fer function H( s) is shown in Fig. 2.9.