Final Thesis Report Rajeesh Sk

Vibration Suppression of Structures using Self Sensing Actuator

Department of Electrical Engineering, College of Engineering Trivandrum 1

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



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



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]



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]



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]



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].



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.



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



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.



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



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



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.



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



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

.



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)



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



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.


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.



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)



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)

]



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.


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.



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



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.



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



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



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.


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.



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



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



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



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



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



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



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:



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



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]


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.



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



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



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



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


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.



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.



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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