Page 855 - The Mechatronics Handbook

0066_frame_C27 Page 16 Wednesday, January 9, 2002 7:10 PM

2. Determine L(jω) as ω → ∞ : L(s) has a relative degree of 1 and

1
Ljw) ≈ ----- = 0 – 90°
(
∠
jw
according to Table 27.2.
+
3. From the Bode plot, draw the polar plot of L(jω) as ω varies from 0 to ∞. Although the magnitude
2
curve of the factor (s − 2s + 2) is the same as the factor (s + 2s + 2), the phase of the factor (s −
2
2
2s + 2) changes from 0° to −180°. Thus, a sketch of the Bode diagram shows that the magnitude
curve varies from infinity to zero and the phase changes from −90° to −450°. Since there are two
points at which the phases are −180° and −360°, there will be two intersections of the L(jω) locus
with the real axis in the L(s)-plane.
−
4. Draw the polar plot of L(jω) as ω varies from 0 to −∞ by reflecting the curve of L(jω) in procedure
3 with respect to the real axis in the L(s)-plane.
5. Determine the contour map of the small detour around the origin of the s-plane to complete the
plot. On the detour,
s = lim ee , – 90° ≤ q ≤ 90°
jq
e→0
The contour map of the detour can then be determined by

jq 2
jq
( ee ) – 2ee + 2 1 1
jq
(
lim L ee ) = lim --------------------------------------------------------- = lim --------- = lim -- –∠ q
jq
(
jq
jq
e→0 e→0 ee ( ee + 1) ee + 2) e→0 ee jq e→0 e
The resulting map is a large semicircle of radius approaching infinity. This semicircle starts at the
− +
point L(j0 ) and swings 180° in the counterclockwise direction to connect the point L(j0 ) in the
L(s)-plane.
6. Calculate the intersections of the L(jω) locus with the real-axis, for these points are related to the
relative stability of the system. Suppose that the L(jω) locus intersects the real axis for some critical
frequency ω cr . Then
 180° + k360°, for K > 0
Ljw cr ) = 
(
 0° + k360°, for K < 0
where k = 0, ±1, ±2, ±3,…. This phase condition at the critical frequency is directly related to the
angle condition of the root locus when the root locus crosses the imaginary axis. Therefore, we can
utilize the Routh–Hurwitz criterion to determine the points where the L(jω) locus crosses the real
axis. The characteristic equation of the system (27.13) can be written as
3
2
s + ( K + 3)s + ( 22K)s + 2K = 0
–
Thus, the Routh array is
3
s 1 2 2K
–
s K + 3 2K
2
1
s c
0
s 2K
where

(
( K + 3) 22K) 2K
–
–
c = ----------------------------------------------------
K + 3
©2002 CRC Press LLC