Table Of ContentChristopher D. Rahn
Mechatronic Control of Distributed Noise and Vibration
Springer-Verlag Berlin Beideiberg GmbH
ONLINE LIBRARY
http://www.springer.de/engine/
Christopher D. Rahn
Mechatronic Control
of Distributed Noise
and Vibration
A Lyapunov Approach
With 59 Figures
~Springer
Associate Professor CHRISTOPHER D. RAHN
The Pennsylvania State University
Department of Mechanical and Nuclear Engineering
150 A Hammond Building
University Park, PA 16802
USA
e-mail: cdrlO@psu.edu
ISBN 978-3-642-07536-0 ISBN 978-3-662-04641-8 (eBook)
DOI 10.1007/978-3-662-04641-8
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To Matthew, Kevin, and Katelin
PREFACE
Vibration and noise reduce the perceived quality, productivity, and efficiency of many
electromechanical systems. Vibration can cause defects and limit production speeds
during manufacturing and produce premature failure of finished products due to fa
tigue. Potential contact with a vibrating system or hearing darnage from a noisy
machine can produce a dangerous, unhealthy, and uncomfortable operating environ
ment.
Recent advances in computer technology have allowed the development of so
phisticated electromechanical systems for the control of vibration and noise. The
demanding specifications of many modern systems require higher performance than
possible with the traditional, purely mechanical approaches of increasing system stiff
ness or damping. Mechatronic systems that integrate computer software and hard
ware with electromechanical sensors and actuators to control complex mechanical
systems have been demonstrated to provide outstanding vibration and noise reduc
tion. The current trends toward higher speed computation and lower cost, higher
performance sensors and actuators indicate the continuing possibilities for this con
trol approach in future applications.
The software control algorithms that act as the brain of the mechatronic sys
tem are based on an accurate model of the system to be controlled. At the scale of
most mechanical applications, the material composing the system components acts as
a continuum. Components that do not deform appreciably under the applied loading
may be approximated as rigid bodies. The remaining, distributed components can
be modeled by partial differential equations (PDEs) or approximated by ordinary
differential equations ( ODEs) using numerical discretization techniques such as the
Finite Element Method (FEM). Discretization may be the only option for distributed
components with complex geometry or built-up assemblies. Many manufacturing,
aerospace, acoustic, robotic, transportation, and power transmission applications,
however, have the geometric simplicity that make PDE models the most accurate
and concise representation of the system dynamics.
The purpose of this book is to present recent advances in the application of
Lyapunov's method for distributed parameter systems to the control of vibration and
noise. There have been many notable books on vibration and noise control that use
a discrete or modal approach (e.g. [98, 54, 103, 45, 46, 43, 94]). In graduate en
gineering textbooks, Inman [54) and Junkins [59) introduce distributed control and
Lyapunov theory for distributed systems, respectively. Edited books [125, 23) present
VIII
contributions to the control of distributed parameter vibration. Much of the mathe
matical treatises on distributed parameter or infinite dimensional systems ( e.g. [28]),
do not address implementation. The recent book by Luo et al. [78] focuses on semi
group theory. In de Queiroz et al. [31], applications of Lyapunov's direct method
are presented including distributed parameter systems. Unlike previously published
work, this book develops adaptive, output feedback controllers that asymptotically
stabilize distributed vibration and noise using few sensors and actuators, compensate
for the inertial and electrical actuator dynamics, and learn system parameters such
as tension and bending stiffness. Visual feedback control using high speed video,
setpoint regulation for systems with rigid body modes, and isolation of bounded dis
turbances are also presented. The intended audience for the book includes academic
and industrial researchers and graduate and advanced undergraduate students in en
gineering and applied mathematics. In addition to being a reference text, the book is
sufficiently broad and self-contained to act as a graduate text in vibration and noise
control. It includes modeling, simulation, control development, and implementation
for distributed vibration and noise problems.
After a short introduction in Chapter 1, Chapter 2 demonstrates distributed
parameter modeling using Hamilton's Principlc. The equations of motion of second
order ( e.g. strings, cables, and acoustic ducts) and fourth order ( e.g. beams and
plates) systems are derived. It is shown that the equations of motion for vibration
and noise problems can be cast in a matrix operator form with the operators having
special properties. Exact solution of the open and closed loop equations of motion
using modal analysis is demonstrated. Galerkin's method is used for time simulation
of systems with nonuniform parameter distributions and/or damping.
Chapter 3 introduces the mathematical tools needed for the application of Lya
punov's method to distributed parameter systems. Pointwise and L2 boundedness,
asymptotic stability, and exponential stability are defined. Key inequalities are intro
duced that aid in the bounding of Lyapunov functionals and their time derivatives.
Lyapunov's method, Barbalat's lemma, and LaSalle's invariance principle are intro
duced as stability proof techniques. The basics of semigroup theory are demonstrated
to determine the existence and uniqueness of a semigroup solution.
Chapter 4 develops passive controllers for distributed parameter systems. The
effects of boundary, distributed, pointwise, and parametric damping are explored.
Passive boundary controllers are developed that provide exponential stability despite
actuator dynamics. Free boundary problems such as the gantry crane and flexible
link robot are studied. Mechatronic implementation experimentally demonstrates the
feasibility and performance of the proposed controllers.
Chapter 5 develops Exact Model Knowledge (EMK) Controllers for second and
fourth order systems. Simple examples of a pinned-controlled string and a clamped
controlled beam are used to demonstrate the techniques. Compensation for compli
cating effects such as actuator dynamics, nonuniform parameter distributions, and
nonlinearities are demonstrated. Complete boundary actuation ( i. e. on all bound-
IX
aries) allows setpoint regulation of rigid body and vibrational motion. Isolation Con
trollers for acoustic noise and axially moving beam models are developed. Visual
feedback controllers are derived and tested on a flexible link robot using a high-speed
video camera and two computers interfaced via a dedicated TCP /IP connection.
Chapter 6 demonstrates how many of the controllers developed in Chapter 5
can be redesigned as adaptive Controllers that compensate for parametric uncertainty.
Adaptive boundary controllers for nonlinear and spatially varying systems and isola
tion controllers are developed. Mechatronic testbeds are constructed to demonstrated
the improved closed loop performance under EMK and adaptive control relative to
passive and classical PID approaches.
I would like to thank my former colleagues and students in the Robotics and
Mechatronics Labaratory at Clemson University for their contributions to this book.
I was fortunate to have worked with Professor Darren Dawson in the Electrical and
Computer Engineering Department without whom many of the approaches presented
in this book would never have been developed. My former students, Catalin Baicu,
Huseyin Canbolat, Fumin Zhang, Siddhartha Nagarkatti, Yugang Li, Aniket Malat
pure, Sushil Singh, and Dan Aron, did much of iterative development of Lyapunov
fundionals and experimental implementation. I greatly appreciate the review of an
early draft of this book by the students in ME 893, Control of Vibration and Noise
at Clemson in Spring of 2000.
Christopher D. Rahn
University Park, PA
CONTENTS
1 Introduction 1
2 Distributed Parameter Models 7
201 Harnilton's Principle 0 0 0 0 7
202 Lumped Systems 0 0 0 0 0 0 8
20201 Mechanical Systems 0 8
20202 Electrical Systems 0 9
203 Distributed Systems 0 0 0 0 9
20301 Second Order Systems 10
20302 Fourth Order Systems 16
2.4 Matrix Operator Representation 0 22
2.401 Second Order Systems 23
2.402 Fourth Order Systems 0 0 26
205 Simulation 0 0 0 0 0 0 0 0 0 0 0 0 0 28
20501 Exact Modal Analysis for Undamped Systems 28
20502 Approximate Simulation - Galerkin's Method 35
3 Mathematical Preliminaries 43
301 Definitions 0 0 0 0 0 0 0 0 0 43
302 Inequalities 0 0 0 0 0 0 0 0 0 0 0 0 0 44
303 Existence of a Sernigroup Solution 0 47
30301 Axially Moving String 48
30302 Gantry Crane 0 0 0 0 0 0 0 0 50
3.4 Stability Theorems 0 0 0 0 0 0 0 0 0 53
30401 Lyapunov's Direct Method 0 53
3.402 Barbalat's Lemma 0 0 0 0 0 53
3.403 LaSalle's Invariance Principle 54
4 Passive Control 55
401 Damping 0 0 0 0 0 0 0 0 0 0 0 0 55
401.1 Boundary Damping 0 0 55
401.2 Distributed Damping 0 66
401.3 Pointwise Damping 0 0 73
XII CONTENTS
4.1.4 Parametrie Damping .... 75
4.2 Passive Boundary Control . . . . . 77
4.2.1 Boundary Controlled String 77
4.3 Free Boundary Problems . . 84
4.3.1 Gantry Crane .... 84
4.3.2 Flexible Link Robot 88
5 Exact Model Knowledge Control 91
5.1 Boundary Control . . . . . . . 91
5.1.1 Seeond Order Systems . . 91
5.1.2 Fourth Order Systems . . 97
5.1.3 Baekstepping Compensation for Aetuator Dynamies . 102
5.1.4 Compensation for Geometrie and Material Nonlinearities 106
5.1.5 Effeets of Nonuniform Parameter Distributions . 118
5.1.6 Rigid Body Setpoint Regulation 123
5.2 Domain Control . . . . . . . . . . . . 137
5.2.1 Aeoustie Noise Isolator . . . . . 137
5.2.2 Axially Moving Beam Isolator . 144
5.3 Modal Control using Distributed Sensing 150
5.3.1 Mathematieal Model . . 150
5.3.2 Control Design . . . . . . 151
5.3.3 Residual Mode Stability . 153
5.3.4 Flexible Link Robot Arm. 155
6 Adaptive Control 165
6.1 Boundary Control . . . . . . . 165
6.1.1 Seeond Order Systems 165
6.1. 2 Fourth Order Systems 171
6.1.3 Compensation for Geometrie and Material Nonlinearities 176
6.1.4 Effeets of Nonuniform Parameter Distributions . 180
6.2 Domain Control . . . . . . . . . . . . 184
6.2.1 Aeoustie Noise Isolator . . . . 184
6.2.2 Axially Moving Beam Isolator 189
7 Bibliography 203
