rti vorlesung 8 - eth z...zeros 5 โข a zero ๐is a frequency at which the transfer function ฮฃ is...
TRANSCRIPT
8.11.2019
RTI Vorlesung 8
Recap
2
BIBO
Stability
Poles
Transfer
Functions
2nd-Order
System
Zeros
Static gain
Transfer function
3
Y ๐ โ ๐ 2 + ๐1 โ ๐ + ๐0๐ ๐
= ๐1 โ ๐ + ๐0๐ ๐
โ ๐ ๐
Origin: differential equation: ๐ฆโฒโฒ + ๐1 โ ๐ฆโฒ + ๐0 โ ๐ฆ = ๐1 โ ๐ข
โฒ + ๐0 โ ๐ข
Y ๐ โ ๐ 2 +๐1 โ Y ๐ โ ๐ + ๐0 โ Y ๐ = ๐1 โ ๐ ๐ โ ๐ + ๐0 โ ๐ ๐
Y ๐ =๐(๐ )
๐(๐ )โ ๐ ๐
The transfer function is on operator which maps the input ๐ ๐ to the output ๐(๐ ) in
the frequency domain
Y ๐ = ๐ด(s) โ ๐ ๐
Poles
4
โข A pole ๐ is a frequency ๐ at which ฮฃ ๐ is infinite.
โข The poles describe the inherent dynamics of the plant.
โข The poles are relevant for the stability of the system
The poles are the roots of the denominator polynomial ๐(๐ ).
Y ๐ โ ๐ 2 + ๐1 โ ๐ + ๐0๐ ๐
= ๐1 โ ๐ + ๐0๐ ๐
โ ๐ ๐
Origin: differential equation: ๐ฆโฒโฒ + ๐1 โ ๐ฆโฒ + ๐0 โ ๐ฆ = ๐1 โ ๐ข
โฒ + ๐0 โ ๐ข
Y ๐ โ ๐ 2 +๐1 โ Y ๐ โ ๐ + ๐0 โ Y ๐ = ๐1 โ ๐ ๐ โ ๐ + ๐0 โ ๐ ๐
Zeros
5
โข A zero ๐ is a frequency ๐ at which the transfer function ฮฃ ๐ is zero.
โข This zero describes the nontrivial dynamics of the system, which for a given initial state
and a given input yield an output of ๐ฆ ๐ก = 0 for all times ๐ก โฅ 0.
The zeros are the roots of the numerator polynomial ๐(๐ ).
Y ๐ โ ๐ 2 + ๐1 โ ๐ + ๐0๐ ๐
= ๐1 โ ๐ + ๐0๐ ๐
โ ๐ ๐
Origin: differential equation: ๐ฆโฒโฒ + ๐1 โ ๐ฆโฒ + ๐0 โ ๐ฆ = ๐1 โ ๐ข
โฒ + ๐0 โ ๐ข
Y ๐ โ ๐ 2 +๐1 โ Y ๐ โ ๐ + ๐0 โ Y ๐ = ๐1 โ ๐ ๐ โ ๐ + ๐0 โ ๐ ๐
Static gain
6
The static gain ฮฃ 0 is the value of the transfer function at ๐ = 0: ฮฃ 0 =๐0
๐0
โข The static gain ฮฃ 0 is the asymptotic value of lim๐กโโ
๐ฆ(๐ก) in response to a step at the
input ๐ข ๐ก = โ(๐ก).
Y ๐ โ ๐ 2 + ๐1 โ ๐ + ๐0๐ ๐
= ๐1 โ ๐ + ๐0๐ ๐
โ ๐ ๐
Origin: differential equation: ๐ฆโฒโฒ + ๐1 โ ๐ฆโฒ + ๐0 โ ๐ฆ = ๐1 โ ๐ข
โฒ + ๐0 โ ๐ข
Y ๐ โ ๐ 2 +๐1 โ Y ๐ โ ๐ + ๐0 โ Y ๐ = ๐1 โ ๐ ๐ โ ๐ + ๐0 โ ๐ ๐
BIBO Stability
7
ฮฃ(๐ )๐ข(๐ก) ๐ฆ(๐ก)
A system is called bounded-input bounded-output (BIBO) stable iff all
โข finite inputs ๐ข ๐ก < ๐ result in
โข finite outputs ๐ฆ ๐ก < ๐
Theorem 1: For linear systems, BIBO stability is given if the following holds
Theorem 2: a system ฮฃ(๐ ) is
โข BIBO stable iff all poles ๐ have negative real parts, and
โข not BIBO stable in all other cases.
๐ข(๐ก) ๐ฆ(๐ก)
๐ ๐ก = Impulse response
2nd Order Systems:
Poles and the Time-Domain Behavior
8
Location of the poles
in the complex plane
Step response
in the time domain
Zeros and the Time-Domain Behavior
9
๐ฟ = 0.5
๐ด ๐ =โ1/๐ โ ๐ + 1 โ ๐0
2
๐ 2 + 2 โ ๐ฟ โ ๐0 โ ๐ + ๐02
20.09. Lektion 1 โ Einfรผhrung
27.09. Lektion 2 โ Modellbildung
4.10. Lektion 3 โ Systemdarstellung, Normierung, Linearisierung
11.10. Lektion 4 โ Analyse I, allg. Lรถsung, Systeme erster Ordnung, Stabilitรคt
18.10. Lektion 5 โ Analyse II, Zustandsraum, Steuerbarkeit/Beobachtbarkeit
25.10. Lektion 6 โ Laplace I, รbertragungsfunktionen
1.11. Lektion 7 โ Laplace II, Lรถsung, Pole/Nullstellen, BIBO-Stabilitรคt
8.11. Lektion 8 โ Frequenzgรคnge (RH hรคlt VL)
15.11. Lektion 9 โ Systemidentifikation, Modellunsicherheiten
22.11. Lektion 10 โ Analyse geschlossener Regelkreise
29.11. Lektion 11 โ Randbedingungen
6.12. Lektion 12 โ Spezifikationen geregelter Systeme
13.12. Lektion 13 โ Reglerentwurf I, PID (RH hรคlt VL)
20.12. Lektion 14 โ Reglerentwurf II, โloop shapingโ
Modellierung
Systemanalyse im Zeitbereich
Systemanalyse im Frequenzbereich
Reglerauslegung
10
Today: all about one property of asymptotically stable LTI transfer functions ๐ด(๐ )
12
The mapping ๐ด ๐๐ โถ โ โ โ is the frequency response of the system ๐ด(๐ )
Definition of frequency response (Frequenzgang)
Time Domain Frequency Domain
Given: ๐ด(๐ )
๐ด ๐๐ = ๐ผ2 + ๐ฝ2
โ ๐ด(๐๐) = arctan โ๐ฝ
๐ผ
๐ด ๐๐ = ๐ผ โ ๐๐ฝ, ๐ผ, ๐ฝ โ โ
<
Im
Re
ฮฃ(๐เท๐)
โ ๐ด(๐เท๐)
๐ผ(๐เท๐)
โ๐ฝ(๐เท๐)
Proof of ๐ฆโ(๐ก) = ๐ โ cos(๐ โ ๐ก + ๐)
13
with
For any asymptotically stable LTI system ฮฃ(๐ ):Laplace Table
๐ฅ(๐ก) ๐(๐ )
โ ๐ก โ cos(๐ โ ๐ก)๐
๐ 2 + ๐2
โ ๐ก โ sin(๐ โ ๐ก)๐
๐ 2 + ๐2
โ ๐ก โ ๐ก๐ โ ๐๐ผโ ๐ก๐!
๐ โ ๐ผ ๐+1
How to get frequency response experimentally
14
โข Excite an asymptotically stable LTI
system with a harmonic input
signal of a specific frequency
โข Measure the steady-state response
โข Compare the amplitude and the
phase of the response with the
amplitude and the phase of the input
signal; save results for chosen
frequency
โข Repeat the experiment for various
other excitation frequencies
Example of a Frequency Response
๐ข(๐ก)
๐ฆ(๐ก)
Linear system ฮฃ ๐ :
damped single mass oscillator,
two states โ 2nd order system
๐๐
[โ]
Frequency ๐๐๐
[deg]
Frequency ๐
General representations of frequency responses
16
The frequency response can be displayed by two different diagrams.
โข The frequency-explicit Bode diagram with two separate curves
โข The frequency-implicit Nyquist diagram.
ยซBodeยป diagram Nyquist diagram
Real component
Ima
gin
ary
co
mp
on
ent
๐๐
[โ]
๐๐
[deg]
17
The mapping ๐ด ๐๐ โถ โ โ โ is the frequency response of the system ๐ด(๐ )
Definition of frequency response (Frequenzgang)
Time Domain Frequency Domain
Given: ๐ด(๐ )
๐ด ๐๐ = ๐ผ2 + ๐ฝ2
โ ๐ด(๐๐) = arctan โ๐ฝ
๐ผ
๐ด ๐๐ = ๐ผ โ ๐๐ฝ, ๐ผ, ๐ฝ โ โ
<
Im
Re
ฮฃ(๐เท๐)
โ ๐ด(๐เท๐)
๐ผ(๐เท๐)
โ๐ฝ(๐เท๐)
18
Time
Domain
Frequency
Domain
๐ก
๐ = ๐ + ๐๐
ฮฃ ๐ โ ๐
๐ 2 + ๐2= ๐t ๐ +
๐ผ โ ๐ + ๐ฝ โ ๐
s2 + ๐2=๐ผ โ ๐ + ๐ฝ โ ๐
s2 + ๐2
ฮฃ ๐ โ ๐
๐ 2 + ๐2= ๐t ๐ + ๐โ ๐ = ๐โ(๐ )
โ for lim๐กโโ
, we must have lim๐ โ๐๐
:
โ ฮฃ ๐ โ ๐ = (๐ 2 + ๐2) โ ๐t ๐ + ๐ผ โ ๐ + ๐ฝ โ ๐ = (๐ผ โ ๐ + ๐ฝ โ ๐)
โ ฮฃ ๐๐ โ ๐๐ = ๐ผ โ ๐๐ + ๐ฝ โ ๐
How to get frequency response from ๐ด(๐ ) (I)aka โproof: lim
๐กโโโ lim
๐ โ๐๐โ
๐ ๐ = ๐โ ๐ + ๐๐ก(๐ )๐ฆ ๐ก = ๐ฆโ ๐ก + ๐ฆ๐ก ๐ก
lim๐กโโ
๐ก โ [0,โ] ๐ = ๐ + ๐ โ ๐
๐ฆ ๐ก = ๐ฆโ ๐ก ๐ ๐ = ๐โ(๐ )lim๐ โ?
Free variableFree variable
19
Re(ฮฃ ๐๐ )
Im(ฮฃ ๐๐ )
How to get frequency response from ๐ด(๐ ) (II)determine ๐ and ๐
20
The mapping ๐ด ๐๐ โถ โ โ โ is the frequency response of the system ๐ด(๐ )
Definition of frequency response (Frequenzgang)
Time Domain Frequency Domain
Given: ๐ด(๐ )
๐ด ๐๐ = ๐ผ2 + ๐ฝ2
โ ๐ด(๐๐) = arctan โ๐ฝ
๐ผ
๐ด ๐๐ = ๐ผ โ ๐๐ฝ, ๐ผ, ๐ฝ โ โ
<
Im
Re
ฮฃ(๐เท๐)
โ ๐ด(๐เท๐)
๐ผ(๐เท๐)
โ๐ฝ(๐เท๐)
20.09. Lektion 1 โ Einfรผhrung
27.09. Lektion 2 โ Modellbildung
4.10. Lektion 3 โ Systemdarstellung, Normierung, Linearisierung
11.10. Lektion 4 โ Analyse I, allg. Lรถsung, Systeme erster Ordnung, Stabilitรคt
18.10. Lektion 5 โ Analyse II, Zustandsraum, Steuerbarkeit/Beobachtbarkeit
25.10. Lektion 6 โ Laplace I, รbertragungsfunktionen
1.11. Lektion 7 โ Laplace II, Lรถsung, Pole/Nullstellen, BIBO-Stabilitรคt
8.11. Lektion 8 โ Frequenzgรคnge (RH hรคlt VL)
15.11. Lektion 9 โ Systemidentifikation, Modellunsicherheiten
22.11. Lektion 10 โ Analyse geschlossener Regelkreise
29.11. Lektion 11 โ Randbedingungen
6.12. Lektion 12 โ Spezifikationen geregelter Systeme
13.12. Lektion 13 โ Reglerentwurf I, PID (RH hรคlt VL)
20.12. Lektion 14 โ Reglerentwurf II, โloop shapingโ
Modellierung
Systemanalyse im Zeitbereich
Systemanalyse im Frequenzbereich
Reglerauslegung
21
Application of frequency responses
22
Development of nonparametric and parametric
models of the system ฮฃ(๐ ) via measurements
of the frequency response (ยซexperimentsยป).
Lecture 9
Modeling
Controller Design
Using the Nyquist theorem to assess the stability of a closed-loop
system based on the frequency response of the open-loop system.
Lecture 10, 11, โฆ|ฮฃ
๐๐|[โ]
โ ฮฃ๐๐
[deg]
23
Examples of frequency responses
Bode diagrams of 1st order and 2nd systems
Bode Diagram Convention
24
โBode diagramโ so far Conventional Bode Diagram
โข Plot phase in degrees
โข Plot frequency axis in log10 base
โข Plot magnitude in dB (decibel) โ also log10 base
Decibel Scale
25
Conversion Table
Decibel Scale:Decimal Scale Decibel Scale
100 40
10 20
5 13.97โฆ
2 6.02โฆ
1 0
0.1 -20
0.01 -40
0 -Inf
Advantages of log10 and dB (I)
26
โข Curved lines become straight asymptotesโข Slope of asymptotes change depending on poles/zeros location of ฮฃ(๐ )
โข Poles and zeros of transfer function define โedgesโโข โ In simple cases, transfer function can be read directly from Bode diagram
โBode diagramโ so far Conventional Bode Diagram
Rules for drawing a Bode diagram
27
TypeMagnitude
Change
Phase
Change
Stable pole @๐๐ -20 dB/dec -90ยฐ
Unstable pole @๐๐ -20 dB/dec +90ยฐ
Minimumphase zero @๐๐ +20 dB/dec +90ยฐ
Non-minimumphase zero @๐๐ +20 dB/dec -90ยฐ
Time delay 0 dB/dec โ๐ โ ๐
Rules for drawing a Bode diagram
๐๐ = ๐ in ๐๐๐
๐
๐๐ = ๐ in ๐๐๐
๐
Example: reading ฮฃ ๐ from bode plot
28
TypeMagnitude
Change
Phase
Change
Stable pole @๐๐ -20 dB/dec -90ยฐ
Rules for drawing a Bode diagram
โ ๐๐1,2= ๐1,2 = ๐0
โ ๐
General Bode diagram of a 2nd order system
29
Example: with ๐ = 1
๐0
๐10 โ ๐0
๐10 โ ๐0
0.1 โ ๐0
๐ ๐
๐(๐)
Static gain
Magnitude change
Phase change
Cut-off frequency
30
Calculate the frequency response of
๐
|ฮฃ(๐๐)|
๐โ ฮฃ(๐๐)
ฮฃ ๐๐ =๐ ๐๐
๐ ๐๐
โ ฮฃ ๐๐ = โ ๐ ๐๐ โ โ ๐ ๐๐
ฮฃ ๐๐ =๐ ๐๐
๐ ๐๐
Drawing frequency response
Example: 1st order system (II)
ฮฃ ๐ =1
๐ + 1
Bode diagram of a 1st order system
31
Example: with ๐ = 1, ๐ = 1
Static gain
Cut-off frequency
Magnitude change
Phase change
TypeMagnitude
Change
Phase
Change
Stable pole @๐๐ -20 dB/dec -90ยฐ
Rules for drawing a Bode diagram
Advantages of log10 and dB (II)
32
โข Multiplication of magnitudes becomes addition
ฮฃtot = ฮฃ1 โ |ฮฃ2|
20 โ log10( ฮฃtot ) = 20 โ log10( ฮฃ1 โ ฮฃ1 )
= 20 โ log10 ฮฃ1 + 20 โ log10( ฮฃ2 )
ฮฃ1 ๐๐ = ๐ด1(๐๐) โ ๐๐๐1(๐๐)
ฮฃ2 ๐๐ = ๐ด2(๐๐) โ ๐๐๐2(๐๐)ฮฃtot ๐๐ = ฮฃ1 ๐๐ โ ฮฃ2 ๐๐
Sidenote: for phase we have addition property anyway!
โ ฮฃtot dB = ฮฃ1 dB + ฮฃ2 dB
โ (ฮฃtot) = โ (ฮฃ1 + ฮฃ2) = โ (ฮฃ1) + โ (ฮฃ2)
Bode Diagram of higher order system
33
Example: ฮฃ ๐ = 0.1 โ ๐ + 10
๐ + 0.5
Decimal
Scale
Decibel
Scale
2 6.02โฆ
1 0
TypeMagnitude
Change
Phase
Change
Stable pole @๐๐ -20 dB/dec -90ยฐ
Minimumphase
zero @๐๐
+20 dB/dec +90ยฐ
Conversion Table
Rules for drawing a Bode diagram
34
Examples of frequency responses
1st order and 2nd nyquist diagrams
Drawing frequency response
Example: 1st order system (I)
35
Calculate the frequency response of
Re ฮฃ(๐๐)
Im ฮฃ(๐๐)
ฮฃ ๐ =1
๐ + 1
Nyquist diagram of a 1st order system
36
Example: with ๐ = 1, ๐ = 1 Static gain
Cut-off frequency
Magnitude change
Phase change
Nyquist diagram of a 2nd order
system
37
Example: with ๐ = 1
Library of Standard Elements
38
See Appendix A in the script
Why is cosine important? (I)
39
Fourier Series: ยซany signal can be written asยป
๐ข ๐ก =
๐=1
โ
๐ด๐ โ cos(๐๐ โ ๐ก + ๐๐)
๐ฆโ ๐ข1 =๐ ๐1 โ ๐ด1โ cos ๐1 โ ๐ก + ๐1 + ๐ ๐1
๐ฆโ ๐ข2 =๐ ๐2 โ ๐ด2โ cos ๐2 โ ๐ก + ๐2 + ๐ ๐2
๐ฆโ ๐ข3 = ๐ ๐3 โ ๐ด3 โ cos ๐3 โ ๐ก + ๐3 + ๐ ๐3
โฎ
Linear System!+
+
+
+
โฎ
๐ข1 = ๐ด1 โ cos(๐1 โ ๐ก + ๐1)
๐ข2 = ๐ด2 โ cos(๐2 โ ๐ก + ๐2)
๐ข3 = ๐ด3 โ cos(๐3 โ ๐ก + ๐3)
๐ข4 = ๐ด4 โ cos(๐4 โ ๐ก + ๐4)
๐ฆโ ๐ก = ๐ฆโ ๐ข1 + ๐ฆโ ๐ข2 + ๐ฆโ ๐ข3 +โฏ
Why is cosine important? (II)
40
๐ด(๐ ) is a linear system โ all linear operations are possible, among which:
โข ๐๐จ๐ฌ(๐๐) or ๐ฌ๐ข๐ง(๐๐) are the only periodic signals that keep their shape when
passing through a system ฮฃ ๐ (they only change in magnitude and phase)
โข All other (periodic) signals change shape in integration or derivation operation
โข System output could not be described by scaled and shifted system input
โข More than two variables (๐,๐) would be necessary to describe output signal
Derivative operationd
d๐ก๐ฅ ๐ก โ ๐ โ ๐(๐ )
โซ ๐ฅ ๐ก โ1
๐ โ ๐(๐ )Integral operation
Closed loop stability with Nyquist diagram
41
๐ถ ๐๐ ๐ ๐ข ๐ฆ
โ