Control Systems
Bode diagramsL. Lanari
Lanari: CS - Bode diagrams 2
Outline
• Bode’s canonical form for the frequency response
• Magnitude and phase in the complex plane
• The decibels (dB)
• Logarithmic scale for the abscissa
• Bode’s plots for the different contributions
Lanari: CS - Bode diagrams 3
Frequency responseThe steady state response of an asymptotically stable system P(s)
to a sinusoidal input is given byu(t) = sin �t
yss(t) = |P (j�)| sin(�t+ \P (j�))
gain curve
(or magnitude)
phase curve
|P (j�)|
\P (j�)
Frequency response P (j�)
(P(j!) is the restriction of the transfer function to the imaginary axis P(j!)|s = j!
� 2 R+ = [0,+1)for
same frequency as inputamplification/attenuation depends on the system at the frequency of the input
a complex number p can always be represented in terms of its magnitude and phase
p = |p|ej\p
(|p|,\p)
Lanari: CS - Bode diagrams 4
F (s) = K 0 1
sm
Qk(s� zk)
Q�(s
2 + 2��⇥n�s+ ⇥2n�)Q
i(s� pi)Q
z(s2 + 2�z⇥nzs+ ⇥2
nz)
Bode canonical form
pole/zero representation of the transfer function
with m such that
• m = 0 if no pole or zero in s = 0
• m < 0 if m zeros in s = 0
• m > 0 if m poles in s = 0
remarks
• numerator and denominator are by hypothesis coprime
• denominator is monic
• K’ is not the system gain
Lanari: CS - Bode diagrams
• the terms and are relative to
‣ complex conjugate zeros (in )
‣ complex conjugate poles (in )
5
Bode canonical form
s = �` ± j⇥`
(s2 + 2��⇥n�s+ ⇥2n�) (s2 + 2�z⇥nzs+ ⇥2
nz)
s = �z ± j⇥z
!n⇤ =p
↵2⇤ + �2
⇤
⇤⇤ = ��⇤/⌅n⇤ = ��⇤/p�2⇤ + ⇥2
⇤
with
• the terms (s - zk) and (s - pi) are relative to
‣ real zeros (in s = zk)
‣ real poles (in s = pi)
‣ natural frequency
‣ damping coefficient
Lanari: CS - Bode diagrams 6
s� zk = �zk(1� 1/zks) = �zk(1 + �ks) with �k = �1/zk
s� pi = �pi(1� 1/pis) = �pi(1 + �is) with �i = �1/pi
F (s) = K 0 1
sm
Qk(�zk)
Q�(⇤
2n�)
Qk(1 + ⇥ks)
Q�(1 + 2��/⇤n�s+ s2/⇤2
n�)Qi(�pi)
Qz(⇤
2nz)
Qi(1 + ⇥is)
Qz(1 + 2�z/⇤nzs+ s2/⇤2
nz)
F (s) = K1
sm
Qk(1 + ⇥ks)
Q�(1 + 2��/⇤n�s+ s2/⇤2
n�)Qi(1 + ⇥is)
Qz(1 + 2�z/⇤nzs+ s2/⇤2
nz)
K = [smF (s)]s=0 for any m R 0K = K 0Q
k(�zk)Q
�(�2n�)Q
i(�pi)Q
z(�2nz)
Bode canonical form
factoring out the constant terms
with ¿i and ¿k being time constants
defining
Bode canonical form
how to compute K
Lanari: CS - Bode diagrams 7
Ks = F (s)���s=0
= F (0)
K = Ks , m = 0
Gains
K = [smF (s)]s=0 for any m R 0generalized gain
Note that
• for a system with no poles in s = 0 (i.e. m negative or zero) we have defined as
dc-gain (or static gain)
if m < 0 (zeros in s = 0) we have F(0) = 0
• static and generalized gain coincide only when m = 0
• for an asymptotically stable system, the step response tends to the static gain Ks = F(0)
F (s) = K1
sm
Qk(1 + ⇥ks)
Q�(1 + 2��/⇤n�s+ s2/⇤2
n�)Qi(1 + ⇥is)
Qz(1 + 2�z/⇤nzs+ s2/⇤2
nz)= 1 for s = 0
Lanari: CS - Bode diagrams 8
Bode canonical form
Examples
F (s) =s� 1
2s2 + 6s+ 4=
s� 1
2(s+ 1)(s+ 2)= �1
4
1� s
(1 + s)(1 + s/2)
F (s) =s(s� 1)
2(s+ 1)2(s+ 2)= �1
4
s(1� s)
(1 + s)2(1 + s/2)
F (s) =s� 1
2s(s+ 1)(s+ 2)= �1
4
1� s
s(1 + s)(1 + s/2)
K = �1
4= Ks
K = �1
4Ks = 0
K = �1
4@ Ks
Lanari: CS - Bode diagrams 9
F (j⇤) = K1
(j⇤)m
Qk(1 + j⇤⇥k)
Q�(1 + 2��j⇤/⇤n� + (j⇤)2/⇤2
n�)Qi(1 + j⇤⇥i)
Qz(1 + 2�zj⇤/⇤nz + (j⇤)2/⇤2
nz)
Bode canonical form
has 4 elementary factors
1. constant K (generalized gain)
2. monomial j! (zero or pole in s = 0)
3. binomial 1 + j!¿ (non-zero real zero or pole)
4. trinomial 1 + 2³(j!)/!n + (j!)2/!n2 (complex conjugate pairs of zeros or poles)
frequency response
so first check which kind of root you have and then factor it out
Lanari: CS - Bode diagrams 10
F (j�) = Re[F (j�)] + jIm[F (j�)]
Bode diagrams
magnitude of the frequency response as a function of the angular frequency !
angle or phase of the frequency response as a function of the angular frequency !
|F (j�)|
\F (j�)
for any real value of the angular frequency ! the frequency response F(j!) is a complex number
|F (j�)|
\F (j�)
F(j!)
F (j�) = |F (j�)|ej\F (j�)
|F (j�)| =p
Re[F (j�)]2 + Im[F (j�)]2 \F (j�) = atan2(Im[F (j�)],Re[F (j�)])
Im[F(j!)]
Re[F(j!)]
Lanari: CS - Bode diagrams 11
Phase
Phase[F.G] = Phase[F ] + Phase[G]
Phase
F
G
�= Phase[F ]� Phase[G]
Phase⇥1G
⇤= �Phase[G]
the phase of a product is the sum of the phases
the phase of a ratio is the difference of the phases
and therefore
since
very useful since we can find the contribution to the
phase of each term and then just do an algebraic sum
Lanari: CS - Bode diagrams 12
atan2(⇥,�) =
8>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>:
arctan⇣
�↵
⌘if � > 0 (I & IV quadrant)
arctan⇣
�↵
⌘+ ⇤ if ⇥ � 0 and � < 0 (II quadrant)
arctan⇣
�↵
⌘� ⇤ if ⇥ < 0 and � < 0 (III quadrant)
⇡2 sign(⇥) if � = 0 and ⇥ 6= 0
undefined if � = 0 and ⇥ = 0
Phase
|F (j�)|
\F (j�)
F(j!)Im[F(j!)]
Re[F(j!)]
principal argument takes on values
in (- ¼, ¼ ] and is implemented by
the function with two arguments
atan2
P = ® + j ¯
for
III
III IV
Lanari: CS - Bode diagrams 13
|F (j�)|dB = 20 log10 |F (j�)|
|F.G|dB = |F |dB + |G|dB
����1
F
����dB
= � |F |dB
|F |dB ⇥ +⌅ if |F | ⇥ ⌅
|F |dB ⇤ �⌅ if |F | ⇤ 0
|0.1|dB = �20 dB
|1|dB = 0dB
|10|dB = 20 dB |100|dB = 40 dB
��p2��dB ⇡ 3 dB
Magnitudein order to have the same useful property we need to go through some logarithmic function
decibels (dB)
same nice properties
as phase
Lanari: CS - Bode diagrams 14
10−3
10−2
10−1
100
101
−60
−40
−20
0
20
20
log
10(|
x|)
x in logarithmic scale
10−3
10−2
10−1
100
101
0
5
10
|x|
x in logarithmic scale
1 2 3 4 5 6 7 8 9 10−60
−40
−20
0
20
20
log
10(|
x|)
x in linear scale
Logarithmic scale
we use a logarithmic (log10) scale for the abscissa (angular frequency !)
a decade corresponds to multiplication by 10
log10(!) becomes a straight line
if ! is in a logarithmic scale
very useful when we add
different contributions
Lanari: CS - Bode diagrams 15
0 2 4 6 8 100
100
200
300
|F(j
ω)|
linear scale
1 2 3 4 5 6 7 8 9 10−20
0
20
40
60
|F(j
ω)|
dB
linear scale
10−1
100
101
−20
0
20
40
60
|F(j
ω)|
dB
logarithmic scale
Logarithmic scaleadvantages
• quantities can vary in large range (both ! and magnitude)
• easy to build the magnitude plot in dB of a frequency response given in its Bode canonical
form from the magnitudes of the single terms
• easy to represent series of systems
same data
different scales
for abscissa and
ordinates
this is the scale we are going to use
Lanari: CS - Bode diagrams 16
Bode diagrams
magnitude in dB of the frequency response as a function of the angular
frequency ! with logarithmic scale for !
angle or phase of the frequency response as a function of the angular
frequency ! with logarithmic scale for !\F (j�)
|F (j�)|dB
we need to find the magnitude (in dB) and phase for the 4 elementary factors
1. constant K (generalized gain)
2. monomial j! (zero or pole in s = 0)
3. binomial 1 + j!¿ (non-zero real zero or pole)
4. trinomial 1 + 2³(j!)/!n + (j!)2/!n2 (complex conjugate pairs of zeros or poles)
� 2 R+ = [0,+1)for
Lanari: CS - Bode diagrams 17
Constant
magnitude
phase
Re
Im
K
20 log10 |K|
\K
K1 =p10K3 = �
p10
K2 = 0.5
10−2
10−1
100
101
102
−10
−6
0
10
20
frequency (rad/s)
Magnitude (dB)
|K1|dB, |K3|dB
|K2|dB
10−2
10−1
100
101
102
−180
−150
−100
−50
0
frequency (rad/s)
Phase (deg)
K1, K2
K3
|0.5|dB ⇥ �6 dB
|�10|dB = 10dB
|�⇥10|dB = 10dB
\�p10 = �180� = �⇡
\p10 = 0�
\0.5 = 0�
Lanari: CS - Bode diagrams 18
Monomial - Numerator
Re
Im
j!j!
90°|j�|dB = 20 log10 �
|j�|dB = 20x
magnitude
log scale
phase
10−2
10−1
100
101
102
−40
−20
0
20
40
frequency (rad/s)
Magnitude (dB)
10−2
10−1
100
101
102
0
20
40
60
8090
frequency (rad/s)
Phase (deg)
20 dB/dec
Lanari: CS - Bode diagrams 19
Monomial - Denominator
magnitude
phase
from properties of log and phase
10−2
10−1
100
101
102
−40
−20
0
20
40
frequency (rad/s)
Magnitude (dB)
10−2
10−1
100
101
102
−90−80
−60
−40
−20
0
frequency (rad/s)
Phase (deg)
-20 dB/dec
Lanari: CS - Bode diagrams 20
Binomial - Numerator
|1 + j⇥� |dB = 20 log10p
1 + ⇥2�2
p1 + ⇥2�2 �
8<
:
1 if ⇥ ⇥ 1/|� |
⌅⇥2�2 if ⇥ ⇤ 1/|� |
1/|� |
|1 + j⇥� |dB �
8<
:
0 dB if ⇥ ⇥ 1/|� |
20 log10 ⇥ + 20 log10 |� | if ⇥ ⇤ 1/|� |
⇥⇤ = 1/|� | |1 + j�/|� | |dB = 20 log10⇥2 � 3 dB
magnitude
approximation wrt the cutoff frequency (or corner frequency)
and therefore
at the cutoff frequency
two half-lines approximation: 0 dB until the cutoff frequency, + 20dB/decade after
1 + j!¿
Lanari: CS - Bode diagrams 21
Binomial - Numerator
phase depends on the sign of ¿
Re
Im
1
1 + j!¿
j!¿
¿ > 0
ReIm
1
j!¿
¿ < 0
see how the phase changes as ! increases
Lanari: CS - Bode diagrams 22
Binomial - Numerator phase depends on the sign of ¿
case ¿ > 0
case ¿ < 0
\(1 + j⇥�) �
8<
:
0 if ⇥ ⇥ 1/|� |
⇡2 if ⇥ ⇤ 1/|� | and � > 0
\(1 + j⇥�) ⇥
8<
:
0 if ⇥ ⇤ 1/|� |
�⇡2 if ⇥ ⌅ 1/|� | and � < 0
the two asymptotes are connected by a segment starting a decade before (0.1/| ¿ | ) the cutoff
frequency and ending a decade after (10/| ¿ |). The approximation is a broken line.
⇥⇤ = 1/|� |at the cutoff frequency \(1 + j�/|� |) =
8<
:
⇡4 if � > 0
�⇡4 if � < 0
1 + j!¿
0
¼/2
-¼/2
1/| ¿ |
0.1/| ¿ | 10/| ¿ |
Lanari: CS - Bode diagrams 23
Binomial - numerator1 + j!¿
magnitude
phase¿ > 0
−10
03
10
20
30
40
frequency (rad/s)
Magnitude (dB)
−90
−45
0
frequency (rad/s)
Phase (deg)
0
45
90
frequency (rad/s)
Phase (deg)
1/| ¿ |
0.1/| ¿ | 10/| ¿ |
phase¿ < 0
0.1/| ¿ | 10/| ¿ |1/| ¿ |
Lanari: CS - Bode diagrams 24
Binomial - denominator1 /(1 + j!¿)
−40
−30
−20
−10
−30
10
frequency (rad/s)
Magnitude (dB)
−90
−45
0
frequency (rad/s)
Phase (deg)
0
45
90
frequency (rad/s)
Phase (deg)
magnitude
phase¿ > 0
1/| ¿ |
0.1/| ¿ | 10/| ¿ |
phase¿ < 0
0.1/| ¿ | 10/| ¿ |1/| ¿ |
Lanari: CS - Bode diagrams 25
Trinomial
����1 + 2�
⇥n(j⇥) +
(j⇥)2
⇥2n
���� =
����1�⇥2
⇥2n
+ j2�⇥
⇥n
����
=
s✓1� ⇥2
⇥2n
◆2
+
✓4�2
⇥2
⇥2n
◆
|TRINOMIAL| ⇥
8>><
>>:
1 if � ⇤ �n
r⇣�2
�2n
⌘2=
�2
�2n
if � ⌅ �n
|TRINOMIAL|dB ⇥
8<
:
0 dB if � ⇤ �n
40 log10 � � 20 log10 �2n if � ⌅ �n
magnitude
approximation wrt !n
Lanari: CS - Bode diagrams 26
Trinomial
|�| 0 0.5 1/⌅2 ⇥ 0.707 1
|TRIN |dB in ⇥n �⇤ 0 dB 3 dB 6 dB
in ! = !n the magnitude | TRINOMIAL | is equal to 2 | ³ |
large variation of the magnitude in ! = !n depending upon the value of the damping coefficient ³
no approximation around the natural frequency !n
Lanari: CS - Bode diagrams 27
Trinomial
\✓1 + 2
�
⇤n(j⇤) +
(j⇤)2
⇤2n
◆=
8>>>>>><
>>>>>>:
0 if ⇤ ⌧ ⇤n
⇥ if ⇤ � ⇤n and � � 0
�⇥ if ⇤ � ⇤n and � < 0
Phase
transition between 0 and ¼ (or - ¼ ) is symmetric wrt !n and becomes more abrupt as
| ³ | becomes smaller. When ³ = 0 the phase has a discontinuity in !n
!"
#$# %
#
! !
&'
!#()
!
#
(
)"
"
"
"
"" " "How does a generic complex root
varies in the plane as a function of !
Lanari: CS - Bode diagrams 28
Trinomial - numerator
−40
−20
0
20
40
60
frequency (rad/s)
Magnitude (dB)
0
1
0.30.1
0.5
0.7
0.30.1
0.5
0.7
0
45
90
135
180
frequency (rad/s)
Phase (deg)
01
0.1
0.3
0.5
0.7
0.1
0.3
0.5
0.7
−180
−135
−90
−45
0
frequency (rad/s)
Phase (deg) 0.01
1
magnitude
phase
phase
⇣ � 0
� < 0
0.1 !n 10 !n!n
0.1 !n 10 !n!n
Lanari: CS - Bode diagrams 29
Trinomial - denominator
−60
−40
−20
0
20
40
frequency (rad/s)
Magnitude (dB) 0
1
−180
−135
−90
−45
0
frequency (rad/s)
Phase (deg) 0
1
0
45
90
135
180
frequency (rad/s)
Phase (deg)
0.011
magnitude
phase
phase
⇣ � 0
� < 0
0.1 !n 10 !n!n
0.1 !n 10 !n!n
Lanari: CS - Bode diagrams 30
Trinomial
roots =
( �!n if ⇣ = 1
!n if ⇣ = �1
When | ³ | = 1 the trinomial reduces to a product of two identical binomials (real roots)
✓1 + 2
�
⇥ns+
s2
⇥2n
◆
�=±1
=
✓1± s
⇥n
◆2
and therefore the magnitude and phase coincides with that of a double binomial with corner
frequency1
|⌧ | = !n
2⇥ (3 dB) = 6dB (numerator)
2⇥ (�3 dB) = �6 dB (denominator)
that is in ! = !n when | ³ | = 1
example: MSD system with critical value for the damping (µ2 = 4km)
Lanari: CS - Bode diagrams 31
Trinomial
|F (j⇥r)| =1
2|�|p1� �2
!r = !n
p1� 2⇣2
|�| < 1/⇥2 � 0.707if the magnitude of a trinomial factor at the denominator has a peak
at the resonance frequency
resonance peak
(similarly for the anti-resonance peak)
−20
15
14
34
frequency (rad/s)
Magnitude (dB)
0.01
−40
−20
0
20
frequency (rad/s)
Magnitude (dB)
0.01
0.3
0.1
0.5
resonancepeak
anti-resonancepeak