Lecture Outline
Systeem- en Regeltechniek II
Lecture 8 – Frequency Domain Design
Previous lecture: Bode plots, non-minimum-phase systems.
Robert Babuška
Today:
• Bode’s gain-phase relation.
Delft Center for Systems and Control
Faculty of Mechanical Engineering
Delft University of Technology
The Netherlands
• Neutral stability.
• Gain and phase margin, performance specs.
• Controller design.
e-mail: r.babuska@dcsc.tudelft.nl
www.dcsc.tudelft.nl/˜babuska
tel: 015-27 85117
Robert Babuška
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Delft Center for Systems and Control, TU Delft
Robert Babuška
Frequency Domain Methods
Bode Diagrams
0
data
(experiment)
Phase (deg); Magnitude (dB)
Bode plot
-20
-30
-40
-5 0
-1 0 0
-1 5 0
-1
10
frequency
response
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Bode’s Gain-Phase Relation
-10
transfer
function
Delft Center for Systems and Control, TU Delft
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0
10
1
Frequency (rad/sec)
Nyquist plot
For any stable minimum-phase system, phase ∠G(jω) is uniquely
related to magnitude |G(jω)|:
Z
1 ∞ dM
W (u)du
∠G(jω0) =
π ∞ du
where M = ln |G(jω)|, u = ln ω/ω0, W (u) = ctanh|u/2|.
For a constant slope, we can approximate the above by:
∠G(jω0) ' n
• Now we now how to sketch and plot Bode diagrams.
• The next step is analysis of system properties and design.
Robert Babuška
Delft Center for Systems and Control, TU Delft
π
2
where n is the slope ( 1 for 20 dB /dec, 2 for 40 dB /dec, etc).
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Robert Babuška
Delft Center for Systems and Control, TU Delft
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Consequence of the Gain-Phase Relation
Bode Plot: Closed-Loop Stability
For open loop stable minimum-phase system, it is sometimes sufficient to look at the magnitude only.
R(s) +
E(s)
-
C(s)
U(s)
G(s)
Y(s)
This property can be used to derive a simple design rule for control.
But first, we must be able determine, from the Bode plot, whether
the system is stable!
L(s) = G(s)C(s)
Can we infer closed-loop stability from a Bode plot of the loop
transfer function L(s)?
Robert Babuška
Delft Center for Systems and Control, TU Delft
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Proportional Controller: Loop Transfer
K>1
Y (s)
= K G(s)
E(s)
For the Bode plot, the following holds:
∠G(jω) = ∠ (KG(jω))
(K is a real number)
|G(jω)| = |K| · |G(jω)|
(multiplication by a gain)
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Delft Center for Systems and Control, TU Delft
K<1
-2
10
0
10
2
10
4
shift the magnitude response of G(jω) by 20 log(K)
phase does not change
|G(jω)| dB = |K| dB + |G(jω)| dB
Robert Babuška
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Delft Center for Systems and Control, TU Delft
Proportional Controller: Loop Transfer
Magnitude
L(s) =
Robert Babuška
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Robert Babuška
Delft Center for Systems and Control, TU Delft
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Example: DC Motor
DC Motor: Open-Loop Bode Plot
G(s) =
Transfer function:
Kt
θ(s)
=
V (s) s[(Ls + R)(Js + b) + Kt2]
J
damping (friction)
b
= 0.1 Nms
back emf
Kt = 0.01 Nm/A
resistance
R = 1Ω
inductance
L = 0.5 H
magnitude (dB)
inertia of the rotor
= 0.01 kg · m2
50
0
−50
−100
−150 −2
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10
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Influence of Proportional Gain
L(s) = KG(s) =
0
10
10
1
10
2
−100
−200
−300 −2
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−1
0
phase (deg)
G(s) =
2
θ(s)
=
V (s) s(s + 10)(s + 2)
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Robert Babuška
−1
0
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frequency (rad/sec)
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1
10
2
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Delft Center for Systems and Control, TU Delft
Let’s See Whether Root Locus Helps . . .
K ·2
s(s + 10)(s + 2)
15
10
Imag Axis
5
Use Matlab: sisotool(’bode’,G)
0
−5
−10
OK, the magnitude moves up and down with the gain and the
phase does not change . . .
−15
−12
−10
−8
−6
−4
Real Axis
−2
Basic properties of RL: ∠G(s) = −180◦
. . . but, is there anything on the Bode plot that would hint on the
stability of the closed loop?
Robert Babuška
Delft Center for Systems and Control, TU Delft
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0
and
2
|KG(s)| = 1
Neutral stability: |KmaxG(jω)| = 0 dB and ∠G(jω) = −180◦
Robert Babuška
Delft Center for Systems and Control, TU Delft
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Back to the Bode Plot
Point of Neutral Stability
100
magnitude (dB)
System is stable if: |KG(jω)| < 0 dB at ∠G(jω) = −180◦
magnitude (dB)
100
0
−200 −1
10
10
0
10
1
2
3
10
−100
phase (deg)
−100
−200
−300 −1
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1
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frequency (rad/sec)
2
0
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1
10
2
10
3
−100
−200
−300 −1
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3
10
10
0
10
0
phase (deg)
0
−200 −1
10
−100
Kmax|G(jω)|
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Delft Center for Systems and Control, TU Delft
Crossover Frequency and Stability Margins
Robert Babuška
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0
1
10
frequency (rad/sec)
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Delft Center for Systems and Control, TU Delft
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10
3
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Crossover Frequency and Stability Margins
50
0
• The crossover frequency ωc is the frequency for which the loop
TF has gain 0 dB.
Gain margin
-50
-100
• The gain margin (GM) is the factor (or amount dB) by which
the loop gain can be raised before instability occurs.
-150
0
-100
• The phase margin (PM) is the amount (in degrees) by which
the phase exceeds 180◦ at ωc.
Phase margin
-200
-300 -2
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-1
10
0
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Delft Center for Systems and Control, TU Delft
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2
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Delft Center for Systems and Control, TU Delft
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Importance of Stability Margins
Robustness: Example
The margins tell us how far the closed-loop system is from the
point of neutral stability. This indicates the robustness w.r.t. uncertainty in the plant model:
• Gain margin: by what factor the total process gain can increase.
• Phase margin: by how much the phase can decrease.
Suppose our model is:
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L̂(s) = 2
s + 0.4s + 1
while the true plant is:
L(s) =
and performance:
• Phase margin: related to closed loop damping (overshoot).
Relatively small mismatch in terms of step-response behavior, major difference in terms of closed-loop stability!
• Crossover frequency: related to response speed
(bandwidth).
Robert Babuška
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Delft Center for Systems and Control, TU Delft
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(s2 + 0.4s + 1)(0.1s + 1)
Robert Babuška
Bode Plot of Model and True Plant
Delft Center for Systems and Control, TU Delft
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Closed-Loop Bandwidth
Bode Diagram
Bandwidth = frequency up to which the input is “well reproduced”
at the output of the closed-loop system.
50
Magnitude (dB)
0
Defined as frequency ωbw at which the magnitude has an
attenuation of 0.707 (3dB) – corresponds to 0.5 power gain.
−50
−100
0
-3 dB
−90
Magnitude
−150
0
Phase (deg)
System: G1
Phase Margin (deg): 7.59
Delay Margin (sec): 0.0401
At frequency (rad/sec): 3.3
Closed Loop Stable? Yes
−180
System: G2
Phase Margin (deg): −10.1
Delay Margin (sec): 1.89
At frequency (rad/sec): 3.23
Closed Loop Stable? No
−270
−1
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Robert Babuška
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10
1
10
Frequency (rad/sec)
10
2
Delft Center for Systems and Control, TU Delft
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3
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Robert Babuška
Delft Center for Systems and Control, TU Delft
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Bandwidth and Crossover Frequency
Phase Margin and Overshoot
The larger PM, the larger damping (less overshoot):
Typically:
ωc ≤ ωbw ≤ 2ωc
ζ≈
The required speed of response (e.g., the settling time or rise time)
can be expressed in terms of ωc. Recall:
tr = 1.8/ωn
PM
100
this holds up to PM = 60◦
See the Franklin et al. for a graphical relationship between the
overshoot and PM (page 357).
(for second-order dominant response).
Robert Babuška
Delft Center for Systems and Control, TU Delft
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Robert Babuška
Recall Specs for Second-Order Systems
tr =
tp =
ts =
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More Complex Plants
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.
ωn
• System unstable for small K and stable for large K , e.g.,:
π
L(s) =
K(s + 2)
s2 − 1
ω n 1− ζ2
4.6
ζωn
Mp = e
−
for
• Conditionally stable systems (unstable for small and large K ,
stable for some intermediate values), e.g.,:
± 1%
πζ
1− ζ
L(s) =
2
K(s + 2)
(s + 10)2(s + 1)(s − 1)
In the sequel, we consider systems with no poles in RHP.
Robert Babuška
Delft Center for Systems and Control, TU Delft
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Robert Babuška
Delft Center for Systems and Control, TU Delft
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The Basic Idea
Bode Plots: Homework Assignments
• Adjust the proportional gain to get the required crossover frequency and/or steady-state tracking error.
• Read Sections 6.1 through 6.6, except for the Nyquist criterion.
• If needed, use the derivative action to add phase in the neighborhood of ωc in order to increase the phase margin.
• If needed, use the integral action to increase the gain at low frequencies in order to guarantee the required steady-state tracking
error.
Robert Babuška
Delft Center for Systems and Control, TU Delft
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• Work out examples in these sections and verify the results by
using Matlab.
• Reproduce the derivation of the frequency response as given on
the overhead sheets.
• Work out a selection of problems 6.3 through 6.9 and verify
your results by using Matlab.
Robert Babuška
Delft Center for Systems and Control, TU Delft
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