2020 lec11 filter

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


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


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


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


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

?


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>


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>


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


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


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>


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


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) |


−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


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


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


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


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


transient: explicit relations for second order system

W (s) =1

1 + 2⇣s

!n+

s2

!2n

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• step response 1� 1p1� ⇣2

e�⇣!nt

"p

1� ⇣2 !nt+ arctan

p1� ⇣2

⇣

#

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0 < ³ < 1

Mp = e� ⇡⇣p

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Mr =1

2⇣p1� ⇣2

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• resonance peak ⇣ 1p2

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B3 = !n

q1� 2⇣2 +

p2� 4⇣2 + 4⇣4

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tr =1

!n

1p1� ⇣2

"⇡ � arctan

p1� ⇣2

⇣

#

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• 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


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

��

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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)


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)


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



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



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


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


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


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


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


�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


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”)

2020 lec11 filter

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