Search for notes by fellow students, in your own course and all over the country.
Browse our notes for titles which look like what you need, you can preview any of the notes via a sample of the contents. After you're happy these are the notes you're after simply pop them into your shopping cart.
Document Preview
Extracts from the notes are below, to see the PDF you'll receive please use the links above
Nyquist Stability Criterion
A stability test for time invariant linear systems can also be derived in the
frequency domain
...
It is based
on the complex analysis result known as Cauchy’s principle of argument
...
By applying
Cauchy’s principle of argument to the open-loop system transfer function,
we will get information about stability of the closed-loop system transfer
function and arrive at the Nyquist stability criterion (Nyquist, 1932)
...
These stability margins are
needed for frequency domain controller design techniques
...
The Nyquist method is used for
studying the stability of linear systems with pure time delay
...
Since the system poles are determined as those values at which its transfer
function becomes infinity, it follows that the closed-loop system poles are
obtained by solving the following equation
which, in fact, represents the system characteristic equation
...
In addition,
it is easy to see that the poles of
time the poles of
are the zeros of
...
The Nyquist stability test is obtained by
applying the Cauchy principle of argument to the complex function
First, we state Cauchy’s principle of argument
...
Cauchy’s Principle of Argument
Let
be an analytic function in a closed region of the complex
plane
given in Figure 4
...
It is also assumed that
point on the contour
...
6), with
where
is analytic at every
and
times (see
given by
stand for the number of zeros and poles (including their
multiplicities) of the function
inside the contour
...
¤
Im{F(s)}
Re{F(s)}
Re{s}
+
+ +
+
+
+
Im{s}
¢
Z=3
P=6
N= -3
¥
¡
s-plane
F(s)-plane
Figure 4
...
7
...
7: Contour in the -plane
The contour in this figure covers the whole unstable half plane of the
complex plane ,
...
Nyquist Stability Criterion
It states that the number of unstable closed-loop poles is equal to the
number of unstable open-loop poles plus the number of encirclements of
the origin of the Nyquist plot of the complex function
...
7
...
At the same time, the
zeros of
are the closed-loop system poles, and the poles of
the open-loop system poles (closed-loop zeros)
...
The number of unstable closed-loop poles (Z) is equal to the number of
unstable open-loop poles (P) plus the number of encirclements (N) of the
point
of the Nyquist plot of
, that is
Phase and Gain Stability Margins
Two important notions can be derived from the Nyquist diagram: phase
and gain stability margins
...
8
...
8: Phase and gain stability margins
They give the degree of relative stability; in other words, they tell how far
the given system is from the instability region
...
8 are obtained as
and
Example 4
...
This contour
has three parts (a), (b), and (c)
...
into
with
%$
"
#!
form by
is represented in the polar
, we easily see that
%$
(a) On this semicircle the complex variable
...
"
&!
Thus, the huge semicircle from the -plane maps into the origin in the
-plane (see Figure 4
...
-1
(b)
(a)
ω= +
-
8
+
Im{G(s)H(s)}
(c)
Re{s}
B
A (b)
A
(a)
'
(c)
ω=0-
§
Im{s}
(c)
Re{G(s)H(s)}
(c)
ω=0+ B
Figure 4
...
23
(b) On this semicircle the complex variable
with
10
10
)
&(
form by
is represented in the polar
, so that we have
)
#(
10
changes from
at point A to
10
Since
at point B,
to
32
32
will change from
...
(c) On this part of the contour
changing from
to
...
We can
4
with
takes pure imaginary values, i
...
find the real and imaginary parts of the function
are given by
, which
3
3
From these expressions we see that neither the real nor the imaginary
parts can be made zero, and hence the Nyquist plot has no points of
B and since the plot at
5
intersection with the coordinate axis
...
9
...
9 is given by
since at those points
...
e
...
The Nyquist plot is drawn by using the MATLAB function nyquist
num=1; den=[1 1 0];
nyquist(num,den);
axis([-1
...
5 —10 10]);
axis([-1
...
2 1 1]);
The MATLAB Nyquist plot is presented in Figure 4
...
It can be seen
, which implies that
Also, from the same figures it follows that
8
7
from Figures 4
...
9 that
...
In order to find
the phase margin and the corresponding gain crossover frequency we use
the MATLAB function margin as follows
[Gm,Pm,wcp,wcg]=margin(num,den)
producing, respectively, gain margin, phase margin, phase crossover frequency, and gain crossover frequency
...
0
...
6
4
0
...
2
Imag Axis
1
8
Imag Axis
10
0
0
−2
−0
...
4
−6
−0
...
8
−10
−1
−0
...
5
Real Axis
0
Figure 4
...
23
A @
gain crossover frequency are obtained as
Example 4
...
For cases (a) and (b) we have the same analyses and conclusions
...
If we find the real and imaginary parts of
, we get
B
B
B
B
B
B
B
It can be seen that an intersection with the real axis happens at
...
11
...
12
...
11: Nyquist plot for Example 4
...
2
8
0
...
1
4
0
...
05
−4
−0
...
15
−8
−10
−1
...
5
Real Axis
0
0
...
2
−1
−0
...
12: MATLAB Nyquist plot for Example 4
...
Thus, we have
, and
so that the closed-
loop system is stable