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Dynamic Stability Analysis of Linear Time-varying Systems ...
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ORIGINAL ARTICLE
Dynamic Stability Analysis of Linear Time-varying Systemsvia an Extended Modal Identification Approach
Zhisai MA1,2• Li LIU1
• Sida ZHOU1• Frank NAETS2 • Ward HEYLEN2
•
Wim DESMET2
Received: 9 June 2016 / Revised: 29 August 2016 / Accepted: 11 October 2016 / Published online: 17 March 2017
� Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2017
Abstract The problem of linear time-varying(LTV) sys-
tem modal analysis is considered based on time-dependent
state space representations, as classical modal analysis of
linear time-invariant systems and current LTV system
modal analysis under the ‘‘frozen-time’’ assumption are not
able to determine the dynamic stability of LTV systems.
Time-dependent state space representations of LTV sys-
tems are first introduced, and the corresponding modal
analysis theories are subsequently presented via a stability-
preserving state transformation. The time-varying modes of
LTV systems are extended in terms of uniqueness, and are
further interpreted to determine the system’s stability. An
extended modal identification is proposed to estimate the
time-varying modes, consisting of the estimation of the
state transition matrix via a subspace-based method and the
extraction of the time-varying modes by the QR decom-
position. The proposed approach is numerically validated
by three numerical cases, and is experimentally validated
by a coupled moving-mass simply supported beam exper-
imental case. The proposed approach is capable of accu-
rately estimating the time-varying modes, and provides a
new way to determine the dynamic stability of LTV sys-
tems by using the estimated time-varying modes.
Keywords Linear time-varying systems � Extended modal
identification � Dynamic stability analysis � Time-varying
modes
1 Introduction
Classical modal analysis in mechanical and aerospace
engineering works under the time invariance assumption
that the structure dynamic characteristics do not change
with time [1]. Hence, the underlying system can be mod-
eled by a linear time-invariant(LTI) model. However,
many systems in the real world are time-varying and their
intrinsic time-varying behavior is increasingly inevitable.
Typical examples include vibration absorbers with variable
stiffness [2], bridges with crossing vehicles [3], launch
vehicles with varying fuel mass [4], airplanes with varying
additional aerodynamic effects in flight [5], manipulators
with deployable joints and flexible links [6], deployable
space structures [7], rotating machinery [8] and many
more. In order to pursue more accurate analysis, and take
advantage of the richness of information included in the
time-varying dynamics, modal analysis of linear time-
varying(LTV) systems is worth more attention and inves-
tigation [9].
Modal analysis is often used to determine the charac-
teristics of dynamical systems, while stability is one of the
most important properties of these systems. In classical
modal analysis, modal parameters including natural fre-
quencies, damping factors and mode shapes are usually
used to describe the dynamics of LTI systems. If we rep-
resent a LTI system in state space, modal parameters can
Supported by the China Scholarship Council, National Natural
Science Foundation of China(Grant No. 11402022), the Interuniversity
Attraction Poles Programme of the Belgian Science Policy Office
(DYSCO), the Fund for Scientific Research – Flanders(FWO), and the
Research Fund KU Leuven.
& Zhisai MA
1 School of Aerospace Engineering, Beijing Institute of
Technology, 100081 Beijing, China
2 Department of Mechanical Engineering, KU Leuven,
3001 Leuven, Belgium
123
Chin. J. Mech. Eng. (2017) 30:459–471
DOI 10.1007/s10033-017-0075-7
be extracted from the corresponding system matrix A. It is
well known that a LTI system is stable if and only if all
eigenvalues of A have negative real parts. However, this is
no longer true for time-varying systems. A LTV system can
be unstable even if all eigenvalues of its system matrix AðtÞare constant and have negative real parts, and the system
can also be asymptotically stable even if all eigenvalues of
AðtÞ are constant and some have positive real parts [10]. In
other words, the instantaneous modal parameters obtained
at each instant of time under the ‘‘frozen-time’’ assumption
(the ‘‘frozen-time’’ assumption consists in modeling a LTV
system as a piecewise LTI system [11, 12]) cannot be
directly used to determine the stability of LTV systems.
Therefore, the ‘‘frozen-time’’ assumption-based modal
analysis needs to be extended before being applied to LTV
systems.
In the past decades, many efforts have been spent in
extending modal analysis of LTI systems to LTV cases,
and several notions of poles or ‘‘eigenvalues’’ of LTV
systems have been proposed. WU [13] introduced a new
concept of ‘‘eigenvalues’’ and ‘‘eigenvectors’’ of the time-
varying system matrix AðtÞ. KAMEN [14] proposed a
notion of poles of LTV systems by using factorizations of
operator polynomials with time-varying coefficients, and
stability was subsequently studied in terms of the compo-
nents of the modal decomposition. O’BRIEN and IGLE-
SIAS [15] defined poles of continuous-time LTV systems
as functions of time. The pole set was obtained through a
stability-preserving state transformation relating a time-
varying state equation to a diagonal state equation. Then
they extended this definition by converting a given state
equation to an upper triangular state equation via a Lya-
punov state transformation [16]. ZENGER and YLINEN
[17] demonstrated that all Lyapunov transformations can
be used to define the pole set, and presented a new way to
calculate poles of LTV systems starting from a canonical
state space representation.
On the one hand, the above definitions are shown to be
generalizations of existing definitions of poles of LTI
systems. On the other hand, poles of LTV systems cannot
be used to determine the system’s stability, as they do not
share the same physical meaning as their LTI counterparts
any more. Fortunately, the time-varying modes (the time-
varying modes are no more the vibration modes or the
eigenvectors of the system. The definition is first given by
the Ref. [16] and further discussed in section 3 of this
paper) contain the information regarding the stability of
LTV systems. Therefore, modal analysis of LTV systems
presented in this paper focuses on the time-varying modes
instead of the classical modal parameters.
It should be further stressed that the calculation of the
time-varying modes requires explicit knowledge of the
equations which describe the system’s time-varying
dynamics. However, it is not easy to build the explicit
dynamic model of an arbitrarily LTV system. There is also
no guarantee that the model will accurately produce results
that are consistent with experimental measurements.
Hence, the time-varying modes of arbitrarily LTV systems
usually cannot be theoretically solved due to the lack of the
explicit form of the state transition matrix. For this reason,
data-based LTV system identification, in a way that takes
time variation explicitly into account, is receiving renewed
attention.
LTV system identification methods are often classified
under the umbrella of time–frequency methods and further
classified as non-parametric or parametric depending on
the type of model adopted [9]. Most current identification
methods in the time domain employ time-dependent
parametric models, mainly of the auto-regressive moving
average [18–20] or state space types. Time-dependent state
space model-based identification methods are here con-
sidered in order to estimate the state transition matrix of
LTV systems. VERHAEGEN and YU [21] identified LTV
systems in a subspace model identification framework
making use of an ensemble of input–output data. LIU
[22–24] defined the pseudomodal parameters of LTV sys-
tems and identified them via subspace-based methods by
using the ensemble data. SHOKOOHI, et al [25], and
MAJJI, et al [26, 27], extended the eigensystem realization
algorithm from LTI systems to LTV cases in discrete-time
domain by using established notions of the generalized
Markov parameters and the generalized Hankel matrix
sequences. BELLINO, et al [28], defined a short-time
stochastic subspace identification approach under the
‘‘frozen-time’’ assumption and applied this approach to a
pendulum undergoing large swinging amplitudes. JHI-
NAOUI, et al [29], proposed a new subspace-based algo-
rithm which can extract the modal parameters of linear
periodically time-varying systems and preserve the stabil-
ity information of the original system. Due to the lack of a
consistent theoretical background, most LTV system
identification methods mentioned above are developed
under the ‘‘frozen-time’’ assumption. The motivation of
this paper is to suggest an extended modal identification
approach to estimate the time-varying modes of arbitrarily
LTV systems without the ‘‘frozen-time’’ assumption.
The remainder of the paper is organized as follows:
section 2 presents the state space representation of LTV
systems. In section 3, classical modal analysis of LTI
systems and current modal analysis of LTV systems under
the ‘‘frozen-time’’ assumption are first reviewed. Modal
analysis of LTV systems is subsequently introduced
including the definitions of the time-varying modes and
poles, uniqueness and stability. Section 4 presents an
extended modal identification consisting of the estimation
of the state transition matrix and the extraction of the time-
460 Zhisai MA et al.
123
varying modes. The proposed extended modal identifica-
tion approach is numerically and experimentally tested in
section 5 and section 6, respectively. Section 7 summa-
rizes the study.
2 State Space Representation
As with LTI systems, any LTV system can be represented
by a linear ordinary differential equation(ODE), but with
time-varying coefficients [12, 30], as
XI
i¼0
aiðtÞdi
dtiqðtÞ ¼
XJ
j¼0
bjðtÞdj
dtjf ðtÞ; ð1Þ
where I� J is the realizability constraint, guaranteeing that
no future points are used in the process. For detailed analyses
including solution, stability, transformation, solvability,
controllability and observability the interested reader is
referred to Refs. [10, 30–32] and the references therein.
The second order ODE with time-varying coefficients is
usually used to represent LTV structures, i.e. I ¼ 2. In
general, the input of the structure is the external force
which is a function of time, i.e. J ¼ 0. The model to
describe the dynamical behavior of a LTV structure with
n=2 degrees-of-freedom is given by Newton’s equation of
motion, as
MðtÞ€qðtÞ þ EðtÞ _qðtÞ þ KðtÞqðtÞ ¼ fðtÞ; ð2Þ
where MðtÞ,EðtÞ and KðtÞ 2 Rn=2�n=2 are respectively the
mass, damping and stiffness matrices, and f ðtÞ and qðtÞ 2Rn=2�1 are respectively the applied force and response
vectors. The notation R denotes the real numbers and Ri�j
denotes the i� j real matrix space.
A state space model for the system of Eq. (2) is given by
_xðtÞ ¼ AðtÞxðtÞ þ BðtÞf ðtÞ;yðtÞ ¼ CðtÞxðtÞ þ DðtÞf ðtÞ;
�ð3Þ
where xðtÞ ¼ qðtÞ; _qðtÞ½ � 2 Rn�1 and yðtÞ 2 Rno�1 are
respectively the state and output vectors. AðtÞ 2 Rn�n,
BðtÞ 2 Rn�n=2, CðtÞ 2 Rno�n and DðtÞ 2 Rno�n=2 are
respectively the system, input, observation and output
matrices. no is the dimension of the output vector. The
system and input matrices AðtÞ and BðtÞ are given by
AðtÞ ¼0 I
�M�1ðtÞKðtÞ �M�1ðtÞEðtÞ
� �;
BðtÞ ¼0
M�1ðtÞ
� �:
ð4Þ
The solution of Eq. (3) is given by
xðtÞ ¼ UAðt; t0Þxðt0Þ þR tt0UAðt; sÞBðsÞf ðsÞds;
yðtÞ ¼ CðtÞxðtÞ þ DðtÞf ðtÞ;
�ð5Þ
where UAðt; t0Þ is the state transition matrix of AðtÞ from
time t0 to time t. The corresponding discrete-time state
space representation becomes
xðk þ 1Þ ¼ UAðk þ 1; kÞxðkÞ þ GðkÞf ðkÞ; xðk0Þ ¼ xðt0Þ;yðkÞ ¼ CðkÞxðkÞ þ DðkÞf ðkÞ;
�
ð6Þ
where UAðk þ 1; kÞ is the discrete-time state transition
matrix from time kDt to time ðk þ 1ÞDt, where Dt is the
sampling interval. Unlike LTI systems, there is no closed-
form of the state transition matrix for arbitrarily LTV
systems.
The solution of Eq. (6) is given by
yðkÞ ¼ CðkÞUAðk; hÞxðhÞ þ DðkÞf ðkÞ
þ CðkÞUAðk; hÞXk
j¼hþ1
U�1A ðj; hÞGðj� 1Þf ðj� 1Þ; ðk[ hÞ:
ð7Þ
If non-singular matrices Tðk þ 1Þ and TðkÞ 2 Rn�n
exist, the equivalence transformation [33] is defined as
UAðk þ 1; kÞ ¼ T�1ðk þ 1ÞUAðk þ 1; kÞTðkÞ;GðkÞ ¼ T�1ðk þ 1ÞGðkÞ;CðkÞ ¼ CðkÞTðkÞ;DðkÞ ¼ DðkÞ:
8>><
>>:ð8Þ
where UAðk þ 1; kÞ, GðkÞ, CðkÞ and DðkÞ are another
realization of the original LTV system. It should be noted
that UAðk þ 1; kÞ preserves the stability of the transformed
system [34].
3 Modal Analysis
Classical modal analysis focuses on LTI systems, because
linearity, time-invariance and observability constitute the
basic assumptions of the modal analysis theories [1]. For
LTV systems, the time-invariance assumption is violated
and responses of these systems are non-stationary. There-
fore, classical modal analysis cannot be directly used to
acquire the dynamic properties of LTV systems, especially
the system’s stability.
3.1 LTI System Modal Analysis
Consider the state equation of LTI systems
_xðtÞ ¼ AxðtÞ; xðt0Þ: ð9Þ
Modal parameters of the above system are defined based
on the eigenvalues and eigenvectors of system matrix A.
According to the Schur decomposition, there exists an
invertible matrix S such that A ¼ SPS�1, where P is an
upper triangular matrix and the diagonal elements of P are
Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal… 461
123
the eigenvalues of A. So the state transformation zðtÞ ¼S�1xðtÞ yields an upper triangular state equation
_zðtÞ ¼ PzðtÞ; zðt0Þ ¼ S�1xðt0Þ: ð10Þ
Stability is preserved in the state transformation because
the eigenvalues of A and P coincide. It is well known that a
LTI system is stable if and only if all eigenvalues of its
system matrix A have negative real parts.
3.2 LTV System Modal Analysis Under
the ‘‘Frozen-Time’’ Assumption
Consider the state equation of LTV systems
_xðtÞ ¼ AðtÞxðtÞ; xðt0Þ: ð11Þ
If we directly extend the classical modal analysis to
LTV systems, the instantaneous eigenvalues and eigen-
vectors of AðtÞ can be computed at each instant of time
under the ‘‘frozen-time’’ assumption, as
kðtÞI � AðtÞ½ �vðtÞ ¼ 0; ð12Þ
where vðtÞ is the eigenvector of AðtÞ and kðtÞ is the
instantaneous eigenvalue (or pole) of AðtÞ corresponding to
vðtÞ. The instantaneous modal parameters can be defined
based on kðtÞ and vðtÞ at each instant of time.
The ‘‘frozen-time’’ assumption is an attempt to apply the
results of LTI systems to LTV cases at each instant of time.
However, these LTI models are the local approximations of
the original LTV system and every LTI model is in isola-
tion from each other. This is the reason why the instanta-
neous modal parameters of LTV systems under the
‘‘frozen-time’’ assumption cannot be used to determine the
system’s stability [10].
3.3 LTV System Modal Analysis
3.3.1 Time-Varying Modes and Poles
The definitions of the time-varying modes and poles are
reviewed following the development in the Refs. [13, 16].
For a LTV system with the state equation of Eq. (11),
defining a Lyapunov transformation SðtÞ and the state
transformation zðtÞ ¼ S�1ðtÞxðtÞ yields an equivalent upper
triangular state equation of the form
_zðtÞ ¼ PðtÞzðtÞ; zðt0Þ ¼ S�1ðt0Þxðt0Þ; ð13Þ
where PðtÞ is
PðtÞ ¼
p1ðtÞ � . . . �0 p2ðtÞ . . . �... ..
. . .. ..
.
0 0 . . . pnðtÞ
0
BBB@
1
CCCA: ð14Þ
The diagonal elements of PðtÞ are defined as poles of the
original LTV system and the pole set piðtÞ is an ordered set
in general.
Obviously, the Lyapunov transformation SðtÞ is not
formed by eigenvectors of AðtÞ, while SðtÞ is the unique
solution of the matrix differential equation
_SðtÞ ¼ AðtÞSðtÞ � SðtÞPðtÞ: ð15Þ
The solution of Eq. (15) can be given by
SðtÞ ¼ UAðt; t0ÞSðt0ÞU�1P ðt; t0Þ; ð16Þ
where Sðt0Þ is an invertible matrix, UAðt; t0Þ and UPðt; t0Þare respectively the state transition matrices of AðtÞ and
PðtÞ.From Eq. (16) we have UAðt; t0Þ ¼ SðtÞUPðt; t0ÞS�1ðt0Þ.
Applying the QR decomposition to the matrix UAðt; t0Þ at
each instant of time yields the decomposition
UAðt; t0Þ ¼ QðtÞRðtÞ. QðtÞ is orthogonal and also a Lya-
punov transformation because QðtÞk k ¼ Q�1ðtÞ�� �� ¼ 1 for
all t� t0 by the orthogonality of QðtÞ, where �k k denotes
the Euclidean norm. Since UAðt0; t0Þ ¼ I and Qðt0Þ ¼Rðt0Þ ¼ I is a QR decomposition, we have UPðt; t0Þ ¼ RðtÞby assuming Sðt0Þ ¼ I in the sequel.
The time-varying modes [16] are defined by the diago-
nal elements of UPðt; t0Þ, as
/piðt; t0Þ ¼ UPðt; t0Þ½ �i; ð17Þ
where, ½��i denotes the ith diagonal element of the matrix in
the square bracket.
The time-varying poles [16] can be obtained by
piðtÞ ¼ _/piðt; t0Þ
./pi
ðt; t0Þ: ð18Þ
3.3.2 Uniqueness
The time-varying modes /piðt; t0Þ and poles piðtÞ defined
by Eqs. (17) and (18) are nonunique due to the
nonuniqueness of the QR decomposition. In this section,
we prove that all the complex valued time-varying modes
/piðt; t0Þ share the same absolute values, and all the com-
plex valued poles piðtÞ share the same real parts.
As we know, the QR decomposition of a real matrix
yields a real orthogonal matrix Q and a real upper trian-
gular matrix R, and the QR decomposition of a nonsingular
matrix is unique if we take the diagonal elements of the
upper triangular matrix R to be real and positive [35]. In
general, the state transition matrix UAðt; t0Þ of a physical
system is real and nonsingular. Therefore, applying the QR
decomposition to UAðt; t0Þ at each instant of time yields
UAðt; t0Þ ¼ QðtÞRðtÞ, where both QðtÞ and RðtÞ are real.
The QR decomposition of UAðt; t0Þ is unique if we further
take the diagonal elements of RðtÞ to be positive, as
462 Zhisai MA et al.
123
UAðt; t0Þ ¼ �QðtÞ �RðtÞ; ð19Þ
where both �QðtÞ and �RðtÞ are real and the diagonal ele-
ments of �RðtÞ are all positive. Therefore, the time-varying
modes �/piðt; t0Þ obtained by �RðtÞ are nonnegative real
valued numbers and the corresponding poles are real val-
ued numbers.
Defining a complex valued matrix OðtÞ and supposing
another QR decomposition exists, as follows:
UAðt; t0Þ ¼ ~QðtÞ ~RðtÞ ¼ �QðtÞOðtÞð Þ O�1ðtÞ �RðtÞ� �
; ð20Þ
where OðtÞ ¼ �Q�1ðtÞ ~QðtÞ which means OðtÞ is an orthog-
onal matrix, and OðtÞ ¼ �RðtÞ ~R�1ðtÞ which means OðtÞ is
an upper triangular matrix. Hence, the complex valued
matrix OðtÞ is an orthogonal and diagonal matrix of the
form
OijðtÞ ¼exp �jhiðtÞð Þ; i ¼ j;0; i 6¼ j;
�ð21Þ
where j is the imaginary unit, hiðtÞ ði ¼ 1; 2; � � � nÞ are
arbitrary functions of time on R.
The relation between the time-varying modes ~/piðt; t0Þ
computed by ~RðtÞ and the time-varying modes �/piðt; t0Þ
obtained by �RðtÞ is
~/piðt; t0Þ ¼ �/pi
ðt; t0Þ exp jhiðtÞð Þ: ð22Þ
Therefore, we have ~/piðt; t0Þ
�� �� ¼ �/piðt; t0Þ, where �j j
denotes the absolute value. In other words, the nonnegative
real valued time-varying modes �/piðt; t0Þ are the absolute
values of a class of complex valued time-varying modes~/pi
ðt; t0Þ.The corresponding poles can be computed by
~piðtÞ ¼_~/piðt; t0Þ
.~/pi
ðt; t0Þ ¼ �piðtÞ þ j _hiðtÞ: ð23Þ
In summary, all the complex valued time-varying modes~/pi
ðt; t0Þ share the same absolute values which are equal to
the nonnegative real valued time-varying modes �/piðt; t0Þ,
and all the corresponding complex valued poles ~piðtÞ share
the same real parts which are equal to the real valued poles
�piðtÞ. Furthermore, both �/piðt; t0Þ and �piðtÞ are unique. In
the subsequent developments of the paper, all the men-
tioned time-varying modes and poles refer to �/piðt; t0Þ and
�piðtÞ, respectively.
3.3.3 Stability
The time-varying modes contain the information regarding
the stability of LTV systems, as the Lyapunov transfor-
mations preserve the stability of LTV systems [34]. The
notation C denotes the complex numbers, f ðCÞ the space of
uniformly bounded functions on C, and Cf ðCÞ the subset of
continuous functions in f ðCÞ. Following the developments
in the Ref. [16], the stability of a LTV system can be
described by its nonnegative real valued time-varying
modes, as follows.
The mode associated with pole p 2 Cf ðCÞ is uniformly
exponentially stable if there exist finite, positive constants cand k such that
�/pðt; t0Þ ¼ c exp �kðt � t0Þð Þ; t� t0 � 0: ð24Þ
The mode associated with pole p 2 Cf ðCÞ is asymptot-
ically stable if for any t0 � 0, there exists a finite, positive
constant c such that
�/pðt; t0Þ ¼ c; t� t0 and �/pðt; t0Þ ! 0; t ! 1: ð25Þ
The mode associated with pole p 2 Cf ðCÞ is uniformly
stable if there exists a finite, positive constant c such that
�/pðt; t0Þ� c; t� t0 � 0: ð26Þ
The mode associated with pole p 2 Cf ðCÞ is non-expo-
nentially stable if there exist finite, positive constants c1
and c2 such that
c1 � �/pðt; t0Þ� c2; t� t0 � 0: ð27Þ
The corresponding unstability is defined by replacing p
with �p.
4 Extended Modal Identification
The time-varying modes are defined based on the state
transition matrix of the system, while the explicit form of
the state transition matrix of an arbitrarily LTV system is
extremely difficult to obtain. Therefore, we have to extract
the time-varying modes from experimental measurements
by using some identification techniques. In this section, an
extended modal identification approach is proposed, con-
sisting of the estimation of the state transition matrix via a
subspace-based method and the extraction of the time-
varying modes by the QR decomposition.
4.1 Estimation of the State Transition Matrix Via
the Subspace-Based Method
For arbitrarily LTV systems, we need to carry out multiple
experiments on the system with the same time-varying
behavior [21–27]. A series of the Hankel matrices are
formed by an ensemble set of input–output data and the
state transition matrices are estimated by the singular value
decomposition(SVD) of two successive Hankel matrices.
Assume that N experiments have been carried out on the
LTV system whose parameters undergo the same variation
Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal… 463
123
during each experiment. The responses from the ith
experiment are yiðkÞ, and the inputs are f iðkÞ, where i ¼1; 2; . . .;N and k ¼ 1; 2; . . .; L, L is the total data length.
The Hankel matrix is formed using the M successive
responses of N experiments, as follows:
HðkÞ
¼
y1ðkÞ y2ðkÞ . . . yNðkÞy1ðk þ 1Þ y2ðk þ 1Þ . . . yNðk þ 1Þ
..
. ... . .
. ...
y1ðk þM � 1Þ y2ðk þM � 1Þ . . . yNðk þM � 1Þ
0
BBBB@
1
CCCCA:
ð28Þ
The input Hankel matrix FðkÞ is formed in the way
similar to HðkÞ using f iðkÞ. By using Eq. (7), HðkÞ can be
factored as
HðkÞ ¼ CðkÞXðkÞ þHðkÞFðkÞ; ð29Þ
where the state matrix XðkÞ 2 Rn�N is given by
XðkÞ ¼ x1ðkÞ x2ðkÞ . . . xNðkÞð Þ , and the observabil-
ity matrix CðkÞ is given by
CðkÞ ¼
CðkÞCðk þ 1ÞUAðk þ 1; kÞ
..
.
Cðk þM � 1ÞUAðk þM � 1; kÞ
0BBB@
1CCCA: ð30Þ
The ith block matrix in the jth column ofHðkÞ is of the form
HijðkÞ ¼0; i\j;Dðk þ i� 1Þ; i ¼ j;Cðk þ i� 1ÞUAðk þ i� 1; k þ jÞGðk þ j� 1Þ; i[ j:
8<
:
ð31Þ
By using Eq. (8), the observability range space of CðkÞbecomes
CðkÞ ¼ CðkÞTðkÞ ¼
CðkÞCðk þ 1ÞUAðk þ 1; kÞ
..
.
Cðk þM � 1ÞUAðk þM � 1; kÞ
0BBB@
1CCCA:
ð32Þ
In order to extract the observability range space CðkÞ,the second term on the right-hand side of Eq. (29) should
be eliminated. Defining a matrix
F?ðkÞ ¼ I � FTðkÞ FðkÞFTðkÞ� ��1
FðkÞ; ð33Þ
which results in FðkÞF?ðkÞ ¼ 0 [23]. Multiplying Eq. (29)
by F?ðkÞ, we have
HðkÞF?ðkÞ ¼ CðkÞXðkÞF?ðkÞ: ð34Þ
In other words, the observability range space CðkÞ can
also be extracted from HðkÞF?ðkÞ.
A successive Hankel matrix Hðk þ 1Þ is formed using
the responses from k þ 1 to kþM of N experiments, and
Hðk þ 1Þ can be factored as
Hðk þ 1Þ ¼ Cðk þ 1ÞXðk þ 1Þ þHðk þ 1ÞFðk þ 1Þ;ð35Þ
where Cðk þ 1Þ has the similar form as Eq. (30) and its
observability range space Cðk þ 1Þ is
Cðk þ 1Þ ¼
Cðk þ 1ÞCðk þ 2ÞUAðk þ 2; k þ 1Þ
..
.
Cðk þMÞUAðk þM; k þ 1Þ
0
BBB@
1
CCCA: ð36Þ
Select the firstM � 1 block rows ofCðk þ 1Þ asCaðk þ 1Þand the last M � 1 block rows of CðkÞ as CbðkÞ, then the
matrix UAðk þ 1; kÞ can be obtained by
UAðk þ 1; kÞ ¼ Caðk þ 1Þ� �þ
CbðkÞ; ð37Þ
where ð�Þþ denotes the Moore–Penrose pseudoinverse.
The question as how to retrieve an estimate of the
observability range space is addressed now. The SVD of
two successive Hankel matrices results in
HðkÞF?ðkÞ ¼ UðkÞRðkÞVðkÞH;Hðk þ 1ÞF?ðk þ 1Þ ¼ Uðk þ 1ÞRðk þ 1ÞVðk þ 1ÞH;
�
ð38Þ
where ð�ÞHdenotes the Hermitian transpose. The matrices
U 2 RnoM�noM and V 2 RN�N are two orthogonal matrices
called the left and right singular vector matrices, respec-
tively. The matrix R 2 RnoM�N is a rectangular diagonal
matrix with nonnegative real numbers on the diagonal and
the diagonal entries of R are known as the singular values
of the Hankel matrix.
The first n columns of UðkÞ form an orthonormal basis
for CðkÞ [22, 26], as follows,
CðkÞ UðkÞð:; 1 : nÞ; Cðk þ 1Þ Uðk þ 1Þð:; 1 : nÞ:ð39Þ
By using Eq. (37), UAðk þ 1; kÞ can be estimated by
UAðk þ 1; kÞ ¼ Uaðk þ 1Þð:; 1 : nÞð ÞþUbðkÞð:; 1 : nÞ:ð40Þ
4.2 Extraction of the Time-Varying Modes
by the QR Decomposition
As we know, the identified system is one of the equivalent
systems of the original system, i.e. the estimated state
transition matrix �UAðk þ 1; kÞ is the equivalence trans-
formed matrix of UAðk þ 1; kÞ. As was noted from the
preceding sections, UAðk þ 1; kÞ preserves the stability of
464 Zhisai MA et al.
123
the transformed system, so we can extract the time-varying
modes from UAðk þ 1; kÞ instead of the original state
transition matrix UAðk þ 1; kÞ.Eq. (8) generates the following equation
UAðk; k0Þ ¼ UAðk; k � 1Þ � . . .�UAðk0 þ 1; k0Þ; ð41Þ
where UAðk; k0Þ ¼ T�1ðkÞUAðk; k0ÞTðk0Þ. Conducting the
QR decomposition to UAðk; k0Þ yields
UAðk; k0Þ ¼ T�1ðkÞSðkÞUPðk; k0ÞS�1ðk0ÞTðk0Þ¼ �QðkÞ �RðkÞ:
ð42Þ
By assuming Tðk0Þ ¼ I, we have S�1ðk0ÞTðk0Þ ¼ I.
T�1ðkÞSðkÞ ¼ �QðkÞ is the Lyapunov transformation which
transforms the identified system matrix UAðk; k0Þ to the
upper triangular matrix UPðk; k0Þ ¼ �RðkÞ at each instant of
time kDt. It demonstrates that all equivalent systems share
the same time-varying modes and poles, as they can be
transformed to a system described by the same upper tri-
angular state equation.
The time-varying modes can be obtained by picking the
diagonal elements of UPðk; k0Þ, as follows:
�/piðk; k0Þ ¼ UPðk; k0Þ½ �i: ð43Þ
5 Numerical Validation
Three numerical cases are considered in this sec-
tion. Firstly, a stable single degree-of-freedom LTV system
with known theoretical state transition matrix is used to
validate the modal analysis theories and the robustness of
the proposed identification approach with respect to noise.
Secondly, an unstable LTV system with known theoretical
state transition matrix is used to be compared to the
stable LTV system. Thirdly, a coupled moving-mass and
simply supported beam time-varying system is used to
show the performance of the proposed approach in deter-
mining the stability of LTV mechanical systems.
5.1 Stable LTV System
Consider a single degree-of-freedom LTV system with
mass MðtÞ¼1, damping EðtÞ¼2t and stiffness
KðtÞ ¼ t2 � 2. By using Eq. (4), the system matrix AðtÞ of
this system is obtained as
AðtÞ ¼ 0 1
2 � t2 �2t
� �: ð44Þ
The theoretical state transition matrix of this system is
given by
UAðt; 0Þ ¼ffiffiffi3
p
6exp � t2
2
� ��
ffiffiffi3
ph1ðtÞ þ
ffiffiffi3
ph2ðtÞ h1ðtÞ � h2ðtÞ
ð3 �ffiffiffi3
ptÞh1ðtÞ � ð3 þ
ffiffiffi3
ptÞh2ðtÞ ð
ffiffiffi3
p� tÞh1ðtÞ þ ð
ffiffiffi3
pþ tÞh2ðtÞ
!;
ð45Þ
where h1ðtÞ ¼ expffiffiffi3
pt
� �, and h2ðtÞ ¼ exp �
ffiffiffi3
pt
� �.
In the actual implementation, the responses of this sys-
tem are computed by numerical integration using the
Runge–Kutta method. To test the proposed identification
approach, six numerical experiments are carried out based
on different initial conditions. The six initial conditions
used in the simulation are given as
x1ðtÞ; x2ðtÞ; . . .; x6ðtÞð Þ ¼ 1 0 1 0:5 �1 0:50 1 0:5 1 0:5 �1
� �:
ð46Þ
The true time-varying modes of this system are first
extracted by the QR decomposition of the theoretical state
transition matrix of Eq. (45). The tracking ability and
robustness of the identification approach are subsequently
validated by adding Gaussian white noise to the original
responses. The signal-to-noise ratio (SNR) is respectively
set to SNR ¼ 20 dB and SNR ¼ 10 dB. The time-varying
modes of this system are estimated based on the responses
contaminated by Gaussian white noise, as shown in Fig. 1.
The instantaneous eigenvalues of AðtÞ under the ‘‘fro-
zen-time’’ assumption are �t ffiffiffi2
p, and they do not always
have negative real parts. However, Fig. 1 demonstrates that
the LTV system given by Eq. (44) is stable as all the time-
varying modes are asymptotically stable. This can also be
seen from the explicit form of the theoretical state transi-
tion matrix in Eq. (45).
5.2 Unstable LTV System
Consider a LTV system, from the classical papers [16, 36],
with the following system matrix
AðtÞ ¼ �1 þ h cos2 t 1 � h sin t cos t
�1 � h sin t cos t �1 þ h sin2 t
� �; ð47Þ
where h is a constant. The theoretical state transition matrix
of this system is given by
UAðt; 0Þ ¼ exp ðh� 1Þtð Þ cos t exp �tð Þ sin t
� exp ðh� 1Þtð Þ sin t exp �tð Þ cos t
� �:
ð48Þ
It can be seen that this system is stable only if h� 1. The
instantaneous eigenvalues of AðtÞ under the ‘‘frozen-time’’
assumption are 0:5h� 1 0:5ffiffiffiffiffiffiffiffiffiffiffiffiffih2 � 4
p, and the ‘‘frozen-
time’’ eigenvalues have negative real parts only if h\2. In
Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal… 465
123
other words, in case 1\h\2, this system is unstable even
if all eigenvalues of AðtÞ are constant and have negative
real parts.
In the actual implementation, the responses of this sys-
tem are computed by numerical integration using the
Runge–Kutta method. To test the proposed identification
approach, six numerical experiments are carried out based
on different initial conditions, as given in Eq. (46). The
tracking ability and robustness of the identification
approach are validated by adding Gaussian white noise to
the original responses. The SNR is respectively set to
SNR ¼ 20 dB and SNR ¼ 10 dB. The time-varying modes
of this system are estimated based on the responses con-
taminated by Gaussian white noise, as shown in Fig. 2.
Obviously, the LTV system reduces to a stable LTI
system when h ¼ 0. Its time-varying modes are uniformly
exponentially stable, as shown in Fig. 2(a). Fig 2(b) shows
that the LTV system with h ¼ 0:8 is stable as both the two
time-varying modes are asymptotically stable. Fig 2(c)
shows that the LTV system with h ¼ 1 is stable as the two
time-varying modes are respectively asymptotically
stable and non-exponentially stable. Fig 2(d) shows that
the LTV system with h ¼ 1:2 is unstable as one of the two
time-varying modes is unstable. Fig 2 demonstrates that
the estimated time-varying modes contain the same infor-
mation regarding the stability of LTV systems as the the-
oretical state transition matrix.
5.3 Coupled Moving-Mass and Simply Supported
Beam Time-Varying System
Consider the straight simply supported beam, shown in
Fig. 3, of length L, having a uniform cross-section with
constant mass per unit length m, the coefficient of viscous
damping c, and flexural stiffness EI, made from linear,
homogeneous and isotropic material. The transverse dis-
placement response yðx; tÞ is a function of the position x
Fig. 1 Time-varying modes of the single degree-of-freedom LTV
system
Fig. 2 Time-varying modes of the LTV system
466 Zhisai MA et al.
123
and time t, Qðx; tÞ is the transverse loading which is
assumed to vary arbitrarily with position x and time t,
Pðx; tÞ is the force acting on the beam by the moving mass,
M0 is the mass of the moving mass, sðtÞ is the moving-mass
instantaneous position on the beam.
The dynamic model of the coupled time-varying system
is given by [37]
MðtÞðtÞ þ EðtÞ _qðtÞ þ KðtÞqðtÞ ¼ FðtÞ; ð49Þ
with
MðtÞ¼diag½Mi�þM0diag½/iðsÞ�UðsÞ;EðtÞ¼ c=mð Þdiag½Mi�þ4M0 _sdiag½/iðsÞ�U0ðsÞ;KðtÞ¼diag½Ki�þ2M0€sdiag½/iðsÞ�U0ðsÞþ4M0 _s
2diag½/iðsÞ�U00ðsÞ;FðtÞ¼
R L0Qðx;tÞ /1ðxÞ;/2ðxÞ; . . .;/IðxÞð ÞT
dx
þM0g /1ðsÞ;/2ðsÞ; . . .;/IðsÞð ÞT;
8>>>><
>>>>:
ð50Þ
where Mi and Ki are the ith modal mass and modal stiffness
of the simply supported beam, g the gravitational acceler-
ation, /iðxÞ the ith eigenfunction of the unloaded and
undamped beam, UðsÞ the eigenfunctions matrix evaluated
at x ¼ sðtÞ, U0ðsÞ and U00ðsÞ respectively the first and
second order partial derivative of UðsÞ with respect to x
evaluated at x ¼ sðtÞ, diag½/i� a square diagonal matrix
with the elements of /1;/2; . . .;/Ið Þ on the main diagonal.
For the simply supported beam, we have
/iðxÞ ¼ sinipLx
� �ði ¼ 1; 2; . . .; IÞ: ð51Þ
The relationship between the transverse displacement
response yðx; tÞ and the structural response vector qðtÞ is
given by
yðx; tÞ ¼XI
i¼1
/iðxÞqiðtÞ: ð52Þ
In the actual implementation, the moving mass slides on
the simply supported beam with uniform speed, with the
motion form sðtÞ ¼ vt, where v is the speed. The numerical
quantities of the parameters are given, as follows: the
length L ¼ 2 m, the mass per length m ¼ 4:71 kg=m, the
coefficient of viscous damping c ¼ 0, the flexural stiffness
EI ¼ 1 050 Nm2, the mass of the moving mass
M0 ¼ 4:866 kg, the moving-mass speed v ¼ 0:2 m=s , and
the gravitational acceleration g ¼ 9:8 m
s2.
The duration is 10 s for the mass to move from one end
of the beam to the other end. If four eigenfunctions are
used to simulate the coupled time-varying system, i.e.
I ¼ 4, the motion equation in Eq. (49) generates 2I ¼ 8
complex valued eigenvalues, appearing in complex con-
jugate pairs. Based on the above dynamic model and
numerical quantities, the instantaneous complex valued
eigenvalues of the coupled time-varying system under the
‘‘frozen-time’’ assumption can be obtained. The real parts
and the corresponding positive imaginary parts of the four
pairs of complex valued eigenvalues are depicted in
Fig. 4.
The complex valued eigenvalues of the coupled time-
varying system exhibit symmetrical variation during the
mass’ movement due to the symmetrical boundary condi-
tion of the simply supported beam. Fig. 4 shows that the
real parts of the eigenvalues of the coupled time-varying
system under the ‘‘frozen-time’’ assumption are not always
negative.
A white noise input is generated to excite the system at
the position x ¼ 0:571 4 m. 30 numerical simulations are
carried out, and the coupled time-varying system under-
goes the same variation in each simulation. Of course, the
random excitation in every simulation is different from
each other. The responses of the coupled time-varying
system are computed by numerical integration using the
Runge–Kutta method. The proposed identification
approach is used to estimate the time-varying modes of the
coupled time-varying system. Fig 5 depicts the eight esti-
mated time-varying modes based on the inputs and the
responses contaminated by Gaussian white noise
(SNR ¼ 20 dB).
As shown in Fig. 5, all the time-varying modes are
asymptotically stable. It demonstrates that the coupled
time-varying system is stable, although the instantaneous
eigenvalues of the system under the ‘‘frozen-time’’
assumption do not always have negative real parts.
6 Experimental Validation
In this section, an experimental system of the coupled
moving-mass and simply supported beam is built to further
validate the proposed extended modal identification
approach.
6.1 Experimental System and its Set-up
The experimental system is composed of the test structure,
an exciter system, force and motion transducers,
Fig. 3 Coupled moving-mass and simply supported beam time-
varying system
Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal… 467
123
measurement and analysis systems, control systems and
boundary conditions. Fig 6 shows the schematic diagram
of the experimental system and its set-up. The test structure
is the coupled time-varying system consisting of a simply
supported beam and a moving mass sliding on it. The
dimensions of the beam are 2 000 � 60 � 10 mm(L �
W � H) and the weight of the moving mass is 4:866 kg.
The exciter system consists of a ModalshopTM2025E
exciter and a SmartAmpTM2100E21�400 power amplifier.
A PCBTM288D01 impedance head and 15 PCBTM333B30
accelerometers are respectively used as the force trans-
ducer and motion transducers. The measurement and
acquisition module is a LMSTMSCADAS III system. Con-
trol systems consist of a FaulhaberTM DC motor and its
motion controller.
6.2 ‘‘Frozen-Time’’ Experiments
The coupled time-varying system is here studied using the
frozen approximation. The time-dependent dynamic char-
acteristics of the experimental system are function of the
position of the moving mass, while the position of the
moving mass is function of time. The time needed by the
mass to move from one end of the beam to the other end
can be partitioned into several discrete segments. When the
moving mass stays at a certain segment, the experimental
system can be considered as a time-invariant system and its
instantaneous modal parameters can be estimated by using
time-invariant system identification techniques.
During the experiment the moving mass starts at the
midpoint of the beam and moves over 0:8 m. We divide
this trajectory, 1:0� 1:8 m, into 80 equal segments of
0:01 m. The mass is placed at the starting edge of each
segment and a ‘‘frozen-time’’ experiment is carried out. In
each experiment, a random excitation is generated to excite
Fig. 4 Complex valued eigenvalues of the coupled time-varying
system under the ‘‘frozen-time’’ assumption
Fig. 5 Time-varying modes of the numerical system within the time
interval t 2 ½0; 1�s
Fig. 6 Coupled moving-mass and simply supported beam experi-
mental system
468 Zhisai MA et al.
123
the system at the location x ¼ 0:571 4 m, and 15
accelerometers measure the acceleration of the beam at 15
uniformly distributed positions along the axial direction of
the beam from left to right, as shown in Fig. 6. The least
squares complex frequency domain method [1] is used to
identify the ‘‘frozen-time’’ experimental system. Fig 7
shows the driving point frequency response functions
(FRFs) of the ‘‘frozen-time’’ experimental system. The
instantaneous complex valued eigenvalues of the ‘‘frozen-
time’’ experimental system are subsequently estimated.
Fig 8 shows the real parts and the corresponding positive
imaginary parts of the first four pairs of estimated complex
valued eigenvalues.
6.3 Time-Varying Experiments
In the experiment a random excitation is generated to
excite the coupled time-varying system at the position
x ¼ 0:571 4 m. 15 accelerometers are used to measure the
acceleration of the beam at 15 uniformly distributed posi-
tions along the axial direction of the beam. The mass slides
from one end of the beam to the other end with uniform
speed v¼0:2 m=s. The accelerometers are automatically
triggered to measure the responses of the beam when the
mass passes through the midpoint of the beam ðx ¼ 1:0 m),
and they stop measuring when the mass passes through the
position x ¼ 1:8 m. In other words, the measuring duration
of the accelerometers is four seconds. 30 tests are carried
out, and the coupled time-varying system undergoes the
same variation in each test. Fig 9 shows the excitation
force and the corresponding response measured by the
impedance head during one of the 30 tests.
The proposed identification approach is used to estimate
the time-varying modes of the experimental system based
Fig. 7 Driving point FRFs of the ‘‘frozen-time’’ experimental system
Fig. 8 Complex valued eigenvalues of the ‘‘frozen-time’’ experi-
mental system
Fig. 9 Excitation force and the corresponding response of a certain
test
Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal… 469
123
on the excitation forces and the responses measured in the
30 tests. Fig 10 depicts the first eight estimated time-
varying modes of the experimental system.
The experimental system is stable, as all the time-
varying modes of the experimental system are asymptoti-
cally stable, as depicted in Fig. 10. The experimental
identification results are coincident with the numerical
case, which further validate the proposed modal analysis
theories and identification approach.
7 Conclusions
(1) LTV system modal analysis via a stability-preserv-
ing state transformation is presented based on time-
dependent state space models. The time-varying
modes, instead of the instantaneous modal parame-
ters, are further interpreted to determine the dynamic
stability of LTV systems.
(2) An extended modal identification is proposed, and is
numerically and experimentally validated to accu-
rately extract the time-varying modes of LTV
systems from input–output measurements.
(3) As the Lyapunov state transformation and the
equivalence transformation are all stability-preserv-
ing, the extended modal identification provides a
new way to determine the dynamic stability of LTV
systems by using the time-varying modes.
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587–596.
Zhisai MA is a PhD candidate at School of Aerospace Engineering,
Beijing Institute of Technology, China. He is currently a visiting
researcher at Department of Mechanical Engineering, KU Leuven,
Belgium. His research interests include linear time-varying system
identification and modal parameter estimation. E-mail:
Li LIU is currently a professor at School of Aerospace Engineering,
Beijing Institute of Technology, China. Her research interests include
flight vehicle conceptual design, flight vehicle structural analysis and
design, and multidisciplinary design optimization. E-mail:
Sida ZHOU is currently an associate professor at School of
Aerospace Engineering, Beijing Institute of Technology, China. His
research interests include flight vehicle conceptual design, and
structural dynamics in aerospace engineering. E-mail:
Frank NAETS is currently a postdoctoral researcher at Department
of Mechanical Engineering, KU Leuven, Belgium. His research
interests include mechatronic simulation, virtual sensing, flexible
multibody simulation, and nonlinear model reduction. E-mail:
Ward HEYLEN is currently an associate professor at Department of
Mechanical Engineering, KU Leuven, Belgium. His research interests
include structural dynamics, experimental modal analysis, finite
element model updating, and material identification based upon
vibration measurements. E-mail: [email protected]
Wim DESMET is currently a full professor at Department of
Mechanical Engineering, KU Leuven, Belgium. His research interests
include numerical and experimental vibro-acoustics, uncertainty
modeling of dynamic systems, aeroacoustics, active noise and
vibration control, and multibody dynamics. E-mail:
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