Nyquist Stability Criterion
A stability test for time invariant linear systems can also be derived in the frequency domain. It is known as Nyquist stability criterion. It is based on the complex analysis result known as Cauchy's principle of argument. Not
Evans Rules for Sketching the Root Locus
Absolute and Relative Stability. A control system is called absolutely stable if the controlled transfer
function Hd (s) from the reference signal yd (s) to the output signal y(s) has all of its poles in the open
l
Matrix exponential
Contents
1 Computing matrix exponential for diagonalizable matrices
2 Computing matrix exponential for general
2.1 Using Jordan normal form . . . . . . . . .
2.2 Using CayleyHamilton theorem . . . . . .
2.3 Using numerical integration .
2
Performance
In this course we will be considering systems of the form of Fig. 2.1.
The transfer function P (s) represents the plant or system to be controlled;
the feedback sensor F (s) represents the system that is measuring the output
signals in the s
32
2 Performance
2.3 The RouthHurwitz Criterion
We have shown that to check for internal stability of a closedloop system,
it is enough to consider the locations of the roots of a single polynomial
the characteristic polynomial. In this section we show
1
Controllability and observability
Consider the system
x(t) = Ax(t) + Bu(t),
x(0) = x0
(1.1)
where the state x(t) is an nsize vector. We will assume that there is only one
input. This means that A Rnn and B Rn1 .
Recall that the solution to Eq. 1.1 is g
16
1 Modeling
1.3 Transfer Functions
In this section we look at nding transfer functions from one signal to another.
We begin by considering the following example:
y (t) + a1 y(t) + a0 y(t) = b1 r(t) + b0 r(t).
(1.13)
Taking Laplace transforms of both sid
WV! 10%
sior. 1
I]  _ l I 1IJ5\n£h%
nn.nr_.\
P qu . \r . If a
 H.300

L I )f 6  v
1 } F '
\. ;
' '7 _,_ _ jet?
. p = a".
am
y C 1" Q
g. ._I i. . 1
'4. n I: a u I .3. c u . . .
Q
a 5 L. o I; I a u or .c A
I, 
km 5: thus a, but
I l a
The Johns Hopkins University
Electrical & Computer Engineering Department
520.353  Control Systems
Spring 2013
Solutions to Homework #2
1. (a) Using Euler's formula, we have
sin(at) =
1 iat iat
(e e )
2i
where i is the imaginary unit.
Let v denote the un
3
Classical analysis
Classical analysis assumes that the transfer function is given by either a rstG(s) =
p
s+p
or secondorder system:
G(s) =
2 + 2
.
(s + )2 + 2
Note that in both cases, the numerator has no zeros. Also, the transfer functions are normal
%=
=
% RouthHurwitz Stability Criterion
%
% The RouthHurwitz criterion states that the number of roots of D(s) with
% positive real part is equal to the number of changes in sign of the first
% column of the root array.
%
% The necessary and sufficient
l l ' 1. (20 points) True for False. Determine if the following statements are true. If true,
_ 23 I give EL proof. If not1 give a counterexample.
c" l x 
(1). Any two noneidentity elements in 83 generate 83.  TRU E .)
l 2 3 l 2 3 ' .
2 I 3
LE;
101. {3
«u3&3
Mann 1 MA
_ 'o.
5cm; w
0 mk Mud
«B \{M hunt 3%?!
1 AW kaak. fa» \ CmrMMkQc
. ..  Lc \
x
I
m, " o o o
o 1 o T o A.
./
¢~¢ . .x~wtew
_, L, a . . L .5 ME So 1";
L a»;  L * . a. ~54
9 _. N 4/ E. Ii. 5 1
L» t h: _ . a '3 9.
= 3N N
Control Systems
Bode plots
L. Lanari
Monday, November 3, 2014
Outline
Bodes canonical form for the frequency response
Magnitude and phase in the complex plane
The decibels (dB)
Logarithmic scale for the abscissa
Bodes plots for the different contributions
Control Systems
Nyquist stability criterion
L. Lanari
Sunday, November 9, 2014
Outline
polar plots when F(j!) has no poles on the imaginary axis
Nyquist stability criterion
what happens when F(j!) has poles on the imaginary axis
general feedback system
st
The Johns Hopkins University
Electrical & Computer Engineering Department
520.353  Control Systems
Spring 2013
Solutions to Homework #3
1. (a) The zeroes of
H(s) =
N (s)
s(s + 1)
=
D(s)
(s + 3)(s + 2)2
are the roots of the numerator polynomial N (s) and