Page 38 - Servo Motors and Industrial Control Theory -

30 2 Feedback Control Theory Continued

Im Im
1 + GH Plane GH Plane

–1
0 1 Re 0 Re
1 + G(jω) H(jω)

1 + G(jω) H(jω) G(jω) H(jω)

Fig. 2.8 Plot of 1 + G( iω)H( iω) and G( iω)H 1

If the mapping theorem is applied to the special case with F( s) = 1 + G( s)H( s),
then we can make the following statement. Let the closed contour in the s plane cov-
ering the complete right half of the s plane, then the number of half side of s plane
zeros of the mentioned function is equal to the number of poles of the mentioned
function in the right half plane plus the number of clockwise direction of encircle-
ment of the origin of the closed loop function plane by corresponding closed curve
in latter plane. Because of this assumed conditions then,

lim (1 Gs+ ( )·H( )) : Constants =
s→ infinity
The function (1 + G( s)H( s)) remains constant as s traverses the semicircle of infinite
radius. Because of this, whether the locus of the above function encircle the origin
of the function plane can be determined by considering only a part of the closed
contour in the s plane, that is, the imaginary axes. Encirclement of the origin, if
there are any, occur only while a representative point moves from iω to –iω along
the imaginary axis, provided that no zeros or poles lie on the imaginary axis.
Note that the portion of 1 + G( s)H( s) contour from ω = −∞ to ω = ∞ is simply
1 + G( iω)H( iω). Since the 1 + G( iω)H( iω) is a vector sum of unit vector and the
vector G( iω)H( iω), 1 + G( iω)H( iω) is identical to the vector drawn from the − 1 + 0i
point to the terminal point of the vector G( iω)H( iω), as shown in Fig. 2.8. Usually j
or i is used as complex operator. In Fig. 2.8 j is used as complex operator.
Encirclement of the origin by the graph of 1 + G( iω)H( iω) is identical to en-
circlement of the − 1 + i0 point by just the G( iω)H( iω) locus. Thus, stability of a
closed loop system can be investigated by examining encirclements of the − 1 + 0i
point by the locus of G( iω)H( iω).
The number of clockwise encirclement of the − 1 point by the locus can be found
by drawing a vector from the − 1 point to the GH locus starting from ω = −∞, going
through ∞ = ∞ and by counting the number of clockwise rotation of the vector. Note
that the locus of the function when ω goes from infinity to 0 is the mirror image of