16.06 Principles of Automatic Control
Lecture 22
Nyquist Plot for Gpsq with jω´axis poles
Consider
Gpsq “
1
sps ` 1q2
Because of pole at s “ 0, must deform “D contour” pC 1 q.
Im(s)
C1
Re(s)
1
Bode:
Magnitude
50
0
−50
−100
−150 −2
10
10
−1
10
0
10
1
10
2
Phase (deg)
0
−100
−200
−300
10
−2
10
−1
0
10
Frequency, ω (rad/sec)
10
1
10
2
Nyquist:
1.5
Im(G)
ω=0-
1
ω=1
0.5
Re(G)
0
ω=∞
0.5
−0.5
−1
−1.5
ω=0+
−2
−1
0
1
2
Note that deformation in contour (small semicircle in C1 ) maps to large semicircle in GpC1 q.
Since there are no open loop poles inside C1 , the number of closed loop poles is
2,
1,
if ´ 0.5 ă ´1{k ă 0
if 0 ă ´1{k ă 8
pk ą 2q
pk ă 0q
This result is of course in agreement with Routh, root locus.
2
A note on drawing the Nyquist diagram:
As ω Ñ 0` , note that the Nyquist diagram is asymptotic to the vertical line Repsq “
´2. Since the phase at zero frequency goes to ´90˝ , it seems that the diagram should be
asymptotic to the imaginary axis. Why isn’t it?
Express Gpsq as:
Gpsq “
1
1
1
“
¨
s s2 ` 2s ` 1
sps ` 1q2
For small s, can express as series around s “ 0:
1
Gpsq « p1 ´ 2s ` Ops2 qq
s
1
“
´ 2 ` Opωq
jω
So the diagram is asymptotic to
1
jω
´ 2.
Nyquist Plot of Open Loop Unstable System
Now consider the proportional control of an unstable system:
r
+
k
-
-
s+1
s(1 - s/10)
The root locus:
Im(s)
10
5
Re(s)
0
10
-1
−5
−10
−20
−15
−10
−5
0
3
5
10
15
Bode:
Magnitude (dB)
10
10
10
2
0
−2
10
−2
10
−1
10
0
10
1
10
2
Phase (deg)
100
0
−100 −2
10
10
−1
0
10
Frequency, ω (rad/sec)
10
1
10
2
Nyquist diagram:
ω=0+
Im(G)
2
1.5
1
0.5
0
Re(G)
ω=∞
-1.1
−0.5
−1
−1.5
−2
ω=0−4
−3
−2
−1
0
1
Note that arc at 8 is clockwise, because deformation at s “ 0 around pole is counter­
clockwise.
Since there is one open loop pole in right hand plane, need one counter-clockwise encirclement
for stability.
Z “N ` P
0“´1`1
4
where 0 means no closed loop poles,
´ 1 means counter-clockwise encirclement,
` 1means right-half-plane open-loop or pole.
So system is stable for:
´1 ă ´ 1{k ă 0
ñ k ą1
Also, note that for ´1{k ă ´1 p0 ă k ă 1q,
poles is:
N “ 1, so the number of unstable closed-loop
Z “N ` P
“1 ` 1
“2
5
MIT OpenCourseWare
http://ocw.mit.edu
16.06 Principles of Automatic Control
Fall 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.