Dynamics of Controlled Mechanical Systems with Delayed Feedback

302 Pages·2002·6.933 MB·English

by  Professor Haiyan Hu, Professor Zaihua Wang (auth.)

Hu -Wang
Dynamics of Controlled Mechanical Systems with Delayed Feedback
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LIBRARY
Engineering
http://www.springer.de/engine/
H. Y. Hu • Z. H. Wang
Dynamics of Controlled
Mechanical Systems
with Delayed Feedback
With 74 Figures and 8 Tables
Springer
Professor Haiyan Hu
Nanjing University of Aeronautics and Astronautics
Institute of Vibration Engineering Research
210016 Nanjing
P. R. China
e-mail: [e-mail]
Professor Zaihua Wang
Nanjing University of Aeronautics and Astronautics
Institute of Vibration Engineering Research
210016 Nanjing
P. R. China
e-mail: [e-mail]
ISBN 978-3-642-07839-2
Library of Congress Cataloging-in-Publication -Data applied for
Die Deutsche Bibliothek - Cip-Einheitsaufnahme
Hu, Haiyan: Dynamics of controlled mechanical systems with delayed feedback:
with 8 tables I H. Hu ; Z. Wang. -Berlin; Heidelberg; New York; Barcelona;
Hong Kong; London; Milan; Paris; Tokyo: Springer 2002
(Engineering online library)
ISBN 978-3-642-07839-2 ISBN 978-3-662-05030-9 (eBook)
DOI 10.1007/978-3-662-05030-9
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Originally published by Springer-Verlag Berlin Heidelberg New York in 2002
Softcover reprint of the hardcover 1st edition 2002
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To Luna and Amanda
for their love and support
over the years
Haiyan Hu
To my wife Jun Shen
and my daughterJ iayi
for their love
Zaihua Wang
Preface
Recent years have witnessed a rapid development of active control of various
mechanical systems. With increasingly strict requirements for control speed and
system performance, the unavoidable time delays in both controllers and actuators
have become a serious problem. For instance, all digital controllers, analogue anti
aliasing and reconstruction filters exhibit a certain time delay during operation,
and the hydraulic actuators and human being interaction usually show even more
significant time delays. These time delays, albeit very short in most cases, often
deteriorate the control performance or even cause the instability of the system, be
cause the actuators may feed energy at the moment when the system does not need
it. Thus, the effect of time delays on the system performance has drawn much at
tention in the design of robots, active vehicle suspensions, active tendons for tall
buildings, as well as the controlled vibro-impact systems. On the other hand, the
properly designed delay control may improve the performance of dynamic sys
tems. For instance, the delayed state feedback has found its applications to the
design of dynamic absorbers, the linearization of nonlinear systems, the control of
chaotic oscillators, etc.
Most controlled mechanical systems with time delays can be modeled as the
dynamic systems described by a set of ordinary differential equations with time
delays. Finite as the number of unknowns in the ordinary differential equations is,
the time delay implies that the change of a system state depends on the previous
history of system. The solution space of such a set of delay differential equations,
hence, is of infinite dimensions. This gives rise to a tough problem to the theoreti
cal analysis of delayed dynamic systems. Over the past decades, numerous
mathematicians have made great efforts to study the existence of solution, the os
cillation property, the stability and the local bifurcation for delayed dynamic sys
tems mainly in the frame of functional differential equations, and published a
number of excellent monographs. Among them, the books such as (Hale 1977),
(Qin et al. 1989), (Gopalsamy 1992), (Kuang 1993), (Hale and Lunel 1993) and
(Diekmann et al. 1995) are a few to name.
VIII Preface
From the viewpoint of an engineer, however, less attention has been paid to the
practical problems associated with delayed dynamic systems, such as the model
ing and parametric estimation, the stability analysis when some system parameters
are to be designed, the dynamic performance of nonlinear delay systems, and so
forth. Except for the works by (Stepan 1989) and (Moiola and Chen 1997), few
monographs have been available for the engineers, who deal with various prob
lems of control coming from mechanical engineering.
Motivated by the dynamics of controlled elastic structures, active vehicle sus
pensions and four-wheel-steering vehicles, the authors have been engaged in the
dynamics of high dimensional mechanical systems with feedback time delays
over the past five years. Summarized in this monograph are mainly recent ad
vances of authors in the system modeling and simplification, the stability analysis
of linear dynamic systems, the periodic vibration and bifurcation analysis of non
linear dynamic systems, as well as the application of new approaches to controlled
elastic structures and ground vehicles. The contents of the book are organized as
following.
In Chapter 1, the models of a number of typical dynamic systems with time
delays are presented first. Then, two parametric estimation techniques are given
for the linear systems with short feedback time delays and the nonlinear systems
with arbitrary feedback time delays, respectively. Afterwards, the identifiability
problem of delayed dynamic systems is addressed.
Chapter 2 serves as an introduction to the theory of delay differential equations.
It begins with the theorem of existence and uniqueness of a solution of initial val
ue problem, and then outlines the fundamental properties of linear delay differen
tial equations. Afterwards, it turns to the stability analysis of delay differential
equations, offers a brief review for the important concepts and available methods,
such as the Pontryakin theorem, the Hassard theorem, the Michailov criterion and
the Nyquist diagram, with help of a number of illustrative examples.
The topics of Chapter 3 are the delay-independent stability of high dimensional
linear systems with multiple time delays and the stability switches of high dimen
sionallinear systems with an increase of a single time delay. Those high dimen
sional systems may have a number of parameters to be designed so that the stabil
ity analysis becomes a tough problem. On the basis of generalized Sturm theory, a
simple, but systematic approach is presented to solve the tough problem. The ap
proach is demonstrated through the stability analysis of a tall building model
equipped with an active tendon, a quarter-car model of vehicle with an active sus
pension, as well as a four-wheel-steering vehicle with driver's delay.
Preface IX
Chapter 4 is devoted to the interval stability of high dimensional linear systems
with a number of commensurate constant time delays. Based on the well-known
edge theorem and the method of Dixon's resultant elimination, a new approach is
presented for testing the interval Hurwitz stability of a non-polytopic family of
quasi-polynomials. To demonstrate the approach, the interval Hurwitz stability is
analyzed for a single-degree-of-freedom system with two commensurate time de
lays in the paths of displacement and velocity feedback, respectively.
In many applications, the time delays are much shorter than the shortest period
of system vibration. If this is the case, the approximate approaches are preferable.
Several approaches to the stability estimation are presented in Chapter 5, on the
basis of perturbation of eigenvalues, for high dimensional linear systems with a
short time delay in feedback. A criterion of interval stability is suggested by ap
plying the Pade approximation to the exponential terms of time delay in the char
acteristic function of a linear system. In engineering, it is very natural and popular
to simplify the controlled systems with a short time delay by replacing the delayed
terms with their Taylor expansions. A detailed analysis in Chapter 5, together with
the examples of both linear and nonlinear systems of single degree of freedom, in
dicates that this simplification must be implemented with great care.
From Chapter 6, the book turns to the nonlinear dynamics of controlled systems
with time delays. To study the nonlinear dynamics of a system effectively, the
mathematical model for the system should be as simple as possible. In Chapter 6,
the theorem of central manifold and the theory of normal form are introduced first.
Then, the central manifold theory is combined with the singular perturbation tech
nique to simplify the nonlinear delay systems composed of a soft component and a
rigid component. A typical example of this system is the quarter car model of ve
hicle with an active suspension.
For a nonlinear dynamic system, the periodic motion is usually the second most
important topic, following the stability of equilibrium positions. Physically
speaking, there are two important causes for the emergence of a periodic motion if
the system is nonlinear. One is the well-known Hopfbifurcation at the equilibrium
of an autonomous system, and the other is the either external or parametric peri
odic excitation in a non-autonomous system. In Chapter 7, the periodic motions
owing to the two causes are discussed in detail. With help of the theory of the
Hopf bifurcation, the periodic motions and their stability of an autonomous dy
namic system under delayed control can be determined. Furthermore, if the gains
of delayed feedback can be scaled as small parameters, the method of multiple
scales can easily be used to analyze the dynamics of systems. In the case of strong
feedback involving time delays, numerical analysis becomes a possibly unique,
X Preface
but useful tool. The chapter presents a numerical approach to locate the periodic
motion of nonlinear systems with a time delay.
In Chapter 8, the delayed control of nonlinear systems is outlined. As an exam
ple, the delayed resonator with velocity feedback is presented first to work as a vi
bration absorber. Then, the stabilization to a critically stable system is presented.
Finally, controlling chaos, an interesting topic in the past two decades, is dis
cussed through an example of the forced Duffing oscillator with delayed feedback.
The first author appreciates very much the kind host of Professors E. H. Dowell
and L. N. Virgin to his sabbatical of 1996 in The Department of Mechanical Engi
neering and Material Science, Duke University, where he began to pay attention to
the dynamics of mechanical systems with delayed control. Most results presented
in this book come from the later projects supported in part by the National Natural
Science Foundation of China under the Grants 59625511 and 19972025, and in
part by the Ministry of Education under the Grant GG-130-10287-1593. The
authors wish to acknowledge all of the help and encouragement they have re
ceived in the development of this book. Special thanks should be due to Dr. H. L.
Wang and Dr. W. F. Zhang, who carefully read the manuscript of the book and
made invaluable suggestions.
Haiyan Hu and Zaihua Wang
Nanjing, March, 2002
Contents
1 Modeling of Delayed Dynamic Systems ............................................................ 1
1.1 Mathematical Models .................................................................................. 1
1.1.1 Dynamic Systems with Delayed Feedback Control ............................. l
1.1.2 Dynamic Systems with Operator's Retardation .................................... 5
1.2 Experimental Modeling ............................................................................... 9
1.2.1 Identification of Short Time Delays in Linear Systems ..................... 10
1.2.2 Identification of Arbitrary Time Delays in Nonlinear Systems .......... 14
1.2.3 Discussions on Identifiability of Time Delays ................................... 21
2 Fundamentals of Delay Differential Equations .............................................. 27
2.1 Initial Value Problems ............................................................................... 27
2.1.1 Existence and Uniqueness of Solution ............................................... 28
2.1.2 Solution of Linear Delay Differential Equations ................................ 33
2.2 Stability in the Sense of Lyapunov ............................................................ 37
2.2.1 The Lyapunov Methods ...................................................................... 38
2.2.2 Method of Characteristic Function ..................................................... 42
2.2.3 Stability Criteria ................................................................................. 47
2.3 Important Features of Delay Differential Equations .................................. 54
3 Stability Analysis of Linear Delay Systems .................................................... 59
3.1 Delay-independent Stability of Single-degree-of-freedom Systems .......... 60
3.1.1 Stability Criteria ................................................................................. 61
3.1.2 Stability Criteria in Terms of Feedback Gains ................................... 66
3.2 The Generalized Sturm Criterion for Polynomials .................................... 70
3.2.1 Classical Sturm Criterion ................................................................... 70
3.2.2 Discrimination Sequence .................................................................... 72
3.2.3 Modified Sign Table ........................................................................... 74
3.2.4 Generalized Sturm Criterion ............................................................... 75
3.3 Delay-independent Stability of High Dimensional Systems ...................... 76
3.4 Stability of Single-degree-of-freedom Systems with Finite Time Delays ....... 86
3.4.1 Systems with Equal Time Delays ....................................................... 86
3.4.2 Systems with Unequal Time Delays ................................................... 90
3.5 Stability Switches of High Dimensional Systems ...................................... 91
3.5.1 Systems with a Single Time Delay ..................................................... 92
3.5.2 Systems with Commensurate Time Delays ........................................ 98