Why use the Nyquist Criterion?
Answers the questions:
Q1. Are there any closed-loop poles in the RHP?
Nyquist Stability Criterion
Q2. If the answer to Q1 is yes, then how many?
M. Sami Fadali
Professor of Electrical Engineering
University of Nevada
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
( )
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1
Component Vectors
Closed-loop Characteristic
Equation
• Zeros of
are closed-loop poles
• Poles of
are open-loop poles
2
• For any value of the complex number ,
each component is a vector
• Pole/zero at 
s

3
Contour Mapping
Contour
• Contour = closed directed simple (does not
cross itself) curve.
• Consider the angle change for
as
traverses a known contour D.
• Change in
= sum of angle changes for its zeros
− sum of angle changes of its poles.
D
5
6
Angle Change for
Pole/Zero Location
• Zero net angle change for pole/zero outside the
contour.
encirclement =
net angle change
for zero/pole inside the contour, respectively.
s+a
b s + b
Zeros/Poles of
=closed-loop/open-loop poles
No. Encirclements of origin = (No. closed-loop
poles inside contour)  (No. open-loop poles
inside contour)
D
a
7
8
Nyquist Contour
Mapping of
• Encloses all RHP poles and zeros of
Assume real coefficients
• Plot
• Use net angle change to determine stability
• Plot
= polar plot.
= complex conjugate of polar plot
= mirror image of polar plot
, large |s|
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10
Principle of the Argument
Stability Results
• If F(s) = 1 + L(s) has Z zeros and P poles
inside the Nyquist contour (in RHP), a plot of
F(s) as s travels once (clockwise) around the
contour encircles the origin of the complex
plane in which it is plotted (N) times where
(N) = Z  P
= no. of clockwise encirclements
i.e. N = P  Z
= no. of counterclockwise encirclements
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(i)
For stability (no closed-loop poles in the RHP)
Z=0 i.e. N = P
For an open-loop stable system (P = 0), N = 0
(ii) For an unstable system (Z  0)
Z=PN
= number of closed-loop poles in the RHP
(iii) Number of encirclements of the origin by
F(s) = 1 + L(s)
= number of encirclements of (1,0) by L(s)
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Nyquist Plot
Nyquist Stability Criterion
Unity shift
Necessary and sufficient condition for the
closed-loop stability of the loop gain L(s) is:
Nyquist Diagram
2.5
2
= no. of counterclockwise encirclements
of (1,0) by L(s)
1
0.5
0
= no. of RHP open-loop poles of L(s)
-0.5
-1
-1.5
Note: Closed-loop stability is when
-2
-2.5
-1
0
1
2
Real Axis
3
4
5
13
14
Open-loop Unstable System
RHP Closed-loop Poles
K  1: N  0
K  4: N 1
1. For an unstable system, the number of closedloop system poles in the RHP = Z = P  N
2. For open-loop stable systems (P = 0)
(a) The system is stable with no (1,0)
encirclements
(b) Number of RHP poles for an unstable system
=  N = number of clockwise encirclements of
(1,0)
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Z 1
Z 0
Nyquist Diagram
1
K=4
0.8
0.6
0.4
Imaginary Axis
Imaginary Axis
1.5
K=1
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-2
-1.8
-1.6
-1.4
-1.2
-1
Real Axis
-0.8
-0.6
-0.4
-0.2
0
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Root Locus: Open-loop Unstable System
Nyquist Plot: 3rd Order, Type 0
Root Locus
1
0.8
Nyquist Diagram
10
0.6
0.2
6
-0.2
-0.4
Stable for
-0.6
= 12
4
0
Imaginary Axis
Imaginary Axis
8
System: g
Gain: 2
Pole: 0.000592
Damping: -1
Overshoot (%): 0
Frequency (rad/sec): 0.000592
0.4
2
0
-2
K=2
-4
-6
-0.8
-1
-2.5
-8
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Real Axis
17
-10
-4
-2
0
2
4
6
8
10
12
18
Real Axis
Root Locus: 3rd Order, Type 0
Simplified Nyquist Criterion
Root Locus
7
• Assume that the loop gain function has no RHP
open-loop poles (i.e. P = 0).
• Observer follows the polar plot of the loop gain in
the direction of increasing frequency.
• The closed-loop system is stable if and only if the
point (1,0) is to the left of the observer.
• Remark: The criterion implies no encirclements
of (1,0) by the complete Nyquist plot.
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System: g
Gain: 10
Pole: 0.000796 + 3.32i
Damping: -0.00024
Overshoot (%): 100
Frequency (rad/sec): 3.32
5
4
Imaginary Axis
System: g
Gain: 12
Pole: 0.12 + 3.53i
Damping: -0.0339
Overshoot (%): 111
Frequency (rad/sec): 3.5
3
2
System: g
Gain: 2
Pole: -0.781 + 1.86i
Damping: 0.388
Overshoot (%): 26.7
Frequency (rad/sec): 2.01
1
0
-1
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Real Axis
19
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Example: Unstable System
=
+1
12
/2 + 1
Imaginary Poles
/3 + 1
• Add a small semicircle around the origin for
systems of type  1
• The small semicircle is mapped to l infinite
semicircles traversed clockwise for a type l
system.
• Similarly, add a small semicircle around
any imaginary axis pole.
Nyquist Diagram
1
0
-1
Imaginary Axis
-2
-3
-4
-5
-6
-7
-8
-9
-4
-2
0
2
4
Real Axis
6
8
10
12
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Large Semicircles
Modified Nyquist Contour
• For small
• Net rotation for
• Denominator angle =
• Transfer function angle =
( clockwise half circles)
Discontinuities
Poles outside contour
Poles inside contour
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Nyquist Plot: Type II, P = 0
Nyquist Plot: Type I, P = 0
=
K
ss 2  1
3
Nyquist Diagram
2
0
1.5
2
1
1
0.5
Imaginary Axis
Imaginary Axis
L( s ) 
 = 0+
Nyquist Diagram
4
0
0
K
L( s )  -0.52
s-1 s 2  1
-1
-2
 = 0+
-3
-4
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-1.5
-0.2
-0.1
 = 0
0
Real Axis
-2
-35
-30
-25
-20
-15
-10
-5
0
Real Axis
N 0
Type I: one clockwise half circle
= 0 (no RHP open-loop poles)
Z 0
Type II: two clockwise half circles
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Simplified Nyquist: Type I
P0
N 2 Z 2
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Simplified Nyquist: Type II
+
0
1.6
-1
1.4
-2
1.2
-3
1
-4
=
-5
P=0
N=0
Stable
/2 + 1
-6
-7
=
0.8
N≠0
Unstable
/2 + 1
0.6
0.4
-8
0.2
-9
-10
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
-10
0
27
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
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Nyquist Plot: Variable K
Nyquist Plot: Variable K
• Plot frequency response for a gain K = 1.
• Count encirclements of the point 1/K.
• Closed-loop characteristic equation
Nyquist Diagram
0.2
0.15
• Divide equation by
0.1
Imaginary Axis
0.05
• Repeating earlier analysis gives the same
replace by the
results with the point
point
• Point moves as varies.
System: g
Real: -0.0164
Imag: 6.88e-005
Frequency (rad/sec): -3.35
0
-0.05
-0.1
1/Kcr = 1/60
-0.15
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-0.2
-0.05
0
0.05
0.1
Real Axis
0.15
0.2
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