Vibration Suppression of Structures using Self Sensing Actuator
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Chapter 1
Introduction
The common sources of mechanical stresses on aerospace structures are dynamic
loads and in fact dynamical load cycles can damage or cause a reduction in the service
life of aerospace structures. Therefore the investigation of vibration characteristics is an
important design phase of aerospace structures which are frequently experiencing
dynamic loading conditions. In essence, there are tremendous amounts of numerical and
experimental studies focused on investigation of the vibration characteristics and
attenuation of vibration levels of aerospace structures. When the frequency of the
dynamic loading matches with the natural frequency of the structure, the resonance
occurs, and it may cause severe structural vibrations. In this situation, severe vibrations
may damage components of aerospace vehicles, as aerospace structures are mostly light
weight and have low-stiffness characteristics.
The undesirable effects of induced-vibration in aerospace vehicles can be
exemplified by research studies for a fighter-jet, a helicopter and a satellite. Over the past
decade, research studies showed that severe vibrations in the form of buffet can damage
the components of a fighter-jet. Since flight envelope of fighter-jets includes many highly
acrobatic maneuvers and certain speeds higher than the speed of sound, severe vibrations
occur and may damage their components. The cracks in the components of fighter-jets
may cost millions to be replaced and maintained. On the other hand, helicopters are the
aerial vehicles whose structures are under dynamic loading in all flight envelopes because
of their rotary elements such as main rotor, tail rotor and transmission units. Their cabin
crew is exposed to the high levels of vibration in all flight zones and therefore there are
researches focusing on the investigation of vibration characteristics of a helicopter seat
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and effects on cabin crew health. Vibration at the helicopter seat causes excitation at the
natural frequency of the spine and abdominal of the cabin crew and exposition to these
types of vibrations for long time causes variety of health problems on cabin crew.
Satellites are other type of aerospace vehicles which are under dynamic loading during
launch and in-orbit operations. The induced-vibrations may cause both reduction of the
precision of pointing accuracy and cracks on the components of small satellites.
It is obvious that vibration suppression of structures is very crucial for better, safer
and easier life. Therefore, engineers are attempting to suppress such vibrations of
structures by using passive and active methodologies. However, passive vibration
suppression techniques are generally not suitable for low frequency applications.
Recently, active and adaptive vibration Control is receiving considerable attention as
alternative solutions to those passive methods. Passive vibration suppression
methodologies have some drawbacks such as not suitable for low frequency application,
increases weight of system and once designed their design parameters cannot be varied
easily The technological advances in piezoelectric materials also motivate scientists and
engineers to use these materials for the active vibration control as well. Piezoelectric
elements such as piezoceramics have excellent electric-mechanical conversion
characteristics. Therefore, they are widely used as sensors, which utilize the voltage
generated by the strain to which they are subjected, i.e., the piezoelectric effect. They are
also used as actuators, which utilize the strain due to the applied voltage, i.e. the inverse
piezoelectric effect.
The research and production of piezoelectric materials and piezoelectric devices
are rapidly developing. In recent years, active vibration control based on smart material
and structures especially piezoelectric smart structures has been dramatically developed
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for both research and engineering applications because of its good electro mechanical
coupling characteristics, preferable dynamic performance and higher sensitivity.
Now a new concept for the application of a piezoelectric element has emerged
from the field of material engineering. A Self-sensing actuator (SSA) is one in which a
single piezoelectric element functions simultaneously as a sensor and an actuator.
Therefore some researchers have tried to apply this idea to vibration control of flexible
structures. SSA is applied to bimorph cantilever beam which is an example of flexible
structure for vibration suppression.
1.1 Literature Survey
Aerospace structures are subjected to mechanical stresses due to dynamic loads
which may lead to damage or reduction in the life of aerospace structures. Particularly
when frequency of dynamic loading matches with the natural frequency of structures.
Therefore investigation of vibration characteristics is an important phase in the design of
these structures. Active vibration control involves use of sensors to sense vibration of
structures, a controller to generate control signal and an actuator which exerts force on
structure to reduce vibrations. Active vibration control techniques using piezoelectric
have advantages such as light structures, low energy consumption, light weight etc. [1]
Now a day piezoelectric such as PZT (Lead Zirconate Titanate) are widely used as
sensor and actuator for active vibration control owing to their better electromechanical
coupling characteristics and high sensitivity. The coupling factor of PZT is in the range
of 0.7 indicating that they are efficient transducers. Normally independent piezoelectric
elements are necessary for sensors and actuators. [2]
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Vibration control is an essential problem in different structure. Smart material can
make a structure smart, adaptive and self-controlling so they are effective in active
vibration control. A smart structure is basically a distributed parameter system that
employs sensors and actuators at different finite element locations on the beam and
makes use of one or more microprocessors that analyze the response obtained from the
sensor and use different control logics to command the actuators to apply localized
strains to plant to response in a desired fashion and bring the system to equilibrium
Piezoelectric elements can be used as sensors and actuators in flexible structures for
sensing and actuating purposes. PZT is used as sensors and actuator to control the
vibration of a cantilever beam. Also studied the effects of different types of controller on
vibration. [3]
The modern technology demands the system to be light and reliable. However,
with the conventional vibration control techniques it is difficult to keep the system light
with all the damping mechanisms. This work deals with the Active Vibration Control
(AVC) of cantilever beam using piezoelectric (PZT) transducers. Active control involves
the use of sensors to sense the vibratory motion of the structure, a controller to generate a
control signal and an amplifier to amplify the control signal and an actuator which exerts
control force on the structure to reduce the vibrations. An experimental setup is made,
consisting of the aluminum cantilever beam with the PZT patches mounted on both the
sides of the beam. A proportional-integral derivative (PID) controller is designed to
generate the required control signal. A high frequency switch mode power converter is
designed to generate high voltage required for the actuator to produce the control force.
Experiments are performed for the active control of vibration [4]
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An important feature of piezoelectric material is their ability to function as
actuator and sensor simultaneously (self-sensing actuator-SSA).In SSA a single
electromechanical device serves as both sensor and actuator which reduces instability
problem that is usually associated with non-collocated sensor/actuator pairs and also
eliminates possible capacitive coupling between sensors and actuators that occurs in the
case of non-collocated sensor/ actuator pairs. In addition SSA system is simple, reduce
weight and size of system and is of low cost [5,6] .
A SSA can be realized by using a RC bridge circuit. When a single piezoceramics
function as SSA, two voltages, the sensing voltage (piezoelectric effect) and the
actuation voltage (inverse piezoelectric effect) are mixed in piezoceramics. So measuring
of sensor voltage is impossible. Therefore a bridge circuit with equivalent piezo model as
one of bridge circuit element makes it possible to detect sensor voltage that indicates
strain in structure [7].
This paper describes the active vibration control of a plate using a self-sensing
actuator (SSA) and an adaptive control method. In a self-sensing actuator, the same
piezoelectric element functions as both a sensor and an actuator so that the total number
of piezoelectric elements required can be reduced. A method to balance the bridge circuit
of the SSA was proposed and its effectiveness was confirmed by using an extra
piezoelectric sensor, which is not necessary for balancing bridge circuits of SSA in future
applications. A control system including the SSA and an adaptive controller using a finite
impulse response (FIR) filter and the filtered-X LMS algorithm was established. The
experimental results show that the bridge circuit was well balanced and the vibration of
the plate was successfully reduced at multiple resonance frequencies below 1.2 kHz.[8]
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A more common approach in the control of vibration of beam involves the
utilization of LQR (Linear Quadratic Regulator) design. The LQR method is based on the
minimization of a quadratic performance index that is associated with energy of the state
variable and control signals. The goal of LQR controller design is to establish a
compromise between the energy state and control by minimizing a cost function [9].
Inverted pendulum has been the subject of numerous studies in automatic control
system. Since it the system is inherently non linear,it is useful to illustrate some of the
ideas in non linear control system. Wheeled mobile robots have in the recent years
become increasingly important in industry, since they provide a large degree of flexibility
and efficiency with respect to transportation and operation. The objective of this paper is
to design linear quadratic controllers for a system with an inverted pendulum on a mobile
robot.to this goal, it has to be determined which control strategy delivers better
performance with respect to pendulum‟s angle and robot‟s position, since it continually
moves towards an uncontrolled state.[10]
Linear Quadratic Gaussian (LQG) design problem is rooted in optimal stochastic
control theory and has many applications in the modern world which ranges from flight
and missile navigation control systems, nuclear power plants etc. It combines both the
concepts linear quadratic regulator for full state feedback and Kalman filters for state
estimation. Thus the configuration and design of a LQG controller for linear systems are
principally involved in both determining optimal process estimation by a linear quadratic
estimator and making an optimal control strategy by a linear quadratic regulator [11].
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1.2 Problem Definition
From the literature, it can be seen that the space structures are extremely flexible and
having low fundamental vibration modes. To effectively suppress the induced vibration poses
challenging tasks for space craft designers. The aim is to suppress the vibration in a bimorph
cantilever beam by combining a control strategy with Self Sensing Mechanism.
1.3 Objectives
Model flexible cantilever beam and Self Sensing Actuator (SSA).
To design control strategies based on PID, LQR and LQG.
Compare results of performance of system with above controllers.
The thesis report is organized in 9 chapters. After a brief introduction of the topic in chapter 1,
chapter 2 describes about piezoelectric materials. Piezoceramics and its properties are discussed
in chapter 3. Chapter 4 deals with the basics of cantilever beam. The concept of Self Sensing
Actuator is explained in chapter 5. Chapter 6 deals with the designing of controllers . Matlab
simulation results are included in chapter 7.Thesis conclusion is given in chapter 8. Finally, the
future scope of thesis is mentioned in chapter 9.
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Chapter 2
Introduction to Piezoelectric Materials
2.1 History
In the year 1880 Pierre Curie and Jacques Curie discovered that some crystals
when compressed in particular directions show positive and negative charges on certain
positions of their surfaces. The amount of charges produced is proportional to the
pressure applied and these charges were diminished when the pressure is withdrawn.
They observed this phenomenon in the following crystals: zinc blende, sodium chlorate,
tourmaline, quartz, calamine, topaz, tartaric acid, cane sugar, and Rochelle salt. Hankel
proposed the name “piezoelectricity”. The word “piezo” is a Greek word which means
“to press”, therefore piezoelectricity means electricity generated from pressure. The
direct piezoelectric effect is defined as electric polarization produced by mechanical
strain in crystals belonging to certain classes. In the converse piezoelectric effect a
piezoelectric crystal gets strained, when electrically polarized, by an amount proportional
to polarizing field.
2.2 Piezo Electric Direct and Converse Effects
The domains of the piezoelectric ceramic element are aligned by the poling
process. In the poling process the piezoelectric ceramic element is subjected to a strong
DC electric field, usually at temperature slightly below the Curie temperature. When a
poled piezoelectric ceramic is mechanically strained it becomes electrically polarized,
producing an electrical charge on the surface of the materials (direct piezoelectric effect),
piezoelectric sensors work on the basis of this particular property. The electrodes
attached on the surface of the piezoelectric material helps to collect electric charge
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generated and to apply the electric field to the piezoelectric element. When an electric
field is applied to the poled piezoelectric ceramic through electrodes on its surfaces, the
piezoelectric material gets strained (converse effect). The converse effect property is used
for actuator purposes. Figure 2.1 shows the converse piezoelectric effect.
Figure 2.1: Piezoelectric material
2.3 Piezoelectric Materials
Based on the converse and direct effects, a piezoelectric material can act as a
transducer to convert mechanical to electrical or electrical to mechanical energy. When
piezoelectric transducer converts the electrical energy to mechanical energy it is called as
piezo-motor/ actuator, and when it converts the mechanical energy to electrical energy it
is called as piezo-generator/ sensor. The sensing and the actuation capabilities of the
piezoelectric materials depend mostly on the coupling coefficient, the direction of the
polarization, and on the charge coefficients (d31 and d33). Figure 2.2 in the form of block
diagrams shows the transducer characteristics of the piezoelectric materials.
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Some of the typical piezoelectric materials include quartz, barium titanante, lead
titanate, cadmium sulphide, lead zirconate titanate (PZT), lead lanthanum zirconate
titanate, lead magnesium niobate, piezoelectric polymer polyvinylidene fluoride (PVDF),
polyvinyl fluoride (PVF). The piezoelectric ceramics are highly brittle and they have
better electromechanical properties when compared to the piezoelectric polymers. This
section gives in brief introduction about the various classes of piezoelectric materials:
single crystal materials, piezo-ceramics, piezo-polymers, piezo-composites, and piezo-
films.
Figure 2.2: Piezoelectric Transducer.
2.3.1 Single Crystals
Quartz, Lithium nibonate (LiNbO3), and Lithium tantalite (LiTaO3) are some of the
most popular single crystals materials. The single crystals are anisotropic in general and
have different properties depending on the cut of the materials and direction of bulk or
surface wave propagation. These materials are essential used for frequency stabilized
oscillators and surface acoustic devices applications.
2.3.2 Piezoelectric Ceramics
Piezoelectric ceramics are widely used at present for a large number of applications.
Most of the piezoelectric ceramics have perovskite structure. This ideal structure consists
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of a simple cubic cell that has a large cation “A” at the corner, a smaller cation “B” in the
body center, and oxygen O in the centers of the faces. The structure is a network of
corner-linked oxygen octahedral surroundings B cations.
Figure 2.3: Crystalline structure of a Barium Titanate (Perovskite structure)
For the case of Barium Titanate ceramic, the large cation A is Barium ion, smaller
cation B is Titanium ion. The unit cell of perovskite cubic structure of Barium Titanate is
shown in figure 2.3. The piezoelectric properties of the perovskite-structured materials
can be easily tailored for applications by incorporating various cations in the perovskite
structure. Barium Titanate and Lead Titanate are the common examples of the
perovskite piezoelectric ceramic materials.
2.3.3 Polymers
The polymers like polypropylene, polystyrene, poly (methyl methacrylate), vinyl
acetate, and odd number nylons are known to possess piezoelectric properties. However,
strong piezoelectric effects have been observed only in polyvinylidene fluoride (PVDF or
PVF2) and PVDF copolymers. The molecular structure of PVDF consists of a repeated
monomer unit (-CF2-CH2-)n . The permanent dipole polarization of PVDF is obtained
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through a technological process that involves stretching and poling of extruded thin
sheets of polymer. These piezoelectric polymers are mostly used for directional
microphones and ultrasonic hydrophones application.
2.3.4 Composites
Piezo-composites comprised piezoelectric ceramics and polymers are promising
materials because of excellent tailored properties. These materials have many advantages
including high coupling factors, low acoustic impedance, mechanical flexibility, a broad
bandwidth in combination with low mechanical quality factor. They are especially useful
for underwater sonar and medical diagnostic ultrasonic transducers.
2.3.5 Thin Films
Both zinc oxide and aluminum nitride are simple binary compounds that have
Wurtzite type structure, which can sputter-deposited in c-axis oriented thin films on
variety of substrates. ZnO has reasonable piezoelectric coupling and its thin films are
widely used in bulk acoustic devices.
2.4 Concluding Remarks
This chapter explains about the history and evolution of piezo electric phenomenon. The
piezoelectric effect and inverse piezoelectric effect is discussed in detail. Also discusses about
the different types of piezoelectric materials.
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Chapter 3
Piezo Ceramics
A piezoelectric ceramic is a mass of perovskite crystals. Each crystal is composed
of a small, tetravalent metal ion placed inside a lattice of larger divalent metal ions and
oxygen. To prepare a piezoelectric ceramic, fine powders of the component metal oxides
are mixed in specific proportions. This mixture is then heated to form a uniform powder.
The powder is then mixed with an organic binder and is formed into specific shapes, e.g.
discs, rods, plates, etc. These elements are then heated for a specific time, and under a
predetermined temperature. As a result of this process the powder particles sinter and the
material forms a dense crystalline structure. The elements are then cooled and, if needed,
trimmed into specific shapes. Finally, electrodes are applied to the appropriate surfaces of
the structure.
Figure 3.1: Crystalline structure of a piezoelectric ceramic, before and after polarization
Above a critical temperature, known as the “Curie temperature”, each perovskite
crystal in the heated ceramic element exhibits a simple cubic symmetry with no dipole
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moment, as demonstrated in Figure 3.1. However, at temperatures below the Curie
temperature each crystal has tetragonal symmetry and, associated with that, a dipole
moment. Adjoining dipoles form regions of local alignment called “domains”. This
alignment gives a net dipole moment to the domain, and thus a net polarization. As
demonstrated in Figure 3.2 (a), the direction of polarization among neighboring domains
is random. Subsequently, the ceramic element has no overall polarization.
The domains in a ceramic element are aligned by exposing the element to a strong,
DC electric field, usually at a temperature slightly below the Curie temperature (Figure
3.2 (b)). This is referred to as the “poling process .After the poling treatment, domains
most nearly aligned with the electric field expand at the expense of domains that are not
aligned with the field, and the element expands in the direction of the field. When the
electric field is removed most of the dipoles are locked into a configuration of near
alignment (figure 3.2 (c)). The element now has a permanent polarization, the remnant
polarization, and is permanently elongated. The increase in the length of the element,
however, is very small, usually within the micrometer range.
Figure 3.2. Poling process: (a) Prior to polarization polar domains are oriented randomly; (b) A very large DC
electric field is used for polarization; (c) After the DC field is removed, the remnant polarization remains
.
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3.1 Electrical Equivalent Model
Figure 3.3 displays an electrically equivalent model of the piezoceramics when a
piezoelectric element is used as self-sensing actuator where 𝐶 ’ is the piezo ceramics
capacitance, 𝑣 ’ is the voltage generated due to the strain on it, and 𝑣 ’ is the applied
control voltage. This equivalent model is valid in the low frequency band at which the
given structure vibrates. When a voltage 𝑣 is applied, the Piezo ceramic can function
simultaneously as a sensor and as an actuator.
Figure 3.3: Equivalent model of piezo ceramics
.
3.2 Sensor Equation
The sensor equation is a formulation of the piezoelectric effect. When a
piezoelectric element is bonded to a beam, the piezoelectric effect is related to the
difference in the slopes of both ends of the piezoelectric element. The 𝑣 ’ voltage due to
strain is generated proportionally with the slope of beam 𝑦 (𝐿, 𝑡) [5].
𝑣 (𝑡) =
𝑦 (𝐿, 𝑡) (1)
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where 𝐾 = 𝐸 𝑑 𝑏(𝑡 + 𝑡 )/2 (2)
Here 𝐸 is the youngs modulus of the piezoelectric element, 𝑑 the piezoelectric
constant, t is the thickness, b is the width, 𝑦 (𝐿, 𝑡) the difference in the slopes of both
ends of the piezoelectric element. In this study 𝑦 (𝐿, 𝑡) is the slope of end of the
piezoelectric element.
Since the slope is proportional to the strain in the beam, 𝑣 is also proportional to
the strain.
3.3 Actuator equation
The Actua tor equation is a formulation of the inverse piezoelectric effect.
The applied voltage 𝑉 causes the distributed bending moment M of the beam in a
proportional manner a s follow,
𝑀(𝑥, 𝑡) = 𝐾 𝑉 (𝑡) (0 ≤ 𝑥 ≤ 𝐿) (3)
𝐾 = 𝐸 𝑑 𝑏(𝑡 + 𝑡 )/2 (4)
Figure 3.4 A piezoceramics bending beam
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Since M(x, t) in equation (3) is independent of x, we simply use M (t) hereafter.
When the stack piezoceramics is driven at high voltage, there is usually
considerable hysteresis between the applied voltage and the bending moment,
which results in the deflection of the beam (figure 3.4). It is made sure beforehand
that this nonlinearity is negligible in this work because a bimorph piezoceramics is
used and applied voltage is lower than usual.
3.4 Concluding Remarks
The chemical composition and manufacturing process is discussed in detail. The
electrical equivalent model of a Piezoceramic crystal is also mentioned. Expression for
sensor voltage when Piezoceramic function as sensor (piezoelectric effect) and
expression for actuator voltage generated on crystal during the inverse piezoelectric
effect is also discussed in this chapter.
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Chapter 4
Beam Model
Figure 4.1: Cantilever Beam
Fig 4.1 shows structure of bimorph cantilever beam fixed at x=0. The beam is
assumed to be a smart structure .In order to make beam bimorph type, piezoceramics are
bonded to the beam on both sides. This beam is composed of a metal shim (phosphor
bronze) between two piezoceramics sheets. The term y(x,t) denotes deflection of beam
and 𝑦 =
, �� =
.It is assumed that deflection is about x axis only. Simple beam
theory [7] is used to model the beam.
The solution of dynamic equation can be split in to space and time components is given
by
𝑦(𝑥, 𝑡) = ∑ ∅ (𝑥)𝑞 (𝑡) (5)
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where ∅ (𝑥) the mode shape function of ith mode and 𝑞 (𝑡) satisfies the following equation:
𝑞 + 𝐶𝜔 𝑞 + 𝜔
𝑞 = 𝑀(𝜙 (𝐿) − 𝜙
(0) (6)
and ∅ (𝑥) = 𝐿[(cosh(𝜆 , 𝑥) − cos(𝜆 , 𝑥)) − 𝑘 (sinh(𝜆 , 𝑥) − sin(𝜆 , 𝑥))] (7)
where M and 𝜔 denote the distributed bending moment and the natural frequency of ith mode
respectively
Where 𝜔 = √
𝜆
State space expression of beam the first 2 modes of vibration is given by [7].
��=𝐴 𝑞 + 𝐵 𝑣 (8)
y = 𝐶 𝑞 (9)
State variable vector and output vector with states 𝑞’ and ��’ respectively deflection and
slope are given by
𝑞 ≅ [𝑞 𝑞 �� �� ]
(10)
𝑦 ≅ [𝑦 (𝐿) �� (𝐿)] (11)
Where
𝐴 = [
00𝜔
0
000𝜔
10
𝐶𝜔
0
010
𝐶𝜔
] 𝐵 = [
00
𝐾 ∅ (𝐿)
𝐾 ∅ (𝐿)
]
𝐶 = [
∅ (𝐿)
∅ (𝐿)00
00
∅ (𝐿)
∅ (𝐿)
]
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This system is called BEAM with only one input, actuator voltage (𝑣 ) and four
outputs 𝑞 ,𝑞 , �� ,�� which represent deflection and slopes of first and second modes of
vibration respectively. All the constants are given in Table 1.
4.1 Concluding Remarks
In this chapter, modeling of a bimorph cantilever beam consisting of a metal shim
(phosphor bronze) between two piezoceramics sheets is discussed. The state space
expression for the beam formed. From the beam model, it can be notice that beam is
having one input and four outputs.
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Chapter 5
Self Sensing Actuator
A Self-sensing actuator (SSA) is one in which a single piezoelectric element
functions simultaneously as a sensor and an actuator. A SSA realizes the
complete collocation of the sensor and the actuator, which is very advantageous
i n terms of control. Therefore some researchers have tried to apply this idea to
vibration control of flexible structures. Furthermore, it can reduce the size and
weight of the system because individual sensors and actuators are not needed.
An electric bridge circuit is essential for self-sensing actuation. Two
voltages, the sensor voltage due to the piezoelectric effect and the actuation
voltage due to the inverse piezoelectric effect, are mixed in a piezoceramics when
it is used as a SSA. It is therefore impossible to measure the sensor voltage
directly. The bridge circuit that includes the piezoceramics as one of the four
elements makes it possible to derive the sensor voltage that indicates the strain in
the structure, if the bridge is well balanced.
5.1 RC Bridge circuit
When a piezoceramics functions as a Self Sensing Actuator (SSA), voltage 𝑣 is
not directly detectable due to the control voltage 𝑣 . Self sensing actuator is based on the
linear system principle that if two signals are added in to a linear system, and the output
and one of the inputs is known, the second input can be determined .The SSA can be
thought of having two added inputs signals, one input voltage from controller and other
input voltage is is due to strain of material (piezoelectric effect). By subtracting from
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output the effects due to control voltage, only the voltage due to strain of the material
remains in the output signal. Therefore a RC bridge circuit (figure 4) which includes
piezoceramics is introduced to discriminate 𝑣 from 𝑣 . In RC bridge circuit, strain rate
(velocity) is sensed and strain rate feedback is effective in suppressing vibration.
Figure 5.1 RC bridge circuit for strain rate (velocity)Sensing
5.2 Sensor Dynamics
Rate of strain sensing is accomplished by putting an equivalent RC circuit in parallel
to RC circuit formed by the series resistor and piezoelectric material. The circuit output
voltage will be sensor voltage 𝑣 ’ and input will be control voltage 𝑣 ’and piezoelectric
voltage 𝑣 ’.From the circuit,
𝑣 (𝑠) =
(𝑣 (𝑠) + 𝑣 (𝑠)) (12)
𝑣 (𝑠) =
𝑣 (𝑠) (13)
Here the difference between 𝑣 and 𝑣 is defined as sensor voltage 𝑣 which is equal to 𝑣 - 𝑣
And if the two RC time constants are equivalent ( 𝐶 𝑅 = 𝐶 𝑅 ) then the circuit output
equation becomes the same as the simple rate of strain sensor.
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𝑣 (𝑠) =
𝑣 (𝑠) (14)
Equation (14) indicates that sensor voltage 𝑉 is determined only by 𝑣 regardless of 𝑣
At frequencies 1𝐶 𝑅
⁄ , 𝐶 𝑅 𝑠 + 1≈1. So 𝑣 can be expressed as
𝑉 (𝑡) = 𝐶 𝑅 𝑣 (𝑡) (15)
Thus strain rate in beam can be detected.
5.3 Actuator Dynamics
When using the material as an actuator, the effect of resistor in the circuit must be
taken into account. The actuation voltage is no longer simply the control voltage. The
moment generated by the piezoceramics is proportional to the voltage applied across it
which is 𝑣 − 𝑣 .
The moment generated by piezoceramics is by
𝑣 = 𝑣 − 𝑣 (16)
But 𝑣 - 𝑣 =
𝑣 (𝑠) −
𝑣 (𝑠) (17)
The last term in the above equation is the same as the sensor signal 𝑣 - 𝑣 .Simplifying above
equation gives the actuator dynamics
𝑣 =
𝑣 (𝑠) (18)
For frequencies 1𝐶 𝑅
⁄ , 𝑣 will be equal to 𝑣
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5.4 Modeling SSA
State space model of a Self Sensing Circuit is given by
[𝑥
𝑥 ] = [
0
0
] *𝑥
𝑥 + + [
0
] [𝑦
𝑣 ] (19)
*𝑣
𝑣 + = *
−1 010 10
+ *𝑥
𝑥 + (20)
This system is called SSA with two inputs and two outputs. The inputs are time
derivative of slope of beam tip, 𝑦 and control voltage 𝑣 from the controller. The outputs
are sensor voltage 𝑣 to controller and actuating voltage 𝑣 applied to piezoceramics .The
two states in SSA are 𝑣 − 𝑣 and 𝑣 + 𝑣 .
5.5 Plant Block Diagram
Figure 5.2 Plant block diagram
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Figure 5 shows the plant including the systems SSA, BEAM and the blocks for
control specification. The dashed box represents the control object, and K is the
controller to be designed. Input disturbance to the beam results in the vibration of the
beam. The strain rate from the beam is sensed by SSA. Sensing voltage developed on
SSA is given to controller to obtain control voltage. The control voltage generated is fed
to SSA which will develop actuator voltage. This actuator voltage is used to suppress
vibration of the beam.
5.6 Concluding Remarks
This chapter gives an overview of the basic concept of self sensing actuator (SSA)
and about the electrical equivalent circuit of SSA .Expressions for sensor and actuator
equations for SSA is mentioned. A complete plant block diagram is also given.
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Chapter 6
Design of Controller
In this section, three control strategies are proposed and described in detail. The
control strategies are PID, LQR and LQG. The main objective of control strategy is to
suppress the vibration of beam due to disturbances.
6.1 PID controller
PID controller has the optimum control dynamics including zero steady state error,
fast response (short rise time),no oscillations and higher stability. The necessity of using
a derivative gain component in addition to PI controller is to eliminate the overshoot and
the oscillations occurring in the output response of the system. The PID controller
transfer function with controller gains , and is given by,
( ) = 𝐾 +
+ 𝐾 (21)
= Proportional gain
= Integral gain
= Derivative gain
Figure 6.1 Plant block diagram with PID Controller
The proportional–Integral -Derivative controller is designed to produce control signal to
reduce vibration of beam. The sensing signal from SSA is processed with gain values of
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PID operations. The optimum values are obtained by tuning these parameters and it
depends on system and its response. Fig 6.1 shows the block diagram of the system with
a PID controller. The control signal from PID controller is again fed to SSA to generate
the actuating voltage to suppress the vibration. The controller parameters values are =
0.9, = 0.05 and = 0.001.
6.2 LQR Controller
With the advent of technology modern control techniques have emerged which
made the controller design more accurate and efficient. In this section a Linear Quadratic
Regulator (LQR) is proposed as a solution. LQR design method converts control system
design problems to an optimization problem with quadratic time domain performance
criteria .In this technique a controller is designed that provides the best possible
performance with respect to some given performance index. At the same time, good gain
and phase margin and stability of the system is expected. The complete schematic of a
LQR controller are given in figure 6.2. Here is the state space system represented with its
matrices A, B, and C with the LQR controller (shown with K).
Figure 6.2 Plant block diagram for LQR
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The LQR problem rests upon the following three assumptions,
1) All the states are available for feedback, i.e. it can be measured by sensors etc.
2) The system a stabilizable which means that all of its unstable mode are controllable.
3) The system is detectable having all its unstable modes observable.
To check whether the system is controllable and observable, we use the functions
obsv(A,C) and ctrb(A,B).
In LQR, a control method is Linear Quadratic Control methods are best choice for
MIMO systems for effective vibration suppression and for damping out disturbances as
quickly as possible. The control voltage for SSA is determined by optimal control
solution of linear quadratic regulator (LQR) provided the full state vector is observable
LQR design is a part of what in the control area is called optimal control. For a LTI
system, this technique involves choosing a control law 𝑢(𝑡) = −𝐾𝑥(𝑡) which stabilizes
the origin (i.e., regulates x(t) to zero) while minimizing the quadratic cost function
= ∫ (𝑥(𝑡)
𝑥(𝑡) + 𝑢(𝑡) 𝑅𝑢(𝑡))dt (22)
where = ≥ 0 and 𝑅 = 𝑅 > 0 . The term “linear-quadratic” refers to the linear
system dynamics and the quadratic cost function. „Q‟ and „R‟ are weighting matrices for
states and control voltages respectively which are positive semi definite matrices. The
matrices Q and R are also called the state and control penalty matrices, respectively. If
the components of Q are chosen large relative to those of R, then deviations of x from
zero will be penalized heavily relative to deviations of u from zero. On the other hand, if
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the components of R are large relative to those of Q , then control effort will be more
costly and the state will not converge to zero as quickly.
Hence, in an optimal control problem the control system seeks to maximize the
return from the system with minimum cost. It is crucial that Q must be chosen in
accordance to the emphasize that wanted to be given the response of certain states, or in
other words; how we will penalize the states. Likewise, the chosen value of R will
penalize the control effort u. As an example, if Q is increased while keeping R at the
same value, the settling time will be reduced as the states approach zero at a faster rate.
This means that more importance is being placed on keeping the states small at the
expense of increased control effort. On the other side, if R is very large relative to Q, the
control energy is penalized very heavily. Hence, in an optimal control problem the
control system seeks to maximize the return from the system with minimum cost. Since
the objective is to suppress the beam vibration, The weighting values corresponding to
these states are kept high that is the first two states in Q matrix corresponds to beam
vibration which is kept at high value.
The solution to above problem, the control voltage is
𝑢(𝑡) = −𝐾𝑥(𝑡) (23)
where 𝐾 = 𝑅 𝐵 , = 0 is the unique positive semi definite solution of
algebraic Riccati equation given by
𝐴 + 𝐴 − 𝐵𝑅 𝐵 + = 0 (22)
The state matrix of plant (Beam and SSA) and weighting matrixes Q and R obtained by
tuning are given by
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6.3 LQG Controller
Figure 6.3 Plant block diagram for LQG
Linear Quadratic Gaussian (LQG) control technique is a modern state space
method for designing optimal dynamic regulators for handling noise inputs. This control
problem is rooted in optimal stochastic control theory and has many applications in the
modern world which ranges from flight and missile navigation control systems, medical
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processes controllers and even nuclear power plants. This is simply a combination of
Kalman filter and a LQR controller. The separation principle guarantees that these can be
designed and computed independently. LQG controller can be used both in linear time
invariant as well as linear time variant systems. Figure 6.3 illustrates schematic of LQG
control approach
In most cases, all states of system will not be available for feedback. This problem
can be solved by Kalman filter (linear quadratic estimator).Thus in LQG control
technique, the control voltage for the actuators are determined by optimal control solution
of LQR problem of system with state estimated by a Kalman filter. It also takes in to
account the disturbances affecting the system.. The system with noise is given by
�� = 𝐴𝑥 + 𝐵𝑢 + 𝑤 (23)
𝑦 = 𝐶𝑥 + 𝐷𝑢 + 𝑣 (24)
where A is the plant state matrix , B is the plant input matrix , C is the plant output matrix
, D is the plant feed forward, and both w and v are modeled as white noises associated
with process and measurement respectively. is the plant noise gain matrix.
In LQG controller, the regulation performance J is measured by a quadratic performance
criterion of the form
= ∫ (𝑥(𝑡)
𝑥(𝑡) + 𝑢(𝑡) 𝑅𝑢(𝑡)) dt (25)
Where „Q‟ and „R‟ are weighting matrices which are positive semi definite matrices. „Q‟
and „R‟ are weighting matrices for states and control voltages respectively. Hence, in an
optimal control problem the control system seeks to maximize the return from the system
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with minimum cost. So in order to suppress the beam vibration, the weighting values
corresponding to these states are kept high.
The control problem is to determine the optimal control input;
u(t) = −Kx(t) (26)
where K the state feedback controller gain given by
𝐾 = 𝑅 𝐵 , = 0, K is the unique positive semi-definite solution of the algebraic
Riccati equation given by
𝐴 + 𝐴 − 𝐵𝑅 𝐵 + = 0 (27)
In real world control design problems, it is rarely possible to have access to all states
of the system which are needed for full state feedback. Instead, access is only possible to
specific measured outputs of the system. If these measurements carry enough information
about the states of the system, then a state observer using Kalman Filter could be
implemented to estimate all states of the system. This observer is capable of rejecting
disturbances of the system by acting as a low pass filter. The main inputs to the observer
are the control input (u(t)) and the system output (y(t)).
The state space equations of the Kalman Filter are shown in equation (28). It should
be noticed that it uses the same state space matrices (A, B and C) as the main system and
the estimated states 𝑥 are used as the system states.
=𝐴 +𝐵𝑢+L (𝑦−𝐶 ) (28)
The estimator gain must minimize the estimation error and is given by = ,
where X is the positive semi definite solution to the algebraic Riccati equation:
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𝐴 + 𝐴 − 𝐵𝐶 𝑅 𝐶 + = 0 (29)
From figure 6.3, it should be noticed that LQG is formed by connecting the system
and the Kalman Filter through the optimal state estimation gain (L) and then creating full
state feedback by using the estimated states ( (𝑡)) which passed through the optimal
feedback gain (-K). Thus Kalman filter can be applied to estimate state vector and
output vector y by using inputs u and measurements „y‟. Because of the stochastic
separation principle, the previously mentioned gain could be designed individually.
The design process starts with checking controllability and observability of the pairs
(A, B) and (A, C), respectively. These criteria are necessary for the existence of the
solutions for the equations used to find the optimal gains. Then, the optimal state
estimation gain L ( = ) is calculated where X is a positive semi-definite matrix
and the solution of the Filter Algebraic Riccati Equation (FARE) shown in equation (29).
This solution ensures a minimum value of the cost function shown by equation (25).
After that, the optimal state feedback gain (K ) is calculated, where X is a positive semi-
definite matrix and the solution of the Control Algebraic Riccati Equation (CARE) shown
in equation(27). This solution ensures a minimum value of the cost function in equation
(25).
The estimator and controller gain matrices can be designed separately by separation
principle. The state matrix of plant (Beam and SSA) and weighting matrixes Q and R
obtained by tuning are given by
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The optimal state estimation gain „L‟ is obtained by considering 1% noise in the state variables
is
The K matrix obtained by tuning Q and R matrix is:-
K= [−0.0037762 −1.20 10 −7. 10 06.45 0.07]
6.4 Concluding Remarks
This chapter gives an overview of the different control techniques that can be used for
vibration suppression of beam with SSA technique. A conventional PID controller is
designed. Also discussed in detail on designing of LQR and LQG optimal control
techniques.
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Chapter 7
Simulation Results
In this chapter, response of the system is analyzed in detail. The performance of the
system in open loop, closed loop, system with PID controller, system with LQR and
system analysis with LQG controller is done using commercial software package,
MATLAB/Simulink R2013b. Disturbance signal is given as input to the system in the
form of chirp signal. The frequency range of signal varies from 10Hz – 100Hz. The beam
under analysis is having first mode resonant frequency of vibration at 91 Hz.
7.1 System Response in open loop
Figure 7.1 shows the open loop response of the system. Here tip vibration of the
beam is very high with initial tip vibration being 0.2mm. SSA is acting only as sensor in
this case. Also the first mode vibration frequency of beam is seen at 91 Hz (9 seconds)
which is having a vibration of very high magnitude (about 1.2mm).
Figure 7.1 Response of system in open loop
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7.2 System Response in closed loop
In closed loop response of the system, first mode frequency of vibration of beam is
completely eliminated. Here SSA is simultaneously acting as sensor and as actuator. Figure 7.2
shows the closed loop response of the system. The initial tip vibration is reduced to 0.15mm
from 0.2mm in open loop response. Also there is improvement in steady state response.
Figure 7.2 Response of system in closed loop
7.3 System Response with PID controller
Figure 7.3 Response of system with PID controller
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With the introduction of PID controller, system performance is twice as compared with
closed loop response. The tip vibration of beam at 10 seconds is only 50% than that of closed
loop. From figure 7.3, it can be inferred that the initial beam tip vibration is also reduced to 0
.1mm (table 2).
7.4 System Response with LQR controller
Introduction of LQR optimal controller increases the overall performance of the system.
Figure 7.4 shows that the initial tip vibration of the beam is 0.07mm,which is far better than the
PID controller.
Figure 7.4 Response of system with LQR controller
7.5 System Response with LQG controller
Figure 7.5 Response of system with LQG controller
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LQG is an optimal controller that take in to account the noises affecting the
system. Figure 7.5 shows the tip vibration of the beam with LQG controller. The system
is showing best performance in terms of initial vibration(0.07mm) and steady state
vibration (table:2). With LQG controller beam tip vibration settles to 0.05 of final
steady value in 10 seconds.
Table 2: Comparison of Results
Type of
System
Initial Tip
Displacement(mm)
Tip
displacement at
10 sec (mm)
Open Loop
system 0.20 0.10
Closed Loop
system 0.15 0.04
System with
PID
Controller
0.10 0.03
System with
LQR 0.07 0.025
System with
LQG 0.02 0.006
7.6 Concluding Remarks
In result analysis, the response of system with different controllers is discussed in
detail. Response with LQG controller shows excellent performance in damping vibration
of beam than that of LQR and PID. Thus from the results it can be inferred that LQG
controller is the best suited for damping vibration of a flexible cantilever beam.
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Chapter 8
Conclusion
This paper presents a theoretical analysis of vibration suppression of bimorph
cantilever beam using SSA. Simple beam theory is used to model the beam. RC bridge
circuit which includes piezoceramics is used as SSA. SSA with conventional PID, LQR
and LQG controls are applied to suppress the vibration of beam. Simulation results
indicate that vibration of beam has been actively suppressed by applying control voltage
to SSA. From the results it is clear that performance of the system with LQG controller is
better than that of the system with LQR controller and conventional controllers.
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Chapter 9
Future scope
The use of Piezoceramic for structural actuation and sensing is a developing area.
But piezoceramics imposes certain restrictions for its practical use in real world
applications. For instance brittle nature of Piezoceramic requires extra attention during
handling and bonding procedures. In addition the conformability to curved surface is
extremely poor requiring extra treatment of the surfaces. Active fiber composites like
macro fiber composites (MFC) provide not only the required durability and flexibility,
but also higher electro-mechanical coupling characteristics. So the effectiveness of MFC
on vibration suppression can be studied further.
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