Mechanical Systems and Signal Processing 139 (2020) 106620
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing
journal homepage: www.elsevier.com/locate/ymssp
Parameter identification for nonlinear time-varying dynamic
system based on the assumption of ‘‘short time linearly varying”
and global constraint optimization
Tengfei Chen a, Huan He a,⇑, Guoping Chen a, Yuxuan Zheng b, Shuo Hou c, Xulong Xi d
a
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Faculty of Science, University of Auckland, Auckland 1010, New Zealand
c
China Nuclear Power Technology Research Institute, Shenzhen 518000, China
d
Aviation Key Laboratory of Science and Technology on Structures Impact Dynamics, Aircraft Strength Research Institute of China, Xi’an 710065, China
b
a r t i c l e
i n f o
Article history:
Received 11 September 2019
Received in revised form 14 December 2019
Accepted 4 January 2020
Available online 23 January 2020
Keywords:
Nonlinear time-varying dynamic system
Parameter identification
Short time invariant
Short time linear varying
Global constraint
a b s t r a c t
A new identification approach based on a new assumption of ‘‘short time linear varying” is
proposed for nonlinear time-varying (NTV) dynamic systems. In the identification procedure, the whole period is divided into a series of shifting windows. In each window, the
NTV system model, which is known a priori, can be represented by regression equations
and all the time-varying (TV) coefficients are determined by a least squares (LS) algorithm.
The proposed approach has better identification precision than the traditional assumption
of ‘‘short time invariant”. To enhance the robustness and stability, the problem of parameter identification is solved by means of constrained optimization in the global identification strategy when the noise level increases. The validity and accuracy are verified by
applying the method to a single degree of freedom (SDOF) numerical example, and a qualitative analysis on the selection of the window size is carried out in this research.
Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction
In practical engineering, many structures exhibit nonlinear and TV characteristics, which cannot be described with linear
or time-invariant (TI) models [1,2]. In recent years, there has been a growing interest in the identification of NTV systems
[3,4]. However, in the field of parameter identification, it is difficult to take both of these two complicated characteristics
into account at the same time [5].
Researchers have made some useful attempts in different areas [6,7]. Among the existing studies, an idea is applying the
basis function to TV parameters. By combining the modulating function and iterative orthogonal forward regression algorithm, Guo et al. successfully detected the model structure and identified the associated TV parameters. The proposed
method works well even when the measurements are severely corrupted by noise [8]. They also used asymmetric basis function to track both smooth and abrupt changes in the TV parameters [9] with high accuracy.
Another big proportion of the identification methods are based on the theory of ‘‘time-freezing”, and the assumption of
‘‘short time invariant” is employed to approximately treat the slow NTV systems as the slow nonlinear time invariant (NTI)
systems in each short time interval [10]. That is, in each short window, all the TV parameters in the systems are supposed as
⇑ Corresponding author.
E-mail address: hehuan@nuaa.edu.cn (H. He).
https://doi.org/10.1016/j.ymssp.2020.106620
0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.
2
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
constant [11]. Then, the conventional identification approaches for NTI systems such as those based on the LS algorithm, can
be introduced into each short window to determine the ‘‘constant” parameters [12–14]. Chen et al. studied the identification
algorithm for multiple-degree-of-freedom (MDOF) TV systems based on the assumption of ‘‘short time invariant” and LS
algorithm, and the identification results with measurement noise were given [4]. Wang et al. identified the transient modal
parameters for weakly damped and slow TV structures online by approximately taking the non-stationary random response
signals and the TV structures as stationary random response time series and TI structures in a short time interval [10].
The performance of such approaches depends heavily on the window size. On one hand, increasing the window size will
add more data to the identification process in each time interval and enhance the robustness of the algorithm. On the other
hand, the true TV parameters of the models will deviate from the initial values to a greater extent in a larger window, which
will cause the inadaptability of the assumption of ‘‘short time invariant” and serious errors. However, the window size is
usually determined by experience and no general criterion for window size selection is established. This severely limits
the practical application of such approaches [15].
To improve the defect of the assumption of ‘‘short time invariant” in relatively larger window, the new concept of ‘‘short
time linearly varying” is proposed in this research. Different from the former assumption, the system parameters are considered to vary linearly with time in each of the intervals. The temporal variation of the parameter in the shifting window
is described by a linear function, which is determined by an initial value and a changing rate. Obviously, linear function is
closer to the reality than a constant in the approximation of a TV parameter within a short period of time, which is similar to
the advantage of trapezoidal integral in precision compared with normal rectangular integral. For this reason, the new
assumption is introduced into our research on NTV system identification, which can help enlarge the window size and
improve robustness, without sacrificing the estimating precision.
In the traditional approaches based on the assumption of ‘‘short time invariant” and the LS algorithm, the parameter identification equations are established separately in each of the shifting windows by the input and output data of the system on
the sampling points. The equations of different windows are independent of each other. This kind of separate strategy for
establishing identification equations combined with the newly proposed assumption of ‘‘short time linearly varying” is
found to perform well on estimation accuracy when there is no or relatively low measurement noise. However, in the case
of high measurement noise, this strategy may give rise to excessive errors or even cannot converge to a correct solution. In
this research, to improve the stability and the robustness of the identification method based on the new assumption, a series
of global constraints are added into the model equations, and the identification equations are considered to be established
globally by the measured input and output data on all of the sampling points when the measurement noise increases. This
modification to the identification strategy integrates all of the shifting windows by adding global constraints. These global
constraints prevent the algorithm from seeking optimal solutions completely freely. This research suggests that this strategy
extracts more information from the input and output data, which helps reduce the influence of measurement noise on the
identification and enhance the stability and robustness of the proposed identification method.
Parameter identification is premised on the basis of modeling of nonlinear systems, including the process of detecting
nonlinearities [16,17], locating the nonlinearities in an MDOF system and characterizing the type of nonlinearities [18–
20], and a lot of research work has been done on these aspects [21–23]. Besides, parameter identification of an MDOF system
can be transformed into that of several connected subsystems in many cases, such as the Ultra-Orthogonal Forward Regression presented by Guo et al. in which the nonlinear parameters of an MDOF system were identified in several SDOF subsystems under generalized coordinates [17]. This paper focuses on the identification algorithm for nonlinear TV parameters,
therefore the theoretical derivation will be based on the SDOF dynamic system and the types of nonlinearities are assumed
to be known.
The major novelty of this research is drawn from the proposed assumption of ‘‘short time linearly varying”, and the global
identification equations across the whole time period. The remainder of this paper is organized as follows. In Section 2 the
parameter identification problem of the TV nonlinear dynamics system is described and formulated. The model of the TV
nonlinear dynamic system in the shifting window with the assumption of ‘‘short time linearly varying” is described in Section 3. It shows that parameter identification can be converted into the LS regression analysis. Section 4 introduces the
method of constructing the regression equations globally in the whole time period. The TV parameters are determined by
the solutions of the regression equations in all the windows. An investigation on the errors caused by the measurement noise
and numerical integration is carried out in Section 5. In Section 6, some numerical results of the implementation of the identification method are discussed. This is followed by the concluding remarks in Section 7.
2. The issue of parameter identification of the TV nonlinear dynamic system
Firstly, for the convenience of explaining the main idea of this research, the new identification method based on the
assumption of ‘‘short time linearly varying”, the theoretical derivation will be based on the SDOF dynamic system. The governing equation of motion of a linear time-varying (LTV) SDOF dynamic system can be expressed as
_ þ kðtÞxðtÞ ¼ f ðtÞ
mðtÞ€xðtÞ þ cðtÞxðtÞ
ð1Þ
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
3
where mðtÞ, cðtÞ and kðtÞ refer to the TV mass, damping and stiffness of the dynamic system respectively, and f ðtÞ is the external force on the system. xðtÞ represents the displacement and a dot over a variable denotes differentiation with respect to
time.
_ tÞ into the linear model (1), the governing equation of motion of a NTV dynamic
By introducing a nonlinear term Nðx; x;
system is given as
_ þ kðtÞxðtÞ þ Nðx; x;
_ tÞ ¼ f ðtÞ
mðtÞ€xðtÞ þ cðtÞxðtÞ
ð2Þ
_ tÞ refers to the nonlinear restoring force which is usually a nonlinear function of the displacement and velocity.
where Nðx; x;
It can be described with nonlinear stiffness, nonlinear damping, or a combination of different types of nonlinearities. Specifically, the nonlinear term can be written as
_ tÞ ¼ s1 ðtÞN1 ðx; xÞ
_ þ s2 ðtÞN 2 ðx; xÞ
_ þ þ sn ðtÞNn ðx; xÞ
_
Nðx; x;
ð3Þ
_ ¼ 1; 2; :::; nÞ are the nonlinear terms which represent different types of nonlinear restoring forces and
in which N i ðx; xÞði
si ðtÞði ¼ 1; 2; :::; nÞ refer to the TV coefficients of the nonlinear terms. For example, a van der Pol damping with a TV coefficient
_ where N i ðx; xÞ
_ ¼ ð1 x2 Þx,
_ and a Duffing type nonlinear stiffness as si ðtÞN i ðx; xÞ
_ where
can be written as si ðtÞN i ðx; xÞ
_ ¼ x3 .
N i ðx; xÞ
Substituting Eq. (3) into model (2), the objective of parameter identification of NTV systems is to obtain the time functions of the coefficients including mðtÞ; cðtÞ; kðtÞ and all the coefficients of nonlinear terms si ðtÞði ¼ 1; 2; :::; nÞ. If the displacements, velocities, accelerations and external force can be accurately measured, the challenge lies in finding the solution of
the underdetermined equations which are composed of the motion and external force in model (2) at every sampling time:
8
n
P
>
>
_ DtÞ þ kð1DtÞxð1DtÞ þ si ð1DtÞNi ð1DtÞ ¼ f ð1DtÞ
mð1DtÞ€xð1DtÞ þ cð1DtÞxð1
>
>
>
i¼1
>
>
>
>
n
P
>
>
< mð2DtÞ€xð2DtÞ þ cð2DtÞxð2
_ DtÞ þ kð2DtÞxð2DtÞ þ si ð2DtÞNi ð2DtÞ ¼ f ð2DtÞ
i¼1
>
>
..
>
>
>
.
>
>
>
n
P
>
>
>
_ DtÞ þ kðNDtÞxðNDtÞ þ si ðNDtÞNi ðNDtÞ ¼ f ðNDtÞ
: mðNDtÞ€xðNDtÞ þ cðNDtÞxðN
ð4Þ
i¼1
in which Dt is the sampling interval. In Eq. (4), N refers to the total number of sampling points and there are ð3 þ nÞ N
unknowns in N regression equations. To overcome this theoretical problem, the related assumption about the NTV dynamic
system needs to be introduced to avoid infinite solutions. In this paper, the assumption of ‘‘short time linearly varying” will
be considered to solve this problem in Section 3.
3. The assumption of ‘‘short time linearly varying” for NTV dynamic systems
Under the traditional assumption of ‘‘short time invariant”, a relatively short time window including N w ðN w > 3 þ nÞ sampling points is designed and the regression equations within the i-th window can be written as
^i 3
€xðiNw DtÞ 3T 2 m
7
6
_ w DtÞ 7 6 ^ci 7
xðiN
7 2 f ð½ði 1ÞN þ 1DtÞ 3
7 6^ 7
w
7
xðiNw DtÞ 7 6 k
7
7 6 i 7 6
.
6
7
.
7¼
6
.
5
N1 ðiN w DtÞ 7
7 6 ^s1;i 7 4
7 6
7
..
7 6 .. 7
f ðiN w DtÞ
5 4 . 5
.
^sn;i
Nn ð½ði 1ÞNw þ 1DtÞ Nn ðiNw DtÞ
2 €
xð½ði 1ÞNw þ 1DtÞ
6 xð½ði
1ÞNw þ 1DtÞ
6 _
6
6 xð½ði 1ÞNw þ 1DtÞ
6
6 N ð½ði 1ÞN þ 1DtÞ
w
6 1
6
..
6
4
.
ð5Þ
^ ; ^s ; ; ^s are the estimates of the parameters. Eq. (5) can also be expressed as a matrix equation:
^ i ; ^ci ; k
in which m
i 1;i
n;i
_
Ui H i ¼ Fi
ð6Þ
All the coefficients are considered as constants and the problem is simplified to the LS solution of N w equations with 3 þ n
unknowns. It is worth noting that this assumption will lead to a significant error for the systems with parameters changing
quickly with the variance of time. In Eq. (6), the residual vector DF i can be written as
_
DF i ¼ U i H i F i
ð7Þ
4
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
As is widely known, the LS solution minimizes the 2-norm of the residual vector in Eq. (7). The LS algorithm finds the
optimal solution within each of the windows, but cannot strictly satisfy the equations on all sampling points in Eq. (5). In
TI dynamic systems without the consideration of noise, DF i is zero. On the contrary, for the dynamic system identification
with rapidly changing parameters, the constant estimates may generate relatively large residual errors on several of the sampling points in Eq. (5).
In some actual dynamic systems, one or several kinds of restoring forces may be much lower than the total force. For
example, as shown in Fig. 1, in underdamped dynamic systems the damping force may be several orders of magnitude lower
than the inertial force and the elastic force, which could even be ignored in some cases. For another more complicated example as shown in Fig. 2, in the Duffing system, the nonlinear restoring force due to the cubic stiffness may be much lower than
the linear elastic force under the condition of small amplitude, and it is opposite when the amplitude is large. In such cases,
the residual errors calculated by Eq. (7) on some sampling points would be too large to be accepted when compared with the
terms of low forces mentioned above, which has a severe impact on the stability of the identification algorithm. The damping
_
force of the lowest level in the NTV system €
xðtÞ þ 0:001xðtÞ
þ sinð0:4ptÞxðtÞ þ x3 ðtÞ ¼ 2sin2pt along with the residual errors
is demonstrated in detail in Fig. 3. Obviously, the residual errors shown in Fig. 3 are comparable to the damping force or even
much greater on some sampling points. However, one cannot predict the exact level of different types of forces or the residual errors on the sampling points in Eq. (5) merely on measured input and output data when the true system parameters are
unknown. This tends to yield notable errors in the identification procedure, especially for the coefficients of the terms of low
level.
Such defect is inevitable in the identification of NTV systems with rapidly changing parameters under the assumption of
‘‘short time invariant”. Increasing the sampling frequency or shortening the width of the windows may help, but it will sacrifice the stability and robustness when there are large measurement noises. This reflects the essential contradiction mentioned above in Section 1 in the traditional assumption of ‘‘short time invariant”. As a result, the top priority is to improve the
modeling accuracy of the regression equations and reduce the residual errors in Eq. (7). The classic assumption has to be
abandoned and a new solution needs to be proposed in this type of problem.
Inspired by the advantage in precision of trapezoidal integral over rectangular integral, the new assumption of ‘‘short
time linearly varying” is introduced into the identification of NTV dynamic systems in this research. Under this new assumption, the TV parameters in the i-th window can be expressed by linear functions as follows:
_
Fig. 1. Underdamped dynamic system (€
xðtÞ þ 0:001xðtÞ
þ xðtÞ ¼ sin2pt).
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
5
_
Fig. 2. Duffing system (€
xðtÞ þ 0:001xðtÞ
þ xðtÞ þ x3 ðtÞ ¼ Asin2pt; A ¼ 2; 20).
_
Fig. 3. Damping force and residual error in NTV system under the assumption of ‘‘short time invariant” (€
xðtÞ þ 0:001xðtÞ
þ sinð0:4ptÞxðtÞ þ x3 ðtÞ ¼ 2sin2pt).
8
m ¼ mi þ kmi ðt T i Þ
>
>
>
>
>
c ¼ ci þ kci ðt T i Þ
>
>
>
< k ¼ k þ k ðt T Þ
i
i
ki
ðT 6 t 6 T iþ1 Þ
> s1 ¼ s1i þ ks1i ðt T i Þ i
>
>
>
>
>
>
>
:
sn ¼ sni þ ksni ðt T i Þ
ð8Þ
where T i is the initial time of the i-th window, and kmi ; kci ; kki ; ks1i ; :::; ksni are the changing rates of the mass, linear damping,
linear stiffness and the coefficients of nonlinear parameters of the NTV system in the i-th window respectively. That is, the
6
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
TV parameters in each of the windows are considered to be linear functions of time and the change of each parameter is
determined by an initial value and a slope. With Eq. (8) substituted into model (2), the equation of motion of the system
in the i-th window can be rewritten as
_ þ ðki þ kki ðt T i ÞÞxðtÞ
ðmi þ kmi ðt T i ÞÞ€xðtÞ þ ðci þ kci ðt T i ÞÞxðtÞ
n
P
ðsmi þ ksmi ðt T i ÞÞNm ðtÞ ¼ f ðtÞ
þ
ðT i 6 t 6 T iþ1 Þ
ð9Þ
m¼1
By substituting the measurement data into Eq. (9), the discrete equations in the i-th window including N wi sampling
points under the assumption of ‘‘short time linearly varying” can be written as
2
€xðT i þ DtÞ
Dt €xðT i þ DtÞ
€xðT i Þ
€xðT i þ 2DtÞ
2Dt €xðT i þ 2DtÞ
6 0
6
6
6 xðT
_ iÞ
_ i þ DtÞ
_ i þ 2DtÞ
xðT
xðT
6
6 0
_ i þ DtÞ
_ i þ 2DtÞ
Dt xðT
2Dt xðT
6
6
6 xðT i Þ
xðT
þ
D
tÞ
xðT
i
i þ 2DtÞ
6
6
Dt xðT i þ DtÞ
2Dt xðT i þ 2DtÞ
6 0
6
6 N1 ðT i Þ
N 1 ðT i þ DtÞ
N1 ðT i þ 2DtÞ
6
6
0
D
t
N
ðT
þ
D
tÞ
2
D
t N 1 ðT i þ 2DtÞ
6
1
i
6
6 .
.
..
..
6 ..
.
6
6
4 Nn ðT i Þ
Nn ðT i þ DtÞ
Nn ðT i þ 2DtÞ
..
.
Dt Nn ðT i þ DtÞ 2Dt Nn ðT i þ 2DtÞ 0
3T 2
3
€xðT iþ1 Þ
mi
7
6
ðT iþ1 T i Þ €xðT iþ1 Þ 7
7 6 kmi 7
7 6
7
7 6 ci 7
_xðT iþ1 Þ
7 6
7 2
3
7
6
_ iþ1 Þ 7
ðT iþ1 T i Þ xðT
f ðT i Þ
7 6 kci 7
7 6
7 6
7 6 ki 7 6 f ðT i þ DtÞ 7
xðT iþ1 Þ
7
7 6
7 6
7
7 6
7
7
ðT iþ1 T i Þ xðT iþ1 Þ 7 6 kki 7 ¼ 6
f
ðT
þ
2
D
tÞ
i
7
6
7 6
7
7
7 6 s1i 7 6
.
N1 ðT iþ1 Þ
7
6
..
7 6
7 4
5
7 6
7
ðT iþ1 T i Þ N1 ðT iþ1 Þ 7 6 ks1i 7
7 6
7
f ðT iþ1 Þ
7 6 .. 7
..
7 6 . 7
.
7 6
7
7 6
7
5 4 sni 5
Nn ðT iþ1 Þ
ksni
ðT iþ1 T i Þ Nn ðT iþ1 Þ
ð10Þ
in which the model can be also represented by a matrix equation as
Ui Hi ¼ F i
ð11Þ
where
2
€xðT i Þ
€xðT i þ DtÞ
Dt x€ðT i þ DtÞ
€xðT i þ 2DtÞ
2Dt €xðT i þ 2DtÞ
_ i þ 2DtÞ
xðT
6 0
6
6
6 xðT
_ iÞ
_ i þ DtÞ
xðT
6
6 0
_ i þ DtÞ
_ i þ 2DtÞ
Dt xðT
2Dt xðT
6
6
6 xðT i Þ
xðT
þ
D
tÞ
xðT
i
i þ 2DtÞ
6
6
0
D
t
xðT
þ
D
tÞ
2
D
t
xðT
6
i
i þ 2DtÞ
Ui ¼ 6
6 N1 ðT i Þ
N
ðT
þ
D
tÞ
N
ðT
þ
2DtÞ
1
1
i
i
6
6
Dt N1 ðT i þ DtÞ 2Dt N1 ðT i þ 2DtÞ
6 0
6
6 .
..
..
6 ..
.
.
6
6
4 Nn ðT i Þ
Nn ðT i þ DtÞ
N n ðT i þ 2DtÞ
..
.
3T
€xðT iþ1 Þ
ðT iþ1 T i Þ €xðT iþ1 Þ 7
7
7
7
_ iþ1 Þ
xðT
7
_ iþ1 Þ 7
ðT iþ1 T i Þ xðT
7
7
7
xðT iþ1 Þ
7
7
ðT iþ1 T i Þ xðT iþ1 Þ 7 ;
7
7
N1 ðT iþ1 Þ
7
7
ðT iþ1 T i Þ N1 ðT iþ1 Þ 7
7
7
..
7
.
7
7
5
N n ðT iþ1 Þ
ð12Þ
Dt Nn ðT i þ DtÞ 2Dt Nn ðT i þ 2DtÞ ðT iþ1 T i Þ Nn ðT iþ1 Þ
0
Hi ¼ ½ mi kmi ci kci ki kki s1i ks1i sni ksni T ;
Fi ¼ ½ f ðT i Þ f ðT i þ DtÞ f ðT i þ 2DtÞ f ðT iþ1 Þ T
In Eq. (11), the regression matrix Ui is formed by the measured output signals including the displacement, velocity and
acceleration at each sampling point, and the vector of external force F i can be directly measured. Then, to determine the NTV
parameters during this time window, the singular value decomposition (SVD) is adopted to the LS equations of the unknown
coefficient vector Hi :
Ui ¼ Ui Si VTi ¼ ½Uin ; Ui Ri
0
VTi
ð13Þ
1
T
Hi ¼ ðRi VTi Þ UTni F i ¼ Vi R1
i Uni F i
Ri
ð14Þ
is a diagonal matrix consisting of the singular values of Ui , Ui ¼ ½Uin ; Ui and Vi are the left and right singular
0
vectors of Ui , respectively. It has been proved that the SVD is robust to measurement noise in solving the LS equations.
If the entire time period is divided into several consecutive windows as ½T 0 ; T 1 ; ½T 1 ; T 2 ; ; ½T terminal - 1 ; T terminal , Eq. (14)
will be solved in each of the time windows, and the coefficient vectors in these time windows H0 ; H1 ; ; Hterminal1 can
be orderly obtained without difficulties. Therefore, all the parameters in the NTV system can be represented by piecewise
where Si ¼
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
7
linear
functions
of
time,
which
are
fitted
with
the
initial
values
of
the
coefficients
mi ; ci ; ki ; s1i ; ; sni ði ¼ 0; 1; 2; ; terminal 1Þ in each of the time windows. The main advantages of this new assumption
of ‘‘short time linearly varying” lie in the higher accuracy and stability in the parameter identification when compared with
the traditional assumption of ‘‘short time invariant”, which will be illustrated by the numerical example in Section 5.
4. Global regression equations in the whole time period for NTV dynamic systems
In Section 3 the new assumption of ‘‘short time linearly varying” is introduced into the identification of the NTV systems.
It is worth noting that the derived regression equations in model are established on the sampling points within the i-th window separately. The whole identification procedure is divided into a number of independent LS equations within different
shifting windows. Due to the advantage of the new assumption, this kind of identification strategy has a better performance
on the identification precision and stability than the traditional ‘‘short time invariant” strategy in the condition of no or low
measurement noise. Especially for the systems in which all the real TV parameters are linear functions of time, there exists
no error when there is no measurement noise. This is because the adopted assumption in this strategy is totally consistent
with the reality of these systems.
In the identification equations of each shifting window, there is no constraint to the initial and terminal values of the NTV
parameters. The LS algorithm in Eq. (14) seeks the optimal solution in the real domain freely. Though this kind of unconstrained equations can locally obtain the solution that is very close to the real value, they do not have a good noise resistance
performance in practical application.
In this section, to improve the stability and robustness of the identification method, a series of constraints will be added
to the identification procedure, which can connect all the separate windows and integrate the measured data of different
windows into the global regression equations of the whole time period.
For this purpose, all the NTV parameters are considered to continuously vary between any two adjacent shifting windows.
In other words, the identification values on any boundary sampling point obtained in two relative windows are the same.
Specifically, for an arbitrary boundary point T iþ1 (i ¼ 0; 1; 2; ; terminal 2), the identification results on this sampling point
obtained in the equations of the i-th window ½T i ; T iþ1 can be calculated from Eq. (8) as
8
mðT iþ1 Þ ¼ mi þ kmi DT i
>
>
>
>
>
cðT iþ1 Þ ¼ ci þ kci DT i
>
>
>
< kðT Þ ¼ k þ k DT
iþ1
i
i
ki
> s1 ðT iþ1 Þ ¼ s1i þ ks1i DT i
>
>
>
>
>
>
>
:
sn ðT iþ1 Þ ¼ sni þ ksni DT i
ði ¼ 0; 1; 2; ; terminal 2Þ
ð15Þ
where DT i ¼ T iþ1 T i is the width of the i-th time window. Meanwhile, the identification equations of the (i + 1)-th window
½T iþ1 ; T iþ2 also give an estimate of the NTV parameters on T iþ1 as
8
mðT iþ1 Þ ¼ miþ1
>
>
>
>
>
cðT iþ1 Þ ¼ ciþ1
>
>
>
< kðT Þ ¼ k
iþ1
iþ1
> s1 ðT iþ1 Þ ¼ s1;iþ1
>
>
>
>
>
>
>
:
sn ðT iþ1 Þ ¼ sn;iþ1
ði ¼ 0; 1; 2; ; terminal 2Þ
ð16Þ
According to the above statement about the continuously varying parameters on boundary sampling points, it can be
obtained from Eqs. (15) and (16) that
8
miþ1 ¼ mi þ kmi DT i
>
>
>
>
>
ciþ1 ¼ ci þ kci DT i
>
>
>
< k ¼ k þ k DT
iþ1
i
i
ki
>
s
¼
s
þ
k
D
T
1;iþ1
1i
s1i
i
>
>
>
>
>
>
>
:
sn;iþ1 ¼ sni þ ksni DT i
ði ¼ 0; 1; 2; ; terminal 2Þ
ð17Þ
Eq. (17) indicates that there exist unique solutions of the parameters on all of the boundary points in the identification
equations. This constraints connect those adjacent windows and keep the identification algorithm from seeking optimal
solutions completely freely when the measurement noise is high.
Then, with the constraints established in Eq. (17), the NTV system identification can be described by a constrained optimization problem as:
8
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
2
6
6
minimize DF ¼ 6
6
4
DF 1
DF 2
..
.
3
2
U1 H1 F 1
U2 H2 F 2
7 6
7 6
7¼6
7 6
5 4
..
.
3
7
7
7
7
5
DF terminal1
Uterminal1 Hterminal1 F terminal1
8
m
¼
m
þ
k
D
T
iþ1
i
mi
i
>
>
>
>
>
ciþ1 ¼ ci þ kci DT i
>
>
>
< k ¼ k þ k DT
iþ1
i
i
ki
to
ði ¼ 0; 1; 2; ; terminal 2Þ
>
s
¼
s
þ
k
D
T
1;iþ1
1i
s1i
i
>
>
>
>
>
>
>
:
sn;iþ1 ¼ sni þ ksni DT i
subject
ð18Þ
in which the regression matrix Ui and the vector of external force F i are given by Eq. (12).
It is noted that Eq. (8) within the last window ½T terminal - 1 ; T terminal gives the relation:
8
mterminal ¼ mðT terminal Þ ¼ mterminal1 þ km;terminal1 DT terminal1
>
>
>
>
>
cterminal ¼ cðT terminal Þ ¼ cterminal1 þ kc;terminal1 DT terminal1
>
>
>
<k
terminal ¼ kðT terminal Þ ¼ kterminal1 þ kk;terminal1 DT terminal1
> s1;terminal ¼ s1 ðT terminal Þ ¼ s1;terminal1 þ ks1;terminal1 DT terminal1
>
>
>
>
>
>
>
:
sn;terminal ¼ sn ðT terminal Þ ¼ sn;terminal1 þ ksn;terminal1 DT terminal1
ð19Þ
When solving this complicated optimization equation, the changing rates of unknown system TV parameters
kmi ; kci ; kki ; ks1i ; :::; ksni can be firstly eliminated by Eq. (17) for 0 6 i 6 terminal 2 and Eq. (19) for i ¼ terminal 1:
8
m mi
kmi ¼ iþ1
>
DT i
>
>
>
ciþ1 ci
>
>
k
¼
ci
>
DT i
>
>
>
< k ¼ kiþ1 ki
ki
DT i
s1;iþ1 s1i
>
>
k
¼
>
s1i
DT i
>
>
>
>
>
>
>
:
s
sni
ksni ¼ n;iþ1
DT i
ði ¼ 0; 1; 2; ; terminal 1Þ
ð20Þ
With Eq. (20) substituted into Eq. (9), the NTV system within the i-th ði ¼ 0; 1; 2; ; terminal 1Þ shifting window can be
represented by
miþ1 mi
ciþ1 ci
kiþ1 ki
_
ðt T i ÞÞ€xðtÞ þ ðci þ
ðt T i ÞÞxðtÞ
þ ðki þ
ðt T i ÞÞxðtÞ
DT i
DT i
DT i
n
X
sm;iþ1 smi
þ
ðsmi þ
ðt T i ÞÞNm ðtÞ ¼ f ðtÞ ðT i 6 t 6 T iþ1 Þ
DT i
m¼1
ðmi þ
ð21Þ
in which the parameters to be identified are the initial and terminal values within the window, without the changing rates of
them. Then, Eq. (21) can be rewritten in a simpler form as
i €
i €
ð1 tT
ÞxðtÞ mi þ tT
xðtÞ miþ1
DT i
DT i
i _
i _
þð1 tT
ÞxðtÞ ci þ tT
xðtÞ ciþ1
DT i
DT i
i
i €
þð1 tT
ÞxðtÞ ki þ tT
xðtÞ kiþ1
DT
DT
i
i
i
i
þð1 tT
ÞN1 ðtÞ s1i þ tT
N1 ðtÞ s1;iþ1
DT
DT
i
ðT i 6 t 6 T iþ1 Þ
ð22Þ
i
þ þ i
i
þð1 tT
ÞNn ðtÞ sni þ tT
Nn ðtÞ sn;iþ1 ¼ f ðtÞ
DT
DT
i
i
or in the vector form:
ui ðtÞ hTi hTiþ1
T
¼ f ðtÞ
ð23Þ
where ui ðtÞ is a 2ð3 þ nÞ order row vector of which the elements are all functions of time, and hi is a 2ð3 þ nÞ order column
vector:
9
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
ui ðtÞ ¼
t Ti
t Ti
t Ti
t Ti
t Ti
€xðtÞ; 1 _
xðtÞ;
1
xðtÞ; 1 N1 ðtÞ; ; 1 N n ðtÞ;
1
DT i
DT i
DT i
DT i
DT i
t Ti
t Ti
t Ti
t Ti
t Ti
T
€xðtÞ;
_
€xðtÞ;
N1 ðtÞ; ;
Nn ðtÞ ; hi ¼ ½ mi ci ki s1i sni xðtÞ;
DT i
DT i
DT i
DT i
DT i
ð24Þ
With measurement data during the interval ½T i ; T iþ1 Dt substituted into Eq. (23), the discrete equations of the i-th window (the point T iþ1 is removed from the i-th window to avoid repeated calculation) can be written as
Ui Hi ¼ F i ð25Þ
in which
i ¼ 0; 1; ; terminal 1
2
€
1 DDTt €xðT i þ DtÞ
1 2DDTt €xðT i þ 2DtÞ
6 xðT i Þ
i
i
6
6 _
Dt _
2Dt _
1 DT i xðT i þ DtÞ
1 DT i xðT i þ 2DtÞ
6 xðT i Þ
6
6
Dt
6 xðT i Þ
1 DT xðT i þ DtÞ
1 2DDTt xðT i þ 2DtÞ
6
i
i
6
6
Dt
2Dt
6 N 1 ðT i Þ
1 DT N1 ðT i þ DtÞ
1 DT N1 ðT i þ 2DtÞ
6
i
i
6
6 .
.
..
..
6 ..
.
6
6
6 Nn ðT i Þ
1 DDTti Nn ðT i þ DtÞ
1 2DDTti Nn ðT i þ 2DtÞ
Ui ¼ 6
6
6 0
Dt €
2D t €
xðT i þ DtÞ
xðT i þ 2DtÞ
6
DT i
DT i
6
6
Dt _
2D t _
6 0
xðT i þ DtÞ
xðT i þ 2DtÞ
DT i
DT i
6
6
6
Dt
2D t
6 0
xðT i þ DtÞ
xðT i þ 2DtÞ
DT i
DT i
6
6
6 0
Dt
2D t
N ðT i þ DtÞ
N ðT i þ 2DtÞ
6
DT i 1
DT i 1
6
6 .
.
..
6 .
..
6 .
.
4
Dt
2Dt
N
ðT
þ
D
tÞ
N
ðT
0
n
n
i
i þ 2DtÞ
DT i
DT i
hi
¼ mi ci ki s1i sni miþ1 ciþ1 kiþ1
Hi ¼
hiþ1
for
Fi ¼ ½ f ðT i Þ f ðT i þ DtÞ f ðT i þ 2DtÞ f ðT iþ1 DtÞ ..
.
..
.
Dt
DT i
€xðT iþ1 DtÞ
Dt
DT i
_ iþ1 DtÞ
xðT
3T
7
7
7
7
7
7
Dt
7
xðT
D
tÞ
iþ1
7
DT i
7
7
Dt
7
N ðT iþ1 DtÞ
DT i 1
7
7
7
7
7
7
Dt
7
N ðT iþ1 DtÞ
DT i n
7
7 ;
Dt €
1 DT xðT iþ1 DtÞ 7
7
i
7
7
_ iþ1 DtÞ 7
1 DDTt xðT
7
i
7
7
Dt
1 DT i xðT iþ1 DtÞ 7
7
7
Dt
1 DT N1 ðT iþ1 DtÞ 7
7
i
7
7
7
7
5
Dt
1 DT Nn ðT iþ1 DtÞ
s1;iþ1
ð26Þ
i
sn;iþ1
T
;
T
The asterisk of superscript is used to distinguish the new matrix equation in Eq. (25) from that in Eq. (11).
However, the unknown vector of parameters Hi cannot be determined only by the identification equations in the i-th
window separately. This vector to be identified is composed of the parameter values at the start and the end of the i-th window. For i ¼ 0; 1; 2; ; terminal 2, the second half of Hi is exactly the same as the first half of Hiþ1 ; while for
i ¼ 1; 2; ; terminal 1, the first half of Hi is exactly the same as the second half of Hi1 . Therefore, to solve any of the equations of the windows, the equations of the closest windows must be included. To determine the parameters in each window,
the equations in all the shifting windows ½T 0 ; T 1 ; ½T 1 ; T 2 ; ; ½T terminal - 1 ; T terminal are going to be taken into account in calculation sequentially. The parameters in a certain window cannot be identified unless all other windows are considered. That
is, the estimates of the NTV parameters in the whole time period can be obtained together.
According to the above identification strategy, a global equation including all the shifting windows needs to be established. Firstly, a total vector of NTV parameters composed of the parameter values on the start and end points of all the shifting windows successively is designed as:
2
h0
6
6
H¼6
6
4
h1
..
.
hterminal
3
2
7 6
7 6
7¼6
7 6
5 6
4
mterminal
c0
k0
s10
sn0 ½ m1
c1
k1
s11
..
.
sn1 cterminal
kterminal
s1;terminal
3
T
½ m0
T
sn;terminal
T
7
7
7
7
7
5
ð27Þ
Based on the discrete equations of all the shifting windows given in Eq. (25) (i ¼ 0; 1; 2; ; terminal 1) and the total vector of NTV parameters given in Eq. (27), the total identification equations for the NTV system in the whole time period are
obtained as:
UH¼F
ð28Þ
10
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
where
h
U ¼ UT 0 UT 1 UT terminal1 UT terminal
iT
;
U0 ¼ U0 0Nw ½ð3þnÞðterminal1Þ ;
Ui ¼ 0Nw ½ð3þnÞi U1 0Nw ½ð3þnÞðterminali1Þ ði ¼ 1; 2; terminal 2Þ;
ð29Þ
Uterminal1 ¼ 0Nw ½ð3þnÞðterminal1Þ Uterminal1 ;
_ iþ1 Þ xðT iþ1 Þ N1 ðT iþ1 Þ N n ðT iþ1 Þ ;
Uterminal ¼ 01½ð3þnÞterminal €xðT iþ1 Þ xðT
T
F ¼ ½ f ðT 0 Þ f ðT 0 þ DtÞ f ðT terminal DtÞ f ðT terminal Þ In Eq. (29), N w refers to the number of the sampling points in each shifting window. The number of rows of the total
regression matrix U is N w terminal þ 1, which is equal to the number of the total sampling points in the whole time period;
and the number of columns is ð3 þ nÞ ðterminal þ 1Þ, which is equal to the number of unknowns of the total identification
equations in H.
The LS estimates of the NTV parameters in the whole time period can be given by the SVD of the total regression matrix U,
which is similar to Eq. (14). Then, the constrained optimization problem in Eq. (18) is solved. The variation of the NTV param
eters is determined by the segmental linear function of time which can be derived from the elements of the total vector H. As
mentioned above, all the parameter values in different windows are determined together, which is completely different from
the separate identification strategy introduced in Section 3.
Due to the constraints added to the NTV system equations, the proposed identification method performs well on reducing
the effects of measurement noise, showing a better stability and robustness compared to the traditional identification
method based on the classical assumption of ‘‘short time invariant” in Section 2 and the separate identification algorithm
in Section 3, which will be demonstrated by some numerical results in Section 5.
5. Error investigation on the measurement noise and numerical integration procedure
In addition to the inaccuracy due to the inconsistency between the assumption of ‘‘short time linearly varying” and the
true variation of the unknown parameters in each window, the measurement noise is inevitable in the practical application
of the identification. Besides, there is also a need for numerical differentiation or integration. They both lead to the signal
errors and can lower the identification accuracy. Thus, in this section, some investigation is performed on the identification
errors caused by these two main reasons.
The signals (external excitations, displacements, velocities, accelerations and nonlinear terms) with errors can be written
as
8
>
>
>
>
>
>
>
>
>
<
f ðtÞ ¼ ef ðtÞ þ f ðtÞ
€xðtÞ ¼ ea ðtÞ þ €xðtÞ
ð30Þ
_
x_ ðtÞ ¼ ev ðtÞ þ xðtÞ
>
>
>
>
>
>
xðtÞ ¼ ed ðtÞ þ xðtÞ
>
>
>
:
Ni ðtÞ ¼ eNi ðtÞ þ Ni ðtÞ; i ¼ 1; 2; ; n
_
in which f ðtÞ, €
xðtÞ, xðtÞ,
xðtÞ and N i ðtÞ are the real signals, and ef ðtÞ, ea ðtÞ, ev ðtÞ, ed ðtÞ and eNi ðtÞ are the errors respectively. By
substituting the above expressions into Eq. (10), the discrete equations in the i-th window can be obtained as
Ui Hi ¼ F i
ð31Þ
where
h
Ui ¼ Ui þ EUi ; F i ¼ F i þ EF i ; EUi ¼ Eu ðT i ÞT
Eu ðT i þ 1ÞT
Eu ðT iþ1 ÞT
iT
;
Eu ðtÞ ¼ ½ea ðtÞ; ea ðtÞðt T i Þ; ev ðtÞ; ev ðtÞðt T i Þ; ed ðtÞ; ed ðtÞðt T i Þ; eNi ðtÞ; eNi ðtÞðt T i Þ; ; eNn ðtÞ; eNn ðtÞðt T i Þ;
EF i ðtÞ ¼ ½ ef ðT i Þ ef ðT i þ 1Þ ef ðT iþ1 Þ ð32Þ
T
The LS solution of Eq. (31) can be written as
T 1 T Hi ¼ ðUi Ui Þ Ui F i
With Eq. (32) substituted, the LS solution can be rewritten as
ð33Þ
11
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
1
1
1
Hi ¼ ðUTi Ui þ EE Þ ðUTi þ EUi ÞðF i þ EF i Þ ¼ ðUTi Ui þ EE Þ UTi F i þ ðUTi Ui þ EE Þ ðEUi F i þ UTi EF i þ EUi EF i Þ
ð34Þ
where EE ¼ UTi EUi þ ETUi Ui þ ETUi EUi . According to the matrix theory about the generalized formula for matrix inversion:
1
ðA þ BÞ1 ¼ A1 A1 ðI þ BA1 Þ BA1
ð35Þ
the following expression can be obtained
1
Hi ¼ ðUTi Ui Þ
1 1
1
ðUTi Ui Þ ½I þ EE ðUTi Ui Þ EE ðUTi Ui Þ
1
1
UTi F i þ ðUTi Ui þ EE Þ ðEUi F i þ UTi EF i þ EUi EF i Þ
1
¼ ðUTi Ui Þ UTi F i þ EI ¼ Hi þ EI
ð36Þ
in which
1 1
1
1
1
EI ¼ ðUTi Ui Þ ½I þ EE ðUTi Ui Þ EE ðUTi Ui Þ UTi F i þ ðUTi Ui þ EE Þ ðEUi F i þ UTi EF i þ EUi EF i Þ
ð37Þ
is the error of parameter identification induced by signal errors in Eq. (30). For simplicity, the subscript ‘‘i” referring to the ith window is omitted in the expressions of EE and EI in Eqs. (34)–(37).
EI is determined by the combination of the measurement noise and numerical calculation error. To clearly show the effect
of the signal errors caused by the measurement noise and differentiation or integration process, a linear oscillator with TI
parameters under simple harmonic excitation which can be solved analytically is considered:
m€x þ cx_ þ kx ¼ Acosxt;
x0 ¼ x_ 0 ¼ 0
where
m ¼ 1; c ¼ 0:01; k ¼ 10; A ¼ 1; x ¼ 2p;
solution :
i
_
x ¼ Xcos½xt u þ enxn t ½ðx0 XcosuÞcosxd t þ x0 X xsinuþfxxn ðx0 XcosuÞ sinxd t ;
ð38Þ
d
where
=xn
1
X ¼ Ak pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
; u ¼ tan1 ð2fxx
; xn ¼
2
=x Þ2
½1ðx=xn Þ þð2fx=xn Þ
2
2
n
qffiffiffi
k
;f
m
¼ 2mcxn ; xd ¼ xn
pffiffiffiffiffiffiffiffiffiffiffiffiffi
1 f2
The analytical expressions of the velocity and acceleration response can also be derived based on Eq. (38). Data during 0
to 100 s with sampling frequency 1000 Hz and window width of 2 s are chosen in this case. It is worth noting that the
assumption of ‘‘short time linearly varying” can coincide perfectly with the true parameters in this system. Therefore, the
errors caused by the measurement noise and numerical computation can be separated for observation.
Firstly, by using the harmonic excitation, analytical displacement, velocity and acceleration response data without noise
pollution in the identification procedure, the identification results of m, c, and k are given in Fig. 4. There is nearly no error in
the identified parameters. Then, to examine the effect of numerical integration on the accuracy of identification results separately, only the acceleration response from the analytical solution is used as the measured signal. The velocity and displacement used in the identification procedure are obtained by trapezoidal integration of the acceleration. The corresponding
Fig. 4. Identification results with analytical signals.
12
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
identification results are shown in Fig. 5. It can be seen that the errors (within 0.01%) occur mainly in the last 10 s. They could
be caused by the increasing cumulative error in the numerical integration. To investigate the errors caused by the measurement noise separately, different levels of noise (1%, 2%, 5%, and 10% of the real signals) are added to the external excitation
and response signals (analytical displacement, velocity and acceleration). The corresponding identification results are given
in Fig. 6. Obviously, the impact of the measurement noise on the identification accuracy, especially on the identified damping
coefficient, is much more serious than that of the numerical integration error. 10% measurement noise can lead to more than
3% error of identified m, 8% error of identified k, and more error of identified c on some of the time points.
In fact, the measurement noise in the acceleration signals combined with the cumulative error in the numerical integration could cause even much more significant errors. To intuitively show the serious consequence of the measurement noise
arising from the numerical integration, the noise in the level of 5% is added to the acceleration signals. One of the simplest
numerical integration, trapezoidal formula, is adopted to this illustration. The polluted acceleration, and the integrated
velocity and displacement response along with the analytical ones are as shown in Fig. 7. The corresponding identification
results are given in Fig. 8. A trend term in the displacement signal as shown in Fig. 7, is generated in the numerical
Fig. 5. Identification results considering the error of numerical integration.
Fig. 6. Identification results with different levels of measurement noise (1%, 2%, 5%, and 10%).
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
13
integration, which causes huge errors in the identification results. Thus it can be seen from Fig. 8 that the measurement noise
combined with the cumulative error can lead to a large EI term in Eq. (36).
To identify the parameters accurately, the trend term needs to be eliminated. Researchers have carried out studies on this
problem and some useful methods have been proposed [24–27]. The elimination of the trend term will not be discussed in
the present paper because it is beyond the scope of this research. While in the case as shown in Fig. 7, the trend term can be
eliminated by simply assuming the mean value of the true displacement to be zero in a period of time. By this way, the
detrended displacement signal can be obtained as shown in Fig. 9. The identification results are shown in Fig. 10. For the
purpose of comparison, the method based on the traditional assumption of ‘‘short time invariant” is adopted to this case
and the identification results are also given.
Fig. 7. Signals with measurement noise and cumulative error.
Fig. 8. Identification results considering 5% measurement noise and the cumulative error.
14
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
Fig. 9. Detrended displacement signal compared with the analytical and integrated displacement.
From Fig. 10, the accuracy of the identification results is improved significantly when compared with that as shown in
Fig. 8. Besides, it can be seen that the identification errors caused in these two methods are at a similar level. In fact, as mentioned in the previous sections, the advantage of the new method is more obvious for TV systems, especially for those with
fast changing parameters. In the next section, this will be illustrated in detail.
The errors in the identification results caused by the measurement noise and numerical integration are investigated in
this section. It is found that the cumulative error in the numerical integration can remarkably increase the identification
errors cause by the measurement noise. The accurate numerical integration method which can eliminate the trend term
in signals is needed. Because this is beyond the scope of this paper, in the following part of this paper, it is reasonable to
assume that all the acceleration, velocity and displacement responses can be measured directly.
6. An illustrative example
_
This section looks into a Duffing oscillator with TV parameters m€
xðtÞ þ cxðtÞ
þ kxðtÞ þ knl x3 ðtÞ ¼ f ðtÞ where the parameters
15; 0 6 t 6 50s
pt, and k ¼
respectively. Among these parameters, m, k and knl
are m ¼ 0:04t þ 4, c ¼ 0:5, k ¼ 200sin 230
nl
30; 50 < t 6 100s
exhibit TV characteristics. Besides, different from common structural dynamic systems, negative stiffness (k < 0) can emerge
from this Duffing oscillator, which results in unstable equilibrium point, bifurcation, and other more complex nonlinear phenomena. In fact, research on parameter identification of this type of system has practical significance for engineering application. For example, the plate in an elevated thermal environment studied by Lee shows similar characteristics [28]. The
€ þ x0 nq_ þ x20 ð1 sÞq þ jq3 ¼ f 0 þ f ðtÞ, in which the
rectangular simply-supported plate equation of the first mode gives q
Fig. 10. Identification results with the detrended signal (with 5% measurement noise, compared with the assumption of ‘‘short time invariant”).
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
15
combined stiffness x20 ð1 sÞ is positive when the thermal loading is weak (s < 1 for pre-buckling), whereas it becomes negative under a strong thermal loading (s > 1 for post-buckling).
2pt
shown in Fig. 11 is considered and the system is simulated by
A variable frequency external excitation f ðtÞ ¼ 50sin 10:005t
xð0Þ
0
the Runge-Kutta method for 100 s with the response data sampled every 0.001 s. With the initial state _
¼
, the
xð0Þ
10
simulated motion of the system is shown in Fig. 12. Fig. 12 shows that there exist multiple equilibrium points as well as
bifurcation phenomena in the system.
In the identification procedure, the whole time period including 100,001 sampling points is divided into 200 windows
and the width of every window is 0.5 s. For simplicity, as mentioned in Section 5, it is assumed that all the response data,
including displacements, velocities and accelerations can be measured directly here.
Firstly, the ideal condition without measurement noise is studied. The system is identified by three different methods
above with the measured input and output data respectively:
Method 1: method based on traditional assumption of ‘‘short time invariant” [4];
Method 2: separate identification algorithm based on the assumption of ‘‘short time linearly varying” introduced in
Section 3;
Method 3: global identification algorithm based on the assumption of ‘‘short time linearly varying” introduced in
Section 4.
The identification results for m, c, k and knl of the three methods are shown in Figs. 13–16. The values of the mean absolute
error (MAE) of the TV parameter estimates are given in Table 1. Note that in this paper MAE is defined as
MAE ¼
N
1 X
j^cðtÞ cðtÞj
N t¼1
ð39Þ
where ^cðtÞ represents the estimates of coefficients cðtÞ in the NTV system and N is the length of the observations.
It is obvious that the calculated MAE of Method 2 and 3 are much smaller than Method 1. As shown in Figs. 13–16, even in
the case of no measurement noise, the errors of Method 1 are too much to be acceptable. Especially for m, c and knl , the MAE
reaches the same order of magnitude as the true value. In comparison in Table 1, Method 2 has the smallest MAE for all the
NTV parameters, which indicates the advantage of the new assumption of ‘‘short time linearly varying” in identification pre-
Fig. 11. The variable frequency external excitation.
Fig. 12. Simulated motion of the system.
16
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
Fig. 13. Identification results for m
cision. The increased error in Method 3 mainly comes around 50 s, where a step change of knl from 15 to 30 happens. It is
caused by the global strategy, in which the constraints of continuous variation between different windows are not consistent
with the truth. Because of the added constraints in the global identification equations, the dramatic changes in parameters
will not only affect the identification results in one window separately, but also involve the close. While in Method 2, this
kind of error does not exist. As a result, to obtain higher accuracy in the condition of no or low measurement noise, Method 2
is better than Method 3.
In order to verify the stability of the methods for different systems in which the variation of parameters is more rapid, a
pt is adopted while other parameters remain the same. It is worth noting that the changing
new linear stiffness k ¼ 200sin 210
rate of k is high and it is usually very difficult to achieve a good identification accuracy in such a system.
The identification results are shown in Figs. 17–20, and the calculated MAE is given in Table 2. In general, compared with
the other two methods, Method 3 has the highest accuracy for the new system. Method 3 is more stable than Method 2 for
the system with more rapidly varying parameters, which can be seen from Figs. 17–20 and Table 2. In fact, the higher changing rate of NTV parameters cannot accord with the original assumption of ‘‘short time linearly varying”, which will remarkably increase the residual error DF i in Eqs. (7) and (18). For Method 2, the increased residual error leads to violent fluctuation
of the parameter values in the free search of different separate windows. This is the main cause of the increased errors of
Method 2. While the constraints between different windows added in Method 3 stop the parameters from arbitrarily
Fig. 14. Identification results for c.
Fig. 15. Identification results for k.
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T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
Fig. 16. Identification results for knl .
Table 1
MAE of the TV parameter estimates.
Method 1
Method 2
Method 3
m
c
k
knl
0.9846
0.0642
0.1037
1.4121
0.1613
0.4368
23.1324
1.9918
3.1404
9.8210
0.5390
0.6410
Fig. 17. Identification results for m (with rapidly varying k).
Fig. 18. Identification results for c (with rapidly varying k).
changing to find the optimal solution, which is affected severely by the residual errors in the identification equations. It is the
strategy of constrained searching that improves the stability of Method 3 in the systems with rapidly changing parameters.
Apart from better stability for the fast changing systems, it is remarkable that the superiority of the proposed global identification strategy is more significant when the measurement noise level increases. To verify this, the identification is carried
out on the above system at different levels of white Gaussian noise (signal-to-noise ratios (SNRs) equal to 30, 25, and 20 dB
respectively) by Method 2 and 3. To investigate the effect of the window size on the robustness under noisy circumstances,
0.5 s, 1 s, 2 s, and 4 s are tested respectively in the three methods. The values of MAE of the TV parameter estimates are given
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T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
Fig. 19. Identification results for k (with rapidly varying k).
Fig. 20. Identification results for knl (with rapidly varying k).
Table 2
MAE of the TV parameter estimates (with rapidly varying k).
Method 1
Method 2
Method 3
m
c
k
knl
2.0463
0.4733
0.3870
5.0450
1.0039
0.6350
66.1064
17.2626
12.8976
33.7092
2.4093
2.7317
in Table 3. Also, for a better illustration, the identification results by Method 3 with noise (SNR = 30) considering different
window widths are given in Figs. 21–24
Obviously, window size being the same, most of the calculated MAE of Method 3 is smaller than that of Method 2. These
results confirm that the proposed global identification strategy is more significant when the noise level increases. Besides,
the robustness of both methods can be enhanced by appropriately increasing the width of each identification window. In
comparison with other different window sizes (0.5, 1, and 2 s), the width of 4 s gives the best robustness and the most accurate estimates for both Method 2 and 3 with the existence of measurement noise, which can be observed in Table 3. For a
better illustration, the identification results by Method 3 with noise (SNR = 30) considering different window widths are
given in Figs. 21–24. Understandably, given the sampling frequency of 1000 Hz, the number of equations in each window
Table 3
MAE of the TV parameter estimates (SNR = 30, 25 and 20 dB).
Method
Window
size (s)
30 dB
m
c
k
knl
25 dB
m
c
k
knl
20 dB
m
c
k
knl
2
0.5
1
2
4
4.9899
2.6084
0.6647
0.3962
13.8488
8.5865
0.5634
0.5747
142.7971
81.7558
31.0366
22.4959
28.5145
13.1145
5.1167
2.5514
5.0047
3.0172
1.1431
0.7001
13.8010
9.7259
1.0392
0.7189
149.2872
98.3802
49.6399
32.8634
30.0837
15.3265
8.4572
4.8343
4.9954
3.4582
1.9522
1.4006
12.6791
11.2063
2.2270
0.8998
150.7619
111.8907
73.3872
50.1662
30.0427
17.4607
12.7234
8.4298
3
0.5
1
2
4
2.9290
1.0758
0.5914
0.4343
10.0087
1.3887
0.5642
0.5241
92.7832
41.5229
24.6460
19.5781
14.6157
7.1406
3.7401
2.4793
3.2707
1.6336
0.9239
0.6767
11.1374
2.3218
0.9625
0.6486
106.1391
62.4183
37.5919
28.2463
16.2701
10.2361
6.0649
4.7443
3.5665
2.2120
1.5067
1.2494
11.8373
3.3273
1.1635
0.8280
114.8740
83.3578
58.6417
46.5821
17.7783
13.0701
9.7232
8.3401
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
Fig. 21. Identification results for m with window width = 1 s, 2 s and 4 s.
Fig. 22. Identification results for c with window width = 1 s, 2 s and 4 s.
Fig. 23. Identification results for k with window width = 1 s, 2 s and 4 s.
Fig. 24. Identification results for knl with window width = 1 s, 2 s and 4 s.
19
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T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
of different sizes is 500, 1000, 2000, and 4000 respectively, among which the window of 4 s involves the most regression
equations, and therefore possesses the strongest capability against noise disturbance. This indicates that appropriately
increasing the width of window within a certain range can effectively improve the identification performance when the
input and output data is disturbed by noise. On the other hand, too large width goes against the wish to track the rapid
changes in system parameters when the noise is in a low level.
7. Conclusions
In this paper a new NTV system identification method based on the assumption of ‘‘short time linearly varying” is proposed, where the TV parameters are identified in different time windows and a global algorithm is adopted when the measurement noise is high. An error investigation is carried out, which shows the impact of the measurement noise and
numerical integration on the identification accuracy. A TV Duffing system is identified to test the performance of the newly
proposed method. In this numerical example, several different types of NTV parameters including both smooth and abrupt
changes are considered. The value of MAE is used to measure the accuracy of the estimated parameters and the classical
identification method based on the traditional assumption of ‘‘short time invariant” is also employed in the comparison
to verify the validity and the accuracy of the proposed method. The simulation results show that the method based on
the new assumption can give more accurate estimates for different types of NTV parameters when the noise level is relatively low; meanwhile, the global identification strategy effectively enhances the robustness and stability of the new
method, which can improve the performance of identification when the signals are badly disturbed by noise. Though the
method is illustrated by an SDOF system numerical simulation, it can also deal with MDOF systems.
An advantage of the newly proposed method over the traditional method based on the assumption of ‘‘short time invariant” lies in that it causes much smaller errors when the window size is increased for better estimates when the noise level is
high. That is, it ensures the robustness and stability without sacrificing the identification precision. The essential contradiction between the robustness and precision mentioned in Section 1 is avoided in the new assumption of ‘‘short time linear
varying”.
The main weakness of this proposed identification method for MDOF systems is its increasing computational load. In
comparison with that based on the assumption of ‘‘short time invariant”, the unknowns in the regression equations of each
identification window of the newly proposed method are the assumed initial values and changing rates of all the parameters,
of which the number is doubled. When applying the proposed method to MDOF systems, one will encounter the heavier
computational load due to the increasing number of parameters and DOFs.
Besides, though the essential contradiction mentioned above in the traditional assumption is greatly cushioned by the
proposed method, this research does not involve the quantitative determination of the size of the identification window.
To a certain extent, it is decided by a priori knowledge or practical experience. In future studies, a detailed quantitative analysis on determining the optimal identification window size is necessary.
CRediT authorship contribution statement
Tengfei Chen: Conceptualization, Methodology, Writing-original draft. Huan He: Conceptualization, Methodology.
Guoping Chen: Supervision. Yuxuan Zheng: Writing-review & editing. Shuo Hou: Software, Validation. Xulong Xi: Funding
acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgements
This research is supported by the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant No. MCMS-I-0118G01) and Nanjing University of Aeronautics and Astronautics PhD short-term visiting scholar project (Grant No. 190609DF01).
References
[1] J.P. Noël, G. Kerschen, Nonlinear system identification in structural dynamics: 10 more years of progress, Mech. Syst. Signal Process. 83 (2017) 2–35,
https://doi.org/10.1016/j.ymssp.2016.07.020.
[2] Q. Liu, J. Mohammadpour, R. Toth, N. Meskin, Non-parametric identification of linear parameter-varying spatially-interconnected systems using an LSSVM approach, Proc. Am. Control Conf. 2016 (2016-July) 4592–4597, https://doi.org/10.1109/ACC.2016.7526076.
T. Chen et al. / Mechanical Systems and Signal Processing 139 (2020) 106620
21
[3] T. Ahmed-Ali, G. Kenné, F. Lamnabhi-Lagarrigue, Identification of nonlinear systems with time-varying parameters using a sliding-neural network
observer, Neurocomputing. 72 (2009) 1611–1620, https://doi.org/10.1016/j.neucom.2008.09.001.
[4] T. Chen, H. He, C. He, G. Chen, New parameter-identification method based on QR decomposition for nonlinear time-varying systems, J. Eng. Mech. 145
(2019), https://doi.org/10.1061/(ASCE)EM.1943-7889.0001555.
[5] Y. Li, W.G. Cui, Y.Z. Guo, T. Huang, X.F. Yang, H.L. Wei, Time-varying system identification using an ultra-orthogonal forward regression and
multiwavelet basis functions with applications to EEG, IEEE Trans. Neural Networks Learn. Syst. 29 (2018) 2960–2972, https://doi.org/10.1109/
TNNLS.2017.2709910.
[6] I.A. Kougioumtzoglou, P.D. Spanos, An identification approach for linear and nonlinear time-variant structural systems via harmonic wavelets, Mech.
Syst. Signal Process. 37 (1–2) (2013) 338–352, https://doi.org/10.1016/j.ymssp.2013.01.011.
[7] X. Zhang, J. Xu, An extended synchronization method to identify slowly time-varying parameters in nonlinear systems, IEEE Trans. Signal Process.
(2018), https://doi.org/10.1109/TSP.2017.2770092.
[8] Y. Guo, L.Z. Guo, S.A. Billings, H.L. Wei, Identification of continuous-time models for nonlinear dynamic systems from discrete data, Int. J. Syst. Sci. 47
(2016) 3044–3054, https://doi.org/10.1080/00207721.2015.1069906.
[9] Y. Guo, L. Wang, Y. Li, J. Luo, K. Wang, S.A. Billings, L. Guo, Neural activity inspired asymmetric basis function TV-NARX model for the identification of
time-varying dynamic systems, Neurocomputing. 357 (2019) 188–202, https://doi.org/10.1016/j.neucom.2019.04.045.
[10] C. Wang, W. Guan, J.Y. Wang, B. Zhong, X. Lai, Y. Chen, L. Xiang, Adaptive operational modal identification for slow linear time-varying structures based
on frozen-in coefficient method and limited memory recursive principal component analysis, Mech. Syst. Signal Process. 100 (2018) 899–925, https://
doi.org/10.1016/j.ymssp.2017.06.018.
[11] K. Yu, Advances in method of time-varying linear/nonlinear structural system identification and parameter estimate, Csb. 54 (2009) 3147, https://doi.
org/10.1360/972008-2471.
[12] J. Prawin, A.R.M. Rao, Nonlinear identification of MDOF systems using Volterra series approximation, Mech. Syst. Signal Process. 84 (2017) 58–77,
https://doi.org/10.1016/j.ymssp.2016.06.040.
[13] S. Ozer, H. Zorlu, S. Mete, System identification application using Hammerstein model, Sadhana – Acad. Proc. Eng. Sci. 41 (2016) 597–605, https://doi.
org/10.1007/s12046-016-0505-8.
[14] F. Ding, T. Chen, Performance bounds of forgetting factor least-squares algorithms for time-varying systems with finite measurement data, IEEE Trans,
Circuits Syst. I Regul. Pap. 52 (2005) 555–566, https://doi.org/10.1109/TCSI.2004.842874.
[15] Y. Li, H.L. Wei, S.A. Billings, P.G. Sarrigiannis, Identification of nonlinear time-varying systems using an online sliding-window and common model
structure selection (CMSS) approach with applications to EEG, Int. J. Syst. Sci. (2016), https://doi.org/10.1080/00207721.2015.1014448.
[16] S. Chen, S.A. Billings, W. Luo, Orthogonal least squares methods and their application to non-linear system identification, Int. J. Control. (1989), https://
doi.org/10.1080/00207178908953472.
[17] Y. Guo, L.Z. Guo, S.A. Billings, H.L. Wei, An iterative orthogonal forward regression algorithm, Int. J. Syst. Sci. (2015), https://doi.org/10.1080/
00207721.2014.981237.
[18] S.A. Billings, H.L. Wei, A new class of wavelet networks for nonlinear system identification, IEEE Trans. Neural Networks. (2005), https://doi.org/
10.1109/TNN.2005.849842.
[19] H.L. Wei, S.A. Billings, J. Liu, Term and variable selection for non-linear system identification, Int. J. Control. (2004), https://doi.org/10.1080/
00207170310001639640.
[20] Y. Guo, L.Z. Guo, S.A. Billings, H.L. Wei, Ultra-orthogonal forward regression algorithms for the identification of non-linear dynamic systems,
Neurocomputing. (2016), https://doi.org/10.1016/j.neucom.2015.08.022.
[21] Y. Li, H.L. Wei, S.A. Billings, Identification of time-varying systems using multi-wavelet basis functions, IEEE Trans. Control Syst. Technol. (2011),
https://doi.org/10.1109/TCST.2010.2052257.
[22] H.L. Wei, S.A. Billings, J.J. Liu, Time-varying parametric modelling and time-dependent spectral characterisation with applications to EEG signals using
multiwavelets, Int. J. Model. Identif. Control. (2010), https://doi.org/10.1504/IJMIC.2010.032802.
[23] Y. Li, M.L. Luo, K. Li, A multiwavelet-based time-varying model identification approach for time-frequency analysis of EEG signals, Neurocomputing.
(2016), https://doi.org/10.1016/j.neucom.2016.01.062.
[24] S. Han, Measuring displacement signal with an accelerometer, J. Mech. Sci. Technol. 24 (2010) 1329–1335, https://doi.org/10.1007/s12206-010-03361.
[25] Y. Yang, D. Jiang, Numerical Integration and Complex Trend Term Elimination of Acceleration Signal in Fault Diagnosis, 501 (2015) 501–506.
doi:10.2991/ipemec-15.2015.93.
[26] G. Mingkun, L. Zhenhua, Identification of a Mechanism ’ s Vibration Velocity and Displacement Based on the Acceleration Measurement, (2011) 5–9.
doi:10.13433/j.cnki.1003-8728.2011.04.001.
[27] E.D. Engineering, Study on the real – time elimination method of random noise and trend terms in acceleration signal, 21 (2013) 18–22. doi:10.14022/
j.cnki.dzsjgc.2013.14.033.
[28] J. Lee, Large-amplitude plate vibration in an elevated thermal environment, Appl. Mech. Rev. 46 (1993) S242–S254, https://doi.org/10.1115/1.3122643.