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Modelling and control
summaries
by Anthony Rossiter
Nyquist 11: Concepts of encirclements
The full definition of a Nyquist diagram is the mapping of G(s) while s describes the D-contour.
We need to decide whether a given point (usually -1) lies inside or outside of a
Nyquist diagram. If inside, how many times does the diagram go around?
Clearly outside
Simple illustration
Think of the Nyquist diagram like a string. Can
a given point escape without cutting the
string? This allows the user to move and bend
the string however they wish.
Number of encirclements is number of times
the point needs to cross the string to escape.
Clearly
inside
OUTSIDE! Can
unwrap string so
point escapes
without crossing.
Encircled
twice
Encircled
once
NEXT: Look at some examples of Nyquist diagrams and count the encirclements of
the ORIGIN and dependence on pole/zero positions.
SYSTEMS WITH STABLE POLES GIVE ZERO ENCIRCLEMENTS OF THE ORIGIN AS SEEN BY NEXT TWO
EXAMPLES. G=1/(s+0.2) AND G=10/(s+2)(s+4)
One can use logic that if (s+a) gives no encirclements of origin (this is D-contour shifted to right by
a), then clearly 1/(s+a) cannot give encirclements as they have opposite arguments.
Right hand
turn
3
2
Represent right
hand turns and skip
around origin.
0.8
0.6
0.4
Right hand turns –
skip around origin
0
0.2
imag
imag
1
0
-0.2
-1
-0.4
-2
-3
0
1
2
3
-0.6
0 encirclements
-0.8
of origin
5
-0.5
4
0
0.5
real
1
1.5
real
SYSTEMS WITH UNSTABLE POLES GIVE ANTI-CLOCKWISE ENCIRCLEMENTS OF THE ORIGIN AS SEEN
BY NEXT TWO EXAMPLES. G=1/(s-0.2) AND G=10/(s-4)(s+2)
One can use logic that if (s-a) gives clockwise encirclement of the origin (this is D-contour shifted
to LEFT by a), then clearly 1/(s-a) gives anti-encirclements as they have opposite arguments.
Represent right
hand turns and
skip around origin
3
2
0.3
0.2
0.1
1 RHP pole and
1 encirclement
imag
imag
1
1 RHP pole and
1 encirclement
0
0
-0.1
-0.2
-1
-0.3
-2
-0.4
-3
-5
-4
-3
-2
-1
0
1
real
-1.2
-1
-0.8
Right hand turns
-0.4 skip
-0.2
0
and
around
real
origin
-0.6
SUMMARY: Allowing s to follow D-contour, it is clear that when plotting the Nyquist diagram:
1. LHP poles and zeros result in no net change in angle and no encirclements (of the origin).
2. RHP zeros give rise to clockwise encirclements of the origin.
3. RHP poles give rise to anti-clockwise encirclements of the origin.
RULES OF COMPLEX NUMBERS
When multiplying complex numbers, the phases add. Therefore the total number of
encirclements N of the origin (equivalent to the total change in phase as s describes the Dcontour) is given by (Note this is an interim result to be extended next):
N = number of RHP zeros minus the number of RHP poles