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Contents
Abstract ... xi
List of Abbreviations ...xii
Certification ... iii
Acknowledgement ... iv
Introduction ... 1
1.1 Motivation ... 1
1.2 Research Methodology ... 2
1.3 Vibration Control of Structure ... 3
1.4 Active Vibration Control of Structure ... 4
1.4.1 openloop and closedloop control ... 4
1.4.2 Feedforward and Feedback Control ... 6
1.4.3 Wave Control and Modal Control ... 7
1.4.4 SISO Control and MIMO Control ... 9
1.4.5 Collocated Control and Uncollocated Control ... 9
1.5 Modal Based Controller for Multimode Vibration Control ... 10
1.5.1 Independent Modal Space Control (IMSC) ... 10
1.5.2 Resonant Control ... 11
1.5.3 Positive Position Feedback (PPF) Control ... 11
1.6 Plate or Shell Structure Vibration Control ... 16
1.7 Aim of the Thesis ... 17
1.8 Outline of the Thesis ... 18
Model of Flexible Plate Structure ... 20
2.1 Introduction ... 20
2.2 Description of Experimental Plant ... 22
2.3 Numerical Solution Using ANSYS ... 25
2.3.1 Resonance Frequency ... 26
2.3.2 Mode Shape ... 28
2.3.3 Harmonic Response Analysis ... 30
2.4 Simulation Model of The plate ... 35
2.5 Summary ... 44
ii
Spatial Norm and Model Reduction ... 45
3.1 Introduction ... 45
3.2 Plate Model Reduction and Balanced Realization ... 45
3.2.1 SISO Plate Model Reduction and Balanced Realization ... 45
3.2.2 MIMO Plate Model Balanced truncation ... 50
3.3 Summary ... 52
Model Correction ... 53
4.1 Introduction ... 53
4.2 Plate Correction Model ... 53
4.2.1 SISO Plate Model Correction ... 54
4.2.2 MIMO Plate Model Correction ... 56
4.3 Summary ... 62
Multimode SISO and MIMO PPF Controller ... 63
5.1 Introduction ... 63
5.2 PPF Controller Structure ... 64
5.3 PPF Controller Closed loop Stability ... 66
5.3.1 Scalar Case ... 66
5.3.2 Multivariate Case ... 68
5.3.3 Multivariate PPF Controller implemented with feedthrough plant ... 69
5.4 Multimode SISO and MIMO PPF Controller Parameter Selection ... 72
5.4.1 MATLAB Optimization toolbox and GA Optimization Search ... 72
5.4.2 Multimode SISO PPF Controller Optimal Parameter Selection ... 73
5.4.3 Multimode MIMO PPF Controller Optimal Parameter Selection ... 76
5.5 Summary ... 77
Simulation ... 78
6.1 Multimode Three SISO PPF Controller Simulation ... 78
6.2 Multimode MIMO PPF Controller Simulation ... 89
6.3 Summary ... 99
Experiment ... 101
7.1 Selfsensing ... 101
7.2 Electronics ... 102
iii
7.2.1 dSpace ... 102
7.2.2 Interfacing circuits ... 103
7.2.3 Additional electronics ... 104
7.3 Multimode SISO PPF Controller Experiment Implemented Result ... 105
7.4 Multimode MIMO PPF Controller Experimental Implemented Result ... 122
7.5 Summary ... 139
Conclusion and Future Work ... 142
8.1 Outcomes of the Research ... 142
8.2 Future Work ... 143
Bibliography ... 146
iv
List of Figures
Figure 2.1 A thin plate in transverse vibration ... 22
Figure 2.2 Transducer cross section (a) and model (b) ... 23
Figure 2.3 System model shown in isometric view and side view ... 26
Figure 2.4 Modal Analysis mode shape of first mode ... 28
Figure 2.5 Modal Analysis mode shape of second mode ... 29
Figure 2.6 Modal Analysis mode shape of third mode ... 29
Figure 2.7 Modal Analysis mode shape of forth mode ... 29
Figure 2.8 Harmonic Analysis Example ... 30
Figure 2.9 Amplitude Response of whole top plate to a harmonic disturbance at shaker
... 31
Figure 2.10 Phase Response of whole top plate to a harmonic disturbance at shaker . 31
(a) Transducer 1 resonant peaks ... 32
(b) Transducer 1 resonance phase ... 32
(c) Transducer 2 resonance peaks ... 33
(d) Transducer 2 resonance phase ... 33
(e) Transducer 3 resonance peaks ... 34
(f) Transducer 3 resonance phase ... 34
Figure 2.11 Simulated harmonic disturbance at shaker and measured resonant peaks
and phase of top plate at transducer 1 (a, b), transducer 2 (c, d) and transducer 3
(e, f). ... 34
Figure 2.12 Block diagram representation of the PPF control problem ... 35
Figure 2.13 three SISO (g11,g22,g33) 47 modes plate model ... 37
Figure 2.14 SISO (g11) 47 modes plate model at transducer 1 ... 38
Figure 2.15 SISO (g22) 47 modes plate model at transducer 2 ... 38
Figure 2.16 SISO (g33) 47 modes plate model at transducer 3 ... 39
Figure 2.17 MIMO 47 modes plate model ... 39
Figure 2.18 MIMO (Gp(1,1)) 47 modes plate model ... 40
v
Figure 2.19 MIMO (Gp(1,2)) 47 modes plate model ... 40
Figure 2.20 MIMO (Gp(1,3)) 47 modes plate model ... 41
Figure 2.21 MIMO (Gp(2,1)) 47 modes plate model ... 41
Figure 2.22 MIMO (Gp(2,2)) 47 modes plate model ... 42
Figure 2.23 MIMO (Gp(2,3)) 47 modes plate model ... 42
Figure 2.24 MIMO (Gp(3,1)) 47 modes plate model ... 43
Figure 2.25 MIMO (Gp(3,2)) 47 modes plate model ... 43
Figure 2.26 MIMO (Gp(3,3)) 47 modes plate model ... 44
Figure 3.1 SISO Plate (g11) at transducer 1 Hankel singular values ... 46
Figure 3.2 SISO Plate (g11) at transducer 1 model reduction and balanced
realization frequency domain compare ... 46
Figure 3.3 SISO Plate (g11) at transducer 1 model reduction and balanced
realization singular values compare ... 47
Figure 3.4 SISO Plate (g22) at transducer 2 Hankel singular values ... 47
Figure 3.5 SISO Plate (g22) at transducer 2 model reduction and balanced
realization frequency domain compare ... 48
Figure 3.6 SISO Plate (g22) at transducer 2 model reduction and balanced
realization singular values compare ... 48
Figure 3.7 SISO Plate (g33) at transducer 3 Hankel singular values ... 49
Figure 3.8 SISO Plate (g33) at transducer 3 model reduction and balanced
realization frequency domain compare ... 49
Figure 3.9 SISO Plate (g33) at transducer 3 model reduction and balanced
Realization singular values compare ... 50
Figure 3.10 MIMO Plate Hankel singular values ... 50
Figure 3.11 MIMO Plate model reduction and balanced realization frequency
domain compare ... 51
Figure 3.12 MIMO Plate model reduction and balanced realization singular values
compare ... 51
Figure 4.1 SISO plate model correction result ... 54
Figure 4.2 SISO plate (g11) model reduction result compare (balreal and correction).. 55
Figure 4.3 SISO plate (g22) model reduction result compare (balreal and correction)
vi
... 55
Figure 4.4 SISO plate (g33) model reduction result compare (balreal and correction)
... 56
Figure 4.5 MIMO plate model reduction result compare (balreal and correction) ... 56
Figure 4.6 MIMO Plate model correction and balanced truncation singular values
compare ... 57
Figure 4.7 MIMO (Gpc(1,1)) plate model correction result ... 57
Figure 4.8 MIMO (Gpc(1,2)) plate model correction result ... 58
Figure 4.9 MIMO (Gpc(1,3)) plate model correction result ... 58
Figure 4.10 MIMO (Gpc(2,1)) plate model correction result ... 59
Figure 4.11 MIMO (Gpc(2,2)) plate model correction result ... 59
Figure 4.12 MIMO (Gpc(2,3)) plate model correction result ... 60
Figure 4.13 MIMO (Gpc(3,1)) plate model correction result ... 60
Figure 4.14 MIMO (Gpc(3,2)) plate model correction result ... 61
Figure 4.15 MIMO (Gpc(3,3)) plate model correction result ... 61
Figure 5.1 Block Diagram of a SecondOrder System with Positive Position Feedback
... 64
Figure 5.2 Bode Plot of a Typical PPF Filter Frequency Response Function ... 65
Figure 5.3 Feedback control system associated with a flexible structure with ... 70
Figure 5.4 multimode SISO PPF controller K11 optimal parameter result through GA
search ... 74
Figure 5.5 multimode SISO PPF controller K22 optimal parameter result through GA
search ... 74
Figure 5.6 multimode SISO PPF controller K33 optimal parameter result through GA
search ... 75
Figure 5.7 multimode MIMO Controller optimal parameter result through GA
search ... 76
Figure 6.1 SISO vibration control at transducer 1 (K11) openloop and closedloop
impulse signal simulation result ... 79
Figure 6.2 SISO vibration control at transducer 2 (K22) openloop and closedloop
impulse signal simulation result ... 79
vii
Figure 6.3 SISO vibration control at transducer 3 (K33)openloop and closedloop
impulse signal simulation result ... 80
Figure 6.4 SISO vibration control at transducer 1 (K11) openloop and closedloop
step signal simulation result ... 80
Figure 6.5 SISO vibration control at transducer 2 (K22) openloop and closedloop
step signal simulation result ... 81
Figure 6.6 SISO vibration control at transducer 3 (K33) openloop and closedloop
step signal simulation result ... 81
Figure 6.7 three SISO controller openloop and closedloop simulation result ... 82
Figure 6.8 SISO vibration control at transducer 1 (K11) openloop and closedloop
simulation result (1) ... 82
Figure 6.9 SISO vibration control at transducer 1(K11) openloop and closedloop
simulation result (2) ... 83
Figure 6.10 SISO vibration control at transducer 2 (K22) openloop and closedloop
simulation result (1) ... 83
Figure 6.11 SISO vibration control at transducer 2 (K22) openloop and closedloop
simulation result (2) ... 84
Figure 6.12 SISO vibration control at transducer 3 (K33) openloop and closedloop
simulation result (1) ... 84
Figure 6.13 SISO vibration control at transducer 3 (K33) openloop and closedloop
simulation result (2) ... 85
Figure 6.14 MIMO Controller openloop and closedloop (Gwo(1,1),Gwc(1,1)) impulse
signal simulation result at transducer 1 ... 90
Figure 6.15 MIMO Controller openloop and closedloop (Gwo(2,1), Gwc(2,1)) impulse
signal simulation result at transducer 2 ... 90
Figure 6.16 MIMO Controller openloop and closedloop (Gwo(3,1), Gwc(3,1)) impulse
signal simulation result at transducer 3 ... 91
Figure 6.17 MIMO Controller openloop and closedloop (Gwo(1,1), Gwc(1,1)) step
signal simulation result at transducer 1 ... 91
Figure 6.18 MIMO Controller openloop and closedloop (Gwo(2,1), Gwc(2,1)) step
signal simulation result at transducer 2 ... 92
Figure 6.19 MIMO Controller openloop and closedloop (Gwo(3,1), Gwc(3,1)) step
signal simulation result at transducer 3 ... 92
Figure 6.20 MIMO Controller openloop and closedloop simulation result ... 93
viii
Figure 6.21 MIMO Controller openloop and closedloop (Gwo(1,1),
Gwc(1,1))simulation result (1) at transducer 1 ... 93
Figure 6.22 MIMO Controller openloop and closedloop (Gwo(1,1),
Gwc(1,1))simulation result (2) at transducer 1 ... 94
Figure 6.23 MIMO Controller openloop and closedloop (Gwo(2,1),
Gwc(2,1))simulation result (1) at transducer 2 ... 94
Figure 6.24 MIMO Controller openloop and closedloop (Gwo(2,1),
Gwc(2,1))simulation result (2) at transducer 2 ... 95
Figure 6.25 MIMO Controller openloop and closedloop (Gwo(3,1),
Gwc(3,1))simulation result (1) at transducer 3 ... 95
Figure 6.26 MIMO Controller openloop and closedloop (Gwo(3,1),
Gwc(3,1))simulation result (2) at transducer 3 ... 96
Figure 7.1 principle of the selfsensing technique used to measure the backemf
voltage
Vemf ... 101
Figure 7.2 block diagram for the calculation of the backemf voltage ... 102
Figure 7.3 The interface circuit used to power the control transducer and to measure
... 104
Figure 7.4 SISO vibration control at transducer 1 first mode 27.1 Hz before control .. 106
Figure 7.5 SISO vibration control at transducer 1 first mode 27.1 Hz after control ... 106
Figure 7.6 SISO vibration control at transducer 1 second mode 34.4 Hz before control
... 107
Figure 7.7 SISO vibration control at transducer 1 second mode 34.4 Hz after control 107
Figure 7.8 SISO vibration control at transducer 1 third mode 40.5 Hz before control. 108
Figure 7.10 SISO vibration control at transducer 1 forth mode 49.2 Hz before control
... 109
Figure 7.11 SISO vibration control at transducer 1 forth mode 49.2 Hz after control . 109
Figure 7.28 SISO vibration control for sweep signal at transducer 1 before control ... 118
Figure 7.29 SISO vibration control for sweep signal at transducer 1 after control ... 118
Figure 7.30 SISO vibration control for sweep signal at transducer 2 before control ... 119
Figure 7.31 SISO vibration control for sweep signal at transducer 2 after control ... 119
Figure 7.32 SISO vibration control for sweep signal at transducer 3 before control ... 120
ix
Figure 7.33 SISO vibration control for sweep signal at transducer 3 after control ... 120
Figure 7.58 MIMO vibration control for sweep signal at transducer 1 before control 136
Figure 7.59 MIMO vibration control for sweep signal at transducer 1 after control ... 136
Figure 7.61 MIMO vibration control for sweep signal at transducer 2 after control ... 137
Figure 7.63 MIMO vibration control for sweep signal at transducer 3 after control ... 138
Figure 8.1 Modal Analysis mode shape of first mode with four transducers ... 144
x
List of Tables
Table 2.1: Mechanical parameters of the transducers ... 23
Table 2.2: Electrical parameters of the three transducers (T 1, 2 and 3) ... 24
Table 2.3: The first four natural frequencies of the experimental model ... 25
Table 2.4 natural frequencies comparison ... 27
Table 5.1 classical algorithm and genetic algorithm comparison ... 73
Table 5.2 multimode SISO PPF controller optimal parameter result ... 75
Table 5.3 multimode MIMO PPF controller optimal parameter result ... 77
Table 6.1 SISO PPF closedloop frequency domain result at transducer 1 ... 86
Table 6.2 SISO PPF closedloop frequency domain result at transducer 2 ... 87
Table 6.3 SISO PPF closedloop frequency domain result at transducer 3 ... 88
Table 6.4 MIMO PPF closedloop frequency domain result at transducer 1 ... 97
Table 6.5 MIMO PPF closedloop frequency domain result at transducer 2 ... 98
Table 6.6 MIMO PPF closedloop frequency domain result at transducer 3 ... 99
Table 7.2 AD and DA converters on the dSpace DS1103 used for signal acquisition and
reference voltage output ... 103
Table 7.3 specific measurement resistor value for each interface circuit ... 104
Table 7.4 SISO vibration control experimental results ... 121
Table 7.5 multimode SISO PPF controller experimental parameter result ... 122
Table 7.6 multimode MIMO PPF controller parameter result ... 123
Table 7.7 multimode MIMO vibration control experimental results ... 139
xi
Abstract
In this thesis, experimental, analytical and numerical analysis three kinds of methods
are used for distributed parameter plate structure modeling, an infinitedimensional
and a very highorder plate mathematical transfer function model is derived based on
modal analysis and numerical analysis results. A feedthrough truncated plate model
which minimizing the effect of truncated modes on spatial lowfrequency dynamics
of the system by adding a spatial zero frequency term to the truncated model is
provided and numerical software MATLAB is used to compare the feedthrough
truncated plate model with traditional balanced reduction plate model which is used
to decrease the dimensions and orders of the infinitedimensional and very highorder
plate model. Active vibration control strategy is presented for a flexible plate
structure with bonded three selfsensing magnetic transducers which guarantee
unconditional stability of the closedloop system similar as collocated control system.
Both multimode SISO and MIMO control laws based upon positive position
feedback is developed for plate structure vibration suppression. The proposed
multimode PPF controllers can be tuned to a chosen number of modes and increase
the damping of the plate structure so as to minimize the chosen number of resonant
responses. Stability conditions for multimode SISO and MIMO PPF controllers are
derived to allow for a feedthrough term in the model of the plate structure which is
needed to ensure little perturbation in the inbandwidth zeros of the model. A
minimization criterion based on the
H
norm of the closedloop system is solved by
a genetic algorithm to derive optimal parameters of the controllers. Numerical
simulation and experimental implementation are performed to verify the
effectiveness of multimode SISO and MIMO PPF controllers vibration suppression
for the feedthrough truncated plate structure.
xii
List of Abbreviations
SISO
Single Input Single Output
MIMO Multiple Input Multiple Output
T.F
Transfer Function
PDE
Partial Derivative Equation
ODE
Ordinary Differential Equation
PPF
Positive Position Feedback
IMSC
Independent Modal
FSS
Flexible Spacecraft Simulator
AVA
Active Vibration Absorber
SRF
Stain Rate Feedback
LQR
Linear Quadric Regulator
NNP
Neural Network Predictive
LQG
Linear Quadratic Gaussian
PACE
Planar Articulating Controls Experiment
RCGA
Real Coded Genetic Algorithm
SIMO
Single Input Multiple Output
FXLMS FilteredX Least Mean Square
AVC
Active Vibration Control
MISO
Multiple Input Single Output
FEM
Finite Element Method
MDF
Medium Density Fibreboard
xiii
Acknowledgement
I am heartily thankful to Associate Professor Fangpo He, whose guidance and
support at the initial and medium level.
To Dr. Lei Chen, who is from the mechanical engineering school of Adelaide
University, thanks a lot for your willingness to help and your availability time for
long discussions and helpful suggestions.
To S. O. R. Moheimani, thanks a lot for your contributions to vibration control
theory. I felt that I was just like under your supervision to finish my thesis after
read your books and thesis.
To Siyang Yu, thanks for your help and discussion the ANSYS model with me.
Finally, I wish to thank my family, my fatherinlaw, motherinlaw, my mum, and my
daughter Hanzhen , Hanen, especially to my wife Clare, whom I mostly in debt, thank
you so much for your constant love, support, patience, and encouragement during the
whole study time of master degree at Flinders University, without whom I would be
unable to complete my degree, thanks a million, love you forever!
Also thanks all the people who giving help:
Miss Natalie Hills
Ms Kylie Sappiatzer
Ms Vanesa Duran Racero
Prof Sonia Kleindorfer,
Professor Paul Calder,
Dr Peter Anderson,
Ms Anne Hayes
Ms Lisa O'Neill
Prof Mark Taylor
Prof John Roddick
Prof Warren Lawrance
Prof Carlene Wilson
Prof Jeri Kroll
1
Chapter 1
Introduction
This chapter discusses the motivation for the research described in this thesis. The
research methodology is presented, followed by a literature review. Finally, an outline
of the thesis is given along with a list of original contributions.
1.1 Motivation
In order to improve dynamic performance, the operating efficiency, and the amount of
material which is used in mechanical structures, many designers employ lightweight
materials to reduce the cross sectional dimensions of those structures [4].
However, one sideeffect of employing lightweight materials and reducing the cross
sectional dimensions is that the structures become more flexible. Flexible structures are
more susceptible to the detrimental effects of unwanted vibration, particularly when
they operate at or near their natural frequencies or when they are excited by
disturbances that coincide with their natural frequencies [4].
The design and implementation of a high performance vibration controller for a flexible
structure can be a difficult task. The difficulty is due to the following factors but not
limited:
i.
A major difficulty in control of flexible structures is due to the fact that they are
distributed parameter systems. Consequently, these structures have a very large
number of vibration modes and their transfer functions contain many poles close to
the j axis [1]. Highfrequency modes also contribute to the dynamics at low
frequencies. Thus, a model may have to include a relatively high number of highf
requency modes to capture the low frequency dynamics with acceptable accuracy.
This yields a model with a relatively high order and the systems are generally
difficult to control [2].
2
ii.
In most cases, a small number of inbandwidth modes of the structure are required
to be controlled, and it is possible that some inbandwidth modes are not targeted to
be controlled at all. The presence of uncontrolled modes can lead to the problem of
spillover. That is, the control energy is channeled to the residual modes of the
system and this process may destabilize the closedloop system. In particular, the
spillover effect is of major concern at higher frequencies where obtaining a
precise model of the structure is rather difficult [3]. Normally there are two types
of spillover: control spillover and observation spillover. Control spillover
occurs when the control force excites unmodeled dynamics. The excitation of
these unmodeled dynamics can degrade the performance of the system.
Observation spillover refers to the measurement error caused by the contribution
of excluded modes to the sensor measurements. While control spillover leads to
poor performance, observation spillover can lead to instability [4].
iii.
To control a flexible structure that has widely separated multimode vibration a
wideband controller is needed. However, the design of a high performance
wideband controller is difficult. The difficulty is due to the design tradeoff
between the error reduction in one frequency band and the increase of sensitivity
at other frequencies, as explained in Bode's theorem [5] .
Given these requirements, the question is how to design and implement a suitable
controller. Seeking the answer to this questions is the motivation for the research
described in this thesis.
1.2 Research Methodology
The research methodology includes of four steps:
In the first step, a literature review overviews the relevant existing methods for
controlling the vibration of flexible structures, and discusses the relevant shortcomings
or gaps in those methods. Based on the gaps that are found in the existing methods, new
control methods are proposed.
In the second step, an experimental plant that can be used as a tool for the design and
evaluation of the effectiveness of the proposed control methods, is designed and
implemented. The plant chosen must represent a real application and have the essential
3
characteristics of a flexible structure. A flexible two dimensions plate with three
transducers which are treated as supporting feet is chosen as the experimental plant.
In the third step, analytical and numerical analysis three kinds of methods are used for
modeling, an infinitedimensional and a very highorder plate mathematical transfer
function model is derived based on modal analysis and numerical analysis results. A
feedthrough truncated system model which minimizing the effect of truncated modes
on spatial lowfrequency dynamics of the system by adding a spatial zero frequency
term to the truncated system model is provided and numerical software MATLAB is
used to compare the feedthrough truncated system model with traditional balanced
reduction model which is used to decrease the dimensions and orders of the
infinitedimensional and very highorder model.
In the fourth step, SISO and MIMO PPF controllers were designed based on the
H
norm of the closedloop system by a genetic algorithm to derive optimal parameters
of the controller.
In the fifth step, simulation models of the experimental plant are implemented, and
computer simulations are exercised. These simulations reduce the design time, increase
the success rate of the realtime implementation, and help in the evaluation of the
performances of the proposed controllers prior to their use with the experimental plant.
In the sixth step, the proposed controllers are used with the experimental plant and the
effectiveness of the proposed control methods are evaluated.
1.3 Vibration Control of Structure
Vibration control is applied with respect to avoid the unwanted vibration of the
structure. Vibration control can be categorized into two major techniques: passive
control and active control.
Passive Control
For passive control, vibration is attenuated or absorbed by traditional vibration dampers,
shock absorbers, and base isolation. However, it has two major drawbacks.:
4
i.
Firstly, it is ineffective at low frequencies. The natural frequency is inversely
proportional to the square root of the spring compliance and to the mass of the
damper. Hence, at low frequencies, the volume and mass requirements are often
impractically large for many applications where physical space and mass loading
are critical.
ii.
Secondly, the passive technique only works effectively for a narrow band of
frequencies and is not easy to modify [6].
Active Control
For active vibration control, it is the active application of force in an equal and opposite
fashion to the forces imposed by external vibration.In contrast with passive control,
active control works effectively over a wide bandwidth where the working band does
not depend on the characteristics of the structure, and is limited only by the bandwidth
of the actuators. Furthermore, the actuators are less sensitive to the characteristics of the
structures and the vibration sources. Therefore, the same actuators can be used even if
the characteristics of the structures or the vibration sources are changed. To maintain
the system performance, the electronic controller might need to be modified, but this
modification is relatively easy, especially with digital controllers [7].
From the above discussion, it is clear that active control shows better potential
comparing with passive control. So this thesis will focus on the design of active
controllers.
1.4 Active Vibration Control of Structure
1.4.1 openloop and closedloop control
Active control can be classified as openloop and closedloop.
Openloop Control
An openloop control system uses a controller and an actuator to obtain the desired
response and it is a system without feedback [8]. In general, an openloop system relies
on the model of the plant to obtain a command input that, supplied to it, causes the
output to follow a desired pattern. This strategy requires very good knowledge of the
5
dynamics of the controlled system and is usually applied only as a feedforward
component in conjunction with a feedback controller [9].
advantages:
simplicity and stability: they are simpler in their layout and hence are economical
and stable too due to their simplicity.
construction: since these are having a simple layout so are easier to construct [9].
disadvantages:
accuracy and reliability: since these systems do not have a feedback mechanism, so
they are very inaccurate in terms of result output and hence they are unreliable too.
due to the absence of a feedback mechanism, they are unable to remove the
disturbances occurring from external sources [9].
Closedloop Control
In contrast to an openloop control system, a closedloop control system utilizes an
additional measure of the actual output to compare the actual output with the desired
output response. The measure of the output is called the feedback signal. A feedback
control system is a control system that tends to maintain a prescribed relationship of
one system variable to another by comparing functions of these variables and using the
difference as a means of control. With an accurate sensor, the measured output is a
good approximation of the actual output of the system [8].
advantages:
accuracy: they are more accurate than openloop system due to their complex
construction. They are equally accurate and are not disturbed in the presence of
nonlinearities.
noise reduction ability: since they are composed of a feedback mechanism, so they
clear out the errors between input and output signals, and hence remain unaffected
to the external noise sources [8].
disadvantages:
6
construction: they are relatively more complex in construction and hence it adds up
to the cost making it costlier than openloop system.
since it consists of feedback loop, it may create oscillatory response of the system
and it also reduces the overall gain of the system.
stability: it is less stable than open loop system but this disadvantage can be striked
off since we can make the sensitivity of the system very small so as to make the
system as stable as possible [8].
In order to achieve better performance, the design control method discussed in this
thesis will concentrate on closedloop control.
1.4.2 Feedforward and Feedback Control
Active control can be classified as feedforward or feedback control depending on the
derivation of the error signal.
Feedforward Control
Feedforward is a term describing an element or pathway within a control system which
passes a controlling signal from a source in its external environment, often a command
signal from an external operator, to a load elsewhere in its external environment. A
control system which has only feedforward behavior responds to its control signal in a
predefined way without responding to how the load reacts; it is in contrast with a
system that also has feedback, which adjusts the output to take account of how it affects
the load, and how the load itself may vary unpredictably; the load is considered to
belong to the external environment of the system [10].
In a feedforward system, the control variable adjustment is not errorbased. Instead it
is based on knowledge about the process in the form of a mathematical model of the
process and knowledge about or measurements of the process disturbances [10].
For the feedforward system[11,12,13], it has some advantages [14]:
wider bandwidth
works better for narrowband disturb
7
also include disadvantages [14]:
reference needed
local method (response may be amplified in some part of the system)
large amount of real time computations
Feedback Control
In feedback control, the error signal, which is the difference between the desired
response and the controlled output, is fed to the controller. The controller then
generates control signals to drive the error signal to zero. With feedback control,
stability becomes a major concern because the feedback modifies the characteristic of
the original plant [14].
advantages [14]:
guaranteed stability when collocated
global method
attenuates all disturbances within
c
(bandwith)
disadvantages [14]:
effective only near resonances
limited bandwidth
disturbances outside
c
(bandwith) are amplified
spillover
Due to the excitation signal in the flexible plate, feedforward control is not suitable for
application with this system. Therefore, the design control method discussed in this
thesis mainly focuses on feedback control.
1.4.3 Wave Control and Modal Control
Active control can also be classified according to the model descriptions upon which
the control design is based. The most common descriptions of the vibration of
continuous systems are in terms of waves and modes of motion [
15
]. These two
descriptions lead to two different approaches for active control: wave control and
8
modal control.
Wave Control
In a structure where the flow of vibrational energy from one part to another is
significant and needs to be reduced, wave control is normally used. Wave control
design makes use of the wave equation of a structure and the local properties at and
around the control region. Since inherently the local properties of the structure are less
sensitive to system properties wave control has a good robustness. However, because it
does not take into account global motion, global behaviour can adversely affect the
amount of control achieved [4].
Difficulty in realizing an active wave control system is that all components in the
system are expressed as noncausal and irrational functions of Laplace variable s.
Therefore, in a practical case, the wave controllers are approximately realized to a
limited extent [16].
Modal Control
Modal control of flexible structures has been of great interest for several decades
among vibration control engineers. In general, modal analysis and control refer to the
procedure of decomposing the dynamic equations of a system into modal coordinates
and designing the control system in this modal coordinate system [
17
]. The principle
behind it is that it somehow extracts a target mode signal from the structural response
and controls it in modal domain in a similar way to controlling a single
degreeoffreedom oscillator. Advantages are as follows: controller design is easy as it
is conducted in modal domain, a global vibration reduction over the whole structure can
be achieved by suppressing modes at a number of discrete positions (i.e., global control
using local feedback), and the designed controller is inherently very robust to the
dynamics of uncontrolled modes [
18
].
Because of the implement limitation, in this thesis, the design control method will
follow on collocated control.
9
1.4.4 SISO Control and MIMO Control
A singleinput and singleoutput (SISO) system is a simple single variable control
system with one input and one output. Systems with more than one input and more than
one output are known as multiinput multioutput (MIMO) systems. Normally SISO
systems are typically less complex than MIMO systems, and SISO system is also easier
to be constructed and implemented.
Due to the increasing complexity of the system under control and the interest in
achieving optimum performance, the importance of control system engineering has
grown in the past decade. Furthermore, as the systems become more complex, the
interrelationship of many controlled variables must be considered in the control
scheme [4].
In order to make the clearly comparison, in this thesis, both SISO and MIMO methods
will be invested later.
1.4.5 Collocated Control and Uncollocated Control
Collocated Control
A collocated control system is a control system where the actuator and the sensor are
attached to the same d.o.f.(degree of freedom). It is not sufficient to be attached to the
same location; they must also be dual, that is, a force actuator must be associated with a
displacement (or velocity or acceleration) sensor, and a torque actuator with an angular
(or angular velocity) sensor, in such a way that the product of the actuator signal and the
sensor signal represents the energy (power) exchange between the structure and the
control system [3].
The structure of the collocated system allows for the design of feedback controllers,
with specific structures, that guarantee unconditional stability of the closedloop
system. Such controllers are of interest due to their ability to avoid closedloop
instabilities arising from the spillover effect [3].
Uncollocated Control
10
Noncollocation of sensor and actuator is often unavoidable due to installation
convenience of transducers or is even recommendable for high degrees of observability
and controllability. However, noncollocated control is generally known to be more
involved than collocated control as the plants are no longer minimum phase [18].
For noncollocated control, one major reason for this is a modeling difficulty, since it is
impossible to model an infinite number of modes existing in a flexible structure. Those
controllers designed based on a few low order modes may thus seriously suffer from the
control spillovers associated with unmodeled but often nonnegligible high order
modes. Another is due to the nonminimum phase characteristic of the plant, and thus
the modes excited by a control actuator will not all be the same in phase, when
measured at a noncollocated sensor location [18].
Based on the information, in this thesis, the design control method will pay close
attention on collocated control.
1.5 Modal Based Controller for Multimode Vibration
Control
There are three main modal control methods that can be found in the literature for
controlling multimode vibration in flexible structures: independent modal space
control (IMSC) , resonant control and positive position feedback (PPF) control.
1.5.1 Independent Modal Space Control (IMSC)
Meirovitch [19,20] established the independent modal space control (IMSC) which
allows the control design for each single mode to be implemented independently, hence
there is little spillover to the residual modes. However, this method requires as many
sensors/ actuators as the number of modes to be controlled, and thus it can only control
a limited number of modes. Furthermore, the control system is vulnerable to
uncertainties such as parameter fluctuation. To overcome this problem, the application
of the robust control techniques to active vibration control problem has been discussed
in the past two decades [16].
11
1.5.2 Resonant Control
The resonant control method proposed by Moheimani et al. [21,22,23,24,25,26] is
based on the resonant characteristic of flexible structures. The controller applies high
gain at the natural frequency and rolls off quickly away from the natural frequency thus
avoiding spillover. It is also described as having a decentralized characteristic from a
modal control perspective [27], thereby making it possible to treat each of the system's
modes in isolation. One of the issues with resonant controllers is their limited
performance in terms of adding damping to the structure [3].
1.5.3 Positive Position Feedback (PPF) Control
Positive position feedback (PPF) was devised by Goh and Caughey [28,29], the
stability was proved by Fanson J. L. [30,31], it has several distinguished advantages
[32]. It has been shown to be a solid vibration control strategy for flexible systems with
smart materials, particularly with the PZT (lead zirconium titanate) type of
piezoelectric material [28,29,30].
PPF controller development
After the PPF controller theory was provided, lots of people did the simulations and
experiments using PPF theory for active vibration, for example:
[31] developed the method further and showed that PPF was capable of controlling the
first six bending modes of a cantilever beam. The second order PPF filter was simple to
implement and had global stability conditions, which were easy to fulfill even in the
presence of actuator dynamics.
[33,34] introduced a first order PPF filter eliminating one of the three filter parameters.
They also combined the positive position feedback with independent modal space
control.
[35] realized that effective vibration control with PPF depends on the accuracy of the
modal parameters used in the control design. They extended the original feedback
technique with an adaptive estimation procedure to identify the structural parameters.
12
[36] implemented PPF in both discrete and continuous systems, showed that an exact
knowledge of the natural frequencies of the structure is not required in order to design
an effective control system.
[37] presented the effectiveness of the optimal modal positive position feedback
algorithm in damping out two vibration modes of a cantilever beam with one
piezoelectric actuator and three position sensors.
[38] illustrated that an adaptive first order PPF filter can successful damp vibration of a
cantilever beam even if the first mode changes by 20%.
[39] provide PPF controller employing multiple actuators instead of a single one for
any particular vibration mode.
[
40
] designed a combined scheme of PD feedback controller for AC servo motor and
PPF controller for PZT actuators to suppress multimode vibration applied to
experimental singlelink flexible manipulator.
[41] implemented a single mode PPF and also a multimode PPF controller under
single channel control scheme for vibration of beam
[
42
] introduced PPF to increase the stability margins and allow higher control
bandwidth, compensated for the coupled fuselagerotor mode of a Rotary wing
Unmanned Aerial Vehicle (RUAV).
[43] PPF controllers are designed based on the identified results and investigated active
vibration control of a beam under a moving mass using a pointwise fiber Bragg grating
(FBG) displacement sensing system.
[44] indicated a multimode controllable SISO PPF controller, a noncollocated sensor/
moment pair actuator, and tuned to different vibration modes of beam based on the
results of the parametric study for the design parameters.
Modified PPF (MPPF)
based on the performance has already achieved, some researchers modify the structure
of PPF, [45] provided a modified compensator which enhanced flexibility actively
13
changing damping and stiffness of flexible structures. [46] constructed a new MPPF
which consists of first order and second order two SISO parallel compensators, and find
the optimal parameters through experimental way. [47] combined PPF and an output
feedback sliding mode control (AOFSMC) for vibration and attitude control. [48]
designed suboptimal positive position feedback (SOPPF) and output feedback sliding
mode control (OFSMC). [49] provided negative position feedback (NPF) and positive
position feedback (PPF) to reduce multimode vibration of a lightly damped flexible
beam using a piezoelectric sensor and piezoelectric actuators. [50] proposed positive
velocity and position feedback (PVPF) controller and achieved highamplitude
actuation of a piezoelectric tube. [51] extended the linear modal control to nonlinear
modal control by using quadratic modal positive position feedback (QMPPF) control
algorithm to suppress forced vibrations in distributed parameter structures.
Robust PPF
Expect the normal advantages, some people also showed the robust ability of PPF
controller.
PPF control system is robust and performs significantly well at the target frequency,
because the high control action can be generated at the resonance of the controller due
to the tuning stated in literatures [52,53]. [54] presented the experimental results
robustness study of vibration suppression of a cantilevered beam with PZT sensors and
PZT actuators using PPF control. PPF controllers were implemented for singlemode
vibration suppression and for multimode vibration suppression. Experiments found
that PPF control is robust to frequency variations for singlemode and for multimode
vibration suppressions. [55] considered PPF which derived from solving a group of
LMIs with adjustable parameters with respect to the inaccurate structure modal
frequencies and a simulation was presented to illustrate the effectiveness of the
proposed robust PPF controller design method.
Adaptive PPF
In order to control vibration of structures with varying parameter, an adaptive PPF
controller was put into the consideration. [56] presented an adaptive modal control
algorithm, utilized only modal position signals, fed through firstorder filters to damp
14
out the vibration, [57,58,59] proposed the use of GA for tuning PPF controller for grid
structure, [60] proposed a new APPF based on gradientdescent approach for beam
structure, [61] provided a SISO PPF controller with RLS and Bairstow combined
online frequency estimator, [62] designed a SISO PPF controller for a simulation study
with RLS estimator for the first two natural frequencies of a beam structure, [63,64]
developed RLS estimator with SISO PPF controller for the beam structure, [65]
through system identification, a twomode SISO PPF controller was designed for beam
structure based on GA which was designed to minimize the Hnorm and choose PPF
optimal parameters.
PPF combine with Genetic Algorithms (GA)
During the PPF controller design, in order to achieve better performance, some
designer used Genetic Algorithms (GA) to choose the placement of sensor and
actuator. [
66
] applied GA to find efficient location of sensor and actuator of a
cantilevered composite plate, showed significant vibration reduction for the first three
modes (controlled modes) has been observed using the coupled PPF in the vibration
control experiment. and the closed loop has been observed to robust with respect to
system parameter variations. [
67
] presented GA method of optimal placement for the
cantilever plate. Simulations and experimental results on the actual process have
shown that the proposed control method by combining PPF and PD can suppress the
vibration effectively, especially for vibration decay process and the smaller amplitude
vibration.
PPF compare with other controllers
In order to compare the performance with other controllers, some researchers did
simulations and experiments, such as [68] compared with AVA controller at SISO and
MIMO situation, [69] conducted simulations and experiments for SISO PPF case
compared with LQG controller for suppressing the single and multi modes vibration of
flexible spacecraft simulator (FSS). [70] showed that a PPF controller may be
formulated as an output feedback controller both in centralized and decentralized
situation. [71,72,73,74] compared PPF with stain rate feedback(SRF), [75]compared
PPF with velocity feedback. [76] compared Positive position feedback (PPF) control,
linear quadric regulator (LQR) control and neural network predictive (NNP) control
15
strategies. [77] compared negative imaginary feedback controllers (PPF, Resonant
Controllers, Integral Resonant Controllers) with state feedback controller. [78]
compared velocity feedback controller, the integral resonant controller (IRC), the
resonant controller, and the positive position feedback (PPF) controller. [79] gave a
PPF controller for controlling the first mode of vibration, a decentralized controller
which used three independent PPF filters for suppressing the first three modes of
vibration and a MIMO linear quadratic Gaussian (LQG) for the vibration of USAF
Phillips Laboratory's Planar Articulating Controls Experiment (PACE)
MIMO PPF
In order to achieve better performance, and because the complicated coupled
relations between controllers, only some designer provided MIMO PPF controller.
For beam structure
[1,3] reported experimental implementation of MIMO PPF controller on an active
structure consisting of a cantilevered beam with bonded collocated piezoelectric
actuators and sensors through pole placement and
H
optimization. [65] designed
MIMO PPF controller on a flexible manipulator and based on GA method to find
controller parameters to optimize
H
result.
For grid structure
[57, 58,59] proposed the use of GA method for tuning MIMO PPF controller for grid
structure. [80] designed MIMO PPF controller for grid structure based on the
blockinverse technique.
For plate structure
[81] designed MIMO PPF controller for plate structure based on the pseudoinverse
technique.
For shell structure
[82] studied MIMO PPF controller for shell structure for the first two vibration mods
based on the blockinverse technique.
16
1.6 Plate or Shell Structure Vibration Control
In the following, we will summarize the vibration control method for plate or shell
structure in the literature. It can be divided to three methods: SISO vibration control,
decentralized MIMO vibration control and MIMO vibration control.
SISO vibration controller for plate or shell structure
[83] studied two SISO control algorithms: constantgain negative velocity feedback
and constantamplitude negative velocity feedback for plate. [84] designed SISO
direct feedback control and Lyapunov control for the vibration of shell. [85] proposed
SISO
H
based robust control for bending and torsional vibration modes of a
flexible plate structure. [86] designed SISO velocity proportional feedback to control
the vibration of a crossstiffened plate with a very small laminated piezoelectric
actuator (LPA) under low voltage. [87] presented the development of an active
vibration control (AVC) mechanism using realcoded genetic algorithm (RCGA)
optimization. The approach is realized with singleinput singleoutput (SISO) and
singleinput multipleoutput (SIMO) control configurations in a flexible plate
structure.
decentralized MIMO vibration controller for plate or shell structure
[88] gave a decentralized MIMO experimental compensation method based on PPF
theory which is applied to switched reluctance machine (SRM). [89] provided the
optimal placements of three acceleration sensors and PZT patches actuators are
performed to decouple the bending and torsional vibration of such cantilever plate for
sensing and actuating. A nonlinear MIMO PPF control method is presented to
suppress both high and low amplitude vibrations of flexible smart cantilever plate. [90]
provided MIMO PPF, PPF and PD combined controller to control the decoupled
bending and torsional modes of plate. [91] implemented decentralized velocity
feedback control on panel structure. [92] studied the decentralized multiple velocity
feedback loops on a flat panel.
MIMO vibration controller for plate structure
17
For MIMO vibration control design, it can be divided to following methods:
[93,94,95] studied MIMO adaptive filteredx least mean square (FXLMS)
feedforward control method for the vibration of plate structure and obtained a good
performance.
[96] presented the development of an active vibration control (AVC) mechanism
using Genetic Algorithms, Particle Swarm Optimization and Ant Colony
Optimization method. The approach is realized with multiplinput multipleoutput
(MIMO) and multiple input singleoutput (MISO) control configurations in a flexible
plate structure.
[97,98,99,100,101,102,103,
104
] designed
H
2
,
H
and
synthesis robust MIMO
controller for the plate vibration. The result of the simulation showed that the control
method and the controller designed in the paper was useful
[105,106,107,108,109,110] gave linear MIMO LQR & LQG controller for the
vibration of plate structure. The control method was verified by experiment and
achieve better performance.
[111,112] provided MIMO Sliding Mode controller to the vibration control of plate
structure. According to MATLAB/Simulink platform, the simulation results clearly
demonstrated an effective vibration suppression.
Based on the literature review, the author will adopt optimal MIMO PPF controller
based
H
optimization through GA searching for multimode vibration control of a
flexible plate structure, similar method application only can be found on beam
structure.
1.7 Aim of the Thesis
The aim for this research can be summarized as:
i.
set up model of the mechanical plate structure using experimental, mathematical
and numerical method ;
18
ii.
truncate the model of the mechanical plate structure comparing with model
reduction and model correction method;
iii.
select an actuator/sensor system suitable for active control;
iv.
increase damping of resonant vibration modes of the mechanical plate structure
using multimode SISO and MIMO PPF controller;
v.
develop an algorithm to tune the multimode SISO and MIMO PPF controller
parameters;
vi.
analyze and compare the performance of closedloop dynamics between shaker
and plate with the openloop dynamics;
1.8 Outline of the Thesis
This thesis presents the design and implementation of the optimal multimode SISO
and MIMO PPF controller based
H
optimization through GA searching for
attenuating vibration of plate structure.
A detailed outline of the thesis structure is given below.
In Chapter 2, the model of the plate structure is derived in three ways: experimental
method, analytical method and numerical method. The author also set up the transfer
function matrix of plate simulation model, MATLAB plot figure about SISO and
MIMO plate simulation model is given in this chapter.
In Chapter 3, the model reduction and spatial
H
,
H
2
norm is given. In order to
truncate the plate model into lower order, the balanced reduction method is proposed.
MATLAB plot figures are also given for SISO and MIMO plate model.
In Chapter 4, model correction method is introduced, and the MATLAB plot figures
are given for SISO and MIMO plate model, comparing with the result of SISO and
MIMO balanced reduction plate model.
In Chapter 5, multimode SISO and MIMO PPF controllers are given. The stability of
controllers are derived. In order to achieve better performance, controller parameters
are selected based on
H
optimization through GA searching, MATLAB plot figures
19
are given.
In Chapter 6, simulation results of multimode SISO and MIMO PPF controllers are
given in time and frequency domain. According to comparing the performance of
multimode SISO and MIMO PPF controllers.
In Chapter 7, experimental implement results of multimode SISO and MIMO PPF
controllers are given in time and frequency domain. According to comparing the
performance result of multimode SISO and MIMO PPF controllers, final conclusion
is given.
In Chapter 8, a summary and a conclusion obtained from the research are presented
and recommendations for further continuation of the research are given.
20
Chapter 2
Model of Flexible Plate Structure
Normally, there are three methodologies, analytical analysis, experimental method and
numerical calculation, could be implemented to describe a system. In this chapter, the
design and implementation of the experimental plant, the analytical and numerical
models used to simulate the experimental plant are discussed. The purpose for the
modeling is outlined and the reasons for choosing the experimental plant, and the
modeling steps are given. The plate used in the experimental plant is followed by a
description of the analytical and numerical method used to obtain the mathematical
model of the plant.
The model derivation itself is not original and can also be found in
[113,4,117,118,119,120].
2.1 Introduction
As mentioned in Chapter 1, in order to design and evaluate the proposed controller,
an experimental plant together with its mathematical representations is need to
obtained.
Modeling methods can be applied to find models which represent the experimental
plant after the experimental plant is decided. According to these models, it is easy for
us to study the dynamics of the plant. As all of the proposed control methods employ
natural frequency as the controller parameter, the most important part of the modeling
is that how to find the models with accurate representations of the natural frequencies
of the systems [4].
Physical and mathematical theories such as Newton's laws, Hooke's laws, Lagrange's
equations, Hamilton's principle, etc will be used to obtain the analytical model. The
mathematical model that is adopted is known as the equation of motion, which is
usually given in the form of a Partial Differential Equation (PDE). To determine the
dynamics of the model, the solution of the PDE needs to be found. Normally there are
21
two common methods used to find the solution of the PDE: the analytical method and
the numerical method [4].
The analytical method gives an exact solution of the PDE. The solution is in a closed
form and is expressed in terms of known functions. Although analytical methods can be
used to very accurately describe the dynamics of structures, the types of applications
where this method can be applied are limited. The analytical method is only applicable
for systems that are characterized by uniformly distributed parameters and simple
boundaries [
114
]. In many cases, even though closedform solutions may be possible,
great effort and time are required to obtain them. Therefore in practice the analytical
method has fewer application areas than the numerical method [4].
In numerical method, a discrete version of the model is produced. The spatial
dependence in the solution of the PDE is eliminated by applying spatial discretization
and the differential eigenvalue problems are transformed into an algebraic form [
115
].
Several methods exist for constructing the discrete model [114,115]: Rayleigh's
method, RayleighRitz's method, Galerkin's method, assumed modes method,
collocation method, Holzer's method, Myklestad's method and the finite element
method (FEM). FEM is currently the most widely used method for representing
discrete models [115]. The FEM package ANSYS is used here to study the dynamics of
the structures. Due to the use of approximation, numerical methods do not give the
same exact results as analytical methods. However, approximation algorithms have led
to accuracy improvements for the numerical method.
Simulation is an important step in the control system design process. It provides a
flexible and relatively inexpensive means by which to study the dynamics of a plant,
design controllers, and evaluate the performances of the controllers prior to their
implementation in an actual system. The simulation tool Mablab Simulink is used for
that purpose. Using MATLAB Simulink, simulation models, which are derived from
the modification of the mathematical models which is obtained from the analytical
method or from the modification of the numerical models which is obtained from the
numerical method, need to be implemented [4].
The implementation of the models is undertaken in four steps. In the first step, the
experimental plant (experimental models), which describes the mechanical system is
22
built. In the second step, analytical models of the experimental models are derived. In
the third step, numerical models of the experimental models are built using ANSYS.
Then the numerical models are compared with the analytical models and the
experimental models to determine their accuracy. In the fourth step, the numerical
models are used to construct the simulation models in MATLAB Simulink.
2.2 Description of Experimental Plant
(Experimental Model)
Plate structure can be served as a basic representative model for a number of flexible
structures such as solar panels of the aerobat and aircraft wings [116]. This plant was
built, analyzed, and designed by [139]. With using energy dissipation structure,
vibration control was achieved in 2010 using the shunt control by [139] in simulation
and experiment study. With using three SISO controllers, vibration control was
achieved in 2012 by [138]in simulation study.
The schematic of the experimental plant is shown in Fig. 2.1. An uniform AL6061T6
plate is mounted with screw on the MDF board using three electromagnetic transducers
(anticlockwise number 1, 2, 3)which are called 'Response CS2277 Power Bass Rocker',
normally used for vibrating car seats while playing music as to enhance the experience
of the low frequency range. Another electromagnetic transducer is mounted on the
MDF board with screw as a disturbance noise shaker. The MDF board is placed on
the table with four rubber legs which are screwed on the MDF board. [139].
Figure 2.1 A thin plate in transverse vibration
23
Figure 2.2 shows a cross section of the transducer. This transducer is quite similar in
structure to a normal loudspeaker. It consists of a coil which produces a magnetic field
when a current is fed through. This will move the core magnet which is mounted inside
the coil and supported by a flexible structure. The model used for this transducer is
shown in Figure 2.2b [139].
Figure 2.2 Transducer cross section (a) and model (b)
It is basically a massspringdamper system with an electrical port consisting of a coil, a
series resistance and a voltage source modeling the backemf voltage. The transducers
massspringdamper system with an electrical port consisting of a coil, a series
resistance and a voltage source modeling the backemf voltage. The transducers were
assumed mechanically identical. Table 2.1 shows the measured mechanical parameters
of the transducers [139].
Table 2.1: Mechanical parameters of the transducers
The frequency range at which actuation is possible was found to be 20 Hz to 200 Hz for
all transducers. However, they were not found to be electrically identical. Table 2.2
shows the measured electrical parameters of the transducers used for control. For a
detailed description of the measurement procedures [139].
24
As stated in 1.5.3 of Chapter 1, PPF combine with Genetic Algorithms (GA), the
optimal control position of the three control transducers can be derived through GA
calculation. The author did not derived that because the limited time, the other reason
is that during the implementation, for some mechanical plant systems, some optimal
positions may not be used due to the physical conditions. Expect the analytical GA
calculation, the other easier method is that set up the model and do the simulation
using the numerical software such as ANSYS. It is easier to find and confirm the
minimum displacement of the observation position according to different control
transducer position.
Parameter
Symbol
T 1
T 2
T 3
Impedance at 50 Hz
Zi[]
3.6170
3.7204
3.8442
Inductance at 50 Hz
L
i
[µH]
518
533
597
Resistance at 50 Hz
R
i
[]
3.6133
3.7166
3.8396
CurrentForce coupling at 26.9 Hz
C
iF
[N/A]
2.0523
3.6361
3.6996
VelocityVoltage coupling at 26.9 Hz
C
V
[Vs/m]
2.0523
3.6361
3.6996
Table 2.2: Electrical parameters of the three transducers (T 1, 2 and 3)
The natural frequencies of the experimental models are the most important parameters
to be considered for the proposed control methods. Therefore, a series of experiments
are undertaken to measure the natural frequencies of the experimental model. The
natural frequencies are obtained by applying a sweep signal to experimental model. The
amplitude of the vibration is then measured against the frequency of the signal. The
frequencies where the amplitude of vibration forms a peak are the natural frequencies
of the experimental model. In the experiment, a sweep signal from a signal generator is
amplified by a 50 W Jay Car amplifier and the output from the amplifier is applied to
the disturbance noise transducer. The attenuation is necessary in order to make the
signal level suitable for input to the analogtodigital converter. The amplitude of the
vibration is measured by accelerometers then it is shown on Agilent 54621A
oscilloscope. Experiments show that the dominant modes for all the experimental
models are the first four modes. The frequencies for the first four modes are shown in
25
Table 2.3 [139].
In the next section analytical models of the experimental model will be de rived.
Table 2.3: The first four natural frequencies of the experimental model
2.3 Numerical Solution Using ANSYS
(Numerical Model)
In practice, especially for complex structures, a software tool such as ANSYS can be
used to find the natural frequencies of the structures. ANSYS uses the finiteelement
method (FEM) to solve the underlying governing equations and the associated
problemspecific boundary conditions. FEM tools are used widely in industry to
simulate the responses of a physical system to structural loading, and thermal and
electromagnetic effects. In this research, ANSYS software version 14.0 is used to find
the natural frequencies of the plate structure. The results are validated through
comparison with the results from the experimental and analytical method [4]. This
numerical model was built, analyzed, and designed by [139], and also studied by [138].
Due to the limitation of the transducers, the author adjusted the thickness of the top
plate and MDF board, and other relevant parts in the numerical model to make sure
that the first natural frequency of the top plate is within the frequency range of the
transducers.
26
Figure 2.3 System model shown in isometric view and side view
2.3.1 Resonance Frequency
Normal mode, an inherent property of the system, is characterized with a specific
natural frequency, damping ratio and mode shape, and it can be applied to describe the
system motion called resonance which is approximate equal to the natural
frequency[117,118,119]. For any system, modes are totally independent with each
other so that they have different modal parameters including natural
frequencies, damping ratios and mode shapes (even if the same value). Those modal
parameters normally depend on the system structure, materials and boundary
conditions. Hence, the excitation of one mode of a system would not cause the motion
of different mode. In ANSYS, the modal analysis module is able to be used for
computing the resonant frequencies and corresponding mode shapes [4].
Resonance frequency (or natural frequency) refers to a certain frequency that is able to
lead the significant magnitude of the oscillation. To determine the dynamics on the top
plate, start with modal analyses so that the resonance frequencies can be found.
In the simulation the model is supposed to be analyzed in range of [0, 200 Hz] due to
27
following three reasons [139].
i.
In practice, the transducers used in the mechanical structure can only sense the
signal which frequency is from 20 Hz to 200 Hz. To guarantee that there is no
mode below 20 Hz is omitted, the simulation should be analyzed the mode from 0
Hz instead of the lower limit of the transducer detecting range.
ii.
More serious, since the modes with low natural frequencies would normally cause
more significant response than the ones with high frequencies, it is necessary to
ensure if there is a mode invisible when the modes were detected experimentally,
especially for the natural frequency below the physical detection range.
iii.
On other hand, as the magnitude of the response declines with the frequency
increasing, it might be worthless to sense any natural frequencies above 200 Hz.
The resonant frequencies from virtual simulation are shown in the right column of
Table 2.5. To verify the simulation results, detecting the natural frequencies of the
physical system has been implemented experimentally. Apply sine swept which was
the disturbance signal from 20 Hz to 200 Hz to the shaker on the bottom plate and
observe the acceleration at the top plate. To measure the acceleration, the PCB
Piezotronics 353B17 accelerometer is used to sense the natural frequencies which have
the acceleration peak values. The physical measurement results are also listed in the
Table 2.4. For comparison purposes, the first four natural frequencies of the model are
shown below. From the comparison, it can be seen that the difference between the
results from the experimental, analytical method and the results from ANSYS [139].
Mode
Experimental Model
natural frequencies (Hz)
Analytical Model
natural frequencies (Hz)
Numerical Model
natural frequencies (Hz)
1
27.1
25.94336
27.195
2
34.4
31.39146
35.01
3
40.5
31.39146
37.467
4
49.2
50.22634
46.202
Table 2.4 natural frequencies comparison
From this comparison it can be concluded that the ANSYS results are more accurate
28
cause the deformation is more complex and the deformation equation which had been
derived from theoretical analysis can not exactly predict the deformation. Based on the
accuracy of the results, numerical model from ANSYS will be used here to form
simulation model for the experimental plant.
2.3.2 Mode Shape
Mode shape, one of the mode parameters, describes a deflection pattern associated with
a particular natural frequency. Same as the natural frequency, mode shape is
independent with the ones with other modes. The actual deformation or displacement at
arbitrary point of the structure is the summation of all the mode shapes of that
system[118,119]. Basically, the pure mode shapes can be generated and observed as the
displacement if the external harmonic excitations have the same frequencies as the
system modes. Conversely, the excitations with random frequencies tend to produce an
arbitrary "shuffling" of all the structural mode shapes superposition. The mode shape
can be treated as the inherent dynamic property when the system is in free vibration
without any external force applying. In another word, the mode shapes can be
considered as the superposed deflection of the all parts in the system [139].
Figure 2.4 Modal Analysis mode shape of first mode
29
Figure 2.5 Modal Analysis mode shape of second mode
Figure 2.6 Modal Analysis mode shape of third mode
Figure 2.7 Modal Analysis mode shape of forth mode
30
In ANSYS, the mode shapes can be obtained by implementing the modal analysis such
as Fig 2.4
2.7. With the given mode which has been determined before, the modal
shape is computed and represented as maximum deformation or displacement.
2.3.3 Harmonic Response Analysis
Instead of computing the transient vibrations which exists at the beginnings the
excitation, the ANSYS harmonic response analysis technique calculates the steady
state response of the linear structure system to loads that vary harmonically [118,119].
That is, harmonic analysis can be applied with given (range of) frequency (frequencies)
when the structure is in forced vibration. In the harmonic analysis, once the input has
been determined, the corresponding output could be computed. Note that the dynamic
between input and output which is characterized by natural frequency, damping ratio
and mode shape, is the inherent property of the existed system. It means that the system
dynamic would not change whatever the external excitation is applied unless the
system itself has been changed. To determine the dynamic between shaker and any
positions on the top plate, apply 1N force through the shaker and collect the results
given as deflections from the top plate. For instance, as shown in Figure 2.8, the 1N
force, which is the userdefined input, has been applied to the shaker with specific
natural frequency. The amplitude and phase response result of the whole top plate is
shown in Figure 2.9 and 2.10. The harmonic analysis is also able to present the
corresponding deflections at the position of each transducer. The same implementation
can be applied to the shaker, while other three transducers (transducer 1, 2, 3 and the
shaker) are considered as outputs [139].
Figure 2.8 Harmonic Analysis Example
31
With applying 1N force via the shaker (input), respectively, the corresponding
deflection of transducer in millimetre (output) only depends on that single input and its
related dynamic. The harmonic analysis results is shown at Figure 2.11.
It is shown that for Fig 2.11(a), at transducer 1, the mode shape of mode 2 and 4 are
very small. For Fig 2.11 (b) and (c), only three modes can be seen because the mode
shape of mode 2 is very small
Figure 2.9 Amplitude Response of whole top plate to a harmonic disturbance at shaker
Figure 2.10 Phase Response of whole top plate to a harmonic disturbance at shaker
32
(a) Transducer 1 resonant peaks
(b) Transducer 1 resonance phase
33
(c) Transducer 2 resonance peaks
(d) Transducer 2 resonance phase
34
(e) Transducer 3 resonance peaks
(f) Transducer 3 resonance phase
Figure 2.11 Simulated harmonic disturbance at shaker and measured resonant peaks
and phase of top plate at transducer 1 (a, b), transducer 2 (c, d) and transducer 3 (e, f).
35
2.4 Simulation Model of The plate
Simulation plant model using Simulink is used to design and evaluate the proposed
controllers. The simulation results are then used as a benchmark for realtime
implementation of the proposed controllers in the experimental plant. To use Simulink
as a simulation platform, the numerical model from ANSYS need to be modified into
simulation model in the form of transfer functions or state space equations [138].
In this section simulation plate model to represent the experimental plant is designed
and implemented. The simulation plate model is created in transfer functions form.
By following the plant physically set up in the laboratory, the system is modeled and
analyzed using FEM simulation software ANSYS. Random disturbance signal is
applied to the shaker, and the signal will be transmitted to the top plate through the base
plate and then subjected to the three coupled transducers labeled as 1, 2 and 3 in
counterclockwise in Figure 2.3. So we use the same locations where the three
transducers connect to the top plate as the observation points [139]. Therefore this
system is considered as MIMO with three inputs and three outputs as shown in Figure
2.12.
Figure 2.12 Block diagram representation of the PPF control problem
Due to the MIMO properties, any one of the outputs on the top plate is induced by
superposing all the inputs and their related dynamics. Hence, the fully description of the
dynamics of the system can be expressed using open loop system superposition
36
technology.
Y
1
(s)=
Y
11
(s) +
Y
12
(s) +
Y
13
(s) +
Y
1d
(s) =
T
11
(s)
U
1
(s) +
T
12
(s)
U
2
(s) +
T
13
(s)
U
3
(s)
(2.1a)
Y
2
(s)=
Y
21
(s) +
Y
22
(s) +
Y
23
(s) +
Y
2d
(s) =
T
21
(s)
U
1
(s) +
T
22
(s)
U
2
(s) +
T
23
(s)
U
3
(s)
(2.1b)
Y
3
(s)=
Y
31
(s) +
Y
32
(s) +
Y
33
(s) +
Y
3d
(s) =
T
31
(s)
U
1
(s) +
T
32
(s)
U
2
(s) +
T
33
(s)
U
3
(s)
(2.1c)
The equations (
2.130
a
2.130
c) can be illustrated using matrix form
Y
1
(s)Y
2
(s)Y
3
(s) = T
11
(s) T
12
(s)
T
21
(s) T
22
(s) T
13
(s)T
23
(s)
T
31
(s) T
32
(s) T
33
(s) * U
1
(s)U
2
(s)U
3
(s) (2.2)
where
U
1
(s)U
2
(s)U
3
(s) = D
1
(s)D
2
(s)D
3
(s) * U
d
(s) (2.3)
U
i
 input through the transducer i to the top plate, i = 1, 2, 3
U
d
 input to shaker
Y
i
 output at the position on the top plate, i = 1, 2, 3
D
i
 dynamic between the shaker and transducer
T
mn
 dynamic between transducer m and n, m = 1, 2, 3, n = 1, 2, 3
When the three transducers are treated as three SISO system, the only difference is
that only the diagonal elements exist, other elements are equal to zero.
Based all the information before, we could set up the simulation model of the plate
37
plant in MATLAB, the model include SISO model and MIMO model.
Due to fact that there is no available spectrum analyzer in the laboratory and
workshop, no budget to buy or rent the equipment, we could not plot the numerical
result and compare with the experimental result, according to [120], we hope that we
can adopt 150 modes plate model as the simulation model, and compare the truncated
model which used balanced truncation method with the 150 modes simulation model.
When simulation model was calculated by the MATLAB software, we found that if
there is more than 47 modes in the plate model, the parameter in the denominator
would be too small that MATLAB could not recognize and treat it as infinite small.
So what we can do is that we adopt 47 modes in the plate model and take it as the
simulation model.
SISO 47 modes plate model is shown in Figure 2.132.16. MIMO 47 modes plate
model is shown in Figure 2.172.26.
Figure 2.13 three SISO (g11,g22,g33) 47 modes plate model
38
Figure 2.14 SISO (g11) 47 modes plate model at transducer 1
Figure 2.15 SISO (g22) 47 modes plate model at transducer 2
39
Figure 2.16 SISO (g33) 47 modes plate model at transducer 3
Figure 2.17 MIMO 47 modes plate model
40
Figure 2.18 MIMO (Gp(1,1)) 47 modes plate model
Figure 2.19 MIMO (Gp(1,2)) 47 modes plate model
41
Figure 2.20 MIMO (Gp(1,3)) 47 modes plate model
Figure 2.21 MIMO (Gp(2,1)) 47 modes plate model
42
Figure 2.22 MIMO (Gp(2,2)) 47 modes plate model
Figure 2.23 MIMO (Gp(2,3)) 47 modes plate model
43
Figure 2.24 MIMO (Gp(3,1)) 47 modes plate model
Figure 2.25 MIMO (Gp(3,2)) 47 modes plate model
44
Figure 2.26 MIMO (Gp(3,3)) 47 modes plate model
2.5 Summary
In this chapter we studied the dynamics of plate structure with specific boundary
conditions. The majority of system in this chapter was flexible structure with regular
shape and welldefined boundary conditions, whose eigenvalue problems can be solved
analytically. We derived transfer functions that represent the behavior and capture the
spatially distributed nature of the system. Comparing the accuracy of the natural
frequencies of the model which obtained from experimental method and numerical
calculation. We also set up the simulation model of the plate.
45
Chapter 3
Spatial Norm and Model Reduction
3.1 Introduction
Modeling of physical systems often results in high order models. It is desirable to
reduce such a highorder model to a simpler model of lower dimension. The problem of
model reduction is important since the majority of modern controller design techniques
(such as
H
and LQG design methods) result in a controller of dimension equal to that
of the plant. Hence, a controller design based on a suitable reduced model would result
in a lower order controller which would reduce implementation problems [113]. In
order to use the model in analysis and synthesis problems, we need to define suitable
performance measures that take into account their spatially distributed nature.
Traditional performance measures, such as
H
2
and
H
norms, only deal with
pointwise models for such systems. In this chapter, we extend these performance
measures to include the spatial characteristics of spatially distributed systems [113].
Another topic that is covered in this chapter is that of model reduction by balanced
truncation. The problem of model reduction for dynamical systems has been studied
extensively throughout the literature; see, for example, [121,122] Here, we address the
problem for spatially distributed linear timeinvariant systems, following [123].
3.2 Plate Model Reduction and Balanced Realization
3.2.1 SISO Plate Model Reduction and Balanced Realization
Based on the knowledge in [113], we can use spatial
H
2
and
H
norm for spatially
distributed systems. This norm can be used as measures of performance in certain
analysis and synthesis problems. In some cases, it can be beneficial to add a spatial
weighting function to emphasize certain regions [113]. In this section, we extend those
definitions to allow for spatially distributed weighting functions. So we can reduce the
46
plate model order by the balanced method: Fig 3.13.8 are shown the SISO plate
model reduction by balanced truncation method.
Figure 3.1 SISO Plate (g11) at transducer 1 Hankel singular values
Figure 3.2 SISO Plate (g11) at transducer 1 model reduction and balanced
realization frequency domain compare
47
Figure 3.3 SISO Plate (g11) at transducer 1 model reduction and balanced
realization singular values compare
Figure 3.4 SISO Plate (g22) at transducer 2 Hankel singular values
48
Figure 3.5 SISO Plate (g22) at transducer 2 model reduction and balanced
realization frequency domain compare
Figure 3.6 SISO Plate (g22) at transducer 2 model reduction and balanced
realization singular values compare
49
Figure 3.7 SISO Plate (g33) at transducer 3 Hankel singular values
Figure 3.8 SISO Plate (g33) at transducer 3 model reduction and balanced
realization frequency domain compare
50
Figure 3.9 SISO Plate (g33) at transducer 3 model reduction and balanced
Realization singular values compare
We can see that, according to Hankel singular values, first 44 states of g11, first 36
states of g22, first 32 states of g33 are all unstable states, so if we want to reduce the
order of the model, we need to obtain all the unstable states. It can be seen that the
"balreal" method is better compare to others.
3.2.2 MIMO Plate Model Balanced truncation
Fig 3.103.12 are shown the MIMO plate model reduction by balanced truncation
method.
Figure 3.10 MIMO Plate Hankel singular values
51
Figure 3.11 MIMO Plate model reduction and balanced realization frequency
domain compare
Figure 3.12 MIMO Plate model reduction and balanced realization singular values
compare
It can be seen that the "balreal" method is better compare to others.
52
3.3 Summary
In this chapter, we showed the method to reduce the model order by balanced
truncation method, and also introduce the spatial norm of model. We gave the
performance of "balred", "balreal" and "modred" for SISO plate model, the
performance of "balred" and "balreal" and for MIMO plate model, and according to
compare the performance, we gave the decision that "balreal" balanced truncation
method is better both for SISO and MIMO plate model, but for some cases caused too
big phase angle.
53
Chapter 4
Model Correction
4.1 Introduction
Dynamics of flexible structures and some acoustic systems consist of an infinite
number of modes. For control design purposes these models are often approximated by
finitedimensional models via truncation. Direct truncation of higher order modes
results in the perturbation of zero dynamics and hence generates errors in the spatial
frequency response of the system. If the truncated model is then used to design a
controller which is implemented on the system, say in the laboratory, the closed loop
performance of the system can be considerably different from the theoretical
predictions. This is mainly due to the fact that although the poles of the truncated
system are at the correct frequencies, the zeros can be far away from where they should
be. Therefore, it is natural to expect that a controller designed for the truncated [113].
Reference [124] discusses the effect of out of bandwidth modes on the lowfrequency
zeros of the truncated model. There, it is suggested that the effect of higher frequency
modes on the low frequency dynamics of the system can be captured by adding a zero
frequency term to the truncated model to account for the compliance of the ignored
modes [133]. The model correction method that is proposed allows for a spatially
distributed DC term to capture the effect of truncated modes in an optimal way. In this
chapter, we take a similar approach in the sense that we allow for a zero frequency term
to capture the effect of truncated modes. However, this constant term is found such that
the
H
2
norm of the resulting error system is minimized [113].
4.2 Plate Correction Model
As discussed in Chapter 3, Fig 3.1, 3.4, 3.7, 3.10 showed that there are unstable states
in the Hankel singular values, so we could not choose the state which we want to save
and truncate as normal, so we just focus on the first four modes in this study.
54
4.2.1 SISO Plate Model Correction
Figure 4.14.4 show the model correction result for SISO plate and compare with
"balreal" balanced truncation method. Compare to "balreal" balanced truncation
method, model correction method is better because
i.
the transfer function of correction model is simple, for example, if we only focus
on the first four modes of the plate, there are only 8 states of correction model
comparing to 94 states of balanced truncation model.
ii.
there is no big phase angle.
iii.
it is very easy to implemented in MATLAB or simulink.
Figure 4.1 SISO plate model correction result
55
Figure 4.2 SISO plate (g11) model reduction result compare (balreal and correction)
Figure 4.3 SISO plate (g22) model reduction result compare (balreal and correction)
56
Figure 4.4 SISO plate (g33) model reduction result compare (balreal and correction)
4.2.2 MIMO Plate Model Correction
Figure 4.5 MIMO plate model reduction result compare (balreal and correction)
57
Figure 4.6 MIMO Plate model correction and balanced truncation singular values
compare
Figure 4.7 MIMO (Gpc(1,1)) plate model correction result
58
Figure 4.8 MIMO (Gpc(1,2)) plate model correction result
Figure 4.9 MIMO (Gpc(1,3)) plate model correction result
59
Figure 4.10 MIMO (Gpc(2,1)) plate model correction result
Figure 4.11 MIMO (Gpc(2,2)) plate model correction result
60
Figure 4.12 MIMO (Gpc(2,3)) plate model correction result
Figure 4.13 MIMO (Gpc(3,1)) plate model correction result
61
Figure 4.14 MIMO (Gpc(3,2)) plate model correction result
Figure 4.15 MIMO (Gpc(3,3)) plate model correction result
62
4.3 Summary
In this chapter, we reported the method to truncate the model order by correction
method. We gave the performance of "balreal" balanced truncation and model
correction for SISO and MIMO plate model. According to compare the performance,
we made the decision that model correction method is better both for SISO and
MIMO plate model.
63
Chapter 5
Multimode SISO and MIMO PPF
Controller
This chapter provides an overview of the positive position feedback (PPF) technique.
In the first two section general information and structure about PPF is given. The third
section shows the PPF controller stability derivation. This derivation itself is not
original and can also be found in [125,1,3]. The following section shows the selection
method of the optimal controller parameters and MATLAB simulation result.
5.1 Introduction
As shown in Chapter 2, one of the characteristics of flexible structures is their highly
resonant nature. For the case of the plate used in this research this characteristic is
made evident by the relatively large vibration at or near to the natural frequencies of
the structure, as shown in Fig. 2.5. From the figure, it is clear that suppressing the
vibration of a structure at or very close to the natural frequencies of the structure is
more important than suppressing the vibration at other frequencies. However,
suppressing the vibration at one or more natural frequencies may excite or amplify
other natural frequencies. Therefore, one important design requirement for flexible
structure control is to achieve high attenuation for modes of interest without driving
the other modes into instability or exciting and amplifying the vibration of other
modes [4].
As stated in Chapter 1, a resonant PPF controller has necessary characteristics that
satisfy the design requirements for multimode vibration attenuation of flexible
structures investigated in this research. Exploiting the highly resonant characteristic of
the flexible structure, the resonant PPF controller only applies high gain at or close to
the natural frequencies of interest, and is therefore able to suppress the vibration at
those frequencies without causing adverse effects at other frequencies. In the
64
following section, a further analysis on the characteristics of a PPF resonant controller
is presented [133].
5.2 PPF Controller Structure
The technique of positive position feedback (PPF) control was first introduced by
Caughey and Goh in a Dynamics Laboratory Report in 1982. Its simplicity and
robustness has led to many applications in structural vibration control [29,30]. Unlike
other control laws, positive position feedback is insensitive to the rather uncertain
natural damping ratios of the structure [29]. The terminology positive position is
derived from the fact that the position measurement is positively fed into the
compensator and the position signal from the compensator is positively fed back to
the structure [31]. This property makes the PPF controller very suitable for collocated
actuator/sensor pairs. Equations 3.1 and 3.2 show the structure and compensator
equations in the scalar case [29]:
Structure:
+ 2 +
2
= g
2
(5.1)
Compensator:
+ 2
f
f
+
f2
=
f2
(5.2)
where g is the scalar gain (positive), is the modal coordinate (structural), is the
filter coordinate (electrical), and
f
are the structural and filter frequencies,
respectively, and and
2
f
are the structural and filter damping ratios, respectively.
This nondynamic stability criterion is characteristic for the positive position feedback
system. Figure 5.1 illustrates the connection of Equations (5.1) and (5.2) in a block
diagram.
Figure 5.1 Block Diagram of a SecondOrder System with Positive Position Feedback
65
The secondorder transfer function in Equation (5.3) also represents the PPF
compensator. The transfer function form will be used in this text for deriving the
properties of the control system [29].
G
ppf
(s) =
g
f2
s
2
+
2
f
f
s+
f2
(5.3)
Another feature of the compensator equation, its secondorder lowpass characteristic,
led to the terminology PPF filter. Figure 5.3 presents a Bode plot of a typical PPF
filter. In effect, a PPF filter behaves much like an electronic vibration absorber for the
structure except that a mechanical vibration absorber can never destabilize a
structure[133,124].
Figure 5.2 Bode Plot of a Typical PPF Filter Frequency Response Function
The rolloff character of the PPF filter is one of the advantages of using a PPF
controller. Structural poles with frequencies higher than the filter frequencies are
hardly affected by the compensator. This is especially useful when complicated
structures with many resonance poles are to be controlled. PPF is able to add damping
to lower frequency poles while leaving the possibly unmodeled high frequency poles
unchanged [133,124].
Since one PPF filter tuned to a certain frequency only adds damping to one certain
pole of the structure, the possibilities of multiple PPF were investigated by many
authors [28]. Multiple PPF filters allow that each filter is tuned to add damping to one
structural pole while only using one actuator/sensor pair.
66
Unfortunately, the stability criterion is not as simple as for the one PPF filter case but
experiments have shown that for widely spaced poles with up to three PPF filters the
stability is usually not a problem [29].
5.3 PPF Controller Closed loop Stability
5.3.1 Scalar Case
Considering the scalar case first, PPF can be described by two coupled differential
equations where the first equation describes structure, and the second describes the
compensator as[29]
++
2
=g
2
(5.4)
+
f
+
f2
=
f2
(5.5)
where is the system coordinate, is the actuator coordinate,
, R, and
f
are the system
damping and actuator damping ratios,
and
f
are the structural natural frequency and filter
frequency, and g is the scalar gain, g > 0.
Theorem 5.1
The combined system and actuator of (5.4) and (5.5) are Liapunov asymptotic stable iff g < 1.
(Also serves as the stability boundary.)
Proof :
Taking the Laplace transform of the secondorder equation (5.4) and (5.5)
(s
2
+
s+
2
) (s)=g
2
(s) (5.6)
(s
2
+
f
s+
f2
) (s)=
f2
(s) (
5.
7)
So we can get
(s)
(s)
=
g
2
s
2
+
s+
2
=
s
2
+
f
s+
f2
f2
(5.8)
67
The closed loop characteristic equation for (1) and (2) is
(s
2
+
s+
2
) (s
2
+
f
s+
f2
)  g
2
f2
=0
(5.9)
Then we can achieve
s
4
+
(
+
f
) s
3
+
(
2
+
f2
+
f
) s
2
+
(
f
2
+
f2
) s
+ (1

g)
2
f2
=0
(5.10)
A sufficient and necessary condition for stability is that all the principal minors of the
corresponding Routh  Hurwitz array be greater than zero. The principal minors of (7) can be
easily show to be
s
4
: 1
2
+
f2
+
f
(1 g)
2
f2
s
3
:
+
f
f
2
+
f2
0
s
2
:
+
f
2
+
f2
+
f
(
f
2
+
f2
)
+
f
(1 g)
2
f2
0
s
1
:
2
+
f
f2
+
f
+
f
f
2
+
f2

(
+
f
)
2
(1 g)
2
f2
2
+
f
f2
+
f
(+
f
)
0
s
0
:
(1 g)
2
f2
M
1
=
+
f
(5.11)
M
2
=
2
+
f
f2
+
f
(+
f
)
(5.12)
M
3
=
f
2

f2
2
++
f
f2
+
f
2
+g(+
f
)
2
2
f2
(5.13)
M
4
=
(1 g)
2
f2
(5.14)
Thus for positive g,
M
1
,M
2
,&
3
are unconditionally positive.
M
4
is positive iff g < 1, and
the proof is complete.
68
5.3.2 Multivariate Case
Having understood the scalar system, the idea of position feedback can be easily generalized
into the multivariate system. The equations governing position feedback (collocated actuators
and sensors) control of a quasidistributer parameter system with actuator dynamics are given
by[1,133,124]
M
y + D y + K y = S
T
C u (5.15)
u +
u +
2
(u
 S y) = 0 (5.16)
where y
N
is the system state vector and u
N
A
the actuator state vector. The
N
A
x
N
A
gain matrix C is positive definite and can be factorized, using its square root
C
12
as
follows:
C
T2
C
12
= C (5.17)
The modal form of the system is obtained by applying the following transformation
y =
(5.18)
=
1
C
12
u (5.19)
so that
+ D + =
T
S
T
C
T2
(5.20)
+
+
2
=
C
12
S
(5.21)
or
+ D 00
I
N
A
+ 
T
S
T
C
T2

C
12
S
2
I
N
A
= 0 (5.22)
Theorem 5.2
The combined system and actuator dynamics as represented by (5.19) are Liapunov
asymptotic stable if
69
 (1+ ) B is positive definite (5.23)
where
is some arbitrarily small positive quantity and B is the modal gain matrix given by
B =
T
S
T
CS
(5.24)
Theorem 5.3
For a second order multivariate dynamical system described by
I
Z+Z+Z = 0 (5.25)
where Z,
, are of appropriate dimension and is positive definite, a sufficient and
necessary condition for Liapunov asymptotic stability is that
is positive definite (5.26)
Theorem 5.4
The system in (5.19) is LAS iff the modiffied stiffenss matrix
P =

T
S
T
C
T2

C
12
S
2
I
N
A
is positive definite (5.27)
Note that the sufficient condition for stability (5.20) is equivalent to
K
 (1+) S
T
CS is positive definite
5.3.3
Multivariate PPF Controller implemented with feedthrough plant
Based on Chapter 4, addition of feedthrough term to the truncated model is quite important,
if the truncation is not to substantially alter openloop zeros of the system. Although the
truncation does not perturb openloop poles of the system, it has the potential to significantly
move the openloop zeros, particularly when the actuators and sensors are collocated.
Consequently, if a feedback controller is designed for the truncated model, and then
implemented on the real system, the performance and stability of the closed loop system
could be adversely affected [1].
Normally PPF techniques which we talked before do not allow for a feedthrough term in the
plant model. However, it is essential to include this in the design phase, if the implemented
70
controller is to perform in a satisfactory manner. Closedloop performance of a controlled
system is highly dependent on openloop zeros of the plant. Inclusion of a feedthrough term
in the model ensures that closedloop performance of the controller, once implemented, is in
close agreement with theoretical predictions [3]. The derivation below are from [1,133,124].
If we are concerned with designing highperformance feedback controllers for multivariable
resonant systems of the form
G(s)=
i
i
s
2
+ 2
i
i
s +
i2
Mi=1
(5.28)
where
i
is an mx1 vector, and M
. In practice, however, the integer is finite, but
possibly a very large number which represents the number of modes that sufficiently describe
the elastic properties of the structure under excitation
Figure 5.3 Feedback control system associated with a flexible structure with
collocated actuator/sensor pairs, and subject to disturbance w
For a system of the form (5.41), a positive position feedback controller is defined as
K
pp
(s) =

i
i
s
2
+ 2
i
i
s +
i2
Ni=1
(5.29)
where
i
mx1
for i = 1, 2, ..., N.
where
i
R
mx1
for i = 1, 2, ...,
N
As illustrated in Fig. 5.3, due to the existence of the negative sign in all terms of (5.35), the
overall system resembles a positive feedback loop. Also, the transfer function matrix (5.34) is
71
similar to that of the force to displacement transfer function matrix associated with a flexible
structure, hence, the terminology positive position feedback.
An important property of PPF controllers is that to suppress one vibration mode requires only
one secondorder term, as articulated in (5.35). Consequently, one can choose the modes,
within a specific bandwidth, that are to be controlled and construct the necessary controller.
To derive stability conditions for this control loop, the series in (5.36) is first truncated by
keeping the first N modes(N
< M) that lie within the bandwidth of interest, and then
incorporating the effect of truncated modes by adding a feedthrough term to the truncated
model. That is, to approximate (5.35) by
G
N
(s)=
i
i
s
2
+ 2
i
i
s +
i2
Ni=1
+ D
(5.30)
To derive stability of (5.43) under (5.42), the following theorem is needed.
Theorem 5.5: Consider the following secondorder multivariable dynamical system:
x(t) + D x(t) + K x(t)= 0 (5.31)
where D, K
R
NxN
and x
R
Nx1
. Furthermore, assume that D = D'
> 0. Then (5.37) is
exponentially stable if and only if K = K'
> 0. Proof same as Theorem 5.3.
The following theorem gives the necessary and sufficient conditions for closedloop stability
under positive position feedback. First, we need to make the following definitions:
Z =
1
2
N
=
1
2
N
= [
1
2
...
N
]
and
=
1
2
N
=
1
2
N
= [
1
2
...
N
]
72
Furthermore, we assume that
> 0
(5.32)
Theorem 5.6: The negative feedback connection of (5.43) and (5.42) with (5.45) is
exponentially stable if and only if
2
 'D > 0
(5.33)
and
2
'(
2
D)
1
> 0
(5.34)
5.4 Multimode SISO and MIMO PPF Controller Parameter
Selection
5.4.1 MATLAB Optimization toolbox and GA Optimization Search
In order to find the minimum value of
H
norm, we need to use MATLAB Optimization
Toolbox, which provides widely used algorithms for standard and largescale optimization.
These algorithms solve constrained and unconstrained continuous and discrete problems. The
toolbox includes functions for linear programming, quadratic programming, binary integer
programming, nonlinear optimization, nonlinear least squares, systems of nonlinear equations,
and multiobjective optimization. We can use them to find optimal solutions, perform tradeoff
analyses, balance multiple design alternatives, and incorporate optimization methods into
algorithms and models [140].
Key Features[140]:
i.
Interactive tools for defining and solving optimization problems and monitoring solution
progress
ii.
Solvers for nonlinear and multiobjective optimization
iii.
Solvers for nonlinear least squares, data fitting, and nonlinear equations
iv.
Methods for solving quadratic and linear programming problems
v.
Methods for solving binary integer programming problems
In the toolbox, it contains several functions, such as GlobalSearch, MultiStart, Genetic
Algorithm, Direct Search, Simulated Annealing. For Genetic Algorithm, it is a method for
73
solving both constrained and unconstrained optimization problems that is based on natural
selection, the process that drives biological evolution. The genetic algorithm repeatedly
modifies a population of individual solutions. At each step, the genetic algorithm selects
individuals at random from the current population to be parents and uses them to produce the
children for the next generation. Over successive generations, the population "evolves"
toward an optimal solution. Genetic algorithm can be applied to solve a variety of
optimization problems that are not well suited for standard optimization algorithms, including
problems in which the objective function is discontinuous, nondifferentiable, stochastic, or
highly nonlinear. The genetic algorithm can address problems of mixed integer programming,
where some components are restricted to be integervalued [140].
The genetic algorithm uses three main types of rules at each step to create the next generation
from the current population [140]:
i.
Selection rules select the individuals, called parents, that contribute to the population at
the next generation.
ii.
Crossover rules combine two parents to form children for the next generation.
iii.
Mutation rules apply random changes to individual parents to form children
The genetic algorithm differs from a classical, derivativebased, optimization algorithm in
two main ways in the following table 5.1 [140]:
Classical Algorithm
Genetic Algorithm
Generates a single point at each iteration.
The sequence of points approaches an
optimal solution.
Generates a population of points at each
iteration. The best point in the population
approaches an optimal solution.
Selects the next point in the sequence by a
deterministic computation.
Selects the next population by computation
which uses random number generators.
Table 5.1 classical algorithm and genetic algorithm comparison
Because the limited space, more details about Genetic Algorithm optimization toolbox please
refer to MATLAB user guide.
5.4.2 Multimode SISO PPF Controller Optimal Parameter Selection
Based the dynamics model between shaker and plate, the results of multimode SISO PPF
74
controller optimal parameters are shown below.
Figure 5.4 multimode SISO PPF controller K11 optimal parameter result through
GA search
Figure 5.5 multimode SISO PPF controller K22 optimal parameter result through
GA search
75
Figure 5.6 multimode SISO PPF controller K33 optimal parameter result through
GA search
Mode 1
Mode 2
Mode 3
Mode 4
Controller K11
control
gain
g1=0.981
g2=0.033
g3=0.016
g4=0.099
damping
zc1=0.029
zc2=0.953
zc3=0.807
zc4=0.987
Controller K22
control
gain
g5=0.997
g6=0.979
g7=0.915
g8=0.962
damping
zc5=0.005
zc6=0.004
zc7=0.441
zc8=0.72
Controller K33
control
gain
g9=0.993
g10=0.462
g11=0.6
g12=0.816
damping
zc9=0.006
zc10=0.356
zc11=0.724
zc12=0.673
Table 5.2 multimode SISO PPF controller optimal parameter result
76
5.4.3 Multimode MIMO PPF Controller Optimal Parameter Selection
Based the dynamics model between shaker and plate, the results of multimode MIMO PPF
controller optimal parameters are shown below.
Figure 5.7 multimode MIMO Controller optimal parameter result through GA
search
Mode 1
Mode 2
Mode 3
Mode 4
Controller 1
control
gain
g1=0.709
g2=0.861
g3=0.674
g4=0.776
damping
zc1=0. 538
zc2=0. 437
zc3=0. 11
zc4=0.703
Controller 2
control
gain
g5=0.791
g6=0.892
g7=0.981
g8=0.327
damping
zc5=0.004
zc6=0.888
zc7=0.818
zc8=0.053
Controller 3
control
g9=0.722
g10=0.974
g11=0.778
g12=0.872
77
gain
damping
zc9=0.683
zc10=0.005
zc11=0.146
zc12=0.596
Table 5.3 multimode MIMO PPF controller optimal parameter result
5.5 Summary
In this chapter, we reported the structure of PPF controller, derived the stability of
scalar case, multivariate case controller and multivariate case controller with
feedthrough plant. We gave the selection method of parameters of controller.
According to MATLAB simulation, we get the final results of controller parameters
which can be used in the next chapter.
78
Chapter 6
Simulation
In the following simulation studies, the resonant multimode SISO and MIMO PPF
controller are applied to control the plate vibration when applying a disturbance sine
wave signal at shaker.
Based the method in Chapter 2, 3 and 4, we can easily derive the dynamics correction
model from shaker to transducer.
The objectives of the simulation study of the PPF control method are to demonstrate
that:
1. PPF control is able to attenuate multimode vibration using only a single
sensoractuator pair(explained before in this experiment test, sensor and actuator is
the same one).
2. PPF control has an independent characteristic, in the sense that the controller is able
to control a particular mode without destabilising other modes.
6.1 Multimode Three SISO PPF Controller Simulation
Three SISO PPF controller are used to control the first four vibration modes of
simulation plate structure model using MATLAB. Firstly, simulation plate structure
model and three SISO PPF controller are connected as feedback closedloop system.
Secondly, dynamics correction model from shaker to transducer is connected to
feedback closedloop system as in Fig2.21. Thirdly, the parameters of the controller
are chosen as Tab 5.2. Fourthly, sinusoid signals which contain resonant frequencies
of first four modes of simulation plate structure model are applied to the shaker.
Fifthly, Openloop dynamics which is between shaker to plate without controller is
compared to closedloop dynamics which is between shaker to plate with controller.
Time domain and frequency domain cases are designed to test the ability of three
79
SISO PPF controller to attenuate multimode vibration when the controller centre
frequencies match the resonant frequencies of the plate.
The simulation results show that the openloop and closedloop dynamics between
shaker and plate.
Figure 6.1 SISO vibration control at transducer 1 (K11) openloop and closedloop
impulse signal simulation result
Figure 6.2 SISO vibration control at transducer 2 (K22) openloop and closedloop
impulse signal simulation result
80
Figure 6.3 SISO vibration control at transducer 3 (K33)openloop and closedloop
impulse signal simulation result
Figure 6.4 SISO vibration control at transducer 1 (K11) openloop and closedloop
step signal simulation result
81
Figure 6.5 SISO vibration control at transducer 2 (K22) openloop and closedloop
step signal simulation result
Figure 6.6 SISO vibration control at transducer 3 (K33) openloop and closedloop
step signal simulation result
82
Figure 6.7 three SISO controller openloop and closedloop simulation result
Figure 6.8 SISO vibration control at transducer 1 (K11) openloop and closedloop
simulation result (1)
83
Figure 6.9 SISO vibration control at transducer 1(K11) openloop and closedloop
simulation result (2)
Figure 6.10 SISO vibration control at transducer 2 (K22) openloop and closedloop
simulation result (1)
84
Figure 6.11 SISO vibration control at transducer 2 (K22) openloop and closedloop
simulation result (2)
Figure 6.12 SISO vibration control at transducer 3 (K33) openloop and closedloop
simulation result (1)
85
Figure 6.13 SISO vibration control at transducer 3 (K33) openloop and closedloop
simulation result (2)
transducer 1
Mode 1
(rad/s) Peak
(dB)
Attenuation
(dB)
Gain
Margin(dB)
Close
loop
Stability
Open
loop
171
(27.1Hz)
16.3
34.3
16.4
stable
Close
loop
171
(24.5Hz)
50.6
86
Table 6.1 SISO PPF closedloop frequency domain result at transducer 1
Mode 2
(rad/s) Peak
(dB)
Attenuation
(dB)
Gain
Margin(dB)
Close
loop
Stability
Open
loop
220
(35.01Hz)
47.3
0.1
stable
Close
loop
220
(35.01Hz)
47.4
47.4
transducer 2
Mode 1
(rad/s) Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
Open
loop
171
(27.1Hz)
24.6
32.9
24.6
stable
Close
loop
156
(25.6Hz)
57.5
68.7
(171Hz)
Mode 2
(rad/s) Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
87
Table 6.2 SISO PPF closedloop frequency domain result at transducer 2
Open
loop
220
(35.01Hz)
57.3
11.6
stable
Closeloop
217
(35.01Hz)
68.9
Mode 4
(rad/s) Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
Openloop
290
(46.2Hz)
66.1
3.9
stable
Closeloop
297
(47.27Hz)
70
transducer 3
Mode 1
(rad/s) Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Closeloop
Stability
Open
loop
171
(27.1Hz)
23.6
36.5
23.3
stable
Close
loop
155
(24.67Hz)
60.1
88
Table 6.3 SISO PPF closedloop frequency domain result at transducer 3
Mode 2
(rad/s) Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Stability
Open
loop
220
(35.01Hz)
58.1
0.8
stable
Close
loop
220
(35.01Hz)
58.9
60.6
Mode 4
(rad/s) Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Stability
Open
loop
290
(46.2Hz)
67.9
4.5
stable
Close
loop
297
(47.27Hz)
72.4
89
6.2 Multimode MIMO PPF Controller Simulation
MIMO PPF controller are used to control the first four vibration modes of simulation
plate structure model using MATLAB. Firstly, simulation plate structure model and
MIMO PPF controller are connected as feedback closedloop system. Secondly,
dynamics correction model from shaker to transducer is connected to feedback
closedloop system as in Fig2.21. Thirdly, the parameters of the controller are chosen
as Tab 5.3. Fourthly, sinusoid signals which contain resonant frequencies of first four
modes of simulation plate structure model are applied to the shaker. Fifthly,
Openloop dynamics which is between shaker to plate without controller is compared
to closedloop dynamics which is between shaker to plate with controller. Time
domain and frequency domain cases are designed to test the ability of MIMO PPF
controller to attenuate multimode vibration when the controller centre frequencies
match the resonant frequencies of the plate.
The simulation results show that the openloop and closedloop dynamics between
shaker and plate.
90
Figure 6.14 MIMO Controller openloop and closedloop (Gwo(1,1),Gwc(1,1))
impulse signal simulation result at transducer 1
Figure 6.15 MIMO Controller openloop and closedloop (Gwo(2,1), Gwc(2,1))
impulse signal simulation result at transducer 2
91
Figure 6.16 MIMO Controller openloop and closedloop (Gwo(3,1), Gwc(3,1))
impulse signal simulation result at transducer 3
Figure 6.17 MIMO Controller openloop and closedloop (Gwo(1,1), Gwc(1,1)) step
signal simulation result at transducer 1
92
Figure 6.18 MIMO Controller openloop and closedloop (Gwo(2,1), Gwc(2,1)) step
signal simulation result at transducer 2
Figure 6.19 MIMO Controller openloop and closedloop (Gwo(3,1), Gwc(3,1)) step
signal simulation result at transducer 3
93
Figure 6.20 MIMO Controller openloop and closedloop simulation result
Figure 6.21 MIMO Controller openloop and closedloop (Gwo(1,1),
Gwc(1,1))simulation result (1) at transducer 1
94
Figure 6.22 MIMO Controller openloop and closedloop (Gwo(1,1),
Gwc(1,1))simulation result (2) at transducer 1
Figure 6.23 MIMO Controller openloop and closedloop (Gwo(2,1),
Gwc(2,1))simulation result (1) at transducer 2
95
Figure 6.24 MIMO Controller openloop and closedloop (Gwo(2,1),
Gwc(2,1))simulation result (2) at transducer 2
Figure 6.25 MIMO Controller openloop and closedloop (Gwo(3,1),
Gwc(3,1))simulation result (1) at transducer 3
96
Figure 6.26 MIMO Controller openloop and closedloop (Gwo(3,1),
Gwc(3,1))simulation result (2) at transducer 3
transducer
1
Mode 1
(rad/s)
Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
Open
loop
171
(27.1Hz)
9.83
34.37
9.86
stable
Close
loop
172
(27.2Hz)
44.2
Mode 2
(rad/s) Peak Attenuation
Mini
Stability
Close
loop
97
Table 6.4 MIMO PPF closedloop frequency domain result at transducer 1
(dB)
(dB)
Margin(dB) Stability
Open
loop
220
(35.01Hz)
43.5
5.8
stable
Closeloop
220
(35.01Hz)
49.7
44.7
(172 rad/s )
transducer
2
Mode 1
(rad/s)
Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
Open
loop
171
(27.1Hz)
12.6
36.8
12.6
stable
Close
loop
173
(27.3Hz)
49.4
49
(172 rad/s )
Mode 2
(rad/s)
Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
Open
loop
220
(35.01Hz)
45
9.2
Closeloop
220
54.2
98
Table 6.5 MIMO PPF closedloop frequency domain result at transducer 2
(35.01Hz)
Mode 4
(rad/s)
Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
Openloop
290
60.4
10.4
Closeloop
308
70.8
Transduce
r 3
Mode 1
(rad/s)
Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
Open
loop
171
(27.1Hz)
12.1
36.6
12.2
stable
Close
loop
173
(27.3Hz)
48.7
48.3
(172 rad/s )
Mode 2
(rad/s)
Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
Open
220
44.8
8.7
99
Table 6.6 MIMO PPF closedloop frequency domain result at transducer 3
6.3 Summary
As explained in Chapter 2, at transducer 1, the mode shape for mode 2 and mode 4 is
too small, but at mode 2, the mode shape of shaker is very big, so only mode 1 and
mode 2 can be seen in the dynamics between shake and transducer 1 in the figure after
superposition. Similar at transducer 2 and 3, only mode 1, 2, 4 can be seen after
superposition.
For multimode three SISO PPF controller, according to compare the openloop and
closedloop dynamics between shaker and plate, the result can be seen through Figure
6.16.13 and Table 6.16.3 which three SISO PPF controller achieved. It is shown a
good result at mode 1 for all of three controllers, but at mode 2 and mode 4 only a
little bit attenuation.
For multimode MIMO PPF controller, according to compare the openloop and
loop
(35.01Hz)
Closeloop
220
(35.01Hz)
53.5
Mode 4
(rad/s)
Peak
(dB)
Attenuation
(dB)
Mini
Stability
Margin(dB)
Close
loop
Stability
Openloop
290
(46.2Hz)
61.3
10.1
Closeloop
308
(49.02Hz)
71.4
100
closedloop dynamics between shaker and plate, the result can be seen through Figure
6.146.26 and Table 6.46.6 which MIMO PPF controller achieved. It is shown a
better result at mode 1to mode 4 comparing with multimode three SISO PPF
controller.
101
Chapter 7
Experiment
Experimental studies are used to verify the results of the simulation studies. The
proposed PPF SISO and MIMO resonant controllers are implemented on a dSpace
DS1103 data acquisition and control board using MATLAB, Simulink and RealTime
Workshop software.
7.1 Selfsensing
To measure the backemf voltage without using any additional sensors, a technique
called 'selfsensing' was used [
126
]. The transducer is then used for actuation and
sensing at the same time, which means that actuation and sensing are performed
collocated. Figure 7.1 shows an electrical model of an electromagnetic transducer in
series with a measurement resistor
R
m
[139].
Figure 7.1 principle of the selfsensing technique used to measure the backemf
voltage
V
emf
From this figure, it can be seen that:
V
t
(t) =
V
Zi
+
V
emf
(7.1)
102
V
emf
is now easily found:
V
emf
(t) =
V
t
(t)
 V
Zi
=
V
t
(t)
 Z
i
I
s
(t) =
V
t
(t)
Z
i
V
Rm
(t)R
m
(7.2)
This is shown in a block diagram in Figure 7.2. When implemented in MATLAB
Simulink this time domain block diagram is converted to the frequency domain.
Figure 7.2 block diagram for the calculation of the backemf voltage
The internal transducer impedance
Z
i
is known as this was characterised (Chapter 2.2)
and because
R
m
is known, the backemf voltage can be calculated by measuring the
voltage across the transducer
V
t
(t) and across the measurement resistor
V
R
m
(t). A 1
measurement resistor with a power rating of 5 W was chosen to be used in series with
the transducer such that the current owing through the transducer would not be limited
severely and thereby also limiting the input voltage
V
p
(t) (Figure 7.2) necessary for
applying a certain voltage
V
t
(t) across the transducer. Three of these block diagrams
are placed in the `Self sensing' block to accomodate for the three control transducers
[139].
7.2 Electronics
7.2.1 dSpace
T
o be able to acquire the backemf voltage and to output the reference transducer
voltage, a dSpace DS1103 Controller Board was used to interface with the system. This
board provides 16 16bit analogtodigital (AD) converters, 4 12bit AD converters and
8 14bit digitaltoanalog (DA) converters. When running a simulation externally in
MATLAB Simulink, the onboard DSP is used for the signal processing and acquisition
103
and output of the various signals via its AD and DA converters. The continuous model
in Simulink is then discretized and run at a sampling rate of 10 kHz, which was found to
be sufficient by looking at the representation frequency sinusoidal signal. A sampling
rate of 1 kHz would show a staircase approximation of the same sinusoid, which is
undesired. For a list of the AD and DA converters used, refer to Table 7.2. Only the 16
bit AD converters were used instead of the 12bit, to minimize the quantisation error
[139].
Table 7.2 AD and DA converters on the dSpace DS1103 used for signal acquisition
and reference voltage output
A lowpass antialiasing and reconstruction filters with cutoff frequencies of 10 kHz
were added to the system.
7.2.2 Interfacing circuits
The MATLAB Simulink model needs the transducer and measurement resistor voltages
to calculate the backemf and a control signal. Subsequently, the output voltage needs
to be put across the transducer. Figure 7.3 shows a block diagram of one of the interface
circuits used to power the control transducers as well as to measure the transducer and
measurement resistor voltage. Because there are three control transducers, this circuit
was built in triplicate. In this figure,
U
1
represents the OPA548 power operational
amplifier and
U
2
and
U
3
represent the generic uA741 operational amplifier
configured as a differential amplifier with a gain of 1.
R
m
indicates a measurement
resistor used for calculating the current owing through the transducer and is specific for
each circuit, because the self sensing measurement depends on an accurate value of this
resistance. Table 7.3 lists the different resistor values [139].
104
Table 7.3 specific measurement resistor value for each interface circuit
Figure 7.3 The interface circuit used to power the control transducer and to measure
its voltage and the measurement resistor voltage for the purpose of self sensing
The transducer voltage
V
t
(t) measured across the transducer which is represented by
Z
L
and measurement resistor voltage
V
R
m
(t) are acquired by the dSpace AD
converters and after calculating the backemf voltage
V
emf
, the control signal is output
by the dSpace DA converter. This is the reference voltage which is placed across the
transducer, using the negative feedback loop via U2. Simple first order lowpass filters
(LPF) with a cutoff frequency of 1500 Hz (using a 1 k
resistor and 100 nF capacitor)
were placed at the input of the transducer voltage measurement and after the dSpace
output to reduce high frequency noise. The
V
R
m
input was not filtered as this was not
necessary and even deteriorated the signal [139].
7.2.3 Additional electronics
To induce vibrations in the system, the disturbance transducer was powered by a 50 W
105
Jaycar amplifier kit. A sinusoid signal was used as input signal, which frequency was
chosen from 20 to 50 Hz.
7.3 Multimode SISO PPF Controller Experiment
Implemented Result
Three SISO PPF controller are used to control the first four vibration modes of
simulation plate structure model. Firstly, Three SISO PPF controller are formed in
MATLAB Simulink. Secondly, sinusoid signals which contain resonant frequencies of
first four modes of simulation plate structure model are applied to the shaker one by
one. Thirdly, Openloop dynamics which is between shaker to plate without controller
is compared to closedloop dynamics which is between shaker to plate with controller.
Time domain cases are designed to test the ability of three SISO PPF controller to
attenuate multimode vibration when the controller centre frequencies match the
resonant frequencies of the plate. The following is the multimode three SISO
vibration control experiment results for every mode at each transducer.
106
Figure 7.4 SISO vibration control at transducer 1 first mode 27.1 Hz before control
Figure 7.5 SISO vibration control at transducer 1 first mode 27.1 Hz after control
107
Figure 7.6 SISO vibration control at transducer 1 second mode 34.4 Hz before control
Figure 7.7 SISO vibration control at transducer 1 second mode 34.4 Hz after control
108
Figure 7.8 SISO vibration control at transducer 1 third mode 40.5 Hz before control
Figure 7.9 SISO vibration control at transducer 1 third mode 40.5 Hz SISO after
control
109
Figure 7.10 SISO vibration control at transducer 1 forth mode 49.2 Hz before control
Figure 7.11 SISO vibration control at transducer 1 forth mode 49.2 Hz after control
110
Figure 7.12 SISO vibration control at transducer 2 first mode 27.1Hz before control
Figure 7.13 SISO vibration control at transducer 2 first mode 27.1Hz SISO after
control
111
Figure 7.14 SISO vibration control at transducer 2 second mode 34.4 Hz before
control
Figure 7.15 SISO vibration control at transducer 2 second mode 34.4 Hz after control
112
Figure 7.16 SISO vibration control at transducer 2 third mode 40.5 Hz before control
Figure 7.17 SISO vibration control at transducer 2 third mode 40.5 Hz after control
113
Figure 7.18 SISO vibration control at transducer 2 forth mode 49.2 Hz before control
Figure 7.19 SISO vibration control at transducer 2 forth mode 49.2 Hz after control
114
Figure 7.20 SISO vibration control at transducer 3 first mode 27.1Hz before control
Figure 7.21
SISO vibration control at transducer 3 first mode 27.1Hz after control
115
Figure 7.22 SISO vibration control at transducer 3 second mode 34.4 Hz before
control
Figure 7.23 SISO vibration control at transducer 3 second mode 34.4 Hz after control
116
Figure 7.24 SISO vibration control at transducer 3 third mode 40.5 Hz before control
Figure 7.25 SISO vibration control at transducer 3 third mode 40.5 Hz after control
117
Figure 7.26 SISO vibration control at transducer 3 forth mode 49.2 Hz before control
Figure 7.27 SISO vibration control at transducer 3 forth mode 49.2 Hz after control
Sinusoid signal which is swept from 20 Hz to 50 Hz is applied to the shaker, sampling
time is 0.1s. Openloop dynamics which is between shaker to plate without controller
is compared to closedloop dynamics which is between shaker to plate with controller.
118
Frequency domain cases are designed to test the ability of three SISO PPF controller
to attenuate multimode vibration when the controller centre frequencies match the
resonant frequencies of the plate . The following is the multimode three SISO
vibration control experiment result at each transducer.
Figure 7.28 SISO vibration control for sweep signal at transducer 1 before control
Figure 7.29 SISO vibration control for sweep signal at transducer 1 after control
119
Figure 7.30 SISO vibration control for sweep signal at transducer 2 before control
Figure 7.31 SISO vibration control for sweep signal at transducer 2 after control
120
Figure 7.32 SISO vibration control for sweep signal at transducer 3 before control
.
Figure 7.33 SISO vibration control for sweep signal at transducer 3 after control
121
The final experimental results for three SISO PPF controller are given in the Table
7.4.
frequency (Hz)
27.1
34.4
40.5
49.2
transduer 1
Acceleration (g)
before
1.44
1.53
0.77
5.06
after
1.28
1.2
0.75
4.44
reduce rate (%)
11.11% 21.569% 2.5974% 12.253%
Max Peak (dB)
before
6.7
6.0
13.7
4.8
after
7.6
8.2
13.7
3.4
reduce(dB)
0.9
2.2
0
1.4
transduer 2
Acceleration (g)
before
0.388
2.75
2.47
0.4
after
0.25
2.16
2.09
0.325
reduce rate(%)
35.57% 21.45% 15.38% 18.75%
Max Peak (dB)
before
20.3
0.8
1.7
22.7
after
26.1
2.9
3.1
26.5
Reduce (dB)
5.8
2.1
1.4
3.8
transduer 3
Acceleration (g)
before
0.381
2.28
2.28
0.556
after
0.381
1.72
1.91
0.45
reduce rate(%)
0
24.561% 16.228% 19.065%
Max Peak (dB)
before
20.4
2.6
2.5
19
after
19.3
4.9
4.2
21.5
Reduce (dB)
1.1
2.3
1.7
2.5
Table 7.4 SISO vibration control experimental results
The parameters of the three SISO PPF controller are given in Table 7.5.
Mode 1
Mode 2
Mode 3
Mode 4
Controller
control
g1=6.9
g2=2.19E02
g3=1.54E03
g4=1.05E03
122
K11
gain
damping
zc1=5.87E02 zc2=0.351
zc3=0.295
zc4=0.243
Controller
K22
control
gain
g5=0.345
g6=6.58E02
g7=1.54E03
g8=1.05E03
damping
zc5=5.87E02 zc6=3.51E01 zc7=2.95E01 zc8=2.43E01
Controller
K33
control
gain
g9=3.45E01 g10=2.19E01 g11=1.54E03 g12=1.05E03
damping
zc9=5.87E02 zc10=3.51E01 zc11=2.95E01 zc12=2.43E01
Table 7.5 multimode SISO PPF controller experimental parameter result
7.4 Multimode MIMO PPF Controller Experimental
Implemented Result
MIMO PPF controller are used to control the first four vibration modes of simulation
plate structure model. Firstly, MIMO PPF controller are formed in MATLAB
Simulink. Secondly, sinusoid signals which contain resonant frequencies of first four
modes of simulation plate structure model are applied to the shaker one by one.
Thirdly, Openloop dynamics which is between shaker to plate without controller is
compared to closedloop dynamics which is between shaker to plate with controller.
Time domain cases are designed to test the ability of MIMO PPF controller to
attenuate multimode vibration when the controller centre frequencies match the
resonant frequencies of the plate. The following is the multimode MIMO PPF
vibration control experiment results.
123
Mode 1
Mode 2
Mode 3
Mode 4
Controller 1
control
gain
g1=0.103E
g2=6.58
g3=1.54E03
g4=1.06
damping
zc1=0.006
zc2=0.818
zc3=0.987
zc4=0.013
Controller 2
control
gain
g5=2.94E04 g6=2.34E02
g7=1.96E02
g8=1.62E02
damping
zc5=0.001
zc6=0.668
zc7=0.029
zc8=0.086
Controller 3
control
gain
g9=1.47E02 g10=2.34E02 g11=3.93E02 g12=1.62E02
damping
zc9=2.94E02 zc10=4.68E02 zc11=1.96E0 zc12=1.62E02
Table 7.6 multimode MIMO PPF controller parameter result
124
Figure 7.34 MIMO vibration control at transducer 1 first mode 27.1 Hz before control
Figure 7.35 MIMO vibration control at transducer 1 first mode 27.1 Hz after control
125
Figure 7.36 MIMO vibration control at transducer 1 second mode 34.4 Hz before
control
Figure 7.37 MIMO vibration control at transducer 1 second mode 34.4 Hz after
control
126
Figure 7.38 MIMO vibration control at transducer 1 third mode 40.5 Hz before
control
Figure 7.39 MIMO vibration control at transducer 1 third mode 40.5 Hz after control
127
Figure 7.40 MIMO vibration control at transducer 1 forth mode 49.2 Hz before
control
Figure 7.41 MIMO vibration control at transducer 1 forth mode 49.2 Hz after control
128
Figure 7.42 MIMO vibration control at transducer 2 first mode 27.1Hz before control
Figure 7.43 MIMO vibration control at transducer 2 first mode 27.1Hz after control
129
Figure 7.44 MIMO vibration control at transducer 2 second mode 34.4 Hz before
control
Figure 7.45 MIMO vibration control at transducer 2 second mode 34.4 Hz after
control
130
Figure 7.46 MIMO vibration control at transducer 2 third mode 40.5 Hz before
control
Figure 7.47 MIMO vibration control at transducer 2 third mode 40.5 Hz after control
131
Figure 7.48 MIMO vibration control at transducer 2 forth mode 49.2 Hz before
control
Figure 7.49 MIMO vibration control at transducer 2 forth mode 49.2 Hz after control
132
Figure 7.50 MIMO vibration control at transducer 3 first mode 27.1Hz before control
Figure 7.51 MIMO vibration control at transducer 3 first mode 27.1Hz after control
133
Figure 7.52 MIMO vibration control at transducer 3 second mode 34.4 Hz before
control
Figure 7.53 MIMO vibration control at transducer 3 second mode 34.4 Hz after
control
134
Figure 7.54 MIMO vibration control at transducer 3 third mode 40.5 Hz before
control
Figure 7.55 MIMO vibration control at transducer 3 third mode 40.5 Hz after control
135
Figure 7.56 MIMO vibration control at transducer 3 forth mode 49.2 Hz before
control
Figure 7.57 MIMO vibration control at transducer 3 forth mode 49.2 Hz after control
Sinusoid signal which is swept from 20 Hz to 50 Hz is applied to the shaker, sampling
time is 0.1s. Openloop dynamics which is between shaker to plate without controller
is compared to closedloop dynamics which is between shaker to plate with controller.
Frequency domain cases are designed to test the ability of MIMO controller to
attenuate multimode vibration when the controller centre frequencies match the
136
resonant frequencies of the plate . The following is the multimode MIMO vibration
control experiment result at each transducer.
Figure 7.58 MIMO vibration control for sweep signal at transducer 1 before control
Figure 7.59 MIMO vibration control for sweep signal at transducer 1 after control
137
Figure 7.60 MIMO vibration control for sweep signal at transducer 2 before control
Figure 7.61 MIMO vibration control for sweep signal at transducer 2 after control
138
Figure 7.63 MIMO vibration control for sweep signal at transducer 3 before control
Figure 7.63 MIMO vibration control for sweep signal at transducer 3 after control
139
The parameters of the MIMO PPF controller are given in Table 7.7.
frequency (Hz)
27.1
34.4
40.5
49.2
transduer 1
Acceleration (g)
before
1.44
1.56
0.7
4.81
after
1.16
1.13
0.45
4.13
reduce rate (%)
19.444% 27.64% 35.714% 14.137%
Max Peak (dB)
before
6.6
6.2
13.9
4.2
after
8.6
8.9
17.8
2.7
reduce(dB)
2
2.7
3.9
1.5
transduer 2
Acceleration (g)
before
0.294
2.75
2.16
0.35
after
0.194
2.22
0.88
0.287
reduce rate(%)
34.01%
19.27% 59.26%
18%
Max Peak (dB)
before
22.6
0.8
3.1
22.5
after
27.6
2.8
11.5
23.5
Reduce (dB)
5
2
7.6
1
transduer 3
Acceleration (g)
before
0.388
2.34
1.84
0.55
after
0.313
1.56
0.91
0.39
reduce rate(%)
19.33% 33.333% 50.543% 29.091%
Max Peak (dB)
before
20.3
2.3
4.3
20.4
after
21.3
5.7
11.2
22.4
Reduce (dB)
1
3.4
6.9
2
Table 7.7 multimode MIMO vibration control experimental results
7.5 Summary
Although we try to derive the mathematical model which has the same dynamics
response as the experimental plant, the parameters of the multimode SISO and
MIMO PPF controllers in the experimental study are still different from the ones in
simulation study because during modelling, we simplified the structure of transducer
which is more complicated with electromagnetic and nonlinear characters in the fact.
140
For multimode three SISO PPF controller, the openloop and closedloop time
domain and frequency domain results can be seen through Figure 7.47.33 and Table
7.4, 7.5.
It can be seen in the figure that at the beginning of turning on controller, there was a
noise happened.
From a mathematical point of view, when the control is off, the transfer function is
simply that of a monic vibrating system subjected to an arbitrary impulse. When the
control comes on, the transfer function that describes the inputoutput relationship
between the sensor and actuator takes over, however, its denominator remains the
same, while its numerator will be altered [64].
Physically, this is due to the fact that for the collocated sensor/actuator pair (in the
experiment, sensor and actuator is the same one) there will be some direct energy
transmission from the actuator to the sensor. Thus the sensor voltage is made up of
two components, some of the response due to the actuator, and some direct
transmission of the force due to the collocation of the actuator [64].
However, this is not a problem for the PPF control law, as it was proven to be stable
even in the presence control law of feed through in Chapter 5.3.
For sweep signal multimode three SISO vibration control result, it did not show the
attenuation at each mode as in simulation.
For multimode MIMO PPF controller, the openloop and closedloop results can be
seen through Figure 7.347.63 and Table 7.5, 7.6. It is shown a better result
comparing with multimode three SISO PPF controller and there is no noise happened
at the beginning of turning on controller.
For the first mode, the performance of multimode MIMO PPF controller is not good,
because there is only one transducer (transducer 1) at that side, but there are two
degrees of freedom at the corners (bending and torsion which could be seen in the
ANSYS result Fig 2.13), one transducer is not enough to control two degrees of
freedom vibration at the same time.
141
Same with mode 3 and mode 4, the performance of multimode MIMO controllers
was decreased because only one transducer is not enough to control two degrees of
freedom vibration at the same time.
For sweep signal multimode MIMO vibration control result, due to the limitation of
oscilloscope which can only read the max(M) result, it could get the exactly
attenuation result at each mode, but the attenuation at each mode still can be seen
from the figure.
142
Chapter 8
Conclusion and Future Work
8.1 Outcomes of the Research
Because the increasing demand for improving dynamic performance, the operating
efficiency, and the amount of material which is used in mechanical structures, many
designers employ lightweight materials to reduce the cross sectional dimensions of
those structures.
Due to the sideeffect of employing lightweight materials and reducing the cross
sectional dimensions, the structures become more flexible and more susceptible to the
detrimental effects of unwanted vibration, particularly when they operate at or near
their natural frequencies or when they are excited by disturbances that coincide with
their natural frequencies.
The design and implementation of a high performance vibration controller for a flexible
structure can be a so important task. In order to complete the task and achieve better
performance, in this thesis, the author did the following things:
Firstly, based on experimental, analytical and numerical three kinds of analysis
methods, an infinitedimensional and a very highorder mathematical transfer
function model for distributed parameter plate structure is derived, using modal
analysis and numerical analysis results. A 47 modes plate model was given as the
simulation model.
Secondly, traditional balanced model reduction method is used to decrease the
dimensions and orders of the plate simulation model.
Thirdly, a four modes feedthrough truncated plate model which minimizing the
effect of other truncated modes on spatial lowfrequency dynamics of the system
by adding a spatial zero frequency term to the truncated model is provided.
143
Numerical software MATLAB is used to compare the feedthrough truncated
plate model with balanced reduction plate model.
Fourthly, both multimode SISO and MIMO PPF active control laws based upon
positive position feedback is proposed, developed and studied for the flexible
plate structure with bonded three selfsensing magnetic transducers which
guarantee unconditional stability of the closedloop system similar as collocated
control system. The proposed multimode SISO and MIMO PPF controllers can
be tuned to the chosen first four modes of the plate structure and increased the
damping of the plate so as to minimize the chosen resonant responses. Stability
conditions for scalar and multivariable case of PPF controllers are derived,
multivariable case which allow for a feedthrough term in the model is also
derived, which is needed to ensure little perturbation in the inbandwidth zeros of
the model. In order to derive optimal controller parameters, a minimization
criterion based on the
H
norm of the closedloop system is solved by a genetic
algorithm.
Fifthly, the proposed multimode MIMO PPF controller, compared with
multimode SISO PPF controller, which is validated in both simulations and
experimental implementation, is an effective solution to suppress the vibration of
distributed parameter plate structure.
According to literature review, the proposed multimode active vibration control
methodology for plate structure using
H
optimization MIMO positive position
feedback based genetic algorithm is the first time, similar method only can be found
such as SISO PPF using
H
optimization based on genetic algorithm applied on
beam structure[65], MIMO PPF using
H
optimization and pole optimization
applied on beam structure[1, 3] .
8.2 Future Work
Within the scope of the research project documented herein, the development and
implementation of multimode active vibration control methodology for plate
structure using
H
optimization positive position feedback based genetic algorithm
is focused on an MIMO basis. Due to the limited time and other restrained conditions,
some parts of the experimental study may need to improve in the future.
144
Firstly, the outcomes of this thesis provide a basis for further research application
and is recommended for future work for the implementation of adaptive active
vibration control methodology using
H
optimization MIMO positive position
feedback based genetic algorithm which can be applied to multimode vibration
control of a large class of flexible structures with varying and unknown
parameters or loading conditions.
Secondly, as discussed in Chapter 7.5, for the first mode, the performance of
multimode MIMO PPF controller is not good enough because there is only one
transducer (transducer 1) at that side, one transducer is not enough to control two
degrees of freedom vibration (bending and torsion which could be seen in the
ANSYS result Fig 2.13) at the same time. So another transducer which could be
added to the plant is recommended for future work. And the author also did a
simulation in ANSYS with four transducers. It is can be seen that the vibration
suppression is better than three transducers.
Figure 8.1 Modal Analysis mode shape of first mode with four transducers
Thirdly, because the whole system was set up several years ago for the research
study of honours degree student, four electromagnetic transducers which the
frequency range is 20
200 Hz were bought before, the author could not find the
similar size and similar frequency range transducer in the market during
experimental study. If the fifth electromagnetic transducer which is added to the
mechanical plant is heavier than others, the parameters of the whole mechanical
145
plant may change hugely, for example, the frequency of the first mode may drop
down and below 20 Hz, which is beyond the frequency range of the transducer, or
just a little bit over 20Hz, which is in the strongly nonlinear output area of the
electromagnetic transducer and very hard to control the vibration of plate . So it is
recommended for future that make a same size and same frequency range
transducer by hand if could not find in the market, either.
Fourthly, because the limitation of electromagnetic transducer, such as the
frequency range only between 20
200 Hz, the limitation of the oscilloscope
which can only read the max(M) result, piezoelectric sensor and actuator, signal
analyzer are recommended for future experimental study to instead of the old
equipments.
Fifthly, as stated in 1.5.3 of Chapter 1, PPF combine with Genetic Algorithms
(GA), the optimal control position of the three control transducers is also
recommended for future work, which could be derived through GA calculation.
Sixthly, although the author try to derive the mathematical model which has the
same dynamic response as the experimental plant, the parameters of the
multimode SISO and MIMO PPF controllers in the experimental study are still
different from the ones in the simulation study because during modelling, the
author simplified the structure of the transducer which is more complicated with
electromagnetic and nonlinear characters in the fact. So it is recommended to
remodel the transducer and minimize the effect to the whole system for future
work.
146
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