Control Systems
System as a filterL. Lanari
Lanari: CS - Filter 2
Outline
• preliminaries
• steady state example for a first order system
• importance of the phase
• steady state example for a second order system
• transient: bandwidth
• transient: resonance peak
• transient: frequency vs time domain characterization
• Mass-Spring-Damper
• other examples
• quarter-car system
Lanari: CS - Filter 3
preliminaries
• moreover recall that a periodic signal can be expanded in a Fourier series which is an infinite sum of weighted sines and cosines
we can compute the steady state response to more complex signals
• system linearity guarantees that
<latexit sha1_base64="u8IbYv0yw8/n4a8+RieR9i+fYkw=">AAACDHicbVDLSgMxFM34rPVVdekmWIR2U2ZEsRuh4sZlBfuAzlAyadqGJpkhuSOUoR/gxl9x40IRt36AO//GtJ2Fth64cHLOvdzcE8aCG3Ddb2dldW19YzO3ld/e2d3bLxwcNk2UaMoaNBKRbofEMMEVawAHwdqxZkSGgrXC0c3Ubz0wbXik7mEcs0CSgeJ9TglYqVsoJiUo4yvsm0R2Ob625RuuStiPJBsQ+4Sy7XIr7gx4mXgZKaIM9W7hy+9FNJFMARXEmI7nxhCkRAOngk3yfmJYTOiIDFjHUkUkM0E6O2aCT63Sw/1I21KAZ+rviZRIY8YytJ2SwNAselPxP6+TQL8apFzFCTBF54v6icAQ4WkyuMc1oyDGlhCquf0rpkOiCQWbX96G4C2evEyaZxXvouLenRdr1SyOHDpGJ6iEPHSJaugW1VEDUfSIntErenOenBfn3fmYt6442cwR+gPn8wd3MZlU</latexit>
u(t) =X
i
Ai sin(!it)asymptotically stable
system F (s)
<latexit sha1_base64="htK3G7e6GnQdjohZIvTLpCRvE6s=">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</latexit>
yss(t) =X
i
Ai|F (j!i)| sin(!it+ \F (j!i))
output at steady state
that is the steady state output of an asymptotically stable system having as input a linear combination of sinusoids coincides with the same linear combination of the steady state responses of the system to each individual sinusoid
input
Lanari: CS - Filter 4
0 2 4 6 8�3
�2
�1
0
1
2
3T
Input
100
101
102
−0.5
0
0.5
frequency (rad/s)
Frequency spectrum of the input
u(t) = �0.6 sin(f1t)�0.4 sin(f2t)+0.5 sin(f3t)+0.5 sin(f4t)�0.3 sin(f5t)�0.2 sin(f6t)+0.2 sin(f7t)�0.2 sin(f8t)
f1 = 2�0.75, f2 = 2�1.25, f3 = 2�1.5, f4 = 2�3, f5 = 2�5, f6 = 2�6, f7 = 2�8, f8 = 2�11
−2
0
2
−2
0
2
−2
0
2
−2
0
2
−2
0
2
−2
0
2
−2
0
2
0 0.5 1 1.5 2 2.5 3 3.5 4−2
0
2
adding one contribution at a time
sameinformationin the frequencydomain time
time
example: a periodic input signal
magnitude of the sine functionat that frequency
period
Lanari: CS - Filter 5
10−2
100
102
−20
−15
−10
−5
0
5
frequency (rad/s)
Magnitude (dB)
F1
F2
F3
F4
F1(s) =1
1 + 10s, F2(s) =
1
1 + s, F3(s) =
1
1 + 0.1s, F4(s) =
1
1 + 0.01s
4 different systems(all first order and with unit gain)
behavior at steady state: example 1
Lanari: CS - Filter 6
10−2
10−1
100
101
102
−0.5
0
0.5
10−2
10−1
100
101
102
−0.1
0
0.1
10−2
10−1
100
101
102
0
0.5
1
Magnitude
System 2
magnitudenot in dB
we need to multiply
input
system
output
1
s+ 1
0 2 4 6 8�3
�2
�1
0
1
2
3T
Input
frequency time
?
Lanari: CS - Filter 7
0 2 4 6 8
−2
−1
0
1
2
System 1
time (s)
Input
Output
10−2
100
102
0
0.2
0.4
0.6
0.8
1
frequency (rad/s)
Magnitude
System 1
10−2
100
102
0
0.2
0.4
0.6
0.8
1
frequency (rad/s)
Magnitude
System 2
0 2 4 6 8
−2
−1
0
1
2
System 2
time (s)
Input
Output
these are plotted for unit magnitude sinusoids
frequency spectrumof the output
…
F1(s) =1
1 + 10s<latexit sha1_base64="zlbAVJNX2eCQdv0oSDWRuHD7lVw=">AAACAXicbVDLSsNAFJ3UV62vqBvBzWARKkLJWNG6EAqCuKxgH9CGMplO2qGTBzMToYS48VfcuFDErX/hzr9xkgbxdeDC4Zx7ufceJ+RMKsv6MApz8wuLS8Xl0srq2vqGubnVlkEkCG2RgAei62BJOfNpSzHFaTcUFHsOpx1ncpH6nVsqJAv8GzUNqe3hkc9cRrDS0sDcuRygijyA57DvCkxilMToEFkyGZhlq2plgH8JykkZ5GgOzPf+MCCRR31FOJayh6xQ2TEWihFOk1I/kjTEZIJHtKepjz0q7Tj7IIH7WhlCNxC6fAUz9ftEjD0pp56jOz2sxvK3l4r/eb1IuXU7Zn4YKeqT2SI34lAFMI0DDpmgRPGpJpgIpm+FZIx1EkqHVspCOEtx8vXyX9I+qqJatXZ9XG7U8ziKYBfsgQpA4BQ0wBVoghYg4A48gCfwbNwbj8aL8TprLRj5zDb4AePtE49clSg=</latexit>
F2(s) =1
1 + s<latexit sha1_base64="y0LJaqKEUpWPeBGAo5FjzHDDgPY=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARKkJJWtG6EAqCuKxgH9CGMJlO2qGTBzMTocQs/BU3LhRx62+482+cpEHUeuDC4Zx7ufceJ2RUSMP41AoLi0vLK8XV0tr6xuaWvr3TEUHEMWnjgAW85yBBGPVJW1LJSC/kBHkOI11ncpn63TvCBQ38WzkNieWhkU9dipFUkq3vXdm1ijiCF3DgcoRjM4nNY5HYetmoGhngPDFzUgY5Wrb+MRgGOPKILzFDQvRNI5RWjLikmJGkNIgECRGeoBHpK+ojjwgrzu5P4KFShtANuCpfwkz9OREjT4ip56hOD8mx+Oul4n9eP5Juw4qpH0aS+Hi2yI0YlAFMw4BDygmWbKoIwpyqWyEeI5WDVJGVshDOU5x+vzxPOrWqWa/Wb07KzUYeRxHsgwNQASY4A01wDVqgDTC4B4/gGbxoD9qT9qq9zVoLWj6zC35Be/8CpnOUtA==</latexit>
Lanari: CS - Filter 8
0 2 4 6 8
−2
−1
0
1
2
System 3
time (s)
Input
Output
0 2 4 6 8
−2
−1
0
1
2
System 4
time (s)
Input
Output
10−2
100
102
0
0.2
0.4
0.6
0.8
1
frequency (rad/s)
Magnitude
System 4
10−2
100
102
0
0.2
0.4
0.6
0.8
1
System 3
frequency (rad/s)
Magnitude
it just shows at whichfrequencies the input
has contributions
F3(s) =1
1 + 0.1s<latexit sha1_base64="E+KH24tv4gB6Giqid1iIZKmS8cc=">AAACAnicbVDLSsNAFJ3UV62vqCtxM1iEihASK1oXQkEQlxXsA9oQJtNJO3TyYGYilBDc+CtuXCji1q9w5984SYP4OnDhcM693HuPGzEqpGl+aKW5+YXFpfJyZWV1bX1D39zqiDDmmLRxyELec5EgjAakLalkpBdxgnyXka47ucj87i3hgobBjZxGxPbRKKAexUgqydF3Lp16TRzAczjwOMKJlSbWoWlYInX0qmmYOeBfYhWkCgq0HP19MAxx7JNAYoaE6FtmJO0EcUkxI2llEAsSITxBI9JXNEA+EXaSv5DCfaUMoRdyVYGEufp9IkG+EFPfVZ0+kmPx28vE/7x+LL2GndAgiiUJ8GyRFzMoQ5jlAYeUEyzZVBGEOVW3QjxGKgqpUqvkIZxlOPl6+S/pHBlW3ahfH1ebjSKOMtgFe6AGLHAKmuAKtEAbYHAHHsATeNbutUftRXudtZa0YmYb/ID29gkEUJVi</latexit>
F4(s) =1
1 + 0.01s<latexit sha1_base64="nZykf7rvEmXqnwu+TTGa5je3Zg8=">AAACA3icbZDLSsNAFIYn9VbrLepON4NFqAglsUXrQigI4rKCvUAbwmQ6aYdOLsxMhBICbnwVNy4UcetLuPNtnKRB1PrDwMd/zuHM+Z2QUSEN41MrLCwuLa8UV0tr6xubW/r2TkcEEcekjQMW8J6DBGHUJ21JJSO9kBPkOYx0ncllWu/eES5o4N/KaUgsD4186lKMpLJsfe/KrlfEEbyAA5cjHJtJbB4bVcMUia2XFWSC82DmUAa5Wrb+MRgGOPKILzFDQvRNI5RWjLikmJGkNIgECRGeoBHpK/SRR4QVZzck8FA5Q+gGXD1fwsz9OREjT4ip56hOD8mx+FtLzf9q/Ui6DSumfhhJ4uPZIjdiUAYwDQQOKSdYsqkChDlVf4V4jFQWUsVWykI4T3X6ffI8dE6qZq1au6mXm408jiLYBwegAkxwBprgGrRAG2BwDx7BM3jRHrQn7VV7m7UWtHxmF/yS9v4Fet2VnQ==</latexit>
Lanari: CS - Filter 9
2 systems with same magnitude but different phase
10−2
100
102
0
0.2
0.4
0.6
0.8
1
frequency (rad/s)
Magnitude
10−2
100
102
−700
−600
−500
−400
−300
−200
−100
0
100
Phase (deg)
frequency (rad/s)
FA
FB
0 10 20 30 40 50−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5System A
time (s)
InputOutput
0 10 20 30 40 50−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5System B
time (s)
InputOutput
FA(s) =1
1 + s/10
FB(s) =(s2 � 0.2s+ 1)(1� 10s)
(s2 + 0.2s+ 1)(1 + 10s)(1 + s/10)
differences in the system phase can lead to noticeable output difference
magnitude vs phase
to replicate an input signal at the output (at steady state) it is not sufficient to require that the system has unitary magnitude at the input frequencies
Lanari: CS - Filter 10
10−2
100
102
−10
−5
0
5
10
frequency (rad/s)
Magnitude (dB)
F1
F2
F3
F4
F (s) =1
(1 + 2�s/⇥n + s2/⇥2n)
³ = 0.2
!n = {1, 8, 10, 100}
4 different systems(all second order with same damping and unit gain)
behavior at steady state: example 2
Lanari: CS - Filter 11
0 2 4 6 8−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5System 1
time (s)
InputOutput
0 2 4 6 8−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5System 2
time (s)
InputOutput
10−2
100
102
0
0.5
1
1.5
2
2.5
frequency (rad/s)
Magnitude
System 1
10−2
100
102
0
0.5
1
1.5
2
2.5
frequency (rad/s)
Magnitude
System 2
(⇣,!n) = (0.2, 1)<latexit sha1_base64="ui8DM0qizZGI64W40zYdirkRkr8=">AAACBXicbVBNS8NAEN3Ur1q/oh71sFiEFkpJWtF6EApePFawH9CGsNlO26WbTdjdCLX04sW/4sWDIl79D978N6ZtELU+GHi8N8PMPC/kTGnL+jRSS8srq2vp9czG5tb2jrm711BBJCnUacAD2fKIAs4E1DXTHFqhBOJ7HJre8HLqN29BKhaIGz0KwfFJX7Aeo0THkmse5jp3oEkBdwIf+sQVeXyBc1axVMB23jWzVtGaAS8SOyFZlKDmmh+dbkAjH4SmnCjVtq1QO2MiNaMcJplOpCAkdEj60I6pID4oZzz7YoKPY6WLe4GMS2g8U39OjImv1Mj34k6f6IH6603F/7x2pHsVZ8xEGGkQdL6oF3GsAzyNBHeZBKr5KCaEShbfiumASEJ1HFxmFsL5FKffLy+SRqlol4vl65NstZLEkUYH6AjlkI3OUBVdoRqqI4ru0SN6Ri/Gg/FkvBpv89aUkczso18w3r8A1V2Vqg==</latexit>
(⇣,!n) = (0.2, 8)<latexit sha1_base64="mAezgZAG76ejr/sZeGx09DNpwLA=">AAACBXicbVBNS8NAEN3Ur1q/oh71sFiEFkpJWtF6EApePFawH9CGsNlO26WbTdjdCLX04sW/4sWDIl79D978N6ZtELU+GHi8N8PMPC/kTGnL+jRSS8srq2vp9czG5tb2jrm711BBJCnUacAD2fKIAs4E1DXTHFqhBOJ7HJre8HLqN29BKhaIGz0KwfFJX7Aeo0THkmse5jp3oEkBdwIf+sQVeXyBc1axVMCVvGtmraI1A14kdkKyKEHNNT863YBGPghNOVGqbVuhdsZEakY5TDKdSEFI6JD0oR1TQXxQznj2xQQfx0oX9wIZl9B4pv6cGBNfqZHvxZ0+0QP115uK/3ntSPcqzpiJMNIg6HxRL+JYB3gaCe4yCVTzUUwIlSy+FdMBkYTqOLjMLITzKU6/X14kjVLRLhfL1yfZaiWJI40O0BHKIRudoSq6QjVURxTdo0f0jF6MB+PJeDXe5q0pI5nZR79gvH8B4ACVsQ==</latexit>
Lanari: CS - Filter 12
10−2
100
102
0
0.5
1
1.5
2
2.5
System 3
frequency (rad/s)
Magnitude
0 2 4 6 8−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5System 3
time (s)
InputOutput
0 2 4 6 8−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5System 4
time (s)
InputOutput
10−2
100
102
0
0.5
1
1.5
2
2.5
frequency (rad/s)
Magnitude
System 4
(⇣,!n) = (0.2, 10)<latexit sha1_base64="ZyFzk9a0dF+G/rp8W8Lws3w/WDE=">AAACBnicbVBNS8NAEN3Ur1q/oh5FWCxCC6UkrWg9CAUvHivYD2hD2Gyn7dLNJuxuhFp68uJf8eJBEa/+Bm/+G9M2iFofDDzem2FmnhdyprRlfRqppeWV1bX0emZjc2t7x9zda6ggkhTqNOCBbHlEAWcC6pppDq1QAvE9Dk1veDn1m7cgFQvEjR6F4PikL1iPUaJjyTUPc5070KSAO4EPfeKKPL7AOatYKmDbyrtm1ipaM+BFYickixLUXPOj0w1o5IPQlBOl2rYVamdMpGaUwyTTiRSEhA5JH9oxFcQH5Yxnb0zwcax0cS+QcQmNZ+rPiTHxlRr5XtzpEz1Qf72p+J/XjnSv4oyZCCMNgs4X9SKOdYCnmeAuk0A1H8WEUMniWzEdEEmojpPLzEI4n+L0++VF0igV7XKxfH2SrVaSONLoAB2hHLLRGaqiK1RDdUTRPXpEz+jFeDCejFfjbd6aMpKZffQLxvsXSyaV5A==</latexit>
(⇣,!n) = (0.2, 100)<latexit sha1_base64="SJwrGxplKhwQ6h0IrSKdQ0h3Tks=">AAACB3icbVBNS8NAEN3Ur1q/oh4FWSxCC6UkrWg9CAUvHivYD2hD2Gyn7dLNJuxuhFp68+Jf8eJBEa/+BW/+G9M2iFofDDzem2FmnhdyprRlfRqppeWV1bX0emZjc2t7x9zda6ggkhTqNOCBbHlEAWcC6pppDq1QAvE9Dk1veDn1m7cgFQvEjR6F4PikL1iPUaJjyTUPc5070KSAO4EPfeKKPL7AOatYKmDbsvKumbWK1gx4kdgJyaIENdf86HQDGvkgNOVEqbZthdoZE6kZ5TDJdCIFIaFD0od2TAXxQTnj2R8TfBwrXdwLZFxC45n6c2JMfKVGvhd3+kQP1F9vKv7ntSPdqzhjJsJIg6DzRb2IYx3gaSi4yyRQzUcxIVSy+FZMB0QSquPoMrMQzqc4/X55kTRKRbtcLF+fZKuVJI40OkBHKIdsdIaq6ArVUB1RdI8e0TN6MR6MJ+PVeJu3poxkZh/9gvH+BcESlh4=</latexit>
Lanari: CS - Filter0 1 2 3 4
−3
−2
−1
0
1
2
3System 3
time (s)
yySS
130 1 2 3 4
−1
−0.5
0
0.5
1System 3 − transient
time (s)
forced response and steady state
transient as the difference between the forced response and the steady state
If a system is asymptotically stable it admits a steady state (not necessarily constant) to any persistent input: for example ramp, parabola, sinusoid. In this case we can also define the transient as the difference between the forced and the steady state response, that is transient exists also for inputs which differ from the step. However we decided to characterize the transient with specific quantities on the step input.
example: transient for a sinusoidal input
transient
Lanari: CS - Filter 14
transient: bandwidthfor the typical magnitude plots encountered so far, we define the bandwidth B3 as the first
frequency such that for all frequencies greater than the bandwidth the magnitude is attenuated
by a factor greater than w.r.t its value in ! = 0 . Recall that 1/p2
20 log10
✓1p2
◆⇡ �3 dB
|W (jB3)| =|W (j0)|�
2
|W (jB3)|dB = |W (j0)|dB � 3
B3 :
and being
B3 :
• characterizes the filtering capacities of the dynamical system with transfer function W (s)
<latexit sha1_base64="INJm9VGNM1g/uMh8tuI8nA9u828=">AAACAXicbVDLSsNAFJ3UV62vqBvBzWARXMVJUdplwY3LCvYBTSiT6aQdOsnEmYlYQt34K25cKOLWv3Dn3zhts9DWAxcO59zLvfcECWdKI/RtFVZW19Y3ipulre2d3T17/6ClRCoJbRLBhewEWFHOYtrUTHPaSSTFUcBpOxhdTf32PZWKifhWjxPqR3gQs5ARrI3Us4/cc0/dSZ1VJtDDSSLFA0ROFVV7dhk5aAa4TNyclEGORs/+8vqCpBGNNeFYqa6LEu1nWGpGOJ2UvFTRBJMRHtCuoTGOqPKz2QcTeGqUPgyFNBVrOFN/T2Q4UmocBaYzwnqoFr2p+J/XTXVY8zMWJ6mmMZkvClMOtYDTOGCfSUo0HxuCiWTmVkiGWGKiTWglE4K7+PIyaVUc99JBNxflei2PowiOwQk4Ay6ogjq4Bg3QBAQ8gmfwCt6sJ+vFerc+5q0FK585BH9gff4ATkeVdw==</latexit>
1/p2 ⇡ 0.707
• relative to the static gain |W (j0)|
1
1
!
! the first system W1 (j!) cuts off more frequencies than the second
B3,1
B3,2
B3,1 < B3,2
| W1 (j!) |/| W1 (j0) |
| W2 (j!) |
| W2 (j!) |/| W2 (j0) |
Lanari: CS - Filter 15
−40
−30
−20
−10
−30
10
frequency (rad/s)
Magnitude (dB)
1/| ¿ |
transient: simplest example
asymptotically stable system (therefore ¿ > 0)
B3 =1
�
W (s) =K
1 + �s
magnitude plot
normalized w.r.t. |K|dB
being
and
|W (j⇥)|dB � |W (j0)|dB = |W (j⇥)|dB � |K|dB= |K|dB + |1/(1 + j⇥�)|dB � |K|dB= |1/(1 + j⇥�)|dB
|1 + j�/|� | |dB = 20 log10⇥2 � 3 dB
for a first order system, the bandwidth coincides with the cutoff frequency
• similarly for higher order systems in the presence of a dominant pole
Lanari: CS - Filter 16
transient: resonant peak
we define the resonant peak Mr as the maximum value of the frequency response magnitude
referred to its value in ! = 0
Mr =max |W (j�)|
|W (j0)|
Mr|dB = max |W (j�)|dB � |W (j0)|dB
or in dB
a high resonant peak indicates that the system behaves similarly to a second order system
with low damping coefficient
10-2 10-1 100 101 102
frequency (rad/s)
0
1
2
3
4
5
6
Magnitude (absolute)
Magnitude trinomial factor at denominator
= 0.5 = 0.3 = 0.1
Lanari: CS - Filter 17
transient: resonant peak
• Note that the resonant peak is defined w.r.t. the value of the magnitude in ! = 0 and it is not
just the maximum value (a constant gain F (s) = K would not give any resonant peak)
• since the presence of a peak in a frequency response is similar to the peak of a second order
system with complex conjugate poles and low values of the damping coefficient, the higher
the peak the smaller the “equivalent damping” value and therefore the higher the overshoot
in the step response
• from the frequency response we get not only information on the steady state but also on
the transient
Lanari: CS - Filter 18
Transient parameters in the frequency domain: on a plot with normalized magnitude (not in dB)
|W (j�)||W (j0)|
!
1
Mr
1p2
B3
transient: frequency domain characterization
magnitudeplot not in dB
transfer functionis strictly properand therefore themagnitude tendsto 0 as ! tends to 1
any sinusoidal signal with frequency greater than B3 will be attenuated at steady state by more than 0.707
• a similar plot can be drawn when the magnitude is in dB
Lanari: CS - Filter 19
transient: relationships in t and !
B3 tr ⇡ constant
1 +Mp
Mr⇡ constant
typically (with some exceptions)
• higher bandwidth (higher frequency components of the input signal are not attenuated and
therefore are allowed to go through) leads to smaller rise time (faster system response)
• higher resonant peak (as if we had a second order system with lower damping
coefficient) leads to higher overshoot (the oscillation damps out slower)
• very useful relationships in order to understand the connections between time and
frequency domain response characteristics
in timein frequency
in timein frequency
Lanari: CS - Filter 20
transient: explicit relations for second order system
W (s) =1
1 + 2⇣s
!n+
s2
!2n
<latexit sha1_base64="IpmbfFDwzOEww6Wda2Is5uajW8U=">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</latexit>
• step response 1� 1p1� ⇣2
e�⇣!nt
"p
1� ⇣2 !nt+ arctan
p1� ⇣2
⇣
#
<latexit sha1_base64="xuIj9+bNPAv7Ozcm4yL0fBZJvw0=">AAACf3icbVHLbtswEKTUV+o+4rbHXogYRQskMaSkaJNbgFxyTIE6CWAqxope2UQoSiVXBVRBv5EPy63/0kNpWejLHYDAcHaWSw7TUitHUfQ9CO/df/Dw0dbjwZOnz55vD1+8vHBFZSVOZKELe5WCQ60MTkiRxqvSIuSpxsv05nRVv/yK1qnCfKa6xCSHhVGZkkBemg1v430uMguyidtGzJUrNdSOao1cuC+WmnhffEOC64O2xetmvRFFjguYGU4tFxozmm6Yudjjv218lwuwksD0wzbObjrm26xaLCmZDUfROOrAN0nckxHrcT4b3ol5IascDUkNzk3jqKSkAUtKamwHonJYgryBBU49NZCjS5ouv5a/8cqcZ4X1yxDv1D87Gsidq/PUO3Ogpfu3thL/V5tWlB0ljTJlRWjkelBWaU4FX30GnyuLknTtCUir/F25XIIPiPyXDboQjlf48OvJm+TiYBwfjg8/vR+dHPVxbLHXbIe9YzH7yE7YGTtnEybZj2An2A32wiB8G47DaG0Ng77nFfsL4fFPV6jChw==</latexit>
0 < ³ < 1
Mp = e� ⇡⇣p
1� ⇣2<latexit sha1_base64="xCe70v2s47+4GU2b6MdHhgmWDQI=">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</latexit>
Mr =1
2⇣p1� ⇣2
<latexit sha1_base64="hhJEz6E42/UuNVikYb5Pe9xwQxg=">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</latexit>
• resonance peak ⇣ 1p2
<latexit sha1_base64="Dhif0sXbH1MBvrMTKCeaOkZUGuM=">AAACBXicbVDJSgNBEO1xjXGLetRDYxA8hRkjGm8BLx4jmAUyIfR0apLGnsXuGiEOc/Hir3jxoIhX/8Gbf2NnQdT4oODxXhVV9bxYCo22/WnNzS8sLi3nVvKra+sbm4Wt7YaOEsWhziMZqZbHNEgRQh0FSmjFCljgSWh61+cjv3kLSosovMJhDJ2A9UPhC87QSN3CnnsHyKgrgbq+Yjx1stTVNwrToyzrFop2yR6DzhJnSopkilq38OH2Ip4EECKXTOu2Y8fYSZlCwSVkeTfREDN+zfrQNjRkAehOOv4iowdG6VE/UqZCpGP150TKAq2HgWc6A4YD/dcbif957QT9SicVYZwghHyyyE8kxYiOIqE9oYCjHBrCuBLmVsoHzISBJrj8OISzEU6+X54ljaOSUy6VL4+L1co0jhzZJfvkkDjklFTJBamROuHknjySZ/JiPVhP1qv1Nmmds6YzO+QXrPcvuaWY1g==</latexit>
B3 = !n
q1� 2⇣2 +
p2� 4⇣2 + 4⇣4
<latexit sha1_base64="+ofO97eFv2GjSgiAn+S9AtGD6gU=">AAACKHicbZDLSsNAFIYnXmu9VV26GSyCIJakLV4WYtGNSwWrQlPDZHraDk4mceZEqKGP48ZXcSOiiFufxLQG7z8MfPznHM6c34+kMGjbr9bI6Nj4xGRuKj89Mzs3X1hYPDVhrDnUeShDfe4zA1IoqKNACeeRBhb4Es78y4NB/ewatBGhOsFeBM2AdZRoC84wtbzC3r5XobvUDQPoME9R11xpTJyNsnsDyC7KdD2zyhvVLyvDar/vFYp2yR6K/gUngyLJdOQVHt1WyOMAFHLJjGk4doTNhGkUXEI/78YGIsYvWQcaKSoWgGkmw0P7dDV1WrQd6vQppEP3+0TCAmN6gZ92Bgy75ndtYP5Xa8TY3m4mQkUxguIfi9qxpBjSQWq0JTRwlL0UGNci/SvlXaYZxzTb/DCEnYE2P0/+C6flklMpVY6rxdp2FkeOLJMVskYcskVq5JAckTrh5JbckyfybN1ZD9aL9frROmJlM0vkh6y3d/wYpDw=</latexit>
tr =1
!n
1p1� ⇣2
"⇡ � arctan
p1� ⇣2
⇣
#
<latexit sha1_base64="y6kXqM0KQBtY6ixR9Au3NznkYJ8=">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</latexit>
• bandwidth
• rise time (up to roughly ³= 0.7 )
for a second order systemsome explicit expressions can be obtained (as an example)
valid for
(1 being = 100%)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
�
0
0.2
0.4
0.6
0.8
1
��
• overshoot
Lanari: CS - Filter 21
0 2 4 6 8 10 12����
0
0.2
0.4
0.6
0.8
1
1.2
��������������� ��� ��
n = 5n = 9n = 13n = 29with Lanczos
detail of the truncated Fourier expansion of a pulse train (square wave): the more components with higher frequency we include in the sum the better the approximation is.
the discontinuous signal (pulse train) is made of infinite sinusoidal components
almost similarly, the step function has an infinite frequency content
example: a discontinuous signal
0 5 10 15 20 25 30 35 40 45 50 55����
-1
-0.5
0
0.5
1
���������
<latexit sha1_base64="Q9zHMzlTt+0cLN4uxE274umEhUU=">AAACEHicbVDJSgNBEO1xjXGLevTSGMQESZgOirkIAS8eI2SDJA49nU7SpGehu0YIw3yCF3/FiwdFvHr05t/YWQ6a+KDg8V4VVfXcUAoNtv1trayurW9sprbS2zu7e/uZg8OGDiLFeJ0FMlAtl2ouhc/rIEDyVqg49VzJm+7oZuI3H7jSIvBrMA5516MDX/QFo2AkJ3NGzksdHXlOLK5JgnMFkr+PRdLpcQnUiQskyUFB1PJOJmsX7SnwMiFzkkVzVJ3MV6cXsMjjPjBJtW4TO4RuTBUIJnmS7kSah5SN6IC3DfWpx3U3nj6U4FOj9HA/UKZ8wFP190RMPa3Hnms6PQpDvehNxP+8dgT9cjcWfhgB99lsUT+SGAI8SQf3hOIM5NgQypQwt2I2pIoyMBmmTQhk8eVl0igVyWXRvrvIVsrzOFLoGJ2gHCLoClXQLaqiOmLoET2jV/RmPVkv1rv1MWtdseYzR+gPrM8fdP2a8A==</latexit>
1 + 2X
i=1
(�1)i��1(t� iT )
we can therefore see in the frequency domain the filtering effect of a system on a step input
!
¼± (!)
continuous spectrumin the frequency domain
t
time domain±—1 (t)
Lanari: CS - Filter 22
1 1
!
!
!
!
!
!
systems with different bandwidths
B3,1 B3,2
B3,1 < B3,2
| F1 (j!) | | F2 (j!) |
output frequency contentis different
t t
faster response
system as a filter (transient)
input frequency content(this is not the frequency content of a step function, it’s just for illustration purposes)
Lanari: CS - Filter 23
Mass - Spring - Damper
k
µ
p
uF (s) =
1
ms2 + µs+ km
0 < µ < 2pkm p1,2 =
� µm ± j
q4�
km
��� µm
�2
2
µ = 2pkm
transfer function
• over-damped
• critically-damped
• under-damped
µ > 2pkm
p1,2 = � µ
2m
p1,2 =� µ
m ±q� µ
m
�2 � 4�
km
�
2
complex conjugate poles
real coincident poles
real distinct poles
asymptotically stable system for µ > 0
Lanari: CS - Filter 24
Mass - Spring - Damper
0 < µ < 2pkm
µ = 2pkm
• over-damped
• critically-damped
• under-damped
µ > 2pkm
trinomial factor
two coincident binomial factors
two distinct binomial factors
�n =
rk
m� =
µ
2pkm
1
�= �p1,2 =
µ
2m=
rk
m
1
�1= �p1 >
rk
m
1
�2= �p2 <
rk
m
gain = 1/k
Lanari: CS - Filter 25
Mass - Spring - Damper
m = 1 kgk = 100 N/m
magnitude in terms of the damping factor µ
the two single binomials are also shown (over-damped case) in order to put in evidence the dominant pole (corresponding to the real pole closest to the origin)
10-1 100 101 102
frequency (rad/s)
-75
-70
-65
-60
-55
-50
-45
-40
-35
-30
-25
Magnitude (dB)
MSD - Magnitude with variable
Lanari: CS - Filter 26
input force on mass 1
7 - MSDsystem
(with damping)
100
101
102
103
−200
−150
−100
−50
Magnit
ude
100
101
102
103
−800
−600
−400
−200
0
Phase
7-mass
output position on mass 3
resonance peaks
anti-resonancepeak
natural frequenciesvery close to each other
119.6311 107.5098 78.7957 73.3411 42.2283 19.0596 11.0478
natural frequencies
Lanari: CS - Filter 27
dynamic bass microphone(tailored for kick drum, works well with any low frequency instrument,
low frequency peak at 100 Hz)
a 10.000 € voice microphone ...
electric guitar microphone
microphones
recall that 1 Hz = 2¼ rad/s and that voice is in the frequency rance roughly from 300 to 3000 Hz
Lanari: CS - Filter 28
quarter-car suspension model
wheel
body
road
mb
mw
xb
xw
r
ks bs
kt
fs
Model of a quarter part of a car with its wheel and tire
The body with mass mb represents the car chassis connected to the wheel by a passive spring (ks), and a shock absorber represented by a damper (bs).The spring (kt) models the compressibility of the tire pneumatic.
In an active suspension a hydraulic actuator (fs) between the chassis and wheel assembly may help in balancing conflicting objectives as passenger comfort, road handling and suspension deflection.
mbxb + ks(xb � xw) + bs(xb � xw) = fs
mwxw � ks(xb � xw) + bs(xb � xw) + kt(xw � r) = �fs
fs acts on both the body and the wheel assembly
r can be seen as an input affecting the evolution of the system through the tire (disturbance)
r(t) genericterrain profile
Lanari: CS - Filter 29
state vector
several outputs of interest
⇥xb xb xw xw
⇤T
C1 =⇥1 0 0 0
⇤
C2 =⇥�ks/mb �bs/mb ks/mb bs/mb
⇤
body (passenger) position
body (passenger) acceleration
suspension deflectionC3 =⇥1 0 �1 0
⇤
two inputs (one, fs, can be controlled, the other is the disturbance r)
by setting one of the two inputs to zero and choosing the output of interest, we have a SISO system with corresponding transfer function
Passenger comfort is associated to small passenger acceleration
Physical limitation of the actuator (limits on maximum displacements) defines a constraint
these are two of the possible outputs
Lanari: CS - Filter 30
�150
�100
�50
0
50
100
To: a
b
100
101
102
�200
�150
�100
�50
0
50To
: sd
Magnitude from road disturbace and actuator force (fs) to body acceleration and suspension travel
Frequency (rad/sec)
Mag
nitu
de (d
B)
Input: road disturbance (r)Input: actuator force (fs)
actuator to body accelerationfrequency response magnitude
actuator to suspension deflectionfrequency response magnitude
road to body accelerationfrequency response magnitude
road to suspension deflectionfrequency response magnitude
rattlesnake frequency
tire-hop frequency
Lanari: CS - Filter 31
rattlesnake frequency: pure imaginary zeros in the transfer function from the actuator to the suspension deflection, anti-resonance at 22.97 rad/s
tire-hop frequency: pure imaginary zeros in the transfer function from the actuator to the body acceleration (also from actuator to body position), anti-resonance at 56.27 rad/s
at these frequencies it is difficult to counteract any effect of the road on acceleration or on the suspension deflection (no control “authority”)