Simplification and Order Reduction of Parametrically Excited
Nonlinear Dynamical Systems
by
Amit P. Gabale
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
December 13, 2010
Keywords: Normal forms, Nonlinear order reduction, Invariant manifold, Solvability and
reducibility conditions, Lyapunov-Floquet transformation, Parametrically excited systems
Copyright 2010 by Amit P. Gabale
Approved by
Subhash C. Sinha, Chair, Professor of Mechanical Engineering
George T. Flowers, Professor of Mechanical Engineering
John Y. Hung, Professor of Electrical and Computer Engineering
Andrew Sinclair, Assistant Professor of Aerospace Engineering
Abstract
System simplification is, by far, the most common theme behind various techniques
available for the analysis and controller design of nonlinear dynamic systems. One such technique
that is widely used by engineers is the direct linearization about an equilibrium point. Most other
techniques, however, involve carefully designed state transformations so that the system in the
transformed domain preserves nonlinear characteristics of the original system and facilitates
efficient analysis and controller design. This dissertation presents a few of such methodologies
for analysis and control of nonlinear dynamic systems. However, the emphasis is on nonlinear
systems with parametric excitations.
The main focus is on the development of techniques for reduced order modeling of
nonlinear dynamic systems that are influenced by certain external inputs. These reduced order
models accurately approximate the dominant dynamics of original systems and are much simpler
to analyze and control. Two types of inputs are considered; namely, external periodic excitations
and a nonlinear, stabilizing state feedback control. The approach is based on the construction of
an invariant subspace such that any motion initiated on this subspace, often called an invariant
manifold, remains confined there for all time, t. As a result, the system dynamics can be modeled
by a smaller number of differential equations which govern system evolution on the manifold,
alone. Techniques to identify suitable state variables which span these manifolds are suggested
and approximate solutions of the resulting partial differential equations, which govern the
geometry of such manifolds, are presented. Reduced order models for systems with periodic
excitation inputs can be integrated directly to analyze the dynamics of original systems. For
ii
situations where there is a state feedback input, nonlinear controllers can be designed in the
reduced order domain. A technique to synthesize such controllers is also discussed in this work.
Another important simplification technique proposed in this work is the construction of
normal forms (simplest forms) of nonlinear systems by direct application of time-periodic near-
identity transformations and Poincar? normal form theory. This is an independent analysis
technique for systems subjected to periodic forces. It also provides a basis for construction of
time-dependent invariant manifolds for reduced order modeling of such systems. Some typical
dynamic systems of engineering interest are considered and the effectiveness of proposed
methodologies is illustrated.
iii
Acknowledgments
The author is gratefully indebted to Dr. S. C. Sinha, Professor, Department of Mechanical
Engineering, for his invaluable guidance, encouragement and patience in all stages of this work.
He would also like to thank Dr. George T. Flowers, Professor, Department of Mechanical
Engineering, Dr. John Y. Hung, Professor, Department of Electrical and Computer Engineering
and Dr. Andrew Sinclair, Assistant Professor, Department of Aerospace Engineering, for serving
on his committee. The author is thankful to Dr. Yandong Zang for his help on numerous
occasions in debugging and improving some of the author?s computer codes. He is also thankful
to Dr. Oluseyi Onawola and Mr. Pregassen Soobramaney for many friendly discussions, both,
technical and non-technical.
Finally, the author would like to dedicate this work to his parents, Pramod and Deepali;
his wife, Uma; and his sister, Anuradha, for all their love and support, without which this
endeavor would never have been successful.
iv
Table of Contents
Abstract ? ?????????????????????????????????... ii
Acknowledgments ??????????????????????????????...iv
List of Figures ???????????????????????????????? vii
Chapter 1 Introduction ??????????????????????????? 1
1.1 Motivation ????????????????????????? 1
1.2 Model Order Reduction Background ?????????????....... 2
1.3 Contributions of this Dissertation ???????????????? 7
Chapter 2 Analysis of Nonlinear Systems Subjected to External Periodic Excitations
via Normal Forms ???????????????????????? 11
2.1 Introduction ???????????????????????? 11
2.2 Mathematical Background ?????????????????..... 12
2.2.1 Time Independent Normal Forms (TINF) ?????????? 12
2.2.2 Time Dependent Normal Forms (TDNF) ?????????? 15
2.3 Construction of Normal Forms for Forced Nonlinear Systems ????. 19
2.3.1 Systems with Constant Coefficients ???????????? 19
2.3.2 Systems with Time-Periodic Coefficients ?????????? 24
2.3.3 An Alternate Approach for Non-resonant Case ???????? 26
2.4 Applications.????????????????????............... 28
2.4.1 Forced Duffing?s Equation ???????????????... 29
2.4.2 Forced Mathieu-Duffing Equation ????????????? 35
Chapter 3 Order Reduction of Nonlinear Systems Subjected to External
Periodic Excitations ???????????????????????. 44
v
3.1 Introduction ???????????????????????? 44
3.2 Order reduction of Systems Subjected to Periodic Excitations ????. 46
3.2.1 Systems with Constant Coefficients ????????????. 46
3.2.2 Systems with Time-Periodic Coefficients ?????????? 54
3.3 Applications ???????????????????????... 57
3.3.1 A 2-DOF Spring-Mass-Damper System ??????????.. 57
3.3.2 A 2-DOF Coupled Inverted Pendulum ??????????? 62
Chapter 4 Reduced Order Controller Design for Nonlinear Systems ????????...76
4.1 Introduction ???????????????????????? 76
4.2 Controller Design for Linear Time-Periodic Systems ???????... 77
4.3 Reduced Order Nonlinear Controller Design ??????????? 78
4.3.1 Systems with Time-Periodic Coefficients ?????????? 78
4.3.2 Systems with Constant Coefficients ????????????. 89
4.4 Applications ???????????????????????... 91
4.4.1 A 4-DOF Coupled Inverted Pendulum ??????????? 91
4.4.2 A 2-DOF Spring-Mass-Damper System ??????????.. 97
Chapter 5 Discussion and Conclusions ???????????????????...110
5.1 Summary of Work ????????????????????? 110
5.2 Suggestions for Future Work ????????????????... 114
References ????????????????????????????????? 117
Appendix A Floquet Theory and Lyapunov-Floquet Transformation ????????... 122
Appendix B Solution of Time-periodic Sylvester Differential Equation ???????... 131
Appendix C Computer Codes ?????????????????????............ 136
vi
List of Figures
2.1 System response for forced Duffing?s equation (non-resonant forcing) ??????...38
2.2 System response for Forced Duffing?s equation ( )3
n
??? ??????????? 39
2.3 System response for forced Duffing?s equation ( )3
n
??? ???????????. 40
2.4 System response for forced Mathieu-Duffing equation (Floquet exponent has
negative real parts) ??????????????????????????? 41
2.5 System response for forced Mathieu-Duffing equation (Floquet exponent are
purely imaginary) ???????????????????????????.. 42
2.6 System response for forced Mathieu-Duffing equation ( 0a ? , no generating solution
exists) ???????????????????????????????? 43
3.1 A 2-DOF spring-mass-damper ??????????????????????.. 65
3.2 System response for 2-DOF system with constant coefficients ( )
122
??=?? ???. 66
3.3 Fast Fourier transforms for 2-DOF system with constant coefficients ( )
122
??=?? ...67
3.4 Amplitude variation vs excitation frequency for 2-DOF system with constant
coefficients ( )
122
??=?? ???????????????????????.. 68
3.5 System response for 2-DOF system with constant coefficients ( )
1222
,3 ???? ?? ?.. 69
3.6 Fast Fourier transforms for 2-DOF system with constant
coefficients ( )
1222
,3 ???? ?? ?????????????????????. 70
3.7 System response for 2-DOF system with constant coefficients ( )
122 2
,3?????? ?. 71
3.8 Fast Fourier transforms for 2-DOF system with constant
coefficients ( )
122 2
,3?????? ?????????????????????.. 72
3.9 A 2-dof coupled inverted pendulum ????????????????????. 73
3.10 System response for 2-DOF system with periodic coefficients ?????????? 74
vii
viii
3.11 Fast Fourier transforms for 2-DOF system with periodic coefficients) ??????? 75
4.1 A 4-DOF coupled inverted pendulums ??????????????????... 100
4.2. Uncontrolled system response for system with periodic coefficients ???????. 101
4.3 Controlled system response for system with periodic coefficients ????????. 102
4.4 Control effort for system with periodic coefficients ?????????????... 103
4.5 Regions of attraction for controlled system (system with periodic coefficients) ???104
4.6 2-DOF spring-mass-damper system ????????????????????105
4.7 Uncontrolled system response for system with constant coefficients ???????.106
4.8 Controlled system response for system with constant coefficients ????????.107
4.9 Control effort for system with constant coefficients ?????????????...108
4.10 Regions of attraction for controlled system (System with constant coefficients) ??...109
Chapter 1
Introduction
1.1 Motivation
Dynamics of many engineering systems is often represented by a set of large number of
nonlinear ordinary differential equations (ODEs). These equations may result from lumped
modeling of systems or finite element discretization of partial differential equations (PDEs)
associated with the system dynamics. In some cases a system itself may have a discrete
representation with a very large number of degrees of freedom. An important class of systems
also results into nonlinear ODEs with periodic coefficients. Systems with structural members
subjected to in-plane loads, torsional systems driven by a Hook?s joint, asymmetric rotor bearing
systems, etc. are few examples which yield equations of motion of this kind. Analysis and
simulations of such large dimensional nonlinear systems, with constant or periodic coefficients,
are costly to perform in terms of computer time and storage. More importantly, it is difficult to
grasp interrelationships among various system parameters and their influence on system
dynamics as these interactions are often too complex due to system nonlinearities; which is true
even for a small scale systems. At the same time, the task of designing nonlinear feedback
controllers for such systems poses serious challenges. Because the nonlinear controller design
techniques typically require symbolic computations (e.g., Deshmukh (2000), Zhang (2007)), the
high dimensionality of the phase-space of large nonlinear systems presents significant challenges
to these efforts. Construction of reduced order models, which are represented by a lesser number
1
of state variables, may overcome some of these difficulties. The main objective of this
dissertation is to address such model order reduction strategies for nonlinear dynamic systems
with constants as well as periodic coefficients which are subjected to external periodic or
stabilizing control input. The approach is based on the construction of an invariant subspace on
which the system dynamics is represented by a fewer number of state equations.
Another simplification technique proposed in this work is aimed at construction of
normal forms for nonlinear systems subjected to external periodic excitation. However, unlike
order reduction, in this case the system dimensions remain the same as that of the original system
in its simplified form. Here also, systems with constant as well as periodic coefficients are
considered.
1.2 Model Order Reduction Background
Consider a dynamic system represented by a set of first order ordinary differential and
algebraic equations given as
( )
()
,,
,
t
t
=
=
xfxu
y ? x
(1.1)
where
n
R?x is the state vector,
m
R?y is the output,
( )
,
s
R sn? ?uu is an input or excitation
function, while and (,,tfxu) ( ),tx? are nonlinear vector fields of an appropriate dimension. The
complexity of such a system is usually indicated by the number of state variables that comprise
the system. Model order reduction in such cases implies construction of a dynamic system of the
form
2
( )
()
,,
,
t
t
=
=
xfxu
y ? x
(1.2)
such that
p
R?x ( )p n< and the output of the reduced order model (1.2)
(
m
)
R?yy is a good
approximation of the original output ( )y . A typical model order reduction method requires one to
define a transformation such that the less important (non-dominant) states of the system are
expressed as functions of the important (dominant) states. This transformation can then be
substituted into the original system to eliminate the less important states. Further, apart from
accurately estimating the dynamics of the original system, a reduced order system may also need
to satisfy some of the following conditions.
1. Stability and bifurcation characteristics of the original model should be preserved.
2. In case the system is subjected to a state feedback, it should be feasible to design a
controller in the reduced order domain.
3. The procedure should be computationally stable and efficient.
The problem outlined above is a generalized order reduction problem and more specific
cases of this problem have been addressed by various researchers in the past. Among them, the
case of linear time-invariant systems (LTI) has received a considerable amount of attention in the
literature. These systems result from the direct linearization of (1.1) about the equilibrium point
when the equation do not contain any time-varying terms. A comprehensive survey of the recent
work on this topic can be found in Antoulas (2005), Antoulas et al. (2004), Benner (2005, 2006)
and Benner et al. (2006). These order reduction techniques, which are based on the idea of linear
3
projecti
The idea behind balanced truncation is to eliminate those states which are difficult to
control and observe. This is achieved by transforming the system equation into the balanced form
for which the controllability and observability grammians are equal and in the diagonal form.
States corresponding to the small diagonal elements are nearly uncontrollable and unobservable
and, therefore, can be eliminated. However, this approach preserves the stability and provides
bounds on approximation error when applied to stable systems. Next, the moment matching is a
frequency domain based order reduction technique. In this technique the transfer functions of a
large system and a reduced order system are expanded into a power series and first coefficients
of the power series are matched to find parameters of the reduced order system. Coefficients of
the power series are called ?moments? and is the order of the reduced system. Modal truncation
technique is, by far, the simplest of the three. The principle of this method is to project the state
space onto the subspace spanned by eigenvectors corresponding to the dominant eigenvalues of
linear state matrix. Thus, the dominant behavior of the original system is preserved in a reduced
order s
Further, these techniques have also been generalized for the linear time-varying (LTV)
systems. Among these, balanced truncation is the most widely used techniques for order
reduction of the LTV systems. Some of the early work and references on balanced truncation of
the LTV systems can be found in Shokoohi et al. (1983) and Verriest and Kailath (1983).
Recently the moment matching type techniques have also been used for order reduction of the
LTV systems arising from large electronic circuits (Phillips, 1998; Roychowdhury, 1999; Wan
on of system?s original state-space onto a smaller dimensional space, fall into three main
categories (Benner, 2006) namely; balanced truncation, moment matching and modal truncation.
k
k
ystem. Typically, eigenvalues which are very close to the stability boundary or those
which are in resonance with external excitations are considered the dominant.
4
and Roychowdhury, 2005). However, a direct modal truncation for LTV systems is not possible
due to the fact that unlike LTI systems, the modes of LTV systems are not defined, in general.
For linear systems with time-periodic coefficients, one may employ the Lyapunov-Floquet (L-F)
transformation which yields time-invariant systems. However, not only the calculation of the L-F
transformation matrix is difficult, it also renders a time-periodic/quasiperiodic input gain matrix.
To author?s best knowledge there is only one reference (Deshmukh et al., 2000), so far, which has
explored this possibility.
lity conditions, is an important tool in understanding interactions between
system?s modes and their interactions with external and/or parametric excitations. Moreover,
POD based techniques require a priori simulation of the original large system to find principle
orthogonal modes.
Although there exists an abundant amount of literature on the order reduction of linear
systems, there is not as much work done for nonlinear systems; more so, for nonlinear systems
with time-varying parameters. The existing nonlinear order reduction techniques can be divided
into two main categories; namely, 1) linear projection based techniques adapted for nonlinear
model reduction and 2) nonlinear projection based techniques which are intrinsically nonlinear in
nature. The former category typically employs either the Guyan or proper orthogonal
decomposition (POD) like order reduction techniques to nonlinear dynamic systems. Some recent
examples of the application of these techniques to nonlinear order reduction problem can be
found in Friswell et al. (1995), Burton and Rhee (2000), Lenaerts et al. (2001), Kerschen et al.
(2005), and Kumar and Burton (2007, 2009). Although, these techniques are computationally
efficient and easy to implement, they do not yield various reducibility conditions in a closed
form. These reducibi
5
The nonlinear projection based order reduction techniques can be broadly lassified into
three categories; namely, center manifold, nonlinear normal modes (NNMs) and singular
perturbation. A common thread which relates these three techniques is the construction of an
invariant subspace in systems phase-space such that any motion initiated on this invariant
subspace, often called an invariant manifold, remains on the subspace for all time t . The system
dynamics on this subspace is governed by a fewer number of equations. First technique, the
?center manifold?, is a useful order reduction tool when the linear system matrix has a special
spectral decomposition; wherein, some of the eigenvalues of the system lie on the imaginary axis
(critical modes) while the remaining are on the left half of the complex plane (stable modes). In
such systems, the evolution of the dynamics is governed by the critical modes alone and damped
modes follow this ?dominant dynamics? passively. As a result, it is possible to construct a
minimal sized model which depends only on the critical modes of the system. A detailed
discussion of the center manifold technique can be found in Carr (1981) and Guckenheimer and
Holmes (1983). The second technique, NNMs (Nonlinear Normal Modes), was originally
introduced by Rosenberg (1966). He defined NNMs for autonomous, conservative nonlinear
systems as a vibration in unison with fixed relations between generalized coordinates. Later,
Shaw and Pierre (1993) generalized the concept of NNMs to non-conservative systems. They
used invariant manifold approach in which NNMs are assumed to be invariant subspaces which
are non-planer and tangential to corresponding normal-modes of the linearized system at the
equilibrium. This nonlinear extension of the modal analysis allows one to generate reduced order
models by projecting non-essential modes on the invariant manifold which is parameterized by a
set of desired modes. NNMs approach has gained popularity among structural/mechanical
engineers in recent years. Agnes and Inman (2001) and Pesheck et al. (2002) have applied this
technique to reduce the order of real engineering problems. The third technique, ?singular
c
6
perturbatio , is more popular in the control engineering community where it has been used
effectively to design reduced order controllers for various nonlinear systems. Systems which
come under the premise of singular perturbation admit (or can be transformed into) a special form
in the state-space. In this form some of the derivatives of states are multiplied by a small positive
parameter
n?
? . Such systems typically display a two time-scale phenomenon; by which, the fast
subsystem approaches a steady state more quickly than the slow subsystem. Consequently, the
long term behavior of such systems can be approximated by the slow subsystems, alone, on a
lower dimensional subspace. The singular perturbation provides a theoretical frame work to
decompose such large systems into two subsystems; the slow subsystems representing the
reduced order models. A detailed discussion on the subject can be found in books by Kokotovi?
et al. (1999) and Khalil (2002). Among the techniques discussed above, the center manifold
technique has been extended to systems with periodic coefficients (Malkin, 1962). An interesting
account of the application of time-periodic center manifold theory for the bifurcation analysis of
parametrically exited systems can be found in Pandiyan and Sinha (1995) and D?vid and Sinha
(2003). Further, Sinha et al. (2004) have also generalized the NNM technique, introduced by
Shaw and Pierre (1993), for systems with periodic coefficients. The treatment for singular
perturbation technique also accommodates time-varying parameters in some of the literature.
1.3 Contributions of this Dissertation
However, to the author?s best knowledge, the practical implementation of this technique to
general time-varying nonlinear systems has not been reported.
The primary goal of this dissertation is to develop some new strategies as well as
practical algorithms for the order reduction of nonlinear dynamic systems which are subjected to
certain external inputs. Two types of inputs are considered; namely, external periodic excitations
7
and a nonlinear, stabilizing state feedback control. The techniques proposed here are based on the
construction of invariant manifolds such that the system dynamics projected on these manifolds is
represented by a smaller number of equations. Another important contribution of this dissertation
is the development of Poincar? normal form technique for nonlinear systems subjected to external
periodic excitations. Apart from being an independent analysis technique, this work also provides
a basis for the construction of time-dependent invariant manifolds for systems subjected to
periodic inputs. The rest of the dissertation is arranged is in the following order.
riodic systems. Comparisons with numerical
solutions show that the proposed techniques provide accurate results for all cases considered. This
is a stand-alone analysis technique for periodically forced nonlinear problems; however, results
obtained in this work also play an important role in the development of methodology for the
construction of time-dependent invariant manifolds.
In Chapter 2, a direct application of the method of normal forms for the analysis of
nonlinear systems subjected to a periodic excitation is discussed. The direct approach presented
here is in contrast to the procedures where such systems are transformed into higher order
homogenous forms by defining additional coordinates. A set of time-periodic transformations,
called near-identity transformations, are applied such that the resulting system of equations is of
simpler form, or even linear if all nonlinear terms can be eliminated. The solvability conditions
for such reductions are also derived in a closed form. Systems with time-periodic coefficients are
handled via an application of the L-F transformation that converts the linear parts of systems into
time-invariant forms. The technique has distinct advantage over averaging or perturbation
techniques because periodic coefficient of the linear term, in this case, does not require a small
parameter assumption. In addition, the question of existence of so called ?generating solution?
does not arise in this approach. The usefulness of the method is demonstrated by applications to
periodically forced time-invariant as well as time-pe
8
In Chapter 3, an order reduction technique for nonlinear systems subjected to external
periodic excitations is proposed via construction of time-dependent invariant manifolds. The
geometry of invariant manifolds is governed by a PDE which is nonlinear and contains time-
periodic coefficients. An approximate solution of this PDE is suggested in terms of a
multivariable Taylor-Fourier series. It is shown that in the case when the external excitations
projected on the dominant dynamics are small, the equation governing the manifold can be
reduced to a set of equations which are simpler to solve and yield the reducibility conditions in a
closed form. These reducibility conditions, which represent interactions among dominant, non-
dominant states and the excitation terms, are indicators of the feasibility and extent of accuracy of
such order reductions. The proposed technique does not put any constraint on the selection of
dominant and non-dominant states; provided that the chosen partitioning satisfies all reducibility
conditions. Therefore, an investigator is free to choose any set of states of practical importance as
dominant states. The resulting reduced order models can be numerically integrated or can be
analyzed using traditional nonlinear analysis techniques. Systems with time-periodic coefficients
are handled via an application of the L-F transformation. Due to the periodic nature of the L-F
transformation matrix, however, the system nonlinearities in the transformed domain contain
additional time-periodic coefficients which warrant a modification in the structure of the invariant
manifold. The reducibility conditions are obtained in terms of Floquet exponents in this case. The
effectiveness of the method is demonstrated by constructing reduced order models for some
typical engineering systems. Results obtained from reduced order models are compared with
numerical simulations of full order models. Based on these results, it is concluded that the
proposed technique provides accurate reduced order models for nonlinear systems of fairly
general form provided that the reducibility conditions are satisfied.
9
In Chapter 4, a technique for the design of reduced order controllers for parametrically
excited nonlinear systems is described first. The approach is restricted to the class of local
controllers where large scale nonlinear systems may need to be stabilized about an equilibrium
point, a periodic motion or desired to be driven to a periodic orbit. Since only the stabilization
problem is addressed in this work, unstable states (Floquet multipliers outside the unit circle) are
considered to be the dominant states and stable states (Floquet multipliers inside the unit circle)
are considered as the non-dominant states. Techniques similar to the Poincar? normal form theory
is used to transform the reduced order nonlinear closed-loop system to a time-periodic linear
system. The linear controller is designed using a symbolic approach that allows one to place the
Floquet multipliers within the unit circle at the desired locations. After the necessary
transformations, a controller can be obtained in the original states. The case of systems with
constant coefficients is treated as a special case of its time-periodic counterpart.
In Chapter 5, a summary of this dissertation is presented. Some suggestions for future
research activities in this area are also outlined.
10
Chapter 2
Analysis of Nonlinear Systems Subjected to External Periodic Excitations
via Normal Forms
2.1 Introduction
The theory of normal forms, originally introduced by Poincar?, is a powerful tool in
simplification and analysis of complex nonlinear systems. Originally the normal form theory was
developed for time-invariant systems only. Some of the early discussions on time-invariant
normal form theory can be found in Birkhoff (1927), Moser (1928) and Arnold (1978). Later, the
technique was extended to systems with parametric forcing (Arnold 1983; Arrowsmith and Place
1990), where the Lyapunov-Floquet theory was used to transform the time-periodic system into a
form where the linear part of the system becomes time invariant. However, the applications to
real engineering problems remained unsuccessful until Sinha and Pandiyan (Sinha and Pandiyan,
1994; Pandiyan and Sinha, 1995) developed a practical technique for the analysis of such systems
via an efficient computation of L-F transformation and use of the time dependent normal form
theory. However, these techniques are not directly applicable when systems are subjected to an
external periodic forcing. Nayfeh (1993) analyzed Duffing?s oscillator with external forcing using
the method of normal forms, where an additional state-variable was introduced to convert the
system into a homogenous form. However, to author?s knowledge, to date, there is no normal
form formulation available for the analysis of nonlinear dynamic systems with time varying
coefficients and external forcing.
11
The work presented in this chapter deals with a quantitative analysis of nonlinear forced
problems via a direct application of near-identity transformations and normal forms. Systems
with constant as well as periodic coefficients are considered. In the former case, the suggested
transformation is applied directly; while, for the latter case, first, the L-F transformation is used to
convert a system into the form where linear part becomes time-invariant. In the following section
the general frame work of the normal form technique for homogeneous systems is discussed.
2.2 Mathematical Background
2.2.1 Time Independent Normal Forms (TINF)
The idea behind the normal form reduction is to simplify nonlinear equations by
eliminating as many nonlinear terms as possible, while keeping the linear part unchanged. This is
achieved by application of successive ?near-identity? transformations to the original nonlinear
system, beginning with the lowest order transformation. Consider a nonlinear dynamic system
given by
() () ()
( )
1
23
k
k
+
=+ + ++ +xAxfx fx fx Oxnull null (2.1)
Typically, such systems results form a Taylor series expansion of nonlinear vector fields about the
equilibrium point. In the above equation is an A nn? constant matrix, ( )
r
fx are vectors
of homogeneous monomials in of order
1n?
x r ( )2,= 3, ,rnull k. An application of the modal
transformation ( )x=Mz to above system yields
() () ()
( )
1
11 11
23
k
k
+
?? ??
=+ + ++ +zJzMfMzMfMz MfMzMOMznull null (2.2)
where matrix is in the Jordan canonical form and is the modal transformation matrix. nn? J M
12
Further, an application of the ?near-identity? transformation of the form
( )
2
=+zvhv (2.3)
where ( )
2
,thv is an homogeneous vector of monomials in of order 2, to equations (2.2)
yields
1n? v
( )
() () () ()
(
1
2
22 3
k
k
+
???
??
=? ? ? + ++ +
??
?
??
hv
vJv JvJhv fv fv fv Ov
v
null null
)
(2.4)
Thus, if one wants to eliminate all the quadratic terms form equation (2.4) the ?near-identity?
transformation (2.3) must satisfy a PDE given by
( )
() ()
2
22
0
?
? ?=
?
hv
Jv Jh v f v
v
(2.5)
Above equation is known as the ?homological equation? associated with ( )
2
hv.
Similarly, it can be shown that the higher order nonlinearities can be eliminated by successive
application of near-identity transformations of the respective degree. The resulting general form
for the homological equation can be written as
( )
() () 0
r
rr
?
? ?=
?
hv
Jv Jh v f v
v
(2.6)
It should be noted that for , 3r > ( )()
r
fv is expressed in terms of the solution of homological
equation of degree . In general, it is not possible to determine an exact solution for the
homological equation (2.6). However, one can obtain an approximate solution in terms of series
expansion. Accordingly,
1r ?
( )
r
hv and ( )
r
fv are expressed as
13
()
()
,,
1
,,
1
n
r j
rrj
r
j
r
n
r j
rrj
r
j
r
h
f
=
=
=
=
??
??
m
m
m
m
m
m
hv ve
fv v e
(2.7)
where ( )
12
,,,
rn
mm m=m null , , ()
1
2,3, ,
n
i
i
mrr k
=
==
?
null
12
12
mmm
n
n
=
m
vvv vnull and
j
e is the
member of the natural basis. Then, substitution of (2.7) into equation (2.6) and comparison of
coefficients of the similar terms on both sides yields
th
j
,,
,,
r
r
rj
rj
rj
f
h
?
=
? ?
m
m
m ?
(2.8)
which are coefficients of any order near-identity transformation. In above equation
th
r
12
(,,,)
T
n
? ??=? null is a vector containing eigenvalues of the linear matrix . It should be noted
that all coefficients of the order near-identity transformation can be found only if the
following solvability condition is satisfied
A
th
r
0
rj
?? ??m ? (2.9)
If the solvability condition is satisfied then equation (2.4) can be reduced to its equivalent linear
form. In case, if the condition is not satisfied then it is called as the ?resonant? case and the
corresponding resonant terms remain in the transformed equation. The resulting equation can be
written in its simplest nonlinear form as
()
( )
1
*
2
k
k
r
r
+
=
=+ +
?
vJv fv Ovnull (2.10)
where are the resonant nonlinear terms.
()
*
r
fv
14
One can also define a linear operator
() ,2,
r
Ar r
L ,k
?
=? =
?
h
hJvJh
v
null (2.11)
which carries homogeneous vector polynomials over the vector polynomials of the same degree.
If the set of eigenvalues of does not contain any zeros then is invertible, and equation
(2.6) can be solved.
A
L
A
L
2.2.2 Time Dependent Normal Forms (TDNF)
Next, consider a nonlinear dynamic system with time-periodic coefficients given as
() () () ()
( )
1
23
,, ,
k
k
ttt t
+
=+++++xA xfx fx fx Oxnull null ,t (2.12)
where matrix is T periodic such that nn? ()tA ( ) ( )ttT= +AA . nonlinear vectors 1n?
( ),
r
tfx are homogenous monomials in x with T periodic coefficients. Typically, such equations
arise due to Taylor series expansion of time-periodic systems about an equilibrium point. They
may also arise in the study of stability and bifurcation of periodic orbits which may represent
steady-state solutions of autonomous/nonautonomous systems. Since the matrix ( )tA in
equation (2.12) is time dependent, the direct application of normal form theory is not possible.
However, if one can find the L-F transformation matrix, which transforms equation (2.12) into its
equivalent dynamic form where the linear part of the transformed equation is time-invariant
(Refer appendix A), an application of time dependent normal forms is possible as shown by
Arnold (1983). For certain class of systems, called commutative systems, it is possible to obtain
the L-F transformation in a closed form. However, for general periodic systems computation of
such L-F transformation matrices is a difficult task. In this case, one has to compute the state
15
transition matrix (STM) as an explicit function of time. Sinha and co-workers (Sinha and Wu,
1991; Sinha et al., 1993; Wu and Sinha, 1994; Sinha and Pandiyan, 1994; Pandiyan and Sinha,
1995; Sinha et al., 1996) introduced one such technique which uses Chebyshev expansion for this
purpose. The technique is not only computationally efficient, it is also suitable for relatively large
problems and it has been used, through out this work, for successful computation of the L-F
transformation matrix.
Application of L-F transformation ( )()tL to system represented by equation (2.12) yields
( ) ( )( ) ( ) ( )( )
() () () ()
( )
11
23
1
,,
k
k
ttt ttt
ttt t tt
??
+
=+ + +
++
yRyL fL y L fL y
LfLyL? Ly
null null
(2.13)
where and is a (( )t=xLy R )nn? constant real matrix which is, in general, complex. However,
it is always possible to find a periodic L-F transformation which results in a real matrix .
Above form of the equation is amenable to the direct application of the time dependent normal
form technique.
2T R
Equation (2.13) can be written in the Jordan canonical form as
() () ()
( )
1
23
,, ,
k
k
tt t
+
=+ + ++ +zJzfz fz fz ? znull null ,t (2.14)
where is Jordan canonical form of matrix , J R
( )
,
r
tfz contain homogeneous monomials in z
of order and their periodic coefficients have commensurate periods. Then, a sequence of time-
periodic near-identity coordinate transformation can be applied, successively, in order to remove
nonlinear terms. A general form for this time-periodic near-identity transformation can be given
as
r
16
( ),
r
t=+zvhv (2.15)
where ( ),
r
thv is a formal power series in of degree . The unknown periodic coefficients of
the near-identity transformation are assumed to have periods commensurate with T . Application
of the near-identity transformation (2.15) to equation (2.14) yields
v r
()
( )
()
( )
()
()
()
*
1
1
1
,,
,,
,,
rr
rr
k
r
tt
tt
t
tt
?
+
+
,
r
t
? ???
? ?
=+ ? ? + ?
? ?
? ?
??
+++
hv hv
vJvf v JvJhv fv
v
fv Ov
null
null
(2.16)
Therefore, the homological equation in this case is defined as
( )
()
( )
()
,,
,
rr
rr
tt
t
t
??
,0t? +?
hv hv
Jv Jh v f v
v
= (2.17)
Similar to the time-invariant case, one can solve these time-periodic homological equations
(Arnold, 1983; Arrowsmith and Place, 1990; Sinha and Pandiyan, 1994; Pandiyan and Sinha,
1995; Sinha et al., 1996; Butcher and Sinha, 2000; Wooden and Sinha, 2007) by expanding
known ( )
(
,
r
tfv and unknown ( )
( )
,
r
thv terms into a multivariable Taylor-Fourier series as
()
()
,, ,
1
,, ,
1
,
,
r
r
r
r
r
r
nk
il t j
rrjl
jlk
nk
il t j
rrjl
jlk
tfe
the
?
?
==?
==?
=
=
???
???
m
m
m
m
m
m
fv ve
hv ve
(2.18)
where 1i =?,
T
?
?= and all other symbols carry same meaning as before. A term by term
comparison of the Taylor-Fourier coefficients of the monomials yields coefficients of ( ),
r
thv as
17
,, ,
,, ,
r
r
rj l
rj l
rj
f
h
il? ?
=
+ ??
m
m
m ?
(2.19)
where
12
(, , , )
T
n
? ??=? null and ? ?s are eigenvalues of matrix and referred as the Floquet
exponents of the system. Clearly, the solvability condition,
J
0
rj
il? ?+ ?? ??m , must be satisfied
to determine all the coefficients of the near-identity transformation. Otherwise corresponding
resonant terms will remain in the transformed equation and it can be written in its simplest
nonlinear form as
()
(
1
*
2
,
k
k
r
r
t
+
=
=+ +
?
vJv fv Ovnull
)
,t (2.20)
where ( )
*
,
r
tfv are the resonant nonlinear terms. It is interesting to note the dependence of the
resonance condition on the Fourier harmonics of periodic coefficients in the nonlinear terms. A
detailed discussion on the structure of these resonance conditions for certain periodic and
periodic-quasiperiodic systems can be found in Butcher and Sinha (2000) and Wooden and Sinha
(2007), respectively.
It can be seen from the above discussion that solution of time-dependent homological
equation requires solution of a large set of algebraic equations, which make application of TDNF
computationally intensive. It has been shown by Rosenblat and Cohen (1980, 1981) that an
approximation can be made, with out any significant compromise in quality of solution, by
retaining only constant terms in
( ),
r
tfz and neglecting all time-periodic terms in equation
(2.14). Since the resulting equation is autonomous, the analysis can be carried out via the time
independent normal form theory as described before.
18
In many cases, the resulting normal form equation can be solved analytically and the
stability characteristics can be analyzed. Also, for these simpler equations, some algebraic
manipulations and/or transformation to polar co-ordinate may render closed form solution. This
final solution can be transformed back into original coordinate system by substituting back all the
transformations.
2.3 Construction of Normal Forms for Forced Nonlinear Systems
This section discusses the quantitative analysis of nonlinear forced problems via a direct
application of near-identity transformations and normal form theory. In the case of systems with
constant coefficients, the suggested transformation is applied directly; while, for time varying
systems, first, the L-F transformation is used to convert the system into the form where linear part
becomes time-invariant. The transformation yields an expression that presents the solution as a
superposition of steady state and transient solutions. Desired steady state solutions are obtained
by the method of harmonic balance. Transient solutions are obtained by solving the homological
equations and the resulting normal form equations.
2.3.1 Systems with Constant Coefficients
Consider the nonlinear system given by
() () () ()
( )
1
21
23
k
kk
k
tO?? ? ?
+
?
=+ + + ++ +xAxF fx fx fx xnull null (2.21)
where ? is a small bookkeeping parameter, ( )tF is a periodic force vector with period T and all
other terms carry similar meaning as described before. It is important to note that, generally
speaking, there is no small parameter restriction on the forcing term ( )tF .
19
Application of modal transformation to equation (2.21) puts the linear part of the
equation in Jordan canonical form as
x=Mz
() () () ()
( )
1
21
23
k
kk
k
t ?? ? ?
+
?
=+ + + ++ +zJzF fz fz fz Oznull null (2.22)
where is the Jordan form of matrix , J A ( ) ( )
rr
? =?
-1
Mf f and ( ) ( )tt=
-1
MF F .
As discussed before, the basic principle behind the method of normal forms is to apply a
series of near-identity transformations in order to reduce a system to its simplest form. Following
this general idea, a sequence of transformations is constructed, beginning with the lowest order
nonlinearity, to successively remove nonlinear terms from equation (2.22). In general, in order to
remove the nonlinear term of order , a near-identity transformation of the form r
( ) ( ) ( ) ( ) ( )
( )
1
012 1
,, , ;
r
rr r r
ttt t?
?
?
=+ + + ++ + ?zvh hvhv hvhvnull 2r (2.23)
is applied, where unknown ( )
0
th is a function of time alone, ( ),
rs
thv constitutes of monomials
in of degree v ( )1, 2,ss=null , 1r? with periodic coefficients, while ( )
rr
hv are monomials in
of degree with constant coefficients. Applying transformation (2.23) to equation (2.22) and
assuming ???< 1, one obtains
v
r
( ) ( ) ( )
() ( ) ()()()
() ( ) ()()
()
() () ()
121
1
01
011
01
12 11
,,
,
,,
,, ,
rr rr
r
rrr
sr
rrr
rr rr
tt
I
tt t
dt
tt t
dt
tt t
?
?
??
?
?
?
??
??
??
?? ?
=? + ++??
??
?
?++ ++ +
?
?
+++ ++ ?
?
???? ?
??? ?
??
?
?
??
?
hv hv hv
v
vv v
Jv h h v h v F
h
fvh h v h v
hv hv h v
null null
null
null
null
(2.24)
20
Expanding equation (2.24) and collecting terms of like powers in results in v
() () ()
()
()
()
() ()
()()
()
()
()
()
()()
()
()
()
()
()
()
10
00
1011
00
11
11
21
11
2011
0
,
,
,,
,,
,
s
r
rrr
r
rr
Ar r
r rr
Ar r
r
r
d
ttt
dt
tdt
ttt
dt
t
Lt t
t
tt
Lt t
t
tdt
tt
dt
?
??
?
?
??
?
??
?
?
??
??
=+ ? + +
??
??
?
+
?
???
?+?
??
?
??
??
++?
?
+
?
h
vJv Jh F f
hv h
Jh F f
v
hv
hv fv
hv hv
hv fv
v
hv h
Jh F f
v
null
()
()()()
()
()()()
()
()
()()()
()
()
()
()()
()
()
()
()
()()
()
()
() ( )
0
211
21 1
1
21 2
22
21 1
11
1
10 1 2
,
,
,
,
,,
,
,,
,
rr
Arr r Arr r
r r
Arr r
rr
r r
Ar r
r rr r
Ar r
rr
r
t
LL
t
Lt
t
t
Lt t
t
t
Lt t
t
tt
??
?
?
?
??
??
?
?
?
?
?
+
??
??
??
??+ ?
?
+?+
?
?
???
++
??
??
??
?
+++ +
hv fv hv fv
hv
hv fv
v
hv
hv
hv fv
v
hv hv
hv fv
v
fvh hv hv
null
null
null
null
() (),,
rr
tt+ ++hvnullnull
(2.25)
where
()
()
( )
()( )
,
,,;1
rs
Ars rs
t
Lt ts r
?
,21= ?=?
?
hv
h v Jv Jh v
v
null
()
()
( )
()
rr
Arr r
L
?
=?
?
hv
h v Jv Jh v
v
21
are the Lie operators which carries homogeneous vector polynomials over to vector polynomials
of the same degree. ( ),
rs
tfv are
1r
?
?
order terms of ( )
r
fz after the transformation and s is the
degree of . Thus, v ( )
0r
tf are pure time terms of ( )
r
fz after the transformation, ( )
1r
tf are linear
in and so on. v
It is seen from equation (2.25) that nonlinear terms of degree can be eliminated if
following set of equations is satisfied.
r
()
( )
() ()()
0 1
00
0
r
r
dt
tt
dt
?
?
t? ++ =
h
Jh F f h (2.26.a)
()()
( )
()()
()()
()
()()
()()()
1
110
1
11
,
,,
,
,,
0
r
Ar r
rr
Arr r
Arr r
t
Lt t
t
t
Lt t
t
L
?
??
?
+? =
?
?
0
+ ?=
?
?=
hv
hv fvh
hv
hv fvh
hv fv
null
(2.26.b)
Equation (2.26.a) involves only temporal terms and the desired steady state solution of
this equation is obtained by the method of harmonic balance. This permits an investigator to
choose the steady state solution of the system in the desired form (i.e., fundamental, sub or super
harmonic). Quasiperiodic solutions are not included in this study.
Equation set (2.26.b) is a set of homological equations with periodic coefficients except
for the last equation, which has constant coefficients. These equations are independent of each
other and can be solved if the set of eigenvalues of ( )/
A
Lt+?? does not contain any zeros. To
solve homological equations with time varying coefficients, approximately, a similar procedure as
22
described in the earlier section can be followed. For this purpose, known ( ),
rs
tfv and unknown
( ),
rs
thv monomials are expressed in a multivariable Taylor-Fourier series as
,, ,
1
,, ,
1
(,)
(,)
s
s
s
s
s
s
nk
il t j
rs r j l
jlk
nk
il t j
rs r j l
jlk
tfe
the
?
?
==?
==?
=
=
???
???
m
m
m
m
m
m
fv ve
hv ve
(2.27)
where 1, 2, , 1r=?null , ( )
12
,,,
s n
mm m=m null , (
1
1, 2, , 1
n
i
i
mss r
=
)= =?
?
null and s ? is the principal
frequency of ( )
0
th .
Substituting these expressions into homological equations (2.26.b) and comparing Taylor-
Fourier coefficients of the expansion, term-by-term, one obtains a set of linear algebraic
equations, which can be solved to determine unknown coefficients of ( ),t
rs
hvas
,, ,
,, ,
s
s
rj l
rj l
s j
f
h
il? ?
=
+ ?
m
m
m ?i
(2.28)
where, again, ( )
12
T
n
? ??=? null
s
il
are the eigenvalues of Jordan matrix . Thus when the
solvability condition
J
0
j
? ? ?+?m ?i is satisfied, then the corresponding term in ( ),
r
tfz can
be eliminated. Otherwise the resonant term will remain in the reduced equation.
Homological equation for the term ( )
rr
hv is time-independent. Following the solution
procedure discussed earlier, one can find
,,
,,
r
r
rj
rj
rj
f
h
?
=
??
m
m
m ?
(2.29)
23
where ( )
12
,,,
rn
mm m=m null and
1
n
i
i
mr
=
=
?
.
Equation (2.29) can also be regarded as a special case of equation (2.28) where 0l = ,
meaning coefficients of terms in are constant. The solvability condition h 0
rj
???? ?m
represents the case of internal resonance (Nayfeh and Mook, 1979, Nayfeh, 1985).
After the transformation, the subsequent reduced equation takes the form
( ) ( ) ( ) ( )
()
()
1* 1* 1*
12 1
1
1
,,
rr rr
k
kk
k
tt
O
?? ??
??
?? ?
+
+
?
=+ + ++ + +
++
vJv f v f v f v f v
fv v
null nullnull
(2.30)
where contains only the resonating terms. Next, ( order terms can be removed in a
similar fashion.
*
f )1
th
r +
2.3.2 Systems with Time-Periodic Coefficients
Consider a nonlinear system represented by
() () () () ()
( )
1
21
23
,, ,
k
kk
k
tt t t t?? ? ?
+
?
=++ + ++ +xA xF fx fx fx Oxnull null ,t (2.31)
where ? , once again, is a small bookkeeping parameter, nn? matrix is T periodic such
that . nonlinear vector
()tA
() ( )tt=+AAT 1n? ( ),
r
tfx contains homogenous monomials in with
periodic coefficients and
x
1
kT ()tF is a force vector with period . and are assumed to
be integers.
2
kT
1
k
2
k
24
Following the approach summarized in appendix A to calculate the real L-F
transformation matrix ( )tL and applying the transformation ( )t=xL y to equation (2.31)
produces
() ( ) ( ) ( )( ) ( ) ( )( )
() () () ()
( )
11 21
23
1
11 1
,,
,,
k
kk
k
tt t tt t tt
ttt ttt
??
??
?? ?
+
?? ?
=+ + + +
++
yRyL F L fL y L fL y
L fLy L OLy
null null
(2.32)
where is typically periodic and R is a real valued ()tL 2T nn? constant matrix. This equation
may be written as
() () () ()
( )
1
21
23
,, ,
k
kk
k
tt t t?? ? ?
+
?
=+ + + ++ +zJzF fz fz fz Oznull null ,t (2.33)
where is the Jordan canonical form of R , J ( ) ( )
1
t
?
=
-1
MLF Ft and
1
(,)(,
rr
tt
?
=
-1
MLf )Mz f zL .
Similar to the time invariant case, in order to remove the order nonlinearity from
equation (2.33), the following transformation is applied
th
r
( ) ( ) ( ) ( )
( )
1
012
,,
r
rr r
ttt?
?
=+ + + ++zvh h v h v h vnull ,t (2.34)
where ( )
0
th is a function of t alone, ( ),
rs
thv constitutes of monomials in of degree v s with,
yet unknown, periodic coefficients. Applying above transformation to equation (2.33) and
collecting terms of like of powers of , one obtains a similar set of equations as in the time-
invariant case (c.f., equations 2.26.a and 2.26.b). The only difference being that the last equation
in (2.26.b) now has both spatial and temporal terms and is given by
v
25
()
()
( )
()
,
,
rr
Arr r
t
Lt t
t
?
,0+ ?
?
hv
hv fv= (2.35)
The coefficients of ( ),
rr
tfv are periodic and the solution of equation (2.35) follows a similar
general procedure as described for periodic equations in (2.26.b).
Finally the transformed equation takes the form
( ) ( ) ( ) ( )
()
()
1* 1* 1*
12 1
1
1
,, ,,
rr rr
k
kk
k
tt tt?? ??
??
?? ?
+
+
?
=+ + ++ + +
++
vJv f v f v f v f v
fv Ov
null nullnull
(2.36)
where
*
sf contain the resonating term. This procedure is continued until all nonlinearities are
eliminated. Later, while discussing an example, it is shown that for a single degree of freedom
system with damping all nonlinearities can be eliminated and the resulting normal form has a
closed form solution.
2.3.3 An Alternate Approach for Non-Resonant Case
In the case, ( )
0
th is limited to be the non-resonant fundamental solution, a simple
modification in the near-identity transformation yields considerable simplification. This
alternative approach results into two linear differential equations instead of a nonlinear equation
(2.26.a). These two equations can be solved easily using convolution integral and the need for
using harmonic balance solution is obviated. In order to remove the order nonlinearity from
equation (2.22), a near-identity transformation of the form
th
r
( ) ( ) ( ) ( ) ( )
( )
1
0012
,,
r
rr r r
tttt?
?
=+ + + + ++zvh h h v h v h vnull (2.37)
26
is applied, where an additional term
( )
0r
th is purely time dependent. Applying this
transformation to equation (2.22) and collecting terms of the order
0
? yield
()
( )
()
0
0
0
dt
tt
dt
? +=
h
Jh F (2.38)
Equation (2.38) is a linear differential equation involving only temporal arguments. The solution
can be determined using the convolution integral as
() ()
()
()
00
0
0
t
tt
te e d
?
? ?
?
=+
?
JJ
hh F (2.39)
Expressing in terms of finite Fourier series as ()tF
()
,
1
nk
il t
j l
jlk
tce
?
==?
=
??
F
j
(2.40)
where ? is the principal frequency of forcing, the solution of integral equation (2.39) (with
( )
0
00h ? ) can be written as
()
0, ,
11
j
t
nk nkil t
j ljjl
jjjlk jlk
e
tc c
il il
?
?
?? ??
==? ==?
=?
??
?? ??
he
j
e
n
(2.41)
where ; 1,2, ,
j
? = ? , are the eigenvalues of . If all eigenvalues of are purely imaginary
then it can be seen that, if
J J
j
l? ?= for any l , ( )
0
th cannot be determined and the system is said
to be in the main resonance.
Collecting the terms of the order
1r
?
?
yields
27
()
( )
()
0
00
0
r
rr
dt
t
dt
t? +=
h
Jh f (2.42)
Equation (2.42) is again a linear differential equation which can be solved using convolution
integral as discussed above. The forcing term ( )
0s
tf consists of all temporal terms of order
1r
?
?
in ( )
s
fz after the transformation. In addition, one gets a set of homological equations similar to
(2.26.b), the solution for which has already been discussed (c.f., section 2.3.1). This methodology,
in general, is applicable for time varying case also.
A similar approach has been used by Wooden (2002) to solve forced nonlinear equations
with quasiperiodic coefficients. Wherein, a near-identity transformation of the form
( ) ( )
( )
1
0
,
r
rr
t?
?
=+ +zv h h vt (2.43)
has been used to reduce the problem to its normal form. Resulting equations, which need to be
satisfied, are equation (2.35) and (2.38). Again, this transformation holds well only if the desired
steady state solution is non-resonant. Moreover, it does not have any provision to include
superharmonic or subharmonic component in the solution.
2.4 Applications
To illustrate applications and demonstrate effectiveness of the suggested techniques, two
examples are considered. First, a forced Duffing?s equation with constant coefficients is
considered. This equation can be solved through a direct application of the proposed technique
suggested in section 2.3.1. In the following, known functions ( ),
rs
tfvare expressed in a closed
form and all possible resonance conditions are derived. It is shown that for time-independent
resonance, the resulting normal form equation is still solvable. For the case of systems with time
28
varying coefficients, a Mathieu-Duffing type equation is considered. Using the L-F
transformation, the system is first reduced to a form where the linear part is time invariant and
then the procedure outlined in section 2.2.3 is applied to obtain an approximate solution. In each
case, normal form solutions are compared with numerical solutions.
2.4.1 Forced Duffing?s equation
Consider a forced Duffing?s equation given by
11
3
22
1
0
01 0
()
xx
F
ad costcx
?
??
?? ??? ??
??
=?+
?? ?? ? ? ? ?
??
?? ?
??
?
? ???? ??
??
null
null
?
(2.44)
where and ,,adc ? are system parameters. It is to be noted here that damping and forcing term
do not contain any small parameter. After applying modal transformation ( )x=Mz the system
becomes
()
11
3
11 1 12 2
0
0
()
F
cos t
cM z M z
?
??
??
? ?
??
=? +
? ??
?
+
??
??
zJz M Mnull
?
(2.45)
where is in the Jordan canonical form. The above equation is of the form J
( ) ( )
3
t ?=+ +zJzF fznull (2.46)
where
29
()
( )
()
()
() () () ()
() () () ()
() ()
1
12
1
22
32 2
11213,1, 3,0 3,1, 2,1 3,1, 1,2 3,1, 0,3
3
32 2
11213,2, 3,0 3,2, 2,1 3,2, 1,2 3,2, 0,3
13 12
12 11 12 11 123,1, 3,0 3,1, 2,1
3,
cos
cos
;
;3 ;
MF t
t
MF t
fzfz fzfz
2
2
f zf zzf zzf z
f cMM f cMMM
f
?
?
??
???
??
=
??
?
??
+++
=?
+++
==
F
fz
() ()
() ()
() ()
12 13
12 11 12 12 121, 1,2 3,1, 0,3
13 12
22 11 22 11 123,2, 3,0 3,2, 2,1
12 13
22 11 12 22 123,2, 1,2 3,2, 0,3
3; ;
;3 ;
3; ;
cM M M f cM M
fcMMf cMMM
fcMMfcMM
??
==
(2.47)
Following equation (2.23), in order to remove cubic nonlinearity from the above
equation, a near-identity transformation of the form
()
()
22
3,1, 0 3,1,(2,0) 1 3,1,(1,1) 1 2 3,1,(0,2) 2
1 1 3,1,(1,0) 1 3,1,(0,1) 2
2 2 3,2,(1,0) 1 3,2,(0,1) 2
3,2,(2,0) 1 3,2,(1,1) 1 2 3,2,(0,2) 23,2, 0
h
hvhvhvz v hvhv
z v hvhvh
hvhvhv
??
??
? ?
??+++
????
??? ??
=+ + +
????? ? ? ? ? ?
+
++
??????
?????
?
??
?
*3*2 * 2*3
3,1,(3,0) 1 3,1,(2,1) 1 2 3,1,(1,2) 1 2 3,1,(0,3) 2
*3*2* 2*3
3,2,(3,0) 1 3,2,(2,1) 1 2 3,2,(1,2) 1 2 3,2,(0,3) 2
hvhvhvhv
hvhvhvhv
?
?
?
+++
+
(2.48)
is applied. Where,
(
12
,, ,rj mm
h
)
are time periodic and
(
12
*
,, ,rj mm
h
)
are constant coefficients. Substituting
this transformation in equation (2.46) and collecting terms of like powers of , one obtains the
following set of equations
v
()
()
() ()
()
()
3,1, 0
0
0300
3,2, 0
0;
h
dt
ttt
dt h
?
? ?
? ?
?++==
? ?
? ?
??
h
Jh F f h (2.49)
()
()
( )
()
31 3,1,(1,0) 1 3,1,(0,1) 2
31 31 31
3,2,(1,0) 1 3,2,(0,1) 2
,
,,0;
A
t hvhv
Lt t
hvhvt
? ?? +
? ?
+?==
? ?
+?
? ?
??
hv
hv fv h (2.50)
()
()
( )
()
22
32 3,1,(2,0) 1 3,1,(1,1) 1 2 3,1,(0,2) 2
32 32 32
3,2,(2,0) 1 3,2,(1,1) 1 2 3,2,(0,2) 2
,
,,0;
A
t hvhvhv
Lt t
hvhvhvt
? ?? ++
? ?
+?==
? ?
?
? ?
??
hv
hv fv h (2.51)
30
()
()()
*3*2 * 2*3
3,1,(3,0) 1 3,1,(2,1) 1 2 3,1,(1,2) 1 2 3,1,(0,3) 2
33 33 33 *3* * *3
3,2,(3,0) 1 3,2,(2,1) 1 2 3,2,(1,2) 1 2 3,2,(0,3) 2
0;
A
hvhvhvhv
L
hvhvhvhv
? ?+++
? ?
?= =
? ?
? ?
??
hv fv h (2.52)
Equation (2.49) is a nonlinear differential equation with
()
() () () () () () () () () ()
() () () () () () () () () ()
32 2
3,1, 2,1 3,1, 0 3,1, 2,1 3,1, 0 3,2, 0 3,1, 1,2 3,1, 0 3,2, 0 3,1, 0,3 3,2, 0
30
3,2, 2,1 3,1, 0 3,2, 2,1 3,1, 0 3,2, 0 3,2, 1,2 3,1, 0 3,2, 0 3,2, 0,3 3,2, 0
fh fhh fhh fh
t
fh fhh fhh fh
??+++
??
=?
??
??
f (2.53)
This equation is solved using the harmonic balance method which yields the steady state value of
( )
0
th .
Equations (2.50) and (2.51) are homological equations with time varying coefficients for
which terms of the form remain when the following resonance conditions is
satisfied
12
,, , 1 2
s
mmil t
rj l j
fevv
?
m
e
11 2 2
0
j
il m m? ???+ +?= (2.54)
where ,
12
mm s+=? is principal frequency of ( )
0
th and
j
e is the member of the natural
basis. It is apparent that time-dependent resonance
th
j
( )0l ? occurs only if eigenvalues of matrix J
are purely imaginary, and therefore only systems with no damping are relevant in this case. Time-
independent resonance occurs when 0l = .
For equation (2.50), ( )
31
,tfvis given as
31
()
() () () () () () ()
( )
() () () () () () ()
() () () () () () ()
() ()
22
13,1, 3,0 3,1, 0 3,1, 2,1 3,1, 0 3,2, 0 3,1, 1,2 3,2, 0
31
13,2, 3,0 3,1, 0 3,2, 2,1 3,1, 0 3,2, 0 3,2, 1,2 3,2, 0
23,1, 0,3 3,2, 0 3,1, 1,2 3,1, 0 3,2, 0 3,1, 2,1 3,1, 0
2
3,2, 0,3 3,2, 0
32
,
32
32
32
f hfhhfhv
t
f hfhhfhv
f hfhhfhv
fh
?
++
?
=
?
?
?
++ +
++
fv
() () () () ()
2
23,2, 1,2 3,1, 0 3,2, 0 3,2, 2,1 3,1, 0
f hh f h v
?
?
?
+ ?
?
(2.55)
and ( ) ( )
1
1, 0 , 0,1=m . ( )
31
,f tv contains quadratic terms in ( )
0
th , and therefore time-
independent resonance ( is always present in this case and the corresponding term in )0l =
( )
31
,f tv can not be eliminated. The resulting normal form equation can be written as
()
()
13,1,0, 1,0
11 1
222 23,2,0, 0,1
0
0
f v
vv
f v
?
?
?
? ?
??? ???
? ?
=+
?? ?? ? ?
??
??? ???
? ?
??
null
null
(2.56)
The resonant terms, being linear in , can be added to matrix which remains in the diagonal
form and the two independent equations can be solved in a closed form. For time-independent
resonances, , and the corresponding normal form is given by
v J
0l ?
()
()
()
()
1 23,1,0, 1,0 3,1, , 0,1
11
222 3,2,0, 0,1
13,2, , 1,0
0
0
il t
l
il t
l
f fev
vv
f
f ev
?
?
??
?
??
?
?
? ?
+??
?? ??
? ?
??=+
?? ?? ? ?
+
?? ??
? ?
??
??
null
null
(2.57)
where
1
2il? ?= .
For equation (2.51), ( )
32
,tfv can be written as
32
()
() () () ()
( )
() () () ()
( )
() () () () () () () ()
() () () ()
() ()
2
113,1, 3,0 3,1, 0 3,1, 2,1 3,2, 0 3,1, 2,1 3,1, 0 3,1, 1,2 3,2, 0
32
2
3,2, 3,0 3,1, 0 3,2, 2,1 3,2, 0 3,2, 2,1 3,1, 0 3,2, 1,2 3,2, 0
2
23,1, 1,2 3,1, 0 3,1, 0,3 3,2, 0
3,2, 1,2 3,1, 0
32
,
3
f hfhvfh fhv
t
f h fhv fh fhv
fh fh v
fh
?
+++
?
=
?
?
?
++
++
fv
() ()
2
23,2, 0,3 3,2, 0
3 fhv
?
?
?
?
?
(2.58)
and ( ) ( ) ( )
2
2,0 , 1,1 , 0,2=m
32
(,)tfv
. Time-independent resonance is absent in this case as periodic
coefficients in are linear in ( )
0
th
32
(,tfv
which eliminate the case . Time-dependent
resonance occurs for Fourier terms of that satisfy the following conditions.
0l =
)
1
1
3
il
il
? ?
? ?
=
=
(2.59)
Homological equation (2.52) is time independent and solution of which is widely
discussed in the literature (Arnold, 1983; Arrowsmith and Place, 1990). In the present example
( )
33
,tfv has the form
()
() () () ()
() () () ()
32 2
11213,1, 3,0 3,1, 2,1 3,1, 1,2 3,1, 0,3
33
32 2
11213,2, 3,0 3,2, 2,1 3,2, 1,2 3,2, 0,3
2
2
f vf vvf vvf v
f vf vvf vvf v
??
+++
??
=
??
+++
??
fv (2.60)
where . For purely imaginary eigenvalues of the system the resonance
terms remain, resulting in the normal form (Sinha and Pandiyan, 1994; Pandiyan and Sinha,
1995; Butcher and Sinha, 2000]
()()()()
3
3,0,2,1,1,2,0,3=m
()
()
2
123,1, 2,1
11 1
2
22
123,2, 1,2
0
0
f vv
vv
f vv
?
?
?
? ?
??? ???
? ?
=+
?? ?? ? ?
??
??? ???
? ?
??
null
null
(2.61)
33
Resonant terms in the above equation are in addition to the terms shown in equation (2.56) and
(2.57).
Above analysis shows that when damping is present in the system, terms that cannot be
eliminated are the linear resonant terms in ( )
31
,thv. The resulting normal form equation is given
by equation (2.56). However, resonant terms being linear in with constant coefficients, solution
can be obtained in a closed form. Resulting solution vector is transformed back to
v
v
12
(, )x x by
back substitution of all the transformations.
In the following, some case studies for various values of and , , ,abdc ? are presented
for the forced Duffing?s equation. For parameter set 3, 0.3, ,ad 1, 2cF 4= ===?= and 0.5? = ,
the undamped natural frequency of the system is 1.732 and thus it is the case of a non-resonant
excitation. After applying the transformations, terms that can not be eliminated are
and . These are linear terms with constant coefficients and
can be added to the corresponding terms in matrix . It is to be noted that these terms, being
purely imaginary, do not affect the stability of the system. Fig. 2.1 shows variations in states
J
11
0.00676442iv e
12
(, )
22
0.00676442iv? e
x x with respect to time. It is seen that the approximate solutions obtained by the proposed
technique matches almost exactly with the numerical solutions, including the transient response.
Next the case where superharmonic component is dominant in the solution is considered.
For this situation, the parameter set is 5, 0.3, 1, 4, 0.745ad cF= ===?= and 0.5? = which is a
superharmonic resonance case of order 3 since the natural frequency 5 2.236
n
? == . All
homological equations are still solvable since the eigenvalues of the system are complex. Time-
independent resonance terms are and
11
245095iv e
22
0.245095iv0. ? e . Fig. 2.2 shows a close
agreement of the proposed solution with the numerical solution.
34
For a typical set of system parameters 4, 0.1, 7.972, 10, 6.6ad c F= == =?=
and 0.5? =
2
n
, response is dominated by the subharmonic component of order 1/ (note
that
3
? = ). The linear leftover terms are and
1
0.96214iv
1
e
22
0.96214iv? e . Fig. 2.3 shows
presence of subharmonic component. The results from the proposed method follow the transient
solution fairly well and the steady state motion matches almost exactly with the numerical
solution.
2.4.2 Forced Mathieu-Duffing equation
Next, a Mathieu-Duffing type equation is analyzed where the linear as well as nonlinear
terms carry periodic coefficients. Consider
(2.62)
11
3
221
0
01 0
cos( )
( cos(2 ) cos(4 )
xx
t
ab t d F tx
??
??
??
?? ???? ?
??
=?+
?? ?? ??? ?
??
?+ ?
???? ??? ??
??
null
null
?
?
where and ,,abd ? are system parameters. In order to apply a near-identity transformation the
system is first transformed to the form where the linear part is time-invariant. This is
accomplished by an application of the L-F transformation. Accordingly, a -periodic real L-F
transformation matrix
2T
( )tL is constructed. After applying L-F and modal transformations, the
system becomes
()(
11
3
11 11 1 12 2 12 21 1 22 2
11
0
()
0
(4 )
cos t
MQzQz MQzQz
Fcos t
??
?
??
??
)
? ?
? ?
=?
? ?
++ +
? ?
??
??
+
??
??
zJz ML
ML
null
(2.63)
where is in the Jordan canonical form. The above equation is of the form J
35
( ) ( )
3
,t?=+ +zJzF fznull t (2.64)
A near-identity transformation of the form
()
()
22
3,1, 0
3,1,(2,0) 1 3,1,(1,1) 1 2 3,1,(0,2) 21 1 3,1,(1,0) 1 3,1,(0,1) 2
2 2 3,2,(1,0) 1 3,2,(0,1) 2
3,2,(2,0) 1 3,2,(1,1) 1 2 3,2,(0,2) 23,2, 0
h
hvhvhvz v hvhv
zv hvhvh hvhvhv
??
??
? ?
??+++
????
??? ??
=+ + +
????? ? ? ? ? ?
+
?????? ?
????
?
??
?
32 2
3,1,(3,0) 1 3,1,(2,1) 1 2 3,1,(1,2) 1 2 3,1,(0,3) 2
32
3,2,(3,0)1 3,2,(2,1)1 2 3,2,(1,2)12 3,2,(0,3)2
hvhvhvhv
hvhvhvhv
?
?
?
+++
+
(2.65)
is applied. It is to be noted that all coefficients
(
12
3, , , )j mm
h are time periodic. Upon substitution of
this transformation into equation (2.64), a similar set of equations as given by (2.49), (2.50),
(2.51) and (2.52) is obtained where the last homological equation (equation (2.52)) now contains
time periodic coefficients. Solution of equations (2.49, 2.50 and 2.51) in this case follows the
same procedure as discussed before. Following a similar solution procedure for equation (2.52)
leads to a specific structure of resonance conditions. Readers are directed to reference Butcher
and Sinha (2000) for a detailed analysis of these resonance sets.
In the following, some case studies for various values of and ,,abd ? are presented. Fig.
2.4 shows a state response of the system when system parameters are
and
3,ab1, 0.3, 1d F=== =
0.9? =
15 1.4107??
. For these parameter values, the eigenvalues of (Floquet exponents) are
. These results are compared with numerical results. In both cases an excellent
comparison was observed. It is interesting to note that in Fig. 2.4, the value of
R
0. 9i
? is 0.9, implying
that the nonlinearity is not small, and yet the accuracy is good.
For system parameters 3, 1, 0, 1abd F= == = and 0.5? = , the eigenvalues of matrix
are . Since eigenvalues of are purely imaginary and non-commensurate with the
excitation frequency, the system solution is quasiperiodic in nature. Fig. 2.5 shows system
J
1.40429i? J
36
response as calculated from the proposed methodology and numerical integration which show
excellent agreement.
Fig. 2.6 shows state trajectories when the time-invariant part of the linear term vanishes
and thus the system does not have a generating solution (i.e., 0a ? ). Perturbation and averaging
techniques are not applicable in this situation. For this case also the proposed technique shows a
close agreement with numerical solutions.
37
Fig. 2.1 System response for forced Duffing?s equation (non-resonant forcing).
, numerical solution; , normal form solution.
38
Fig. 2.2 System response for Forced Duffing?s equation ( )3
n
??? .
, numerical solution; , normal form solution.
39
Fig. 2.3 System response for forced Duffing?s equation ( )3
n
??? .
, numerical solution; , normal form solution.
40
Fig. 2.4 System response for forced Mathieu-Duffing equation (Floquet
exponent has negative real parts).
41
Fig. 2.5 System response for forced Mathieu-Duffing equation (Floquet
exponent are purely imaginary).
42
Fig. 2.6 System response for forced Mathieu-Duffing equation ( , no
generating solution exists).
0a ?
43
Chapter 3
Order Reduction of Nonlinear Systems Subjected to External Periodic Excitations
3.1 Introduction
This chapter discusses a methodology for determining reduced order models of
periodically excited nonlinear systems with constant as well as periodic coefficients. Approach is
based on construction of a time-periodic invariant manifold such that the dynamics of a system,
when projected on the invariant manifold, is governed by a fewer number of ordinary differential
equations. The idea of using invariant manifolds for model order reduction of nonlinear
homogeneous systems has been exploited by quite a few researchers, recently. As discussed in
Chapter 1, these techniques can be broadly classified into three categories; namely, center
manifold, nonlinear normal modes (NNMs) and singular perturbation. A detailed discussion and
comparison of some these techniques can be found in Steindl and Troger (2001) and Rega and
Troger (2005).
However, nonlinear systems subjected to external excitations require a special attention
as none of the above techniques can directly be applied to this class of problems. Following the
general idea of introducing a new state variable to represent the forcing term, such that the
augmented system is homogeneous, Shaw et al. (1999) first applied the concept of NNMs to
reduce the order of periodically excited autonomous systems. They proposed polynomial
expansion to approximate the solution to a set of PDEs that governs the geometry of the invariant
manifold. Later, Jiang et al. (2005) used a similar approach and employed Galerkin based
44
numerical solution procedure to solve these PDEs. Their objective was to improve the validity of
reduced order models in the large amplitude region. Further, Agnes and Inman (2001) used the
method of multiple scales to construct NNMs for such systems; however, they also employed
augmented state-space to deal with the periodic forcing. The idea of introducing new state
variables to transform the system into a homogeneous form, however, is not very appealing for
order reduction of systems which are subjected to multifrequency external and/or parametric
excitations. Since every excitation frequency, present in the system, increases the degrees of
freedom of the equivalent homogenous system by one. This may make the construction of the
invariant manifold a cumbersome process and may outweigh the advantages of order reduction all
together. To overcome this drawback, recently, Redkar and Sinha (2008) proposed an innovative
approach where they directly constructed a time-varying invariant manifold in system?s modal
coordinates. They assumed the invariant manifold as time modulated nonlinear functions of the
dominant states which has a certain structure in terms of spatial and temporal terms. Their
approach is quite general, in the sense that it can also be used for systems with periodic
coefficients after the application of L-F transformation (Redkar, 2005; Redkar and Sinha, 2009).
However, they observed that the formulation does not yield promising results when the
unmodeled dynamics of the systems is near secondary resonance with external excitation
frequencies. The work presented here draws on the idea of Redkar and Sinha (Redkar, 2005;
Redkar and Sinha, 2008; Redkar and Sinha, 2009) but significantly differs in the way the time-
varying manifolds are constructed. It is expected that reduced order models obtained with the
proposed methodology will accurately estimate the response of the original system even if the
unmodeled dynamics of the system is near primary or secondary resonance with external
excitations. In the following section, the construction of such invariant manifolds and ensuing
model order reduction for systems with constant as well as periodic coefficients is discussed.
45
3.2 Order Reduction of Systems Subjected to Periodic Excitations
3.2.1 Systems with Constant Coefficients
It is proposed to construct an approximation of the forced nonlinear system of the form
( ) ( )
r
t?=+ +xAx fx Fnull (3.1)
by a set of differential equations with smaller dimension. In the above equation ? is a small
positive number, is an time-invariant matrix, A nn? ( )
r
fx is an dimensional vector of
nonlinear terms containing monomials in up to order and
n
x r ( )tF is a T periodic force vector
of commensurate dimension.
The proposed technique follows a two step procedure in which the system is first
transformed into a state-space realization in which the new state variables can be grouped
according to some measure of importance. The second step involves truncation of less important
state variables in order to get a lower dimensional model. Starting with the modal transformation
, equation (3.1) becomes =xMz
( ) ( )
r
t?=+ +zJz fz Fnull (3.2)
where is in Jordan canonical form, J ( ) ( )
1
rr
?
=fz MfMz and ( ) ( )
1
t
?
=FMFt. Without any loss
of generality, Equation (3.2) can be partitioned as
( )
()
()
()
,
0
0
,
r
r
t
t
?
??
? ???? ???
?? ?? ? ?? ?
+
?? ?? ? ?? ???
?? ?? ? ?
??? ??? ????
??
ppq
pp p p
qqq q
qpq
fzz
zJ z F
=+
F
fzz
null
null
(3.3)
46
where is a
p
J ( )p pp n? null Jordan block associated with the dominant states
(
. These are
the states to be retained in the reduced order model. is
)
p
z
q
J ( )qqq n?=p?Jordan block
associated with the non-dominant states
( )
q
z . These are the states to be eliminated from the
reduced order model. The proposed technique does not put any restrictions on the selection of the
dominant and non-dominate states provided that they satisfy certain reducibility conditions,
which are derived later in the discussion. As a result, one can choose any set of states, which are
of practical interest, as the dominant states. For systems which are subjected to periodic
excitations, the response is usually dominated by the evolution of states which are in or near
resonance (primary or secondary) with the external excitations or those which corresponds to
eigenvalues of with small negative real parts. Therefore, these states usually become natural
choice of the dominate states.
J
Once the dominant and non-dominant dynamics of the system is identified in Equation
(3.3), the non-dominant states can be eliminated from the dominant dynamics by expressing them
in terms of the dominant states. A nonlinear, time-varying constraint relationship between the
dominant and non-dominant states is proposed in form:
() () ( ) ( ) ( )()( )
01 2 1
,,,,
r r rr rr
tt t t t?
?
==+ + ++ +
qp p p p p
z Hz h hz hz h z hznull (3.4)
where the unknown ( )
0
th is assumed to be a function of time alone, constitutes of
monomials in of degree
(
,
rs
t
p
hz
)
p
z ( )1, , 1ss r= ?null with unknown time-periodic coefficients, while
( )
rr p
hz are monomials in of degree with unknown constant coefficients. The constraint
relationship represents a
p
rz
p dimensional manifold. The idea of constructing the manifold in this
specific structural form emanates from the construction of near-identity transformation for the
47
forced nonlinear systems as described in the previous chapter (c.f., Equation (2.23)). There it has
been shown that the near-identity transformation with the particular structural form facilitates an
investigator to correctly express the system response in the desired form (i.e., fundamental, sub or
super harmonic solution). Similarly, here the invariant manifold of this special structural form
allows an investigator to correctly express the interaction between the external forcing terms with
the non-dominant dynamics of the system. This is why the proposed methodology accurately
estimates the response of the original system even if the unmodeled dynamics of the system is
near primary or secondary resonance with external excitations.
For the manifold (Equation (3.4)) to be invariant (i.e., given ( ) ( )
( )
00=
qp
zHz,0, the
solution ( ) ( )
( )
,tt
pq
zz lies on the manifold 0t? ? ), it must satisfy Equation (3.3). Substituting
equation (3.4) into equation (3.3) yields
( )( ) ()
,,
r
t?=+ +
ppp pp p p
zJz fzHz Fnull t (3.5.a)
( ) ( )
()()()
,,
,,
r
tt
tt
t
?
??
+=+ +
??
pp
pqq qp p q
p
Hz Hz
zJz fzHz F
z
null (3.5.b)
Dynamics of the system on the invariant manifold is described by p ( )p n< dimensional
differential system (3.5.a), which is a desired reduced order model of the original system (3.1).
The geometry of the invariant manifold is governed by PDE (3.5.b). Thus, the solution of
equation (3.5.b) is essential for a successful order reduction which is a non-homogenous,
nonlinear PDE with time-varying coefficients. To author?s best knowledge this PDE is not
tractable analytically and one has to use numerical techniques or a series expansion to solve this
equation approximately. In this work, the series expansion is proposed for this purpose.
48
Since the invariant manifold is assumed in a specific temporal and spatial form (3.4), a
combined Fourier and multivariable Taylor-Fourier series is used. This technique can not only be
easily automated for a nonlinearity of any degree, it also yields various reducibility conditions in
a closed form in certain cases. However, it should be noted that such an approximation is only
valid for systems with weak nonlinearities and small amplitudes of motion.
Accordingly, expansion of nonlinear terms in equation (3.5.b) and collection of terms
which are alike in powers of yields
p
z
()
()
() ()()
( )( )
()
()
()
()()
()
()
()
()
()()
()
()
() ()
10
0
000
12
11
1
110
,
0
,,
,, , , 0
,,
,, , , 0
,, , 0
r
r
rr
rr
rr rr
rr rr
rr rr
t
dt
ttt t
dt
tt
Lt t t
t
tt
Lt t t
t
Lt
??
?
??
?
?++ ? =
?
??
+? + =
??
+ ?+
?=
p
qqq p
p
pp
ppq qp p
p
pp
ppq qp p
p
ppqqp
hzh
h
Jh F f h F
z
hz hz
hz JJ f zh F
z
hz hz
hz JJ f zh F
z
hz JJ f z
null
=
(3.6)
where terms up to the order of ? are retained in the expansion. Lie operators ( )
(
L i are defined
as
()
( )
()()
,
,, , ,; 1,2, , 1
rs
rs rs
t
Lt ts r
?
= ?=
?
p
ppq ppqp
p
hz
hz JJ Jz Jhz
z
null ?
()
( )
()
,,
rr
rr rr
L
?
=?
?
p
ppq pp q p
p
hz
hz JJ Jz Jhz
z
49
and ( )
sq
f i represents
th
s degree terms in arising from the non-dominant dynamics in equation
(3.3).
p
z
The elements of invariant manifold are expanded as
()
()
()
0,
1
,, ,
1
,,
1
,
,
q k
il t j
jl
jlk
q k
s
il t j
rs r j l
s
jlk
s
q
r
j
rr r j
r
j
r
the
the
th
?
?
==?
==?
=
=
=
=
??
???
??
m
pm
m
m
pmp
m
h
hz z e
hz z e
p
(3.7)
where 1, 2, , 1s r=?null ,
( )
12
,,,
s p
mm m=m null , , ()
1
1, 2, ,
p
i
i
mss r
=
==
?
null
12
12
ms
mm p
p
=
m
p
zzzznull ,
1i =?,
T
?
?= and
j
e is the
th
j member of the natural basis. Substituting equation (3.7) into
equation set (3.6) and comparing coefficients of similar terms of the expansion one gets a set of
coupled nonlinear algebraic equations. These equations then can be solved numerically to obtain
the Fourier and Taylor-Fourier coefficients of (3.7).
However, depending upon the dimension of original system, the type of external
excitation (single frequency/multi-frequency) and the degree of the nonlinearity, finding a
solution to this set of coupled, nonlinear, algebraic equations may prove to be a difficult task.
Further, these algebraic equations typically have multiple solutions which give rise to multiple
invariant manifolds. As a result, an investigator faces the challenge of choosing one of these
manifolds (out of many) that yields the most accurate reduced order model of the system.
Moreover, this technique does not reveal the reducibility conditions; which, if not satisfied can
lead to singularities in the solution of the nonlinear algebraic equations. In the case, however,
50
when the external periodic excitations projected on the dominant dynamics are small
( ) ( )( )t??
pp
FFt, the above drawbacks can be overcome as described next.
Assuming ( ) ( )t??
pp
FFt, segregation of terms in equation (3.5.b) according to their
degree in yields
p
z
()
( )
() ()()
0
00
0
r
dt
tt
dt
? t? ++ =
qqq
h
Jh F f h (3.8.a)
()
( )
()()
()
()
()()
1
11
1
,
,, , , 0
,
,, , , 0
r
Ar r
rr
Ar r
t
Lt t
t
t
Lt t
t
?
?
?
+? =
?
?
+ ?=
?
p
ppq qp
p
ppq qp
hz
hz JJ f zh
hz
hz JJ f zh
null (3.8.b)
( )( ) ( )
,, 0
rr rr
L ? =
ppq q p
hz JJ f z (3.8.c)
where, again, terms up to order of ? are retained in the expansion. One can notice that this set of
equations is very similar to the set obtained in Chapter 2 which determines coefficients of the
near-identity transformation (c.f., equations 2.26.a and 2.26.b).
In the above set of equations, equation (3.8.a) involves only temporal terms ( )
(
0
th and
is decoupled from rest of the equations. Therefore, any conventional technique such as
perturbation, averaging, normal form or harmonic balance can be used to obtain a solution to this
non-homogeneous nonlinear differential equation. It also permits an investigator to choose the
form of desired solution, i.e., fundamental, sub or superharmonic to correctly express the
interaction between non-dominant dynamics and the external excitations. In the present work the
51
method of harmonic balance is used for this purpose and the discussion is limited to the periodic
solutions only.
Substitution of ( )
0
th in equation set (3.8.b) yields a set of decoupled equations with
periodic coefficients. To solve these equations approximately, the known ()
()( )
0
,
rs
t
qp
fzh and
the unknown
(
terms are expanded in multivariable Taylor-Fourier series and the
coefficients of the expansion are compared term-by-term. Equation (3.8.c) is time independent
and similar to homological equations associated with the standard Poincar? Normal form theory.
Solution of this equation can be approximated by a polynomial expansion of know and unknown
terms and a comparison of coefficients of the similar terms as discussed in earlier chapter. In
either case a set of linear algebraic equations are obtained which are solvable only if the
corresponding linear coefficient matrices are invertible. It is important to note that this solution
technique presents a unique solution to PDE (3.5.b) for any chosen solution of (3.8.a). It also
provides a greater control over the way an investigator want to describe the interaction between
external excitations and non-dominant dynamics.
(
,
rs
t
p
hz
)
If matrix has n linearly independent eigenvectors, matrix J and linear coefficient
matrices corresponding to (3.8.b) and (3.8.c) can be obtained in a diagonal form. Consequently, a
general forms for the Taylor-Fourier coefficients of
A
( )
,
rs
t
p
hz and the polynomial coefficients of
( )
rr p
hz are given as
,, ,
,, ,
rj l
s
rj l
s
s j
f
h
il? ?
=
+ ??
qm
m
dom
m ?
(3.9.a)
52
,,
,,
rj
r
rj
r
rj
f
h
?
=
? ?
qm
m
dom
m ?
(3.9.b)
where
(
12
T
p
)
? ??=
dom
? null is a vector of eigenvalues of the Jordan matrix ,
p
J
j
? ( )1, 2, ,jq= null
are eigenvalues of the Jordan matrix ,
q
J ()
()
0,,
1
,
q k
s
t j
jl
jl
s
tfe
==?
=
???
m
qp p
m
fzh z e
il
s
?
mss
k
q
and
()
,,
1
q
r
j
rrj
r
j
r
f
=
=
??
m
qp qmp
m
fz z e. It can be easily noticed from equations (3.9.a) and (3.9.b) that
one can find all Taylor-Fourier and polynomial coefficients of the expansion (or the linear
coefficient matrices are invertible) only if the following reducibility conditions are satisfied.
0
sj
il? ?+ ???
dom
m ? (3.10.a)
rj
?? ??
dom
m ? (3.10.b)
In the event when condition (3.10.a) and/or (3.10.b) are not satisfied, it is called a
?resonant? case and the corresponding non-dominant states can not be expressed in terms of the
dominant states.
Reducibility condition (3.10.a), based on the value of l , represents either the case of
?true internal resonance? ( )0= or ?true combination resonancel ? ( )0l ? ; while, the reducibility
condition (3.10.b) always represents the case of ?true internal resonance?. In the event of a true
internal resonance, eigenvalues corresponding to the states involved in the resonance satisfy an
integral relationship governed by (3.10.a) or (3.10.b). However, in the event of true combination
resonance, these eigenvalues satisfies an integral relationship governed by (3.10.a) which
includes one of the external excitation frequencies. In such cases the proposed dominant and non-
53
dominant dynamics can not be separated by the suggested constraint relationship. Near resonance
cases are also not desirable as they yield excessively large coefficients in the series expansion.
Nevertheless, for ?resonance? and ?near resonance? cases one can always construct a reduced
order model by making all resonating states a part of either the dominant or the non-dominant
state vector, in its entirety. In doing so, the unsatisfied reducibility conditions which arise due to
the particular choice of dominant and non-dominant states, can be avoided.
Once the constraint relationship between dominant and non-dominant states is obtained,
it can be substituted into equation (3.5.a) to get the desired reduced order model. The reduced
order model can then be solved by any conventional nonlinear solution technique such as normal
form, perturbation etc., or can be integrated numerically. Using the modal transformation the state
response in the reduced order domain can be transformed into the original domain. It should be
noted that if the external excitations are absent and the eigenvalues of are critical, then the
formulation reduces to the well known center manifold reduction.
p
J
3.3.2 Systems with Time-Periodic Coefficients
Next, consider an order reduction problem for nonlinear systems with periodic external
and parametric excitations given by
( ) ( ) ( ),
r
tt?=+ +xA x fx Fnull t (3.11)
In the above equation, ? is a small positive parameter, nn? matrix is periodic such
that
()tA T
( ) ( )tT=+AAt. ( ),
r
tfx is an dimensional vector of nonlinear terms containing
monomials in up to the order with periodic coefficients and
n
x r
1
kT ( )tF is a periodic
force vector of appropriate dimensions. and are assumed to be integers. However, the
2
kT
1
k
2
k
54
technique is not limited by this assumption and can be extended to systems with non-
commensurate excitations, also.
Proceeding along similar lines as for center manifold and Poincar? normal form theories
for periodic systems (Malkin, 1962; Arnold, 1983; Arrowsmith and Place, 1990; Sinha and
Pandiyan, 1994; Pandiyan and Sinha, 1995), the system is first transformed into its equivalent
dynamic form for which the linear part of the system is time-invariant.
Application of the transformation ( )t=xL y to (3.11) yields
( ) ( )
( ) ( ) ( )
1
,
r
ttt t?
??
=+ +yRy L fL y L Fnull
1
t (3.12)
where is a real valued constant matrix. For this system a modal transformation can be
found such that the linear part of (3.12) takes the Jordan canonical form.
R nn?
The resulting equation can be partitioned similar to equation (3.3) and written as
( )
()
()
()
,,
0
0
,,
r
r
t
t
t
t
?
??
? ???? ???
?? ?? ? ?? ?
+
?? ?? ? ?? ???
?? ?? ? ?
??? ??? ????
??
ppq
pp p p
qqq q
qpq
fzz
zJ z F
=+
F
fzz
null
null
(3.13)
where is a
p
z p dimensional vector of dominant states and is a dimensional (
q
z q )p qn+ =
vector of non-dominant states. Selection of dominant states, in this case, essentially follows
similar guidelines as described in the previous section. For this purpose, eigenvalues of matrix
, which have similar properties as ?Floquet exponents?, can be used. Accordingly, one can
select these states based on to the relative magnitudes of the real part of Floquet exponents, or
their resonances with external excitations. However, unlike systems with constant coefficients, in
this case, the nonlinearities contain periodic coefficients. Therefore, in addition to the primary
R
55
and secondary resonances one has to check for the possibility of combination resonances, too
(Deshmukh et al., 2006). Moreover, the external excitations are inherently multifrequency due to
the periodic nature of ( )tL .
Equation (3.13) differs from the (3.3) in nonlinear terms
( )
,,
r
t
ppq
fzz and
( )
,,
r
t
qpq
fzz ,
which now contains time periodic coefficients. As a result, a constraint relationship between
dominant and non-dominant states is proposed in the following form
() () ( ) ( ) ( )()( )
01 2 1
,,,,
r r rr rr
tt t t t?
?
+ + ++ +
qp p p p p
z Hz h hz hz h z hznull ,t (3.14) ==
in which all the terms are similar to the one represented by (3.4) except the degree terms in
; which are assumed to have time-periodic coefficients. This constraint relationship is a
th
r
p
z p
dimensional invariant manifold and must also satisfy the PDE (3.5.b). The same techniques are
used as described before to solve this PDE, approximately. However, in this case, segregation of
equation (3.5.b), in terms of power of , yields equation with time-periodic coefficients for the
degree terms. In the case, when external excitation terms projected on the dominant dynamics
are small
p
z
th
r
( )( )( )t??
p
F t
p
F one can approximate PDE (3.5.b) by equations similar to (3.8.a),
(3.8.b) and (3.8.c); except that, equation (3.8.c) now contains time-periodic terms. Consequently,
the reducibility conditions (3.10.a) and (3.10.b) can be represented by a single generalized
reducibility condition as
0
sj
il? ?+ ???
dom
m ? (3.15)
56
where
( )
12
,,,
s p
mm m=m null , . If condition (3.15) is not satisfied, it
corresponds to the ?resonant? case and states involved can not be decoupled. Nonetheless, in
certain cases, one can avoid the resonance by proper choice of dominant and non-dominant state
vectors as pointed out earlier.
(
1
1, 2, ,
p
i
i
mss r
=
==
?
null )
3.3 Applications
3.3.1 A 2-DOF Spring-Mass-Damper System
To demonstrate possible applications, first, a reduced order model is constructed for a 2-
DOF spring-mass-damper system with nonlinear springs, as shown in Fig. 3.1. The equations of
motion (with constant coefficients) can be written in the state-space form as
()
()
()
()
()
()
1 1
2 2
11 1 113 3
222 224 4
3
3
12 1 1 11 1 1
3
3
22
22 1 2 12
0010
01
// /0
0/
0
0
0
0
//cos
cos
x x
x x
kkm km cmx x
km k k m c mx x
bm x x b mx f t
ft
bm x x b mx
?
???? ??
???? ??
?? ??
=
?? ??
?+ ?
?? ??
?? ??
?+ ?
?? ????
??
??
??
??
++
???
?? ?
?
?
???
??
??
null
null
null
null
?
?
(3.16)
The nonlinearities are assumed to be of the cubic form. A similar system was considered by Jiang
et al. (2005) for order reduction via construction of NNM. It involved augmentation of the system
to its higher order homogenous form before the order reduction procedure can be applied.
Moreover, the analysis was limited to a single frequency excitation ( )
2
0f = and they considered
only the case of primary resonance near the linear modes. In the present example, a two
dimensional time-varying invariant manifold is directly constructed using the proposed technique
57
and the 2-DOF system is approximated by a single-DOF system. Cases of multifrequency
excitations are also considered which include secondary resonances near one of the linear modes.
Application of modal transformation ( )=xMzto equation (3.17) yields an equation
similar to (3.3). For the present case, 4n = , r 3= , and ( )t
p
F and ( )t
q
F are given by
()
() ( )
() ()
11
13 1 1 14 2 2
23 1 1 24 2 2
cos cos
cos cos
M ftMf
t
t
M ftMf
??
???+ ?
??
=
??
?+ ?
??
p
F
t
and ()
( )(
() ()
11
33 1 1 34 2 2
43 1 1 44 2 2
cos cos
cos cos
M ftMf t
t
M ftMf t
?
?
??
? ?+?
? ?
=
? ?
? ?
??
q
F .
1
ab
M
?
represents elements .
1?
M
Since the system contains cubic nonlinearity, an invariant manifold of the following form
(c.f., equation (3.4)) is proposed.
( ) () ( ) ( ) ( )( )
031 32 3
,,,tt t t?==+ + +
qp p p p
zHz h hz hz hz (3.17)
As discussed before, equation (3.17) must satisfy equation set (3.6) or (3.8). If all
elements of (3.17) can be determined, then a reduced order model of the system can be obtained
by replacing the non-dominant states from the dominant dynamics (3.5.a) as functions of
dominant states, alone.
In the following, some case studies for various system parameter values are presented.
Parameters are chosen such that the system satisfies all the reducibility conditions and the order
reduction using proposed constraint relationship is possible. In contrast to guidelines proposed
earlier for the selection of dominant/ non-dominant states, states which are near resonance
(primary or secondary) with the external excitation are assumed as the non-dominant states. This
58
is intentionally done in order to highlight the effectiveness of the technique in expressing un-
modeled dynamics of the system via reduced order models.
First a case of primary resonance is considered. For system parameters
12
1mm= = ,
, , , , 1k =
1
4k =
2
13k =
1
0.2c =
2
0.2c = , 1b = ,
1
1b = ,
2
5b = ,
1
0.5f = , , the undamped
natural frequencies of the system are
2
1.1f =
1
2.2114 rad/sec? = ,
2
3.7563 rad/sec? = and matrix in
equation (3.2) can be written as
J
0.1 3.7550 ]i[diag 0.1 2.2091 ,i 0.1 2.2091 , 0.1i 3.7550 ,i? +?
4.25 ra
?
d/s
?+?+.
The frequency of external excitation chosen as
( )
12
?=?=4.25 rad/sec . The states
corresponding to eigenvalues 0.1 2.2091i? ?
.1 3.755??
are taken as the dominant states while the states
corresponding to eigenvalues are assumed as the non-dominant states 0 0i ( )2pq= =
and the original 2-DOF system is approximated by a 1-DOF system. Invariant manifolds are
calculated by solving, both, equation set (3.6) as well as (3.8) using proposed techniques. Since
frequency of excitation is away from the (4.25 )rad/sec
2
3? and
2
/3? terms, a fundamental
solution of equation (3.8.a) is sought, which is found by using the method of harmonic balance.
Of course, one can use any other analytical method such as perturbation, averaging or normal
form to find solution of (3.8.a). The reduced order model is integrated numerically and using the
modal transformation all states in domain are recovered. The original system is also integrated
numerically for a comparison. The initial conditions are taken such that the system?s trajectories
are initiated on the invariant manifold (i.e.,
x
() () ()( )
{ }
0.z0,
pp
Hz0,0
T
=xM ). Fig. 3.2 shows
time traces for states
1
x and
2
x as obtained from reduced and original models which show very
good agreement. In order to gain insight into the frequency content of the response, fast Fourier
transforms (FFTs) of the time traces of the original and reduced order systems are also presented
(Fig. 3.3). In this case the total response is dominated by the fundamental solution of the system,
59
alone. Therefore, only one dominant peak is observed (frequency ) in the FFTs.
Variation in amplitude of stable, steady-state response of the system near main resonance
frequency is also shown in Fig. 3.4, which shows close agreement between the reduced and full-
order model response. Thus both reduced order models, as derived by solving equation set (3.6)
and (3.8), are found to be equally successful in approximating response of the original model in
this case.
4.25 rad/sec
For parameter values
12
1mm= = , 0k = ,
1
4k = ,
2
10k = , , ,
1
0.2c =
2
c 0.2= 4b = ,
, , ,
1
2b =
2
3b =
1
0.5f =
2
f 3.4= , the undamped natural frequencies of the system are
1
2 rad/sec? = ,
2
3.1623 rad/sec? =
975 , 0.1i ??
and matrix can be written in the form
. The external excitation
frequencies are chosen as and
J
.1 3.160??
2
1.2
[diag 0.1 1.9?+ 1.9975 , 0.1 3.i ?+
1
3.0 rad/sec?=
1607 , 0i 7 ]i
rad/sec? =
( )
22
3 ??? such that the
system response contains a dominant superharmonic component of order 3. States corresponding
to eigenvalues are taken as the dominant states and to those corresponding to
as the non-dominant states. The non-dominant dynamics of the system is near
super-harmonic resonance with the external excitation. Therefore, a third order superharmonic
solution is sought for equation (3.8.a), which is obtained using the method of harmonic balance. It
should be noted that the use of equation (3.6) and the following numerical solution technique,
does not offer such control on the construction of the invariant manifold. Fig. 3.5 and 3.6 show
time traces and FFTs, respectively, of the system response as obtained from simulation of the
reduced and full order models.
0.1 1.9?+975i
0.1?+3.1607i
It is observed form Fig. 3.6 that the system response is dominated by three distinct
harmonics at 1.2 , and . The first two harmonics corresponds to the rad/sec 3 rad/sec 3.6 rad/sec
60
external excitation frequencies while the third harmonic is due to superharmonic resonance of
non-dominant states with the external excitation. It is evident that the both reduced order models
capture these components of the system dynamics successfully. Fractional harmonics at
and 1 are also rendered well by these models. The reduced order model
obtained by assuming small dominant forcing, however, overestimates harmonics at 2.4
in and at and in . Since amplitudes of these harmonics are much
smaller compared to the dominant one it has negligible effect on the overall system response.
0.6 rad/sec
1
x
.8 rad/sec
8 rad/sec
rad/sec
4. 5.4 rad/sec
1
2
x
2
Next, the parameter values 1mm , 0k = ,
1
43k = , , ,
2
4k =
1
0.1c =
2
0.1c == = ,
, 3, 23b =
1
b =
2
7.97b = ,
1
2f ,
2
f 7.5= = are chosen such that the undamped natural
frequencies of the system are
1
6.5574? = ,
2
2? = and the ma ix J can be written in the form
. The forcing frequencies
are chosen as ?= and
tr
1.9994?[ 0.05diag ?+6.5572 , 0.05i ?
1
4.4 r
6.5572 , 0.0i?
sec
5 .99
2
6
1?+
6.
9
r
4 , 0.05i ?
ad/sec
]i
ad/ ? =
( )
22
3??? , which, for the above
parameter values, introduce subharmonic components of order
1
3
in the system response. The
states corresponding to the eigenvalues 0. 6.5572i05? ? are assumed as the dominant states and
those corresponding to are assumed as the non-dominant states and once again
the system is approximated by a 1-DOF system. Since the non-dominant undamped natural
frequency
0.05 1.9994i??
is close to
2
3
?
2
? , a sub harmonic solution of order
1
3
is sought for equation (3.8.a).
Fig. 3.7 shows the system response as calculated from the original and reduced order models for
the current parameter values, while Fig. 3.8 shows FFTs of the steady state response. In this case,
too, both reduced order models estimates response of the original system quite well.
61
3.3.2 A 2-DOF Coupled Inverted Pendulum
Next the reduced order modeling of a dynamic system with periodic coefficients is
considered. Fig. 3.9 shows a 2-DOF coupled inverted pendulum in the horizontal plane. The
pendulum is subjected to periodic loading and torques as shown in the figure. The equations of
motion for this system, when expanded up to the cubic terms, about the fixed point
{ }
{
1212
, , , 0,0,0,0???? =
nullnull
} can be shown to be (Sinha et al., 2005)
()()()
()
()
()()()
()
()
23
11
111 1121212131222
3
1
1
111
23
22
222 1211221132122
3
2
2
222
4
cos
6
4
cos
6
t
t
kh k
cc c
ml ml m
Pt
Mt
ml
kh k
cc c
ml ml m
Pt
Mt
ml
??? ???? ??
?
?
??? ???? ??
?
?
? ?
+++ ?+?+?
? ?
??
??=?
??
??
? ?
+++ ?+?+?
? ?
??
??=?
??
??
nullnull null
nullnull null
(3.18)
where ( ) ( )
12
cos
iii
Pt P P t?=+
(
. and denote torsional stiffness and damping of the system,
respectively
ti
k
i
h
)1, 2i = . Above set of equations contains time-periodic coefficients in linear as well
as nonlinear terms. Typically, nonlinear decoupling of its states using traditional NNM technique
requires transformation of this system into a higher order system which is homogenous as well as
autonomous. With proposed techniques, however, one can directly construct a 2-dimensional,
time-varying invariant manifold and above 2-DOF time-periodic system can be approximated by
a 1-DOF system as illustrated below.
For a typical parameter set, equation (3.18) can be written in the state-space form as
62
()()
()()
() ( )
() ( )
()
()
1 1
2 2
3 3
4 4
3
3
112
3
3
212
0010
11cos2 0 1 0
032cos201
0
0
0
0
cos 2 / 6 2.5 0.5cos
0.8cos 0.5
2cos 2 /6 2.5
xx
t
t
tx x x t
t
tx x x
?
?
??
?
?
??
? ??
??
?
? ???
?
=
? ??
?+ ?
?
?
?
?+ ?
?? ??
??
??
? ?
??
? ?
? ?
++
???
???
?+
??
??
null
null
null
null
?
(3.19)
where {}
{ }
1234 1212
,,, ,,,
T
T
xx xx ? ???=
nullnull
,
1
0
i
P = and
2
0
i
c = . As described in section 3.3.2, the L-F
transformation ( )
()
t=xL y is applied first such that the linear part becomes time-invariant.
Application of the modal transformation ( )=yMzto this equation transforms it into the form
represented by equation (3.13) as shown below.
(3.20)
()
() ( )
11
22
33
44
3222
222 4
24 24
0.5 0.8739 0 0
0.8739 0.5 0 0
0 0 0.5 1.4621
0 0 1.4621 0.5
1.4111 1.2409 1.0020 3.1826 cos
0.0106 cos 0.0101 cos 3
zz
zzzz zt
zz t zz t
?
??
???? ????
?? ????
?
?? ??
=
?? ??
??
?? ??
?? ??
?
???? ??
?? ? + +
?+
+
null
null
null
null
null
()
() ()
() ()
() ( )
2
23
223 3
24 34 4 2 2
2323
23 3 24 2 2
0.0115 sin
0.9912 0.1496 0.1131 0.3610 cos 0.8672 sin
1.2276 0.1892 0.1495 + 1.0494 cos 0.2983 sin
0.8296cos 0.0113cos 3
0.0031sin
zz t
zz zz z z t z t
zz z zz z t z t
tt
?
??
??
? ?
? ?
+
? ?
? ?
+++ ? +
? ?
? ?
??? +
? ?
++
+
null
null
null
null
() ( )
() () ()
() () ()
0.0031sin 3
0.1615cos 0.5 0.1615cos 1.5 0.4212sin 0.5
0.4696 cos 0.5 0.4696cos 1.5 0.1449sin 0.5
ttt???
??
??
??
+?+
++
??
null
null
null
The approach described in the appendix A is used to calculate the real L-F transformation
matrix ( )tL which is periodic in this case. The states corresponding to exponents
(eigenvalues of matrix )
2T
R 0.5 0.8739i? ? are assumed as the dominant while those
63
corresponding to are assumed as the non-dominant. Due to the cubic nature of the
nonlinearity, a constraint relationship between the dominant and non-dominant states is proposed
in a form similar to (3.17) with the exception of last term
0.5 1.4621i??
( )( )
33 p
hz which, in this case, is
assumed to have time-periodic coefficients.
As described in section 3.3.2, this constraint relationship is governed by equations similar
to equation set (3.6) or (3.8). In the present example, the invariant manifolds are constructed by
solving both sets of governing equations and thus two separate reduced order models are
obtained. equation (3.8.a) is solved using the method of harmonic balance where the solution is
expanded up to 10 harmonic terms. Simulated response of the original system and its reduced
order models in shown in Fig. 3.10. Fig. 3.11 shows FFTs of the steady state trajectories.
It is interesting to note that, in this case, a typical multivariable Taylor-Fourier expansion of the
governing PDEs similar to (3.6), in which 10 Fourier terms are kept in the expansion, yields a set
of 420 coupled nonlinear algebraic equations. An identical expansion of the terms, when used in
equation set (3.8), results in 42 coupled nonlinear algebraic equations and 378 linear algebraic
equations, where the nonlinear equations are decoupled from the linear equations. Thus, if ( )t
p
F
is assumed to be small, the effort required to construct the invariant manifold is considerably
smaller. However, such simplification may come with a tradeoff (as seen in this example), where
the reduced model calculated assuming small ( )t
p
F overestimates the 2? frequency component
in
2
x .
64
( )
22
cos( )
11
cos f tf t ??
2
b
b
1
b
Fig. 3.1 A 2-DOF spring-mass-damper.
2
c
2
k
1
c
1
k
k
m
1
m
2
65
Fig. 3.2 System response for 2-DOF system with constant coefficients
( )
122
??=?? .
, full order model; +, reduced order model; ?, reduced order
model assuming to be small.
p
F
66
Fig. 3.3 Fast Fourier transforms for 2-DOF system with constant
coefficients ( )
122
??=?? .
? , full order model; +, reduced order model; ?, reduced order model
assuming to be small.
p
F
67
Fig. 3.4 Amplitude variation vs excitation frequency for 2-DOF system with
constant coefficients ( )
122
??=?? .
? , full order model; +, reduced order model; ?, reduced order model
assuming to be small.
p
F
68
Fig. 3.5 System response for 2-DOF system with constant coefficients
( )
1222
,3 ???? ?? .
? , full order model; +, reduced order model; ?, reduced order model
assuming to be small.
p
F
69
Fig. 3.6 Fast Fourier transforms for 2-DOF system with constant
coefficients ( )
1222
,3 ???? ?? .
? , full order model; +, reduced order model; ?, reduced order
model assuming F to be small.
p
70
Fig. 3.7 System response for 2-DOF system with constant
coefficients ( )
122 2
,3?????? .
? , full order model; +, reduced order model; ?, reduced
order model assuming to be small.
p
F
71
Fig. 3.8 Fast Fourier transforms for 2-DOF system with
constant coefficients ( )
122 2
,3?????? .
?, full order model; +, reduced order model; ?, reduced order
model assuming
p
F to be small.
72
()
22
22
,
cos
t
kh
M t?
( )
11 12 1
cosPP t?+
2
l
2
l
k
()
11
11
,
cos
t
kh
M t?
2
?
1
?
( )
21 22 2
cosPP t?+
Fig. 3.9 A 2-dof coupled inverted pendulum.
73
Fig. 3.10 System response for 2-DOF system with periodic coefficients.
? , full order model; +, reduced order model; ?, reduced order model
assuming to be small.
p
F
74
Fig. 3.11 Fast Fourier transforms for 2-DOF system with periodic
coefficients).
? , full order model; +, reduced order model; ?, reduced order model
assuming to be small.
p
F
75
Chapter 4
Reduced Order Controller Design for Nonlinear Systems
4.1 Introduction
In this chapter a methodology for designing a reduced order controller for nonlinear
dynamic systems with constant as well as periodic coefficients is discussed. The approach is
based on a similar concept as for the construction of invariant manifolds discussed in the last
chapter. However, compared to the open-loop order reduction, the problem of closed-loop order
reduction is much more complex in nature. This is because, any nonlinear controller, which is
designed in the reduced order domain, in tern, alters the geometry of the invariant manifold,
itself. As a result, one has to construct an invariant manifold which is parameterized in terms of
nonlinear gains. The complexity of the problem becomes multifold for systems with time-
periodic coefficients, as invariant manifolds for such systems are typically time-varying.
A class of systems which requires a stabilizing controller design is considered in this
work. The system equations are represented by quasi-linear differential equations in state space.
The controller is assumed to be a scalar; however, it is not a limitation on the methodology and
one can extend the technique to systems with multiple inputs. Since the readers are already
acquainted with the application of L-F transformation and construction of time-periodic invariant
manifolds, this chapter addresses the case of time-periodic systems in detail, first. The case of
systems with constant coefficients is treated as a subset of its periodic counterpart.
76
4.2 Controller Design for Linear Time-Periodic Systems
In this section, a symbolic controller design technique for linear time-periodic systems as
introduced by Sinha et al. (2005) is outlined, first. Consider an n-dimensional linear periodic
system of the form
( )t=xA xnull (4.1)
where matrix nn? ( )tA is T periodic such that ( ) ( )tTt= +AA . Let be the state
transition matrix (STM) and
()t?
( )? T=? be the Floquet transition matrix (FTM) of the linear time-
periodic system (4.1). Then, according to the Floquet theory (appendix A), the stability of
equation (4.1) can be completely characterized by the FTM of the system. Moreover, system (4.1)
is asymptotically stable if all eigenvalues of the FTM, called the Floquet multipliers, lie within
the unit circle of the complex plane. The system is unstable if at least one of the eigenvalues of
the FTM has magnitude greater than one. Floquet theory can effectively be used for design of a
linear controller for systems with periodic coefficients. To illustrate this let us consider a full state
feedback control problem,
( ) ( )tt=+xA xBnull u (4.2)
where ( ) ( )tT t+=AA and ( ) ( )tT t+=BB. With ( )ut=?Kx, assuming (4.2) is controllable,
the closed-loop system is given by
( ) ( ) ( )ttt? ?=?
? ?
xA BKnull x (4.3)
where is a time-periodic matrix with unknown gains . Since equation (4.3) is
periodic, the FTM is computed symbolically using the method introduced by Sinha and Butcher
()tK
12
,,,
m
kk knull
77
(1997). Generally speaking, such FTMs contain high degree polynomial expressions of
. Eigenvalues of the FTM (Floquet multipliers) can then be placed at desired
locations in the unit circle with a technique similar to the ?Pole placement method?, as described
in the following.
12
,,,
m
kk knull
The characteristic polynomial of ? is given by,
()(
1
Det ?
n
i
i
)? ??
=
?= ?
?
I (4.4)
where
i
? are the desired Floquet multipliers. The coefficients of
k
? in the left-hand side of
equation (4.4) are dependent on the control gains and coefficient of
12
,,,
m
kk knull
k
? in the right-
hand side are known. A term by term comparison of the coefficients provides the values of
unknown gains.
4.3 Reduced Order Nonlinear Controller Design
4.3.1 Systems with Time-Periodic Coefficients
The idea is based on the premise that under certain conditions, a nonlinear, time-periodic
system of the form
( ) ( ),,tt=+xfx gxnull u (4.5)
can be transformed into a reduced order linear time-periodic system
( )
,0
t ?=+
pp
yJygnull (4.6)
78
through the construction of an invariant manifold by finding a suitable state feedback and
coordinate transformations. In above equations
n
R?x , ,
p
R?y ()( ),,tT t+=fx fx
0=x
and
are nonlinear vector fields. It is assumed that and ()(,,tT t+=gx gx ) 1n? 0u = is the
equilibrium point. is a
p
J pp? constant matrix and ( )tT+=
p
g (
,0
t
p
g)
,0
n
is a control
influence matrix for the reduced order system ( ). For simplicity,
1p?
pnull u and ? are restricted to
be scalars; the generalization to the vector case is straight forward. A linear time-periodic
controller can be designed for system (4.6) by the symbolic controller design technique discussed
in section 4.2 which guaranties asymptotic stability.
The technique described here is local in nature, and assuming sufficient smoothness
equation (4.5) can be expanded in into Taylor series about the equilibrium point as
( ) ( ) ( ) ( )
() () ()
23
02 1
,, ,
r
r
ttt t
tt t
?
=+++++
??++ ++ +
??
x ? xfx fx fx
ggx gx
null nullnull
nullnullu
(4.7)
where
a
f and
b
g are and order monomial forms of with T periodic coefficients,
respectively; and is the perturbation variable. Alternatively, the system represented by equation
(4.7) may also result if autonomous or time-periodic system is driven to a desired periodic orbit.
th
a
th
b x
x
Following the approach described in appendix A to calculate the L-F transformation
matrix ( )tL and applying the transformation ( )t=xL Mzto (4.7) produces
( ) ( ) ( ) ( ) ( ) ( )
23 02 1
,, , , ,
rr
tt t t t t
?
u? ?=+++++++++ +
? ?
zJzfz fz fz g gz g znull nullnull null null (4.8)
where is the modal matrix and is in the Jordan canonical form. Equation (4.8) can be
partitioned and written in a block diagonal form as
M J
79
( )
()
01
01
0,, ,,
,,
r r
r r
t tt
u
t
?
?
? ?+??? ???? ?
?? ??? ?? ?
?? ??? ?? ???
+
?? ??? ?? ?
??? ???? ???
pp pppq pppq
qqqq qqpq
zJ zf(zz)gg(zz)
=++
gg(zz)
null
null
(4.9)
where is a
p
J ( )p pp n? null Jordan block associated with the dominant states
( )
p
z . These are
the states to be kept in the reduced order model. is
q
J ( )qqq n p?=?Jordan block associated
with the non-dominant states . These are the states to be eliminated from the reduced order
model. As shown earlier, construction of an invariant manifold and application of a near-identity
transformation do not affect the lower order terms. Therefore, without any loss of generality, only
order nonlinearity is kept in the equation (4.9). In practice, one can start with the lowest order
nonlinearity in equation (4.8) and construct the invariant manifolds and apply the near-identity
transformations sequentially to eliminate higher order nonlinearities, successively.
()
q
z
th
r
Selection of dominant and non-dominant states can be arbitrary, in general. However, for
the stabilizing controller design, unstable states of a system should be chosen as the dominant
states and stable states as the non-dominant states. For time-periodic systems, this implies that
states corresponding to Floquet multipliers outside the unit circle are the dominant states while
those corresponding to inside the unit circle are the non-dominant states. However, in the case,
one would like to increase the stability margin of a system because some of the Floquet
multipliers inside the unit circle but close to the boundary, then those states could also be chosen
as the dominant states.
As stated earlier, in closed-loop systems the nonlinear state feedback affects the geometry
of the invariant manifold. To account for this effect a controller in the following specific form is
proposed
80
( )
,
nl
uu?=+
p
z t
,
(4.10)
where
()
()
()
()
()
()
()
111
000
,,
rrr
nl rlr rlr rlr
ll
ut ? ?
+? +? +??
===
=++
???pp p
K ? z ? z ? z ??
p
(4.11)
Later, the invariant manifold is constructed in terms of the linear and nonlinear control gains. In
equation (4.10) the linear control term ( )t? =
p
Kz
p
is such that the Floquet multipliers of
( ) ( )( )
,0
tt+
pp p
Jg K
(
can be placed at desired locations to guarantee asymptotic stability.
)( ),
a
t?
p
? z is a 1p?
(
vector of order monomials in with unknown time-periodic
coefficients.
th
a
p
z
)( ),
a p
? z t? and ( )
( )
,?
pb
z t
th
a? are and order scalar monomials in with
unknown time-periodic coefficients, respectively. It should be noted that the proposed controller
contains only the dominant states.
th
b
p
z
The resulting closed-loop equation takes the form
() ()
() ()
( )
()
()
()
() ()
() ()
,0
,0
101
,,
0
,,
,, ,,
,, ,,
r
r
rr
nl
t
tt
tt
t
tt t
u
tt t
?
??
? ?
??+?? ??
?? ??? ?
+
???? ??? ?
?? ??
?? ??? ???
?
???
+
???
++
???
+
???
ppq
pppp p
qqqp q
qpq
ppq p ppq
qpq q qpq
fzz
zzJg K
=
gK J
fzz
gzz g gzz
gzz g gzz
null
null
?
?
?
?
?
(4.12)
Above equation is coupled in linear terms; moreover, the linear part now contains time-
periodic terms. As a result, equation (4.12) is not amenable for the construction of an invariant
manifold. A linear state transformation, ( )t=+
qq
zvPz
p
, decouples the linear part, provided that
( )tP satisfies the following time-periodic Sylvester differential equation.
81
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
,0 ,0
ttt tt t=? + +
qpppq
PJPPJgK gK
null
t
p
(4.13)
Equation (4.12) then yields
() ()
( )
()
()
()
() ()
() ()
0
101
1, 0 1
,,
0
0
,,
,, ,,
,, ,,
r
r
rr
nl
t
tt
t
tt t
u
tt t
?
??
??
??+?? ??
?? ??? ?
+
???? ??? ?
?? ??
?? ??????
?
???
+
???
++
???
+
???
ppq
pppp p
qqq
qpq
ppq p ppq
qpq q qpq
fzv
zzJg K
=
vvJ
fzv
gzv g gzv
gzv g gzv
null
null null
nullnullnull
?
?
?
?
?
(4.14)
where
()()( ) ( )
,, ,, ,,
rr r
ttt=?
qpq qpq ppq
f zv f zv P f zv
null
t, ( ) ( ) ( ) ( )
00 0
ttt=?
qq p
ggPgnull t and
()()( ) ( )
11 1
,, ,, ,,
rr r
ttt
?? ?
=?
q qpq ppq
v g zv P g zvt
qp
g znull . A detailed discussion on the solution of
equation (4.13) is given in appendix B.
Application of transformation ( )t=
pp p
zL Mv
p
to (4.14) produces,
( )
()
()
()
() ()
() ()
101
1, 0 1
,,
0
0
,,
,, ,,
,, ,,
r
r
rr
nl
t
t
tt t
u
tt t
?
??
??
???? ??
?? ??? ?
+
???? ??? ?
?? ??
?? ??????
??
???
+
???
++
???
+
???
ppq
ppp
qqq
qpq
ppq p ppq
qpq q qpq
fvv
vvJ
=
J
fvv
gvv g gvv
gvv g gvv
null
null
?
?
?
?
(4.15)
where and are the L-F and modal transformations for the matrix ()t
p
L
p
M ( ) ( )
0
tt+
pp p
Jg K ,
respectively,
p
J is in the Jordan canonical form. Expressions for the terms with over-bars are
obvious and are omitted here for brevity. Equation (4.15) is in the form where construction of an
invariant manifold is possible. As shown by Sinha et al. (2004), a nonlinear relationship (invariant
manifold) between dominant and the non-dominant states is given by
82
( )
,
r
t=
qp
vhv (4.16)
where,
( )
,
r
t
p
hv is a vector of order monomials in with unknown time-periodic
coefficients.
1q?
th
r
p
v
Substituting equation (4.16) into equation (4.15) yields the manifold equation
()
( )
()
0
,
,, , , () 0
r
rr
t
Lt ttu
t
?
nl
+ ??
?
p
ppq qp q
hv
hv JJ f v g = (4.17)
where Lie operator ( )
(
L i is defined as
()
( )
(
,
,, , ,
r
r
t
Lt
?
=?
?
p
ppq ppqp
hv
hv JJ Jv Jhv
x
)
r
t (4.18)
A formal power series solution for the above PDE (4.17) is sought by expanding the known and
unknown terms into a multivariable Taylor-Fourier series and comparing coefficients of the
similar terms. The known and unknown terms are expanded as
()
,, ,
1
q,, ,
1
,0 ,
1
,,
,,
11,
,
,
,
,
,
q k
r
il t j
rrjl
r
jlk
r
q k
r
il t j
rrjl
r
jlk
r
q k
il t j
jl
jlk
k
r
il t
rrl
r
lk
r
k
r
il t
rrl
r
lk
r
rr
the
tfe
tge
te
te
t
?
?
?
?
?
?
?
?
==?
==?
==?
=?
=?
??
=
=
=
=
=
=
???
???
??
??
??
m
pm
m
m
qp m p
m
q
m
pmp
m
m
pmp
m
p
h(v ) v e
f(v ) v e
ge
? (v ) v
? (v ) v
? (v )
p
1
,
1
1
k
r
il t
l
r
lk
r
e
?
?
?
=?
?
??
m
mp
m
v
(4.19)
83
where
( )
11
,,,
s p
mm m=m null , ,
()
1
1,
p
l
l
mssr r
=
==?
?
12
12
m
mm p
s
p
vv v=
m
p
v null , /T? ?= , 1i = ?
and
j
e is the member of the natural basis. If
th
j
p
J and are diagonal, the general form for the
coefficients of
q
J
( ),
r
thv is given as
,, ,
,, ,
rj l
r
rj l
r
rj
w
h
il? ?
=
+ ?
m
m
dom
m ?i
(4.20)
where
(
12
T
p
)
? ??=
dom
? null is a vector of eigenvalues of the Jordan matrix
p
J ,
j
? ( )1, 2, ,jq= null
are eigenvalues of the Jordan matrix and
q
J
()
0,
k
w
,
1
,()
r
r
r
q k
il t j
rnljl
jl
ttu e
?
==?
+=
???
m
qp q m p
m
fv g v e.
It is rather obvious from equation (4.20) that in order to find coefficients of the invariant manifold
the following ?reducibility condition? must be satisfied.
0
rj
il? ?+ ??
dom
m ?i (4.21)
In the event equation (4.21) is not satisfied then it is called a ?resonant? case and the
corresponding non-dominant states can not be expressed in terms of the dominant states.
Depending upon the value of , it is either called a ?true internal resonance? k ( )0k = , or a ?true
combination resonance? ( )0k ? . Also, a near resonance condition is not desirable as it results in
very large values of the manifold coefficients
( )
,l,,
0,
r
r dom j r j
il h??+ ??
m
m ?i ??. While, the
resonant and near resonant cases can pose problems in the order reduction of open-loop systems,
the problem may be avoided here by changing the eigenvalues of
p
J by the linear feedback
. In doing so, the denominator of equation (4.20) is moved away from zero. The
resonant case can also be avoided by retaining the resonant terms in the reduced order model.
()t
p
Kz
p
84
Thus one can still successfully reduce the order of the system; however, the reduced order model
will be of higher dimension in this case. For no resonance, the invariant manifold can be
constructed, and it is in terms of the unknown time-periodic nonlinear gains
,,
il t
rl
r
e
?
?
m
,
,,
il t
rl
r
e
?
?
m
and
1, ,
1
il t
rl
r
e
?
?
?
?
m
. After the inverse transformation ( )
11
t
??
=
ppp
vML z
1
,, 1, ,
, , 1,
,
, ,
rr r
rr
il t il t
rl r l
il t il t
rl r
e
e
??
??
?
??
?
?
?
m m
()
()
1
,
rlr
t
+?
y
y
p
, the resulting reduced order
equation can be obtained as
(4.22)
()
()
() 1
,,1
0
,0 , , ,11
0
,, ,
,,
r
il t
rlrlr
l
r
il
rl lrlr
l
te e
tte
?
??
?
?
+?
=
+??
=
=+
++
?
?
p p mp
ppp
z f z
ggz
r
t
e
?
?
??
??
??
()
nl
u+
pp
zJnull
(),
a
t? y
The equation is nonlinear and contains time-periodic terms. Designing a nonlinear controller for
this system is still a challenging task. However, system (4.22) has a special structure and lends
itself to application of a series of near-identity transformations and the Poincar? normal form
theory which can transform it into a linear, time-periodic, closed-loop system given by equation
(4.6). A stabilizing controller for this linear time-periodic system can be designed using the
Floquet theory. A similar technique was originally introduced by Zhang and Sinha (2007) as a
feedback linearization methodology for parametrically excited nonlinear systems.
Application of the near identity transformation
(4.23)
0
r
l=
=+
?p
zy ?
(where are order monomial in y with unknown time-periodic coefficients) to
equation (4.22) and collecting terms of like powers of results in
th
a
85
() () ()() ()()
( )
() ( ) ( )
()
()
( ) () () ()()
(
() ()
() ()
()
00
01 1 0
21 1 0 21 21
21
,
,,,
,
,,
,, , , , ,
,, ,
,,
r
rrAr
r
rr
rrrr r Ar
rr r
rr
t
ttttLt
t
t
tt t t
tttLt
tt t
tt
?
?
???
??
????
?
???
=+ + + ? ?
??
?
??
?
++?
?
?
???? ?
?? +??
??
?
??
??
?
+
pp p p
pp p
pp
? y
yJyg f y g ? y ? y
? y
g ? ygy g
y
fy g? y ? y
? y ? y ? y
J ? yfy
y
g
null
() ( ) ( )
()
()
()
() ( )
()()()()()
()
()
() ( )
()
2
22
2
22
2
21
022 22 0
01
10 21
1
1
11
,
,,
,
,
,
,, , , , ,
,,
,,,,
r
rr
r
r
r
rrr A r
r
rr
rrr
rr rr
rr
t
tt t t
t
tt
t
tttLt
t
tt
tt
t
?
???
???
?
??
?
??
?+
?
?+ ?+
?+
? ?
+?
?
?
?
?
??
?+
?
?
?
?
? ?
++??
?
?
?
?
????
??
??
??
?
?
pp p
p
p
p
p
p
? y
? yg y g
y
? y
g ? y
y
? y
fy g ? y ? y
? y? y
J ? yf y
y
?
nullnull
null
()
?
() ()
()
() ( ) ( )
()
()
()
() ( )
()
() ( )
2
22
2
2
1
00
11
1
, ,
,,
,
,,
,,
r
rr
r
rr
r
rr
r
rr
t t
tt
t
t
tt t t
tt
tt tt?
??
?+
?
?
?
?
+? ?
?
?
? ?
++?
?
?
?
?
???
??
?
?
??
?
pp
pp
p
y ? y
J ? yfy
y
? y
g ? ygy g
y
? y? y
g ? yg? y
yy
null
1
(4.24)
where the Lie operator ( )
( )
A
L i is defined as
()
()
( )
()()
2
,
,,;,
s
As s
t
Lt tsrrr
?
=?=?
?
pp
? y
? yJyJ? y
v
null21
and
,,
il t
rl
r
e
?
?
m
,
,,
il t
rl
r
e
?
?
m
,
1, ,
1
il t
rl
r
e
?
?
?
?
m
are shown as
r
? ,
r
? and
1r
?
?
for brevity.
()t? =
p
Ky
)
and is chosen such that it satisfies
()
(
1
0
,
r
rlr
l
t
+?
=
? p
? z
86
(4.25)
()
(
1
0
,
r
rlr
l
t
+?
=
=+
?p
yz ? z
)
p
It can be seen from equation (4.24) that equation (4.22) can be reduced to equation (4.6)
if the following sets of equations are satisfied
()()
( )
() ()
()
() () ( ) ( )
0
001 1
,
,,
,
,,
r
Ar r r
r
rr
t
Lt t t
t
t
tt t t
??
?
,0
0
+ ??
?
?
??
?
pp
pp p
? y
? yfyg? y
? y
gg? ygy
y
=
=
(4.26.1)
()()
( )
()
() ( )
()
() ()
()
()
() () ( ) ( )
()
() ( )
21
21 21
021
21
0 0 22 22 0 1
,
,,,
,,
,0
,, ,
r
A r r rrr
rr
rrr
r r
rr r
t
Lt t
t
tt
tt tt
t
tt t t t t
???
?
??
?
?
?? ?
?
+?
?
????
?+ +?=
??
??
??
??+
p
ppp
pp p p
? y
? yfy
? y ? y
g ? yJ? yfy
y
? y ? y
gg? yg y g? y
yy
0=
(4.26.2)
null
null
()()
( )
()()()
()
() ( )
()
()
() ()
()
()
() () ( ) ( )
2
22 2
2
22
2
2
22
0
1
11
1
00
11
,
,,,
,,
,,,,
, ,
,, 0
,
,,
r
Arr
rr r
r
rr
rrr
rr rr
r
rr
rr
r
rr
r
t
Lt t tt
t
tt
tt
t
t t
tt
t
t
tt t t
???
???
?+
?+ ?+
?+
??
?
+? ?
?
????
++
??
??
??
? ?
?=
?
??
?
?
+
p
p
p
p
pp
pp
p
? y
? yfyg? y
? y? y
J ? yf y
y
? y ? y
J ? yfy
y
? y
gg? ygy
y
? y
null
()
+
() ( )
()
() ( )
2
2
1
,,
rr
r
rr
tt
tt tt
?+
?
?
?
++ =
??
? y
g ? ? y
yy
null
1
0
(4.26.r)
87
Finding an analytical solution for the above sets of coupled PDEs is a difficult task, to
say the least. Therefore, once again an approximate analytical solution for these equations are
attempted by a multivariable Taylor-Fourier series approximation. Further, it can be noticed that
equation set (4.26.1) is independent of sets (4.26.2-26.r); while equation set (4.26.2) is
independent of the higher order sets. Therefore, an approximate solution to these PEDs can be
found by solving these equation sets sequentially starting with the lowest order set (4.26.1). The
first equation in (4.26.1) is identical to equation (4.17) whose solution can be approximated by
expanding known and unknown terms into a multivariable Taylor-Fourier series and comparing
coefficients of the expansion. It also yields a solvability condition similar to (4.21). This solution,
which is in terms of the unknown
r
? , then can be substituted in the second equation of (4.26.1).
Upon expanding the terms of this equation into a multivariable Taylor-Fourier series and
comparing coefficients of the expansion, typically, a set of underdetermined algebraic equations
is obtained. Generally speaking, such systems may have no solutions or have an infinite number
of solutions. If a solution can be found, then equation set (4.26.1) is solved. The remaining PDEs
can be solved in a similar way and the possible solution contains coefficients of the near-identity
transformations and nonlinear feedback gains. Even in the case of resonance, it may be possible
to find that cancel the resonant terms via feedback. Also, depending upon the
degree of sophistication needed in the controller design, one may choose only first few terms of
the controller and the near-identity transformation
()1
:0,,
rlr
l
+?
?? null r
( )0 lr? ? in equation (4.11) and (4.23), thereby
reducing the number of PDEs in (4.26) to be solved. Since equation (4.25) is polynomial
inversion of the near-identity transformation (4.23), any conventional algorithm for the inversion
of formal power series (e.g., Arfken (1996)) can be adopted for finding the inverse of
multivariable truncated Fourier series. The original controller u in equation (4.5) is obtained by
substituting the L-F and modal transformations.
88
4.3.2 System with Constant Coefficients
The technique described above can be easily adapted for systems with constant
coefficients. In this case, of course, one does not have to use the L-F transformation. Moreover,
the controller u in equation (4.10) contains only the spatial terms. For briefness, the entire flow
of transformations for the time-invariant case is not repeated but only the important points are
noted below.
For the time-invariant case, equation (4.12) takes the following form
( )
()
()
()
()
()
0
0
101
,
0
,
,,
r
r
rr
nl
vu
??
??
+??? ???
?? ??? ?
+
?? ??? ???
?? ??
??? ?????
??
? ?? ?
+
? ?? ?
++
? ?? ?
+
? ?? ?
???
ppq
pppp p
qqpqq
qpq
ppq pppq
qpq qqpq
fzz
zJgK z
=
zgKJz
fzz
gzz ggzz
gzz ggzz
null
null
?
(4.27)
where is a constant gain matrix that places eigenvalues of
p
K
0
+
pp
JgK
p
at the desired locations
in the complex plane. A linear transformation, = +
qq
zvPz
p
, decouples the linear part, where
must satisfy the algebraic Sylvester equation given by
P
( )
0
?+ =?
qpppq
JP PJ gK gK
0p
j
(4.28)
Algorithms to find a solution for the algebraic Sylvester equation have been widely
discussed in the literature; some of the important references are Gantmacher (1959) and Bartels
and Stewart (1972). Further, commercially available software MATLAB
?
has a routine to solve
the Sylvester equation. It should be noted that equation (4.28) has a unique solution only
if
i
? ?? , where
i
? (1, 2, ,= null )q are the eigenvalues of and
q
J
j
? ( )1, 2, ,= null p are the
89
eigenvalues of . This provides a condition under which such a decoupling
transformation is possible. The resulting equation, which is decoupled in the linear terms, can be
used to construct an invariant manifold. Here, an invariant manifold of the form
0
+
pp
JgK
p
( )
r
=
q
vhz
p
(4.29)
is proposed, where, ,
()
,
1
||
q
r
rj
r
j
r
h
=
=
??
m
pmp
m
hz z e
j
( )
12
,,,
p
mm m=m null
,
1
p
l
l
mr
=
=
?
,
, and
12
12
||
mm
zz=
m
p
z null
m
p
p
z
j
e is member of natural basis. The manifold in this case must
satisfy the following equation.
th
j
( )( ) ( )
0
,, 0
rr
L
nl
u? ?=
ppq qp q
hz JJ f z g (4.30)
which is time independent. An approximate solution for above PDE can be obtained by a term-
by-term comparison of coefficients of the monomials. It can also be noticed that, in order to
express the non-dominant states in terms of the dominant states, the following reducibility
condition must be satisfied.
0
rj
?? ?
dom
m ?i (4.31)
(
12
T
p
)
? ??nullwhere =
dom
? is a vector of eigenvalues of the Jordan matrix
p
J and
j
?
( )1,j= 2, ,null q are eigenvalues of the Jordan matrix . If an invariant manifold can be
constructed, then a reduced order model can be successfully obtained. The procedure for
transforming this equation into a linear closed-loop equation essentially follows the same steps as
described before and the resulting linear closed-loop equation takes the following form
q
J
90
0
?= +
pp
yJygnull (4.32)
Any traditional linear method can be used for designing a controller for the above
equation and the results may be transformed back into the original coordinates.
4.4 Applications
The technique proposed above can be effectively applied to design a reduced order
controller for practical engineering structures modeled by a set of nonlinear differential equations
with or without periodic coefficients. To demonstrate possible applications, a reduced order
controller is designed for a system consisting of four inverted coupled pendulums moving in the
horizontal plane with time dependent loads acting on each pendulum. Such a system yields
equations of motion with periodic coefficients. A 2-DOF spring-mass-damper system with
nonlinear springs is also considered for which the equations of motion have constant coefficients.
4.4.1 A 4-DOF Coupled Inverted Pendulum
The structural diagram of the inverted pendulums with axial periodic loads is shown in
Fig. 4.1. The equations of motion for this system, when expanded up to cubic terms, about the
fixed point
( )
(
12341234
, , , , , , , 0,0,0,0,0,0,0,0???????? =
nullnullnullnull
) can be shown to be
91
()()()
( )
()()()
()()()
() ()
3
23
11
1 1
111 11212121312 122
23
22
222 1211221132122
3
23
2
2
2123 223 2323 2 2
3
3
0
46
4
6
t
t
Pt
kh k
cc c
ml ml m ml
kh k
cc c
ml ml m
Pt ut
cc c
ml ml
h
ml
?
??? ???? ?? ?
??? ???? ??
?
?? ?? ?? ?
?
??
??
+++ ?+?+?? ?=
??
??
??
?
+++ ?+?+?
?
??
?
+?+?+?? ?=
??
?
??
+
nullnull null
nullnull null
nullnull
()()()
()()()
()
()()()
()
23
3
3 3 21 3 2 22 3 2 23 3 222
3
23
3
3
3134 3234 334 3
3
23
44
4 4
444 31433243343 422
4
0
6
0
46
t
t
k k
cc c
ml m
Pt
cc c
ml
Pt
kh k
cc c
ml ml m ml
? ? ?? ?? ??
?
?? ?? ?? ?
?
??? ???? ?? ?
?
++ ?+?+?
?
??
?
+?+?+?? ?=
??
?
??
??
??
+++ ?+?+?? ?
??
??
??
null
nullnull null
=
(4.33)
where ( ) ( )
12
cos
iii
Pt P P t?=+ . , , , and denote mass, length, torsional stiffness,
coupling stiffness and torsional damping, respectively
m l
ti
k k
i
h
( )1, 2, 3, 4i = , and ( )ut is the scalar
controller. Setting , and selecting some typical parameter values, equation (4.33)
can be written in the state space form as
1
0
i
P =
2i
c 0=
()()
()()
()()
()()
1
2
3
4
5
6
7
8
1
2
3
0000
0000
1 16.5cos 2 1 0 0
1202cos21
012cos21
00192.5c
1 000
0100
0010
0001
0.5000
0700
007.50
0007
x
x
x
x
t
x
t
x
x
t
x
t
x
x
x
?
?
?
os2?
?
??
?
??
?
??
?
??
?
??
?
??
?
=
?? ??
?
??
?
????
?
??
?
????
?
??
???
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
null
null
null
null
null
null
null
null
() ()
()() ()
()() ()
() ()
4
3
3
12 1
5
33
3
23 12 26
3
7
34 23 3
3
3
8
34 4
0
0
0
0
3.3 2.75cos 2
2 3.3 0.33cos 2
2 2 0.17cos 2
20.42cos2
x
xx txu
x
xx xx txx
x
xx xx tx
x
xx tx
?
?
?
?
??
??
??
??
??
??
??
??? ?
+
??? ?
??? +
??? ?
??? ?
?+ ??
??? ?
??? ?????
??? ?
??
???
(4.34)
92
where {}
{ }
12345678 12341234
,,,,,,, ,,,,,,,
T
T
xxxxxxxx ? ???????=
nullnullnullnull
and
( )
2
ut
u
ml
= .
After applying the L-F transformation to equation (4.34) one gets a dynamically
equivalent system whose linear part is time-invariant. The eigenvalues of this time-invariant
matrix (Floquet exponents) are obtained as
( )0.2656, 0.3355, 3.5214 0.4108 , 3.5809 0.1825 , 3.6376 0.7214ii???????i
and the corresponding Floquet multipliers are
( )1.3043, 0.7149, 0.0271 0.0118 , 0.0273 0.0051 , 0.0197 0.0174ii???? ?? ??i. The states
corresponding to Floquet multipliers ( )1.3043, 0.7149??
22
are chosen as the dominant states to
approximate the closed-loop system dynamics. For the resulting ? system a stabilizing
feedback controller is designed. As discussed in section 4.3.1, selection of the dominant states is
governed by the relative location of corresponding Floquet multipliers in the unit circle. In the
present case the first Floquet multiplier lies outside the unit circle, which makes the system
unstable. Therefore, the state corresponding to this Floquet multiplier is chosen as one of the
dominate states. The second Floquet multiplier, though inside the unit circle, is very close to the
boundary. The state corresponding to this Floquet multiplier, if not accounted in the reduced order
model, will results in a very limited region of asymptotic stability. Therefore the state
corresponding to Floquet multiplier 0.7149? is chosen as the second dominant state. Following
section 4.3.1, the linear and nonlinear feedback controllers are selected in the following forms
93
(4.35)
() (){}
() (){}
() () () ()
() () () ()
() ()
1
12
2
32 3
112 13,1, 3,0 ,0 3,1, 2,1 ,0 3,1, 3,0 ,1 3,1, 0,5 ,0
12
32 3
112 13,2, 3,0 ,0 3,2, 2,1 ,0 3,2, 3,0 ,1 3,2, 0,5 ,0
32
1123, 3,0 ,0 3, 2,1 ,0
,
,
it
nl
it
z
vKtKt
z
zzz ez
uKtKt
zzz ez
zzz
?
?
?? ? ?
?? ? ?
?? ?
??
=
??
??
+++ +
??
=
++ ++
nullnull
nullnull
null
() ()
() () () ()
35
123, 3,0 ,1 3, 0,5 ,0
224
112 1 22, 2,0 ,0 2, 1,1 ,0 2, 2,0 ,1 2, 0,4 ,0
it
it
ez z
zzz ez zv
?
?
?
?? ? ?
++
++++ ++
null
nullnull
2
5
2
z
z
where and are the dominant states. Nonlinear feedback up to
1
z
2
z 5
th
( )1l = order is used in this
example. The linear control gains ( )
1
Kt and ( )
2
Kt are chosen such that the Floquet multipliers
of the dominant system are placed at the desired locations inside the unit circle (c.f., section 4.2).
For the given example
0.2656 0
00.35
?
=
?
?
??
p
J
()
()
?
?
and
( ) ( )()
( ) () ( )
0.0416cos 3 2.4594sin 0.2293sin 3
0.0434cos 3 2.4694sin 0.2301sin 3
ttt
t
???????+ +?
??
??
?+
??
null
null()
0
0.2883cos
0.2974cos
t =
p
g
+
Choosing ( ) ( ) ( ) ( ) ( )
1
1.7184cos 0.4807cos 3 0.2006sin 0.0871sin 3Kt t t t t?? ?=? +?+nullnull?+
and ( ) ( ) ( ) ( ) ( )
2
Kt 1.7511cos 0.4841cos 3 0.1205sin 0.0599sin 3t t t t?? ??=? + + + + +nullnull
0.6657 0.2163i
the
linear feedback places the Floquet multipliers of the dominant system at ?
(Floquet exponent at ) which guarantees the linear asymptotic stability. After
decoupling the linear part of the system, an invariant manifold of the form given by equation
(4.16) is assumed. For this particular example,
0.3566 0.3141i??
{ }0.3566 0.3141 , 0.3566 0.3141ii=? + ? ?
dom
?
21 0.4115 , 3.5826 0.1826 , 3.6381 0.7213iii
,
3.52
j
? =? ? ? ? ? ? ( and ) ( ) ( ) ( )3,0 , 2,1 , 1,2 , 0,3=m
therefore, the system does not undergo ?true internal resonance? (c.f., equation (4.21)). Also,
94
since none of the eigenvalues of
p
J and are purely imaginary the system does not undergo
?true combination resonance?, either (it can be seen from equation (4.21) that in order to have a
true combination resonance at least one of the eigenvalue of
q
J
p
J and must be purely
imaginary). Therefore, an approximate solution for the PDE (4.17) can be successfully obtained.
The resulting reduced order closed-loop equation is obtained in the following form
q
J
( )
()
( )
()
2
2
,
, ,
t?
?
()
()t
3
1
3
1
()
1 2 5,1 1 2 3 311
1 2 5,2 1 2 3 322
7,112332 9,112332
7,2 1 2 3 3 2
,, ,, , ,0.2656 0
,, ,, ,0 0.3355
,, , , , ,, , , ,
,, , , ,
fzzt fzzzz
fzzt fzz t
fzztfzzt
fzz
??
??
??? ???
???
??? ?
?? ????
??? ?
=++
?? ?? ?? ?
??
?
???? ??
??? ?
??
??
??
??
null
null
()
()
0,1
9,2 1 2 3 3 2 0,2
,, , , ,
nl
gt
vu
fzz t g???
???
???
++
???
???
3,1
3,2
t
?
++
()
3
3,
,1, 4,1 ,0
h
?
?
?
?
(4.36)
The symbolic computations were carried out using Mathematica
TM
and the explicit long
expressions are omitted here for brevity. Next, a near-identity transformation of the form
() () ()
() () ()
() ()
()
32
1123,1, 3,0 ,0 ,1, 2,1 ,0 3,1, 3,0 ,1
11
32
22 112
3,2, 3,0 ,0 2, 2,1 ,0 3,2, 3,0 ,1
54 5
112 15,1, 5,0 ,0 5 5,1, 5,0 ,1
5
15,2, 5,0 ,0
it
it
it
hyhy hey
zy
zyhyhy hey
hy y hey
hy
?
?
?
? ?++++
????
? ?
?
=+
?????
????
? ?
+++ +
+
+
nullnull
null
nullnull
()
45
12 15 0 5,2, 5,0 ,1
it
hyy hey
?
??
++ +nullnull
null
(4.37)
(),2, 4,1 ,
is used such that the nonlinear terms up to order five are eliminated from the reduced order model
(4.36). This is consistent with the order of the nonlinear controller chosen. Application of the
near-identity transformation (4.37) to equation (4.36) produces two sets of PDEs similar to
equations (4.26.1) and (4.26.2), which are solved by expanding the known and unknown terms
into a multivariable Taylor-Fourier series and comparing Taylor-Fourier coefficients term-by-
term. For the present set of eigenvalues of
p
J , the system does not undergo any kind of resonance
and corresponding nonlinear controller is obtained as
95
( ) ( ) ( ) ( )
() () () ()
() () ()
32 3
112 1
54 5
32 3
112 1
0.0231cos 0.0622cos 0.0104sin 0.0336sin
0.0136cos 0.0355cos 0.0116sin 0.0487sin
0.2961cos 0.5631cos 0.5761sin 1.7708sin
nl
u tz tzz tz tzz
tz tzz tz tzz
tz tzz tz
?? ??
?? ??
=? ? + + ? +
?+
+ ?
nullnull
nullnull
null ()
() () () ()
((
() ()
2
22
54 5
222
1122 1 1
43 4 3
112 1 1
0.0611cos 0.4504cos 0.0877sin 0.1672sin
0.0498 0.0762 0.0317 0.0349cos 2 0.0737sin 2
0.0145 0.1367 0.0594cos 2 0.0334cos 2
tzz
tz tzz tz tzz
z zz z tz tz
zzz tz tz
??
+
?+
+?++ + +
?? ++ +
null
nullnull
nullnull
nullnull
())
2
4
1
0.1121sin 2 tz??
+
+
null
2
1
4
4
(4.38)
The controller is then transferred back into the original coordinates. The states for the
uncontrolled and controlled dynamics are shown in Fig. 4.2 and 4.3, respectively. Fig. 4.4 shows
the control effort in the original ( )x coordinates. Dotted lines in Fig. 4.3 and 4.4 show the state
response and control effort, respectively, when only the linear controller is applied to the
system. It is evident from the figures. that the proposed duel controller performs better than the
linear controller in terms of time required for convergence to the steady state as well as the
amount of control effort required. Further, Fig. 4.5 shows the region of attraction (
()v
)? for the
linear and duel controller on ( )
151
x xxnull? plane. ? is defined as
( ){ }
00
|, 0
n
Rt as??= ? ?xxt?? , where ( )
0
,t? x is the solution of the closed-loop system
that starts at initial state at time
0
x 0t = . The regions of attraction were found by numerical
integration of the closed-loop systems at various initial conditions and observing the state
trajectories. The stability of closed-loop system is most sensitive to the initial angular
displacement and velocity of the first pendulum (this is the pendulum which causes instability of
the system). Therefore, the states
1
x and ( )
51
x xnull are chosen to find the regions of attraction while
all other states are fixed at . It can be seen that use of the nonlinear controller greatly
increases the region of attraction for the closed-loop system.
0.5
96
4.4.2
Fig. 4.5 shows a 2-DOF sprig-mass-damper system with cubic stiffness terms. For this
system the equation of motion in state-space form can be written as
A 2-DOF Spring-Mass-Damper System
()
()
()
1 1
2 2
11 1 113 3
222 224 4
3
3
12 1 11
3
3
0010
01
// /0
0/
0
0
0
0
/ 0
()
22 1 22
1
/
x x
x x
kkm km cmx x
km k k m c mx x
u
bm x x bx
???? ??
???? ??
?? ??
=
?? ??
?+ ?
?? ??
?? ??
?+ ?
?? ????
??
??
??
??
??
++
????
??+
?
??
null
null
null
null
bm x x bx??+??
??
y eter values, , 3k
(4.39)
Typically, systems modeled by these equations may have very small damping and the
main purpose of a controller in such cases is vibration suppression. However, since the problem
under consideration involves stabilizing controller design some negative damping is introduced in
the system, without any loss of generalit . For some typical param
12
1mm== = ,
, 4b
1
2k = ,
2
11k = ,
1
1c =? ,
2
7c = = ,
1
2b = ,
2
3b =?
1.1700i
, the eigenvalues of linear part of the
equation are 0.4189? 2.1203i and 3.4189? ?
; where, the states belonging to
1
. Therefore, the
unstabl
he form
2-DOF syste
e eigenvalues are
of equation (4
m is
.10) but
approxim
assu
ated by
med to
a sing
be the dom
le-DOF sy
inant states.
stem
s. By
The controlle
taking l
r is still in t
contains only time-invariant term = , a 5
th
order nonlinear controller is proposed.
Linear control gains { } { }
12
After finding
, 1.4617, 2.2102KK =? ? place the eigenvalues of the dominant states
at 0.9? and ?1. decoupling transformation between the dom
dominant states by the methodology suggested earlier, the resulting reduced order model is
obtained as
inant and non-
97
(4.40)
() () ()
()
() ()
32
11112
22
5
13, 3,0 2, 2,0 3, 2,1 ,1
3, 0,3 2, 0,2
0.4189 2.1203 0.1160 0.1011
2.1203 0.4189 0.3909 0.3258
0.0192 0.0248 0.0186 0.0117
0.0522 0.0189 0.0225
zzzzz
z???
??
??? ?+ +?? ????
??
=+
?? ??? ?
??
?+ +
???? ??
?++ +
+
+?
null
null
null
null
nullnull
()
()
5
13, 0,3 ,1
0.0277
0.2641
1.1545
nl
z
u
?
?
??
?+
??
++
??
??
nullnull
+
+
?
?
+
?
+
It should be noted that for the present set of eigenvalues of the system, in equation
(4.28) has a unique solution. Also, the system reducibility condition (4.31) is satisfied. In order to
transform equation (4.40) into its equivalent linear form, a near-identity transformation of the
form
P
(4.41)
() ()
() ()
() ()
() ()
32 54
112 1123,1, 3,0 3,1, 2,1 5,1, 5,0 5,1, 4,1
11
32 54
22 112 112
3,2, 3,0 3,2, 2,1 5,2, 5,0 5,2, 4,1
hyhyy hyhyy
zy
zyhyhyy hyhyy
???++ ++
????
???
=+ +
????? ?? ?
????
???
nullnull
nullnull
is applied. Substituting equation (4.41) into equation (4.40) and colleting the terms of like powers
of result in PDEs similar to equations (4.26.1) and (4.26.2), which are time-invariant.
Approximate solutions are obtained in the form of power series expansion. Once again, since the
dominant eigenvalues are complex, the reduced order system does not exhibit the resonance and a
nonlinear feedback and coordinate transformation (near-identity transformation) can be found to
transform equation (4.40) into a dynamically equivalent linear equation. The corresponding of
nonlinear feedback is obtained as
y
(4.42)
()
32 235
1121221
22
11221
0.3660 0.4400 0.2254 0.0336 0.0244
0.0297 0.0326 0.0215 0.0034
nl
uzz z zz
zzzzz?
=? + ??
+ ? +?+
null
null
98
The uncontrolled and controlled response of the system is shown in Fig. 4.7 and 4.8, respectively.
It is evident from Fig. 4.8 that the duel controller performs better than the linear controller ( )v ,
alone, in terms of the speed of convergence to the steady state. Fig. 4.9 shows control efforts by
the linear and duel controller. Again, the duel controller needs much less effort compared to the
purely linear controller. Fig. 4.10 shows the region of attraction ( )? for the two controllers on
( )
131
x xxnull? plane for some fixed initial values of
2
x and
4
x . In this case too, the nonlinear
controller has a greater region of attraction as compared to the linear controller.
99
( )
21 22
cosPP t?+
44
,
t
kh 22
,
t
kh 11
,
t
kh
33
,
t
kh
2
l
2
l
k k k
( )ut
( )
31 32
cosPP t?+
2
?
1
?
3
?
4
?
( )
11 12
cosPP t ()
31 32
cosPP t+ ? + ?
Fig. 4.1 A 4-DOF coupled inverted pendulums.
100
Fig. 4.2. Uncontrolled system response for system with periodic coefficients.
101
Fig. 4.3 Controlled system response for system with periodic coefficients.
, linear + nonlinear control; , linear control
102
Fig. 4.4 Control effort for system with periodic coefficients.
, linear + nonlinear control; , linear control
103
Fig. 4.5 Regions of attraction for controlled system
(system with periodic coefficients).
104
c
2
k
2
b
2
c
1
k
1
b
1
b
k
u
m
1
m
2
Fig. 4.6 A 2-DOF spring-mass-damper system.
105
Fig. 4.7 Uncontrolled system response for system with constant coefficients.
106
Fig. 4.8 Controlled system response for system with constant coefficients.
, linear + nonlinear control; , linear control
107
Fig. 4.9 Control effort for system with constant coefficients.
, linear + nonlinear control; , linear control
108
Fig. 4.10 Regions of attraction for controlled system
(System with constant coefficients).
109
Chapter 5
Discussion and Conclusions
5.1 Summary of Work
This dissertation proposes some new methodologies for simplification of nonlinear
dynamic systems such that the analysis and control problems can be carried out in a simpler and
efficient way. The systems under consideration may include parameters that are constant or
periodically excited and they are subjected to either external periodic excitations or stabilizing
state feedback. The main focus of this work is on order reduction of nonlinear dynamical systems
which is accomplished through the construction of invariant manifolds. Another important
simplification technique presented in this work is a direct application of near-identity
transformations and construction of the normal forms for nonlinear systems subjected to external
periodic excitations. It is rather important to note that the methodologies developed here are
equally applicable to nonlinear systems with constant as well as periodic coefficients. Moreover,
for systems with periodic coefficients, the techniques are not limited by the size of time-periodic
component which is a distinct advantage over the traditional approaches.
Chapter 2 presents a direct methodology for a quantitative analysis of nonlinear dynamic
systems with external periodic forcing via an application of the theory of normal forms. Rather
than introducing a new state variable to reduce the problem to a homogenous one, a set of time-
dependent near-identity transformations is applied to construct the normal forms. In the process
the total response of the system is expressed as superposition of a steady state solution and a
110
transient solution. A steady state solution of the system is obtained by the method of harmonic
balance and the transient solution is obtained by solving a set of time periodic homological
equations. After discussing the time-invariant case, the methodology is extended to systems with
time-periodic coefficients. The case of time periodic systems is handled through an application of
the L-F transformation. Application of the L-F transformation produces a dynamically equivalent
system in which the linear part of the system is time-invariant, making the system amenable to
near-identity transformations. An example for each type of system, viz., constant coefficients and
time-varying coefficients, is included to demonstrate effectiveness of the method. Various
resonance conditions are discussed. It is observed that the linear parametric excitation term need
not be small as generally assumed in perturbation and averaging techniques. Results obtained by
proposed methods are compared with numerical solutions. Close agreements are found in some
typical applications.
In Chapter 3, a methodology for determining reduced order models of periodically
excited nonlinear systems is presented. The approach is based on the construction of an invariant
manifold. Due to the existence of external and parametric periodic excitations, however, the
geometry of the manifold varies with time. As a result, the manifold is constructed in terms of
temporal and dominant state variables. The governing PDE for the manifold is nonlinear and
contains time-varying coefficients. An approximate technique to find a solution of this PDE using
a multivariable Taylor-Fourier series is suggested. It is shown that, in certain cases, it is possible
to obtain various reducibility conditions in a closed form. The case of time-periodic systems is
handled through the use of L-F transformation. Application of the L-F transformation produces a
dynamically equivalent system in which the linear part of the system is time-invariant, however,
the nonlinear terms get multiplied by a truncated Fourier series containing multiple parametric
excitation frequencies. This warrants some structural changes in the proposed manifold, but the
111
solution procedure remains the same. Two examples; namely, a 2-DOF mass-spring-damper
system and an inverted pendulum with periodic loads, are used to illustrate applications of the
technique for systems with constant and periodic coefficients, respectively. Results show that the
dynamics of these 2-DOF systems can be accurately approximated by equivalent 1-DOF systems
using the proposed methodology.
Chapter 4 provides a methodology for reduced order stabilizing controller design for
nonlinear dynamic systems. The equations of motion are represented by quasi-linear differential
equations in state space, containing a time-periodic linear part and nonlinear monomials of states
with periodic coefficients. The L-F transformation is used to transform the time-varying linear
part of the system into a time-invariant form. Eigenvalue decomposition of the time-invariant
linear part can then be used to identify the dominant/ non-dominant dynamics of the system. The
non-dominant states of the system are expressed as a nonlinear, time-periodic, manifold
relationship in terms of the dominant states. As a result, the original large system can be
expressed as a lower order system represented only by the dominant states. A reducibility
condition is derived to provide conditions under which a nonlinear order reduction is possible.
Then a proper coordinate transformation and state feedback can be found under which the
reduced order system is transformed into a linear, time-periodic, closed-loop system. This permits
the design of a time-varying feedback controller in linear space to guarantee the stability of the
system. The case of systems with constant coefficients is treated as a subset of its periodic
counterpart. The proposed methodology is illustrated by designing a two dimensional reduced
order controller for a 4-degrees of freedom (DOF), inverted pendulum subjected to a periodic
force for which the equations of motion are time-periodic. An example involving a 2-DOF
spring-mass-damper system is also presented which yields the equations of motion with constant
coefficients.
112
In some sense, the order reduction technique presented here is a generalization of the
center manifold theory. Similar to the center manifold theory, where stable states are expressed as
nonlinear functions of critical states, in this approach the stable states are expressed as nonlinear
time-periodic functions of the unstable states. From the results obtained by numerical simulation
of the 4-DOF system with periodic coefficients, it is concluded that a satisfactory control system
may be designed using a single-DOF model. Advantages of using the nonlinear controller over
the linear controller are evident from the improved system response (in terms of time required for
convergence to the steady state) and the lesser amount of control effort needed. It should also be
noted from Figures 4.5 and 4.10 that the regions of attraction for the examples with proposed
nonlinear controllers are much larger as compared to those with the linear controllers.
In summary, the main contributions and salient features of this work can be listed as
follows:
? Development of a methodology for the construction of normal forms for nonlinear
systems with parametric and external periodic excitations via direct application of
near-identity transformations and the Poincar? normal forms theory.
? Development of an invariant manifold based methodology for the construction of
reduced order models for nonlinear systems with parametric and external periodic
excitations.
? Development of an invariant manifold based methodology for the construction of
reduced order models for parametrically excited nonlinear systems subjected to a
nonlinear stabilizing state feedback.
113
? For systems subjected to external periodic excitations, the proposed methodologies
accurately capture interactions between external excitations and systems?
nonlinearities.
? A transformation for the diagonalization of linear time-periodic systems with lower
triangular linear matrices is proposed. The proposed transformation is governed by a
time-periodic Sylvester differential equation. A methodology for the construction of a
solution for this equation is also proposed.
? It is demonstrated that normal form equations and reduced order models derived
using proposed methodologies capture the important dynamics of original systems
accurately. Also, nonlinear controllers developed for reduced order models have
distinct advantages over linear controllers.
Suggestions for Future Work
There are several opportunities to extend the work presented here beyond its current
precincts and some of those are highlighted here.
1) In this work, reduced order models are obtained for small to medium order systems. Although
techniques presented here are not limited by the size of a system (number of DOF), their
effectiveness for large systems needs to be assessed.
2) Nonlinear dynamic systems with periodic-quasiperiodic coefficient appear naturally in
various branches of science and engineering. For example, a continuous cantilever beam
subjected to periodic axial and transverse load with dissimilar frequencies yields such
equations. The techniques presented in this work can be adapted for simplified analysis and
114
controller design for such systems. Work by Wooden and Sinha (2007) and Redkar (2005)
are few such examples which propose normal form and order reduction methodologies for
periodic-quasiperiodic, homogeneous systems. Techniques presented can be employed for
periodic-quasiperiodic systems subjected to external inputs.
3) Many mechanical and structural dynamic systems are usually modeled as a set of second
order differential equations given by ( ) ( ),,t+++ =Mx Cx Kx f x x F t. Where, M , C , K are
mass, damping and stiffness matrix, respectively and x is the displacement vector. Rather
than transforming these equations into a state-space form, developing order reduction
techniques in second order form is more intuitive in nature. References Burton and Rhee
(2000) and Deshmukh et al. (2006) are few such examples, among others, which use second
order formulation for nonlinear order reduction. The class of problems considered in this
work yet needs to be explored for second order formulation.
4) Reduced order controller design methodology proposed in the Chapter 5 assumes availability
of all system?s states. In practice, however, it may not be always possible to obtain all states
of a dynamic system using direct measurements. Therefore, there is a great potential for
designing reduced order observers which can estimate the desired states of a system.
5) The author believes that the work described here lays foundation for the development of a
reduced order controller design methodology for systems subjected to periodic disturbances.
In the case, if system is subjected to known periodic disturbance the techniques described in
Chapter 3 and 4 can be used directly, in conjunction, with the reduced order controller design.
However, in the case of unknown periodic disturbance one has to device a methodology to
estimate the disturbance. This could be a challenging task but has tremendous potential for
applications to some real life control problems.
115
6) In the present work, regions of attraction for the controlled systems are obtained by actual
numerical simulation of systems in the domain of interest. Development of an analytical
technique for the estimation of such regions of attraction is something the future work should
look into.
116
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121
Appendix A
Floquet Theory and Lyapunov-Floquet Transformation
A.1 Floquet Theory
Floquet theory is an important tool in predicting stability and response of linear
differential equations with periodic coefficients. The idea is based on the fact that, once the
solution of problem is known for the principal period, it is known for all time t . Stability of the
system is determined by eigenvalues of the State Transition Matrix (STM) calculated at the end of
one period. These eigenvalues are called Floquet multipliers of the system and the STM evaluated
at the end of the principle period is called as Floquet Transition Matrix (FTM). Another important
outcome of the Floquet theory is Lyapunov-Floquet (L-F) transformation which essentially gives
methodology to convert time-periodic linear system into equivalent time independent linear
system.
Consider a linear time-periodic system given as
() () ()ttt=xAx (A.1)
where matrix is a T periodic such that nn? ( )tA ( ) ( )tT t+ =AA. Let be the state
transition matrix of (A.1) so that
( )t?
(0) =? I , where I is the identity matrix. Then, according to the
Floquet theory
1. and hence ()()()0tT t T tT+= ?????
122
2. ()()()0 ,2,3
n
tnT t T tTn+= ??=??? "
3. () () (0) 0tt t=?x ? x
This implies that if the solution is known over the principal period then it can be constructed for
all time t .
Floquet theory also addresses the stability of linear time periodic system in terms of
eigenvalues of FTM which are called characteristic multipliers. Let (1,,
k
)n? = " denote
eigenvalues of . Then system (A.1) is asymptotically stable if all
k
( )T? ? lie inside the unit
circle of the complex plane. The system is unstable if at least one of the eigenvalues of the FTM
has magnitude greater than one. Alternatively, stability can also be expressed in terms of
characteristic exponents of the system. If
kkR k
i
I
? ??= + are characteristic multipliers of the
system, which, in general, are complex. Then, characteristic exponents of the system are defined
as
k
i
k
? ?? where
1
1
ln
1
tan
kk
kI
k
kR
T
T
??
?
?
?
?
=
?
=
??
??
?
(A.2)
The system is stable if 0
k
? < for 1, 2, ,kn= " . The detailed treatment of Floquet theory can be
found in Yakubovich and Starzhinskii (1975).
Another important result of Floquet theory is the Liapunov-Floquet theorem (Yakubovich
and Starzhinskii 1975)
123
Theorem: Each fundamental matrix of equation (A.1) can be written as the product of
two matrices as
nn? ( )t?
nn?
(A.3) () ()
t
tte=
C
? Q
where is T periodic and is a constant matrix. and C , in general, are complex. ( )tQ C ( )tQ
Corollary-1: Each fundamental matrix can also be factored as ( )t?
(A.4) () ()
t
tte=
R
? L
where is real and periodic with period and is an appropriate real matrix. ( )tL 2T R
Corollary-2: The Liapunov-Floquet transformation
() () ()ttt=xQz (A.5)
reduces the original system (A.1) to
() ()tt=zCz (A.6)
Moreover, the periodic transformation 2T
() () ()ttt=xLz (A.7)
produces a real representation given by
() ()tt=zRz (A.8)
However, though the Liapunov-Floquet transformation reduces a non-autonomous linear
differential equation to an autonomous one, there is no simple way to construct either ( )tQ
124
or ; exception being to the case of commutative system for which there is analytical
procedure to calculate L-F transformation matrix in close form (Lukes, 1982).
( )tL
, ]T
{
A.2 Calculation of STM and L-F Transformation Matrix
It has been shown by Sinha and associates (Sinha & Wu 1991; Sinha et al. 1993; Wu &
Sinha 1994) that the STMs for linear periodic systems can be obtained in terms of the shifted
Chebyshev polynomials of the first kind. This procedure is suitable for calculating STM for any
general time-periodic matrix . The STM is expressed in terms of time t and Liapunov-
Floquet theory can be applied further to find the L-F transformation matrix for the system. The
main idea of this technique is to expand the solution vector and time periodic matrix
of the linear time-periodic system in terms of the shifted Chebyshev polynomials in the interval
. Then using recursive integration and multiplication properties of Chebyshev polynomials,
original time periodic differential equation can be reduced to a set of algebraic equations with
unknown Chebyshev coefficients which can be solved to find the solution of the system.
( )tA
( )tx ( )tA
[0
()tx and are expanded in terms of Chebyshev polynomials in the interval [ as ( )tA 0, ]T
(A.9)
1
**
0
1
**
0
() () () , 1,2, ,
() () () , , 1,2, ,
T
T
m
ii
irr
r
m
ij ij
ij r r
r
tbst ti n
At dst t ij n
?
=
?
=
?? =
?? =
?
?
xsb
sd
"
"
where
} { } { }
****
01 1 0 1 1 0 1 1
, , () () () ()
TTT
i i ij ij ij ij
mm
bb b d d d t s ts t s t
??
== =bd s""
i i
m?
m" and is number of
terms in Chebyshev expansion.
125
Here are unknown expansion coefficients of
i
r
b ()
i
x t , are known expansion
coefficients of and
ij
r
d
( )
ij
At
*
()
r
s t are the shifted Chebyshev polynomials of the first kind. Let
Chebyshev polynomial matrix be defined as
(A.10)
*
?
() ()
T
t=?SIst
where represents Kronecker product, and I is (? )nn? identity matrix. Using equation (A.9)
and (A.10), and can be written as ( )tx ( )tA
??
() () ; () ()tt tt==xSbASD (A.11)
and the product
?
() () ()tt t=Ax SQb (A.12)
where
123
{},{ },1,23
Tiiij
ij n===
123 n
bbbb b Dddd d""2" and matrix Q is a ( )nm nm?
product operation matrix. Q is given as
01 2 1
102 13 2
213 04
12 022
/2 /2 /2
/2 ( )/2 ( )/2
( )/2 /2
()/2 /
m
mm
mmm m
dd d d
ddd dd dd
ddd dd
ddd dd
?
?
?? ?
????
??
++?+
?
=? ? ? ?? ?
??
?? ?
+? +
??
Q
The detailed development of this matrix can be found in Wu (1991). The integral form of the
equation (A.1) can be written as
126
0
() (0) ( ) ( )
t
td? ???=
?
xx Ax (A.13)
Substituting equation (A.11) and (A.12) into this equation and using recursive integral properties
of Chebyshev polynomials Sinha and Wu (1991) have shown that one can obtain a set of linear
algebraic equations of the form
[](0?=IZbx) (A.14)
where b is a set of unknown coefficients as defined before, matrix is a constant matrix given
by
Z
0
[]
TT
A
=?+?ZA G C GQ and
1/2 1/2 0 0 0
1/8 0 1/8 0 0
1/6 1/4 0 1/12
1/16 0 1/8 0
1
4( 1)
(1) 1
0
2 ( 2) 4( 2)
T
m
G
m
mm m
?? ???
??
??
?? ?? ?
??
= ?????? ?
??????
?
??
??
?????
??
Above equation can be solved for b and hence solution can be found out by equation
(A.11).
( )tx
State transition matrix for the given linear system is given by set of solutions for n
initial conditions as . It can be seen that all
( )t?
,0,0,(0) (1 0), (0,1,0, 0), , (0,0,0, ,1)
i
=x """"
i
b ?s
corresponding to the above set of initial conditions can be determined simultaneously by defining
127
right hand side of the equation (A.14) in the matrix form. Then the state transition matrix is given
by
?
() ()tt=? SB (A.15)
where
123
[, , , , ]
n
bbb b=B " . It should be noted here that this STM is valid only over the interval
as the shifted Chebyshev polynomials are defined over the interval [ . When ,
the STM can be evaluated using Floquet theory as explained earlier. Once is known, the T -
periodic complex matrix or the 2 - periodic real matrix can be computed as follows
(Arrowsmith & Place, 1990). Since
0 tT?? 0, ]T
( )t
tT>
?
( )tQ T
( )t
( )tL
=? I , (0) ( )T= =QQ I, the Floquet transition matrix can
be written as
(A.16) ()
T
Te=
C
?
where is a constant complex matrix which can be computed by performing eigenvalue
analysis on ? and T - periodic L-F transformation matrix is given as
C ( )nn?
( )T
() ()
t
tte
?
=
C
Q ? (A.17)
It is also to be noted that (Yakubovich and Starzhinskii, 1975)
(A.18)
*
2
(2 ) ( )
TT T
TTeee== =
CC R
??
2
/2where is conjugate matrix of C and . Then periodic L-F transformation
matrix is given as
*
C
*
{}=+RCC 2T
128
(A.19)
() () ; 0
()()(); ()2;0
t
t
tte tT
TtTeTT T??
?
?
=??
+= ?+? ??
R
R
L ?
L ? L T?
t
t?
When it is necessary to find and one can possibly invert and
through symbolic software like MATHEMATICA/MAPLE or one can first find the STM of
the adjoint system
1
()t
?
Q
1
()t
?
L ( )tQ ( )tL
( )t?
(A.20) ()
T
t=?wAw
and use the following relationship (Yakubovich and Starzhinskii, 1975)
(A.21)
1
() ()
T
t
?
=??
Knowing , and can be computed using properties of adjoint system. e.g.
can be computed as
1
()t
?
?
1
()t
?
Q
1
()t
?
L
1
()t
?
Q
(A.22)
111
() [ () ] () ()
tt tT
tte ete
????
===
CC C
Q ??
The technique described here has been successfully employed by Pandiyan et al. (1993)
and Bibb (1992) to compute L-F transformation matrices. Such approximation of L-F
transformation matrix has been found to be extremely convergent and since L-F transformation
matrix is periodic, the elements and has truncated Fourier series representation as
ij
Q
ij
L
(2 / )
0
11
,1
cos sin
2
q
intT
ij n
nq
qq
ij n n
nn
ce i
a nt nt
ab
TT
?
? ?
=?
==
?=?
?+ +
?
??
Q
L
(A.23)
129
1
ij
?
Q and has similar Fourier series representation.
1
ij
?
L
130
Appendix B
Solution of Time-periodic Sylvester Differential Equation
Time-periodic Sylvester differential equation presented in Chapter 4 has a very special
structure such that if the equation is written in the expanded form, then the states in each row of
the equation decouple form the states in other rows. Moreover, each decoupled equation has the
same time-periodic linear matrix. Due to these properties, construction of a solution for this
equation is simple and computationally efficient. One such technique based on an application of
the L-F transformation and use of convolution integral is presented in the following.
Consider a time-periodic Sylvester differential equation given by equation (4.14):
() ( ) ( ) ( ) ( )
( ) ( ) ( )
,0 ,0
ttt ttt=? + +
q ppp qp
PJPPJgK gK
null
t (B.1)
It is desirable to find a closed form solution ( )tP for the above equation. Typically, the above
equation can be written in the component form as
131
11 12 1 1 11 12 1
21 22 2 2 21 22 2
12 12
11 12 1 1 1 1
21 22 2
12
000
00
00
pp
qq qp qqq qp
p
p
qq qp
pp p pp p
pp p pp p
pp p pp p
pp p Bk
pp p
pp p
?
?
?
?
??????
??????
=
??????
??????
+??
??
?
??
??
q
q
q
pp
nullnull null??
nullnull null ?
nullnullnullnull nullnullnullnullnullnullnullnull
nullnull null???
?
?
nullnullnullnull
?
12 1
21 2 22 2
12
11 12 1
21 22 2
12
p
p
prp p
p
p
qq qp
Bk Bk
Bk Bk Bk
p
B kBk B
Bk Bk Bk
Bk Bk Bk
Bk Bk Bk
?
? k
? ?
? ?
+
? ?
? ?
? ?
+
? ?
? ?
??
??
+
??
pp
ppp
qq q
qq q
?
?
nullnullnullnull
?
?
?
nullnullnullnull
?
(B.2)
where
qp
pnull and
qp
p are the elements of ( )tP
null
and ( )tP , respectively.
q
?
q
are eigenvalues of
Jordan block ,
q
J
p
?
p
are eigenvalues of Jordan block ,
p
J
q
B
q
are elements of the control
influence matrix corresponding to the non-dominant states ( )
( )
,0
t
q
g ,
p
B
p
are elements of the
control influence matrix corresponding to the dominant states ( )
( )
t
,0p
g and are the elements
of control gain vector
p
k
( )t
p
K .
After multiplication equation (B.2) yields
132
(B.3)
()
()
11 12 1 1 11 1 12 1 1
21 22 2 2 21 2 22 2 2
12 1 2
11 1 1 1 12 2 1 1 1
11 11 221
pp
q q qp qq qq qqp
rp
qqqp
pp p p p p
pp p p p p
pp p p p p
pBkpBkpBk
pBkpBk Bk
?? ?
?? ?
?? ?
?
?
??? ?
??? ?
=
??? ?
??? ?
++ ++
?
++ ++
qq q
qq q
q
pp
pp p p
nullnull null??
nullnull null
nullnullnullnull null nullnullnull
nullnull null??
null?
nullnull
null
()
()
1
11 1 12 2 1
11 2 2
11 12 1
21 22 2
12
ppppp
qpq p qpppp
p
p
qq qp
pBk pBk p Bk
pBk pBk p Bk
Bk Bk Bk
Bk Bk Bk
Bk Bk Bk
?
?
?
?
?
?
?
?
?
++++
?
?
?
++++
?
?
??
??
+
??
??
pp
qq q
qq q
?
?null
nullnull
?null
?
?
nullnullnullnull
?
1p
It can be noticed that rows of equation (B.3) are decoupled and each row forms a set of
coupled differential equations given by
11 1 1 1 1 2 1 1 11
12 1 1 2 2 2 2 2 12
11
11
12
1
00
00
00
p
p
ppp
p
p Bk Bk Bk p
p Bk Bk Bk p
p Bk Bk Bk p
Bk
Bk
Bk
??
??
??+
??? ?? ???
??
??? ?? ???
+
?? ??
??
=?
?? ??
?? ??
? ?
?? ??
+
??
??? ?? ???
??
?
+
?
qppp
qp p
q
q
q
null ??
null
nullnullnullnullnull null nullnullnull null
null
null
?
??
?
??
??
??
(B.4.1)
null
null
133
11121
21
12
1
2
00
00
00
qq pq
qp q p p p p p qp
q
q
qp
1
2
p Bk Bk Bk p
p Bk Bk Bk p
p Bk Bk Bk p
Bk
Bk
Bk
??
??
??+
??? ?? ???
??
??? ?? ???
+
?? ??
??
=?
?? ??
?? ??
?
?? ??
+
??
??? ?? ??
??
?
+
?
qpppp
qp
q
q
q
null ??
null
nullnullnullnullnull null nullnullnull null
null
null
?
??
?
??
??
??
?
?
(B.4.q)
Above set of equations is of the form
( ) ( )
ii ip
t??=+ +
??
pAApBKnull t
)q
(B.5)
where is a diagonal matrix with identical constant elements, (1, 2, ,
i
=A null ()tA is a time-
periodic matrix, which remains same for all equations in the set. is a
periodic forcing term. This time-periodic equation can be solved using the L-F transformation and
convolution integral. Since is a diagonal matrix (with identical constant elements) the state
transition matrix for equation (B.5) is given as
()( )1, 2, ,
ip
ti p=BK null
i
A
( )
i
At
te=??, where ( )t? is the state transition
matrix corresponding to ()tA . By using the method suggested by Sinha et al. (1996), one can
compute ()t? and the L-F transformation matrix ( )tL . The L-F transformation matrix ( )tL
together with the modal transformation ( )M , transforms equation (B.5) into the form
( )
iii i
t=+pJpG
null
(B.6)
134
where is in the Jordan canonical form,
i
J ( )
11
i
t
??
=GMLBK
ip
. Here, it should be noted that
()t? is computed only once which makes this solution procedure computationally less
demanding. Using the convolution integral, the solution of equation (B.6) is given as
()
()
()
0
0
ii
t
tt
ii i
ee
?
d? ?
?
=+
?
JJ
pp G (B.7)
Expressing in terms of a finite Fourier series as ()
i
tG
(B.8) ()
1
,
1
p k
il t
ijl
jlk
tce
?
==?
=
??
G
j
(where 1i =?, ? is principle frequency of forcing ( )
i
tG and
j
e is
th
j member of natural
basis), the solution of integral (B.7) (with ( )00?p ) can be written as
,,
11
j
t
il tppkk
ijljjl
jlk jlk
jj
ee
cc
il il
?
?
?? ??
==? ==?
=?
?
?? ??
pe
j
?
e (B.9)
where ;1,2,
j
j? = null p are the eigenvalues of . It can be seen from equation (B.9) that
i
J
i
p
can not be determined if any of the eigenvalues of have positive real part (unstable case) or if
i
J
j
ik? ?= .
135
Appendix C
Computer Codes
C.1 Mathematica code (Finding Q and R.nb) for computation of periodic L-F
transformation matrix
2T
( )( )tL and its inverse ( )( )
1
t
?
L for a 4-DOF system with periodic
coefficients.
C.2 Mathematica code containing supporting functions required by ?Finding Q and R.nb?.
C.3 Mathematica code to compute the near-identity transformation for the forced Mathieu-
Duffing equation.
C.4 Mathematica code to find the harmonic balance solution for a 1-DOF parametrically
excited nonlinear system.
C.5 Mathematica code to compute the dynamic response of a system using its normal form
equation. The code also compares this response with the original system response.
C.6 Mathematica code to construct the invariant manifold for a 2-DOF parametrically excited
nonlinear system assuming ( )t
p
F to be large.
C.7 Mathematica code to construct the invariant manifold for a 2-DOF parametrically excited
nonlinear system assuming ( )t
p
F to be small.
136
C.8 Mathematica code to compute the dynamic response of a system using the reduced order
equation. The code also compares a reduced order system response with a full order system
response.
C.9 MATLAB code to find Fast Fourier Transform of a steady state periodic solution.
C.10 Mathematica code to design a nonlinear reduced order controller for a 4-DOF
parametrically excited system.
C.11 Mathematica code to find shifted Chebyshev coefficients of the first kind for a periodic
function.
C.12 Mathematica code (Symbolic FTM.nb ) to find the FTM, symbolically.
C.13 Mathematica code containing supporting functions required by ?Symbolic FTM.nb?.
C.14 Mathematica code to find stabilizing feedback gains for a 1-DOF linear closed-loop system
with periodic coefficients.
C.15 Mathematica code to find a solution for the time-periodic Sylvester differential equation.
C.16 Mathematica code to find an inverse of a polynomial containing 3 and 5 degree
monomials with periodic coefficients.
rd th
C.17 Mathematica code to compute the system response a 4-DOF (full order) system subjected
to the reduced order controller.
137
C .1
H? File name "Finding R and Q.nb ?L
H? Program to find 2T periodic L?
F tranformation matrix and its innvers for for a 4?DOF time?periodic system ?L
H? Execute file "LFtransform_functions.nb" before
execution of this set of functions ?L
a1 = 1.0; a2 = 20; a3 = 21; a4 = 19; b1 = 16.5; b2 = 2; b3 = 1;
b4 = 2.5; c1 = 1; c2 = 1; c3 = 1; h1 = 0.05; h2 = 7; h3 = 7.5; h4 = 7;
A = 880, 0, 0, 0, 1, 0, 0, 0<, 80, 0, 0, 0, 0, 1, 0, 0<, 80, 0, 0, 0, 0, 0, 1, 0<,
80, 0, 0, 0, 0, 0, 0, 1<, 8?Ha1?b1 Cos@2 ? tDL, c1, 0, 0, ?h1, 0, 0, 0<,
8c1, ?Ha2?b2 Cos@2 ? tDL, c2, 0, 0, ?h2,0,0<, 80, c2, ?Ha3?b3 Cos@2 ? tDL,
c3, 0, 0, ?h3, 0<, 80, 0, c3, ?Ha4?b4 Cos@2? tDL,0,0,0,?h4<<;
t0 = TimeUsed@D;
chcoef =80.30424, 0, 0.97087, 0, ?0.30285, ?1.1369`?^?013, 0.029092, 0,
?0.0013922, ?6.9122`?^?011, 4.019`?^?005, ?4.7148`?^?009, ?7.6044`?^?007,
2.9802`?^?008, ?4.7684`?^?007, 3.3379`?^?006, 1.1444`?^?005<;
H? chcoef:= CHebyshev COEFicients of periodic terms ?L
n = 8;H? null states ?L
m = 18; H? null Chebyshev coefficients in the expansion ?L
fGT@mD;
fGG@Transpose@MGpD,m,nD;
tensorP := Array@tP, 8n, n, m> "b.txt";
R >> "R.txt";
Qft >> "Q.txt"; invQft >> "invQ.txt";
6 Finding R and Q.nb
143
C .2
H? File name "LFtransform_functions.nb" ?L
H? Functions required by "Finding R and Q.nb" ?L
fQp@p_D := Module@8i, j<,
MQp := Array@Qp, 8Length@pD, Length@pD> "J1.txt";
H??used here is just a book keeping parameter
it doesnt take any value during execution of the code ?L
H? ni := near?identity transformation ?L
nt30 =8h1, h2<;
nt31 =? 8h110 v1+h101 v2, h210 v1+h201 v2<;
nt32 =? 8h120 v1^2+h111 v1 v2+h102 v2^2, h220 v1^2+h211 v1 v2+h202 v2^2<;
nt33 =? 8h130 v1^3+h121 v1^2 v2+h112 v1 v2^2+h103 v2^3,
h230 v1^3+h221 v1^2 v2+h212 v1 v2^2+h203 v2^3<;
nt =8v1, v2<+nt30+nt31+nt32+nt33;
fv = Expand@
? 8fz@@1DD?. 8z1 ? nt@@1DD,z2? nt@@2DD<,fz@@2DD?. 8z1 ? nt@@1DD,z2? nt@@2DD<> "fv30.txt";
uv >> "uv.txt";
H? ????????? Use routine "Harmonic solution ?.nb" to find nt30 ???????????? ?L
H? ???????????????????? Finding nt31 ?????????????????????????? ?L
v1nt30 = TrigToExp@ReadList@"v1nt30.txt"D@@1DDD;
hh =??3;
fv31 = Expand@8coeffv1@@2, 3, 1, 2, 1DD h1^2 v1 +coeffv1@@2, 2, 2, 2, 1DD h1 h2 v1+
coeffv1@@2, 1, 3, 2, 1DD h2^2 v1+coeffv1@@2, 3, 1, 1, 2DD h1^2 v2 +
coeffv1@@2, 2, 2, 1, 2DD h1 h2 v2+coeffv1@@2, 1, 3, 1, 2DD h2^2 v2,
coeffv2@@2, 3, 1, 2, 1DD h1^2 v1 +coeffv2@@2, 2, 2, 2, 1DD h1 h2 v1+
coeffv2@@2, 1, 3, 2, 1DD h2^2 v1+coeffv2@@2, 3, 1, 1, 2DD h1^2 v2 +
coeffv2@@2, 2, 2, 1, 2DD h1 h2 v2+coeffv2@@2, 1, 3, 1, 2DD h2^2 v2> "lnresterm.txt";
H? ??????????????????? Finding nt32 ????????????????????????? ?L
fv32 = Expand@8coeffv1@@2, 2, 1, 3, 1DDh1 v1^2+coeffv1@@2, 2, 1, 2, 2DDh1 v1 v2+
coeffv1@@2, 2, 1, 1, 3DDh1 v2^2+coeffv1@@2, 1, 2, 3, 1DDh2 v1^2+
coeffv1@@2, 1, 2, 2, 2DDh2 v1 v2+coeffv1@@2, 1, 2, 1, 3DDh2 v2^2,
coeffv2@@2, 2, 1, 3, 1DDh1 v1^2+coeffv2@@2, 2, 1, 2, 2DDh1 v1 v2+
coeffv2@@2, 2, 1, 1, 3DDh1 v2^2+coeffv2@@2, 1, 2, 3, 1DDh2 v1^2+
coeffv2@@2, 1, 2, 2, 2DDh2 v1 v2+coeffv2@@2, 1, 2, 1, 3DDh2 v2^2> "v1nt.txt"
H? v1nt30.txt is generated by Harmonic solution ?.nb ?L
Chop@v1nt31, 10^?3D>> "v1nt31.txt"
Chop@v1nt32, 10^?3D>> "v1nt32.txt"
Chop@v1nt33, 10^?3D>> "v1nt33.txt"
152
C .4
Clear@"Global`?"D
H? File name "Step 2 Harmonic solution.nb" ?L
H? This routine finds harmonic balance solution to 0th order equation,
No Resonance case ?L
SetDirectory@NotebookDirectory@DD;
J1 = ReadList@"J1.txt"D@@1DD;
fv30 = ReadList@"fv30.txt"D@@1DD;
uv = ReadList@"uv.txt"D@@1DD;
?=ReadList@"omega.txt"D@@1DD;
hh = 12;
h0 =8a1+Sum@c1@iD Cos@i?tD+s1@iD Sin@i?tD, 8i, 1, hh> "v1nt30.txt";
2 Step 2 Harmonic solution.nb
154
C .5
H? File name "Step 3 fplot.nb" ?L
H? This function finds system response using the normal
form equation and compares with original system response ?L
Clear@"Global`?"D
SetDirectory@NotebookDirectory@DD;
R1 = ReadList@"R1.txt"D@@1DD;fx= ReadList@"fx.txt"D@@1DD;
ux = ReadList@"ux.txt"D@@1DD;Q1= ReadList@"Q1.txt"D@@1DD;
invQ1 = ReadList@"invQ1.txt"D@@1DD;M1= ReadList@"M1.txt"D@@1DD;
v1nt = ReadList@"v1nt.txt"D@@1DD; lnresterm = ReadList@"lnresterm.txt"D@@1DD;
invM1 = Inverse@M1D;
J1 = Chop@invM1.R1.M1D;
J2 = J1+lnresterm;
ti = 0;
tf = 10;
s0 =80.1, 0.1<;
sol = NDSolve@8s1'@tDnull J2@@1, 1DD s1@tD+J2@@1, 2DD s2@tD,
s2'@tDnull J2@@2, 1DD s1@tD+J2@@2, 2DD s2@tD,
s1@0Dnull s0@@1DD,s2@0Dnull s0@@2DD<, 8s1, s2<, 8t, ti, tf8Red, Green8Red, Green> "v1im.txt"
4 Step 1 Initialize.nb
159
C .7
H? File name "Step 1 Initialize.nb" ?L
Clear@"Global`?"D
SetDirectory@NotebookDirectory@DD;
R1 = ReadList@"R1.txt"D@@1DD;Q1= ReadList@"Q1.txt"D@@1DD;
invQ1 = ReadList@"invQ1.txt"D@@1DD;
?=ReadList@"omega.txt"D@@1DD;fx= ReadList@"fx.txt"D@@1DD;
ux = ReadList@"ux.txt"D@@1DD;M1= ReadList@"M1.txt"D@@1DD;
invM1 = Inverse@M1D;
J1 = Chop@invM1.R1.M1D;
z =8z1, z2, z3, z4<;
x = Q1.M1.z;
fz = Chop@Expand@
TrigToExp@invM1.invQ1.fx ?. 8x1 ? x@@1DD,x2? x@@2DD,x3? x@@3DD,x4? x@@4DD> "q.txt";
H? Construction of invariant manifold ?L
H? im :? Invariant Manifold ?L
im30 =8h1, h2<;
im31 =? 8h110 z1+h101 z2, h210 z1+h201 z2<;
im32 =? 8h120 z1^2+h111 z1 z2+h102 z2^2, h220 z1^2+h211 z1 z2+h202 z2^2<;
im33 =? 8h130 z1^3+h121 z1^2 z2+h112 z1 z2^2+h103 z2^3,
h230 z1^3+h221 z1^2 z2+h212 z1 z2^2+h203 z2^3<;
im = im30+im31+im32+im33;
fzq = Expand@
? 8fz@@3DD?. 8z3 ? im@@1DD,z4? im@@2DD<,fz@@4DD?. 8z3 ? im@@1DD,z4? im@@2DD<> "Jq.txt";
fzq30 >> "fzq30.txt";
uzq >> "uzq.txt";
H? ???????? Use routine "Harmonic solution ? .nb" to find im30 ??????????? ?L
H? ???????????????????? Finding im31 ?????????????????????????? ?L
v1im30 = TrigToExp@ReadList@"v1im30.txt"D@@1DDD;
fzq31 = Expand@8coeffzq1@@2, 3, 1, 2, 1DD h1^2 z1 +coeffzq1@@2, 2, 2, 2, 1DD h1 h2 z1+
coeffzq1@@2, 1, 3, 2, 1DD h2^2 z1+coeffzq1@@2, 3, 1, 1, 2DD h1^2 z2 +
coeffzq1@@2, 2, 2, 1, 2DD h1 h2 z2+coeffzq1@@2, 1, 3, 1, 2DD h2^2 z2,
coeffzq2@@2, 3, 1, 2, 1DD h1^2 z1 +coeffzq2@@2, 2, 2, 2, 1DD h1 h2 z1+
coeffzq2@@2, 1, 3, 2, 1DD h2^2 z1+coeffzq2@@2, 3, 1, 1, 2DD h1^2 z2 +
coeffzq2@@2, 2, 2, 1, 2DD h1 h2 z2+coeffzq2@@2, 1, 3, 1, 2DD h2^2 z2> "v1im.txt"
H? v1im30.txt is generated by Harmonic solution ?.nb ?L
Chop@v1im31, 10^?3D>> "v1im31.txt"
Chop@v1im32, 10^?3D>> "v1im32.txt"
Chop@v1im33, 10^?3D>> "v1im33.txt"
Step 1 Initialize.nb 7
166
C .8
H? File name "Step 3 fplot.nb" ?L
Clear@"Global`?"D
SetDirectory@NotebookDirectory@DD;
H? ???????????????? fft data ??????????????????????????? ?L
?=ReadList@"omega.txt"D@@1DD;
?
max
= 7?;
?
min
=?;
H?tf
fft
=N@ti+2???
min
D; ?L
tint = N@H2????
min
L?Ceiling@N@?
max
?PiD?H2????
min
LDD;
H? ???????????????????????????????????????????????????? ?L
SetDirectory@NotebookDirectory@DD;
R1 = ReadList@"R1.txt"D@@1DD;fx= ReadList@"fx.txt"D@@1DD;
ux = ReadList@"ux.txt"D@@1DD;Q1= ReadList@"Q1.txt"D@@1DD;
invQ1 = ReadList@"invQ1.txt"D@@1DD;
M1 = ReadList@"M1.txt"D@@1DD; v1im = ReadList@"v1im.txt"D@@1DD;
invM1 = Inverse@M1D;
J1 = Chop@invM1.R1.M1D;
z =8z1, z2, z3, z4<;
x = Q1.M1.z;
fz = Chop@Expand@
TrigToExp@invM1.invQ1.fx ?. 8x1 ? x@@1DD,x2? x@@2DD,x3? x@@3DD,x4? x@@4DD v1im@@2DD8Red, Green, 8Red, Thick<, 8Green, Thick<8Red, Green, 8Red, Thick<, 8Green, Thick<> bx.txt;
ux = k1 x1+k2 x2+k3 x3+k4 x4+k5 x5+k6 x6+k7 x7+k8 x8;
invM = Inverse@MD;
J = Chop@invM.R.M, E^?6D;
Jp =88J@@1, 1DD,J@@1, 2DD<, 8J@@2, 1DD,J@@2, 2DD<<;
Jq =88J@@3, 3DD,J@@3, 4DD,0,0,0,0<, 8J@@4, 3DD,J@@4, 4DD,0,0,0,0<,
80, 0, J@@5, 5DD,J@@5, 6DD,0,0<, 80, 0, J@@6, 5DD,J@@6, 6DD,0,0<,
80, 0, 0, 0, J@@7, 7DD,J@@7, 8DD<, 80, 0, 0, 0, J@@8, 7DD,J@@8, 8DD<<;
z =8z1, z2, z3, z4, z5, z6, z7, z8<;
x = Q.M.z;
bz = Chop@Expand@TrigToExp@invM.invQ.bxDDD;
g0p =8bz@@1DD,bz@@2DD<;
g0q =8bz@@3DD,bz@@4DD,bz@@5DD,bz@@6DD,bz@@7DD,bz@@8DD<;
uz = Expand@TrigToExp@ux ?. 8x1 ? x@@1DD,x2? x@@2DD,x3? x@@3DD,
x4 ? x@@4DD,x5? x@@5DD,x6? x@@6DD,x7? x@@7DD,x8? x@@8DD>
Kr.txt;
H? Find system nonlinearities using "Nonlinear parameter selection.nb" ?L
fx = ReadList@"fx.txt"D@@1DD;
H?nonlinear term in z?L
fz = Chop@
Expand@TrigToExp@invM.invQ.fx ?. 8x1 ? x@@1DD,x2? x@@2DD,x3? x@@3DD,x4? x@@4DD,
x5 ? x@@5DD,x6? x@@6DD,x7? x@@7DD,x8? x@@8DD> "ft.txt";
H? ???????????????????????????????????????????????????????????? ?L
H? Find P using FindingP.nb ?L
P = TrigToExp@ReadList@"P.txt"D@@1DDD;
H? ???????????????????????????????????????????????????????????? ?L
zs = Expand@8v3, v4, v5, v6, v7, v8<+P.8z1, z2 coef?3@@1DD, ?c121@0D?> coef?3@@2DD,
?c112@0D?> coef?3@@3DD, ?c103@0D?> coef?3@@4DD, ?c230@0D?> coef?3@@5DD,
?c221@0D?> coef?3@@6DD, ?c212@0D?> coef?3@@7DD, ?c203@0D?> coef?3@@8DD<,
Table@8?c130@jD?> coef?3@@8 H2 j?1L+1DD, ?s130@jD?> coef?3@@8 H2 j?1L+2DD,
?c121@jD?> coef?3@@8 H2 j?1L+3DD, ?s121@jD?> coef?3@@8 H2 j?1L+4DD,
?c112@jD?> coef?3@@8 H2 j?1L+5DD, ?s112@jD?> coef?3@@8 H2 j?1L+6DD,
?c103@jD?> coef?3@@8 H2 j?1L+7DD, ?s103@jD?> coef?3@@8 H2 j?1L+8DD,
4 Step 1 Initialize.nb
174
?c230@jD?> coef?3@@8 H2 j?1L+9DD, ?s230@jD?> coef?3@@8 H2 j?1L+10DD,
?c221@jD?> coef?3@@8 H2 j?1L+11DD,
?s221@jD?> coef?3@@8 H2 j?1L+12DD, ?c212@jD?> coef?3@@8 H2 j?1L+13DD,
?s212@jD?> coef?3@@8 H2 j?1L+14DD, ?c203@jD?> coef?3@@8 H2 j?1L+15DD,
?s203@jD?> coef?3@@8 H2 j?1L+16DD<, 8j, 1, ft coef?3?2@@1DD, ?c21@0D?> coef?3?2@@2DD, ?c12@0D?> coef?3?2@@3DD,
?c03@0D?> coef?3?2@@4DD<, Table@8?c30@jD?> coef?3?2@@7 H2 j?1L+1DD,
?s30@jD?> coef?3?2@@7 H2 j?1L+2DD, ?c21@jD?> coef?3?2@@7 H2 j?1L+3DD,
?s21@jD?> coef?3?2@@7 H2 j?1L+4DD, ?c12@jD?> coef?3?2@@7 H2 j?1L+5DD,
?s12@jD?> coef?3?2@@7 H2 j?1L+6DD, ?c03@jD?> coef?3?2@@7 H2 j?1L+7DD,
?s03@jD?> coef?3?2@@7 H2 j?1L+8DD<, 8j, 1, ft coef?3?2@@5DD, ?c11@0D?> coef?3?2@@6DD, ?c02@0D?> coef?3?2@@7DD<,
Table@8?c20@jD?> coef?3?2@@7 H2 j?1L+9DD, ?s20@jD?> coef?3?2@@7 H2 j?1L+10DD,
?c11@jD?> coef?3?2@@7 H2 j?1L+11DD,
?s11@jD?> coef?3?2@@7 H2 j?1L+12DD, ?c02@jD?> coef?3?2@@7 H2 j?1L+13DD,
?s02@jD?> coef?3?2@@7 H2 j?1L+14DD<, 8j, 1, ft coef?3?2@@1DD, ?c21@0D?> coef?3?2@@2DD, ?c12@0D?> coef?3?2@@3DD,
?c03@0D?> coef?3?2@@4DD<, Table@8?c30@jD?> coef?3?2@@7 H2 j?1L+1DD,
?s30@jD?> coef?3?2@@7 H2 j?1L+2DD, ?c21@jD?> coef?3?2@@7 H2 j?1L+3DD,
?s21@jD?> coef?3?2@@7 H2 j?1L+4DD, ?c12@jD?> coef?3?2@@7 H2 j?1L+5DD,
?s12@jD?> coef?3?2@@7 H2 j?1L+6DD, ?c03@jD?> coef?3?2@@7 H2 j?1L+7DD,
?s03@jD?> coef?3?2@@7 H2 j?1L+8DD<, 8j, 1, ft> "alpha3.txt";
v1?2 >> "beta2.txt";
v2?3 >> "phi3.txt";
6 Step 1 Initialize.nb
176
H? ???????????????? Construction of Invariant Manifold ???????????????? ?L
v1fptrunc3 =80, 0<;
H? This truncate all the terms of order greater than 3 from v1fp ?L
v1fptrunc3@@1DD=
Chop@Expand@coefLv1fp@@1DD@@4, 1DD z1^3+coefLv1fp@@1DD@@3, 2DD z1^2 z2+
coefLv1fp@@1DD@@2, 3DD z1 z2^2+coefLv1fp@@1DD@@1, 4DD z2^3DD;
v1fptrunc3@@2DD= Chop@Expand@coefLv1fp@@2DD@@4, 1DD z1^3+coefLv1fp@@2DD@@3, 2DD
z1^2 z2+coefLv1fp@@2DD@@2, 3DD z1 z2^2+coefLv1fp@@2DD@@1, 4DD z2^3DD;
Pdotv1fptrunc3 = Chop@Expand@P.v1fptrunc3DD;
v1fq = Chop@Expand@8fq@@1DD?Pdotv1fptrunc3@@1DD,fq@@2DD?Pdotv1fptrunc3@@2DD,
fq@@3DD?Pdotv1fptrunc3@@3DD,fq@@4DD?Pdotv1fptrunc3@@4DD,
fq@@5DD?Pdotv1fptrunc3@@5DD,fq@@6DD?Pdotv1fptrunc3@@6DD v2im@@1DD,v4?> v2im@@2DD,
v5 ?> v2im@@3DD,v6?> v2im@@4DD,v7?> v2im@@5DD,v8?> v2im@@6DD y2+v2?3@@2DD y2 y2+v2?3@@2DD y2 y2 coef?5@@1DD, ?c141@0D?> coef?5@@2DD, ?c132@0D?> coef?5@@3DD,
?c123@0D?> coef?5@@4DD, ?c114@0D?> coef?5@@5DD, ?c105@0D?> coef?5@@6DD,
?c250@0D?> coef?5@@7DD, ?c241@0D?> coef?5@@8DD, ?c232@0D?> coef?5@@9DD,
?c223@0D?> coef?5@@10DD, ?c214@0D?> coef?5@@11DD, ?c205@0D?> coef?5@@12DD<,
Table@8?c150@jD?> coef?5@@12 H2 j?1L+1DD, ?s150@jD?> coef?5@@12 H2 j?1L+2DD,
?c141@jD?> coef?5@@12 H2 j?1L+3DD, ?s141@jD?> coef?5@@12 H2 j?1L+4DD,
?c132@jD?> coef?5@@12 H2 j?1L+5DD, ?s132@jD?> coef?5@@12 H2 j?1L+6DD,
?c123@jD?> coef?5@@12 H2 j?1L+7DD, ?s123@jD?> coef?5@@12 H2 j?1L+8DD,
?c114@jD?> coef?5@@12 H2 j?1L+9DD, ?s114@jD?> coef?5@@12 H2 j?1L+10DD,
?c105@jD?> coef?5@@12 H2 j?1L+11DD,
?s105@jD?> coef?5@@12 H2 j?1L+12DD, ?c250@jD?> coef?5@@12 H2 j?1L+13DD,
?s250@jD?> coef?5@@12 H2 j?1L+14DD, ?c241@jD?> coef?5@@12 H2 j?1L+15DD,
?s241@jD?> coef?5@@12 H2 j?1L+16DD, ?c232@jD?> coef?5@@12 H2 j?1L+17DD,
?s232@jD?> coef?5@@12 H2 j?1L+18DD, ?c223@jD?> coef?5@@12 H2 j?1L+19DD,
?s223@jD?> coef?5@@12 H2 j?1L+20DD, ?c214@jD?> coef?5@@12 H2 j?1L+21DD,
?s214@jD?> coef?5@@12 H2 j?1L+22DD, ?c205@jD?> coef?5@@12 H2 j?1L+23DD,
?s205@jD?> coef?5@@12 H2 j?1L+24DD<, 8j, 1, ft y2+v2?3@@2DD coef?5?4@@1DD, ?c41@0D?> coef?5?4@@2DD, ?c32@0D?> coef?5?4@@3DD,
?c23@0D?> coef?5?4@@4DD, ?c14@0D?> coef?5?4@@5DD, ?c05@0D?> coef?5?4@@6DD<,
Table@8?c50@jD?> coef?5?4@@11 H2 j?1L+1DD, ?s50@jD?>
coef?5?4@@11 H2 j?1L+2DD, ?c41@jD?> coef?5?4@@11 H2 j?1L+3DD,
?s41@jD?> coef?5?4@@11 H2 j?1L+4DD, ?c32@jD?> coef?5?4@@11 H2 j?1L+5DD,
?s32@jD?> coef?5?4@@11 H2 j?1L+6DD, ?c23@jD?> coef?5?4@@11 H2 j?1L+7DD,
?s23@jD?> coef?5?4@@11 H2 j?1L+8DD, ?c14@jD?> coef?5?4@@11 H2 j?1L+9DD,
?s14@jD?> coef?5?4@@11 H2 j?1L+10DD, ?c05@jD?> coef?5?4@@11 H2 j?1L+11DD,
?s05@jD?> coef?5?4@@11 H2 j?1L+12DD<, 8j, 1, ft coef?5?4@@7DD, ?c31@0D?> coef?5?4@@8DD, ?c22@0D?>
coef?5?4@@9DD, ?c13@0D?> coef?5?4@@10DD, ?c04@0D?> coef?5?4@@11DD<, Table@
8?c40@jD?> coef?5?4@@11 H2 j?1L+13DD, ?s40@jD?> coef?5?4@@11 H2 j?1L+14DD,
?c31@jD?> coef?5?4@@11 H2 j?1L+15DD,
?s31@jD?> coef?5?4@@11 H2 j?1L+16DD, ?c22@jD?> coef?5?4@@11 H2 j?1L+17DD,
?s22@jD?> coef?5?4@@11 H2 j?1L+18DD, ?c13@jD?> coef?5?4@@11 H2 j?1L+19DD,
?s13@jD?> coef?5?4@@11 H2 j?1L+20DD, ?c04@jD?> coef?5?4@@11 H2 j?1L+21DD,
?s04@jD?> coef?5?4@@11 H2 j?1L+22DD<, 8j, 1, ft coef?5?4@@1DD, ?c41@0D?> coef?5?4@@2DD, ?c32@0D?> coef?5?4@@3DD,
?c23@0D?> coef?5?4@@4DD, ?c14@0D?> coef?5?4@@5DD, ?c05@0D?> coef?5?4@@6DD<,
Table@8?c50@jD?> coef?5?4@@11 H2 j?1L+1DD, ?s50@jD?>
coef?5?4@@11 H2 j?1L+2DD, ?c41@jD?> coef?5?4@@11 H2 j?1L+3DD,
?s41@jD?> coef?5?4@@11 H2 j?1L+4DD, ?c32@jD?> coef?5?4@@11 H2 j?1L+5DD,
?s32@jD?> coef?5?4@@11 H2 j?1L+6DD, ?c23@jD?> coef?5?4@@11 H2 j?1L+7DD,
?s23@jD?> coef?5?4@@11 H2 j?1L+8DD, ?c14@jD?> coef?5?4@@11 H2 j?1L+9DD,
?s14@jD?> coef?5?4@@11 H2 j?1L+10DD, ?c05@jD?> coef?5?4@@11 H2 j?1L+11DD,
?s05@jD?> coef?5?4@@11 H2 j?1L+12DD<, 8j, 1, ft> "alpha5.txt";
v1?4 >> "beta4.txt";
v2?5 >> "phi5.txt";
Step 1 Initialize.nb 17
187
C .11
H? File name "Find_Cheb. Coefficients.nb" ?L
Module@8m = 20, at =?1.0453752591269074`k1+0.5277492082264343`k5+
1.2572916861075216`k1 Cos@2? tD?0.03386552195838752`k5 Cos@2? tD?
0.20020462591781812`k1 Cos@4? tD?0.013645923171801314`k5 Cos@4? tD+
0.009442291809787024`k1 Cos@6? tD+0.0025080443350063166`k5 Cos@6? tD?
3.9498983953756985`k5 Sin@2? tD+1.2579246931574315`k5 Sin@4? tD?
0.08899154020493226`k5 Sin@6? tD,w=Ht?t^2L^H?0.5L<,
p11 = Table@0, 8m> "Rbarp.txt";
Qbarpft >> "Qbarp.txt";
invQbarpft >> "invQbarp.txt";
Mbarp >> "Mbarp.txt";
invMbarp >> "invMbarp.txt";
4 Symbolic FTM.nb
193
C .13
H? File name "Functions symbolic FTM.nb" ?L
fQp@p_D := Module@8i, j<,
MQp := Array@Qp, 8Length@pD, Length@pD> P.txt;
Finding P.nb 7
203
C .16
H? File name "Step 2 Inverse transformation.nb" ?L
Clear@"Global`?"D
H? ????????????????? Polynomial inverse of 3 rd order terms ????????????????? ?L
SetDirectory@NotebookDirectory@DD;
v2?3 = ReadList@"phi3.txt"D@@1DD;ft= ReadList@"ft.txt"D@@1DD;
f3 = TrigToExp@8Sum@Hfc130@iD Cos@i? tD+ fs130@iD Sin@i? tDL z1^3+
Hfc121@iD Cos@i? tD+ fs121@iD Sin@i? tDL z1^2 z2+
Hfc112@iD Cos@i? tD+ fs112@iD Sin@i? tDL z1 z2^2+
Hfc103@iD Cos@i? tD+ fs103@iD Sin@i? tDL z2^3, 8i, 0, ft coeff3@@1DD, fc121@0D?> coeff3@@2DD, fc112@0D?> coeff3@@3DD,
fc103@0D?> coeff3@@4DD, fc230@0D?> coeff3@@5DD, fc221@0D?> coeff3@@6DD,
fc212@0D?> coeff3@@7DD, fc203@0D?> coeff3@@8DD<,
Table@8fc130@jD?> coeff3@@8 H2 j?1L+1DD, fc121@jD?> coeff3@@8 H2 j?1L+2DD,
fc112@jD?> coeff3@@8 H2 j?1L+3DD, fc103@jD?> coeff3@@8 H2 j?1L+4DD,
fc230@jD?> coeff3@@8 H2 j?1L+5DD, fc221@jD?> coeff3@@8 H2 j?1L+6DD,
fc212@jD?> coeff3@@8 H2 j?1L+7DD, fc203@jD?> coeff3@@8 H2 j?1L+8DD,
fs130@jD?> coeff3@@8 H2 j?1L+9DD, fs121@jD?> coeff3@@8 H2 j?1L+10DD,
fs112@jD?> coeff3@@8 H2 j?1L+11DD,
fs103@jD?> coeff3@@8 H2 j?1L+12DD, fs230@jD?> coeff3@@8 H2 j?1L+13DD,
fs221@jD?> coeff3@@8 H2 j?1L+14DD, fs212@jD?> coeff3@@8 H2 j?1L+15DD,
fs203@jD?> coeff3@@8 H2 j?1L+16DD<, 8j, 1, ft> "v1f3.txt";
H? ??????????????????????????????? Polynomial
inverse of 5th order terms ???????????????????? ?L
SetDirectory@NotebookDirectory@DD;
v2?5 = ReadList@"phi5.txt"D@@1DD;
f5 = TrigToExp@8Sum@Hfc150@iD Cos@i? tD+ fs150@iD Sin@i? tDL z1^5+
Hfc141@iD Cos@i? tD+ fs141@iD Sin@i? tDL z1^4 z2+
Hfc132@iD Cos@i? tD+ fs132@iD Sin@i? tDL z1^3 z2^2+
Hfc123@iD Cos@i? tD+ fs123@iD Sin@i? tDL z1^2 z2^3+
Hfc114@iD Cos@i? tD+ fs114@iD Sin@i? tDL z1 z2^4+
Hfc105@iD Cos@i? tD+ fs105@iD Sin@i? tDL z2^5, 8i, 0, ft z1+v1f3@@1DD,y2?> z2+v1f3@@2DD coeff5@@1DD, fc141@0D?> coeff5@@2DD, fc132@0D?> coeff5@@3DD,
fc123@0D?> coeff5@@4DD, fc114@0D?> coeff5@@5DD, fc105@0D?> coeff5@@6DD,
fc250@0D?> coeff5@@7DD, fc241@0D?> coeff5@@8DD, fc232@0D?> coeff5@@9DD,
fc223@0D?> coeff5@@10DD, fc214@0D?> coeff5@@11DD, fc205@0D?> coeff5@@12DD<,
Table@8fc150@jD?> coeff5@@12 H2 j?1L+1DD, fc141@jD?> coeff5@@12 H2 j?1L+2DD,
fc132@jD?> coeff5@@12 H2 j?1L+3DD, fc123@jD?> coeff5@@12 H2 j?1L+4DD,
fc114@jD?> coeff5@@12 H2 j?1L+5DD, fc105@jD?> coeff5@@12 H2 j?1L+6DD,
fc250@jD?> coeff5@@12 H2 j?1L+7DD, fc241@jD?> coeff5@@12 H2 j?1L+8DD,
fc232@jD?> coeff5@@12 H2 j?1L+9DD, fc223@jD?> coeff5@@12 H2 j?1L+10DD,
fc214@jD?> coeff5@@12 H2 j?1L+11DD,
fc205@jD?> coeff5@@12 H2 j?1L+12DD, fs150@jD?> coeff5@@12 H2 j?1L+13DD,
fs141@jD?> coeff5@@12 H2 j?1L+14DD, fs132@jD?> coeff5@@12 H2 j?1L+15DD,
fs123@jD?> coeff5@@12 H2 j?1L+16DD, fs114@jD?> coeff5@@12 H2 j?1L+17DD,
fs105@jD?> coeff5@@12 H2 j?1L+18DD, fs250@jD?> coeff5@@12 H2 j?1L+19DD,
fs241@jD?> coeff5@@12 H2 j?1L+20DD, fs232@jD?> coeff5@@12 H2 j?1L+21DD,
fs223@jD?> coeff5@@12 H2 j?1L+22DD, fs214@jD?> coeff5@@12 H2 j?1L+23DD,
fs205@jD?> coeff5@@12 H2 j?1L+24DD<, 8j, 1, ft> "v1f5.txt";
4 Step 2 Inverse transformation.nb
207
C .17
H? File name "Step 3 fplot.nb" ?L
SetDirectory@NotebookDirectory@DD;
A = ReadList@"A.txt"D@@1DD;
R = ReadList@"R.txt"D@@1DD;Q= ReadList@"Q.txt"D@@1DD;
invQ = ReadList@"invQ.txt"D@@1DD;
fx = ReadList@"fx.txt"D@@1DD;bx= ReadList@"bx.txt"D@@1DD;
M = ReadList@"M.txt"D@@1DD;Kr= ReadList@"Kr.txt"D@@1DD;
v1?3 = Chop@ReadList@"alpha3.txt"D@@1DD,10^?2D;
v1?5 = Chop@ReadList@"alpha5.txt"D@@1DD,10^?2D;
v1?2 = Chop@ReadList@"beta2.txt"D@@1DD,10^?2D;
v1?4 = Chop@ReadList@"beta4.txt"D@@1DD,10^?2D;
v1f3 = Chop@ReadList@"v1f3.txt"D@@1DD,10^?2D;
v1f5 = Chop@ReadList@"v1f5.txt"D@@1DD,10^?2D;
invM = Inverse@MD;
x =8x1, x2, x3, x4, x5, x6, x7, x8<;
z = Chop@Expand@TrigToExp@invM.invQ.xDDD;
uxln = Chop@ExpToTrig@Expand@HKr@@1DD z1+Kr@@2DD z2L?. 8z1 ? z@@1DD,z2? z@@2DD> "uxln.txt";
uxnln5 = Chop@ExpToTrig@
Expand@HKr@@1DD Hz1+v1f3@@1DD +v1f5@@1DDL+Kr@@2DD Hz2+v1f3@@2DD+v1f5@@2DDL+
v1?3+v1?5+v1?2 HKr@@1DD Hz1+v1f3@@1DDL+Kr@@2DD Hz2+v1f3@@2DDLL+
v1?4 HKr@@1DD z1+Kr@@2DD z2LL?. 8z1 ? z@@1DD,z2? z@@2DD> "uxnln5.txt";
uxnln3 =
Chop@ExpToTrig@Expand@HKr@@1DD Hz1+v1f3@@1DDL+Kr@@2DD Hz2+v1f3@@2DDL+v1?3+
v1?2 HKr@@1DD z1+Kr@@2DD z2LL?. 8z1 ? z@@1DD,z2? z@@2DD> "uxnln3.txt";
208
SetDirectory@NotebookDirectory@DD;
A = ReadList@"A.txt"D@@1DD;
R = ReadList@"R.txt"D@@1DD;Q= ReadList@"Q.txt"D@@1DD;
invQ = ReadList@"invQ.txt"D@@1DD;
fx = ReadList@"fx.txt"D@@1DD;bx= ReadList@"bx.txt"D@@1DD;
M = ReadList@"M.txt"D@@1DD;Kr= ReadList@"Kr.txt"D@@1DD;
v1fx = fx ?. 8x1 ? l1@tD,x2? l2@tD,x3? l3@tD,
x4 ? l4@tD,x5? l5@tD,x6? l6@tD,x7? l7@tD,x8? l8@tD<;
ti = 0;
tf = 18;
l0 =80.21, 0.2, 0.2, 0.2, 0.3, 0.2, 0.2, 0.2<;
uxln = ReadList@"uxln.txt"D@@1DD;
v1uxln = uxln ?. 8x1 ? l1@tD,x2? l2@tD,x3? l3@tD,
x4 ? l4@tD,x5? l5@tD,x6? l6@tD,x7? l7@tD,x8? l8@tD<;
H? ?????????? System response with linear controller ??????????? ?L
sol2 =
NDSolve@8l1'@tD null A@@1, 1DD l1@tD+A@@1, 2DD l2@tD+A@@1, 3DD l3@tD+A@@1, 4DD l4@tD+
A@@1, 5DD l5@tD+A@@1, 6DD l6@tD+A@@1, 7DD l7@tD+
A@@1, 8DD l8@tD+v1fx@@1DD+bx@@1DD v1uxln, l2'@tDnull
A@@2, 1DD l1@tD+A@@2, 2DD l2@tD+A@@2, 3DD l3@tD+A@@2, 4DD l4@tD+A@@2, 5DD l5@tD+
A@@2, 6DD l6@tD+A@@2, 7DD l7@tD+A@@2, 8DD l8@tD+v1fx@@2DD+bx@@2DD v1uxln,
l3'@tDnull A@@3, 1DD l1@tD+A@@3, 2DD l2@tD+A@@3, 3DD l3@tD+
A@@3, 4DD l4@tD+A@@3, 5DD l5@tD+A@@3, 6DD l6@tD+A@@3, 7DD l7@tD+
A@@3, 8DD l8@tD+v1fx@@3DD+bx@@3DD v1uxln, l4'@tDnull
A@@4, 1DD l1@tD+A@@4, 2DD l2@tD+A@@4, 3DD l3@tD+A@@4, 4DD l4@tD+A@@4, 5DD l5@tD+
A@@4, 6DD l6@tD+A@@4, 7DD l7@tD+A@@4, 8DD l8@tD+v1fx@@4DD+bx@@4DD v1uxln,
l5'@tDnull A@@5, 1DD l1@tD+A@@5, 2DD l2@tD+A@@5, 3DD l3@tD+
A@@5, 4DD l4@tD+A@@5, 5DD l5@tD+A@@5, 6DD l6@tD+A@@5, 7DD l7@tD+
A@@5, 8DD l8@tD+v1fx@@5DD+bx@@5DD v1uxln, l6'@tDnull
A@@6, 1DD l1@tD+A@@6, 2DD l2@tD+A@@6, 3DD l3@tD+A@@6, 4DD l4@tD+A@@6, 5DD l5@tD+
A@@6, 6DD l6@tD+A@@6, 7DD l7@tD+A@@6, 8DD l8@tD+v1fx@@6DD+bx@@6DD v1uxln,
l7'@tDnull A@@7, 1DD l1@tD+A@@7, 2DD l2@tD+A@@7, 3DD l3@tD+
A@@7, 4DD l4@tD+A@@7, 5DD l5@tD+A@@7, 6DD l6@tD+A@@7, 7DD l7@tD+
A@@7, 8DD l8@tD+v1fx@@7DD+bx@@7DD v1uxln, l8'@tDnull
A@@8, 1DD l1@tD+A@@8, 2DD l2@tD+A@@8, 3DD l3@tD+A@@8, 4DD l4@tD+A@@8, 5DD l5@tD+
A@@8, 6DD l6@tD+A@@8, 7DD l7@tD+A@@8, 8DD l8@tD+v1fx@@8DD+bx@@8DD v1uxln,
l1@0Dnull l0@@1DD,l2@0Dnull l0@@2DD,l3@0Dnull l0@@3DD,l4@0Dnull l0@@4DD,
l5@0Dnull l0@@5DD,l6@0Dnull l0@@6DD,l7@0Dnull l0@@7DD,l8@0Dnull l0@@8DD<,
8l1, l2, l3, l4, l5, l6, l7, l8<, 8t, ti, tf