dynamic stability analysis of linear time-varying systems

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

[email protected]

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

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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


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


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


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


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


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


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


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


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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The ISMA2014 International Conference on Noise and Vibration

Engineering, Leuven, Belgium, September 15–17, 2014:

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:

[email protected]

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:

[email protected]

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:

[email protected]

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:

[email protected]

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:

[email protected]


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dynamic stability analysis of linear time-varying systems

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