ee c128 / me c134 { feedback control systems...i advantages of fr techniques over rl i de ne fr i de...
TRANSCRIPT
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EE C128 / ME C134 – Feedback Control SystemsLecture – Chapter 10 – Frequency Response Techniques
Alexandre Bayen
Department of Electrical Engineering & Computer ScienceUniversity of California Berkeley
September 10, 2013
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Lecture abstract
Topics covered in this presentation
I Advantages of FR techniques over RL
I Define FR
I Define Bode & Nyquist plots
I Relation between poles & zeros to Bode plots (slope, etc.)
I Features of 1st- & 2nd-order system Bode plots
I Define Nyquist criterion
I Method of dealing with OL poles & zeros on imaginary axis
I Simple method of dealing with OL stable & unstable systems
I Determining gain & phase margins from Bode & Nyquist plots
I Define static error constants
I Determining static error constants from Bode & Nyquist plots
I Determining TF from experimental FR data
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Chapter outline
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.1 Intro
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.1 Intro
Advantages of frequency response (FR) methods, [1, p.534]
In the following situations
I When modeling TFs from physical data
I When designing lead compensators to meet a steady-state errorrequirements
I When finding the stability of NL systems
I In settling ambiguities when sketching a root locus
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10 FR techniques 10.1 Intro
The concept of FR, [1, p. 535]
I At steady-state, sinusoidalinputs to a linear systemgenerate sinusoidal responsesof the same frequency withdifferent amplitudes and phaseangle from the input, each ofwhich are a function offrequency.
I Phasor – complexrepresentation of a sinusoid
I ||G(ω)|| – amplitudeI ∠G(ω) – phase angleI M cos(ωt+ φ) ... M∠φ Figure: Sinusoidal FR: a. system; b.
TF; c. IO waveforms
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10 FR techniques 10.1 Intro
The concept of FR, [1, p. 535]
I Steady-state output sinusoid
Mo(ω)∠φo(ω)
= Mi(ω)M(ω)∠(φi(ω) + φ(ω))
I Magnitude FR
M(ω) =Mo(ω)
Mi(ω)
I Phase FR
φ(ω) = φo(ω)− φi(ω)
I FRM(ω)∠φ(ω)
Figure: Sinusoidal FR: a. system; b.TF; c. IO waveforms
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10 FR techniques 10.1 Intro
Analytical expressions for FR, [1, p. 536]
I General input sinusoid
r(t) = A cos(ωt) +B sin(ωt)
=√A2 +B2 cos
(ωt− tan−1
(BA
))I Input phasor forms
I Polar, Mi∠φi
Mi =√A2 +B2
φi = − tan−1(BA
)I Rectangular, A− jBI Euler’s, Mie
jφi
Figure: System withsinusoidal input
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10 FR techniques 10.1 Intro
Analytical expressions for FR, [1, p. 536]
I Forced response
C(s) =As+Bω
s2 + ω2G(s)
I Steady-state forced response after partial fraction expansion
Css(s) =MiMG
2 e−j(φi−φG)
s+ jω+
MiMG2 ej(φi−φG)
s− jωwhere MG = ||G(jω)|| and φG = ∠G(jω)
I Time-domain response
c(t) = MiMG cos(ωt+ φi + φG)
I Time-domain response in phasor form
Mo∠φo = (Mi∠φi)(MG∠φG)
I FR of systemG(jω) = G(s)|s→jω
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10 FR techniques 10.2 Asymptotic approximations: Bode plots
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.2 Asymptotic approximations: Bode plots
History interlude
History (Hendrik Wade Bode)
I 1905 – 1982
I American engineer
I 1930s – Inventor of Bode plots,gain margin, & phase margin
I 1944 – WWII anti-aircraft(including V-1 flying bombs)systems
I 1947 – Cold War anti-ballisticmissiles
I 1957 – Served on NACA (nowNASA) with Wernher vonBraun (inventor of V-1 flyingbombs & V-2 rockets)
Figure: Hendrik Wade Bode
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10 FR techniques 10.2 Asymptotic approximations: Bode plots
General Bode plots, [1, p. 540]
G(jω) = MG(ω)∠φG(ω)
I Separate magnitude and phase plots as a function of frequencyI Magnitude – decibels (dB) vs. log(ω), where dB = 20 log(M)I Phase – phase angle vs. log(ω)
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10 FR techniques 10.2 Asymptotic approximations: Bode plots
Bode plots approximations, [1, p. 542]
I TFG(s) = s+ a
I Low frequencies
G(jω) ≈ a∠0◦
I High frequencies
G(jω) ≈ ω∠90◦
I Asymptotes – straight-lineapproximations
I Low-frequencyI Break frequencyI High-frequency
Figure: Bode plots of s+ a: a.magnitude plot; b. phase plot
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10 FR techniques 10.2 Asymptotic approximations: Bode plots
Simple Bode plots, [1, p. 542]
0
10
20
30
40
Magnitude (
dB
)
10−2
10−1
100
101
102
0
30
60
90
Phase (
deg)
Frequency (Hz)
Figure: Bode plot of s+aa
−40
−30
−20
−10
0
Magnitude (
dB
)10
−210
−110
010
110
2−90
−60
−30
0
Phase (
deg)
Frequency (Hz)
Figure: Bode plot of as+a
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10 FR techniques 10.2 Asymptotic approximations: Bode plots
Simple Bode plots, [1, p. 545]
0
10
20
30
Magnitude (
dB
)
10−1
100
101
8989.289.489.689.8
9090.290.490.690.8
91
Phase (
deg)
Frequency (Hz)
Figure: Bode plot of s
−30
−20
−10
0
Magnitude (
dB
)10
−110
010
1−91
−90.8−90.6−90.4−90.2
−90−89.8−89.6−89.4−89.2
−89
Phase (
deg)
Frequency (Hz)
Figure: Bode plot of 1s
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10 FR techniques 10.2 Asymptotic approximations: Bode plots
Simple Bode plots, [1, p. 549]
0
20
40
60
80
Magnitude (
dB
)
10−2
10−1
100
101
102
0
45
90
135
180
Phase (
deg)
Frequency (Hz)
Figure: Bode plot ofs2+2ζωns+ω
2n
ω2n
−80
−60
−40
−20
0
Magnitude (
dB
)10
−210
−110
010
110
2−180
−135
−90
−45
0
Phase (
deg)
Frequency (Hz)
Figure: Bode plot ofω2
n
s2+2ζωns+ω2n
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10 FR techniques 10.2 Asymptotic approximations: Bode plots
Detailed 2nd-order Bode plots, [1, p. 550]
Figure: Bode plot ofs2+2ζωns+ω
2n
ω2n
Figure: Bode plot ofω2
n
s2+2ζωns+ω2n
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10 FR techniques 10.3 Intro to Nyquist criterion
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.3 Intro to Nyquist criterion
History interlude
History (Harry Theodor Nyquist)
I 1889 – 1976
I American engineer
I 1917 – 1934 AT&T
I 1934 – 1954 Bell TelephoneLabs
I 1924 – Nyquist-Shannonsampling theorem
I 1926 – Johnson–Nyquist noise
I 1932 – Nyquist stabilitycriterion Figure: Harry Theodor Nyquist
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10 FR techniques 10.3 Intro to Nyquist criterion
Introduction, [1, p. 559]
I Relates the stability of a CLsystem to the OL FR and theOL poles and zeros
I # CL poles in RHP
I Provides information on thetransient response andsteady-state error
Figure: CL control system
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10 FR techniques 10.3 Intro to Nyquist criterion
Derivation concepts, [1, p. 560]
G(s) =NG
DGand H(s) =
NH
DH
G(s)H(s) =NGHG
DGDH
F (s) = 1 +G(s)H(s) =DGDH +NGNH
DGDH
T (s) =G(s)
1 +G(s)H(s)=
NGNH
DGDH +NGNH
I Poles of 1 +G(s)H(s) are the same as the poles of the OL system,G(s)H(s)
I Zeros of 1 +G(s)H(s) are the same as the poles of the CL system,T (s)
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10 FR techniques 10.3 Intro to Nyquist criterion
Derivation concepts, [1, p. 560]
I Map – function
I Contour – collection of points
I For our particular scenario,assume
F (s) =(s− z1)(s− z2)...(s− p1)(s− p2)...
and a clockwise direction formapping the points on thecontour A
Figure: Mapping contour A throughfunction F (s) to contour B
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10 FR techniques 10.3 Intro to Nyquist criterion
Derivation concepts, [1, p. 561]
I If F (s) has only zeros or onlypoles that are not encircled bythe contour then contour Bmaps in a clockwise direction
Figure: Contour mapping – withoutencirclements
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10 FR techniques 10.3 Intro to Nyquist criterion
Derivation concepts, [1, p. 561]
I If F (s) has only zeros that areencircled by the contour thencontour B maps in a clockwisedirection
I If F (s) has only poles that areencircled by the contour thencontour B maps in acounterclockwise direction
I If F (s) has only poles or onlyzeros that are encircled by thecontour then contour B mapdoes encircle the origin
Figure: Contour mapping – withencirclements
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10 FR techniques 10.3 Intro to Nyquist criterion
Derivation concepts, [1, p. 562]
I If F (s) has #poles = #zeros
that are encircled by thecontour then contour B mapdoes not encircle the origin Figure: Contour mapping – with
encirclements
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10 FR techniques 10.3 Intro to Nyquist criterion
Derivation concepts, [1, p. 562]
I Each pole or zero of1 +G(s)H(s) whose vectorundergoes a complete rotationof contour A must yield achange of 360◦ in theresultant, R, or a completerotation of contour B
I A zero inside a CW contour Ayields a CW rotation ofcontour B
I A pole inside a CW contour Ayields a CCW rotation ofcontour B
I N = P − ZI N , # CCW rotations of
contour B about the originI P , # poles of 1 +G(s)H(s)
inside contour AI Z, # zeros of 1 +G(s)H(s)
inside contour A
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10 FR techniques 10.3 Intro to Nyquist criterion
Derivation concepts, [1, p. 562]
Adjustment – extend the contour A to includethe entire RHP
I Z, # RHP CL polesI CL stability!
I P , # RHP OL polesI Easy
I N , # CCW rotations of contour B aboutorigin
I Difficult
Adjustment – map G(s)H(s) instead of1 +G(s)H(s)
I N , # CCW rotations of contour B about−1
I Less difficult
Figure: Contour enclosingRHP to determinestability
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10 FR techniques 10.3 Intro to Nyquist criterion
Definition, [1, p. 563]
Definition (Nyquist stability criterion)
I If a contour, A, that encircles the entire RHP is mapped through theOL system, G(s)H(s), then the # of RHP CL poles, Z, equals the #of RHP OL poles, P , minus the # of CCW revolutions, N , around−1 of the mapping.
Z = P −N
I The mapping is called the Nyquist diagram of G(s)H(s).
I FR technique because the mapping of points on the positive jω-axisthrough G(s)H(s) is the same as substituting s = jω into G(s)H(s)to form the FR function G(jω)H(jω).
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10 FR techniques 10.3 Intro to Nyquist criterion
Applying the Nyquist stability criterion, [1, p. 563]
I No RHP CL polesI P = 0I N = 0I Z = 0I CL system is stable
I 2 RHP CL polesI P = 0I N = −2I Z = 2I CL system is unstable Figure: Mapping examples – with
encirclement: a. contour does notenclose CL poles; b. contour doesenclose CL poles
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10 FR techniques 10.4 Sketching the Nyquist diagram
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.4 Sketching the Nyquist diagram
Method
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10 FR techniques 10.4 Sketching the Nyquist diagram
Example, [1, p. 564]
Example (general)
I G(s) = 500(s+10)(s+3)(s+1)
Figure: Vector evaluation of theNyquist diagram: a. vectors on contourat low frequency, b. vectors on contouraround ∞; c. Nyquist diagram
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10 FR techniques 10.4 Sketching the Nyquist diagram
Example, [1, p. 567]
Example (poles on contour)
I G(s) = s+2s2
Figure: a. contour, b. Nyquist diagram
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10 FR techniques 10.5 Stability via the Nyquist diagram
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.5 Stability via the Nyquist diagram
Example, [1, p. 569]
Example (general)
I G(s) = K(s+3)(s+5)(s−2)(s−4)
Figure: a. system; b. contour, c.Nyquist diagram
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10 FR techniques 10.5 Stability via the Nyquist diagram
Example, [1, p. 570]
Example (general)
I G(s) = Ks(s+3)(s+5)
Figure: a. contour; b. Nyquist diagram
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10 FR techniques 10.5 Stability via the Nyquist diagram
Stability via mapping only the positive jω-axis, [1, p. 571]
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10 FR techniques 10.6 Gain margin & phase margin via the Nyquist diagram
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.6 Gain margin & phase margin via the Nyquist diagram
Definitions, [1, p. 574]
Two quantitative measures of howstable a system is
I Gain margin, GM – the changein OL gain, expressed in dB,required at 180◦ of phase shiftto make the CL systemunstable
I Phase margin, ΦM – thechange in OL phase shiftrequired at unity gain to makethe CL system unstable
Figure: Nyquist diagram showing gainand phase margins
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10 FR techniques 10.7 Stability, gain margin, & phase margin via Bode plots
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.7 Stability, gain margin, & phase margin via Bode plots
Stability via Bode plots, [1, p. 576]
Method
I Draw a Bode log-magnitude plot
I Determine the range of the gain that ensures that the magnitude isless than 0 dB (unity gain) at that frequency where the phase is±180◦
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10 FR techniques 10.7 Stability, gain margin, & phase margin via Bode plots
Example, [1, p. 577]
Example (general)
I Use Bode plots to determinethe range of K within whichthe unity FB system is stable.
G(s) =k
(s+ 2)(s+ 4)(s+ 5)
Figure: Bode log-magnitude and phasediagrams
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10 FR techniques 10.7 Stability, gain margin, & phase margin via Bode plots
Gain & phase margin via Bode plots, [1, p. 578]
MethodI Gain margin
I Phase plot →ωGM
= ω|Φ=180◦
I At ωGM, magnitude plot →
gain margin, GM , which isthe gain required to raise themagnitude curve to 0 dB
I Phase marginI Magnitude plot →ωΦM
= ω|G=0dB
I At ωΦM, phase plot → phase
margin, ΦM , which is thedifference between the phasevalue and 180◦
Figure: Gain and phase margins on theBode diagrams
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10 FR techniques 10.7 Stability, gain margin, & phase margin via Bode plots
Example, [1, p. 579]
Example (general)
I If K = 200, find the gain andphase margins.
G(s) =k
(s+ 2)(s+ 4)(s+ 5)
Figure: Bode log-magnitude and phasediagrams
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10 FR techniques 10.8 Relation between CL transient & CL FRs
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.8 Relation between CL transient & CL FRs
Damping ratio & CL FR, [1, p. 580]
Peak magnitude of the CL FR
Mp =1
2ζ√
1− ζ2
Frequency of the peak magnitude
ωp = ωn√
1− 2ζ2
Figure: 2nd-order CL system
Figure: CL FR peak vs. %OS for a 2pole system
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10 FR techniques 10.8 Relation between CL transient & CL FRs
Response speed & CL FR, [1, p. 581]
I Bandwidth of a 2-pole system
ωBW = ωn
√(1− 2ζ2) +
√4ζ4 − 4ζ2 + 2
I ωn–Ts relation
ωn =4
Tsζ
I ωn–Tp relation
ωn =π
Tp√
1− ζ2
I ωn–Tr relationI Found using look-up table
Figure: Representativelog-magnitude plot
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10 FR techniques 10.8 Relation between CL transient & CL FRs
Response speed & CL FR, [1, p. 582]
Figure: Normalized bandwidth vs. damping ratio for:a. Ts, b. Tp; c. Tr
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10 FR techniques 10.9 Relation between CL & OL FRs
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.9 Relation between CL & OL FRs
Constant M circles & constant N circles, [1, p. 583]
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10 FR techniques 10.10 Relation between CL transient & OL FRs
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.10 Relation between CL transient & OL FRs
Damping ratio from M circles, [1, p. 589]
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10 FR techniques 10.10 Relation between CL transient & OL FRs
Damping ratio from phase margin, [1, p. 589]
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10 FR techniques 10.10 Relation between CL transient & OL FRs
Response speed from OL FR, [1, p. 591]
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10 FR techniques 10.11 Steady-state error characteristics from FR
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.11 Steady-state error characteristics from FR
Position constant, [1, p. 593]
Type 0 system
G(s) = K
∏ni=1(s+ zi)∏mj=1(s+ pj)
Initial log-magnitude value
20 logM = 20 logKp
Position constant
Kp = K
∏ni=1 zi∏mj=1 pj
Figure: Typical unnormalized andunscaled Bode log-magnitude plotsshowing the value of static errorconstants
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10 FR techniques 10.11 Steady-state error characteristics from FR
Velocity constant, [1, p. 594]
Type 1 system
G(s) = K
∏ni=1(s+ zi)
s∏mj=1(s+ pj)
Initial log-magnitude value
20 logM = 20 logKv
ω0
Velocity constant
Kv = K
∏ni=1 zi∏mj=1 pj
Frequency axis intersect
ω = Kv
Figure: Typical unnormalized andunscaled Bode log-magnitude plotsshowing the value of static errorconstants
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10 FR techniques 10.11 Steady-state error characteristics from FR
Acceleration constant, [1, p. 595]
Type 2 system
G(s) = K
∏ni=1(s+ zi)
s2∏mj=1(s+ pj)
Acceleration constant
Ka = K
∏ni=1 zi∏mj=1 pj
Initial log-magnitude value
20 logM = 20 logKa
ω20
Frequency axis intersect
ω =√Ka
Figure: Typical unnormalized andunscaled Bode log-magnitude plotsshowing the value of static errorconstants
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10 FR techniques 10.12 System with time delay
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.12 System with time delay
Modeling time delay, [1, p. 597]
Time delay – delay between thecommanded response and the startof the output response
G′(s) = e−sTG(s)
FR
G′(jω) = e−jωTG(jω)
= |G(jω)|∠[−ωT + ∠G(jω)]Figure: Effect of delay upon FR
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10 FR techniques 10.13 Obtaining TFs experimentally
1 10 Frequency response techniques10.1 Introduction10.2 Asymptotic approximations: Bode plots10.3 Introduction to Nyquist criterion10.4 Sketching the Nyquist diagram10.5 Stability via the Nyquist diagram10.6 Gain margin and phase margin via the Nyquist diagram10.7 Stability, gain margin, and phase margin via Bode plots10.8 Relation between closed-loop transient and closed-loopfrequency responses10.9 Relation between closed- and open-loop frequency responses10.10 Relation between closed-loop transient and open-loopfrequency responses10.11 Steady-state error characteristics from frequency response10.12 System with time delay10.13 Obtaining transfer functions experimentally
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10 FR techniques 10.13 Obtaining TFs experimentally
Obtaining transfer functions experimentally, [1, p. 602]
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10 FR techniques 10.13 Obtaining TFs experimentally
Method
1. Estimate the pole-zero configuration from the Bode diagramsI Initial slope → system typeI Phase excursions → #poles & #zeros
2. Look for obvious 1st- & 2nd-order pole or zero FR characteristics
3. Peaking & depressions → underdamped 2nd-order pole & zero,respectively
4. Extract 1st- & 2nd-order characteristicsI Overlay ±20 or ±40 dB/decade lines on magnitude curve &±45◦/decade lines on the phase curve
I Estimate break frequenciesI For 2nd-order poles & zeros, estimate ζ & ωn
5. Form a TF of unity gain using the poles & zeros foundI Subtract the FR of the model from the measured FR and repeat the
process if necessary
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10 FR techniques 10.13 Obtaining TFs experimentally
Bibliography
Norman S. Nise. Control Systems Engineering, 2011.
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 64 / 64