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4K410 MOTION CONTROL
EXERCISES AND EXPERIMENTS
S. Colak
0754862
J.C. Pérez Muñoz
0755654
Professor M. Steinbuch
Technische Universiteit Eindhoven
Mechanical Engineering Department
Systems & Control
2010/2011
1
CONTENTS
CONTENTS................................................................................................................................................... 1
LIST OF FIGURES ........................................................................................................................................ 2
1. SYSTEMS AND CONTROLLER DESIGN ................................................................................................ 4
1.2. Interpreting a Bode Diagram .............................................................................................................. 4
1.3. Estimating Transfer Functions ........................................................................................................... 5
1.4. Control of a motion system ................................................................................................................ 8
1.5. Load feedback exercise ................................................................................................................... 12
2. STABILITY .............................................................................................................................................. 14
2.1. Inverted pendulum ........................................................................................................................... 14
2.2. Non-minimum phase systems .......................................................................................................... 17
3. DESIGN FOR PERFORMANCE ............................................................................................................. 19
3.1. Non-collocated plant ........................................................................................................................ 19
3.2. Performance vs. Robustness ........................................................................................................... 21
3.3. Wafer stage exercise ....................................................................................................................... 23
4. FREQUENCY RESPONSE MEASUREMENTS ..................................................................................... 27
4.2. Frequency response function of a mass system ............................................................................. 27
4.3. Closed loop FRF measurement ....................................................................................................... 29
5. FEEDFORWARD CONTROL ................................................................................................................. 33
5.1. Mass Feedforward design ............................................................................................................... 33
5.4. Feedforward design ......................................................................................................................... 34
6. The Digital Environment .......................................................................................................................... 35
6.1. Delay and Sampling ......................................................................................................................... 35
6.2. Digital Control Systems .................................................................................................................... 36
MOTION CONTROL EXPERIMENTS ........................................................................................................ 39
Experiment 2. Frequency Response Measurements .............................................................................. 39
Experiment 3. Feedforward tuning .......................................................................................................... 43
Experiment 4. Loop shaping game ......................................................................................................... 45
REFERENCES ............................................................................................................................................ 51
2
LIST OF FIGURES
Figure 1. Frequency response for system H1 ............................................................................................... 4
Figure 2. Frequency response for system H2 ............................................................................................... 5 Figure 3. Frequency response of systems H1 and H2 ................................................................................... 6 Figure 4. Frequency response for system H1 and the fitted model S1 .......................................................... 7 Figure 5. Frequency response for system H2 and the fitted model S2. ......................................................... 7 Figure 6. Mass-spring-damper-mass system................................................................................................ 8
Figure 7. Bode diagram for systems H1 and H2. ........................................................................................... 9 Figure 8. Model of the system H1 controlled with a lead filter with gain. ....................................................... 9 Figure 9. Error of the system using an unitary step at 0.1 seconds. .......................................................... 10 Figure 10. Time specifications at different phase margins. ........................................................................ 11 Figure 11. Time specifications at different Bandwidths. ............................................................................. 12
Figure 12. Nyquist plot for the system H with a PD controller (Left) and closed loop response (Right). .... 13 Figure 13. Nyquist plot (Left) and close loop response (Right) for the system with PD and Notch controller.
.................................................................................................................................................................... 13 Figure 14. Bode diagram (Left) and control effort (Right) of the system H with a Notch+Lead filter
controller...................................................................................................................................................... 14
Figure 15. Bode diagram for the inverted pendulum. ................................................................................. 15 Figure 16. Nyquist diagram for the inverted pendulum with controller C1=1 (Left) and C2=-0.55 (Right). .. 15 Figure 17. Step response of the closed loop inverted pendulum with controller C1 and C2. ...................... 16 Figure 18. Nyquist plot (Left) and closed loop step response (Right) of the inverted pendulum using
controller C. ................................................................................................................................................. 16
Figure 19. Bode diagram of systems H1 and H2. ........................................................................................ 17 Figure 20. Bode diagram (Left) and Nyquist plot (Right) for the system H2 using the controller C1. .......... 18 Figure 21. Nyquist plot (Left) and close loop step response (Right) for the system C2H2. ......................... 19 Figure 22. Bode diagram (Left) and Time response (Right) of the d to e transfer function. ....................... 20 Figure 23. Time response for the d to e transfer function when the controller has 2 poles at 2Hz with
damping factor 0.01 (Left) and 0 (Right). .................................................................................................... 20 Figure 24. Mass-spring-damper system. .................................................................................................... 21 Figure 25. Bode diagram (Left) and control sensitivity function (Right) for the system H with control C1. . 22 Figure 26. Step response for the nominal system and controller C1 (Left) and systems with different mass
value using controller C2 (Right). ................................................................................................................ 22
Figure 27. Bandwidth (Top left), Modulus margin (Top Right), settling time (Bottom left) and maximum
overshoot (Bottom right) vs. mass. ............................................................................................................. 23 Figure 28. Measured data and fitted model for the wafer stage. ................................................................ 24 Figure 29. Diagram for the closed loop wafer stage with disturbances in the measurements and the
output. ......................................................................................................................................................... 24
Figure 30. Complementary sensitivity of the system. ................................................................................. 25 Figure 31. Complementary sensitivity obtained with controller C2. ............................................................. 26 Figure 32. Measurement error (Top), FFT of the measurement error (Center) and system output with
unitary step reference (Bottom). ................................................................................................................. 26 Figure 33. PD-controller mass model with noise signal after the controller block. ..................................... 27
Figure 34. Signals in time domain (Left) and Frequency domain (Right). .................................................. 27 Figure 35. FRF of the sensitivity of the system and Coherence function. .................................................. 28 Figure 36. Computed open loop FRF from the sensitivity. ......................................................................... 28 Figure 37. FRF of the PD controller and the plant. ..................................................................................... 29 Figure 38. Simulink model used for the closed loop FRF measurement. ................................................... 29
3
Figure 39. Closed loop sensitivity and its coherence. ................................................................................. 30 Figure 40. Process sensitivity and its coherence. ....................................................................................... 30 Figure 41. FRF measurement of the plant. ................................................................................................. 31
Figure 42. Sensitivity and process sensitivity coherences, FRF of the plant with 60s simulation time. ..... 32 Figure 43. Sensitivity and process sensitivity coherences, FRF of the plant with 120s simulation time. ... 32 Figure 44. Sensitivity and process sensitivity coherences, FRF of the plant with 240s simulation time. ... 33 Figure 45. Acceleration for the 3
rd order reference (Top), error without feedforward (Center) and error with
acceleration feedforward (Bottom). ............................................................................................................. 34
Figure 46. Feedforward model. ................................................................................................................... 34 Figure 47. Acceleration, error without feed forward and error with feedforward. ........................................ 35 Figure 48. Represents the phase drop due to delay ................................................................................... 36 Figure 49. Magnitude Frequency response of a lead filter and its discrete time equivalents. ................... 37 Figure 50. Magnitude Frequency response of a lead + notch filter and its discrete time equivalents. ....... 38
Figure 51. Simulink scheme of the closed loop FRF measurement. .......................................................... 39 Figure 52. FRF of the plant. ........................................................................................................................ 40 Figure 53. FRF of the plant measured with 3 different sample frequencies. .............................................. 41 Figure 54. FRF measured with different PD controllers. ............................................................................. 42 Figure 55. Open-loop FRF using QadScope and Closed-loop FRF. .......................................................... 43
Figure 56. Motor feedback scheme for feedforward tuning. ....................................................................... 43 Figure 57. Setpoint trajectory used for the feedforward tuning. .................................................................. 44 Figure 58. Tracking error and acceleration without using feedforward. ...................................................... 44 Figure 59. Tracking errors obtained with and without feedforward. ............................................................ 45 Figure 60. Expected motor and load feedback responses. ........................................................................ 46
Figure 61. Simulink scheme for the FRF measurements. .......................................................................... 46 Figure 62. measured closed loop sensitivity and coherence. ..................................................................... 47 Figure 63. Controller frequency response. .................................................................................................. 47 Figure 64. Plant FRF measurement. ........................................................................................................... 48 Figure 65. Frequency response for C2(s)H(s) (left) and Sensitivity (Right). ............................................... 49
Figure 66. Measured open loop and sensitivity for the plant. ..................................................................... 49 Figure 67. Time response of the load feedback setup. ............................................................................... 50
4
1. SYSTEMS AND CONTROLLER DESIGN
1.2. Interpreting a Bode Diagram
(a) System H1(s)
Figure 1. Frequency response for system H1
The magnitude diagram starts with constant value so there is no integrator in the system. It is assumed
that diagrams start at frequency 10-1
Hz. At frequency 100, System H1(s) has 2 stable poles because of
the peak in the magnitude and approximately -180° phase change. At a frequency between 2Hz and
3Hz, there is a zero in the system and it is very close to first two stable poles and that’s why phase
diagram could not reach -180°. At frequency 20Hz, there is a zero because it changes phase diagram
+90°. At frequency 30Hz, there are two stable poles which causes -180° phase change in the system. If
slope of magnitude and phase are checked for higher frequencies, it could be seen that relative degree is
2. Also, in the system there are 4 poles and 2 zeros which shows relative degree is 2.
(b) System H2(s)
It is assumed that frequency at the beginning of the diagram is 10-1
Hz. Magnitude diagram starts with a
negative slope which indicates that the system has integrators. Every integrator adds -90° phase change,
so the phase diagram starts in -360° instead of 0° which indicates system has 4 integrators. At frequency
0.9 Hz, system has 2 zeros because of +180° phase change and temporary decreasing of the magnitude
diagram, the real phase value of the system is -360° + 180° = -180° and this agrees with phase diagram.
At frequency 4 Hz, system has again 2 zeros which causes +180° phase change. At frequency 20 Hz and
40 Hz, there is a stable pole which results -90° phase change in the system. Finally, the system has 5
poles and 4 zeros which shows relative degree is 1. Also, the slope for higher frequencies is -1 which
indicates relative degree is 1.
5
Figure 2. Frequency response for system H2
1.3. Estimating Transfer Functions
The frequency response of both systems is presented in the Figure 3. It’s possible to determine that
system H1 is unstable because it has 2 unstable poles located between 7.25 Hz and 7.5 Hz, while system
H2 has two stable poles at the same frequency.
From the bode diagram, it can be inferred that System H1 has 6 poles, 4 zeros and no integrators, then
the command used to fit the model was:
[num,den] = frfit(H1,hz,[6,4,0],1)
The fitted model is ����� = ������ with N1(s) and D1(s) approximated as:
N��s� = 0s� − 1.0658 × 10���s� + 9.1995s� + 6.2172s� + 4.5961 × 10 s� + 7.8594 × 10 s + 1.3927 × 10��
D��s� = 1s� + 5.8698s� + 8.0999 × 10�s� − 6.8218 × 10�s# + 3.4707 × 10�$s� − 7.1509 × 10�$s+ 6.9253 × 10�#
In the other hand, it can be inferred that System H2 has 8 poles, 2 zeros and no integrators, then the
command used to fit the model was:
[num,den] = frsfit(H2,hz,[8,2,0],1)
6
The fitted model is ����� = �%���%�� with N2(s) and D2(s) approximated as: N��s� = 0s& + 6.3949 × 10��#s − 2.3283 × 10�'s� + 2.1234 × 10� s� − 1.0376 × 10�#s�− 2.9297 × 10�#s# + 7.4363 × 10�#s� + 1.1860 × 10��s + 2.4101 × 10�& D��s� = 1s& + 4.4030 × 10s + 1.2447 × 10�s� + 2.7233 × 10 s� + 3.9497 × 10��s�+ 2.1569 × 10��s# + 1.5483 × 10��s� + 2.8260 × 10��s + 3.0534 × 10��
Figure 3. Frequency response of systems H1 and H2
From the model of system H1(s) we have that the unstable poles of the model are:
1.1182e+000 +4.5786e+001i, 1.1182e+000 -4.5786e+001i
Which are located at the frequency 45.786/2π = 7.2871 Hz, this result matches with the previous results.
In the Figure 4 and Figure 5 the frequency response of the fitted models are presented and compared
with the data. For frequencies higher than 200 Hz the fitted magnitudes aren’t close to the data, but this
could be explained if the data contains noise. Also, the phase diagrams of the fitted models don’t have
the phase delay present on the data, this is because the fitted model are in continuous time but the data
presents the phase delay caused by the sampling process.
10-1
100
101
102
103
-160
-140
-120
-100
-80
-60
-40
-20
0
20
Hz
Magnitude
20|H1|
20|H2|
10-1
100
101
102
103
-200
-150
-100
-50
0
50
100
150
200
Hz
Phase a
ngle
H1 phase
H2 phase
7
Figure 4. Frequency response for system H1 and the fitted model S1
Figure 5. Frequency response for system H2 and the fitted model S2.
The plant S2 is easy to control at 20 Hz, because is stable and closed-loop stable, so to achieve a good
performance at 20Hz is enough to make the gain higher and increase the phase using a lead filter.
10-1
100
101
102
103
-140
-120
-100
-80
-60
-40
-20
0
Hz
Magn
itud
e
20|H1|
20|S1|
10-1
100
101
102
103
-200
-150
-100
-50
0
50
100
150
200
Hz
Ph
ase a
ngle
H1 phase
S1 phase
10-1
100
101
102
103
-200
-150
-100
-50
0
50
Hz
Magnitude
20|H2|
20|S2|
10-1
100
101
102
103
-200
-150
-100
-50
0
50
100
150
200
Hz
Phase a
ngle
H2 phase
S2 phase
8
1.4. Control of a motion system
We consider the classic mass-spring-damper-mass system depicted in Figure 6.
Figure 6. Mass-spring-damper-mass system.
The transfer function from F to x1, H1 is given by:
����� = (����)��� = 1*� +�� + ,*� � + -*�.�� +�� + + ,*� + ,*�. � + -*� + -*�.
The resonance frequency of the system is / 01 + 01% and the anti-resonance frequency is / 01%.
The transfer function from F to x2, H2 is given by
����� = (����)��� = ,*�*� +� + -,.�� +�� + + ,*� + ,*�. � + -*� + -*�.
Given the values m1=0.015 kg, m2=0.045kg, d=0.4Ns/m and k=2200N/m, the bode diagrams of H1 and H2
are presented in the Figure 7, it’s possible to identify the zeros of the system H1 at 35.1 Hz. For
frequencies lower than 10 Hz the two systems have the same response, then both systems have a
bandwidth of 0.65 Hz, also the two system have two poles at 71.03 Hz as expected. For frequencies
higher than 1000 Hz, system H1 has slope -2 meanwhile system H1 has slope -3, this is because the
system H2 has only one zero.
9
Figure 7. Bode diagram for systems H1 and H2.
The model of the system with the designed controller, a Lead filter with gain, is presented in Figure 8. The
gain is 429, the zero and pole of the lead filter are located at 20/3 Hz and 60 Hz respectively. The error of
the system with a unitary step at 0.1 seconds is presented in Figure 9, the overshoot in this case is 0.35
(or 35%), the rising time is close to 0.025 seconds ant the settling time is about 0.08 seconds. In Figure
10, the time specifications obtained with a bandwidth of 20 Hz and different phase margin, as presented
in Table 1, are plotted. In this figure it’s possible to distinguish that the Maximum overshoot diminish when
the phase margin increases, this is because bigger phase margins mean that the phase at the crossing
frequency is closer to -90 degrees and, then, the poles of the system have lower real parts which
translates in less overshoot.
We also can see that the rising time it’s almost constant in all the cases, this is because the controller has
a faster response that the system dynamics, then the rising time depends more in the system.
Figure 8. Model of the system H1 controlled with a lead filter with gain.
10-1
100
101
102
103
104
105
-300
-250
-200
-150
-100
-50
0
50
Hz
Magnitude
20|H1|
20|H2|
10-1
100
101
102
103
104
105
-200
-150
-100
-50
0
50
100
150
200
Hz
Phase a
ngle
H1 phase
H2 phase
t
t
Step
s +(d/m2+d/m1)s +(k/m1+k/m2)s 4 3 2
1/m1*[1 d/m2 k/m2]
H1
e
Error
K/(2*pi*20/3).s+K
1/(2*pi*f2)s+1
Controller
Clock
10
Figure 9. Error of the system using an unitary step at 0.1 seconds.
Table 1. Phase margin specification and parameter of the controller
Phase Margin Gain Lead filter pole
53.6 429 60
56.1 423 70
58 419 80
59.5 417 90
60.8 415 100
62.6 412 120
64.5 410 150
65.5 409 175
66.4 408.7 200
67 408.2 225
67.9 407.9 275
0 0.05 0.1 0.15 0.2 0.25 0.3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time (s)
Err
or
(m)
11
Figure 10. Time specifications at different phase margins.
The same experiment was done maintaining the phase margin with the value 53.6 degrees and changing
the bandwidth with the specification presented in Table 2, in this case the settling time doesn’t have
significant changes, as expected. The overshoot in this case increases because the changes in the
bandwidth with constant phase margin lead to lower modulus margin which in inversely related to the
maximum overshoot. The settling time diminish as the bandwidth increases because higher bandwidth
means faster responses of the system, leading to lower settling times.
Table 2. Bandwidth specifications and parameters of the controller.
Bandwidth Gain Lead filter
pole
10.47 150 150
12.71 200 82
16.42 300 63.5
20 429 60
20.48 450 59.8
21.53 500 59.5
52 54 56 58 60 62 64 66 680
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Phase margin (Degrees)
Time specifications vs Phase margin at 20 Hz of Bandwidth
Rising Time
Maximum overshoot
Settling time
12
Figure 11. Time specifications at different Bandwidths.
1.5. Load feedback exercise
The transfer function for this system is:
���� = 1.711 × 10�� + 1.566 × 10'����� + 16.71� + 9.073 × 10��
We used a PD (Gain + Phase lead) controller with K=0.0035 and D=0.0035/(2*pi*1.2) so the zero of the
controller is in the frequency 1.2 Hz. The 2 stable poles are limiting the Bandwidth because they create a
loop in the Nyquist plot than can be corrected with only a phase lead controller, so we can only reduce
the loop so it doesn't encircle the -1 point by diminishing the gain and then we can add the phase lead to
shift the phase angle to obtain a stable behavior of the plant+controller. The Nyquist plot of the system
with the PD controller is presented in Figure 12 along with the closed loop response, in th Nyquist plot is
possible to see that there are not encirclements of the (0,-1) point and the closed loop response displays
a stable behavior.
10 12 14 16 18 20 220
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Bandwidth (Hz)
Time specifications vs Bandwidth with Phase Margin 53.6
Rising Time
Maximum overshoot
Settling time
13
Figure 12. Nyquist plot for the system H with a PD controller (Left) and closed loop response
(Right).
Now, we add a notch filter to the controller which has the goal to reduce the peak caused by the poles of
the plant so the bandwidth can be modified by increasing the gain, and the phase lead helps to stabilize
the plant adding the phase shift necessary close to the bandwidth. The notch has the zeros and poles at
50 Hz with damping 0.1 and 0.9 for the zeros and the poles respectively. In Figure 13 both nyquist plot
and closed loop response are plotted, in this case the system has a faster response but the maximum
overshoot is increased.
Figure 13. Nyquist plot (Left) and close loop response (Right) for the system with PD and Notch
controller.
Now, assuming that the control effort can’t be higher than 10V for a step setpoint of 1mm, we remove the
PD controller and we add a lead filter with zero at 5 Hz and pole at 16 Hz. In this case the notch has its
zeros and poles at 50 Hz but the damping is 0.1 for both. In the Figure 14 the bode diagram and the
control effort after a step of 1 mm are plotted, the bandwidth of the system is 12.09 Hz and the maximum
control effort is 0.5 V.
-1.5 -1 -0.5 0 0.5
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Am
plit
ude
Closed Loop Response
-3 -2 -1 0 1 2
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
Am
plit
ude
Closed Loop Response
14
Figure 14. Bode diagram (Left) and control effort (Right) of the system H with a Notch+Lead filter
controller.
2. STABILITY
2.1. Inverted pendulum
Given the inverted pendulum linear model around the equilibrium point and the parameter values, the
bode diagram is presented in the Figure 15, we observe on the Bode diagram that we have 1 stable pole
between 0.02 and 0.03 Hz, and another 2 poles close to 2 Hz but it seems like one is stable and the
other not because the phase diagram doesn’t exhibit changes, then also knowing about the zero on the
origin, we can say that the system is unstable.
The poles of the system are:
2� = 1.2076 × 10 2� = −1.2187 × 10
2# = −0.22218
They are located at the frequencies 1.922 Hz, 1.9397 Hz and 0.035362 Hz respectively, and these
frequencies agree with the previous results.
Using the simpler controllers C1=1 and C2=-0.55, we obtain the Nyquist diagram presented in Figure 16,
because we have an unstable pole the Nyquist diagram needs to have one counterclockwise
encirclement of the point (-1,0), neither of the both controllers satisfies this, so both closed loop are
unstable. In Figure 17 the step response of the closed loops are presented, this responses agree with the
previous result: the closed loops are unstable.
-400
-300
-200
-100
0
100M
agn
itude
(dB
)Bode Diagram
Frequency (Hz)
10-1
100
101
102
103
104
105
106
-360
-270
-180
-90
Ph
ase
(de
g)
0 0.2 0.4 0.6 0.8 1-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Time (s)
Contr
ol effort
[V
]
Control effort - Reference response
15
Figure 15. Bode diagram for the inverted pendulum.
Figure 16. Nyquist diagram for the inverted pendulum with controller C1=1 (Left) and C2=-0.55
(Right).
-200
-150
-100
-50
0
50
Magnitu
de (
dB
)
10-2
10-1
100
101
102
103
104
105
0
45
90
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Nyquist Diagram
Real Axis
Imagin
ary
Axis
-1 -0.5 0 0.5 1 1.5-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4Nyquist Diagram
Real Axis
Imagin
ary
Axis
16
Figure 17. Step response of the closed loop inverted pendulum with controller C1 and C2.
In order to obtain a stabilizing controller, we use a negative gain, a lead filter and a weak integrator. The
weak integrator help to counter the action of the zero of the system. The model of the controller is:
3��� = −18 4� + 25 × 0.5� 6 + �25 × 10 + 1.+ �25 × 60 + 1.
The nyquist plot and the step response of the closed loop using the controller C, are presented in the
Figure 18, where it’s possible to check that the Nyquist diagram has one counterclockwise encirclement
of the (-1,0) point, giving the stable behavior showed in the time response.
Figure 18. Nyquist plot (Left) and closed loop step response (Right) of the inverted pendulum
using controller C.
For stable open loop systems we know that the system doesn't have unstable poles, so we don't need to
create a encirclement of the (-1,0) point in the Nyquist diagram, in other words: we only need to be sure
that our plant+controller have phase angle between -180° and -90° in the bandwidth frequency.
0 0.2 0.4 0.6 0.8 1-1000
-800
-600
-400
-200
0
200
400
600
800
1000
Time (s)
C1
C2
-20 -15 -10 -5 0
-4
-3
-2
-1
0
1
2
3
4
Nyquist Diagram
Real Axis
Imagin
ary
Axis
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6Step Response
Time (sec)
Am
plit
ude
17
2.2. Non-minimum phase systems
The Bode diagram for the systems H1 and H2 is presented in the Figure 19, we can see that magnitudes
are equal, but the phase diagrams differ because H2 has a unstable zero, this unstable zero can be
written as -(6-s) and that's the reason for the phase diagram to start at +90 degrees. The name non-
minimum phase corresponds to the fact that the phase for this system is
∠����� = ∠ 81000 4 6 − ��# + 20�� + 5000�69 + 180°
In other words, the unstable zero introduces a phase-shift of +180 in the phase diagram.
Figure 19. Bode diagram of systems H1 and H2.
As the system H1 is stable, we only need to mode the phase diagram to the right in order to increase the
bandwidth, that's possible with a lead-lag controller, then is possible to finally adjust the exact bandwidth
using the gain. The controller has the expression:
3���� = 18 �25 × 20 + 1�25 × 90 + 1
The modulus margin is 3.2864 and the phase margin is 43.088°. The open loop Bode diagram for system
C1xH2 is presented in Figure 19 along with the Nyquist plot of the system, the phase margin in this case is
46.740°. The Nyquist plot of this system has a (clockwise) encirclement of the point (-1,0), which means
that the system has one unstable closed loop pole. Because the integrator on the system, the
encirclement is a infinite loop, and the position of the encirclement (right hand or left hand plane) should
10-2
10-1
100
101
102
-180
-90
0
90
Phase (
deg)
Bode Diagram
Frequency (Hz)
-60
-40
-20
0
20
40
Magnitu
de (
dB
)
H1
H2
H1
H2
18
be derived by replacing s in the transfer function for :;<= with −90° ≤ ? ≤ 90° and a really small ε, then
it’s possible to check the resulting angle, as presented in [1].
3@:;<=A�@:;<=A = 18 :;<=25 × 20 + 1:;<=25 × 90 + 1 × 1000 B :;<= − 6:;<=�:;<�= + 20:;<= + 5000�C
Then, if : is close to 0,
3@:;<=A�@:;<=A~ − E: ;�<=
And using the fact that – E = E;<�&$°, we have then that the angle of the infinite loop is 180° − ? as θ
goes from -90° to 90°, in other words the infinite loop goes from 270° to 90° crossing through 180°
meaning that is located at the left hand plane.
Figure 20. Bode diagram (Left) and Nyquist plot (Right) for the system H2 using the controller C1.
The new stabilizing controller for H2 has the form
3���� = −2.22 14+ �25 × 12.� + 2�25 × 12 + 16
This controller give us |I| = 5.9dB, a phase margin of 58.4° and a Bandwidth of 0.47 Hz. In this case, the
bandwidth of the controller + plant system is limited by the frequency of the non-minimum phase zero
which is located at 6 25 = 0.9549 Hz⁄ . In Figure 21 the Nyquist plot and the time response of the closed
loop for the system C2H2 are plotted, and again the Nyquist plot exhibits a infinite loop but in this case this
loop is located at the right hand plane, thus the close loop system has 0 unstable poles.
-100
-50
0
50
100
Magnitu
de (
dB
)
10-2
10-1
100
101
102
103
104
-180
-90
0
90
Phase (
deg)
Bode Diagram
Frequency (Hz)
-4 -2 0 2 4 6 8 10 12-60
-40
-20
0
20
40
60Nyquist Diagram
Real Axis
Imagin
ary
Axis
19
Figure 21. Nyquist plot (Left) and close loop step response (Right) for the system C2H2.
3. DESIGN FOR PERFORMANCE
3.1. Non-collocated plant
For the given plant, the stabilizing controller is constructed with a gain, a lead filter and a notch. In this
case the notch is used to attenuate the resonance of the system at 52.3 Hz and the lead filter helps to
achieve the desired phase at the desired bandwidth (10Hz). In this case the controller has the structure:
3��� = 0.756 �25 × 6 + 1�25 × 30 + 1+ �25 × 52.3.� + 2 × 0.1 × �25 × 52.3 + 1+ �25 × 52.3.� + 2 × 0.5 × �25 × 52.3 + 1
The transfer function from the disturbance to the error is given by the expression
K���L��� = −����1 + ����3���
The bode diagram of the transfer function from d to e and the time response of this system are plotted in
Figure 22, when the sine disturbance with amplitude 1 and frequency 2 Hz is applied, the error has a
periodic response with an amplitude larger than 1 and frequency 2 Hz. Looking at the bode diagram, any
disturbance signal with a frequency lower than 7 Hz will have a direct effect on the error, because the
transfer function has magnitude larger than 0 for those frequencies.
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0
-1
-0.5
0
0.5
1
Nyquist Diagram
Real Axis
Imagin
ary
Axis
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
1
1.2Step Response
Time (sec)
Am
plit
ude
20
Figure 22. Bode diagram (Left) and Time response (Right) of the d to e transfer function.
To achieve the disturbance rejection specification error less than 0.01 within 0.5 seconds, we need to
create an anti-resonance at the 2 Hz frequency for the d to e transfer function. This can be possible
adding two poles at 2 Hz in the controller with a damping factor of 0.01, but then the phase will change
drastically making the system unstable, then we add two zeros at 2 Hz in the controller with a damping
factor of 0.95, this works like adding a notch at the d to e transfer function. In order to make the error
converge to zero, we need to made the damping factor of the two poles in the controller equal to zero. In
the Figure 23 the time response of the system with the improved controller with the poles with damping
factor 0.01 and 0, we can see that the error in the second case goes to zero.
Figure 23. Time response for the d to e transfer function when the controller has 2 poles at 2Hz
with damping factor 0.01 (Left) and 0 (Right).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.5
0
0.5
1
Time (s)
d
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
Time (s)
e
100
101
102
103
104
-270
-180
-90
0
90
180
Ph
as
e (
de
g)
Bode Diagram
Frequency (rad/sec)
-200
-150
-100
-50
0
50
Mag
nitu
de (
dB
)
0 0.5 1 1.5-1
-0.5
0
0.5
1
Time (s)
d
0 0.5 1 1.5-0.3
-0.2
-0.1
0
0.1
Time (s)
e
0 0.5 1 1.5-1
-0.5
0
0.5
1
Time (s)
d
0 0.5 1 1.5-0.3
-0.2
-0.1
0
0.1
Time (s)
e
21
3.2. Performance vs. Robustness
We consider here the mass-spring-damper system presented in Figure 24, this system has the model:
���� = 1*�� + ,� + -
Figure 24. Mass-spring-damper system.
We’ll assume that m = 0.5kg, d = 0.5Ns/m and k = 40N/m. We need to design a controller that achieve a
module margin lower than 6 dB and that has steady state error equal to 0 for stepwise references,
furthermore, this controller must accomplish |3I| < 50dB, open loop bandwidth lower or equal to 3 Hz
and maximum order equal to 2.
To achieve the |3I| < 50dB constraint we need a lowpass filter to limit the magnitude of CS, in the other
hand, we need a integrator to achieve the 0 steady state error. A PD controller will help to improve the
settling time. We are interested in achieve the constraints on the Control Sensitivity function, so the first
performance measure is the magnitude of CS for large frequencies. After that, the steady state error, the
maximum overshoot and the settling time are the other 3 measures of the performance of the controller,
we expect a maximum overshoot of 20% and a settling time lower than 2 seconds, with the performance
constraint that steady state error should be 0 for stepwise references. The finally designed controller is:
3���� = 60 �25 × 2 + 1�25 × 10 + 1 � + 25 × 0.6�
The bode diagram and control sensitivity function of this controller are presented in Figure 25, the
bandwidth for the open loop is 2.6440 Hz, the modulus margin is 5.6377 dB, the phase margin is 30.1447
degrees, the settling time with an unitary step is 1.3351 seconds and the maximum overshoot is 18.55%
while we have 0 steady state error. We can see that the control sensitivity function is lower than 50 dB for
frequencies larger than 2Hz. This controller has the lower settling time achieved meanwhile accomplish
all the design and performance constraints.
22
Figure 25. Bode diagram (Left) and control sensitivity function (Right) for the system H with
control C1.
Now, taking in account that the mass has an uncertainty of 0.1 kg, we change the parameters of the
controller in order to have a maximal performance for every possible value of the mass parameter. In this
case, the new controller is:
3���� = 58 �25 × 2 + 1�25 × 10 + 1 � + 25 × 0.45�
The time response of the nominal system with the controller C1 is presented in Figure 26 along with the
responses using controller C2 of the systems with mass 0.4kg, 0.5kg and 0.6kg. The settling time with the
robust controller is increased but it’s still lower than 2 seconds, the maximum overshoot also remains
lower than 20%. In the nominal case, the system has a bandwidth of 2.5910 Hz, a module margin of
5.0306 dB, a phase margin of 32.9847 degrees, a settling time of 1.8585 seconds and a maximum
overshoot of 12.99%.
Figure 26. Step response for the nominal system and controller C1 (Left) and systems with
different mass value using controller C2 (Right).
To illustrate the behavior of the robust controller, Figure 27 presents the graph for the bandwidth,
modulus margin, settling time and maximum overshoot for different values of the mass. It’s is possible to
10-2
10-1
100
101
102
103
-180
-135
-90
-45
0
Phase (
deg)
Bode Diagram
Frequency (Hz)
-100
-50
0
50
Magnitu
de (
dB
)
10-2
10-1
100
101
102
103
-180
-135
-90
-45
0
45
Phase (
deg)
Sensitivity function
Frequency (Hz)
20
40
60
80
100
120
Magnitu
de (
dB
)
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (sec)
Am
plit
ude
Controller C1 and m=0.5kg
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (sec)
Am
plit
ude
m = 0.4
m = 0.5
m = 0.6
23
check that every performance constraint is accomplished, but also that the settling time is minimum for a
mass value close to 0.5kg.
Figure 27. Bandwidth (Top left), Modulus margin (Top Right), settling time (Bottom left) and
maximum overshoot (Bottom right) vs. mass.
When comparing C1 and C2, we can conclude that the more robust controller can’t achieve the
performance achieved by a controller designed for a particular system without uncertainty, normally some
performance measures will be increased when the robust controller is used, like the maximum overshoot
or the settling time, but other measures could be reduced as for example the bandwidth, leading to a
slightly slower response of the system.
3.3. Wafer stage exercise
After loading the wafer stage FRF measurements, we plotted the Bode diagram of the data and we
identified the presence of 2 integrators, 8 zeros and 10 poles. This can be inferred after observing the
phase changes between 100Hz and 200Hz. Then using the command
[num,den] = frsfit(frf,hz,[12,8,2],1);
We obtain the model of the fitted wafer stage. In Figure 28 the bode diagram for the real data and the
fitted model are plotted, the fitted model is used to design the controller for the plant.
0.4 0.45 0.5 0.55 0.62.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
Mass (kg)
(dB
)
Bandwidth
0.4 0.45 0.5 0.55 0.64
4.5
5
5.5
6
Mass (kg)
(dB
)
Modulus margin
0.4 0.45 0.5 0.55 0.61.84
1.86
1.88
1.9
1.92
1.94
Mass (kg)
(s)
Settling time
0.4 0.45 0.5 0.55 0.610
11
12
13
14
15
16
Mass (kg)
(%)
Maximum overshoot
24
Figure 28. Measured data and fitted model for the wafer stage.
The controller has a PD controller with the zero at 400 Hz, an notch at 1210 Hz and a lead filter to
increase the phase around 400 Hz, its structure is:
3���� = 5.5 + �25 × 400 + 1. N+ �25 × 1210.� + 2 × 0.01�25 × 1210 + 1+ �25 × 1210.� + 2�25 × 1210 + 1O N �25 × 120 + 1�25 × 1200 + 1O
The achieved bandwidth is 395 Hz, the modulus margin 7.5dB and the phase margin is 38.8 degrees and
the gain margin is infinite. Now, assuming there are disturbances in the output and in the measurement
as depicted in Figure 29, the transfer function, called complementary sensitivity, from η to e is given by
K������ = 3���P���1 + 3���P���
Figure 29. Diagram for the closed loop wafer stage with disturbances in the measurements and
the output.
In this case, if the measurement error has frequencies of 100 Hz and higher, non all these frequencies
aren’t suppressed because the complementary sensitivity has magnitude higher than 0dB for frequencies
between 100Hz and 600 Hz as showed in Figure 30, this is caused because the open loop bandwidth is
close to 400Hz.
100
101
102
103
104
-100
-50
0
50
100
Frequency (Hz)
Ma
gn
itu
de
(d
B)
100
101
102
103
104
-200
-100
0
100
200
Frequency (Hz)
Ph
ase
(D
eg
ree
s)
Data
Fitted model
Data
Fitted model
25
Figure 30. Complementary sensitivity of the system.
In order to suppress the error measurements with frequencies equal or higher than 100 Hz, we add a
notch filter at 100 Hz and we change the desired bandwidth to a value lower than 100 Hz in this case
25Hz. The new controller has the structure:
3���� = 0.05 + �25 × 120+ 1. N+ �25 × 1210.� + 2 × 0.01�25 × 1210 + 1
+ �25 × 1210.� + 2�25 × 1210 + 1O N+ �25 × 100.� + 2 × 0.01�25 × 100 + 1+ �25 × 100.� + 2 × 0.95�25 × 100 + 1O N �25 × 12 + 1�25 × 150 + 1O
The complementary sensitivity obtained with this controller is presented in Figure 31, it’s possible to
observe that measurement errors with frequencies equal or higher than 100Hz are suppressed. To check
this, we implemented the fitted system and the designed controller in simulink and we add a sinusoidal
measurement error with frequencies 100Hz, 200Hz and 400Hz, as reference we used an unitary step. In
Figure 32 we plot the measurement error η, its fast Fourier transform and the obtained output y, we can
see that the measurement error is suppressed and that the error is close to 0.
-150
-100
-50
0
50
Magnitu
de
(d
B)
101
102
103
104
105
180
360
540
720
Ph
ase (
de
g)
Bode Diagram
Frequency (Hz)
26
Figure 31. Complementary sensitivity obtained with controller C2.
Figure 32. Measurement error (Top), FFT of the measurement error (Center) and system output
with unitary step reference (Bottom).
-150
-100
-50
0
50
Magnitu
de (
dB
)
100
101
102
103
104
105
180
360
540
720
900
Phase (
deg)
Complementary sensitivity
Frequency (Hz)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.5
0
0.5
1
Time (s)
η
0 50 100 150 200 250 300 350 400 450 5000
0.2
0.4
0.6
0.8
Frequency (Hz)
FFT(η)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
0
0.5
1
1.5
Time (s)
y
27
4. FREQUENCY RESPONSE MEASUREMENTS
4.2. Frequency response function of a mass system
We need to create a simulink model with a PD-controlled mass of 1kg, with bandwidth of 10Hz, which can
track a prescribed setpoint as function of time. We injected a noise signal just behind the controller block
as presented in Figure 33.
Figure 33. PD-controller mass model with noise signal after the controller block.
We simulated the model over 120s with the fixed step Euler solver with step size 0.001s. Using an
reference equal to 0, we obtain the signals presented in Figure 34, where we can see that e=-y. From the
time domain plot, we can’t obtain more information about the signals. On the other hand, frequency
domain plots let us see that even when the disturbance is white noise, the control signal only has
frequency components between 0 Hz and 100 Hz and 400 Hz and 500 Hz. Meanwhile the error and
output signals only have low frequency components.
Figure 34. Signals in time domain (Left) and Frequency domain (Right).
y
To Workspace4
d
To Workspace3
u
To Workspace2
e
To Workspace1
xs
To Workspace Random
Number
1
s
Integrator1
1
s
Integrator
DCT pd
Dctpd
0
Constant
0 20 40 60 80 100 120-5
0
5x 10
-3
Time (s)
e
y
0 20 40 60 80 100 120-5
0
5
Time (s)
d
0 20 40 60 80 100 120-2
0
2
Time (s)
u
0 50 100 150 200 250 300 350 400 450 5000
0.01
0.02
Frequency (Hz)
U
0 50 100 150 200 250 300 350 400 450 5000
0.5
1x 10
-3
Frequency (Hz)
E
0 50 100 150 200 250 300 350 400 450 5000
0.01
0.02
Frequency (Hz)
D
28
We obtained the FRF of sensitivity of the system along with the coherence function, and we plotted them
in Figure 35. Before 10 Hz, magnitude of sensitivity has negative values which agree with expected
bandwidth because controller is designed with bandwidth 10 Hz. The coherence function is close to 1 as
expected because system is linear and signals are not correlated. After 400 Hz, magnitude dropped
because of sampling process.
Figure 35. FRF of the sensitivity of the system and Coherence function.
We computed the open loop FRF from the sensitivity FRF. Bandwidth is equal to 10 Hz. Phase margin is
close to 90°. Gain margin is approximately 1/24.03 dB and the modulus margin is 0.6dB. For high
frequencies (larger than 10 Hz), the phase of the FRF exhibits a drop caused by the sampling process.
Figure 36. Computed open loop FRF from the sensitivity.
10-2
10-1
100
101
102
103
-100
0
100
Frequency (Hz)
Magnitude S
10-2
10-1
100
101
102
103
-200
0
200
Frequency (Hz)
Phase S
10-2
10-1
100
101
102
103
0
0.5
1
Frequency (Hz)
Coherence S
10-2
10-1
100
101
102
103
-50
0
50
100
Frequency (Hz)
Magnitude CP
10-2
10-1
100
101
102
103
-200
-100
0
100
200
Frequency (Hz)
Phase CP
29
After measuring the FRF response of the PD controller, we used that FRF along with the FRF of the open
loop system to compute the FRF response of the double integrator (The plant), the results are plotted
together in the Figure 37. We can see that the plant FRF resembles a double integrator, because its
magnitude has a -2 slope and the phase is close to 180° until the phase starts to drop because the
sampling process.
Figure 37. FRF of the PD controller and the plant.
4.3. Closed loop FRF measurement
We used the controller and plant model given in the simulink model frf_ex3.mdl, we introduce a noise
signal between the controller and the plant, as showed in Figure 38. To measure the sensitivity of the
closed loop system, we use the signals d and u.
Figure 38. Simulink model used for the closed loop FRF measurement.
10-2
10-1
100
101
102
103
-200
-100
0
100
200
Frequency (Hz)
Magnitude PD controller
Magnitude P
10-2
10-1
100
101
102
103
-200
-100
0
100
200
Frequency (Hz)
Phase PD Controller
Phase P
e
To Workspace3
y
To Workspace2
d
To Workspace1
u
To Workspace
Random
Number
PlantController
0
Constant
30
In Figure 39, the closed loop sensitivity and its coherence are plotted, The coherence is not equal to 1 for
frequencies lower than 2 Hz because the magnitude of the sensitivity as such frequencies is lower than
0dB, then noise measurements are dominant. The measurement is reliable at frequencies between 2 Hz
and 400 Hz.
Figure 39. Closed loop sensitivity and its coherence.
In order to measure the process sensitivity of the closed loop, we need to use the signals d and e, in
order to obtain its FRF measurement and coherence. This measurements are plotted in Figure 40, where
we can see again that the coherence has problems when the process sensitivity is lower than 0 dB,
making the noise more dominant in this case. We can conclude that measurement is reliable for
frequencies between 2 Hz and 100 Hz.
Figure 40. Process sensitivity and its coherence.
10-1
100
101
102
103
-20
0
20
Frequency (Hz)
Magnitude (
dB
)
Magnitude S
10-1
100
101
102
103
-200
0
200
Frequency (Hz)
Phase (
Degre
es)
Phase S
10-1
100
101
102
103
0
0.5
1
Frequency (Hz)
Magnitude (
dB
)
Coherence S
10-1
100
101
102
103
-100
0
100
Magnitude PS
10-1
100
101
102
103
-200
0
200
Phase PS
10-1
100
101
102
103
0
0.5
1
Coherence PS
31
To determine the plant dynamics using the closed loop and process sensitivities, we divide the process
sensitivity by the sensitivity, obtaining the FRF measurement of the plant:
PI�Q�I�Q� = ��Q�1 + 3�Q���Q� 1 + 3�Q���Q�1 = ��Q�
The FRF results are plotted in Figure 41. Bad amplitude estimate could be caused by correlated signals
or nonlinearities. In this case, because we use r=0 and white noise, we are sure that the signals are
uncorrelated, but we aren't sure about the nonlinearities. The phase drop is mainly caused by the
sampling process and the way to diminish this problem is using higher sampling frequencies.
Figure 41. FRF measurement of the plant.
To study the effects of the different values of nfft and the simulation time, we repeat this process with time
simulations of 60s, 120s and 240s, while we use for nfft the values needed to obtain sample resolutions
of 0.0625Hz, 0.125Hz and 0.25Hz, this values are 16000, 8000 and 4000 respectively. In Figure 42,
Figure 43 and Figure 44 we plotted all these results. We can see that the sensitivity has higher coherence
when the nfft is higher, but the process sensitivity coherence shows that higher values of nfft make the
noise more dominant in the process sensitivity. We also see that higher simulation times result in FRF
measurements that aren’t dominated by the noise, which means that in order to have reliable FRF
measurements, we need to have a high nfft and a high simulation time.
10-1
100
101
102
103
-100
-50
0
50
100
Magnitude P
10-1
100
101
102
103
-200
-100
0
100
200
Phase P
32
Figure 42. Sensitivity and process sensitivity coherences, FRF of the plant with 60s simulation
time.
Figure 43. Sensitivity and process sensitivity coherences, FRF of the plant with 120s simulation
time.
10-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1Coherence S
10-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1Coherence PS
10-2
10-1
100
101
102
103
-150
-100
-50
0
50
100
150Magnitude P
10-2
10-1
100
101
102
103
-200
-150
-100
-50
0
50
100
150
200Phase P
Time = 60, nfft = 16000
Time = 60, nfft = 8000
Time = 60, nfft = 4000
Time = 60, nfft = 16000
Time = 60, nfft = 8000
Time = 60, nfft = 4000
Time = 60, nfft = 16000
Time = 60, nfft = 8000
Time = 60, nfft = 4000
Time = 60, nfft = 16000
Time = 60, nfft = 8000
Time = 60, nfft = 4000
10-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1Coherence S
10-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1Coherence PS
10-2
10-1
100
101
102
103
-100
-50
0
50
100
150Magnitude P
10-2
10-1
100
101
102
103
-200
-150
-100
-50
0
50
100
150
200Phase P
Time = 120, nfft = 16000
Time = 120, nfft = 8000
Time = 120, nfft = 4000
Time = 120, nfft = 16000
Time = 120, nfft = 8000
Time = 120, nfft = 4000
Time = 120, nfft = 16000
Time = 120, nfft = 8000
Time = 120, nfft = 4000
Time = 120, nfft = 16000
Time = 120, nfft = 8000
Time = 120, nfft = 4000
33
Figure 44. Sensitivity and process sensitivity coherences, FRF of the plant with 240s simulation
time.
5. FEEDFORWARD CONTROL
5.1. Mass Feedforward design
We want to control a system with transfer function
In this case we use a stabilizing controller to achieve an stable closed loop with bandwidth of 10 Hz, the
controller has the structure:
3��� = N+ �25 × 8.78.� + 2 × 0.1�25 × 8.78 + 1+ �25 × 8.78.� + 2 × 0.19�25 × 8.78 + 1O N �25 × 3.9 + 1�25 × 38 + 1 O N �25 × 4.2 + 1�25 × 52 + 1 O R0.018� + 1S
In the Figure 45 the acceleration pattern used for the 3rd
order reference and the error obtained with this
controller are plotted, the maximum error is 2.1x10-3
, we also observed that the error is 0 when the
acceleration is 0 (When we have constant velocity). The acceleration feedforward parameter corresponds
to the mass of the system if we thought of this as an analogue of the mass-damper-spring system, then
the feedforward gain is Ka=1/1000. In Figure 45 we plotted the new error which is close to 0 for all the
10-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1Coherence S
10-2
10-1
100
101
102
103
0
0.2
0.4
0.6
0.8
1Coherence PS
10-2
10-1
100
101
102
103
-100
-50
0
50
100
150Magnitude P
10-2
10-1
100
101
102
103
-200
-150
-100
-50
0
50
100
150
200Phase P
Time = 240, nfft = 16000
Time = 240, nfft = 8000
Time = 240, nfft = 4000
Time = 240, nfft = 16000
Time = 240, nfft = 8000
Time = 240, nfft = 4000
Time = 240, nfft = 16000
Time = 240, nfft = 8000
Time = 240, nfft = 4000
Time = 240, nfft = 16000
Time = 240, nfft = 8000
Time = 240, nfft = 4000
34
interval when the acceleration is constant and has a maximum value of 2x10-4
after the acceleration
changes.
Figure 45. Acceleration for the 3rd
order reference (Top), error without feedforward (Center) and
error with acceleration feedforward (Bottom).
5.4. Feedforward design
We use the model given in the simulink model feedforward.mdl, presented in Figure 46. We need a
relatively low bandwidth controller for feedforward design because we need a controller that allows us to
see the phenomena of mass acceleration, viscous damping and dry friction, and a high bandwidth
controller will have such fast response that wouldn't allow us to see those phenomena.
Figure 46. Feedforward model.
0 5 10 15 20 25 30 35 40-0.5
0
0.5
Time (s)
Acceleration
0 5 10 15 20 25 30 35 40-4
-2
0
2
4x 10
-3
Time (s)
Error with Ka=0
0 5 10 15 20 25 30 35 40-4
-2
0
2
4x 10
-4
Time (s)
Error with Ka=1/1000
Lead / lag
t
To Workspace3
pos
To Workspace2
acc
To Workspace1 e
To Workspace
acc
vel
pos
Ref3
Sign
Plant
In1 Out1
Kfv
Gain3
Kfc
Gain2
Kfa
Gain
Clock
35
In Figure 47 we plotted the acceleration pattern used and the error without feedforward controller, in this
case the maximum error is 1.4x10-2
. After tuning the Kfc value to 1.6x10-4
the maximum error diminished
to 8.1x10-4
, then with the Kfv value tuned to 5x10-6
we obtain an error close to 0 when the acceleration is 0
and a maximum error of 4x10-4
. After tuning the Kfa value to 9.6x10-6
, we obtain a maximum error of
8.4x10-6
.
Figure 47. Acceleration, error without feed forward and error with feedforward.
6. The Digital Environment
6.1. Delay and Sampling
It is known that sampling process makes the continuous time systems discrete. Also, sampling introduces
the delay to the system as a typical non linear phenomenon. The delay frequency response Hd (jw) is
given below with a total delay time Td.
�T�UV� = ;�<WXY
�T�UV� = ;�<WXY = cos�V\T� − Usin�wTa�
|�T�UV�| = bc;� + d*� = b�ef� V\T�� + �sin V\T�� = 1
∠�T �UV� = ghi�� 4d*c;6 = ghi�� 4− sin V\Tcos V\T 6 = ghi���− tan V\T� = −V\T
As expressed above equation, -Td is seen in the phase equation. It causes decrease of phase due to
delay. Also, sampling and calculation cause delay which is independent from frequency of the input.
Resulting phase delay calculation is shown below:
0 2 4 6 8 10 12 14 16 18 20-0.5
0
0.5
Time (s)
Acceleration
0 2 4 6 8 10 12 14 16 18 20-0.02
0
0.02
Time (s)
Error without feedforward
0 2 4 6 8 10 12 14 16 18 20-1
0
1x 10
-3
Time (s)
Kfc tunned
Kfc K
fv tunned
Kfc K
fv K
fa tunned
0 2 4 6 8 10 12 14 16 18 20-1
0
1x 10
-5
Time (s)
Kfc K
fv K
fa tunned
36
\T = \2 + \l = �1 ∗ 10�#� + �100 ∗ 10��� = 1.1 ∗ 10�# = 1.1 *�
At 10 Hz , the resulting phase delay given the total delay time is:
∠�10Hz� = −10 ∗ 25 ∗ 1.1 ∗ 180°25 = −3.96°
In Figure 48, we plotted the phase delay obtained with the given. total delay time.
Figure 48. Represents the phase drop due to delay
6.2. Digital Control Systems
Firstly a lead controller with a zero at 10 Hz and a pole at 100 Hz is designed as shown below,
3 = �2π × 10 + 1�2π × 100 + 1 = 0.01592 � + 10.001592 � + 1
In Figure 49 we presented the frequency response of the lead controller and different discrete time
equivalents
10-1
100
101
102
-40
-35
-30
-25
-20
-15
-10
-5
0
Frequency (Hz)
Pha
se
de
lay (D
eg
ree
s)
37
Figure 49. Magnitude Frequency response of a lead filter and its discrete time equivalents.
Firstly, discrete functions are valid up to half of the sampling frequency. As seen in the Figure 49, zero
order method gives the worst approximation. On the other hand, other methods give approximately same
solution. In addition, before 3 Hz, all methods are accurate. Between 3 Hz and 300 Hz, it can be said that
except zero order method, all methods give accurate approximation to continuous time system but Tustin
and Prewarping approximations give a little bit better solution compared to First Order and Matched. After
300 Hz, none of them gives close solution to continuous time system. Finally, it can be said that all
approximations provide best solutions for low frequencies.
Additional to lead controller, a notch added to the system with a depth of 20 dB and a damping of the
zeros of 0.01 at 200 Hz.
3 = � �200 2π�� + +2.0.01 �200 2π . + 1� �200 2π�� + + 2.0.1 �200 2π. + 1 = 1.008. 10�& �# + 8.866; − 007 �� + 0.01593 � + 11.008. 10�' �# + 8.866; − 007 �� + 0.001751 � + 1
In … the frequency response of the continuous time controller and its discrete time equivalents is plotted,
as mentioned above, discrete functions are only valid up to half of the sampling frequency. After that
point, due to aliasing, data is lost.
10-1
100
101
102
103
104
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Frequency (Hz)
Ma
gn
itu
de
(d
B)
C
zoh
foh
Tustin
Prewarping
Matched
10-1
100
101
102
103
104
-80
-60
-40
-20
0
20
40
60
80
Frequency (Hz)
Ph
ase
(D
eg
ree
s)
C
zoh
foh
Tustin
Prewarping
Matched
38
Figure 50. Magnitude Frequency response of a lead + notch filter and its discrete time equivalents.
When magnitude and phase diagrams are examined, it is seen that Zero-order is the worst one. If the
controller was designed to stabilize a plant with a real resonance at 200 Hz, the best controller option is
the matched controller, because it's the only one that show the same anti-resonance at 200 Hz and a
similar phase with the 200 Hz sampling frequency. If the frequency to 1KHz is increased, the First-Order
hold controller could be used too, at 2KHz we can use all the controller but the zero order hold controller,
because its gain and phase doesn't match the designed controller.
Discretizing a continuous time controller can sometimes yield closed loop instability because several
reasons:
• The discretization process always causes a phase lag, which can change the response the
controller at the desired frequency.
• The discretization method also create anti resonances that the original controller doesn't exhibit,
depending on the sampling frequency. 3, some discretization method need high sampling
frequency in order to match all the resonances and anti resonances of the designed controller.
• ‘Aliasing’ is the another problem for closed loop instability because after some point data is lost.
10-1
100
101
102
103
104
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency (Hz)
Mag
nitu
de (
dB
)
C
zoh
foh
Tustin
Prewarping
Matched
10-1
100
101
102
103
104
-80
-60
-40
-20
0
20
40
60
80
Frequency (Hz)
Ph
ase
(D
eg
ree
s)
C
zoh
foh
Tustin
Prewarping
Matched
39
MOTION CONTROL EXPERIMENTS
Experiment 2. Frequency Response Measurements
For parts a, b and c use the closed loop Simulink scheme. Connect the setup in the motor-feedback case.
a. Determine the plant FRF using a sampling frequency of 2000 Hz. What do you observe? Try to
relate all (anti-)resonances and phase phenomena to the setup.
Measurements are made in closed-loop, so the relevant signals are the control effort, u, and the
disturbance, d. In this case the simulink scheme is presented in the Figure 51. We used 240 seconds of
measurements with the given sample frequency.
Figure 51. Simulink scheme of the closed loop FRF measurement.
Using the equations presented on the lecture, we first calculate the Sensitivity function of the closed loop
using the command:
nfft = 2^14;
[S hz]=tfestimate(d,u,hann(nfft),nfft/2,nfft,fs);
In order to calculate the FRF of the system, we need to calculate first the FRF of the used controller which
has the next transfer function:
3��� = 0.005� + 0.2+ �25200.� + + �25200. + 1
This controller is a PD controller with a second order low-pass filter with 200 Hz frequency. The equation
that allows us to find the FRF of the system is:
��Q� = 3���Q�RI���Q� − 1S So we can obtain the FRF of the system using the next commands:
CFR=squeeze(freqresp(C,hz));
H=CFR.^-1.*((S).^(-1)-1);
The response of the system is shown in the Figure 52, we can observe that after 1 Hz the phase of the
plant is -180° and the magnitude has a -2 slope, which is caused by the 2 integrators that theoretically the
3
Out4
2
Out3
1
Out2
Input Output
simple01
Random
Number
DAC0 (motor) [V]
ENC0 [rad]
ENC1 [rad]
Pato setup1
0
Constant
40
system has. We also observe the existence of 2 stable zeros close to 40 Hz and 2 stable poles close to
50 Hz, and this behavior is the expected from the theory. The phase lag between 0.1 and 1 Hz could be
explained by the friction present in the motor and the setup, this friction causes the motor to be stopped
when the applied force is not high enough. The system seems to have another dynamic close to 1 KHz
which is the maximum frequency we can obtain with the sample frequency of 2 KHz, this is a good reason
to change the sample frequency to a higher one.
Figure 52. FRF of the plant.
b. Change the sampling frequency to 4000 Hz and 8000 Hz (or 1000 HZ) respectively and again
determine the plant FRF. Compare the measurements. What do you observe?
Using the same commands used in part a. with the sampling frequencies 4000 Hz and 1000 Hz, we
obtain two new FRF for the plant. These FRF are presented in Figure 53, the position of the zeros and
poles of the plant are the same for the 3 measured FRF, and the phase diagrams are similar too. We can
observe that the system exhibits two stable poles near to 1000 Hz, this poles may be part of the motor
system because the motor is the actuator of the system and we are measuring all the setup together.
Also, it’s clear that the poles at 1000 Hz can only be measured when the sampling frequency is higher
than 2000 Hz.
The most important difference between the measured FRFs of the plant is that the phase diagrams look
more like expected when the sampling frequency increases, for example the 180 degrees shift caused by
the 2 poles of the plant is more evident with the 4000 Hz sampling frequency. Also, the lag between the
phase diagrams can be explained by the sampling process, because this process introduces a lag in the
system of the half of the sample time, then when the sampling frequency increases the sampling time
diminishes as the lag in the system.
10-1
100
101
102
103
-150
-100
-50
0
50
100
Frequency (Hz)
Mag
nitude (
dB
)
10-1
100
101
102
103
-200
-100
0
100
200
Frequency (Hz)
Phase
Magnitude of the plant
Phase of the plant
41
Figure 53. FRF of the plant measured with 3 different sample frequencies.
c. Design 2 or 3 different PD-controllers with significantly different bandwidths and measure the
sensitivity and plant FRF of the system for each controller. Interpret results, and determine the
influence of the bandwidth on your measurements.
The bandwidth of the system is approximately 8 Hz and, looking at the FRF of the system, we can
conclude that a PD controller will change the bandwidth of the system increasing the bandwidth when the
Proportional part increases and the Derivative part remains constant and decreasing the bandwidth as
the Proportional part decreases. We designed 3 different PD controllers and using the FRF of the system
we calculated the new Bandwidth as presented in the Table 3.
Table 3. Designed PD controllers and approximated bandwidth
CONTROLLER P D Bandwidth (Hz)
1 2 0.005 54
2 0.05 0.005 9
3 4 0.005 75
The measured FRF of the system with the different PD controllers is presented in Figure 54, we can
observe how the controllers with higher bandwidth are more sensitive to lower frequencies, causing the
measured FRF to be affected on those lower frequencies and the measured FRFs are closer for
frequencies higher than 12 Hz. This could be caused by the fact that the controllers with higher bandwidth
have, in this case, a higher stiffness and the disturbance has to be higher in order to move the mass. As
conclusion, we could claim that in order to obtain good closed loop FRF measurement in our setup we
need to use controllers with low bandwidth.
10-2
10-1
100
101
102
103
104
-200
-150
-100
-50
0
50
100
Frequency (Hz)
Mag
nitude (
dB
)
10-2
10-1
100
101
102
103
104
-200
-100
0
100
200
Frequency (Hz)
Phase
4 KHz
2 KHz
1 KHz
4 KHz
2 KHz
1 KHz
42
Figure 54. FRF measured with different PD controllers.
d. Measure again the plant FRF, but now in open-loop using QadScope. Compare the result with
the closed-loop measurement.
For the open-loop measurement, we used 4000 Hz as sampling frequency and a white noise signal as
input for the system, the FRF is presented in the Figure 55. We can observe that the measurements are
similar, but looks like the open-loop FRF has a higher gain, causing the bandwidth of the system to be
close to 200 Hz, when the closed-loop bandwidth was at 8 Hz. In the other hand, the open-loop phase
diagram has a 180 degrees shift compared with the closed-loop measurement, this can be explained by a
-1 factor in the transfer function, caused by a wrong wire connection of the output or the input of the
system, meanwhile this wrong connection doesn’t affect the closed-loop measurement because the
closed-loop measurement takes account for every gain in the system. As another conclusion, we can be
more confident of FRF measurements in closed-loop.
10-1
100
101
102
103
104
-200
-150
-100
-50
0
50
100
Frequency (Hz)
Mag
nitude (
dB
)
10-1
100
101
102
103
104
-200
-100
0
100
200
Frequency (Hz)
Phase
C1
C2
C3
C1
C2
C3
43
Figure 55. Open-loop FRF using QadScope and Closed-loop FRF.
Experiment 3. Feedforward tuning
For this experiment we use the motor feedback case with the pato01.mdl scheme, we use the outputs
presented in the Figure 56, at the beginning we set the feedforward gains at 0 and we tuned the controller
to obtain a tracking error suitable for the feedforward tuning process. The setpoint used is presented in
Figure 57.
Figure 56. Motor feedback scheme for feedforward tuning.
100
101
102
103
-200
-100
0
100
200
freq [Hz]
scope 1
: [c
oun
ts/V
in d
B]
100
101
102
103
-200
-100
0
100
200
freq [Hz]
sco
pe 2
: [d
eg
]
Open-Loop with QadScope
Closed-Loop with Simulik
6
Out3
5
Out2
4
Out1
3
e
2
u
1
w
start
acc
vel
pos
Ref3
Sign
On/Of f
Ref power
-K-
Ref acc scal ing
DAC0 (motor) [V]
ENC0 [rad]
ENC1 [rad]
Pato setup
Noise
Kfv
Kfv
Kfc
Kfc
Kfa
Kfa
Input Output
Controller
ClockActual time in s
44
Figure 57. Setpoint trajectory used for the feedforward tuning.
The PD controller used has proportional gain Kp=1 and derivative gain Kv=0.1. In Figure 58 we plotted the
tracking error along with the acceleration with the feedforward gains set to 0. In this case we can see that
the error is different from 0 even when the velocity is constant (Acceleration equal to 0).
Figure 58. Tracking error and acceleration without using feedforward.
We used the real time QadScope in order to check the error while tuning the feedforward gains. After the
tuning process, the feedforward gains were set to the values:
Kfc=0.0035, Kfv=0.0005, Ka=4.7
0 2 4 6 8 10 12 14 16 18 20-50
0
50
100
150
Time (s)
po
sitio
n (
rad
)
Set point
0 2 4 6 8 10 12 14 16 18 20-100
-50
0
50
100
Time (s)
Ve
locity (
rad
/s)
Velocity
0 2 4 6 8 10 12 14 16 18 20-400
-200
0
200
400
Time (s)
Acce
lera
tio
n (
rad
/s2)
Acceleration
0 2 4 6 8 10 12-0.4
-0.2
0
0.2
0.4
0.6
Time (s)
Err
or
(ra
d)
Error
0 2 4 6 8 10 12-300
-200
-100
0
100
200
300
Time (s)
Acce
lera
tio
n (
rad
/s2)
Acceleration
45
The tracking error obtained with the feedforward controller is presented in Figure 59 and compared with
the error without feedforward and the acceleration pattern, the maximum error was diminished until 10%
the original tracking error and is close to 0 when the acceleration is 0. We can say that the oscillations on
the error are caused by the position encoding process.
Figure 59. Tracking errors obtained with and without feedforward.
Experiment 4. Loop shaping game
We are using the load feedback setup for this experiment, based in the results from exercise 1.4. Control
of a motion system, we can expect that the new system has the same frequency response that the motor
feedback case without the 2 zeros close to 30 Hz, in this case we expect only one zero close to 2000 Hz,
as plotted in Figure 60. This figure was obtained using the FRF measurements of the motor feedback
case in Experiment 2. Frequency Response Measurements, but eliminating the 2 zeros and adding the
expected zero.
Based in this expected result, we designed a stabilizing controller for the plant composed by a lead filter,
a notch and a second order lowpass filter, the two last ones block were added by the experience working
with the motor setup. The resulting controller has the structure:
3���� = N �25 × 1 + 1�25 × 9 + 1O N+ �25 × 800.� + 2�25 × 800 + 1+ �25 × 800.� + 2�25 × 800 + 1O N 1
+ �25 × 200.� + 2 × 0.2�25 × 200 + 1O
254 256 258 260 262 264-0.05
0
0.05
Time (s)
Err
or (r
ad
)
Error with Feedforward
0 2 4 6 8 10 12-0.4
-0.2
0
0.2
0.4
0.6
Time (s)
Err
or
(ra
d)
Error without feedforward
254 256 258 260 262 264-400
-200
0
200
400
Time (s)
Accele
ratio
n (
rad
/s2)
Acceleration
46
Figure 60. Expected motor and load feedback responses.
We used DCtools in shapeit to obtain the discrete time equivalent of our controller and implemented this
in matlab to perform a FRF measurement of the system, the simulink scheme is presented in the Figure
61, in this case we introduce a noise signal between the controller and the plant in order to measure the
sensitivity of the closed loop system.
Figure 61. Simulink scheme for the FRF measurements.
After measuring the disturbance and control signal, we calculated the sensitivity of the closed loop and its
coherence and plotted them in Figure 62, we observe that the coherence is lower than 1 for the
frequencies lower than 2 Hz, this could be caused by the low sensitivity at those frequencies, making the
noise the dominant signal. Our first conclusion is that our measurements are reliable for the frequencies
between 2 and 2000 Hz. Using the measured response of the controller, presented in Figure 63, and the
measured sensitivity, we obtain the plant FRF using the expression:
��Q� = 3���Q�RI���Q� − 1S
-400
-300
-200
-100
0
100
Magnitu
de (
dB
)
100
101
102
103
104
105
-180
0
180
360
Phase
(d
eg)
Bode Diagram
Frequency (Hz)
Motor feedback
Load feedback
Motor feedback
Load feedback
4
Out4
3
Out3
2
Out2
1
Out1Random
Number
DAC0 (motor) [V]
ENC0 [rad]
ENC1 [rad]
Pato setup1
Input Output
Controller
0
Constant
47
Figure 62. measured closed loop sensitivity and coherence.
Figure 63. Controller frequency response.
The resulting plant FRF measurement in plotted in Figure 64, where we can see that the frequency start
approximately at -180° (We should remember that the measurements are reliable after 2 Hz) and we
have the phase 180° drop caused by the 2 poles close to 52.5Hz. Also we can see how the noise becomes dominant for high frequencies where the magnitude of the plant is lower than 0 dB and that the phase has a constant drop caused by the sampling process.
10-1
100
101
102
103
104
-20
-15
-10
-5
0
5
Frequency (Hz)
Ma
gn
itu
de
(d
B)
Magnitude S
10-1
100
101
102
103
104
-50
0
50
100
Frequency (Hz)
Ph
ase
(D
eg
ree
s)
Phase S
10-1
100
101
102
103
104
0.4
0.6
0.8
1
1.2
1.4
Frequency (Hz)
Co
he
ren
ce
Coherence
-200
-150
-100
-50
0
Magnitu
de (
dB
)
100
101
102
103
0
90
180
270
360
450
Phase (
deg)
Bode Diagram
Frequency (Hz)
48
Figure 64. Plant FRF measurement.
Using this measurement, we designed a controller with a higher bandwidth keeping in mind the constraint
|S|<6dB. In this case we use a PD controller with a notch filter and a lead filter to achieve stability and a
higher bandwidth, we also include the notch filter at 800Hz and the low pass filter at 200Hz trying to avoid
the high frequency resonances of the plant. The final controller has the structure:
3���� = n �25 × 20 + 1o N �25 × 4 + 1�25 × 36 + 1O ×
N+ �25 × 52.� + 2 × 0.1�25 × 52 + 1+ �25 × 52.� + 2 × 0.5�25 × 52 + 1O N+ �25 × 800.� + 2�25 × 800 + 1
+ �25 × 800.� + 2�25 × 800 + 1O N 1+ �25 × 200.� + 2 × 0.2�25 × 200 + 1O
We achieve a Bandwidth of approximately 17.3 Hz, as showed in the Figure 65, and the modulus margin
is approximately 5.873dB which is lower than the given constraint. Both open loop frequency response
and sensitivity for the plant and the controller are plotted in the Figure 65.
10-1
100
101
102
103
104
-300
-250
-200
-150
-100
-50
0
50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
Magnitude plant
10-1
100
101
102
103
104
-200
-150
-100
-50
0
50
100
150
200
Frequency (Hz)
Ph
ase
(D
eg
ree
s)
Phase plant
49
Figure 65. Frequency response for C2(s)H(s) (left) and Sensitivity (Right).
We implemented the controller in an experiment and we measured all the variables like in the previous
parts, in this case our plant is still stable. The plots for the achieved open loop and sensitivity are
presented in Figure 66. The obtained bandwidth is approximately 7.6Hz and the modulus margin is
4.833dB, both lower than the designed values. This could be explained by the discretization of the
controller, because we use shapeit to obtain the discrete time equivalent of our designed controller, but
we don’t have completely control on the discretization method.
Figure 66. Measured open loop and sensitivity for the plant.
In order to check the performance of the designed controller, we plotted the time response of the output,
the disturbance between the controller and the plant and the error of the system in Figure 67. We can
observe that the maximum error without disturbances is 0.19rad meanwhile with noise disturbance the
maximum error increases to 0.29rad. This allows us to conclude that our controller has a good
performance even in presence of noise.
100
101
102
103
-200
-100
0M
agnit
ude [
dB
]
open loop
X: 17.33
Y: 0.04125
100
101
102
103
-180
-90
0
90
180
Frequency [Hz]
Phase [
º]
100
101
102
103
-40
-20
0
20 X: 25.63
Y: 5.873
Magnitude [
dB
]
sensitivity
100
101
102
103
-180
-90
0
90
180
Frequency [Hz]
Phase [
º]
10-1
100
101
102
103
104
-200
-100
0
100
Frequency (Hz)
Ma
gn
itu
de
(d
B)
X: 7.568
Y: -0.3088
Measured Open loop
10-1
100
101
102
103
104
-200
-100
0
100
200
Frequency (Hz)
Ph
ase
(D
eg
ree
s)
Measured Open loop
10-1
100
101
102
103
104
-40
-20
0
20
X: 52
Y: 4.833
Frequency (Hz)
Ma
gn
itu
de
(d
B)
Measured Sensitivity
10-1
100
101
102
103
104
-50
0
50
100
150
Frequency (Hz)
Ph
ase
(D
eg
ree
s)
Measured Sensitivity
50
Figure 67. Time response of the load feedback setup.
0 10 20 30 40 50 60 70 80 90 100-100
0
100
200
300
Time (s)
xs
y
0 10 20 30 40 50 60 70 80 90 100-2
-1
0
1
2
Time (s)
d
0 10 20 30 40 50 60 70 80 90 100-0.4
-0.2
0
0.2
0.4
Time (s)
e
51
REFERENCES
[1] Witvoet, G. 4K410 Motion Control Lectures, Mechanical Engineering Deparment, Technische
Universiteit Eindhoven, 2010.
[2] Matlab R2010a Reference. The Mathworks.
[3] Ogata, K. Modern Control Engineering, Fourth edition. Pearson Education International, 2002.
