ME 304
CONTROL SYSTEMS
Mechanical Engineering Department,
Middle East Technical University
Radar Dish
Armature
controlled
dc motor
Outside
θD output
Inside
θr input
p
θm
Gearbox
Control
Transmitter
θD
dc amplifier
ME 304 CONTROL SYSTEMS
Control
Transformer
Prof.
Prof Dr.
Dr Y.
Y Samim Ünlüsoy
Prof. Dr. Y. Samim Ünlüsoy
1
CH IX
COURSE OUTLINE
I.
II.
III.
INTRODUCTION & BASIC CONCEPTS
MODELING DYNAMIC SYSTEMS
CONTROL SYSTEM COMPONENTS
IV.
V.
VI.
VII.
VIII.
STABILITY
TRANSIENT RESPONSE
STEADY STATE RESPONSE
DISTURBANCE REJECTION
BASIC CONTROL ACTIONS & CONTROLLERS
IX.
FREQUENCY RESPONSE ANALYSIS
X.
XI.
SENSITIVITY ANALYSIS
ROOT LOCUS ANALYSIS
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
2
FREQUENCY RESPONSE - OBJECTIVES
In this chapter :
„
„
„
A short introduction to the steady
state response
p
of control systems
y
to
sinusoidal inputs will be given.
Frequency domain specifications for
a control system will be examined.
Bode plots and their construction
g asymptotic
y p
approximations
pp
using
will be presented.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
3
FREQUENCY RESPONSE – INTRODUCTION
Nise
Ni Ch
Ch. 10
„
„
„
In frequency response analysis of control
systems, the steady state response of the
system to sinusoidal input is of interest.
The frequency response analyses are
carried out in the frequency
q
y domain,
domain,
rather than the time domain.
It is to be noted that, time domain
properties of a control system can be
predicted from its frequency domain
characteristics..
characteristics
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
4
FREQUENCY RESPONSE - INTRODUCTION
„
For an LTI system the Laplace transforms
off the
th input
i
t and
d output
t t are related
l t d to
t each
h
other by the transfer function, T(s).
Laplace Domain
Input
„
R(s)
T(s)
C(s)
Output
In
I the
th frequency
f
response analysis,
l i the
th
system is excited by a sinusoidal input of
fixed amplitude and varying frequency.
frequency.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
5
FREQUENCY RESPONSE - INTRODUCTION
„
Let us subject
j
a stable LTI system
y
to a
sinusoidal input of amplitude R and
frequency
q
y
ω in time domain.
r(t)=Rsin(ωt)
„
The steady state output of the system will
be again a sinusoidal signal of the same
frequency,, but probably with a different
frequency
amplitude and phase.
phase.
c(t)=Csin(ωt+φ)
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
6
FREQUENCY
Q
RESPONSE - INTRODUCTION
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
7
FREQUENCY RESPONSE - INTRODUCTION
„
To carry
y out the same process
p
in the
frequency domain for sinusoidal steady
state analysis, one replaces the Laplace
variable
i bl s with
i h
s=jjω
in the input output relation
C(s)=T(s)R(s)
C(s) T(s)R(s)
with the result
C(jω)=T(j
) T(jω)R(jω)
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
8
FREQUENCY RESPONSE - INTRODUCTION
„
The input, output, and the transfer
function have now become complex and
thus they can be represented by their
magnitudes and phases.
„
Input :
R(jω)= R(jω) ∠R(jω)
„
Output :
C(jω)= C(jω) ∠C(jω)
C(jω)
Transfer
Function :
T(jω)=
(j ) T(jω)
(j ) ∠Τ(jω)
(j )
„
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
9
FREQUENCY RESPONSE - INTRODUCTION
„
With similar expressions for the input
and
d th
the ttransfer
f
ffunction,
ti
th
the iinputt
output relation in the frequency
domain consists of the magnitude and
phase expressions :
C(jω)=T(jω)R(jω)
C(jω) = T(jω)
(jω) R(jω)
(jω)
∠C(jω)= ∠T(jω)+ ∠R ( jω)
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
10
FREQUENCY RESPONSE - INTRODUCTION
„
For the input and output described by
r(t)=Rsin(ωt)
c(t)=Csin(ωt+φ)
the amplitude and the phase of the
output can now be written as
C =R T(jω)
φ=∠
∠T(jω)
T(jω)
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
11
FREQUENCY RESPONSE
„
Consider the transfer function for the
general closed loop system.
system
C(s)
G(s)
T(s)=
=
R(s) 1+G(s)H(s)
For the steady state behaviour,
behaviour insert s=jω.
C(jω)
G(jω)
T(jω)
)=
=
R(jω) 1+G(jω)H(jω)
T(jω) is called the Frequency Response
Function (FRF) or Sinusoidal Transfer
Function..
Function
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
12
FREQUENCY RESPONSE
„
The frequency response function can be
written in terms of its magnitude and
phase.
T(jω)
)= T(jω) ∠T(jω)
Since this function is complex,
p
, it can also
be written in terms of its real and
imaginary parts.
T(jω)= Re [ T(jω)] + jIm[ T(jω)]
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
13
FREQUENCY RESPONSE
„
„
Remember
R
b
that
th t for
f
a complex
l
number
b
be
b
expressed in its real and imaginary parts :
the magnitude is given by :
z=
„
z = a+bj
( a+bj)( a - bj) =
the phase is given by :
2
2
a +b
-1 b
∠z
∠
z = tan
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
a
14
FREQUENCY
Q
RESPONSE
„
The magnitude and phase of the
frequency response function are
given by :
G(jω)
G(jω)
T(jω) =
=
1+G(jω)H(jω) 1+G(jω)H(jω)
∠T(jω)= ∠G(jω) - ∠ [1+G(jω)H(jω)]
These are called the gain and phase
characteristics.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
15
FREQUENCY RESPONSE – Example
p 1a
„
For a system described by the
diff
differential
ti l equation
ti
+2x = y(t)
x
determine the steady state response
xss(t) for a pure sine wave input
y(t)= 3sin(0.5t)
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
16
FREQUENCY RESPONSE – Example
p 1b
„
The transfer function is given by
X(s)
1
T(s)=
=
Y(s) s ( s +2 )
+2x = y(t)
x
Insert s=jω to get :
For
1
T(jω)=
jω ( jω +2 )
ω=0.5
0 [rad/s]:
[ d/ ]
1
1
T(0.5j)=
=
0.5j ( 0.5j+2 ) -0.25 + j
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
17
FREQUENCY RESPONSE – Example
p 1c
„
Multiply and divide by the complex conjugate.
⎛
⎞ ⎛ -0.25 - j ⎞
1
-0.25 - j
T(0.5j) = ⎜
⎟⎜
⎟=
⎝ -0.25 + j ⎠ ⎝ -0.25 - j ⎠ 1+0.0625
T(0.5j) = -0.235 - 0.941j
„
Determine the magnitude and the angle.
T(0.5j) =
cos - cos +
sin + sin +
76o
cos sin -
cos +
sin -
2
( -0.235 )
2
+ ( -0.941) = 0.97
-1
1
∠T(0.5j)
T(0 5j) = tan
t
-0.941
0.941
= -104
104o
-0.235
-104o
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
18
FREQUENCY RESPONSE – Example
p 1d
„
The steady state response is then given by :
(
xss (t)= 3 ( 0
0.97
97 ) sin 0
0.5t
5t -104
104o
(
= 2.91sin 0.5t -104o
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
)
)
19
FREQUENCY
Q
RESPONSE – Example
p 2a
„
Express the transfer function (input : F,
output : y) in terms of its magnitude and
phase.
+cy +ky =F
my
F
k
c
m
F
ME 304 CONTROL SYSTEMS
y
G(s)=
1
2
ms +cs +k
Prof. Dr. Y. Samim Ünlüsoy
20
FREQUENCY
Q
RESPONSE – Example
p 2b
„
Insert s=jω in the transfer function to
obtain the frequency response function.
G(s)=
T(jω)=
„
1
2
m ( ωjj) +c
+ ( ωjj) +k
=
(
1
ms2 +cs +k
1
)
k - mω2 +cωj
+
j
Write
W i the
h FRF iin a+bj
bj form.
f
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
21
FREQUENCY
Q
RESPONSE – Example
p 2c
„
Multiply and divide the FRF expression with
the complex conjugate of its denominator.
(
)
(
)
T(jω)=
=
2
2
2
2
2
k
mω
+cωj
k
mω
cωj
(
)
(
)
k
mω
+
cω
(
)
(
)
k - mω2 - cωj
1
(
k - mω2 - cωj
)
⎡
⎤ ⎡
⎤
2
k
mω
⎢
⎥ ⎢
⎥
cω
-cω
T(jω)= ⎢
⎥+⎢
⎥j
2
2
⎢ k - mω2 + ( cω )2 ⎥ ⎢ k - mω2 + ( cω )2 ⎥
⎢⎣
⎥⎦ ⎢⎣
⎥⎦
(
)
(
)
T(jω)=Re [ T(jω)] +Im[ T(jω)] j
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
22
FREQUENCY
Q
RESPONSE – Example
p 2d
„
Obtain the magnitude and phase of the
frequency response function.
z = a2 +b2
T(jω) =
(
(
k - mω
)
)
2
2
+ ( cω )
2
⎡
2⎤
2
+ ( cω ) ⎥
⎢ k - mω
⎣
⎦
b
∠z = tan-1
a
ME 304 CONTROL SYSTEMS
2
2
=
1
(
2
k - mω
-1
∠T(jω)= tan
Prof. Dr. Y. Samim Ünlüsoy
)
2
(
2
+ ( cω )
-cω
k - mω2
23
)
FREQUENCY
Q
RESPONSE – Example
p 3a
„
„
„
The open loop transfer
function of a control
system is given as :
G ( s) =
300 ( s + 100 )
s ( s + 10 )( s + 40 )
Determine an expression for the phase angle
of G(jw)
(j ) in terms of the angles
g
of its basic
factors.. Calculate its value at a frequency of
factors
28.3 rad/s.
Determine the expression for the magnitude
of G(jw) in terms of the magnitudes of its
basic factors . Find its value
al e in dB at a
frequency of 28.3 rad/s.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
24
G ( s) =
300 ( s + 100 )
FREQUENCY RESPONSE
– Example 3b
s ( s + 10 )( s + 40 )
∠G(jω) = ∠300 + ∠G(jω + 100) - ∠G(jω) - ∠G(jω + 10) - ∠G(jω + 40)
-1 ⎛
ω ⎞
-1 ⎛ ω ⎞
-1 ⎛ ω ⎞
-1 ⎛ ω ⎞
= 0 + tan ⎜
⎟ - tan ⎜ ⎟ - tan ⎜ ⎟ - tan ⎜ ⎟
⎝ 100 ⎠
⎝0⎠
⎝ 10 ⎠
⎝ 40 ⎠
-1 ⎛
ω ⎞
o
-1 ⎛ ω ⎞
-1 ⎛ ω ⎞
= 0 + tan ⎜
⎟ - 90 - tan ⎜ ⎟ - tan ⎜ ⎟
⎝ 100 ⎠
⎝ 10 ⎠
⎝ 40 ⎠
o
o
-1 ⎛ 28.3 ⎞
∠G(28.3j) = 0 + tan ⎜
o
-1 ⎛ 28.3 ⎞
-1 ⎛ 28.3 ⎞
⎟ - 90 - tan ⎜
⎟ - tan ⎜
⎟
⎝ 100 ⎠
⎝ 10 ⎠
⎝ 40 ⎠
= 0o + 15.8o - 90o - 70.5o - 35.3o = -180o
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
25
G ( s) =
300 ( s + 100 )
s ( s + 10 )( s + 40 )
G ( jω ) =
=
FREQUENCY RESPONSE
– Example 3c
300 jω
j + 100
jω jω + 10 jω + 40
2
2
2
ω + 40
300 ω + 100
2
ω ω + 10
2
2
2
300 28
28.3
3 + 100
G ( 28.3j ) =
2
28.3 28.3 + 10
2
2
2
28.3 + 40
( 300 )(103.9 )
=
= 0.749
( 28.3 )( 30.0 )( 49.0 )
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
26
2
FREQUENCY RESPONSE
„
Typical gain and phase characteristics of a
closed loop system.
|T(jw)|
∠T (jω)
0
Mr
1
0.707
ωr
ME 304 CONTROL SYSTEMS
BW
ω
Prof. Dr. Y. Samim Ünlüsoy
ω
27
FREQUENCY
Q
DOMAIN SPECIFICATIONS
„
Similar to transient response
specifications in time domain,
frequency response specifications are
defined.
- Resonant peak, Mr,
- Resonant frequency, ωr,
- Bandwidth, BW,
- Cutoff
ff Rate.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
28
FREQUENCY
Q
DOMAIN SPECIFICATIONS
„
Resonant p
peak, Mr :
peak,
This is the maximum value of the
transfer function magnitude
|T(jω)|.
|T(jω)|
Mr
1
Mr depends on the damping ratio
ξ only and indicates the relative
stability of a stable closed loop
system.
A large Mr results in a large
overshoot of the step response.
As a rule of thumb, Mr should be
between 1.1 and 1.5.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
ω
ωr
Mr =
1
2ξ 1 - ξ2
29
FREQUENCY
Q
DOMAIN SPECIFICATIONS
„
Resonantt
R
frequency,, ωr :
frequency
|T(jω)|
Mr
1
This is the frequency
at which the resonant
peak is obtained.
ωr = ωn 1 -2ξ2
ωr
ω
Note that resonant frequency is different than both the
undamped
p
and damped
p
natural frequencies!
q
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
30
FREQUENCY DOMAIN SPECIFICATIONS
„
Bandwidth,, BW :
Bandwidth
This is the frequency at
which the magnitude of
the frequency response
function, |T(jω
|T(jω)|, drops to
0 707 of its zero frequency
0.707
value.
„
|T(jω)|
Mr
1
0.707
ωr
BW
ω
BW is directly proportional to ωn and gives an
indication of the transient response
characteristics of a control system.
system The larger
the bandwidth is, the faster the system
responds.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
31
FREQUENCY
QU C DOMAIN
O
S
SPECIFICATIONS
C C
O S
|T(jω)|
„
„
Bandwidth,, BW :
Bandwidth
Mr
1
It is also an indicator off
robustness and noise
filtering characteristics
of a control system.
0 707
0.707
ωr
ωBW = ωn
ME 304 CONTROL SYSTEMS
(
BW
ω
)
1 − 2ξ 2 + 4ξ 4 − 4ξ 2 + 2
Prof. Dr. Y. Samim Ünlüsoy
32
FREQUENCY DOMAIN SPECIFICATIONS
„
Cut--off Rate :
Cut
|T(jω)|
This is the slope of the
magnitude of the frequency
response function,
f
i
|
|T(j
(jω)|,
)|
at higher (above resonant)
frequencies.
frequencies
„
„
It indicates the ability of a
system
t
to
t distinguish
di ti
i h
signals from noise.
Mr
1
0.707
ωr
BW
ω
Two
T
systems having
h i
the
h same bandwidth
b d id h
can have different cutoff rates.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
33
BODE PLOT
Dorf & Bishop Ch.
Ch 8,
8 Ogata Ch
Ch. 8
„
„
The Bode plot of a transfer function is a useful
graphical
hi l tool
t l for
f
the
th analysis
l i and
d design
d i
off
linear control systems in the frequency domain.
The Bode plot has the advantages that
- it can be sketched approximately using
straightline segments without using a
computer.
- relative stability characteristics are easily
determined, and
- effects of adding controllers and their
parameters are easily visualized.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
34
BODE PLOT
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
35
BODE PLOT
Nise Section 10
10.2
2
„
The Bode plot consists of two plots drawn
on semisemi-logarithmic paper.
paper
1. Magnitude of the frequency response
f
function
ti
i decibels,
in
d ib l i.e.,
i
20 log|T(j
g| (jω)|
on a linear scale versus frequency on a
logarithmic scale.
scale.
2. Phase of the frequency response
function on a linear scale versus
frequency on a logarithmic scale.
scale.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
36
BODE PLOT
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
37
BODE PLOT
„
„
„
It is possible to construct the Bode plots
of the open loop transfer functions, but
the closed loop frequency response is not
so easy to plot.
It is also possible, however, to obtain the
closed loop
p frequency
q
y response
p
from the
open loop frequency response.
Thus, it is usual to draw the Bode plots of
the open loop transfer functions.
functions. Then the
closed loop frequency response can be
evaluated from the open loop Bode plots.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
38
BODE PLOT
„
„
It is p
possible to construct the Bode p
plots by
y
adding the contributions of the basic factors
of T(jω
T(jω) by graphical addition.
Consider the following general transfer
function.
P
(
p=1
1
K ∏ 1+Tps
T(s)=
)
⎛
2 ⎞
s
s ⎟
⎜
N
s ∏ (1+ τms ) ∏ ⎜1+2ξ q
+ 2 ⎟
ωnq ω ⎟
m=1
q=1 ⎜
nq ⎠
⎝
M
ME 304 CONTROL SYSTEMS
Q
Prof. Dr. Y. Samim Ünlüsoy
39
P
(
K ∏ 1+ Tp s
T(s)=
p=1
)
⎛
s
s2
⎜
N
s ∏ (1+ τms ) ∏ ⎜1+2ξ q
+
ωnq ω2
m=1
q=1 ⎜
nq
⎝
Q
M
„
BODE PLOT
⎞
⎟
⎟
⎟
⎠
The logarithmic magnitude of T(jω
T(jω)
can be obtained by summation of the
l
logarithmic
ith i magnitudes
it d
off individual
i di id l
terms.
P
log T ( jω ) =logK + ∑ log 1+ jωτp p
N
-log ( jω )
ME 304 CONTROL SYSTEMS
⎛ jω
- ∑ log 1+ jωτm - ∑ log 1+
jω+ ⎜
ωnq
⎜ ωnq
m
q
⎝
M
Q
Prof. Dr. Y. Samim Ünlüsoy
2ξ q
2
⎞
⎟
⎟
⎠
40
P
(
K ∏ 1+ Tp s
T(s)=
p=1
)
⎛
s
s2
⎜
s ∏ (1+ τms ) ∏ ⎜1+2ξ q
+
ωnq ω2
m=1
q=1 ⎜
nq
⎝
N
Q
M
„
BODE PLOT
⎞
⎟
⎟
⎟
⎠
Similarly, the phase of T(jω
T(jω) can be
obtained by simple summation of the
phases of individual terms.
⎛ 2ξ ω ω ⎞
q nq ⎟
-1
o
-1
-1 ⎜
φ = ∠ T ( jω ) = ∑ tan ωτp -N 90 - ∑ tan ωτm - ∑ tan ⎜
2
2 ⎟
ω
ω
p
m
q
⎜ nq
⎟
⎝
⎠
P
ME 304 CONTROL SYSTEMS
(
)
M
Prof. Dr. Y. Samim Ünlüsoy
Q
41
BODE PLOT
„
Therefore, any transfer function can be
constructed from the four basic factors :
1. Gain
Gain,, K - a constant,
2. Integral
Integral,, 1/jω, or derivative factor
factor,, jω –
pole or zero at the origin,
3. First order factor – simple lag, 1/(1+jωT),
or lead 1+jωT (real pole or zero),
zero)
4. Quadratic factor – quadratic lag or lead.
2
2
⎡
⎡
⎛ ω ⎞ ⎛ ω ⎞ ⎤
⎛ ω ⎞ ⎛ ω ⎞ ⎤
1 ⎢1+2ξ ⎜ j
⎟ +⎜ j
⎟ ⎥ or ⎢1+2ξ ⎜ j
⎟ +⎜ j
⎟ ⎥
⎢
⎢
⎝ ωn ⎠ ⎝ ωn ⎠ ⎥⎦
⎝ ωn ⎠ ⎝ ωn ⎠ ⎥⎦
⎣
⎣
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
42
BODE PLOT
Some useful definitions :
„
The magnitude is normally specified in
decibels [dB].
The value of M in decibels is given by :
M[dB]=20logM
[ ]
g
„
Frequency ranges may be expressed in
terms of decades or octaves.
Decade : Frequency band from ω to 10ω.
Octave : Frequency band from ω to 2ω
2ω..
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
43
BODE PLOT
Gain Factor K.
„
„
The gain factor multiplies the overall gain
by a constant value for all frequencies.
It has no effect on phase.
M[dB]
G(s)=
( ) K
G(jω)=K
20logK
M = 20log G(jω)
= 20log(K) [dB]
φ =0
0
φ[o]
M : magnitude, φ : phase.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
0
ω
ω
44
BODE PLOT
Integral Factor 1/jω – pole at the origin.
„
„
Magnitude is a straight line with a slope of -20
dB/decade becoming zero at ω=1 [rad/s].
Phase is constant at -90o at all frequencies.
M[dB]
1
1
1
G(s) = , G ( jω ) =
=- j
s
jω
ω
20
0
M= 20log G(jω)
⎛1⎞
= 20log ⎜ ⎟= -20logω
⎝ω⎠
o
φ = -90
ω
0.1
1
10
-20
-20 dB/decade slope
Im
-1/ω
ME 304 CONTROL SYSTEMS
decade
φ
Re
φ[o]
0
-90
Prof. Dr. Y. Samim Ünlüsoy
ω
45
-1/ω2
BODE PLOT
Re
φ
Double pole at the origin.
„
Im
Simply double the slope of the magnitude
and the phase, i.e., -40 dB/decade becoming
zero at ω=1
1 [rad/s] and -180o phase.
G(s)=
1
s2
, G ( jω ) =
1
( jω )2
=-
1
ω2
M = 20log G(jω)
⎛ 1
= 20log ⎜
⎝ ω2
φ = -180
o
ME 304 CONTROL SYSTEMS
M[dB]
decade
40
0
ω
0.1
1
10
-40
40
⎞
⎟= -40logω
⎠
-40 dB/decade slope
φ[o]
0
-180
Prof. Dr. Y. Samim Ünlüsoy
ω
46
BODE PLOT
Derivative Factor jω – zero at the origin.
„
„
Magnitude is a straight line with a slope of 20
dB/decade becoming zero at ω=1 [rad/s].
Phase is constant at 90o at all frequencies.
M[dB]
G(s)= s , G ( jω ) = ωj
20
0
M 20log
M=
20l
G(jω)
G(j )
= 20log ( ω )
φ = 90o
ME 304 CONTROL SYSTEMS
decade
ω
0.1
1
10
-20
φ[o]
20 dB/decade slope
90
0
Prof. Dr. Y. Samim Ünlüsoy
ω
47
BODE PLOT
Double zero at the origin.
„
Simply double the slope of the magnitude
and the phase, i.e., 40 dB/decade
becoming zero at ω=1
1 [rad/s] and 180o
phase.
M[dB]
G(s)= s2 , G ( jω ) = -ω2
M = 20log G(jω)
= 40log ( ω )
φ =180o
40
0
0
ω
0.1
1
10
-40
φ[o]
40 dB/decade slope
180
0
ME 304 CONTROL SYSTEMS
decade
Prof. Dr. Y. Samim Ünlüsoy
ω
48
BODE PLOT – First Order Factor
Simple lag (Real pole) 1/(1+jωT).
1
G(s)=
1+ Ts
G(jω)=
1
1 - jωT
1
ωT
=
j
2
2
2
2
1+ jωT 1 - jωT 1+ω T
1+ω T
⎛
1
M = 20log G(jω) = 20log ⎜
⎜
2 2
⎝ 1+ω T
⎞
⎟
⎟
⎠
M= -20log 1+ω2 T2 [dB]
φ = tan-1 ( -ωT ) = -tan-1ωT
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
49
BODE PLOT – First Order Factor
Simple lag (Real Pole) 1/(1+jωT).
M = -20log 1 + ω2 T2 [dB]
M[dB]
0
0.1
T
1
T
10
T
100
T
ω
F
For
ω <<
-20
M ≅ -20log1 = 0 [dB]
For ω >>
-40
ME 304 CONTROL SYSTEMS
1
T
1
T
M ≅ -20log
l ω T [dB]
[d ]
Prof. Dr. Y. Samim Ünlüsoy
50
BODE PLOT – First Order Factor
It is clear that the actual magnitude curve
can be approximated
pp
by
y two straight
g
lines.
M[dB]
0.1
0 T
1
T
10
T
100
T
ω
-3
M ≅ -20log1
20l 1 = 0 [dB]
-20
M ≅ -20log ω T [dB]
-40
For ω <<
ME 304 CONTROL SYSTEMS
1
T
For ω >>
1
T
Prof. Dr. Y. Samim Ünlüsoy
51
BODE PLOT – First Order Factor
ωc=1/T is called the corner (break) frequency.
frequency.
Maximum error between the linear
approximation and the exact value will be at
the corner frequency.
M[dB]
0
M= -20log 1+ω2 T2 [dB]
0.1
T
1
T
10
T
100
T
ω
-3
1⎞
⎛
M ⎜ ω = ⎟ = -20log 2
T⎠
⎝
≅ -3[dB]
-20
-40
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
52
BODE PLOT – First Order Factor
ω
0.1ωc 0.5ωc
ωc
2ω c
10ωc
Error
[dB]
0.04
3
1
0.04
M[dB]
0
0.1
T
1
1
T
10
T
100
T
ω
-3
-20
-40
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
53
BODE PLOT – First Order Factor
„
„
„
„
Transfer function G(s)=1/(1+Ts) is a
low pass filter.
filter.
At low frequencies the magnitude ratio
is almost one,, i.e.,, the output
p
can follow
the input.
For higher frequencies, however, the
output cannot follow the input because
a certain amount of time is required to
build up output magnitude (time
constant!).
Thus, the higher the corner frequency
the faster the system response will be.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
54
BODE PLOT – First Order Factor
Simple lag 1/(1+jωT).
φ[o]
0
φ = tan-1 ( -ωT ) = -tan-1ωT
0.1
T
1
T
10
T
100
T
ω
For ω <<
φ ≅ 0 [o ]
-45
For ω >>
10
T
φ ≅ -90 [o ]
-90
ME 304 CONTROL SYSTEMS
0.1
T
Prof. Dr. Y. Samim Ünlüsoy
55
BODE PLOT – First Order Factor
It is clear that the actual phase curve can
be approximated by three straight lines.
φ[o]
0
o
φ≅0[ ]
0.1
T
1
T
10
T
-5.7
100
T
ω
Linear variation
in the range
0.1
10
≤ω≤
T
T
-45
-90
φ ≅ -90 [o ]
In this case corner frequencies
q
are : 0.1/T and 10/T
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
56
BODE PLOT – First Order Factor
ω
0 01ωc
0.01
0 1ω c
0.1
ωc
10ωc
100ωc
φ [o]
-0.57
-5.7
-45
-84.3
-89.4
Error
[o]
0.6
5.7
0
-5.7
-0.6
Thus the maximum error of the linear
approximation is 5.7o.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
57
BODE PLOT – First Order Factor
Simple lead (Real zero) 1+jωT.
G(s)=1+ Ts
G(j ) 1
G(jω)=1+ωTj
Tj
M = 20log G(jω) = 20log ⎛⎜ 1+ω2 T2 ⎞⎟
⎝
⎠
M=20log 1+ω2 T2 [dB]
φ = tan-1 ( ωT ) = tan-1ωT
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
58
BODE PLOT – First Order Factor
Simple lead (Real zero) 1+jωT.
M = 20log 1 + ω2 T2 [dB]
M[dB]
For ω <<
40
1
T
M ≅ 20log1 = 0 [dB]
For ω >>
20
0
1
T
M ≅ 20log ω T [dB]
ω
0.1
T
ME 304 CONTROL SYSTEMS
1
T
10
T
100
T
Prof. Dr. Y. Samim Ünlüsoy
59
BODE PLOT – First Order Factor
It is clear that the actual magnitude curve
can be approximated
pp
by
y two straight
g
lines.
1
For
ω
<<
M[dB]
T
For ω >>
1
T
40
M ≅ 20log
20l ω T [dB]
20
M ≅ 20log1 = 0 [dB]
0
ω
0.1
T
ME 304 CONTROL SYSTEMS
1
T
10
T
100
T
Prof. Dr. Y. Samim Ünlüsoy
60
BODE PLOT – First Order Factor
Simple lead 1+jωT.
φ = tan-1 ( ωT )
φ[o]
For ω <<
90
0.1
T
φ ≅ 0 [o ]
For ω >>
45
10
T
φ ≅ 90 [o ]
0
ω
0.1
0
1
T
ME 304 CONTROL SYSTEMS
1
T
10
T
100
T
Prof. Dr. Y. Samim Ünlüsoy
61
BODE PLOT – First Order Factor
It is clear that the actual phase curve can
pp
by
y three straight
g
lines.
be approximated
φ[o]
φ ≅ 90 [o ]
90
Linear variation
Li
i ti
in the range
0.1
10
≤ω≤
T
T
45
5.7
φ≅0[ ] 0
o
ME 304 CONTROL SYSTEMS
0.1
T
1
T
10
T
Prof. Dr. Y. Samim Ünlüsoy
100
T
ω
62
BODE PLOT – Quadratic Factors
As overdamped systems can be replaced by
two first order factors,, only
y underdamped
p
systems are of interest here.
G(s)=
ω2
n
s2 +2ξωns +ω2
n
G(jω)=
A set of two complex
conjugate poles.
1
2
⎛ ω ⎞
⎛ ω ⎞
⎜j
⎟ +2ξ ⎜ j
⎟ +1
⎝ ωn ⎠
⎝ ωn ⎠
2
2
⎡ ⎛ ω ⎞2 ⎤
⎛
⎞
ω
M=20log G(jω) = -20log
20log ⎢1 - ⎜
⎟ ⎥ + ⎜ 2ξ
⎟ [dB]
ωn ⎠
⎢ ⎝ ωn ⎠ ⎥
⎝
⎣
⎦
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
63
BODE PLOT – Quadratic Factors
2
2
⎡ ⎛ ω ⎞2 ⎤
⎛
ω ⎞
⎥
M 20log G(jω) = -20log
M=
20log ⎢1 - ⎜
+
2ξ
⎟
⎜
⎟ [dB]
ωn ⎠
⎢ ⎝ ωn ⎠ ⎥
⎝
⎣
⎦
Low frequency asymptote,
asymptote ω<<ωn :
M ≅ −20log (1) = 0 [dB]
High frequency asymptote, ω>>ωn :
2
⎛ ω ⎞
⎛ ω ⎞
M ≅ -20log ⎜
⎟ = -40log ⎜
⎟ [dB]
⎝ ωn ⎠
⎝ ωn ⎠
Low and high frequency asymptotes intersect at
ω=ωn, i.e. corner frequency is ωn.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
64
BODE PLOT – Quadratic Factors
Therefore the actual magnitude curve can
be approximated by two straight lines.
lines
M[dB]
20
LF
Asymptote
ξ (increasing)
0
HF
Asymptote
-20
40
-40
-40dB/decade
slope
-60
ωn/100
ωn/10
ωn
10ωn
100ωn
Frequency
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
65
BODE PLOT – Quadratic Factors
φ = ∠G(jω)= -tan-1
ω
2ξ
ωn
2
⎛ ω ⎞
1-⎜
⎟
ω
⎝ n⎠
o
φ
≅
0
[
]
At low frequencies,
frequencies ω→0 :
o
φ
≅
−
90
[
]
At ω=ωn :
At high frequencies, ω→ :
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
φ ≅ -180 [o ]
66
BODE PLOT – Quadratic Factors
Thus, the actual phase curve can be
approximated by three straight lines.
lines
0
φ[o]
ξ (increasing)
-90o/decade
slope
-90
-180
ωn/10
ωn
10ωn
100ωn
Frequency
Corner frequencies
q
are : ωn/
/10 and 10ωn.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
67
BODE PLOT – Quadratic Factors
„
„
It is observed that,, the linear
approximations for the magnitude and
phase will give more accurate results
f
for
damping
d
i
ratios
i
closer
l
to 1.0.
10
The peak magnitude is given by :
Μr =
„
1
2ξξ 1− ξ 2
The resonant frequency :
ωr = ωn 1 -2ξ2
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
68
BODE PLOT – Quadratic Factors
„
For ξ=0.707 :
(or M=20log1=0
g
dB).
)
Mr=1 (
Thus, there will be no peak on the
magnitude plot.
„
Note the difference that in transient
response for step input,
input there will be
no overshoot for critically or
overdamped
p
systems,
y
, i.e.,, for ξ ≥ 1.0.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
69
BODE PLOT – Example 1a
„
Sketch the Bode plots for the given
open loop transfer function of a control
system.
100000 (1+s )
T(s)=
„
(
s ( s +10 ) 0
0.1s
1s2 +14s +1000
)
First convert to standard form.
100000 (1+ jω )
T(jω)=
⎡⎛ ω ⎞2
⎤
⎛ ω ⎞
( jω )(10 )(1+0.1ωj)(1000 ) ⎢⎜ j
⎟ +1.4 ⎜ j
⎟ +1⎥
⎝ 100 ⎠
⎢⎣⎝ 100 ⎠
⎥⎦
T(jω)=
T(jω)
ME 304 CONTROL SYSTEMS
10 (1+ jω )
⎡⎛ ω ⎞2
⎤
⎛ ω ⎞
( jω )(1+0.1ωj) ⎢⎜ j
⎟ +1.4 ⎜ j
⎟ +1⎥
⎝ 100 ⎠
⎢⎣⎝ 100 ⎠
⎥⎦
Prof. Dr. Y. Samim Ünlüsoy
70
BODE PLOT – Example 1b
T(jω)=
„
10 (1+ jω )
⎡⎛ ω ⎞2
⎤
⎛ ω ⎞
1+0 1ωj) ⎢⎜ j
+1 4 ⎜ j
( jω )(1+0.1ωj
⎟ +1.4
⎟ +1⎥
⎝ 100 ⎠
⎢⎣⎝ 100 ⎠
⎥⎦
Identify the basic factors and corner frequencies :
-Constant gain K : K=10, 20log10=20 [dB]
-First order factor (simple lead – real zero) :
T=1 (ωc1=1/T=1) - for magnitude plot
-Integral factor : 1/jω
-First order factor (simple lag – real pole) :
T=0.1
T=0 1 (ωc1
1=1/T=10) - for magnitude plot
-Quadratic factor (complex conjugate poles) :
ωn=ωc1=100, ξ=0.7 - for magnitude plot
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
71
BODE PLOT – Example 1c
T(jω)=
„
10 (1+ jω )
⎡⎛ ω ⎞2
⎤
⎛ ω ⎞
1+0 1ωj) ⎢⎜ j
+1 4 ⎜ j
( jω )(1+0.1ωj
⎟ +1.4
⎟ +1⎥
⎝ 100 ⎠
⎢⎣⎝ 100 ⎠
⎥⎦
Identify the basic factors and corner frequencies :
-Constant gain K : K=10, 20log10=20 [dB]
-First order factor (simple lead – real zero) : T=1
(ωc2=0.1/T=0.1, ωc3=10/T=10) – for phase plot
-Integral factor : 1/jω
-First order factor (simple lag – real pole) : T=0.1
(ωc2=0.1/T=1, ωc3=10/T=100) – for phase plot
-Quadratic factor (complex conjugate poles) : ωn=100
(ωc2=ωn/10=10, ωc3=10ωn=1000) – for phase plot
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
72
BODE PLOT – Example 1d
1+jω
M[dB]
40
K=10
20
0
1
10
100
1000
ω[rad/s]
-20
Quadratic
factor
-40
1/(1+0.1jω)
1/jω
-60
Bode (magnitude) plot
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
73
BODE PLOT – Example 1e
90
0
1+jω
φ[o]
1
10
100
1000 K=10
ω
1/(1+0.1jω)
-90
90
1/jω
Quadratic
factor
-180
180
-270
Bode (phase) plot
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
74
BODE PLOT – Example 1f
Matlab plot:
full blue lines
num=[100000 100000]
den=[0 1 15 1140 10000 0]
den=[0.1
bode(num,den)
grid
Approximate
plots: dashed
lines
ME 304 CONTROL SYSTEMS
40
Magnittude (dB)
(just 4 lines
to plot !)
Bode Diagram
60
20
0
-20
-40
-60
60
0
Phase ((deg)
„
-90
-180
-270
-1
10
10
0
1
10
Frequency
Prof. Dr. Y. Samim Ünlüsoy
10
2
10
75
3
STABILITY ANALYSIS
Nise Sect
Sect. 10
10.7,
7 pp
pp.638
638-641
638-
„
„
„
Transfer functions which have no poles or
zeroes on the right hand side of the
complex plane are called minimum phase
transfer funtions.
Nonminimum p
phase transfer functions,, on
the other hand, have zeros and/or poles on
the right hand side of the complex plane.
The major disadvantage of Bode Plot is that
stability of only minimum phase systems
can be determined using Bode plot.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
76
STABILITY ANALYSIS
„
From the characteristic equation :
1 + G(s)H(s) = 0 or G(s)H(s)= -1
Then the magnitude and phase for the
open loop transfer function become :
20log G(jω)H(jω) =20log1= 0 dB
∠G(jω)H(jω)= -180
o
Thus, when the magnitude and the phase
angle of a transfer function are 0 dB and
-180
80o, respectively,
i l then
h
the
h system iis
marginally stable.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
77
STABILITY ANALSIS
„
„
„
If at the frequency, for which phase
becomes equal to -180o, gain is below 0
dB, then the system is stable (unstable
otherwise).
otherwise)
Further, if at the frequency, for which
gain
i becomes
b
equall to
t zero, phase
h
is
i
above -180o, then the system is stable
(unstable otherwise).
otherwise)
Thus, relative stability of a minimum
phase system can be determined
according to these observations.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
78
GAIN and PHASE MARGINS
Nise
Ni pp. 638638-641
„
„
Gain Margin : Additional gain to make
the system marginally stable at a
frequency
q
y for which the p
phase of the
open loop transfer function passes
through -180o.
Phase Margin : Additional phase angle
to make the system marginally stable at
a frequency for which the magnitude of
the open loop transfer function is 0 dB.
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
79
GAIN and PHASE MARGINS
Mag
gnitude (dB))
50
Gain
G
i
Margin
0
-50
-100
Phase ((deg)
-150
-90
-135
-180
Phase
M i
Margin
-225
-270
-1
10
10
0
10
1
10
2
10
3
Frequency (rad/sec)
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
80
BODE PLOT
„
Can you
identify the
transfer
function
approximately
if the
measured
meas red
Bode diagram
is available ?
ME 304 CONTROL SYSTEMS
Prof. Dr. Y. Samim Ünlüsoy
81