Introduction of Cauchy Argument Principle in Nyquist Stability Analysis By Yong-Nien Rao In Nyquist stability criterion, a very famous and key equation is N = Z − P N
…(eq.1.1)
is the net number of clockwise encirclement, Z is the number of zeros and
P is the number of poles
We will show how we could get this equation mathematically. mathematically. Winding number 12 The intuitive description in winding number of the curve is equal to the total number of counterclockwise rotations that the object makes around a point. C
C
Z0
C
Z0
Z0
N(C,Z 0)= -2
N(C,Z0)= -1
N(C,Z 0)= 0
C
C
Z0
N(C,Z 0) is the winding number of curve C about a point Z 0
Z0
N(C,Z0)= 2
N(C,Z0)= 1
In complex analysis, the winding number of a closed curve
C
in the complex
plane ( z -plane) about a point z 0 , denoted n( C , z 0 ) , is defined by n( C , z 0 ) ≡
1 2π i
dz
∫ z − z C
0
…(eq.2.1)
For simplicity, simplicity, we consider the z 0 = 0 firstly. Let 1 2
http://en.wikipedia.org/wiki/Winding_number http://mathworld.wolfram.com/ContourWindingNumber.html
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n( C ,0) =
1
dz
2π i
∫ z
…(eq.2.2)
C
Since z = re dz = e iθ dr + ire iθ d θ iθ
Therefore dz dr = + id θ = d [ ln ( r ) ] + id θ z r Thus right side of (eq.2.1) could be 1 n( C ,0) = d [ ln( r ) ] + id θ …(eq.2.3) C 2π i C
{∫
∫ }
Because C is a closed curve, the total change of d [ ln ( r ) ] is zero. Thus (eq.2.3) could be further reduced to 1 1 n( C ,0) = id θ = d θ 2π i C 2π C
∫
∫
The graphic interpretation of could be
m * 2π
…(eq.2.4)
∫ d θ θ C C
is just the total argument change of curve
for m is any integer because
∫ d θ = m * 2π C
C
It
is a closed curve. That is,
…(eq.2.5)
(Note: m could be negative. For example, when the clockwise,
C .
C
is a circle oriented
m = −1 )
From (eq.2.4) and (eq.2.5), we get 1 n * 2π n( C ,0 ) = d θ = =m …(eq.2.5) 2π C 2π
∫
This is just the intuitive description of winding number. We prove (eq.2.2) To generalize the case to z 0 ≠ 0 Let p = z − z 0 Then dp = dz 1 ( ) , = n C z 0 And (Eq.2.1) could be 2π i
© Copyright 2009 by Yong-Nien Rao
dp
∫ p C
= m (from (eq.2.2)).
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The Argument Principle34 Let f be analytic in a domain D except at a finite number of poles. Let CA be a piecewise-smooth simple closed curve oriented counterclockwise in D not passing through any zeros or poles of f . Then 1 f ′( z ) dz = Z − P …(eq.3.1) 2π i CA f ( z )
∫
where Z is the number of zeros of f enclosed by CA in the z -plane and P is the number of poles of f enclosed by CA in the z -plane.
Get the N=Z-P equation Check the left side of (eq.3.1), it is just the case of (eq.2.2) in the f ( z ) -plane (not z -plane). It is the winding number of a closed curve C (Note: C is the curve of f ( z ) over CA in the f ( z ) -plane; In contrast, CA is the curve in the z -plane.) Thus we get n( C ,0 ) = Z − P
…(eq.3.2)
(eq.3.2) is very similar to (eq.1.1) except the orientation. The difference is the encirc encirclem lement ent is clockw clockwise ise in (eq.1. (eq.1.1). 1). If we define define the counte counterclo rclockw ckwise ise encirclement
N ′ ,
we get
− N ′ = N
…(eq.3.3)
The N ′ is just the winding number, number, that is, N ′ = n( C ,0) …(eq.3.4) In (eq.3.2), curve is counterclockwise. This, however, would not be a problem. It is because in (eq.1.1), an unmentioned precondition is the Nyquist path, which is also oriented clockwise. Nyquist path is just as the And it refers to
− CA
in (eq.3.1).
− C
From the complex analysis, the winding number holds n( − C ,0 ) = −n( C ,0) …(eq.3.5) We derive − n( C ,0 ) = Z − P
…(eq.3.6)
from (eq.3.2) and (eq.3.5) for the Nyquist N yquist analysis case And from (eq.3.3), (eq.3.4) and (eq.3.6), we get N = Z − P Q.E.D. 3
Detail prove involves many complex analysis theory, please refer to Peter V. V. O’Neil, “Advanced Engineering Mathematics”, Third Edition, Chap 19~Chap22. 4 http://en.wikipedia.org/wiki/Argument_principle
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