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I . LIBRARY I
mg Michigan State
University

 

 

 

This is to certify that the
dissertation entitled

HYBRID CONTROL OF FLEXIBLE STRUCTURES

presented by
SHAHIN SABOKDAST NUDEHI

has been accepted towards fulfillment
of the requirements for the

Ph.D D. degree. in MECHANICAL ENGINEERING

wmfl.

Major Professor’s Signaturé’
guy/m1 /5, 2005
a ,

Date

 

MSU is an Affirmative Action/Equal Opportunity Institution

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE
I? U 4 0 a.
.123 (t 2 200?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

HYBRID CONTROL OF FLEXIBLE STRUCTURES
By

Shahin Sabokdast N udehi

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

2005

ABSTRACT
HYBRID CONTROL OF FLEXIBLE STRUCTURES
By
Shahin Sabokdast Nudehi

Active vibration control remains a topic of significant relevance and
importance due to high performance demands of certain space structures as well as
growing interest in the development of terrestrial structures using feedback control.
This thesis presents two new approaches, to control system design for flexible
structures. In the first approach, piezoelectric transducers are continually switched
between actuator and sensor modes to enhance controllability and observability of the
system. This approach can potentially reduce the number of piezoelectric transducers
and associated hardware by 50%. In the second approach, piezoelectric transducers
are used as sensors to estimate modal displacements and cables are used for the
purpose of actuation. It is shown that tension in cables can be applied and released to
directly suppress vibration of structures or vary the stiffness of the structure which
results in modal energy redistribution. By properly designing switching strategies for
the cable tension, modal energy can be redistributed, and specifically energy
associated with higher modes can be funneled to the lower modes. This enables
vibration suppression using a simple controller that can potentially sidestep the

spillover problem.

 

TABLE OF CONTENTS

LIST OF FIGURES v
1 Introduction 1
1.1 Motivation ................................. 1
1.2 Literature review ............................. 2
1.2.1 Flexible structures with piezoelectric transducers ....... 2

1.2.2 Flexible structures with end-force ................ 3

1.2.3 Switched systems ......................... 5

1.3 Scope and content of the dissertation .................. 7

2 Mathematical Preliminaries 9
2.1 Piezoelectric materials and properties .................. 9
2.2 Asymmetric configuration of piezo transducer ............. 10
2.3 State space modelling of beams with piezoelectric transducers . . . . 13
2.3.1 Beam dynamics .......................... 13

2.3.2 State equation with piezoelectric actuator ........... 15

2.3.3 Output equation with piezoelectric sensor ........... 17

2.4 Beam dynamics in the presence of an end-force ............. 19

3 Switching Piezos between Actuator and Sensor Modes 21
3.1 Background ................................ 21
3.2 Effect of switching on controllability and observability ......... 22
3.3 Switching requirement for observer-based controller .......... 25
3.4 Vibration suppression in a flexible beam ................ 29
3.5 Simulation of observer-based control design ............... 30
3.6 Sub-optimal switching schedule ..................... 33
3.7 Experimental verification ......................... 37
3.7.1 Apparatus ........................ - ..... 37

3.7.2 Results ............................... 40

iii

4 Vibration Control of a Flexible Beam using an End Force

4.1
4.2
4.3
4.4
4.5
4.6
4.7

Background ................................

Mathematical model of cantilever beam with end-force ........

Rayleigh-Ritz approximation .......................

Preliminary feedback control design ...................

Modified control design . . .

OOOOOOOOOOOOOOOOOOOOOOO

Adding bias tension in the cable .....................

Experimental verification . .

4.7.1 Hardware description .......................

4.7.2 The effect of bias tension on structural damping ........

4.7.3 Observer design . . .
4.7.4 Results ........

5 Modal Disparity

5.1
5.2
5.3
5.4
5.5
5.6

Background .........
Proof of concept .......
Dynamic analysis ......
Numerical example .....

Modified switching strategy

Experimental Results . . . .

6 Conclusion

BIBLIOGRAPHY

OOOOOOOOOOOOOOOOOOOOO

iv

45
45
46
49
51
55
59
60
60
62
63
64

68
68
68
73
75
77
81

83

86

2.1
2.2
2.3

3.1

3.2
3.3

3.4

3.5

3.6

3.7
3.8

3.9

4.1
4.2

4.3
4.4

4.5

LIST OF FIGURES

Piezoelectric asymmetric configuration and associated distribution . .
A pined-pined Beam instrumented with PZT transducers .......
Free body diagram of element d1: with an end load ..........

Plot of state variables in Example 1 for (a) xf = (O, O)T and (b)
:1: f = (—1 , -2)T .............................
A simply supported flexible beam with two piezoelectric elements

Third and fourth mode shapes of beam in Figure (3.2) and location of
PZT elements in relation to these mode shapes .............
Amplitude of vibration of the (a) first, (b) second, (c) third, and (d)
fourth modes of the beam with observer-based control design .....
Amplitude of vibration of the (a) first, (b) second, (c) third, and (d)
fourth modes of the beam with sub-optimal switching .........
Experimental test-rig ...........................
Schematic of switching between actuator and sensor modalities . . . .
Experimental results: Plot of ypzt with time for (a) uncontrolled sys-
tem, and controlled system with (b) fixed switching schedule, (0) vari-
able switching schedule, and (d) sub-optimal switching schedule. . . .
(a) An observepbased controller (b) a discontinuous and (c) a con-
tinuous controller obtained from using a fixed-time and variable-time
switching schedule ..............................

A flexible cantilever beam with an end force ..............
Simulation of decay in modal amplitude a1 due to structural damping,
(b), (c) decay in modal amplitudes a1 and a2 due to control in the
presence of structural damping, and (d) plot of the control action. . .
Control design based on output filtering. ................
Plot of modal amplitudes a1 and a2, and the control action u for the
modified control design when 6 = 0 ....................
Control design based on bias tension and output filtering ........

11
14
19

25
29

30

32

36

37
41

43

44

46

54

56

58
59

4.6
4.7

4.8
4.9

5.1
5.2
5.3

5.4
5.5

5.6

Experimental setup ............................
Free vibration: (a) in the absence of bias tension, and (b) in the pres-
ence of 20 N bias tension .........................
Vibration suppression using active control ...............
Vibration suppression using a one mode dynamic model results a)-spill
over problem b)- the role of low pass filter in reducing the effect of
spillover ..................................

cantilever beam mode shapes for P = 0 and P = 40N .........
Concept of the modal disparity .....................
plot of the modal amplitude for the described switching strategy in
Equation (5.20) ..............................
Modified switching strategy .......................
Plot of the modal amplitude and the end-force in the presence of the
low pass filter in the loop .........................
Plot of the modal amplitude and the end-force in the presence of the
low pass filter and first order controller in the 100p ...........

vi

61
65

66

67

70
73

76
78

79

81

CHAPTER 1

Introduction

1. 1 Motivation

Active vibration control remains a topic of significant relevance and importance due
to high performance demands of certain space structures as well as growing interest
in the development of terrestrial structures using feedback control. The traditional
approach in active control of these structures is based on a linear system model in
modal coordinates and estimation and control of the significant modes of the system
using piezoelectric actuators and sensors. The problem of vibration suppression in
highly flexible structures, such as space structures, require a large number of sensors
and actuators, and piezoelectric transducers are commonly used as dedicated sensors
and actuators. The hardware associated with piezoelectric actuators and sensors,
such as power amplifiers and data acquisition boards, add significantly to the weight
and cost of the system and motivates the development of control systems that require
fewer transducers. It also motivates the development of viable control strategies that
will be more effective than those based on the traditional approach, such that it can
meet the high performance demands of the structures.

This thesis presents two new approaches, to control system design for flexible struc-

tures. In the first approach, piezoelectric transducers are continually switched be-

tween actuator and sensor modes to enhance controllability and observability of the
system. This approach can potentially reduce the number of piezoelectric transducers
and associated hardware by 50%. In the second approach, piezoelectric transducers
are used as sensors to estimate modal displacements and cables are used for the pur-
pose of actuation. It is shown that tension in cables can be applied and released to
directly suppress vibration of structures or vary the stiffness of the structure which
results in modal energy redistribution. By properly designing switching strategies for
the cable tension, modal energy can be redistributed, and specifically energy associ-
ated with higher modes can be funnelled to the lower modes. This enables vibration
suppression using a simple controller that can potentially sidestep the spillover prob-

lem.

1.2 Literature review

1.2.1 Flexible structures with piezoelectric transducers

There have been extensive studies on the use of piezoelectric transducers in structural
control. Some of the early work on piezoelectric elements was aimed at developing
actuator and sensor mathematical models (1991 Alberts and Colvin [1]; 1995 Alberts,
et. a1 [2]; 1996 Fuller, et. a1, [3]; 1998 Clark, et. a1 [4]). These models have been
effectively used in control of flexible structures, (1987 Rawley and de Luis [5]; 1990
Hagood, et. a1 [6]; 1991 Garcia, et. a1 [7], 1991 Lazarus et. a1. [8]), for example.

The transfer function of flexible structures often have a large number of lightly
damped modes and feedback control problems for systems of this nature are diffi-
cult to handle (1994 Skelton [9]). The control design for such systems are based on
a finite number of modes and the resulting closed-loop system is prone to instabil-

ity due to spillover (1978 Balas, [10], [11]). One way to avoid spillover is to have

a collocated structure which guarantees closed-loop system stability despite model
truncation (2001 Halim and Moheimani [12]). In order to achieve perfect collocation,
Anderson, et. a1 1992, [13] and Dosch, et. a1. 1992, [14] proposed the “sensori-
actuator” or “self-sensing actuator”. In their approach, a capacitor with capacitance
identical to the piezoelectric element is used to resolve the mechanical strain of the
structure. Some of the drawbacks of self-sensing are crosstalk (2001 Holterman and de
Vries [15]) and lack of stability robustness due to capacitance uncertainty (1994 Cole
and Clark, [16]; 2000 Acrabelli and Tonoli, [17]; 2003 Moheimani, [18]). Although
switching piezoelectric elements between actuator and sensor modes has not been
proposed earlier, Demetriou 2001, [19] and Murugavel, 2002 [20] proposed switching
piezoelectric actuators between active and dormant states for optimization of a cost
functional and improvement of closed-loop response. In our approach, presented in
Chapter 3, a piezoelectric element is used both as an actuator and sensor, but not
simultaneously. This can potentially avoid the problems encountered in self-sensing.
Besides, it results in reduction of hardware and consequently reduction of cost and
weight of the control system.

For flexible structures, an alternative way to reduce control system hardware is to
use cables for vibration suppression. In Chapter 4, we show that a cable can be used
to apply an end-force on a cantilever beam, and it can be switched on and off by a

controller to suppress vibration.

1.2.2 Flexible structures with end-force

To the best of our knowledge, there has been no work reported in the literature
on vibration control of flexible structures using end forces. However, the dynamics
of structural elements, such as beams and plates have been investigated under the
application of end forces (e.g., 1963 Boltin, [21]; 1971 Herrmann, [22]; 1989 Higuchi,
[23]; 1992 Dowel] [23]; 2002 Hodges, et. a1. [24]; 2000 Langthjem and Sugiyama,

[25]; 1996 O’Reilly, et. a1, [26]; 2000 Park and Kim, [27]; 1996 Zuo and Schreyer,
[28]). Most of the these results are related to the flutter instabilities associated with
follower forces, which are nonconservative in nature. In our studies in Chapter 4 and
5, we use a cable to apply a conservative end force in a cantiliver beam.

In the literature, cables have been primarily used for increasing the stiffness of
lightweight structures but there has been few studies on their use for vibration con-
trol. Achkire and Preumont (1996, [29]) investigated multiple control strategies for
control of cable-stayed bridges and Preumont and Bossens (2000, [30]) used tendons
to introduce active damping in truss structures. Skidmore and Hallauer (1985, [31])
demonstrated active damping in a beam-cable structure and Magana, et. a1 (1997,
[32]) proposed a nonlinear control design to reduce vibartion introduced by external
disturbances. In contrast of these approaches, Thomson et. a1. (1995, [33]) proposed
a passive method to significantly increase structural damping. They used shape mem-
ory alloy wires to constrain the motion of a beam and experimentally demonstrated
that damping increases significantly when the wires are cyclically stressed with their
pseudoelastic hysteresis loop.

In our approach, we do not restrict our control design to active damping. We use
a cable to apply an end-force on a cantilever beam and demonstrate two different
approaches for vibration control. The tension in the cable is switched on and off
for direct vibration suppression of all the modes of the system and this approach is
presented in Chapter 4. In Chapter 5, cable tension is switched to transfer energy
from the higher modes to the lower modes of the system, which enables a simpler
control system design. In both Chapters 4 and 5, the stability of the system is taken
into consideration while designing switching strategies for cable tension. In the next
section, we present a concise literature review of switched systems which has been a

topic of considerable interest over the last decade.

1.2.3 Switched systems

A switched systems consists of two or more continuous subsystems and a rule that
orchestrate switching between them (1999 Liberzon and Morse [34]). In most control
systems, switching between different subsystems occur because of changing dynamics
or operating conditions of the plant, or change in the control law invoked for en-
hanced performance of the system. In Chapter 3, we propose switching piezoelectric
transducers between actuator and sensor modes to enhance the controllability and
observability of the system. The switching strategies developed in Chapters 4 and 5
are aimed at developing new control methodologies for flexible structures.

For switched systems, stability is an important consideration since the switching
can introduce instability even when the individual subsystems are stable. Some of
the important concepts in stability, such as “dwell time”, and “common Lyapunov
function” were proposed by [34]. Other approaches to the study of stability include the
Lie-algebraic approach for linear systems (1999 Liberzon, Hespanha and Morse, [35];
2001 Agrachev and Liberzon, [36]), extension of the invariance principle (2001, [37]),
and the approach based on the Lyapunov functions and linear matrix inequalities
(2000 DeCarlo, et. al, [38]). In our study in Chapter 3, where time interval between
switchings is fixed, the issue of stability is addressed by designing stable subsystems
and choosing the switching intervals to greater than the dwell time.

Some of the recent work on switched linear systems have addressed the issues of reach-
ability and controllability. Ge, et. a1 (2002, [39]) and Zhenyu (2002,[40]) investigated
controllability and observability of systems for pre—assigned switching sequences and
Egerstedt (2002, [41]) investigated the complicity of the reachability problem between
two given states for fixed number of switchings. In Chapter 4, we present observer-
based control designs for a flexible beam in which controllability is guaranteed when

all the piezoelectric transducers are used as actuators and observability is guaranteed

when all of them are used as sensors.

A part of our study in Chapter 3 is related to designing optimal switching se—
quences and intervals. Some of the early work on optimization of switched systems
(1995 Branicky and Mitter, [42]; 1998 Branicky, et. al., [43]) used the maximum
principle and dynamic programming to address general problems. In recent years,
the focus has moved to specific switched systems. For example, Xu and Antsaklis
(2002, [44]) addressed the problem of determining optimal switching instants for lin-
ear switched systems with fixed number of switchings and pre—specified sequence of
autonomous subsystems. The total time was assumed to be finite in their approach
but the infinite time horizon problem was investigated by Giua, eta] (2001, [45]).
They assumed a quadratic performance index and considered discontinuities after
switchings. The optimal control problem for switched linear systems with a known
switching sequence and fixed switching intervals was addressed by Xu and Antsak-
lis (2002, [46]). They also addressed the more general problem of determining both
optimal switching instants and optimal inputs for fixed number of switchings and
pre—specified sequence of subsystems (2002 Xu and Antsaklis, [47]).

Most of the results on optimal control of switched systems provide open loop solu-
tions. In contrast, Bemporad et. a1 (2002, [48]) obtained a solution to the problem of
switching between finite number of autonomous subsystems based on state feedback.
The time horizon was assumed to be infinite but Giua, et.al (2002,[49]) addressed
the fixed final time problem. For both the problems, the number of switchings were
assumed to be fixed but the sequence of subsystems were assumed not assigned apri-
ori. The approach adopted in both cases applies well to two-dimensional systems but
poses significant computational challenges for problems of higher dimensions. Fur-
thermore, the approach is based on complete state information. The optimal control
problem, where complete state information is not available and output variable de-

scription switches along with state variable description, has been investigated only by

a few researchers, such as Rantzer and Johansson (2000,[50]). The focus of this work,
however, is to analyze the performance of the optimal control system and generalize
concepts such as Grammians and LQR using the framework of piecewise quadratic

Lyapunov functions.

1.3 Scope and content of the dissertation

This thesis is organized as follows. In chapter 2, we present some background material
that includes mathematical modelling of piezoelectric transducers. The state space
model of an Euler-Bernoulli beam is derived for both cases where actuation is provided
by a piezoelectric transducer and a cable providing an end force. These models are
used in Chapters 3, 4 and 5 to design controllers for our flexible beam.

Chapter 3 starts with a discussion of the effect of switching on controllability and
observability of linear time-variant systems. The requirement for observer-based con-
trol design, in terms of the number of switchings, is presented next. We establish
the merit of introducing under-actuation and under-sensing with the objective of re-
ducing the total number of piezoelectric transducers and associated hardware needed
for vibration control. The feasibility of switching the piezoelectric transducers be-
tween actuator and sensor modalities was demonstrated by simulations as well as
experiments.

In chapter 4, we study the dynamics of a cantilever beam with a buckling-type end
force and derive its mathematical model for the purpose of feedback control design.
Following that, we design a preliminary feedback control strategy for vibration sup-
pression and demonstrate its efficiency through numerical simulations. We modify our
control design to meet actuator bandwidth limitations and provide both simulation
and experimental results based on the modified control design.

In Chapter 5, we introduce the concept of modal disparity. Modal disparity is a mea-

sure of the difference between modes in two stiffness states and can be exploited to gain
control authority over the significant flexible modes of a system using a low dimen-
sional state space model. Although stiffness variation in a structure can be achieved
in many ways, we used cables to apply an end force for the cantilever beam problem.
The control methodology relies on variation in stiffness of the beam to achieve modal
energy redistribution, from higher modes to the lower modes, and dissipating the en-
ergy associated with the lower modes. Since the lower modes are only estimated and
controlled, this approach has the potential to sidestep spillover problem. We present
an analytical framework for control design exploiting the concept of modal disparity
and verify the results through simulations and preliminary experiments. Chapter 6,

provides concluding remarks and directions for future research.

CHAPTER 2

Mathematical Preliminaries

2.1 Piezoelectric materials and properties

The piezoelectric effect was first discovered in 1880 by Pierre and Jacques Curie who
demonstrated that when a stress field was applied to certain crystalline materials
an electric charge was produced on the material surface [3]. It was subsequently
demonstrated that the converse effect is also true; when an electric field is applied
to a piezoelectric material it changes its size and shape. This effect is due to the
electric dipoles of the material that spontaneously align themselves with the electric
field. Due to the stiffness of the material, piezoelectric elements generate relatively
large forces when their expansion is constrained. The relationship between applied
forces and the resultant responses of piezoelectric materials depend upon a number
of parameters, such as the material properties, size and shape, and direction in which
forces or electrical fields are applied. The constitutive equations for a linear piezo-
electric material, when the applied electric field and the generated stress are not very

large, can be written as

5" = Sgaj+dmiEm, (2.1)
Dm = dmi0i+€ikEka (2-2)

where the indices 2', j = 1, 2, ..., 6 and m, k = 1, 2, 3 refer to different directions
within the material coordinate system. In Equation (2.1) e , o , D and E are the
strain, stress, electrical displacement (charge per unit area) and the electric field (volts
per unit length), respectively. In addition, SE,d and 5 are the elastic compliance
(the inverse of elastic modulus), piezoelectric strain constant, and permittivity of the
material, respectively. For many structural applications, certain stress and strain
terms in Equation (2.1) are negligible and in these cases the constitutive equations
reduce to scalar equations. An example of such a structure is a flexible beam with a
laminated piezoelectric. For this application, Equation (2.1) boils down to two scalar
equations

5 = —01 + d31E3, (2.38.)
03 = 613101 + {33133, (23b)

In the absence of stress, Equation (2.3) can be simplified to

d V
51 = —31 (2.4)
ha
where, d31, the piezoelectric strain constant, is equal to the ratio of the developed
free strain to the applied electric field E = V/ha. In this equation V is the input

voltage to the piezoelectric and ha is the element thickness, as shown in Figure (2.1).

2.2 Asymmetric configuration of piezo transducer

One common form of arrangement of a piezoelectric actuator is the asymmetric con-
figuration shown in Figure (2.1). In this arrangement, the actuator is bonded to the
surface of the structure and when a voltage is applied across the electrodes (in direc-

tion of polarization) the actuator induces surface strains to the beam. It is assumed

10

that the beam is covered by a layer of thin piezoelectric material of thickness ha (see
Figure 2.1) which is perfectly bonded to the beam and produces a strain in the :1:
direction only. When a voltage is applied across the bonded piezoelectric element it
will attempt to expand but will be constrained due to stiffness of the beam. Due to
symmetric nature of the load, the beam will both bend and stretch, leading to an
asymmetric strain distribution as shown in Figure (2.1). In this figure, the origin of
the z axis lies at the center of the beam. In the linear region, the strain distribution

can be written as [3]

 

Figure 2.1. Piezoelectric asymmetric configuration and associated distribution

5(2) = 02 + 50 (2.5)

where C is the slope and 50 is the z intercept. Equation (2.5) can be decomposed
into the sum of an antisymmetric distribution 02 (i.e. flexural component) about the
center of the beam and a uniform strain distribution 50 (i.e longitudinal component)
as shown in Figure( 2.1). Using the strain distribution of Figure( 2.1) and Hook’s

law, the stress distribution within the beam can be written as

ob(z) = Eb(Cz + 80) (2.6)

11

where Eb is the Young’s modules of the beam.
The stress distribution within the piezoelectric actuator, 0198(2), is a function of
the unconstrained actuator strain, the Young’s modules of the material, Epe, and the

strain distribution shown in Figure 2.1. Mathematically it can be expressed as follows

apea) = Epe(Cz + 50 — 5,...) (27)

Applying force and moment equilibrium conditions about the center of the beam at

the origin of the x axis, we get

hb hb+ha
/ ope(z)dz+/ ope(z)dz = 0 (2.8)
0 hb
hb hb-l'ha
[0 ope(z)zdz+/h 0pe(z)zdz = 0 (2.9)
b

where hb is the half—thickness of the beam. In order to solve for the unknowns C and

50, we integrate Equation (2.8) to get
50 = Kche (2.10)

where K L is the material geometric constant, given by the expression [3]

 

16E§h§ + EbEpe(32hgha + 24h§hg + 8hbhg) + E38123 '
and C is the slope, given by the expression
C = Kfspe (2.12)

12

where K f is a constant [3] given by

16233123 + EbEpe(32h§ha + 24123123 + Shbhg) + 33.2.3

 

(2.13)

Equations (2.12 and 2.13) imply that the induced moment distribution in the beam

beneath the actuator, mm, is

f
= ——E”IbK d31v (2.15)
ha

where Eb and 1b are the elasticity and moment inertia of the beam respectively. The
response of the beam to the asymmetric actuator, as shown in Figure 2.1, consists of a
moment distribution mg; proportional to the excitation voltage V, specified by Equa-
tion (2.16), and a longitudinal strain distribution 5(2), specified by Equation (2.5).
The longitudinal strain is also proportional to the voltage and can be ignored in

comparison to the flexural component.

2.3 State space modelling of beams with piezo-

electric transducers

2.3.1 Beam dynamics

In this section we derive the equation of a flexible beam governing the dynamics of
a flexible beam using a piezoelectric actuator. In this derivation, the effects of shear
deformation and rotary inertia are not considered and it is assumed all the displace—

ments are small (Euler-Bernoulli beam). In Figure (2.2) the lateral vibration of the

13

beam in the my plane is shown wherein a piezoelectric actuator and a piezoelectric
sensor patch are bonded between locations 51:1 and 11:2, and x3 and 11:4, respectively.

Consider a free-body of an element of the beam shown in Figure (2.2). From Newton’s

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ily
~x PETactuator PZ/T sensor y
(\Iuif) M-l-dM
x a s— dx 4) M

x1 : +dq

x2 =

x3 _ y (M)

x4 7 : x

Figure 2.2. A pined-pined Beam instrumented with PZT transducers
second law, the dynamic force in the lateral direction is [3]

a27’0””)21 — aqd 2 17
or
2
t
mm = _@ (2.18)

6t2 3:1:

where m is mass per unit length of the beam and q is the shear force. Summing the

moments M about any point on the right face of the element yields

_@

M—Ba:

(2.19)

Where M is comprised of two parts, namely, the distributed moment produced by

the piezoelectric actuator (equal to KaV between .231 and x2 and zero anywhere else),

14

and the moment due to beam curvature. Thus

321/0? t)

M = KaV(Heatiiside(r — 2:1) — Heaviside(:c —- 132)) + EbIb Bt2, (2.20)

By combining Equations (2.18 to 2.20) we get the following equation for lateral

vibration of the beam

84110:, t) + 323100, t)

82:4 m 6t2 = Kai/(5%“: “ 1’2) — 5,013 — 1131)) (2.21)

Eblb
where 6’(.) is the derivative of the Dirac function with respect to :12.

2.3.2 State equation with piezoelectric actuator

In this section a state-space model of Equation (2.21) is presented from [51]. In
this representation the input to the system is the voltage V applied to the piezoelec-
tric actuator. we assume that the effects of the laminated piezoelectric actuator on
the mode shapes is negligible, which is a valid assumption if the dimensions of the
piezoelectric are small compared with those of the beam.

From the theory of vibrations, we know that the lateral displacement of a beam can

be written in modal coordinates as follows
ya, t) = Z ¢.~<x)n.~(t> (2.22)

where (tn-(:17) are the normalized orthogonal mode shapes and the 77,-(t) are the modal
amplitudes. Substituting Equation (2.22) into Equation (2.21) and projecting on to

the 2th mode yields the 2th decoupled modal equation

mm) + EMWWU) = We: — x2) — ¢’<x — motKavc) (2.23)

15

where, 1,0,- is given by
Lb

1.92 = 0 ¢t($)¢§m($l (2.24)

and where, Lb is the beam length. If we define

/ .
“122 g —Eb1b% (2.25)
m
A 1 z 2
Bi - RM (iv-$2) -¢> (iv—$1llK (2-25)
Equation (2.23) can be written as
(W) + wEm-c) = Bit/(t) (2.27)

It is clear from this equation that the ith mode is controllable if and only if B,- is
nonzero. If we truncate our representation to n modes, meaning that we are interested

in the first n modes of oscillation only, Equation (2.27) can be written as

2(t) = Az(t) + BV(t) (2.28)
where
T
A . . .
Z: 771 722 77a 771 772 (in (2329)
_ 0 I
Aé " n (2.30)
422 On

16

 

 

I -
0n
31

B 3 32 (2.31)
Bn

where f2 3- diag{w1, ....,wn}, and On and In are zero and identity matrices, respec-

tively, of dimension n x n. In this derivation, structural damping of the beam is not
taken into the consideration. One might approximate the effect of structural damping
by viscous damping through modifying the A matrix as follows

_ 0 I
A e n n (2.32)

w 422 —2(n

where C is the viscous damping coefficient.

2.3.3 Output equation with piezoelectric sensor

For a piezoelectric material, the ratio of the strain in the material to the charge

density is constant and is denoted by

Strain Developed

: 2.33
Applied Charge Density ( )

 

931

The incremental charge dQ generated on an infinitesimal area of the piezoelectric
(assuming the width of the piezoelectric to be equal to the width of the beam, b) is

therefore

 

2 2
(162 = a “22/8": hbbda: (2.34)
31

where b and 2hb are the width and thickness of the beam, respectively. Equation (2.34)

can be integrated over length of the beam covered by the piezoelectric sensor to yield

17

the expression for the output voltage, Vs,

 

Vs(t) = (ii/(934, 15) - #03 It)) (235)

v.0) é Kat/(24¢) — y’(:c3,t)) (2.36)

where C19 is the capacitance, and .223 and 11:4 are the start and the end locations of
the piezoelectric sensor on the beam (see Figure (2.2)). Substituting Equation (2.22)

into Equation (2.36) yields
00
Vs(t) = Ks Z 77i(t)ld>§($4) - ¢§($3)l (2-37)
i=1

For an n mode approximation, we have

Vs(t) = Ks Z 772(t)l¢>§(5v4) - 253333)] (2-38)
321
Byt defining C,- as
Ci = Ks[¢§(x4) - 453163)] (2-39)
we get
V3(t) = Ciz(t) (2.40)

It should be noted that the 2th mode is observable if and only if C, is nonzero.
Equation (2.28) together with Equation (2.40) provide a state space representa-
tion of the dynamics of an Euler-Bernoulli beam with a piezoelectric actuator and a

piezoelectric sensor.

18

 

2.4 Beam dynamics in the presence of an end-force

In this section we present the mathematical model of a flexible beam in the presence
of an end-load. The nature of the end-load can be either conservative such as an
axial force (buckling force) or non-conservative such as a follower force, or it can be
a combination of both. Consider the free-body diagram of an element dz along the
length of beam shown in Figure (2.3). The force P reflects the end load and it is
assumed to be constant for small deflections of the beam. Now, let m = mass/ length,

6 = 6y(:z:, t) / 63: = slope, and q=shear force, shown in Figure (2.3).

 

 

 

Figure 2.3. Free body diagram of element d1: with an end load

From Newton’s second law, we have

82y(:r,t) 39 86
—5t—2—dz — —(q + grim) —I- q — P(t9 + Eda?) + P9 (2-41)
or
2 2
m6 WI”) = -92 _. RM) (2.42)

8132 8:1: 6:32

Substituting Equation (2.18) in Equation (2.42) results in

64m, t) + mazya, t) + Page,»

Eb 1b 8:134 (9t2 8:132

= 0 (2.43)

19

which is the equation of the beam in the presence of the end load P. It should be
noticed that the governing equation is independent of the type of the end load (axial,
follower, etc). The boundary conditions for these different cases, however, will be
different. We will discuss the boundary conditions for our particular case in Chapters

4 and 5.

20

CHAPTER 3

Switching Piezos between Actuator

and Sensor Modes

3. 1 Background

In the first part of this chapter we develop general results for gaining controllability
and observability in under-actuated and under—sensed systems through switching. In
our study, as different from the work in [52], [53] , where controllability and observ-
ability for pre—assigned switching sequences is studied, we investigate the minimum
number of switchings required to achieve controllability, observability, and to design
controllers based on observers. In the second part of this chapter, we present a novel
application of switching control for vibration suppression in flexible structures. We
consider a flexible Euler-Bernoulli beam instrumented with PZTs for our simulation
and experiments. The PZTs are attached to the beam such that the system is com-
pletely controllable when all of them are used as actuators and completely observable
when all of them are used as sensors. The underlying objective is to switch piezo
transducers between actuator and sensor modes and thereby reduce the number of

transducers and associated hardware required for vibration suppression.

21

3.2 Effect of switching on controllability and ob-
servability

Consider a multi—input linear time-invariant system with the minimum number of
inputs required for complete controllability. Now imagine a situation where all the
inputs cannot be used at the same time, meaning that there are some inputs that can
be used in the control loop (active) while the rest cannot be used (inactive). This
system will not be completely controllable. If the roles of the active and inactive
inputs are reversed, the new system will still lack complete controllability since it will
have a set of inactive inputs. The time-varying system, comprised of the two time-
invariant systems, with the inputs switching between their active and inactive modes
will however be controllable. This is stated with the help of the following Theorem.
Theorem 1: Consider the linear time-varying system that switches between the

two time-invariant systems
:i? = An: -I- Blul, it = Ail: + 3211.2, 513(t0) = $0 (3.1)

where a: E R” and {A, [81, 82]} is completely controllable, but neither {A, Bl} nor
{A, 32} are completely controllable. The switched time-varying system is completely
controllable on the interval [t0, t2] if and only if the number of switchings within the
interval is one or greater.

Proof: To prove sufficiency, consider one switching at t = t1, to < t1 < t2. Then,

the controllability grammian can be written as

t1 t2

W<to.t2> = /. ¢<t0,t)st§QT(to,t>dt (3.2)

0

<I>(t0, t)BlB[F<I>T(t0, t) dt + f
t

1

where <I>(t0, t) is the state transition matrix. We prove sufficiency, zle. show that

22

the grammian is full-rank, by contradiction. If the grammian is rank deficient, we

can find a vector 23a, 33a 75 0, such that rgWUO, t2):ra = 0. This implies

xZo(t0,t)Bl=0, t0§t<t1 (3.3a)
z£©(to.t)32=0. (11992 (33b)

Considering the time-invariant nature of the individual systems in (3.1), we can claim

$51313A31,4231, ° “An—lBll = 0, $ZIB2IAB2IA2B2, ' ' ' A71—132] = 0
(3.4)

which can be written as
sci-{11131. 32]. 21181.32], - - - Ail-1181, 8211 = o (3.5)

The above equation violates the assumption that {A, [81, 82]] is controllable and this
proves that the grammian is full-rank, or the time-varying system is controllable. To
prove necessity, we simply show the grammian to have rank deficiency for t1 > t2,
which corresponds to the case of no switching. o o o

The result in Theorem 1 can also be deduced from Theorem 2 in [52]. A simple
example is presented next to illustrate the result in Theorem 1.

Example 1: We assume the two time-invariant systems in Theorem 1 to have

the description

1 0 1 0
A é , Bl é , B2 g (3-6)

0 2 0 1

Clearly, {A, BI} and {A, 32} are not completely controllable, whereas {A, [81, 32]}

is completely controllable. For to = 0.0, t1 = 0.5, t2 = 1.0, and initial system

23

description :1: = A2: + Blul, the state transition matrix and controllability grammian

are described by the relations

<I>(O,t) = , W(0,1) =

(3.7)
To converge the system states to a: = 23f at t2 = 1.0, we designed a controller for the

time-varying system as follows [54]

u(t) = BT(t)<I>T(O, t)W‘1(0, 1) {—a:0 + <I>(0,1):rf} , 0.0 g t g 1.0 (3.8)

where r
1 0
for 0 S t < 0.5
0 0
30) = < (39)
0 0
for 0.5 S t < 1.0
k 0 1

 

The simulation results for :1: f = (0, 0)T and 2f = (—1, —2)T are shown in Figure (3.1),
respectively. The results show that all the states are converged to their desired values.
It should be noted from Figure (3.1a) that 3:1 is converged to zero at t = 0.5 since 5131
is uncontrollable thereafter. For 0.5 S t S 1.0, 11:1 remains at zero since 1:1 = O is an
equilibrium point. In the case of Figure (3.1b), 2:1 is converged to an intermediate
value at t = 0.5. Starting at this intermediate value, 1:1 converges to its desired value
of —-1 over the interval 0.5 S t S 1.0, simply by virtue of being unstable. In both
cases, the state 2:2 is uncontrollable during the first interval but is converged to its
desired value by proper choice of input 112 over the interval 0.5 S t S 1.0.

Since observability is a dual property of controllability, we extend the result in

24

 

 

 

 

 

 

 

 

 

3.0 . . . . a . . . . 3.0 . . . . , . . . .
: (a) . ’ i (b) ‘
§ 2.0 3 i 3
2.0 ~ x . x ,
5 1.0 E .
1.0 3 i
+ X] i 0.0 . X
0.0 . ‘11)]
_05 . . - . t . . . L _2.0 - . . - i . . . .
0 0.2 0.5 0.8 1.0 0 0.2 0.5 0.8 1.0
time(sec) time(sec)

Figure 3.1. Plot of state variables in Example 1 for (a) xf = (0, 0)T and (b)
x, = (—1, —2)T

Theorem 1 to the problem of state estimation using Theorem 2, stated next.
Theorem 2: Consider the linear system :i: = A2: + Bu, whose output is time-

varying and switches between the relations
01 = C113, :12 = C213 (310)

where {A,C} is observable for C = [C{,Cg]T, but {A, C1} and {A,Cg} are not
observable. The switched system is observable on an interval if and only if the number
of switchings within the interval is one or greater.

Proof: The proof is very similar to the proof of Theorem 1. o o o

3.3 Switching requirement for observer-based con-
troller

The controller used in Example 1, given by the expression in Equation (3.8), assumes

knowledge of the initial state 51:0. If the initial state is unknown, it has to be first

25

estimated and if Equation (3.10) represents the output description of the system, one
switching will be required for state estimation. This follows from Theorem 2 and
motivates the next theorem on observer-based control design.

Theorem 3: (Observer-based Controller) Consider the linear time-varying system

that switches between the two time-invariant descriptions
21: i: = A112 + Blula y1 = 01:1: (3.113.)

22 : (it = Ax + Bgug, yg = C22: (3.11b)

If a: = (strip, 23;, 23%“)? where .731 6 R10 is controllable and observable (CC) for both
231 and 22, $2 6 Rq, q 74 0, is controllable but unobservable (CC) for El and
uncontrollable but observable (CC) for 22, and 2:3 6 RT, 7' 7f 0, is CO for 21 and
CC for 22, then

(i) {A,Bl}, {A,Bz} are not completely controllable but {A, [81,82]} is com-

pletely controllable.

(ii) {A,Cl}, {A,Cg} are not completely observable but {A, [C¥1,C52T]T} is com-

pletely observable.

(iii) All the states of the switched system can be steered to the origin in finite time

using estimated states if and only if the number of switchings is two or more.

Proof: Since 7* aé 0, {A, Bl} is not completely controllable. Similarly, {A, 82} is not
completely controllable since q 75 0. The states 2:1 and $2 are controllable with input
matrix Bl whereas states 1:1 and 2:3 are controllable with input matrix 32. Therefore,
all the states, 271, 2:2, :53, are controllable with input matrix [81, 82]. This completes
the proof of (i). The proof of (ii) is very similar to that of (i) and is skipped. We

prove (iii) next, as follows:

26

Sufficiency: Let the initial time be to and the initial state be 230 = x(t0). For the
1,3211}? y = {it/ff, yg}T, and without loss

of generality we assume the initial system description to be 21. To prove sufficiency

purpose of convenience, we define u = {14"

for two switchings, we assume the switching instants to be t = t1 and t = t2, the
final time to be t f, and tlg to be an intermediate time, all of which satisfy to < t1 <
t12 < t2 < t f. We now claim that the following observer-based control design moves

all states of the switched system to the origin:

0, to S t < tn
”0) =

in
-BT(t)¢T(t12,t)W’1(t12yt;)¢'>(t12,toll/"100812) f ¢T(T.t0)CT(T)y(T)dTa t12 S t S tf

t

0 (3.12)
In the above equation, W(., .) and V(., .) are the controllability and observability

grammians, respectively, gb(., .) is the state transition matrix, and B(t) and C(t) are

defined as follows:

(Bl, 0), for to _<_ t < t1

B“) = (0, 32), for t1 S t < t2 (3.13a)
(81,0), for thtStf
(CT, 0)T, for to St < t1

C(t) = (0, 023’)? for t1 g t < t2 (3.13b)
(Ci-r, 0)T, for t2 3 t g tf

To prove our claim, we first note that y(t) = C(t):1:(t) and y(t) = C(t)¢(t, t0):ro on the
interval to S t < t12 since u = 0 on this interval. From (ii) we know that the switched
system in Equation (3.11) satisfies the conditions of Theorem 2. Furthermore, there

is one switching in the interval [to, t12). Therefore, the observability grammian

t12
V(tott12) = ¢T(T,t0)CT(T)C(T)¢(T,toldT (3-14)

to

27

is nonsingular and its inverse exists. Hence :r(t12) can be estimated as follows

W12, t0)'~100 (3.153.)

= ¢>(t12,t0)V_1(t0,t12)V(t0,t12)fE0 (3.151))
t
= ¢(t12,t0)V_1(t0,t12) t12¢T(T,t0)CT(T)C(T)¢(T,t0):1:0d7'(3.15c)

0

in
= ¢(t12,t0)V71(t0,t12) ¢T(T, t0)CT(T)y(T)dT (3-15d)

0

$012)

Equation (3.15) is true since since u = 0 over the interval [to, t12). Using this equa-

tion, our claim in Equation (3.12) can be simplified to the form
=—BTt Tt tW‘lt t t t <t<t 316
u ()¢(12,) (12,f)$(12) 12— —f (-)

From (i) we know that the switched system in Equation (3.11) satisfies the conditions
of Theorem 1. Since there is one switching within the interval [t12, t f], the control-
lability grammian W(t12,tf) is nonsingular and hence the control input given by
Equation (3.16) moves the states from :z:(t12) at t = t12 to their final values xf = 0
at t = t f. This can be easily deduced from Equation (3.8) in Example 1. The above
proof can be easily modified to establish sufficiency when the number of switchings
is greater than two.

Necessity: Let the initial and final times be to and t f, respectively. We prove
necessity by contradiction. Suppose all the states, 5: = {2%, mg, 2%}? can be
steered to the origin at t = t f after one switching at t = t1 (to < t1 < t f) using
estimated states. Let the initial system description be 21 and it switches to 22
after the switching. Since q 31$ 0, $2 is the nonzero set of uncontrollable states of
22. Since 1:2 is not controllable, 22(tf) = 0, and (t f — t1) is finite, we must have
22(t) = 0, Vt E [t1,tf]. Although 2:2 is controllable in 231, it is not observable.

Therefore, it cannot be steered to 2:2(t1) = 0 from an arbitrary initial condition,

28

$2030), in finite time. This contradicts our earlier claim that 2:2(t1) = 0. If the initial
system description is 232 and it switches to 21 after the switching, 2:3 is the nonzero
set of uncontrollable states of 231, since 1" ¢ 0. We can again prove by contradiction
that it is not possible to steer 253, and hence all the states, to the origin in finite time.
o o o

In our discussion so far, we did not address the issue of stability. It is well known,
[34] for example, that arbitrary switching between asymptotically stable systems can
result in instability. For our observer-based control designs, we will avoid this problem

by simply ensuring that the switching interval is greater than the dwell time [34].

3.4 Vibration suppression in a flexible beam

In this section we consider the problem of vibration suppression in the flexible beam
shown in Figure (3.2). Our objective is to design an observer—based feedback controller

that will suppress the vibration in the first four modes of the beam using the two

PZT elements shown in the figure.

 

 

 

veiw sectional view
PZT-1 ’ PZT-2 pzr
...4 / L - -- - 1
l

7/7//// 5mm

41> ] 41>~~1

 

 

 

 

 

50mm

 

--Q

 

 

 

2: 1 m t:

Figure 3.2. A simply supported flexible beam with two piezoelectric elements

In conformity with our discussion in Section 2, we define the linear time-invariant
system (A, B1, Cl) based on use of PZT-1 as the actuator and PZT-2 as the sensor,

and the system (A, 82, C2) based on use of PZT-1 as the sensor and PZT-2 as the

29

 

actuator. For the goal of vibration suppression in the first four modes (n = 4), it
can be easily inferred from Figure (3.3) that the fourth mode is uncontrollable and
the third mode is unobservable for (A, Bl, C1) since PZT-1 lies at the node of the
fourth mode and PZT-2 lies at the node of the third mode. For (A, 82, Cg), we
can similarly conclude that the third mode is uncontrollable and the fourth mode is
unobservable. In relation to our discussion in Section 3.2, we have p = 4, q = 2, and
r = 2 for both systems (A, B1, C1) and (A, 32, C2). For example, the controllable
and observable states corresponding to p = 4 are comprised of the displacement and

velocity of the first and second modes.

 

    
   

 

 

 

 

I "\ 3rdmode 4thmode ,’"\
1.0 P / \ / \ s .2
I s \
I \ \
r I s \ 1
\

0.0 /
1-0 * center location ,’ center location ‘

I' PZT 1 ~ - a ’ PZT 2 d

0.0 0.2 0.4 0.6 0.8 1.0
length (m)

Figure 3.3. Third and fourth mode shapes of beam in Figure (3.2) and location of
PZT elements in relation to these mode shapes

3.5 Simulation of observer-based control design

The two linear time-invariant systems (A, Bl, C1) and (A, 32, C2), discussed in Sec-

tion 3.4, have the explicit form

2': = A2: + Blul, y1 = C12 (3.17a)

30

2': = A2: + 32112, yg = C22 (3.17b)

where :c E R8, and 111,112, y1, y2 E R. Using the beam dimensions in Figure (3.2), C =
0.01, material properties of aluminum, and mathematical relations given in Chapter
2, the entries of the matrices A, Bl, 82, Cl, and C2, defined in Equations (3.17),

were computed as follows

A = 0" 1" (3.18a)
422 —2§n

r22 = «4 diag (58.3, 933.3, 4725, 14933.3) (3.18b)

Bl = [01% —0.006, —0.032, —0.044, 0.000]T (3.180)

32 = [01x4, —0.004, —0.016, 0.000, 0.058]T (3.18d)

01 = [—0.606, 2.369, 0.000, —8.659, 01X4]x105 (3.18e)

(:2 = [—0.910, —4.757, —6.592, 0.000, 01x4]x105 (3.18f)

It is clear from the zero entries of 31 and C1 in Equations (3.18) that the fourth mode
is not controllable and the third mode is not observable for (A, 81,01). Similarly, the
third mode is not controllable and the fourth mode is not observable for (A, Bg, Cg).

For both systems, we designed observer-based controllers as follows

111 = —K15’5 I? = A5? + Blul + L1(y1 - 01:13) (3.198.)

82 = _ng 2 = A5? + 321.2 + L2(y2 — 022) (3.191))

where the matrices of controller and observer gains, K1, K2, L1, L2 were designed
using standard pole—placement techniques. Although these matrices are not shown
here for the sake of brevity, it should be noted that the entries of K1 and K2 (L1

and L2), that correspond to the uncontrollable (unobservable) states, are zero. Our

31

 

simulation results, for initial conditions are shown in Figure (3.4).

12(0) = (0.2, —-0.4, 0.2, 0.2, 0.08, 0.0, —0.2, —0.4)T, 55(0) 2 (01x4, 01x4)T (3.20)

 

 

 

 

 

 

 

 

 

 

 

 

 

I=(A.Bl.C1)

II= (A.B2.C2)
0.2 I II : I : II I t 0.4 1 ' II I II I «
0.1 I ] fir” l 0.2 ] 5mm]
0.0 - < 0.0 "1“»: W, ,,,,,,,,,,,, -
-0.1, 1 l . -02 ]
-02’ - - - ‘ -o.4 - 2 -

o 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

time (sec) time (sec)
(3) (b)

 

 

    

 

 

 

 

 

 

 

 

0.2 I 0.4‘ 0.6 A. . ' 0.2 0.4 06‘ 0.8 1.0
time (sec) time (sec)

(0) (d)

Figure 3.4. Amplitude of vibration of the (a) first, (b) second, (c) third, and (d)
fourth modes of the beam with observer-based control design

The time duration of simulation is 1.0 sec and switching occurred at 0.2, 0.4, 0.6
and 0.8 secs, starting with the system with the description (A, Bl, C1). These results
indicate that the first and second modes of vibration, which are controllable and ob-
servable for both system descriptions in Equations (3.17), are attenuated rapidly and
continuously. The fourth mode is observable for the system description (A, Bl, C1)
and controllable for the description (A, B2, C2). Therefore it is rapidly attenuated
during all time intervals where the system description is (A, 82,02), namely, sec-
ond interval, fourth interval, and so on. The attenuation during the first, third, and

fifth intervals are due to structural damping. The third mode is observable for the

32

system description (A, 32, C2) and controllable for the description (A, 81,01). This
mode is rapidly attenuated during all time intervals where the system description is
(A, 81,01), except the first interval since observability is required prior to controlla-
bility for the purpose of control.

The results in this section indicate that we are able to sense and control all four
modes of the beam using two piezoelectric elements only. With dedicated and col-
located sensors and actuators, we would have used two piezoelectric elements as
actuators and an additional two elements as sensors. This would require two power
amplifiers for the two actuators and two analog-to-digital conversion channels in our
data acquisition hardware for the two sensors. Clearly, our approach has halved the
number of piezoelectric elements, the number of power amplifiers, and the number of

data acquisition channels, required for sensing and control.

3.6 Sub-optimal switching schedule

By substituting the feedback laws of Equation (3.19) into Equation (3.17), we get the

two closed-loop systems

X = Ach, X = .4ch (3.21)

where X é (2T, ET)T and

(3.22)

 

Given a total time duration of t f secs and m switchings, our goal is to find the
switching times t,, 2' = 1,2,---m, that satisfy 0 S t1 S t2~~ S tm S tf and
minimizes the cost function

tr
J = / XTQX dt (3.23)
0

33

where Q is a constant positive definite matrix. Both AC1,A02 E R2(p+q+r) have
2;) + q + r eigenvalues with strictly negative real parts by virtue of pole placement.
The q + 7‘ remaining eigenvalues correspond to the uncontrollable and unobservable
states and have zero real parts or small negative real parts depending on the level of

structural damping. Consequently, we can solve the Lyapunov equations
ATP PA =— ATP PA =— 324
c11+lc1 Q, c22+262 Q (-l

to obtain unique solutions for P1 and P2. Assuming the closed-loop system matrix

 

to be A01 at the initial time, the cost function in Equation (3.23) can be written as

t t t t
/<>/<>/—<>/—<>
(3.25)

and then simplified to the form

J = X(O) P1X(O) — X(tf)TP.X(tf) + x21)T (P2 — P1) X(t1) (3.26)

— X<t2>T (P2 - P1) X02) + - - - X<tm>T (P2 — P1) X(tm)

where P... equals P] for even number of switchings and P2 for odd number of switch-
ings, and the sign of the last term in Equation (3.26) is positive for odd number of

switchings and negative for even number of switchings. Since the states and their

estimates at t = t1, t2, - - - can be defined iteratively as follows
X01) = eXPIAc1t1IX(t0) (337a)
X02) = eXpIAc2(t2 - t1)IX(t1) (13-271))

34

the cost function in Equation (3.23) can be expressed as
J = XT(t0)HX(t0) (3.28)

where H is a positive definite matrix that is a function of t1,t2, - -- ,tm. With the

objective of minimizing J, we rewrite Equation (3.27) as

 

 

J = trace[J] (3.29a)
= trace 'XT(t0)HX(t0)] (3.2%)
= trace 'HXT(t0)X(tO)1 (3.29c)
= trace 'HXT(t0)X(t0)] (329(1)
_<_ trace[H]||X(t0)]|2 (3.29e)

Since the initial state, X (to), is unknown, we propose to minimize J by minimizing
trace [H]. This approach, which has been proposed earlier ([55], [56], for example)

results in upper-bound minimization since
XT<to>HX<to) 3 WM) I|X(to)ll2 s traceIHI IIX(t0)|I2 (3.30)
We minimize the upper bound by solving
t1,t2, - -- ,tm = arg {min(trace[H])} (3.31)

subject to 0 < t1 < t2 < < tm S t f. We choose to minimize the upper bound
rather than the maximum eigenvalue of H since trace [H] can be easily expressed as
a function of the switching times, t1,t2, - -- ,tm, unlike Anna-(H) which cannot be
expressed in terms of the switching times. However, since we minimize the upper

bound of XT(t0)HX(t0), our solution is sub-optimal and not optimal. The sub-

35

optimal switching instants are also solved in an open-loop fashion and is not based on
state feedback. The simulation results for sub-optimal switching, based on the identity
Q matrix and the initial conditions in Equation (3.20), are shown in Figure(3.5). The
time duration of simulation is 1.0 sec and the number of switchings is set to four.
This allows us to compare the results with the simulation results in Figure(3.4). The
sub-optimal switching instants were obtained as t1 = 0.01, t2 = 0.02, t3 = 0.1, and
t4 = 0.37. A close look at the results obviates that the transient response is better for

the case with sub-optimal switching than the case with switching at regular intervals.

 

 

         
   
 
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l= (A.Bl,C1)
II: (A,Bz,C2)
0.2 l ] [I II 1 4 04 I ' II I II I .
i first - second -
0.0 - 0.0 ,1
l ] l
-02 ' . -o.4 *
0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
time (sec) time (sec)
(:0 (b)
I 0 7
third '7

   
   

 

 

 

 

 

l
I
0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
time (sec) time (sec)
(C) d

Figure 3.5. Amplitude of vibration of the (a) first, (b) second, (c) third, and (d)
fourth modes of the beam with sub—optimal switching

36

3.7 Experimental verification

3.7.1 Apparatus

In order to demonstrate feasibility of switching piezoelectric transducers between ac-
tuator and sensor modalities and obviate the merit of introducing under-actuation
and under-sensing, we performed experiments with the flexible cantilever beam shown
in Figure(3.6). Although we used a simply supported beam in our simulations in Sec-
tions 3.5 and 3.6, we used a cantilever beam in our experiments since the experimental
setup is simpler and the modal frequencies are lower for the cantilever beam. With
lower modal frequencies, the task of real-time data acquisition and control becomes

easier.

  
    
     
   
   
 

 

switching circuit box

amplifier

piezoelectric transducer
strain gage sensor

steel beam: 0.68m X 0.035m X 0.001m

 

Figure 3.6. Experimental test—rig

In our experiments, we used a single piezoelectric element* to sense and control
the first two modes of the beam. Both the modes are controllable but unobservable
when the element is used as an actuator, and observable but uncontrollable when it

is used as a sensor. Our experimental results will establish that we are able to obtain

 

”manufactured by Mide Technology Corporation [57]

37

complete controllability and observability by continuously switching the modality of
the single piezoelectric element. This reduces the number of piezoelectric elements
required by 50% since two elements, one dedicated as an actuator and the other
dedicated as a sensor, would be typically used for sensing and controlling the two
modes. With the dedicated actuator and sensor pair, we would have required one
power amplifier and one data acquisition channel. Our experiments also required
one amplifier and one data acquisition channel but both the amplifier and the data
channel were used only 50% of the time. In a setup with two piezoelectric elements,
similar to the one used in simulations, a single amplifier and a single data acquisition
channel would suffice since they would be time-shared between the two piezoelectric
elements. In such a setup, under-actuation and under-sensing would contribute to
greater cost and weight savings. In our experiments, the strain gage sensor I was only
used for experimental determination of the transfer functioni of the dynamic system
representative of the first two flexible modes of the beam. The procedure for transfer
function determination is described next. The first three natural frequencies of the
beam were analytically determined to be 1.8 Hz, 11.3 Hz, and 31.6 Hz, respectively.
To model the first two modes of the beam only, we excited the piezoelectric element
sinusoidally over the range 0-20 Hz and used the voltage output of the strain gage
sensor, Ysg(s), to compute the gain and phase lag of the system. Using Matlab, the

transfer function that best fit the gain and phase plots was determined to be

01(3) : M = 2475 32 + 0.77753 + 748.5
UIS) (.92 + 0.4523 + 127.7)(s2 + 2.63 + 4225)

 

(3.32)

From the structure of the transfer function in Equation (3.32), it is clear that the

system is completely controllable and observable with the piezoelectric element as

 

lmanufactured by PCB Piezotronics, Inc. [58]

1between voltage input to the piezoelectric element (used as actuator) and voltage output of the
same element (used as sensor)

38

actuator and the strain gage sensor. A complete modal state space representation of

the system was therefore obtained as follows

- j (- '1 r- - T
0.0 0.0 1.0 0.0 0.0 0.15
0.0 0.0 0.0 1.0 0.0 0.21
—127.69 0 —0.45 0 2.5 0.0
_ 0.0 —4225.0 0.0 —2.60 _ L 10 J _ 0.0 ]

 

 

 

 

 

 

which obviously satisfies C1(s) = C(sI — A)—lB. Our desired transfer function,
between voltage input to the piezoelectric element (used as actuator) and voltage
output of the same element (assuming it is simultaneously used as sensor), szt(s),

was obtained using the relation

szt (5)

0(3) ZmSI‘AVIB’ 5310-11811. 0.21a2, 0.0, 0.0] (3.34)

C(s) =

where 01 and a2 denote the ratios of the output of the piezoelectric sensor and the

strain gage, corresponding to the first and second modes of vibration, respectively.
In continuation first, we discuss the combination of analytical and experimental

methods used in determining al and 02 in Equation (3.34). The voltage output of

the piezoelectric sensor, szt, and the strain gage sensor, ng, can be expressed as

823/(xpzta t)

T’ (3.35)

szt = szt

where szt and K 39 are constants that are characteristic of the piezoelectric sensor
and strain gage sensor, respectively, and Cszt and 239 are the locations of the two
sensors, respectively. In Equation (3.34), 01 and ag are equal to the ratio (szt/ng)
when the beam is oscillating purely in the first and second modes, respectively. By

deflecting the tip of the beam and releasing it, the beam was made to oscillate purely

39

in the first mode. During such motion, szt and V39 were measured and their ratio
computed. This ratio, which was found to be approximately constant, is equal to
(11. With the knowledge of a1 and by numerically evaluating 62y(:cpzt,t)/8t2 and
82y(zsg,t)/8t2 for the first mode, we were able to obtain the ratio szt / K 39. We
next evaluated 82y(:rpzt,t)/8t2 and 62y(2¢sg,t)/6t2 for the second mode and with
the knowledge of szt / K 39, we computed 02.

In our case, these ratios were found to be al 2 3.33 and a2 = 33.33. Therefore,
C(s) was described by the A and B matrices in Equation (3.33) and the new output
description matrix 6 = (0.5, 7.0, ,0.0, 0.0)T.

We complete this section with a discussion of the hardware used for data acquisi-
tion, control, and switching. We programmed our control law and switching algorithm
in Matlab/SimulinkTM environment and downloaded it to our dSPACE DSP board,
shown in Figure(3.7). The DSP Board reads the measured signals when the piezo—
electric transducer is in the sensor mode and estimates the states of the flexible beam.
When the transducer is switched to actuator mode, the estimated states are used to
compute the control input required for suppressing the vibration in the beam. The
computed control signal is sent to the power amplifier which provides the voltage
required for actuating the piezoelectric transducer. A trigger signal is used to switch
the piezoelectric transducer between actuator and sensor modes. For a very short
time interval, prior to triggering the transducer from actuator mode to sensor mode,
the voltage commanded to the transducer is set to zero. This enables us to switch to

the sensor mode with zero initial conditions.

3.7 .2 Results

The piezoelectric transducer in our experimental hardware guarantees observability
of several modes of the beam, including the first two, and hence the the voltage

output of the transducer, ypzt(t), is a good measure of vibration of the beam. Our

40

trigger

 

7

 

 

 

I

G r AID ;
power in J) computer
- signal out D8315: d
@T‘ A power D / A 4 ar
amplfier“ ‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

switching
c1rcu1t

Figure 3.7. Schematic of switching between actuator and sensor modalities

experimental results are shown in Figure(3.8). In each of these experiments, the beam
was set into vibration by displacing the beam—tip 5 cms (corresponds to maximum
sensor output of 10 volt in dynamic mode) and releasing it at the initial time. The
initial condition (in modal coordinates) was therefore a nonzero value for the first
state 721 in Equation (2.28)and zero values for the other three states, 172, in, and fig.

The first experiment, shown in Figure(3.8)(a), pertains to free vibration of the
beam. In the absence of control, the beam vibration is attenuated slowly by the
structural damping. The time required for the amplitude of vibration to decay by a
factor of 25 is approximately 50 secs.

The second experiment, shown in Figure(3.8) (b), pertains to vibration suppression
using the single piezoelectric transducer, switching on a fixed-time schedule. In this
experiment, the transducer was switched between actuator and sensor modes with a
time period of 1 sec. Within the 1 sec duration, the transducer was used as a sensor
for 300 msecs and as an actuator for 700 msecs. Figure(3.8) (b) shows the transducer
output only during the time intervals it was used as a sensor. The effectiveness of
the switching observer-based controller is clear from the time required for vibration
attenuation, which is only 12 secs as compared to 50 secs in Figure(3.8)(a).

A closer look at A closer look at Figure(3.8)(b) indicates that the switching

observer-based controller excites the higher modes of the beam. Though these modes

41

die out because of structural damping, they can have detrimental effects. The excita-
tion of the higher modes is attributed to the discontinuous controls profile generated
by the fixed-time switching schedule of our algorithm. This is further explained with
the help of F igure(3.9)(b). To avoid discontinuous inputs, we adopted a variable-
time switching schedule where the piezoelectric transducer is switched (both from
sensor mode to actuator mode and from actuator mode to sensor mode) at the first
instance when the control input is zero, beyond the time proposed by the fixed-
time switching schedule. The variable-time switching schedule is explained with the
help of F igure(3.9)(c). The results obtained with variable-time switching are shown
in Figure(3.8)(c) and here the switching times, t = 0.61, t = 1.08, t = 1.51, and
t = 2.21, for example, are the first instants of time when u = 0 after t = 0.30,
t = 1.00, t = 1.30, and t = 2.00, respectively. It can be seen from the Figure(3.8)(c)
that the higher modes are not excited.

The experimental results in F igure(3.8)(d) pertains to a sub—optimal switching
schedule with m = 10 switchings and t f = 5 secs. We chose m = 10 and t f = 5
to compare the results with those obtained from the fixed—time switching schedule
(Figure(3.8)(b)), which also switches 10 times during 5 secs. The optimal switching

instants were obtained apriori using the Nelder—Mead simplex search in Matlab as

t1,t2,--- ,th = 0.254, 0.924, 1.459, 1.911, 2.267, 2.966, 3.472, 3.938, 4.267, 4.745
(3.36)

A comparison of the amplitudes of vibration in F igure(3.8)(b), F igure(3.8)(c), and
Figure(3.8)(d) indicate that the sub-optimal switching schedule is more effective in
attenuating vibration that the fixed-time and the variable-time switching schedules.
We conclude this section with a discussion on the variable—time switching schedule
which was adopted in our experiments to avoid excitation of higher unmodeled modes

of the beam. Consider an observer-based feedback control law, it = K ff, which has

42

 

8]] 8 :r; ------- t=l.0
1&0 """ go:
>‘ >5

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
   

 

 

 

 

 

-4 .4 5
] damped free vibration ‘ 34:20 fixed ‘
‘8 ' ‘ '8 'm ""t=0.3 switching schedulet
0 10 20 30 40 50 2 4 6 8 10 12
time(sec) time(sec)
(a) (b)
3 , ,.I-t= 1.51 ' ' ' . 3 lultéoi92' 'gtib-tptimti ' .
’ § ‘ . t= 1 46sw1tch1ng schedqu
4 We , 4 ,---t='2.27
E. 0 i , ii. 0 L '
>4 ; , >t .
.4 , -4 , damped free vibration .
] ;_._-t=221 . bl 4 §t=19‘ : ‘
8 ---~t= 1.08 . [Yima god 1 ] -8 :___- ' ‘ ‘
' *""t.= 06.1 SW?“ “.3“ . “° ~-~-.---.=.0-2.5. . . . - . .
0 2 4 6 8 10 12 0 2 4 6 8 10 12
time(sec) time(sec)
(C) (d)

Figure 3.8. Experimental results: Plot of ypzt with time for (a) uncontrolled sys-
tem, and controlled system with (b) fixed switching schedule, (0) variable switching
schedule, and (d) sub-optimal switching schedule.

the profile shown in Figure(3.9)(a). For our experiments, where a single piezoelectric
transducer was used, 2? was obtained using a closed-loop observer when the trans-
ducer was in the sensor mode, and obtained using an open-loop observer when the
transducer was in the actuator mode. Now consider the fixed-time switching schedule
used in our experiments, where the piezoelectric transducer was alternately used as a
sensor for 300 msecs, and as an actuator for 700 msecs. For this switching schedule,
our control input would have the profile shown in Figure(3.9)(b). The control input
would be discontinuous due to the fixed-time nature of the switching and this would
excite higher unmodeled modes of the beam. This has been seen in our experimental
results in Figure(3.8)(b). To alleviate this problem, which is caused by discontinuity
in the input, we delayed switching (both from actuator mode to sensor mode and

sensor mode to actuator mode) till the first instance of time when the control in-

43

put is zero. Such a switching schedule results in the control input profile shown in
Figure(3.9)(c). Although the control input is non-smooth, it is continuous and there-
fore does not excite the higher-order unmodeled dynamics of the beam. This can be

verified from our experimental results in Figure(3.8)(c).

-4 s‘ifilfcliliig
-8

 

time(sec)

Figure 3.9. (a) An observer-based controller (b) a discontinuous and (c) a continuous
controller obtained from using a fixed-time and variable-time switching schedule.

44

CHAPTER 4

Vibration Control of a Flexible

Beam using an End Force

4. 1 Background

In this chapter we propose a novel approach in vibration suppression of a flexible
beam. In a deviation from the traditional approach, we propose to estimate the
significant modes of the system using piezoelectric sensors but control them using
an end-force instead of piezoelectric actuators. Our goal is to introduce a control
strategy to switch the end-force on and off to suppress vibrations of the beam. The
equation governing the lateral vibration of the beam is nonlinear with respect to
the end force. This was shown in Chapter 2. Phrthermore, the end force can only
be applied unidirectionally. These two facts necessitate the use of nonlinear control
tools for design of a feedback law. In this work we use Lyapunov stability theory
along with passivity-based methods to design a stabilizing control for the closed loop

system.

45

4.2 Mathematical model of cantilever beam with

end-force

Consider the cantilever beam of length L and uniform cross-sectional area A, shown
in Figure (4.1). Let P be the force acting at the free end of the beam such that its line

of action always passes through the fixed end of the beam. It was shown in Chapter 2

 

Figure 4.1. A flexible cantilever beam with an end force

that under the assumption of Euler-Bernoulli beam theory and small deflections the

equation of motion of the beam can be written as follows

E1y”” + Py” + my = 0 (4.1)

where E, I, and p are the Young’s modulus of elasticity, area moment of inertia, and
density of the beam, respectively, and y’ and 6 denote the partial derivatives of y(2:, t)
with respect to 2: and t, respectively. Equation (4.1), is identical to the equation of a
beam with a follower end force [21] and a beam with an axial end force whose line is
action remains parallel to the undeformed axis of the beam. The boundary conditions
of the beam in Figure (4.1) are however different from beams with follower and axial

end forces. The geometric boundary conditions, which are related to zero deflection

46

and zero slope at the fixed end, are
11(0, t) = 0, y'(0, t) = 0 (4-2)

The natural boundary conditions corresponding to zero moment and nonzero shear
force at the free end are given by the relations

y”(L, t) = 0, E1 y”’(L, t) + P [y’(L, t) — %y (L, a] = 0 (4.3)

For small deflections, the end force in Figure (4.1) can be decomposed into a force of
magnitude P along the negative 2: axis and a force of magnitude Py(L,t) / L along
the negative y axis. The component along the 2: axis is constant and is therefore con-
servative. The component along the y axis is also conservative since it is proportional
to the displacement of the point of application, similar to a spring force. Our end
force is therefore conservative despite is close resemblance to a follower force, which is
nonconservative in nature [21]. This distinction is important since the Euler method
of determining elastic stability is applicable for conservative external forces only [21].
Now consider the elastic stability problem for the beam with end force, as shown in
Figure (4.1). For a small static deformation, the differential equation of the beam

can be obtained by substituting y(2:, t) = Y(2:),

at:

E] Y”” + PY” = 0, Y’ e (4.4)
d2:
The solution to this differential equation has the form
. A P
Y(r) = 61 + 322 + 63 s1n(a2:) + 63 cos(a:r), a = E—I (4.5)

47

where 61', 2' = 1,2,3,4, are constants that can be determined from the boundary

conditions in Equation (4.2, 4.3), namely,

Y(0) = o, Y’(O) = 0, Y”(L) = 0, Y’”(L) + a2 {Y’(L) — @} = 0 (4.6)

Equation (4.6) can be explicitly written in terms of the constants 62-, i = 1, 2, 3,4, as

 

 

 

 

 

 

follows _ _ q _ _
i 1 0 0 1 fil 0
0 1 a 0 ,3 0
2 = (4.7)
0 0 sin(aL) cos(aL) B3 0
. 1 0 sin(aL) cos(aL) . _ 64 J L 0 .

and the non-trivial solutions can be obtained by equating the determinant of the
matrix in Equation (4.7) equal to zero. The determinant of the matrix is sin(aL) and

by equating it to zero, we get

n71" 2N2 EI

071:1”, n=i1,i2,"' : Pcr=n L2 (4.8)

The above result establishes that the beam in Figure (4.1) first buckles when P =
7r2 EI/Lz. This load is four times larger than the buckling load for a cantilever beam
with an axial end force [21], [60]. This implies that we can use a relatively large
end force on our beam for vibration control without creating instability. Before we
complete this section, we would like to point out that the buckling nature of the
end force in Figure (4.1) is deduced from the fact that the matrix in Equation (4.7),
looses rank and has an eigenvalue at zero when P assumes a critical value. For
a follower type end force, which is non-conservative in nature, the corresponding
matrix is always nonsingular. As the magnitude of the follower force is increased, two
distinct eigenvalues of the matrix approach each other, assume the same value at the

critical load, and then become complex resulting in flutter instability [21].

48

4.3 Rayleigh-Ritz approximation

In order to obtain an approximate solution to Equation (4.1) subject to the boundary
conditions in Equation (4.2, 4.3), we multiply Equation (4.1) with a weight function

111(2), and integrate it over the length of the beam

L
P 111(2) y”(2;, t)d2: +/(; pA 111(2) 37(15, t)d2: = 0 (4.9)

L L
/ EI w(x)y””(2:,t)d2: +/
0

0
We simply assume that 111(23) is continuous, twice differentiable, and satisfies the two
geometric boundary conditions in Equation (4.2), namely

111(0) = 0, w’(0) = 0 (4.10)

In order to distribute the derivatives equally between y(:r, t) and 111(1):), we integrate
the first integral in Equation (4.9) twice by parts, and the second integral once by

parts, to get
L L
(EI'LU y’” — Erw’ y” + Pwy’) [0 + / (E1w”y”— Pw’y’+pAwy) da: = 0 (4.11)
0

Equation (4.11) is called the weak form [61] of Equation (4.1). Using the natural
boundary conditions of y in Equation (4.3) and the geometric boundary conditions

of w in Equation (4.10), we get

 

L

t

P w(L)z(L, ) +/ (E1 111” y” — P w, yI + pA w y) d2: = 0 (4.12)
0

We now use Rayleigh-Ritz approximation [61] to express y(2;, t) as a linear combina-
tion of N suitable functions that satisfy the geometric boundary conditions in Equa-

tion (4.2). Specifically, we use the first N normalized mode shapes of the clamped-free

49

cantilever beam, as follows

N
yN(93tt) = Z attt) 4.14) (4.13)

121
In the above equation, (15,-(27), 1' = 1, 2, - - - , N, are the assumed modes, N is the number
of desired modes, and ai(t), 1' = 1,2, - -- ,N, are the corresponding modal displace-

ments. Now, by choosing y(2:,t) z yN(2:,t) and 111(2) = ¢j(2:), j = 1,2,--- ,N,
consistent with Equation (4.8), we get N differential equations from Equation (4.12),

as follows
N L N L ¢'(L)¢'(L)
II II ,_ I_ I, _ Z J .
EI 22:1 (/0 25,- 2)]- dx) az P z.§=1(‘/0 d), (1)] dz L ) a2 + (4.14)

N L
MEMO ¢t¢jdx)ét=0, j=1.2,-~- ,N
i=1

Using the orthogonality property of the assumed mode shapes, the above N equations

can be written as follows

a+(K—PC)a=O (4.15)

where a Q (a1, a2, - -- , a N)T, K 6 RN XN is a diagonal positive definite matrix
with elements Kit: C 6 RN XN is a positive definite symmetric matrix [62] with
elements Cij, and P is assumed to be positive in the direction shown in Figure (1).

The elements Kit and Cij have the following expressions

K..- 2 fi- OL [¢;’(:c>]24x, Cz’j e 5;,- /0L [18(4) 4314:) 44— W] (4.16)

Earlier, we claimed in Section 4.2 that the end force in Figure (4.1) is conservative
in nature. This can now be claimed [63] from the symmetric nature of the matrix

(K — PC) in Equation (4.15).

50

When P = 0, the eigenvalues of (K - PC) are the same as the eigenvalues of K,
which are all positive. As P is increased, all the eigenvalues start moving towards
the origin. The end force P assumes a critical value, given by Equation (4.8), every
time one of the eigenvalues reaches the origin before crossing the imaginary axis and
becoming negative. This can be claimed from the physics of the buckling problem
and has also been verified using numerical simulations.

In the derivation of Equation (4.15), structural damping was assumed absent. If

structural damping is present, we can use

a+Da+(K—PC)a=O (4.17)

RNXN

where where D E is a diagonal positive definite matrix of modal damping.

4.4 Preliminary feedback control design

In order to design a feedback controller for vibration suppression, we first rewrite

Equation (4.17) in state space form

2'21 = :62 (4.18a)

2:2 = —K:cl — D232 + C21 11 (4.18b)

where 2:1 é a 6 RN and 2:2 9- a 6 RN are the state variables, and 11 3 P E R is
the control input. The task of vibration suppression in the beam can be posed as a

problem of design of the control input 11 that satisfies the constraint

0 g 11 < 7T2 131/L2 (4.19)

51

and guarantees asymptotic stability of the equilibrium point (2:1, 2:2) 2 (0, 0). The
constraints on the input are necessary since u 2 «2 EI/L2 results in buckling instabil-
ity and 11. < 0 cannot be physically applied using a cable. We present our preliminary

control design next with the help of the following Theorem.

Theorem 1: (Asymptotic Stability) The origin of the system described by Equa-

tion (4.18) is rendered globally asymptotically stable by the following choice of input

P if xTCx < 0
6(1) = 0 2 1 0 < P0 < 22 EI/L2 (4.20)
0 if 2:30:51 2 0,

independent of the amount of structural damping present in the system.

Proof: Consider the Lyapunov function candidate

V1(2:1,22) = ($5111le + 233232) (4.21)

Nth—I

It is positive everywhere other than the origin where it is equal to zero. The derivative

of the Lyapunov function candidate is

V1 = 2:?K232 + 2:3(—K2:1 - D22 +C2:111)

= —x§D2:2 + (3350171)“ (4.22)

For the choice of control input in Equation (4.20), it can be easily shown that V1 S 0
and V1 = 0 iff 2:2 = 0. Using LaSalle’s Theorem [59] we can therefore claim that the
origin is asymptotically stable. Since V1 is radially unbounded, the origin is globally

asymptotically stable. 0 o 0

Remark : The control law in Equation (4.20) essentially implies that the buckling-

type end force should be turned “on” whenever it can do negative work or remove

52

energy from the system, and kept “off” at all other times.
We now investigate the efficacy of the control design in Equation (4.20) using

simulation. We assumed the material and geometric properties of the beam to be:

Material Alluminum

Young’s modulus 70 GPa

Mass density 2730 kg/m3
Dimensions 1.00 x 0.05 x 0.003 m

For a two-mode approximation of beam dynamics, the K, C, and D matrices of our

mathematical model in Equation (4.18) were computed as

97.38 0 ( 1 —5.28 0.098 0
K = , C = K D

7

0.0 3824.39 —5.28 44.41 0.0 0.618

(4.23)
The square root of the diagonal entries of the K matrix are the natural frequencies of
the beam and are equal to 9.87 rad / s and 61.84 rad / s, respectively. The critical buck-
ling load of the beam was computed to be approximately 50N using Equation (4.8).
It can also be obtained by computing the minimum eigenvalue of —C—1K. For the
sake of simplicity, we assumed proportional damping (no modal coupling). The diag-
onal entries of the D matrix in Equation (4.23) correspond to C = 0.005. We chose
P0 in Equation (4.20) less than the critical buckling load, and equal to 35 N. The
simulation results are shown in Figure (4.2) for the following initial conditions in SI
units

a 0 0.1 a 0 0.0
21(0)= 1” = , 22(0): 1” = (4.24)

52(0) 0.0 62(0) 0.0

In Figure (4.2)(a) we plot the amplitude of the first mode in the absence of control. It

53

decays to zero very slowly due to low structural damping. The amplitude of the second
mode, in the absence of control, is not shown in Figure (4.2) since it is identically
zero. This is true since the modal dynamics in Equation (4.18) are decoupled (K
and D matrices are diagonal) in the absence of control. The plots in Figure (4.2)(b)
and (0) show the modal amplitudes as a function of time for the control action in
Equation (4.20). The control action itself is plotted in Figure (4.2)(d). A number of

observations can be made from the simulation results in Figure (4.2)(b), (c), and (d):

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 I I I r I I I I I
E
8 O - -
t?
“0 0 ‘2 ‘ 4 (‘10 6 ‘ it 1 IO
10 I I I I I I I I I
E
3 0 _ __v
8‘
'10 0 2 4 (b) 6 8 10
l I I I r I f I I I
”a“
8 0 #
s. W
_1 1 1 1 1 1 1 1 1 l
0 2 4 (c) 6 8 10
40' I I I I I I I I I
' |
:1 . t . I
0
0 2 4 (d) 6 8 10
time (sec)

Figure 4.2. Simulation of decay in modal amplitude a1 due to structural damping, (b),
(c) decay in modal amplitudes a1 and a2 due to control in the presence of structural
damping, and (d) plot of the control action.

54

1. The modal amplitudes, a1 and a2, rapidly decay to zero, as expected, but the
control input keeps on switching. This can be attributed to small numerical
errors causing frequent change in sign of the term 233C271 and the particular
nature of our control law in Equation (4.20). This problem can be easily rectified

during actual implementation.

2. Due to the particular structure of matrix C, the modal dynamics in Equa-
tion (4.18) is coupled. This is evident from the vibration of the second mode

despite zero initial conditions.

3. The main advantage of the control law in Equation (4.20) is that it can be
implemented using state feedback for as many modes as we desire to model.
However, it has one major drawback. The term ngxl will have many fre-
quency components (the highest frequency component will be twice the highest
modal frequency) and it will change sign rapidly. This may easily exceed the
bandwidth of the actuator used to switch the end force. If the actuator does not
have the requisite bandwidth, the closed-loop system will most likely become
unstable since incorrect timing of switching will tantamount to adding energy

to the system.

One way to circumvent the requirement of high actuator bandwidth is to discard
the high frequency components using a low-pass filter in the control loop Without
adversely affecting the stability of the system. This is achieved using our control

design presented in the next section.

4.5 Modified control design

Our dynamical system described by Equation (4.18) can be represented by the input-

output mapping shown in Figure (4.3)

55

 

_ T
= — ' xz C X1
11 h(z) > flexible cantilever beam

 

 

 

>0

 

 

 

 

 

 

 

 

 

memoryless nonlinearity low pass filter
It
Pow: h(z) A z 1
i z ts+l
e

 

 

 

 

 

 

 

 

Figure 4.3. Control design based on output filtering.

where, 11 represents our buckling-type end force and y represents the output, defined

as follows

y = —r§C2:1 (4.25)

Now consider the feedback connection in Figure (4.3) where the input to our
dynamical system is obtained by feeding the output of our system through a low-
pass filter (of time constant T and unity dc gain) and then through a memoryless

nonlinearity h(.) which satisfies

P0 for z 2 e
h(z) = p0 z). for 0 s z < 6 , P0 < 7T2 131/12 (4.26)
0 for z < 0

For this feedback connection, we now have the following result:

Theorem 2: (Asymptotic Stability of Feedback Connection) The origin of the dy-

namical system in Figure (4.3) is globally asymptotically stable.

Proof: The states of the feedback connection are comprised of the states of the
dynamical system in Equation (4.18), namely, 2:1 6 RN, 2:2 6 RN, and the state

of the low-pass filter, 2 E R. Let us therefore consider the positive definite radially

56

unbounded Lyapunov function candidate

2

(23%le 2:1 + 2:322) + T/O h(o) do (4.27)

[\DIH

z
V2(2:1,2:2,z) = V1+ T/ h(o) do =

0
Using Equation (4.22) , the derivative of V2 is found to be

V2 = —x§D 2:2 + (15C 2:1)11 + Th(z) z'

= —x§D 2:2 — yu + h(z) (-z + y)
= :12?ng — zh(z)

3 ago 332 g 0 (4.28a)

Since D is positive definite, V2 = 0 implies (231,232, 2) belongs to the set {:12 = 0}.
In this set, y = —ng:rl = 0 which implies z —-> 0 as t —1 00. This in turn implies
11 = h(z) = 0. Since 2:2 remains identically zero, we can use Equation (4.18) to claim
2:1 = 0 and establish that the maximum invariant set in {2:2 = 0} contains only the
origin (21,22,17.) = (0,0,0). Since V2 is radially unbounded, we can use LaSalle’s

Theorem [59] to claim global asymptotic stability. o <> 0

Remark 2: The modified control design in Figure (4.3) incorporates a low-pass filter
to attenuate high frequency components of y = —2:%wa1. The bandwidth of the
filter can be chosen such that the control input does not exceed the bandwidth of
the actuator. The memoryless nonlinearity is incorporated in the control system to
guarantee that the control input is always positive since a negative end force cannot
be applied by a cable, as in our set up. We repeat the simulation of Section 4.4 to
investigate the efficacy of the modified control design. The simulation results are
presented in Figure (4.4) for our choice of low-pass filter bandwidth 0.13 = l/r = 15

rad/ sec. The results in Figure (4.4) indicate that the modal amplitudes decay slowly

57

for the control design in Figure (4.3) in comparison to those obtained with the control
design in Equation (4.20), shown in Figure (4.2). The control action in Figure (4.4),
however, switches less frequently as compared to the control action in Figure (4.3).
Clearly, the bandwidth of the filter provides a tradeoff between switching frequency
of the control input and speed of vibration suppression. A higher bandwidth (smaller
value of 7) results in faster vibration suppression but causes the input to switch very
frequently, whereas a lower bandwidth results in less switching of the control input

but requires longer time for vibration suppression.

 

 

 

 

 

 

 

 

 

10 I l T T I I r I I
E
3 0 _
E?
"0 0 ' ' ' 4 ' 6 ' t ' 10
(a)
0.5 I I l I T I I I I
g 0
9-05 -
0 2 4 6 8 10
(b)
40 I M TT 1 I l l I l
A ] I j ] I ‘ A i . l
a j i , ; ‘ , 1 1
O l 1 t l
0 2 4 6 8 10
(C)
time (sec)

Figure 4.4. Plot of modal amplitudes a1 and a2, and the control action 11 for the
modified control design when 6 = 0.

58

4.6 Adding bias tension in the cable

In general, cables have a certain amount of slack in their state of zero tension. There-
fore, an actuator providing the end force in Figure (4.3) will have to first overcome
the slack when it switches the tension from zero value to the positive value P0. This
displacement of the actuator will require finite time and cause delay in switching the
end force, which is likely to result in instability. To circumvent this problem, we

propose to incorporate bias tension in the cable, as shown in Figure (4.5).

 

 

 

_ T
u =h z + — -x2 C X}
ii+>CB b ( ) P; flexible cantilever beam
I

 

 

 

memoryless nonlinearity low pass filter

 

 

II
P h(z) A z 1 A
ts+l

 

 

 

 

 

 

 

 

=2

 

 

 

 

 

 

 

Figure 4.5. Control design based on bias tension and output filtering.

Of course, the bias tension can only be positive and the sum of the bias tension and

the controlled tension should not exceed the buckling load for stability, i.e.
P0 > 0, (Pb + P0) < t? EI/L2 (4.29)

By replacing 11 in Equation (4.18) with 11), = h(z) + Pb, as shown in Figure (4.5), the

dynamic equations revert to the form

5171 = 1132 (4.3081)

2:2 = —I-{ 271 — D32 + C1131 11, K 2 (K - PbC) (4.301))

59

where R is positive definite since K and C are both positive definite matrices and

Pb is less than the buckling load. We now state a corollary of Theorem 2.

Corollary 1: The origin of the dynamical system in Figure (4.5) is globally asymp—

totically stable.

Proof: It is clear from Equation (4.30) that if we replace the beam in Figure (4.3)
(with natural frequencies equal to eigenvalues of K) with an identical beam with
natural frequencies equal to the eigenvalues of K, Figure (4.5) becomes equivalent
to Figure (4.3). Therefore, Theorem 2 can be applied to establish global asymptotic

stability of the origin of the dynamic system in Figure (4.5). 0 o 0

Remark 3: We know from Corollary 1 that stability of the system is not adversely
affected by bias tension. On the contrary, bias tension increases structural damping
and results in faster vibration suppression. This will be established in the next section,

through experiments.

4.7 Experimental verification

4. 7. 1 Hardware description

In our experimental setup, the end force was applied using a Kevlar cable, wrapped
around the front face of the beam. The free ends of the cable were wrapped around
pulleys fixed to the base of the beam and then tied together to a pulley on a motor
shaft. The motor, manufactured by MicroMo Electronics [64], was driven by a power
amplifier manufactured by Advanced Motion Control [26], in current mode. A piezo—
electric transducer, manufactured by Mide Technology Corporation [57], was used to
sense the displacement of the beam. It was placed approximately 0.38 m from the

fixed end of the beam wherefrom states :31 and 152 corresponding to both the first

60

and second modes of vibration of the beam are observable.

We programmed our observer-based control law in the Matlab/SimulinkTM environ-
ment and downloaded it to our dSPACE DSP* board, not shown in Figure (4.6).
The DSP Board resides in our control computer and reads the signal provided by the
piezoelectric sensor. This signal is used to estimate the states, 11:1, $2 (the observer
design is discussed in Section 4.7 .3) and compute the signal 2:2 T031 (see Figure (4. 5)).

The signal is filtered and rectified to generate the control action, ub. The DSP board
provides a reference signal to the power amplifier that is proportional to the control
action ub; the proportionality constant depends on the gain of the amplifier, the mo-
tor torque constant, and other mechanical parameters of our setup, such as radius of
the pulley on the motor shaft.

, l 1’ "l "I l I"”lllll :‘llllll "‘llllfl ’3
lllllll‘ .M; “.2; III.» “IlIW‘ .,

I‘v-. am. Illlnit' ”"07 ‘

ll1ull| ll] lllll?er ll'll

ll llllllll
ng- Illllllw III llllmrlll IIllllIIIII 15'" ,

will.) .
windmill! :rlmlnlul. ”W I , llllll
lllWlllmlememM: 1

'm
. .1!!!" li "(MI

 

beam tip with pulleys Kevlar cable
for cable wrap around

Figure 4.6. Experimental setup

 

*Digital Signal Processor

61

4.7 .2 The effect of bias tension on structural damping

In this section we present experimental data on free vibration of the cantilever beam
in the presence of bias tension. The data, provided in Table 1, indicates that the first
natural frequency, wl, decreases, and the damping ratio, C, increases, with increase in

bias tension, Pb- The first natural frequency computed from our theoretical model,
1/2

)‘mian — PbC], differs slightly from the values obtained experimentally but shows the
same trend as cal and provides confidence in our results. Although wl decreases and
C increases, the product, (cal, increases with increase in bias tension. This, evident
from the last column of data in Table 1, indicates that structural damping for the
first mode increases with increase in bias tension. To the best of our knowledge there
has been no studies reported on the variation in structural damping due to variation

of end force.

Table 1. Effect of bias tension on natural frequency and damping ratio

 

 

 

 

 

 

 

Pb (N) Afifniki (rad/s) wl (rad/s) C can (rad/s)
0 9.87 9.42 0.0058 0.0546
4 9.65 9.28 0.0070 0.0649
8 9.43 9.13 0.0078 0.0712
12 9.17 9.04 0.0083 0.0750
16 8.88 8.85 0.0085 0.0752
20 8.57 8.70 0.0090 0.0783

 

 

The same overall trend has also been observed for the second mode of vibration,
i.e., the natural frequency, wg, decreases, and modal damping, CW2, increases, with
increase in bias tension. We do not provide the results here but would like to comment

that the experimental procedures were different for the first and second modes. For

62

the first mode, the beam was given an initial displacement corresponding to its first
mode shape and the data collected from free vibration of the beam. For the second
mode, the natural frequency was determined by sinusoidally exciting the piezoelectric
transducer attached to the beam and identifying the excitation frequency that causes

resonance in the second mode.

4.7 .3 Observer design

In this section we briefly discuss the procedure adOpted for the design of a stable

observer. For our observer, we used Equation (4.30) and the output equation
37 = 012:, (4.31)

where Cll can be computed [51] from the dimensions of the piezoelectric sensor and its
location on the beam. Since the state equations can be written in the form :i: = A(t) x,

a: g (ml, x2)T, the observer is designed as follows

a = Ame + L(g — 01a), A(t) = 0 I (4.32)

-[K — UbCl D(Ub)
where I is the identity matrix and L is the vector of observer gains. The time
dependence of A(t) can be attributed to the fluctuation of ub(t) between the values
Pb and (Pb + P0), as well as variation in structural damping due to variation in ub.
The dependence of structural damping on the end force was mentioned in Remark

3 and conclusively established from experimental data in section 4.7.2. The state

variable description in Equation (4.30) and the observer equation in Equation (4.32)

 

lCl should not be confused with the positive definite square matrix C in Equation (4.30)

63

gives us the error equations

e = [A(t) — L01] 6 (4.33)

wherein A(t) fluctuates between the two fixed descriptions

0 I o 1
A1 = A2 = (4.34)

-[K - PbCl D(Pb) -[K - (Pb + P0)Cl D(Pb + P0)
In equation (4.34), D (.) denotes the functional dependence of D on the end force.
Equation (4.33) represents a switched linear system and hence stability of the observer
cannot be ensured by simply choosing L that guarantees [A1 — LCl] and [A2 — LCl]
are Hurwitz. This is true since switching between two stable systems can potentially
result in instability [34] We avoid this problem by designing a high-gain observer [59]
where large values of the gains, L, minimize fluctuation in A(t) due to change of the

end force.

4.7 .4 Results

The voltage output of the piezoelectric sensor provides a measure of the residual
vibration in the beam and we plot this voltage to compare vibration attenuation in
the presence and absence of control. For proper comparison, we provided the same
initial conditions in all our experiments; the beam was deflected purely in its first
mode and the initial deflection corresponded to a PZT output of 10 volts.

The plots in Figure (4.7)(a) and (b) depict free vibration of the beam in the
absence of bias tension and presence of 20 N bias tension, respectively. These results
indicate that structural damping increases due to the application of an end force and
supports the data in Table 1. The plots in Figure (4.7) also indicate that vibration
is attenuated very slowly in the absence of active control.

The results in Figure (4.8) correspond to active vibration suppression using our

64

 

Pb¥0,Po=0

 

 

 

l . . n
-10 0

 

 

 

 

 

40 time (sec) 80 120
(a)
Pb = 20, P0 = 0
time (sec) 80 120
(b)

Figure 4.7. Free vibration: (a) in the absence of bias tension, and (b) in the presence
of 20 N bias tension

buckling-type end force and two—mode approximation of the beam dynamics. The
different plots in Figure (4.7) were obtained with different combinations of bias ten-
sion, Pb, and the maximum control force, P0, as shown in Table 4.2. For all three

experiments, the value of 'r was kept fixed at 0.2 secs.

Table 2. Different values of bias tension, Pb, and maximum control force, P0, in

experiments with active control

 

Pb (N) P0 (N)

 

Fig.8(a) 20.0 4.0
Fig.8(b) 20.0 2.0
Fig.8(c) 12.0 4.0

 

 

 

 

 

65

 

 

 

 

 

 

 

 

 

 

 

 

 

Pb = 20, P0 = 4
3
'5 0 v‘
>
-10 1 1 1 1 1 1 1 1 1
0 4 time (sec) 16 20
(a)
10 1 r r
Pb = 20, P0 = 2
3
'3 0
>
_10 1 1 1 1 1 1 1 1 1
O 4 time (sec) 16 20
(b)
10 1 t
Pb = 12, P0 = 4
£9
'5 0
>
- 10 . . 1
4 time (sec) 16 20

(C)

Figure 4.8. Vibration suppression using active control

It is clear from Figure (4.8)(a) and (b) that a larger value of the maximum control
force, P0, leads to faster vibration suppression. A larger value of P0 also requires
us to change the slope of the saturation function h(.) but we do not discuss this
here. A comparison of Figure (4.8)(a) and (0) indicates that a higher value of bias
tension, Pb, leads to faster vibration suppression. This can be attributed to higher
structural damping associated with higher bias tension, which was established earlier
through experiments. A comparison of the time required for vibration suppression in
Figure (4.7) and Figure (4.8) demonstrates the efficacy of our control strategy. We
complete this section with one more set of experimental results. These results, which
were obtained with a one-mode approximation of the beam dynamics. The results,
shown in Figure (4.9), illustrate the role of the low-pass filter in reducing spillover [10].
For both experiments shown in Figure (4.9), we used Pb = 8N and P0 = 4N. The

low-pass filter was not used for the experiment in Figure (4.9a) but was used for the

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 v r 10 . . .
5 decay in first mode 5 decay in first mode
.‘3 1 3
E 0 “"iiull “ll _éol . . .
-5 Ella ation o ‘5 * flgggrfdliflggg
-10 . unmo e e ‘secon mode _10 . . . .
0 5 time (sec) 10 15 0 5 time (sec) 20 25
(a) (b)
A 12 1
E 10
3 8
0 5 time (sec) 10 15 0 5 time (sec) 20 25
(a) (b)

Figure 4.9. Vibration suppression using a one mode dynamic model results a)-spill
over problem b)- the role of low pass filter in reducing the effect of spillover

experiment in Figure (4.9b) with 7' = 0.23603 . From the results in the Figure (4.9),
we can make the following observations:

1. The modal controllers in both experiments were effective in attenuating vibration
associated with the first mode.

2. The unmodeled second mode was excited in the experiments corresponding to
Figure (4.9a). This phenomenon of spillover [10] was significantly reduced by the
low-pass filter in the feedback loop, in the experiment corresponding to Figure (4.9b).
3. Although the low-pass filter reduces spillover, it increases the time required for vi-
bration attenuation. This is evident from the experimental results in the Figure (4.9)

and corroborates our earlier simulations.

67

CHAPTER 5

Modal Disparity

5. 1 Background

In this chapter we utilize a buckling-type end force to modify the modal characteristics
of a cantilever beam. The objective is to change the mode shapes of the beam and
to exploit these changes in control design. Specifically, we vary the frequencies and
mode shapes by applying an end load, and by switching between different sets of
modal characteristics, funnel energy from the higher modes to the lower modes. The
energy associated with the lower modes can then be dissipated by employing a low
dimensional state space model. The success of this strategy relies on the difference in
modal characteristics under different levels of end loads, which we refer to as modal

disparity.

5.2 Proof of concept

Consider the cantilever beam with a buckling type end force studied in Chapter 4.

For different levels of the end load, the mode shapes of the beam can be determined

68

using Equation (4.15). For this purpose we do a coordinate transformation as follows
a = [T ]z (5.1)

where columns of [T], the transformation matrix, are formed using eigenvectors of

[K] - P[C']. Substituting for a from Equation (5.1) into Equation (4.15) results
2 + [Q]z = 0 (5.2)

where [Q] = [T] _1([K] -P[C])[T]. In this equation [0] is a diagonal matrix consists of
squared of natural frequencies of the beam. Now, plugging for a from Equation (5.1)

in Equation (4.13), results

N
(we, 0 = 2T1T1T4 = Z 2.0) 4(4) (5.3)
221
in which 0) = [T]T¢ are the true mode shapes of the beam. For P = 40N the mode
shapes are shown in Figure (5.1) and compared with those with P = 0. From the
figure it is clear that the application of the end load changes the mode shapes.
Now, consider an idealized static problem wherein the end-force is instantaneously
switched between two values 0 and P0, and the fundamental modal components are
repeatedly removed from the system after each switch. We will show that such a
strategy removes energy from the beam, including higher mode, in a systematic man-
ner, and requires that one be able to control the fundamental mode corresponding to
the free beam and the beam with end load P0. For the calculations we denote the
attendant mode shapes for the free beam and the beam with end load P0 as ¢j($)

and (03- (2:), respectively. If one starts with a beam deflection y0(:c, t) and no end-load,

69

 

 

1.5»

 

 

 

 

 

0 lst mode ’ 2‘19 mode

 

 

0 05 1 0 0.5 1

 

 

 

H

L 3rd mode

 

 

 

 

 

 

0 0.5 1

 

Figure 5.1. cantilever beam mode shapes for P = 0 and P = 40N

then we can write

N
yum) = E: «SJ-(04m) (54)
3'21

where N denotes the modal truncation level and 6j(t)s are the modal amplitude
components. Assuming that one can remove the first mode, the resulting shape is

given by
N
111(35, t) = 310(3310 - 510M103) = Z 6j(t)¢j(33) (5-5)
'=2

At this point the end-load is switched to P0 and the shape is now conveniently

expressed as

N
311(3)” = Z fllel/Jfll‘) (5-5)
j=1

70

It is assumed that the first mode again removed while P = P0, resulting in the shape

N
120:, t) = 2: 20143-0) (5.7)

N
3203,15) = Z 7j(t)¢j(z) (5-8)

This completes one cycle of the process, and one is interested in how the new modal
coefficients, the 'yj(t)s are related to the originals, the (SJ-((5)3. This is conveniently

described by a linear mapping

r = MA (5.9)

where F and A are the vectors of modal coeflicients.

P = (11, 12, 1n1T (5.10)

A = (51, 62, 6n)T (5.11)

and M, the mapping matrix, can be developed by a sequence of calculations that use

modal projections for each level of the end- force as follows

N N
%(t)=<yz($1)t1¢1($r-(jz:fij(t)¢j($)1¢1($=j§323j( (1521” )) (5'12)
and similarly

N
[3]-(t) = Z 61.01 <4j(:c1,41.<x>>*. (5.13)

k=2

 

*( f, g) is the inner product of functions f and g denoted by fol“ f 9 d2:

71

Now, Substituting for Bj(t) from Equation (5.13) in Equation (5.12), results

N N
110) = Z Z (¢1($),wj($)> (wj(:v)1¢k(~’v) > 5121 i=11~~1N (5-14)
j=2k=2

Comparing this equation with Equation (5.9), reveals that the structure of the map-
ping matrix M. It should be noticed the convergence of this process depends on the
N x N linear operator M, which can be constructed as follows: the first column con-
tains all zeros since the first modal coefficient was zeroed out (note that this implies
that M will always have at least one zero eigenvalues). The remaining columns are
filled in by the coefficient (¢i(x),¢j(x)) (¢j(x),¢k(a:)), 2' = 1,2, ...,N, k = 2,3, ...,N.
If all eigenvalues of M lie inside the unit circle, the process will converge, implying
that all modes consideration die out under repeated cycling and removal of the first
relevant mode. In fact rate of convergence (or divergence) is dictated by these eigen-
values. For better understanding, in Figure (5.2) a schematic of the concept of the
modal disparity is depicted.

As an example, the cantilever beam studied in Chapter 4 is used to demonstrate
the methodology for P0 = 40N. For this case the eigenvalues of the mapping matrix
M is

A,- = {0,051,087, 0.92} (5.15)

We can see that the eigenvalues corresponding to the higher modes is close to unity.
This implies that the rate of convergence for the higher modes will be smaller com-
pared to the lower modes. In this calculation it is assumed that the end-force does
not add or remove energy to or from the beam in the time interval that is switched
on, which is very unlikely. In the next section we will consider the effect of switching
on the total energy of the beam by investigating the subsystems corresponding to

different levels of the end load.

72

Initial after control switch on after control switch off
the force the force
A A B B A

energy of .
1st mode

energy of
2nd mode

 

 

 

 

 

 

 

 

energy of
3rd mode

 

energy of
4th mode

 

 

a0 a1

\ Mapping M/l

Figure 5.2. Concept of the modal disparity

5.3 Dynamic analysis

From Equation (4.17) in Chapter 4, we have the following mathematical model for

the beam subjected to the end load P
{5} ‘l‘ lDli‘S} + (lKlnxn " PlClnxnli5} + 1111051151) = 0 (5-16)

where P is always less than Pbuckling' In this equation [D] is the diagonal modal
damping matrix, assumed to be constant, and H1 is the vector of control laws corre-
sponding to the different modal coordinates. All the enterers of H1 are zero except

the first, which is assumed to be hl. Our goal here is to design a switching strategy

73

for the end force to transfer energy of the higher modes to the first mode such that
it can be subsequently dissipated by the control law hl. We expand Equation (5.16)

as follows

51 + D1151+ K1151+ h1(51151) = Pl0111xn{5}
52 + D2252 + K2252 = PlC2l1xn{<5}
(5.17a)

(5n + 131112512 + Knn5n = PlCnllxnf‘s}

In the above equation Di,- is the 2th diagonal element of the [D] matrix and [0,] refers
to the 2th row of the [C] matrix. By multiplying the first row of Equation (5.17) by

6.1, the second row by 6.2, and so on, we get

E1 = 4101151 - 5'1h1(51,51) + P10105115)
E =-5’D 6' +Pw 5,5
2 2 22 2 2( 2 ) (5.18)

 

where E,- 2 £05? + 192-612), is the modal energy corresponding to the 2th mode and
“12(5215) = 5ilCz'l5-
Summing up all the rows of this equation, results in

d n ‘ ' n . . .
a(131+ E2 + . . . + En) = _ Z 5,0,,6, + FEW-(523(5) _ 511,105,,51) (5.19)

The first term at the right hand side of Equation (5.19) is negative definite whereas
the third term can be made negative semi definite. The second term can be made
negative definite if we apply the end load when 221:1 211,6), 6) is negative. This would

constitute the approach taken in Chapter 4. In order to clearly illustrate the concept

74

of modal disparity, one can design a switching strategy such that the end load does
not change the overall energy of the system. This can be done, for example, if we
apply the end load over any interval in which f (227-;1 20,-(5i, 6)) = 0.

We now present a simulation that combines the approach of Chapters 4 and that of
modal disparity. In this combined approach, we design the switching strategy such
that the end load reduces the over all energy of the system (approach of Chapter 4)
as well as funnels energy from the higher modes to the first mode (approach based
on the concept of modal disparity). This switching strategy can be chosen as follows:

a) apply end load when the following conditions are satisfied

2111 > 0
& (5.20)

w1+w2+...+wn<0

b) remove end load otherwise.

In the next section we provide a numerical simulation to illustrate this approach.

5.4 Numerical example

In our simulation we assume the material and geometric properties of the beam as
follows:

Material Aluminum

Young’s modulus 70 GPa

Mass density 2730 kg/m3

Dimensions 1.00 X 0.05 x 0.003 m

A four-mode approximation of the beam dynamics is considered. To better illustrate

energy dissipation by the end load as well as energy transfer from the higher modes

75

to the first mode due to the end load, we assume modal damping to be absent. For
the same reason, we do not remove the energy of the first mode and therefore choose
the control law 2.6. h1(6,51) = 0. The simulation results are shown in Figure (5.3)

for the following initial conditions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2(0) = [1, -1, —0.5, 0.5] (5.21a)
23(0) 2 [0, 0, 0, 0] (5.21b)
and P0 is equal to 40 N.
81 0 W
-5 - , , , , , .
0 1 2 3
1 I I I fl 4
82 0 .
'1 0 ‘ i ‘ 2 L 3
0.4 ' ' :
53 0
'0.4 1 1 1 1 1
0 1 2 3
0.4 . v r
11111111 1
84 -00. 4 ll l MW” 1 1 1 1 :
0 1 2 3
0 x104 I I . . .
wp -1 [L j
-1.6" 1 . - . I
0 l 2 3
40 ' ' T ' .
P .
0 [1111111111111 1111 1 1 1
0 1 , 3
time(sec)

Figure 5.3. plot of the modal amplitude for the described switching strategy in
Equation (5.20)

76

In Figure (5.3) the first four plots show the amplitude of first four modes, respectively.
The fifth plot shows the work done by the end-force and the last plot is the time
history of the end load. As we can seen in this simulation, amplitude of the first
mode increases. This indicates that some of the energy associated with the higher
modes is pumped into the first mode and consequently the amplitude of the second,
third and the fourth modes decrease. In fact, decrement in the amplitude of the
higher modes occurs for two reasons: first, their energy is funnelled to the first mode,
and second, the end load is doing negative work.

The main concern regarding this simulation is that the end load needs to switch on
and off quite rapidly. This implies that if the actuator does not have the requisite
bandwidth, the switching will not occur at the right time and this can result in
energy being pumped into the system . One way to circumvent the requirement of
high actuator bandwidth is to discard the high frequency components of 2122 +. . .+wn

using a low pass filter in the loop that is described in the next section.

5.5 Modified switching strategy

We propose the control system block diagram in Figure (??) to reduce the actuator
bandwidth requirement.
In this diagram, h1(z) and h2(w) are two memoryless nonlinearities described

below

P0 forz>0 1forw>0
h1(z) = h2(w) = (5.22)
OforzSO OforwSO

To reduce the speed of switching of the end force, we remove the high frequency
components of w,- by feeding the signal through a low pass filter (of time constant 7'

and unity DC gain) and then through the memory less nonlinearity h1(..)

77

 

“’1 fill—20”)

 

 

 

 

 

 

J

 

 

P = h1(z) h2(w1)
9‘9 > flexible cantilever beam
l

 

 

'w2 ‘ W3 ' 'wn

 

 

memoryless nonlinearity low pass filter

IPl h1(Z) Z I
< ts+1

 

 

 

 

 

 

ll

 

 

 

 

 

—'Z

 

 

 

 

Figure 5.4. Modified switching strategy

To prove that the energy associated with the higher modes decreases, we look at the

time derivative of the following Lyapunov function [59]

Z
Etotal = E2 + E3 + . . . + T/O (11(0) d0 (5.23)

where, Etotal consists of two positive terms, namely, energy of the second and higher

modes and the area under the positive function h1(z). From Equation (5.19) we have

dEtotal : _
dt

72 n
52'Dz'i5i + P Z 10,- (51°, (5) + 72h1(z), (5.24)
i=2 i=2

Using Figure (5.4), we can say

(1E 12. . . n , n .
Slim: 2 _ Z con-61+}? Z wimi, 6)+h1(z) (—z—h2(w1) Z w,(5,-, 5)) (5.25)
i=2 i=2 i=2

where

P = h1(z)h2(w1) (5.26)

78

 

Therefore, we have

ELM“: —Z 6 ,-D,-,-5— zh1(z) (5.27)

Using LaSalle’s Theorem [59] and the analysis similar to that used in Chapter 4, we
can claim asymptotic stability of the origin. Hence, the energy of the higher modes
will eventually decay to zero.

We repeat the previous simulation to investigate the usefulness of the modified switch-
ing strategy. In this simulation we assume small value for modal damping. The con—
trol law for removing oscillations of the fundamental frequency is considered absent.
The simulation results are presented in Figure (5.5) for our choice of low pass filter

bandwidth of Lab = 1/7' = 50 rad/ sec.

 

 

 

 

 

20 . v T
610 l
'200 ‘ 21 ' i; ' 12 16 ' 20
”WW
53004
-0.4
O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time(sec)

Figure 5.5. Plot of the modal amplitude and the end-force in the presence of the low
pass filter in the loop

79

 

These results indicate that the modal amplitudes of the higher modes decay slowly
in this case as compared to that shown in Figure (5.4). However, in this case the end-
force switches less frequently as compared to that shown in Figure (5.3). Clearly, the
bandwidth of the filter provides a trade off between switching frequency of the end-
force and the effectiveness of the end force in vibration suppression and modal energy
redistribution. A higher bandwidth (smaller value of 7') results in faster vibration
suppression and higher modal energy redistribution but causes the force to switch
very frequently, whereas a lower bandwidth results in less switching of the force but
requires longer time for vibration suppression.

Figure (5.4) also shows that when time is less than 10 sec, the work done by the
end-force is negative and in next 10 sec, force starts doing positive work. This clearly
indicates that the choice of the controller is very important to avoid instability of the
first mode. If the controller associated with the first mode can not reject energy that
is being pumped by the end-force fast enough, the amplitude of the first mode will
keep growing that is not desirable. We show the result of another simulation where
an observer-based controller is utilized to remove modal energy of the first mode as

well. This controller, chosen as

h1(5'1, 51) = 3(5.1 (528)

adds damping to the first mode for energy removal. The simulation results for this
case is shown in Figure (5.6). In this simulation the initial conditions and low pass
filter are the same as the previous simulation. The amplitudes of all modes decay to

zero, as expected.

80

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15

15

15

o 5 10 15

4o . ~
. H l ' I HI I ll ll ll I Ill
2 l . I

o * i w

5 10 15

time(sec)

Figure 5.6. Plot of the modal amplitude and the end-force in the presence of the low
pass filter and first order controller in the loop

5.6 Experimental Results

For a proof of principal, we conducted two simple experiments with a cantilever beam
and an end force. The beam is the same as that used in our experiments in Chapter 4.
The end force was applied by the cable and motor mechanism described in Chapter 4.
In our experiments, we used a piezoelectric actuator to excite the beam at its natural
frequencies. For this purpose, we input the piezoelectric actuator with a sinusoidal
signal at the same frequency as the natural frequency of the beam. The beam was
excited at its second natural frequency in the first experiment, and its first natural

frequency in the second experiment. After excitation of the beam we disconnected

81

the input to the piezoelectric actuator and applied the end force by energizing the
DC motor.

In the first experiments, the oscillations of the beam prior to application of the end
force consists only the second mode but consists of both the first and second modes
after application of the end force. Clearly, the end force causes energy associated with
the second mode to be redistributed among the first and second modes of the modified
(beam with non-zero end force) system. The results in the second experiment are also
similar; in this case, energy associated with the first mode is redistributed among the
first and second modes of the modified system.

The natural frequencies of the beam were not very low and as a result the vibrations
decay out rapidly in the presence of damping. The beam is therefore not an ideal
experimental platform for demonstration of modal disparity. An ideal platform would
be required to have low natural frequencies of vibration, as in the case of large space

structures to which this control methodology is targeted.

82

 

CHAPTER 6

Conclusion

We developed three new control methodologies for vibration suppression of flexible
structures. Our main objective was to reduce control system hardware that will in
turn reduce the cost and weight of the overall system. This has significant benefits
for space applications from cost, weight, and payload considerations.

Our first control methodology, presented in Chapter 3, is based on continuous
switching of piezoelectric transducers between actuator and sensor modes. In Chap-
ter 3 we first showed that it is possible to continuously reverse the roles of actuator
and sensor transducers in specific dynamical systems to significantly reduce the total
number of transducers and the weight and cost of the system without any loss in con-
trollability and observability. We adopted this idea to design an observer-based con-
troller for suppressing vibration in under-actuated and under-sensed Euler-Bernoulli
beams. Using simulations, we first demonstrated vibration suppression in first four
modes of a flexible beam whose actuator and sensor configurations individually do not
provide complete controllability and observability. Our experiments were less exten-
sive than simulations but they sufficiently demonstrated feasibility of controllability
and observability enhancement through switching. In our experiments, vibration of
the first two modes of a cantilever beam were suppressed using a single piezoelec-

tric transducer, switching between actuator and sensor modes. In general, switching

83

generates discontinuous control inputs which can excite unmodeled dynamics of a
system. This was observed in our experiments where higher unmodeled modes of the
beam were excited when we used a fixed-time switching schedule. The problem was
remedied by adopting a variable-time switching schedule which generates continu-
ous inputs. We also addressed the problem of optimal switching for faster vibration
suppression. For a fixed number of switchings, we determined the optimal switching
times apriori, and demonstrated improved performance through experiments.

Our second control methodology for vibration suppression, presented in Chapter
4, is based on the choice of piezoelectric transducers as sensors and motor-driven cable
actuators. Although the idea is quite general and is applicable to large structures,
we restricted or analysis and experiments to a simple cantilever beam. We demon-
strated the use of a compressive buckling-type end load in active vibration control of
a cantilever beam. The control process involves the use of piezoelectric transducers
for vibration measurements of the beam, an observer to estimate modal vibration am-
plitudes of the beam, filtering the data to restrict the bandwidth requirement of the
cable actuator, and switching the cable actuator on and off to remove the vibration
energy of the beam. The stability of the control system is established mathemati-
cally and both simulation and experimental results are provided for verification of
the theoretical results. The main limitation of this approach is that the number of
modes that can be handled is restricted by the bandwidth of the actuator, and hence
one can actively control only those modes below a certain frequency threshold. The
main advantage of the approach is that all modes below this threshold can be con-
trolled by a single actuator, at least in the particular case of the cantilever beam. The
low-pass filter in the feedback loop sidesteps the spillover problem while maintaining
stability. However, it adversely affects the settling time of the controlled system and
therefore the use of high-bandwidth actuators is desirable. The controller is more

effective when both the level of the bias load and the magnitude of the control force

84

are increased. Of course, this has a limitation, due to the fact that the sum of these
forces must remain below the structural buckling threshold.

The third control methodology is described in Chapter 5. In this chapter, we
introduce the concept of modal disparity. Modal disparity is a measure of the dif-
ference between modes in two stiffness states and can be exploited to gain control
authority over the significant flexible modes of a system using a low dimensional state
space model. In our study, the control methodology relies on variation in stiffness
of the beam to achieve modal energy redistribution from higher modes to the lower
modes and dissipating the energy associated with the lower modes. Since the lower
modes are only estimated and controlled, this approach has the potential to sidestep
spillover problem. We present an analytical framework for control design exploiting
the concept of modal disparity and verify the results through simulations and very

simple experiments.

85

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