1. Control Systems
LEC 4 DESIGN VIA FREQUENCY RESPONSE
BEHZAD FARZANEGAN
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2. Frequency Response
Frequency response methods, developed by Nyquist and Bode in the 1930s, are
older than the root locus method, which was discovered by Evans in 1948
(Nyquist,1932; Bode, 1945). The older method, is not as intuitive as the root locus.
However, frequency response yields a new vantage point from which to view
feedback control systems.
This technique has distinct advantages in the following situations:
1. When modeling transfer functions from physical data.
2. When designing lead compensators to meet a steady-state error requirement and
a transient response requirement
3. When finding the stability of nonlinear systems
4. In settling ambiguities when sketching a root locus
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3. Sinusoidal frequency response
The steady-state output sinusoid is
• M is called Magnitude Freq Response
• Ф is called Phase Freq Response
The system function is given by
( ) ( ) ( ) ( ) [ ( ) ( )]i iM M M o o
( )
( )
( )
( ) ( ) ( )
i
i
M
M
M
o
o
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4. Analytical Expression for
Frequency response
2 2
( ) ( )
( )
As B
C s G s
s
1 2
( )ss
K K
C s
s j s j
2 2
( )
and ( )
i
G
where M R j A B
M G j
= ( )
( )( )
As B
G s
s j s j
2
( )
1
(= )
ss
C s
partial fraction terms from
j s j
G
K
s
s
K
1 4 4 2 4 4 3
( ) ( )
2 2
i G i Gj ji G i GM M M M
e e
s j s j
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5. Plotting Frequency Response
𝐺(𝑗𝜔) = 𝑀 )𝐺(𝜔 < 𝛷 )𝐺(𝜔 can be plotted in several ways; two of them
are
• As a function of frequency, with separate magnitude and phase plots;
• As a polar plot, where the phasor length is the magnitude and the phasor
angle is the phase.
When plotting separate magnitude and phase plots, the magnitude
curve can be plotted in decibels (dB) vs. logω, where dB = 20 log M. The
phase curve is plotted as phase angle vs. log ω.
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6. Frequency response plots Example
Problem: Find the analytical expression for the magnitude and phase
frequency response for a system G(s)=1/(s+2). Plot magnitude, phase and
polar diagrams.
Solution: substituting s=jw, we get
The magnitude frequency response is
The phase frequency response is
2
1 (2 )
( )
( 2) ( 4)
j
G j
j
2
( ) ( ) 1/ ( 4)G j M
2
20log ( ) 20log(1/ 4) vs. logM
1
( ) tan ( / 2) vs. log
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7. Frequency response plots Example
Problem: Find the analytical expression for the magnitude and phase
frequency response for a system G(s)=1/(s+2). Plot the polar diagrams.
Solution: substituting s=jw, we get
2 1
( ) ( ) 1/ ( 4) tan ( / 2)M
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9. Asymptotic Approximations:
Bode plots
The straight-line approximations are called asymptotes.
We have low frequency asymptote and high frequency asymptote.
a is called the break frequency.
Bode plots for G(s) = (s + a)
• At low frequency when ω approaches zero 𝐺(𝑗𝜔) ≈ 𝑎
• The magnitude response in dB is 20log M=20log from ω =0.01a to a
• At high frequencies where ω ˃˃ a
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𝐺(𝑗𝜔) ≈ 𝑎(
𝑗𝜔
𝑎
) = 𝑎(
𝜔
𝑎
)∠90 𝑜 = 𝜔∠90 𝑜
20log𝑀 = 20log𝑎 + 20log
𝜔
𝑎
= 20log𝜔
9
10. Normalized and scaled Bode
plots for different systems
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11. HW!
Problem : Draw the Bode plot for the system defined as
2
( 3)
( )
[( 2)( 2 25)]
s
G s
s s s
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12. Nyquist diagram showing gain
and phase margins
Gain margin, GM, The gain margin is the change in open-loop gain,
expressed in decibels (dB), required at 180⁰ of phase shift to make
the closed-loop system unstable.
Phase margin, 𝛷 𝑀 , The phase margin is the change in open-loop
phase shift required at unity gain to make the closed-loop system
unstable.
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13. HW!
Problem: find the gain and phase margins of system
2
6
( ) ( )
( 23 2)( 2)
G j H j
s s
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14. Relation between Closed-Loop Transient
and Closed-Loop Frequency Responses
Using the open loop transfer function
And closed-loop transfer function
We evaluate the magnitude of the closed-loop freq response as
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𝐺(𝑠) =
𝜔 𝑛
2
)𝑠(𝑠 + 2𝜉𝜔 𝑛
𝑇(𝑠) =
)𝐶(𝑠
)𝑅(𝑠
=
𝜔 𝑛
2
𝑠2 + 2𝜉𝜔 𝑛 𝑠 + 𝜔 𝑛
2
𝑀 = | )𝑇(𝑗𝜔 | =
𝜔 𝑛
2
𝜔 𝑛
2
− 𝜔2 2 + 4𝜉2 𝜔 𝑛
2
𝜔2
14
15. Relation between Closed-Loop Transient
and Closed-Loop Frequency Responses
To relate the peak magnitude to the damping ratio we find
Since damping ratio is related to percent overshoot we can plot MP vs.
percent overshoot
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𝑀 𝑃 =
1
2𝜉 1 − 𝜉2 𝜔 𝑃 = 𝜔 𝑛 1 − 2𝜉2
15
16. Response Speed and Closed-
Loop Frequency Response
We can relate the speed of the time response ( as measured in settling
time, rising time, and peak time) and the bandwidth , 𝜔 𝐵𝑊 of the
Frequency response.
The Bandwidth is defined as the frequency at which the magnitude
response curve is 3 dB down from its value at zero frequency.
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𝜔 𝐵𝑊 = 𝜔 𝑛 1 − 2𝜉2) + 4𝜉4 − 4𝜉2 + 2
𝜔 𝐵𝑊 =
𝜋
𝑇𝑃 1 − 𝜉2
1 − 2𝜉2) + 4𝜉4 − 4𝜉2 + 2
𝜔 𝐵𝑊 =
4
𝑇𝑠 𝜉
1 − 2𝜉2) + 4𝜉4 − 4𝜉2 + 2
𝜔 𝑛 =
𝜋
𝑇𝑃 1 − 𝜉2 𝜔 𝑛 =
4
𝑇𝑠 𝜉
16
17. Relation between Closed-Loop and
Open-Loop Frequency Responses
These circles are called
constant M circles and are the
locus of the closed-loop
magnitude frequency
response for unity feed back
systems.
If the polar frequency
response of an open-loop
function, G(s), is plotted and
superimposed on top of the
constant M circles, the closed-
loop magnitude frequency
response is determined by
each intersection of this polar
plot with the M circles.
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18. Relation between Closed-Loop and
Open-Loop Frequency Responses
These circles are called
constant N circles. If the
polar frequency response
of an open-loop function,
G(s), is plotted and
superimposed on top of
the constant N circles, the
closed-loop phase
response is determined by
each intersection of this
polar plot with the N
circles.
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19. Example
Problem: Find the closed-loop frequency response of the unity feedback
system using G(s)=50/[s(s+3)(s+6)].
Solution
Evaluate the open-loop frequency function
The Polar plot of the open-loop freq response ( Nyquist diagram) is
shown superimposed over the M and N circles in the Figure.
The closed-loop magnitude frequency response is depicted as
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𝐺(𝑗𝜔) =
50
)−9𝜔2 + 𝑗(18𝜔 − 𝜔3
19
20. Nichols Charts
Nichols Charts
displays the constant
M and N circles in dB,
so that changes in gain
are as simple to handle
as in the Bode plots.
The chart is a plot of
the open-loop
magnitude in dB vs.
open-loop phase angle
in degrees.
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21. Example
Nichols chart with frequency response for G(s) = K/[s(s + 1)(s + 2)]
superimposed. Values for K = 1 and K = 3.16 are shown.
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22. Time vs. Freq. Domain Analysis
Control system performance generally judged by time domain
response to certain test signals (step, etc.)
• Simple for < 3 OL poles or ~2nd order CL systems. systems.
• No unified methods for higher-order.
Freq. response easy for higher order systems.
• Qualitatively related to time domain behavior.
• More natural for studying sensitivity and noise susceptibility.
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23. Root locus vs. Frequency
Stability and transient response design via gain adjustment
◦ Frequency response design methods, unlike root locus methods, can be
implemented conveniently without a computer or other tool except for
testing the design.
Transient response design via cascade compensation
• Frequency response methods are not as intuitive as the root locus
Steady-state error design via cascade compensation
• With frequency response techniques, we build the steady-state error
requirement right into the design of the lead compensator. in using root
locus there are an infinite number of possible solutions to the design of a
lead compensator
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25. Performance Specifications
Three requirements enter into the design of a control system: transient response,
stability, and steady-state errors.
The Nyquist criterion tells us how to determine if a system is stable.
• Typically, an open-loop stable system is stable in closed-loop if the open-loop magnitude
frequency response has a gain of less than 0 dB at the frequency where the phase frequency
response is 180°.
Percent overshoot is reduced by increasing the phase margin.
the speed of the response is increased by increasing the bandwidth.
Steady-state error is improved by increasing the low-frequency magnitude
response, even if the high- frequency magnitude response is attenuated.
High frequency region indicates complexity.
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e
Gp(s)
+ Gc(s)
r y
25
26. Performance Specifications
Requirements on open-loop frequency response
The gain at low frequency should be large enough to give a high value for
error constants.
At medium frequencies the phase and gain margins should be large
enough.
At high frequencies, the gain should be attenuated as rapidly as possible
to minimize noise effects.
Compensators
lead: Improves the transient response.
lag: improves the steady-state performance at the expense of slower
settling time.
lead-lag: combines both
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27. Transient Response via Gain
The root locus demonstrated that we can often design
controllers for a system via gain adjustment to meet a
particular transient response.
We can effect a similar approach using the frequency
response by examining the relationship between phase
margin and damping.
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28. Gain Adjustment and the
Frequency Response
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20dB
Wc2
Wc1
𝝎 𝑩𝑾 𝟐
𝝎 𝑩𝑾 𝟏
𝑮(𝒔) =
𝝎 𝒏
𝟐
)𝒔(𝒔 + 𝟐𝝎 𝒏 𝜻
28
29. Design using Phase Margin
Recall that the Phase Margin is closely related to the damping ratio of
the system.
For a unity feedback system with open-loop function
We found that the relationship between PM and damping ratio is given
by
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𝑮(𝒔) =
𝝎 𝒏
𝟐
)𝒔(𝒔 + 𝟐𝝎 𝒏 𝜻
𝜁 ≈
𝑃𝑀
100
, 𝑃𝑀 < 70∘Why?!
29
30. Design using Phase Margin
Given a desired
overshoot, we can
convert this to a
required damping
ratio and hence PM
Examining the
Bode plot we can
find the frequency
that gives the
desired PM
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31. Design using Phase Margin
The design procedure therefore consists of
• Draw the Bode Magnitude and phase plots.
• Determine the required phase margin from the percent
overshoot.
• Find the frequency on the Bode phase diagram that
yields the desired phase margin.
• Change the gain to force the magnitude curve to go
through 0dB.
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32. Phase Margin Example
Problem: For the following position control system shown here, find the
preamplifier gain K to yield a 9.5% overshoot in the transient response
for a step input
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33. Phase Margin Example
Solution:
Draw the Bode plot
For 9.5% overshoot, 𝜁 = 0.6 and PM must be 59.2∘
Locate frequency with the required phase at 14.8 rad/s
The magnitude must be raised by 55.3dB to yield the cross over
point at this frequency
This yields a K = 583.9
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34. HW!
PROBLEM: For a unity feedback system with a forward transfer function
use frequency response techniques to find the value of gain, K, to yield
a closed-loop step response with 20% overshoot.
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35. Designing Compensation
As we saw previously, not all specifications can be met via simple
gain adjustment.
We examined a number of compensators that can bring the root locus
to a desired design point.
A parallel design process exists in the frequency domain.
In particular, we will look at the frequency characteristics for the.
• PI Controller
• Lag Controller
• PD Controller.
• Lead Controller
Understanding the frequency characteristics of these controllers allows
us to select the appropriate version for a given design.
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36. PI Controller
We saw that the integral compensator takes on the form
This results in infinite gain at low frequencies which reduces steady
state error.
A decrease in phase at frequencies lower than the break will also
occur.
The Bode plot for a PI controller looks like this.
The break frequency is usually located at a frequency substantially
lower than the crossover frequency to minimize the effect on the
phase margin.
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𝑮 𝒄(𝒔) = 𝑲
𝒔 + 𝒛 𝒄
𝒔
36
37. Steady-State Error Characteristics
from Frequency Response
For type 0 system (the initial slope is 0)
For un-normalized un-scaled Bode plot, the low frequency magnitude is
20 log KP
For type 1 system (the initial slope is -20dB/decade)
For un-normalized un-scaled Bode plot, the intersection of the initial -
20dB/decade slope with the frequency axis is KV
For type 2 system (the initial slope is -40dB/decade)
For un-normalized un-scaled Bode plot, the intersection of the initial -
40dB/decade slope with the frequency axis is √Ka
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38. Lag Compensation
Lag compensation approximates PI control
This is often rewritten as
where α is the ratio between zero-pole break points
The name Lag Compensation reflects the fact that this compensator
imparts a phase lag.
The Bode plot for a Lag compensator looks like this.
This compensator effectively raises the magnitude for low frequencies
without affecting the stability of the system.
The effect of the phase lag can be minimized by careful selection of the
center frequency.
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𝑮 𝒄(𝒔) = 𝑲
𝒔 + 𝒛 𝒄
𝒔 + 𝒑 𝒄
, 𝒛 𝒄 > 𝒑 𝒄
38
39. Lag Compensation
The function of the lag compensator as seen on Bode diagrams is to
• improve the static error constant by increasing only the low-frequency gain
without any resulting instability,
• increase the phase margin of the system to yield the desired transient
response
Nise suggests setting the gain K for S.S error, then designing a lag
network to attain desired PM.
• Franklin suggests setting the gain K for PM, then designing a lag
network to raise the low freq. gain without affecting system stability.
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40. Lag Compensation Design
The design procedure (Franklin et al.) consists of the following steps:
1. Find the open-loop gain K to satisfy the phase margin specification
without compensation
2. Draw the Bode plot and evaluate low frequency gain
3. Determine a to meet the low-frequency gain error requirement
4. Choose the corner frequency ω=1/T to be one octave to one decade
below the new crossover frequency
5. Evaluate the second corner frequency ω=1/αT
6. Simulate to evaluate the design and iterate as required
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41. Lag Compensation Example
Problem: Returning again to the previous example, we will now design a
lag compensator to yield a ten fold improvement in steady-state error
over the gain compensated system while keeping the overshoot at 9.5%
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42. Lag Compensation Example
In the first example, we found the gain K=583.9 would yield our desired
9.5% overshoot with a PM of 59.2o at 14.8rad/s.
For this system we find that
We therefore require a Kv of 162.2 to meet our specification
We need to raise the low frequency magnitude by a factor of 10 (or 20dB)
without affecting the PM.
First we draw the Bode plot with K=583.9
Set the zero at one decade, 1.48rad/s, lower than the PM frequency
The pole will be at 1/α relative to this so
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𝑮 𝒄(𝒔) =
𝒔 + 𝟏. 𝟒𝟖𝟑
𝒔 + 𝟎. 𝟏𝟒𝟖
42
44. HW!
PROBLEM: For a unity feedback system with a forward transfer function
use frequency response techniques to design a lag controller to yield a
closed-loop step response with 20% overshoot, and improve the steady-
state error tenfold.
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45. PD Controller
The ideal derivative compensator adds a pure differentiator, or zero, to the
forward path of the control system
The root locus showed that this will tend to stabilize the system by
drawing the roots towards the zero location.
We saw that the pole and zero locations give rise to the break points in
the Bode plot.
The Bode plot for a PD controller looks like this.
The stabilizing effect is seen by the increase in phase at frequencies above
the break frequency
The stabilizing effect is seen by the increase in phase at frequencies above
the break frequency.
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)𝑮 𝒄(𝒔) = 𝑲(𝒔 + 𝒛 𝒄
45
46. Lead Compensation
Introducing a higher order pole yields the lead Compensator
This is often rewritten as
where 1/B>1 is the ratio between pole-zero break Point
We want to
• increase the phase margin to reduce the percent overshoot
• increase the gain crossover to realize a faster transient response.
The name Lead Compensation reflects the fact that this compensator
imparts a phase lead.
The Bode plot for a Lead compensator looks like this.
The high frequency magnitude is now limited
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𝑮 𝒄(𝒔) =
)𝑲(𝒔 + 𝒛 𝒄
𝒔 + 𝒑 𝒄
𝒛 𝒄 < 𝒑 𝒄
46
47. Lead Compensation
The lead compensator can be used to change the damping ratio of the
system by manipulating the Phase Margin
The phase contribution consists of
The peak occurs at with a phase shift and magnitude of
This compensator allows the designer to raise the phase of the system
in the vicinity of the crossover frequency
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Why?!
47
48. Lead Compensation Design
The design procedure consists of the following steps:
1. Find the open-loop gain K to satisfy steady-state error or bandwidth
requirements.
2. Evaluate phase margin of the uncompensated system using the value
of gain chosen above.
3. Find the required phase lead to meet the damping requirements.
4. Determine the value of β to yield the required increase in phase
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49. Lead Compensation Design
5. Determine the new crossover frequency
6. Determine the value of T such that ωmax lies at the new crossover
frequency
7. Draw the compensated frequency response and check the resulting
phase margin.
8. Check that the bandwidth requirements have been met.
9. Simulate to be sure that the system meets the specifications (recall
that the design criteria are based on a 2nd order system).
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50. Lead Compensation Example
Problem: Returning to the previous example, we will now design a lead
compensator to yield a 20% overshoot and Kv=40, with a peak time of
0.1s.
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51. Lead Compensation Example
From the specifications, we can determine the following requirements
• For a 20% overshoot we find ζ=0.456 and hence a Phase Margin of 48.1⁰
• For peak time of 0.1s with the given ζ, we can find the require closed loop
bandwidth to be 46.6rad/s.
• To meet the steady state error specification
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52. Lead Compensation Example
From the Bode plot, we evaluate the PM to be 34⁰ for a gain of 1440
We can’t simply increase the gain without violating the other design
constraints
We use a Lead Compensator to raise the PM We require a phase margin of
48.1 ⁰ The lead compensator will also increase the phase margin frequency so
we add a correction factor to compensate for the lower uncompensated
system’s phase angle.
The total phase contribution required is therefore
48.1⁰ – (34 ⁰ – 10 ⁰) = 24.1 ⁰
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53. Lead Compensation
Based on the phase requirement we find
the lead compensator’s magnitude is 3.76 dB at ωmax.
Examining the Bode magnitude, we find that the frequency at which
the magnitude is -3.77dB is ωmax=39rad/s
The break frequencies can be found at 25.3 and 60.2.
The compensator is
The resulting system Bode plot shows the impact of the phase lead.
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54. HW!
PROBLEM: Design a lead compensator for the previous system to meet
the following specifications: %OS = 20%, Ts =0.2s and Kv = 50.
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55. Lag-Lead Compensation
As with the Root Locus designs we considered previously, we often
require both lead and lag components to effect a particular design
This provides simultaneous improvement in transient and steady-state
responses
One method is to design the lag compensation to lower the high-
frequency gain, stabilize the system, and improve the steady-state
error and then design a lead compensator.
In this case we are trading off three primary design parameters
• Crossover frequency ωc which determines bandwidth, rise time and settling
time
• Phase margin which determines the damping coefficient and hence
overshoot
• Low frequency gain which determines steady state error characteristics
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56. Lag-Lead Compensation
The transfer function of a single, passive lag-lead network is
Design Procedure
1. Using a second-order approximation, find the closed-loop
bandwidth required to meet the settling time, peak time, or rise
time requirement
2. Set the gain, K, to the value required by the steady-state error
specification.
3. Plot the Bode magnitude and phase diagrams for this value of gain.
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57. Lag-Lead Compensation
4. Using a second-order approximation, calculate the phase margin to meet the
damping ratio or percent overshoot requirement.
5. Select a new phase-margin frequency near ωBW.
6. At the new phase-margin frequency, determine the additional amount of phase
lead required to meet the phase-margin requirement. Add a small contribution
that will be required after the addition of the lag compensator.
7. Design the lag compensator by selecting the higher break frequency one decade
below the new phase-margin frequency.
8. Design the lead compensator. Using the value of γ from the lag compensator
design and the value assumed for the new phase-margin frequency, find the
lower and upper break frequencies for the lead compensator and obtaining for
T.
9. Check the bandwidth to be sure the speed requirement in Step 1 has been met.
10. Redesign if phase-margin or transient specifications are not met, as shown by
analysis or simulation.
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62. HW!
PROBLEM: Design a lag-lead compensator for a unity feedback system
with the forward-path transfer function
to meet the following specifications: %OS=10%; Tp = 0:6 s, and Kv=10.
Use frequency response techniques.
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63. Design procedure via Nichols chart
1. Calculate the damping ratio from the percent overshoot
requirement .
2. Calculate the peak amplitude, Mp, of the closed-loop response
3. Calculate the minimum closed-loop bandwidth to meet the peak
time requirement by peak time and the damping ratio from (1).
4. Plot the open-loop response on the Nichols chart.
5. Raise the open-loop gain until the open-loop plot is tangent to the
required closed-loop magnitude curve, yielding the proper Mp.
6. Place the lead zero at this point of tangency and the lead pole at a
higher frequency. (Zeros and poles are added in SISOTOOL by
clicking either one on the tool bar and then clicking the position on
the open-loop frequency response curve where you desire to add
the zero or pole.)
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64. Design procedure via Nichols chart
7. Adjust the positions of the lead zero and pole until the open-loop
frequency response plot is tangent to the same Mp curve, but at the
approximate frequency found in (3). This yields the proper closed-
loop peak and proper bandwidth to yield the desired percent
overshoot and peak time, respectively.
8. Evaluate the open-loop transfer function, which is the product of
the plant and the lead compensator, and determine the static error
constant.
9. If the static error constant is lower than required, a lag
compensator must now be designed. Determine how much
improvement in the static error constant is required.
10. Recalling that the lag pole is at a frequency below that of the lag
zero, place a lag pole and zero at frequencies below the lead
compensator and adjust to yield the desired improvement in static
error constant
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65. Lag-Lead Design Using the
Nichols Chart Example
PROBLEM: Design a lag-lead compensator for the plant
to meet the following requirements:
• a maximum of 20% overshoot
• a peak time of no more than 0.5 seconds
• a static error constant of no less than 6.
SOLUTION: We follow the steps enumerated immediately above,
1. ζ . 0=456 for 20% overshoot.
2. Mp =1.23=1.81 dB for ζ = 0.456.
3. ωBW =9.3 r/s for ζ = 0.456 and Tp = 0.5.
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66. Lag-Lead Design Using the
Nichols Chart Example
4. Plot the open-loop frequency response curve on the Nichols chart for
K = 1.
5. Raise the open-loop frequency response curve until it is tangent to
the closed-loop peak of 1.81 dB curve
6. Place the lead zero at this point of tangency and the lead pole at a
higher frequency.
7. Adjust the positions of the lead zero and pole until the open-loop
frequency response plot is tangent to the same Mp curve, but at the
approximate frequency found in 3.
8. Checking Designs/Edit Compensator . . . Shows
which yields a Kv = 3.
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67. Lag-Lead Design Using the
Nichols Chart Example
9. We now add lag compensation to improve the static error constant
by at least 2. Now add a lag pole at 0.004 and a lag zero at 0.008.
Readjust the gain to yield the same tangency as after the insertion of
the lead.
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68. HW!
PROBLEM: Design a lag-lead compensator using the Nichols chart for a
unity feedback system with the forward-path transfer function
to meet the following specifications: %OS=10%; Tp = 0:6 s, and Kv=10.
Use frequency response techniques.
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69. Conclusions
We have looked at techniques for designing controllers
using the frequency techniques
There is once again a trade-off in the requirements of the
system
By selecting appropriate pole and zero locations we can
influence the system properties to meet particular design
requirements
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70. Antenna Control: Gain Design
PROBLEM: Given the antenna azimuth position control system shown
below, Configuration 1, use frequency response techniques to do the
following:
a. Find the preamplifier gain required for a closed-loop response of 20%
overshoot for a step input.
b. Estimate the settling time.
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73. Antenna Control: Cascade
Compensation Design
PROBLEM: Given the antenna azimuth position control system block
diagram shown before, Configuration 1, use frequency response
techniques and design cascade compensation for a closed-loop
response of 20% overshoot for a step input, a fivefold improvement in
steady-state error over the gain-compensated system operating at 20%
overshoot, and a settling time of 3.5 seconds.
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Editor's Notes
Solution: to find the gain margin, first find the frequency where the Nyquist diagram crosses the negative real axis.
The Nyquist diagram crosses the real axis at frequency of . The real part is calculated to be -0.3. Thus, the gain can be increased by (1/0.3) =3.33 before the real part becomes -1. Hence the gain margin is GM = 20log 3.33 = 10.45 dB.
To find the phase margin, find the frequency for which the magnitude is unity. We need to solve the equation to find frequency = 1.253 rad/s. at this freq. the phase angle is -112.3o. The difference between this angle and -180o is 67.7o, which is the phase margin.
we learned that phase margin is related to percent overshoot (Eq. (10.73)) and
bandwidth is related to both damping ratio and settling time or peak time (Eqs. (10.55) and
(10.56)).
An advantage of using frequency
design techniques is the ability to design derivative compensation, such as lead compensation,
to speed up the system and at the same time build in a desired steady-state error requirement
that can be met by the lead compensator alone. Recall that in using root locus there are an
infinite number of possible solutions to the design of a lead compensator. One of the
differences between these solutions is the steady-state error. We must make numerous tries
to arrive at the solution that yields the required steady-state error performance. With frequency
response techniques, we
An unstable system cannot be designed for a specific transient response or steady-state error requirement.
There are many definitions for stability, depending upon the kind of system or the point of view. In this section, we limit ourselves to linear, time-invariant systems.
These, then, are the basic facts underlying our design for stability, transient response,
and steady-state error using frequency response methods, where the Nyquist criterion and
the Nyquist diagram compose the underlying theory behind the design process. Thus,
even though we use the Bode plots for ease in obtaining the frequency response, the
design process can be verified with the Nyquist diagram when questions arise about
interpreting the Bode plots. In particular, when the structure of the system is changed with
additional compensator poles and zeros, the Nyquist diagram can offer a valuable
perspective.
we see that if we desire a phase margin, ΦM, represented
by CD, we would have to raise the magnitude curve by AB. Thus, a simple gain adjustment
can be used to design phase margin and, hence, percent overshoot.
1. Draw the Bode magnitude and phase plots for a convenient value of gain.
2. Using Eqs. (4.39) and (10.73), determine the required phase margin from the percent
overshoot.
3. Find the frequency, ωΦM , on the Bode phase diagram that yields the desired phase
margin, CD, as shown on Figure 11.1.
4. Change the gain by an amount AB to force the magnitude curve to go through 0 dB at
ωΦM . The amount of gain adjustment is the additional gain needed to produce the
required phase margin.
1. Choose K . 3:6 to start the magnitude plot at 0 dB at ω . 0:1 in Figure 11.3.
2. Using Eq. (4.39), a 9.5% overshoot implies ζ . 0:6 for the closed-loop dominant
poles. Equation (10.73) yields a 59.2° phase margin for a damping ratio of 0.6.
3. Locate on the phase plot the frequency that yields a 59.2° phase margin. This
frequency is found where the phase angle is the difference between 180° and
59.2°, or 120:8°. The value of the phase-margin frequency is 14.8 rad/s.
4. At a frequency of 14.8 rad/s on the magnitude plot, the gain is found to be 44.2 dB.
This magnitude has to be raised to 0 dB to yield the required phase margin. Since
the log-magnitude plot was drawn for K . 3:6, a 44.2 dB increase, or K .
3:6 162:2 . 583:9, would yield the required phase margin for 9.48% overshoot.
The function of the lag compensator as seen on Bode diagrams is to (1) improve the static
error constant by increasing only the low-frequency gain without any resulting instability,
and (2) increase the phase margin of the system to yield the desired transient response. These
concepts are illustrated in Figure 11.4.
In (a) system is type 0, since the initial slope is 0. And 20 log KP =25, or KP = 17.78
In (b) system is type 1, since the initial slope is -20dB/decade. And the value of Kv is the value of frequency that the initial slope intersects at the zero dB crossing of the frequency axis. Hence Kv = 0.55
In (c) system is type 2, since the initial slope is -40dB/decade. And the value of √Ka is the value of frequency that the initial slope intersects at the zero dB crossing of the frequency axis. Hence Ka = 32 = 9
For second-order systems, we derived the relationship between phase margin and percent
overshoot as well as the relationship between closed-loop bandwidth and other time-domain
specifications, such as settling time, peak time, and rise time
The design of the lag compensator is not critical, and
any design for the proper phase margin will be relegated to the lead compensator. The
lag compensator simply provides stabilization of the system with the gain required for
the steady-state error specification. Find the value of γ from the lead compensator’s
requirements. Using the phase required from the lead compensator, the phase response
curve of Figure 11.8 can be used to find the value of γ . 1=β. This value, along with the
previously found lag’s upper break frequency, allows us to find the lag’s lower break
frequency.
. As an example, recall from
Eq. (9.5) that the improvement in static error constant for a Type 1 system is
equal to the ratio of the lag zero value divided by the lag pole value. Readjust the
gain if necessary.