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Nonparametric frequency response function estimates for switching piecewise linear systems
Author’s Accepted Manuscript Nonparametric Frequency Response Function Estimates for Switching Piecewise Linear Systems Tao Song, Fubiao Zhang, Defu Lin
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To appear in: Signal Processing Received date: 3 November 2015 Revised date: 3 May 2016 Accepted date: 11 May 2016 Cite this article as: Tao Song, Fubiao Zhang and Defu Lin, Nonparametric Frequency Response Function Estimates for Switching Piecewise Linear Systems, Signal Processing, http://dx.doi.org/10.1016/j.sigpro.2016.05.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Nonparametric Frequency Response Function Estimates for Switching Piecewise Linear Systems Tao Songa, Fubiao Zhangb*, Defu Lina a
School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 10086, China
b
Institute of Flight System Dynamic, Technical University of Munich, Garching, 85748, Germany
*
Corresponding author. Fubiao Zhang, PHD candidate, E-mail: [email protected]
Abstract This paper proposes two algorithms (ATIRMM and ALPM) for estimating the time-varying behavior of single input single output (SISO) switching piecewise linear systems. Walsh basis functions are used to capture the non-smooth fast varying dynamics. The piecewise timevarying frequency response function (TV-FRF) is approximated by the sum of a series of LTI FRFs multiplied with a set of Walsh functions. The best linear time invariant approximation (BLTIA) of the TV-FRF is estimated with small uncertainty. Besides the BLTIA, the two methods are capable of estimating the noise power spectrum and the TV-FRF. The error analysis shows that ATIRMM delivers more accurate TV-FRF and BLTIA estimations, while ALPM has better performance in noise power spectrum estimation. The conclusions are illustrated by simulations. Keywords: frequency response function, switching piecewise time-varying system, nonparametric identification, estimation uncertainty
1 Introduction The linear time-invariant (LTI) model has been generally accepted to be an effective approximation of many physical systems in practice. Nonparametric frequency response function (FRF) describes the steady-state response of a LTI dynamic model in the frequency domain. It provides a fast and easy approach to: (a) get the first impression of model dynamic properties such as stability; (b) evaluate the quality (Signal to noise ratio, SNR) of
input-output measurements at the early stage of the experiment, providing the opportunity to improve the experiment; (c) have possibility to detect the nonlinear behavior [1]. The major disadvantage of the FRF estimation is that the accuracy is corrupted by the measurement noise and leakage error. Leakage error is generated with the truncated time series of measurements whose past inputs and future outputs are unknown [2-5]. Traditionally the FRF estimation was calculated by the spectral analysis method (SAM) [6]. In SAM, leakage error is suppressed via time domain windowing at a cost of a loss in frequency resolution [7]. Then piecewise average of the entail time series is calculated for reducing the measurement noise. In recent years, two new methods have been developed to significantly reduce the effects of leakage and measurement noise in FRF estimations. The first method is the local polynomial method (LPM) proposed by Pintelon and Schoukens [6, 8-10]. In this approach, the smooth behavior of the FRFs of system and leakage over the frequency is utilized. Then the smooth functions are approximated with truncated Taylor series expansions at the local frequency. A local least square (LS) estimator is applied to estimate the coefficients of these expansions. This procedure is repeated until all the frequencies of interest are covered. The second approach is the transient impulse response modeling method (TIRMM) [11, 12]. In this method, the information of the leakage structure is used, the leakage term is assumed as a function of global parameters, and a global LS problem is solved as against many local LS problems in LPM. In practice, many physical systems exhibit time-varying behaviors. Based on the behavior of the time variation, linear time variant (LTV) systems can be naturally classified into two categories: (1) Slow time-varying systems whose dynamics or parameters vary smoothly as functions of time, such as magnetic fields [13], aeroelastic flutter [14, 15], and draft of the power amplifier at different temperatures [16]; (2) switching piecewise time-varying systems which shift between several LTI systems, such as regime switching of system dynamics [17],
change of econometric regime [18], and hybrid systems of power electronics [19]. This paper focus on switching piecewise time-varying systems. Some researchers have paid attention on control synthesis problems of switching piecewise systems [20-22]. These studies are all based on dynamic models which are given beforehand. How to estimate an accurate dynamic model from experiment data is an important issue. Several approaches have been introduced for parametric identification of switching piecewise systems [19, 23-25]. Compared to parametric identification methods, nonparametric identification methods can estimate nonparametric models directly from measurement data, without needing any priori information of model structures [26]. Therefore, nonparametric methods are especially suitable for the systems with unknown or complex model structures, such as morphing or damaged aircrafts [27, 28], hybrid systems of power electronics [19], and biological systems [29]. The artificial neural network method (ANN) [30] has been used for nonparametric estimation, but it needs some priori data for training the input-output mapping. In this paper, the specific aspect we are dealing with is nonparametric estimation of switching piecewise systems from experiment data without needing any other required knowledge (e.g., either system structure or priori data). For nonparametric identification, Zadeh [31, 32] first introduced the definition of timevarying frequency response function (TV-FRF) to describe the time-varying behavior of system dynamics. The TV-FRF reflects frequency responses that evolve with time, providing both the frequency and time dependent spectral. Some approaches have proposed to estimate the TV-FRF. A widely used method is the short-time Fourier transform (STFT) [33, 34]. In STFT, a short sliding time window is applied, and the system is assumed to be time invariant within the window, hence STFT is only capable of tracking slow time variant behavior. The time width of the window should be selected carefully to balance the time tracking accuracy (smaller width) and the frequency resolution (larger width). Recently, Lataire [35-38] and Pintelon [39, 40] developed a new method (hereinafter to be referred as
LPM-Legendre) to estimate the TV-FRF by parameterizing the time variation using Legendre polynomial basis functions. In this method, the LTV system is equivalent to a timedependent weighted sum of a series of LTI systems, and LPM is applied to estimate FRFs of LTI systems. LPM-Legendre can achieve a better description of time variation without reducing frequency resolution compared to STFT. However, this method is only suitable for slow time-varying systems whose dynamics or parameters vary smoothly as functions of time, and its performance degrades for switching piecewise systems. To the best of our knowledge, the nonparametric TV-FRF estimation of a switching piecewise system has not been addressed yet. The aim of this paper is to fill this gap. The paper follows the idea of Lataire [35-38] and Pintelon [39, 40], but focuses on the TVFRF estimation problem of switching piecewise LTV systems. Compared to the existing methods (STFT and LPM-Legendre), there are mainly two motivations of the two proposed approaches (ATIRMM and ALPM): (1) By using non-smooth Walsh basis functions, the TVFRF of a switching piecewise system can be estimated with the much higher accuracy compared to the existing methods; (2) The approach can be selected based on the object to be estimated, ATIRMM delivers more accurate TV-FRF and BLTIA estimations, while ALPM has better performance in noise power spectrum estimation. The paper is organized as follows: in Section 2, the estimation problem of the piecewise LTV system is formulated; the assumptions and definitions of the excitation signal, the measured output, and the BLTIA are described. Section 3 proposes two new algorithms to estimate the BLTIA, the noise spectrum and the TV-FRF from the known input and the output which is corrupted by measurement noise in one experiment, followed by the estimation error analysis and the comparison of the two algorithms in Section 4. In section 5, the selection rules of design parameters in the two methods are discussed. Finally, the algorithms are illustrated on two simulations in Section 6.
2 Problem Formulation and Assumptions
Definitions and properties of the switching piecewise TV-FRF and its BLTIA are recalled, Walsh basis functions are introduced, and some assumptions which are needed in the following of this paper are given. 2.1 Model for linear switching piecewise time-varying system Definition 1 (Switching Piecewise LTV System). The dynamic system is a SISO linear open loop causal system with switching piecewise (non-smooth) time-varying stationary behavior. The LTV system switches between a few linear time-invariant (LTI) subsystems. Definition 2 (TV-FRF). A LTV causal system can be fully characterized by the TV-FRF ( which is the Fourier transform of the shifted response ( (
)
(
∫
)
) to a Dirac impulse [31,32]
)
(1)
The system function can also be expressed as the sum of a truncated series expansion of FRFs of LTI systems multiplied with a set of basis functions [40] ( where, (
(
) for
)
∑
(
)
( )
(2)
are FRFs of LTI systems which are asymptotically stable,
) is the number of LTI systems, and
( ) for
are basis functions
satisfying the following assumption. Assumption 1 (Basis Functions). Basis functions are a set of series which are functions of time, satisfying the constraints [40] ( ) with
and
∫
( )
[
for
]
(3)
the experiment time.
Basis functions are selected to characterize the system time variation. Legendre polynomial functions have perfect performance for slowly time-varying dynamics [37-39], whereas Walsh functions are adopted here to capture the switching time-varying behavior. Walsh functions are defined as [41] ( )
∏
(
(
))
(4)
where,
,
is the order,
integer satisfying
, and
∑
,
is a
.
First 5 orders of Walsh functions are shown in Figure 1. It is easy to prove that Walsh functions satisfy the constraints in (3). 1
w0 (t )
T
t
T
t
3T/4
T
t
3T/4
T
t
7T/8 T
t
1
w1 (t )
T/2
-1 1
w2 (t )
T/4
-1 1
w3 (t )
T/4
-1
T/2
1
w4 (t )
T/8
-1
3T/8
5T/8
Figure 1. First 5 orders of Walsh functions 2.2 Excitation and output Assumption 2 (Excitation). The excitation signal ( ) satisfies following assumptions: a.
The excitation is a band-limited signal with the length , and the signal bandwidth less than ½ the sampling rate
is
of measurements, such that the aliasing error can be
eliminated. b.
DFTs of the excitation at different frequencies are uncorrelated [36].
c.
DFTs of the excitation should be rough enough at all excited frequencies (vary wildly over the frequency) in order to separate the system term from the smooth transient behavior [9].
d.
The excitation signal is known exactly without noise.
Assumption 3 (Output). The output ( ) is corrupted by the stationary filtered white noise ( ), and
( ) is independent with ( ).
( ) where,
( )
( )
(5)
( ) is the noiseless output which can be expressed as (Theorem 2 of [39])
( )
[∑
( ) ( )]
()
(6)
denotes the inverse Laplace operation, ( ) is the Laplace transform of ( ).
where,
The DFTs of input-output are given as
( ) where,
, and
,
√
∑
(
is sample period,
)
(7)
is the number of sampling data.
Remark 1: a.
Assumption 2 can be naturally satisfied with band-limited random phase multisine excitation or filtered white noise excitation, both of the signals have been widely used in the frequency identification.
b.
Assumption 3 implies that the experiment operates in the open loop.
c.
Assumptions 2 and 3 indicate that the approaches proposed in this paper are developed within the generalized output-error framework (noiseless input and noisy output). In practice, the input signal is always corrupted by noise, resulting an error-in-variable framework. However, the case with noisy input-output can be equivalent to an outputerror framework with a noiseless reference signal ( ) and two noisy outputs { ( ) ( )} satisfying Assumptions 2 and 3, shown in figure 2. The detail discussion is given in [42].
( )
( )
Generator /actuator 𝑎
Dynamics
( , ) ( ) ( )
0( )
( , )
0( )
+
( )
( )
Generator /actuator
( , )
𝑎
( ) 0(
)
( )
+
( )
+ ,
( , )
0(
)
+
( )
Figure 2. Equivalent model of the LTV system with noisy input-output. Left: the LTV system with noisy input ( ) and output ( ); Right: the LTV system with the noiseless input ( ) and two noisy outputs { ( ) ( )}. The LTV system (
)
(
)⁄
(
).
Theorem 1: By using Walsh basis functions, equation (6) is equivalent to
( )
∑ { ( )
( )
where,
{ ( ) ( )} ( )} , and
nonzero states of the systems
[∑
()
( )]
( )
∑
()
(8)
( ) are transient (leakage) terms that stem from
at the time points that basis functions
( ) change.
Proof: see Appendix A. Under Theorem 1, the DFT of the output is expressed as
( ) where, (
∑
(
)
(
( )
) is the DFT of the output noise
)
( ), (
(
)
) is the rational function of the (
transient (leakage) term, which is the sum of system leakage (
), and ∑
( (
) with
(
)
)
(
)
(9)
), noise leakage
( ), i.e., (
)
∑
(
)
2.3 BLTIA Definition 2 (BLTIA). The BLTIA of the LTV system satisfying Definition 1 over the time interval [
] is defined as [39] (
)
∫
(
)
(10)
Combining (10) with (2) and (3) gives (
)
(
)
(11)
Under assumptions 2 and 3, the output DFT ( ) can be written as
( ) where,
( ) ( )
( )
(12)
( ) is the residual which is uncorrelated (but not independent) with ( ).
By using equation (12), it can be easily known that the FRF ̂ ( ̂ (
)
[ ( ) ] [| ( )| ]
) estimated by SAM [6] (13)
equals to the BLTIA (10), i.e., ̂ (
)
(
)
(14)
Equation (14) holds once the assumption 2(b) is satisfied, i.e., [ ( )
(
)]
. If the DFT of the input is correlated over the frequency, the FRF ̂ (
for
) is given
as ̂ (
)
(
)
∑
(
)∑
[
( )
(
)
| ( )|
]
( )
(15)
3 Estimation Algorithms For switching LTV systems, two new algorithms are proposed here to estimate: (1) the BLTIA and its variance; (2) the noise power spectrum at each frequency; (3) the TV-FRF. Two algorithms are extensions of TIRMM and LPM, respectively.
3.1 Algorithm based on the TIRMM (ATIRMM) In TIRMM, Hägg [11] has proved that the FRFs of both the system and leakage term are functions of global parameters for a LTI system, and the global parameters can be estimated by a global LS estimator. The FRF of the LTI system (
) at frequency
can be
extended round the frequency ( with (𝑎)
)
(
and
(
)
∑
(𝑎) (
)
)
(16)
.
The leakage term can be written as ( with
()
(
)
√
()
(17)
).
According to equation (9), the output DFT ( (
∑
)
∑
(
)
) at frequency
(
)
(
)
is written as (
)
Inserting (16) and (17) into (18) gives (
)
∑
{[
(
)
∑
(𝑎) (
)]
(
)}
(18)
√
with
(𝑎)
∑
(
()
)
(19)
.
Under the assumption that
(
()
to infinity, and also
(𝑎) decays to 0 when 𝑎 tends
) is asymptotically stable, if
. The sums in (19) can be approximated by the
truncations ∑
(𝑎) (
∑
()
)
∑ ∑
(𝑎) (
)
(20)
()
(21)
Equation (19) is approximated to (
)
∑ √
{[
(
)
∑
∑
(𝑎) (
)]
(
()
(
)}
)
(22)
The LS problem is obtained by assembling equation (22) for {
} , and at each
(
)(
neighboring frequencies are used with
. The global LS estimator consists of )
DFT frequencies
(
(
) equations for
) unknown global parameters (23)
where,
is the [(
)(
)
( [(
estimated,
is the
(
)
regressors,
is the
(
)
(
)
)]
vector of unknown parameters to be
)(
)
(
)] matrix of known
vector consisting output DFTs, and
is the
vector of residuals. Complete expressions of the matrices are given in
Appendix B. Assumption
4
regression matrix
(Identifiability).
The
(
[(
)
)(
)
(
)]
is of full column rank.
It can be verified that with the excitation using random phase multisine or filtered white noise, assumption 4 is fulfilled if the degree of freedom (DOF) (
)
[(
)(
)
of (
satisfies )]
(24)
The BLTIA estimation can be solved ̂
(
) ̂
̂
(25)
[
(26)
]
The vector of residuals is ̂
(
The noise power spectrum ̂ ̂
( (
(
)
)
) at frequency
is estimated by |̂ (
∑
)
In practice, the matrix inversion (
)
(27)
)|
(28)
in (25) and (27) is avoided by using some
numerically stable methods, such as singular value decomposition (SVD) [6].
3.2 Algorithm based on the Local Polynomial method (ALPM) (
) and (
) are expanded at frequency (
)
(
)
∑
(
)
(
)
∑
by applying Taylor’s formula (
)
(
)
(
)
(29) (30)
Inserting (29) and (30) into (18) gives (
)
∑
(
{[
∑
)
(
∑ (
)
The frequency domain window with a width ( {
the local LS estimator with
]
(
)}
(
)
)
(31) ) around
is selected to generate
}. At each frequency
, equation (31)
can be expressed in the matrix form ( )
( )
(
)(
)
(
in Appendix C.
)
) (
is ) [(
(
( ) ( )
[(
( )
where,
)(
( ) ( ) )]
(32)
is the column vector of to-be-identified paramters;
the ) (
( )
column )]
is
the
vector matrix
of of
known
output regressors,
DFTs; and
is the column vector of residuals. Expressions of matrices in (32) are given
Under assumption 4, the identifiability condition of (32) is fulfilled if the DOF of
( )
satisfies (
)(
The BLTIA estimated by ALPM at frequency
)
(33)
is the first element of estimated parameter
vector ̂ ( ), i.e., ̂ (
̂ ( )
)
(34)
The vector of residuals is ̂( The noise power spectrum ̂ ̂
(
)
( )(
(
) (
( )
( ))
) at frequency ) [(
(
)(
( ))
( )
(35)
is estimated by ) (
)]
This procedure is repeated at each central frequency
|̂ (
∑
)|
(36)
over the frequency range of
interest.
3.3 TV-FRF estimation In both ATIRMM and ALPM, the FRFs of LTI systems estimators. The TV-FRF estimation ̂ ( ̂(
(
) in (2) can be estimated by LS
) is elaborated by )
∑
̂ (
)
( )
(37)
The time resolution of the TV-FRF is closely related with the order the experiment time . The minimum resolvable time range
of Walsh functions and
is expressed as (38)
with
satisfying
.
Equation (38) shows that the time resolution can be enlarged with larger However, the reduction of measurement time
would result in an undesired loss of
estimation accuracy and decrease the frequency resolution. The number by the estimated uncertainty which is analyzed in the next section.
4 Error analysis
and smaller .
is also limited
4.1 Bias of estimated BLTIA The bias of estimated BLTIA in ATIRMM is (̂ ( With
(
))
⁄ )
) (
(
)
(39)
the absolute value of the largest eigenvalue of the stability matrix of
.
Proof: see Appendix D. The bias of estimated BLTIA in ALPM is (̂ ( with
(
)
(
) the (
(
)) )
)
(
) ((
)
(
deviation of
)
((
)
)
(40)
).
Proof: see Appendix E.
4.2 Variance of estimated BLTIA With the LS estimator, the variance of residuals is ̂ ̂
̂ In ATIRMM, ̂
(41)
̂ is the variance of residuals, ̂
̂ which is given in (27), and
the degrees of freedom (DOF) of the regressive matrix In ALPM, ̂
is
, shown in (24). ; ̂
̂ ( ) is the spectrum of residuals at frequency
̂ (
) which is
( ) is given in (33).
shown in (35), the DOF of the regressive matrix The covariance of ̂ is ( ̂)
̂ (
)
(42)
In ATIRMM, the variance of estimated BLTIA is given as 𝑎 (̂ )
( ̂ ))}|
{ 𝑎 (
[
]
(43)
In ALPM, the variance of estimated BLTIA is 𝑎 (̂ (
))
( ̂ ( ))|
[
]
for
(44)
By expanding the variance analyses results given in [43], the variances of estimated BLTIA in the two methods can be respectively approximated as
𝑎 (̂ ) 𝑎 (̂ (
(
(
)(
) ( (
(
))
(
)
) )(
) (
)̂
(45)
)
)̂ ( )
(46)
4.3 Bias of estimated noise spectrum The bias of estimated noise spectrum in ATIRMM is (̂
(
))
(∑
)|
(
|∑
((
) )
)
(
(
(
)| )
∑
)
(
|∑
)) (
(
) (47)
The bias of estimated noise spectrum in ALPM is
(̂
(
))
(
∑ (| ∑ (
∑ (|∑ (
((
)
(
)
)
)
(
(
)
((
))| )
(
)
))| )
((
)
)
)
(48)
4.4 Discussion (1) With same values of
,
and
in both ATIRMM and ALPM, the estimated BLTIA in
ALPM has smaller bias and larger variance compared to the one in ATIRMM. (2) ATIRMM has the larger bias of the estimated noise spectrum than ALPM. If the goal is to estimate the noise spectrum, ALPM may be preferred. (3) In both ATIRMM and ALPM, with the increase of
and decreases of
and
, the
biases of the BLTIA and the noise spectrum increase, while the variance of the estimated BLTIA decreases. The larger
increases the variance of the BLTIA, while inversely
decreases the bias of the noise spectrum.
5 Selection of design parameters In both of the two algorithms, the estimation accuracy is highly related to the values of design parameters
,
,
and
.
The leakage term (
) has exactly the same poles with (
( ) decay at the same rate
). It means that ( ) and
. Hence, a reasonable choice is
Based on the analyses of pervious section, the values of
.
and
are selected based on the
purpose of the identification, shown in table 1. Table 1. Selection of
and Cases
Estimation purpose
Small noise , ,
BLTIA Noise power spectrum
large noise , , ,
,
Based on extensive simulations, ranges of design parameters are given as follows to achieve good performance in many cases: a. In ATIRMM,
;
for highly damped systems, and
for lowly damped systems. b. In ALPM,
;
for highly damped systems, and
for lowly damped systems. The regression matrix of the LS estimator should be of full column rank (assumption 4), which implies
, therefore, in ATIRMM, (
)
[(
)(
)
(
)]
(49)
and in ALPM, [( In practice, the number
)(
)
(
)]
(50)
of time-variant branches are unknown, and should be selected
based on input-output data. The estimation procedure starts with is gradually increased until the absolute value of the smaller than its standard deviation, i.e., | ̂ (
)|
, and the value
FRF estimation | ̂ ( {̂ (
)| is
)}.
6 Simulation The goals of the simulation are to: (1) illustrate the estimation accuracy of each FRF and its variance; (2) prove the relationship between the BLTIA
(
) and
(
(
)
); (3)
evaluate the selection procedure of
; (4) estimate the noise spectrum; (5) identify the TV-
FRF and the time point that the switching change of the LTV dynamic occurs; (6) compare two algorithms proposed (ATIRMM and ALPM) with other methods (STFT and LPMLegendre). Two different structures of switching piecewise systems are simulated: (1) A LTV model with parallel time-varying branches; (2) A piecewise change dynamic model of a small morphing unmanned aerial vehicle (MUAV) which changes its sharp in flight.
6.1 Model with parallel structure The LTV system consists of 4 parallel time-varying branches, each
is a second-order stable
( ) are Walsh functions satisfying equation (4). The
transfer-function, shown in figure 3.
TV-FRF of this LTV system is expressed as ( Hence, the BLTIA
(
)
) (
∑
(
)
(51)
). G0
- 200 s 2 20 s 400
y0 (t )
G1
y1 (t )
- 625 s 2 3 0 s 625
u (t )
( )
G2
y2 (t)
- 454 s 2 25 s 500 G3
y3 (t )
- 796 s 2 45 s 1000
w0 (t )
w1(t)
y (t )
w2 (t)
w3 (t )
Figure 3. LTV system with parallel structure The excitation is a random phase multisine signal, which satisfies the assumption 2. ( ) where,
(
)
is the flat amplitude at each frequency,
distributed in the range (
for
). The sampling rate
, the number of sampling excited.
∑
,
(52) .
is uniformly
, the experiment time
, the frequency band [0.01Hz,10Hz] is
The output ( ) is corrupted by filtered white noise
( ), and
( ) has the following
expression ( )
( )
( )
(53)
with is the convolution operator, the noise spectrum is given in figure 6 (black line). Values of design parameters are chosen as follows: (1) The width of window is chosen as (2) For ATIRMM: The estimated FRFs
for both ATIRMM and ALPM;
; For ALPM: (
.
) are shown in figure 4. In figure 4(e), the estimated FRF ̂ (
)
has the same order of amplitude with its standard deviation. By following the procedure of selection in Section 5, the number
is selected as
, which coincides with the
number of parallel branches in (51). Figures 4 (a)-(d) show that the standard deviation
and the estimation error in ATIRMM are smaller than the ones in ALPM.
(a)
(
)
(b)
(
)
(c)
(
)
(d)
(
)
(
(e)
)
) of the model (51). Black thin line: true value of |
Figure 4. Estimated
(
gray thin line: | ̂ (
)| estimated by ATIRMM; light gray thin line: | ̂ ( {̂ (
ALPM; black thick line:
)} by ATIRMM; gray thick line :
dark gray crosses: estimated error | ̂ ( estimated error | ̂ (
)
(
)
(
(
)|; dark
)| estimated by
{̂ (
)} by ALPM;
)| by ATIRMM; light gray squares:
)| by ALPM.
For comparison, the BLTIA is also computed via SAM in (13). 500 runs are repeated to perform a Monte Carlo simulation. The simulation result ̂ (
) is given by the light gray
crosses in figure 5. It is obvious that ATIRMM and ALPM achieve higher BLTIA estimation accuracy than SAM, moreover, the reliable estimation can be obtained only in one experiment.
Figure 5. Estimated BLTIA ̂ ( line: | ̂ (
). Black line: | ̂ (
)| estimated by ALPM; gray crosses: ̂
)| estimated by ATIRMM; gray thick
estimated by SAM.
Figure 6 shows that the noise power spectrum estimated by ALPM has the smaller bias than the one estimated by ATIRMM.
Figure 6. Estimated noise spectrum line: ̂
(
(
). Black line: true value of
) estimated by ATIRMM; dark gray line: ̂
(
(
); light gray
) estimated by ALPM.
6.2 Model of small MUAV Morphing aircrafts can change their shapes or configurations substantially to achieve high performance for a changing mission environment in flight [27]. The methods proposed in the paper can be used to estimate the nonparametric model of a small MUAV. In the simulation, the MUAV has two shapes which can be changed in flight: a) the wing is telescoped in the body for high speed cruise, and b) the wing is unfold for economic cruise and landing. The MUAV can be considered as a switching piecewise system under the assumption that the change between two sharps is instantaneous. The linearized dynamic model (
) of the
( ) and pitch rate output ( ) at the specified velocity and
MUAV with elevator input altitude is given as follows.
(
where,
)
{
(
)
(
)
is the time point that the sharp of aircraft changes,
The simulation time
, and the sampling rate
(54)
. . The excitation is a
random phase multisine signal shown in (52). The frequency range [0.1Hz,15Hz] is excited, over the excited frequencies. The output ( ) is disturbed
and the amplitude
by the measurement noise with the SNR 20dB. The designed parameters are chosen as: a) for both ATIRMM and ALPM, b) for ATIRMM:
; for ALPM:
. The BLTIA is easy to be calculated ( ) The FRFs ̂ (
∫
(
)
(55)
) estimated by ATIRMM are shown in figure 7. ̂ (
of amplitude with
{̂ (
the amplitude | ̂ (
)| gradually decreases, and it becomes close to
)}, hence
) has the same order
. It can be seen that with the increase of {̂ (
)}.
,
(a) ̂ (
)
(b) ̂ (
)
(c) ̂ (
)
(d) ̂ (
)
(e) ̂ (
)
(f) ̂ (
)
(g) ̂ (
)
(h) ̂ (
)
(i) ̂ (
)
(j) ̂ (
)
(k) ̂ (
)
(l) ̂ (
)
Figure 7. ̂ ( {̂ (
)
(n) ̂ (
)
(o) ̂ (
)
(p) ̂ (
)
) estimated by ATIRMM. Black solid line: | ̂ (
)| ; black dash line:
)}.
The FRFs ̂ ( (2) is
(m) ̂ (
) estimated by ALPM are given in figure 8. The number of significant FRFs in
. For
,|̂ (
)| falls within its uncertainty
{̂ (
)}. It shows that
the variance estimated by ALPM is slightly larger than the one estimated by ATIRMM.
(a) ̂ (
)
(b) ̂ (
)
(c) ̂ (
)
(d) ̂ (
)
(e) ̂ (
)
(f) ̂ (
)
(g) ̂ ( Figure 8. FRFs ̂ ( |
(̂ (
(h) ̂ (
)
)
) estimated by ALPM. Black solid line: | ̂ (
)|; black dash line:
))|.
Estimated ̂ (
) and ̂ (
(
) are compared with the true value of
) in (55), shown
in figure 9. Both ATIRMM and ALPM can achieve high accurate BLTIA estimations. Compared to ALPM, ̂ (
) estimated by ATIRMM has smaller error and uncertainty.
(a) Figure 9. |̂ (
(b)
Estimated BLTIA. Figure (a): | ̂ (
)| by ALPM (top light gray thin line),
thick line),
(̂ (
(dark line), | ̂ (
)| by ATIRMM (top dark gray thin line), {̂ (
)} by ATIRMM (bottom dark gray
)) by ALPM (bottom light gray thick line). Figure (b): true |
)
(
)| by ATIRMM (dark gray line), | ̂ (
)
(
(
)|
)| by ALPM
(light gray line). In ATIRMM, the number
of parallel time-varying branches is selected as 15, and the
minimum resolvable time range FRF occurs between
⁄
and
is ⁄ ⁄
. In figure 10(b), the switching change of the TV-
, which implies that the dynamic changes between
two LTI subsystems within the time range of (36.75s-42.875s). In ALPM, the time resolution is ⁄ , and the switching change occurs between in figure 10 (c).
⁄ and
⁄ (i.e., 36.75s-49s), shown
ATIRMM and ALPM are also compared to the existing methods which are STFT and LPMLegendre. In STFT, the width of the sliding time window is selected as 30s. Under the assumption that the system dynamic is time invariant within the window, STFT is not capable to capture the switching time variation, shown in figure 10(d). By using smooth basis functions, LPM-Legendre also has poor performance with larger estimation error and uncertainty, shown in figure (e).
a) True TV-FRF
c) TV-FRF estimated by ALPM
b) TV-FRF estimated by ATIRMM
d) TV-FRF estimated by STFT
e) TV-FRF estimated by LPM_Legendre Estimated TV-FRF ̂ (
Figure 10.
{ ̂(
bottom surface:
) . Upper surface: estimated TV-FRF ̂ (
)}.
It is also possible to identify two subsystems estimated TV-FRF ̂ ( equal to ̂ (
);
(
) and
(
). In this simulation, the estimated ̂
) and ̂ (
), respectively. Estimated ̂
(
) in (54) from the ) and ̂
(
) and ̂
(
(
) are
) are given
in figure 11. It shows that the two algorithms also have ability to estimate the LTI subsystems of the switching time-variant system.
a) ̂
(
b) ̂
)
(
)
Figure 11. Estimated FRFs of LTI subsystems between which the LTV system switches. Black line: true values of | and | ̂ |̂
(
(
)
)
7 Conclusions
( (
(
)| and |
(
)|; dark gray line: | ̂
)| by ATIRMM; light gray line: | ̂
)| by ALPM.
(
( )
)
( (
)|
)| and
Two new approaches which are ATIRMM and ALPM are proposed for estimating the nonparametric TV-FRF and its BLTIA of a switching piecewise LTV system. The TV-FRF is approximated by a sum of multi FRFs of LTI systems multiplied with basis functions. The Walsh functions are adopted to capture fast transients and switching piecewise dynamics. The proposed methods can estimate the TV-FRF with higher accuracy and smaller uncertainty compared to the existing methods. The noise spectrum is also estimated by the two methods. Results of error analysis and simulations show that ATIRMM has a better performance on estimating the TV-FRF and the BLTIA, whereas ALPM is more suitable to estimate the noise power spectrum.
Appendix A. Proof of Theorem 1 Theorem 1 is proved for For
, the proof of the cases for
follows the same lines.
, the equivalent model of the LTV system is given in Figure A.1. 0(
( )
0(
1(
2(
)
)
0(
)
1(
)
2(
)
) 0(
1(
)
2(
)
)
)
Figure A.1 Equivalent model of the LTV system for We have ( )
{
( ) ( )}
( )
{
( ) ( )}
( )
{
( ) ( )}
( )
(A.1)
(a) It is obvious that the first term of (A.1) is equal to
{ ( ) ( )} with
( )
.
(b) The second term of (A.1) is
()
{ ( ) { ()
( )}}
(A.2)
{ ( ) ( )} {
( )
with
( )
Inserting
{
()
()
(A.3)
( ) ( )} ⁄
{
into (A.3) gives
⁄
{ ( ) ( )} with ̅ ( )
()
()
̅ ()
()
(A.4)
⁄
( ) ⁄
We define that ̃ ( )
{
( ) { ̅( )}},
with ̅( )
{
⁄
( )
(A.5)
⁄
̅ ( ) can be expressed as ̅ ( ) where, at
̃ ( )
( )
(A.6)
( ) is the transient term that stems from the nonzero state of the system
( )
, and ( )
{
The relation of ̃ ( ) ̅ ( ), and
(
)
(A.7)
( ) is shown in figure A.2.
~ y 1(t)
y 1(t)
y 1(t) T/2
T
Figure A.2 Relation of ̃ ( ) ̅ ( ), and
( )
Insert (A.5) and (A.6) into (A.4)
{ ( ) ( )}
()
{ ( ) { ̅( )}}
()
()
( ) { ̅( )}}
{
( ) ( )}
{ {
( )(
{ ̅( )}
( ))}
( )
( )
{
( ) { ̅( )
{
( ) { ( )
( )}}
{
( ) { ( )
( )}}
( ) the transient term of LTI system
with
( )}}
( ) ( ) ( ) ( ).
( )
, and
(A.8)
(c) The third term of (A.1) is
{ ( ) ( )}
()
()
()
(A.9)
⁄ ( )
Inserting
⁄
{
into (A.9) gives ⁄
{ ( ) ( )} ( )
With ̅
{
⁄
( )
()
and ̅
⁄
̅ () ( )
̅ ()
(A.10)
⁄
( )
{
()
⁄
We define that
̅
̃
( )
{
( ) {̅
( )}}, with ̅
( )
{
̃
( )
{
( ) {̅
( )}}, with ̅
( )
{
( ) and ̅
⁄
( )
(A.11)
⁄ ⁄
( )
(A.12)
⁄
( ) can be expressed as
̅
( )
̃
( )
( )
with
( )
{
(
̅
( )
̃
( )
( )
with
( )
{
(
)
(A.13)
)
(A.14)
Inserting (A.13) and (A.14) into (A.10) gives {
( ) ( )}
( )
( )
̃
( )
( )
{
( )(
{̅
( )}
{̅
{
( ) { ̅
{
( ) { ( )
( )}}
{
( ) { ( )
( )}}
̃
( )
̅
( )
( ) ( )}
( ))}
( )}} ( )
()
( ) ( ) ( )
( ) ( )
( )
(A.15)
with
( ) the transient term of LTI system
( )
( )
, and
( ).
Combining equations (A.1), (A.2), (A.8) and (A.15) gives ( )
( ) { ( )
{
( ) { ( )
{ The cases for
{
( ) { ( )
( )
()
( )}} ( )}}
( )}} (A.16)
exactly follow the same lines. Theorem 1 is proved.
Appendix B. The expressions of the matrices in (23) ( [
(
]
)
) ( )
with
(B.1)
[ (
[(
( )
)(
) (
)]
)]
with
(
)
(
)
[ ( )]
(
[
[
]
(B.2)
( )
(
) [(
)(
(
) (
)
)] ) (
)]
(B.3)
with
[
] (
) (
( ) (
[ (
) (
[
(
(B.4)
( )
)
)
(
(
( )
( (
( )
) )
(B.5)
)] )
( )
)
)
(
( ) (
) (
)
(
( )
)
) ( )
(
)]
(B.6)
[
(
]
) (
)
(B.7)
with √
√
√
√
√
√
[√
√
√
(
) (
)
(B.8)
]
Appendix C. The expressions of the matrices in (32) ( ( )
[
)
(
) (
(
)
)(
) (
]
(C.1)
)
( ) ( ) [(
( )
( ( (
( )
( )
(
(
[
)
( [
)
(
( )]
(C.4)
) [(
(
( )
)(
) )
(
)
(
( )
)
)
(
(
(C.3)
)]
( ) (
) (
)
(
( )
)
(
( )
) ) )
(
[
(
)]
( ( (
( )
( )
) ) )
(
[
[
(C.2)
( ) ( )]
[
( )
)]
) (
( (
)]
) )
]
)
(C.6) (
(
(C.5)
)
(
) )
) (
) )
(
]
) (
)
(C.7)
Appendix D. Proof of (39) The
(
(
)
) in (16) can be expressed as (
)
(
(
⁄ )
)
(D.1)
( ⁄ ) .
with
(
Using Taylor formula to expand ( ⁄
)
⁄ )
∑
(
(
⁄ )𝑎)
(D.2)
(
(
(D.3)
Inserting (D.2) into (D.1) (
)
∑
The accurate expression of output DFT ( (
)
∑
{[
(
)
{
with
) at frequency )]
(
)}
√
is
∑
()
(
)}
(
) (D.4)
} (
The terms (
(𝑎) (
∑
⁄ )𝑎)
)
) neglected in (22) are expressed as {[
∑ ∑
(
)
∑ (𝑎) (
{[∑
)]
(𝑎) ( )]
(
)}
√
∑
()
(D.5)
)
(D.6)
Insert (D.3) into (D.5) (
)
(
)
(
)
(𝑎)
∑
(
(
where, (
(
)
∑
)
∑ ∑
( with
)
{[
√
(
)
(𝑎)
{[∑ { ∑
∑
( ⁄ )
(
()
∑
(
(𝑎 ⁄ ))
(𝑎 ⁄ ))
(
]
]
(
)} (D.7)
)}
)}
(D.8) (
) (
)
the absolute value of the largest eigenvalue of the stability matrix of
(D.9) [36].
(D.4) can be rewritten into matrix forms as (D.10)
where, )(
, )
, and
(
are the matrices
] th components are
(
),
)
of which the [(
(
) , and
(
),
respectively. The expert value of ̂ is written as {̂ }
{(
{(
)
where, {(
)
}
)
(D.11)
} ,
)
{(
} ,
and
}.
The expect value of estimation ̂ is {̂ } |[
where, Basis ( )
]
(D.12) |[
,
( ) ,
functions ( )
,
]
|[
, and
are linearly
]
independent,
hence
inputs
( ) are linearly independent for non-zero inputs ( ) (lemma4 of [39]). It
is easy to deduce that (D.13) ( ⁄ ) Based on Theorem 6.15 in [6],
(D.14)
is equal to ) (
(
)
(D.15)
Inserting (D.13), (D.14), and (D.15) into (D.12) gives {̂ }
(
)
( ⁄ )
) (
(
)
(D.16)
Equation (39) is proved. Appendix E. Proof of (40) Taylor expansions of (
) and (
) can be written as
(
)
(
)
∑
(
)
(
)
∑
( (
(
) )
√
) (
(E.1) )
(E.2)
The remainder of (E.1) is depends on the (
)th derivation of the system FRF, collect
equations (E.1) and (E.2) into (31) (
)
∑
{[
(
)
(
∑
)
∑
(
(∑
)
(
{
with
(
)
)
]
(
)}
(
)
(
)
(
)) ((
(
)
)
)
∑
(
((
√
)
)
) (E.3)
)
)
}
(E.3) can be written into matrix forms as ( )
( ) ( )
√
( )
((
)
(
)
( ))
((
(E.4) (
∑
(
)
( (
)
(
)
∑
[
)
)
∑ with
(
(∑
)
(
)
)
(E.5) ]
The expert value of ̂ is { ̂ ( )}
{(
( )
( ))
( ) ( )}
( )
( )
( )
( ) (E.6)
( ) is the true vector of parameters,
where,
( ) ( )
{(
( )
( ))
( )
{(
{(
( )
(
( ) (∑ ( )
( ))
( ))
( ) )
( )
(
√
}
(E.7)
)
( )) ((
((
)
)
)}
(E.8)
)}
(E.9)
( )
(E.10)
The expect value of BLTIA can be written as {̂ ( where,
( ),
)}
( ), and
(
)
( )
( )
( ) are the first components of
( ). ( )
( )
( ) are independent, hence
( )
( ), and
( ) ( )
(
(E.11)
)
(
) ((
)
)
(E.12)
Pintelon (appendix 7.C of [6]) has proved that ( )
{(
( )
( ))
( )
√
((
)
)}
((
)
)
(E.13)
Inserting (E.11), (E.12), and (E.13) into (E.10) gives {̂ (
)}
(
)
(
)
(
) ((
)
)
((
)
)
(E.14)
Equation (40) is proved.
Acknowledgements The authors would like to thank the reviewers for their valuable suggestions. References [1] Rik Pintelon, Johan Schoukens, Wendy Van Moer, Yves Rolain, Identification of linear systems in the presence of nonlinear distortions, IEEE Transactions on Instrument and Measurement, 50 (2001), no.4, 855-863. [2] Rabiner L., Allen J.B., On the implementation of a short time spectral analysis method for system identification, IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-28 (1980) 69-78. [3] Peter E. Wellstead, Non-parametric methods of system identification, Automatica, 17 (1981) 55-69. [4] Schoukens J., Rolain Y. and Pintelon R., Improved frequency response function measurements for random noise excitations, IEEE Transactions on Instrumentation and Measurement, 47 (1998) 322-326. [5] Widanage W.D., Douce J.L., and Godfrey K.R., Effects of overlapping and windowing on frequency response function estimates of systems with random inputs, IEEE Transactions on Instrumentation and Measurement, 58 (2009) 214-220. [6] Rik Pintelon, Johan Schoukens, System Identification: A Frequency Domain Approach, second ed., IEEE Press, 2012. [7] Daniel Belega, Dario Petri, Frequency estimation by two- or three-point interpolated Fourier algorithms based on cosine windows, Signal Processing, 117 (2015), 115-125.
[8] Rik Pintelon, Johan Schoukens, Gerd Vandersteen, Kurt Barbé, Estimation of nonparametric noise and FRF models for multivariable systems- part I: theory, Mechanical Systems and Signal Processing, 24 (2010) 573-595. [9] Johan Schoukens, Gerd Vandersteen, Kurt Barbé, Rik Pintelon, Nonparametric preprocessing in system identification: a powerful tool, European Control Conference (ECC), Budapest (2009) 1-14. [10] Johan Schoukens, Rik Pintelon. High quality frequency response function measurements without user interaction, UKACC International Conference on Control, Coventry (2010) 1-5. [11] Per Hägg, Hakan Hjalmarsson, A least squares approach to direct frequency response estimation, 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, USA (2011) 2160 – 2165. [12] Per Hägg, On structured system identification and nonparametric frequency response estimation, Doctoral Thesis, KTH School of Electrical Engineering (2014). [13] Easwar Magesan, Alecandre Cooper, Honam Yum, Paola Cappellaro, Reconstructing the profile of Time-varying magnetic fields with quantum sensors, Physical Review 88 (2013) 1-16. [14] Dimitriadis, G., Cooper, J.E., Flutter prediction from flight flutter test data, Journal of Aircraft, 38 (2001) 355-367, DOI: 10.2514/2.2770. [15] J. Ertveldt, J. Lataire, R. Pintelon, S. Vanlanduit, Frequency-domain identification of time-varying systems for analysis and prediction of aeroelastic flutter, Mechanical systems and signal processing, 47 (2014) 225-242. [16] Chen, S.Y., Yuan, J. S., Adaptive gate bias for power amplifier temperature compensation. IEEE Transactions on Device and Material Reliability, 11 (2011) 442-449. [17] Jahanshah Kabudian, Mohammad Mehdi Homayounpour, Seyed Mohammad Ahadi, Bernoulli versus Markov: Investigation of state transition regime in switching-state acoustic models, Signal Processing, 89 (2009), 662-668. [18] James D. Hamiltion, Analysis of time series subject to changes in regime, Journal of Econometrics, 45 (1990) 39-70. [19] R.P. Aguilera, B.I. Godoy, J.C. Agüero, An EM-based identification algorithm for class of hybrid systems with application to power electronics, International Journal of Control, 87 (2014) 1339-1351. [20] Peng shi, Xiaojie Su, Fanbiao Li, Dissipativity-based filtering for fuzzy switched systems with stochastic perturbation, IEEE Transactions on Automatic Control, (2015).
[21] Xiaojie Su, Peng Shi, Ligang Wu, Yong-Duan Song, Fault detection filtering for nonlinear switched stochastic system, (2015). [22] Peng Zhang, Jiangtao Cao, Guoliang Wang, Mode-independent guaranteed cost control of singular markovian delay jump systems with switching probability rate design, International Journal of Innovative Computing, Information and Control, 10(2014) 12911303. [23] Jose Ragot, Gilles Mourot, Didier Maquin, Parameter estimation of switching piecewise linear system, Proceedings of the 42nd IEEE conference on Decision and Control Maui, Hawaii USA, 6 (2003) 5783-5788, Vol.6. [24] Xiaojie Su, Ligang Wu, Peng Shi, C.L. Philip Chen, Model approximation for fuzzy switched systems with stochastic perturbation, IEEE Transactions on Fuzzy Systems, 23(2015) 1458-1473. [25] Andras Hartmann, Susana Vinga, Joao M. Lemos, Hybrid identification of time-varying parameter with particle filtering and expectation maximization, Mediterranean Conference on Control and Automation, 6 (2013) 884-889. [26] Matthew S. Holzel, Eugene A. Morelli, Real-time frequency response estimation from flight data, Journal of Guidance, Control, and Dynamics, 36 (2012) 1406-1417. [27] Terrence A. Weisshaar, Morphing aircraft technology – new sharps for aircraft design, Meeting Proceedings RTO-MP-AVT-141, (2006) 1-20. [28] Jared Grauer, Eugene Morelli, Method for real-time frequency response and uncertainty estimation, Journal of Guidance, Control, and Dynamics, 37 (2014) 336-343. [29] Kaustubh R. Joshi, Natasha Neogi, William H. Sanders, Dynamic partitioning of large discrete event biological systems for hybrid simulation and analysis, Hybrid Systems: Computation and Control, 7th international workshop, HSCC, (2004) 463-476. [30] Iman Sadeghkhani, Abbas Ketabi, Rene Feuillet, An intelligent switching overvoltages estimator for power system restoration using artificial neural network, International Journal of Innovative Computing, Information and Control, 10(2014) 1791-1808. [31] Zadeh, L. A, The determination of the impulsive response of variable networks, Journal of applied Physics, 21 (1950) 642-645. [32] Zadeh, L. A, Frequency analysis of variable networks, Proceedings of the IRE, 38 (1950) 291-299. [33] Allen, J.B., Rabiner, L., A unified approach to short-time Fourier analysis and synthesis, Proceedings of the IEEE, 65 (1977) 1558-1564.
[34] J.K. Hammond, P.R. White, The analysis of non-stationary signals using time–frequency methods, Journal of Sound and Vibration, 190 (1996) 419-447. [35] John Lataire, Rik Pintelon, Estimating a nonparametric colored-noise model for Linear slowly time-varying systems, IEEE Transactions on Instrumentation and Measurement, 58 (2009) 1535-1545. [36] John Lataire, Ebrahim Louarroudi, Rik Pintelon, Detecting a time-varying behavior in frequency response function measurements, IEEE Transactions on Instrumentation and Measurement, 61 (2012) 2132-2143. [37] John Lataire, Rik Pintelon, Ebrahim Louarroudi, Non-parametric estimate of the system function of a time-varying system, Automatica, 48 (2012) 666-672. [38] John Lataire, Ebrahim Louarroudi, and Rik Pintelon, Non-parametric best linear time invariant approximation of a linear time-varying system, IEEE Instrumentation and Measurement Technology Conference (I2MTC), (2012) 1050-1055. [39] Rik Pintelon, Ebrahim Louarroudi, John Lataire, Detection and quantification of the influence of time variation in frequency response function measurements using arbitrary excitations, IEEE Transactions on instrumentation and measurement, 61 (2012) 33873395. [40] Rik Pintelon, Ebrahim Louarroudi, John Lataire, Nonparametric time-variant frequency response function estimates using arbitrary excitations, Automatica, 51 (2015) 308-317. [41] Rui Zou, Hengliang Wang, Ki H. Chon, A robust time-varying identification algorithm using basis functions. Annals of Biomedical Engineering, 31 (2003) 840-853. [42] Rik Pintelon, Ebrahim Louarroudi, John Lataire, Detection and quantification of the influence of time variation in closed-loop frequency-response-function measurements, IEEE transactions on instrumentation and measurement, 62 (2013) 853-863. [43] Michel Gevers, Per Hägg, Hakan Hjalmarsson, Rik Pintelon, Johan Schoukens, The transient impulse response modeling method and the local polynomial method for nonparametric system identification, 16th IFAC Symposium on System Identification, Brussels, Belgium, 16 (2012) 55-60.
Highlights
Two new approaches (ATIRMM and ALPM) are proposed for estimating the nonparametric TV-FRF of a switching piecewise LTV system in one experiment; The switching time variation is modelled by Walsh functions;
Besides the TV-FRF, the best linear time invariant approximation (BLTIA) and the noise spectrum are also estimated with small uncertainty; Error analyses show that ATIRMM delivers more accurate TV-FRF and BLTIA estimates, while ALPM has better performance on noise power spectrum estimate.
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PII: DOI: Reference:
S0165-1684(16)30075-5 http://dx.doi.org/10.1016/j.sigpro.2016.05.012 SIGPRO6135
To appear in: Signal Processing Received date: 3 November 2015 Revised date: 3 May 2016 Accepted date: 11 May 2016 Cite this article as: Tao Song, Fubiao Zhang and Defu Lin, Nonparametric Frequency Response Function Estimates for Switching Piecewise Linear Systems, Signal Processing, http://dx.doi.org/10.1016/j.sigpro.2016.05.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Nonparametric Frequency Response Function Estimates for Switching Piecewise Linear Systems Tao Songa, Fubiao Zhangb*, Defu Lina a
School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 10086, China
b
Institute of Flight System Dynamic, Technical University of Munich, Garching, 85748, Germany
*
Corresponding author. Fubiao Zhang, PHD candidate, E-mail: [email protected]
Abstract This paper proposes two algorithms (ATIRMM and ALPM) for estimating the time-varying behavior of single input single output (SISO) switching piecewise linear systems. Walsh basis functions are used to capture the non-smooth fast varying dynamics. The piecewise timevarying frequency response function (TV-FRF) is approximated by the sum of a series of LTI FRFs multiplied with a set of Walsh functions. The best linear time invariant approximation (BLTIA) of the TV-FRF is estimated with small uncertainty. Besides the BLTIA, the two methods are capable of estimating the noise power spectrum and the TV-FRF. The error analysis shows that ATIRMM delivers more accurate TV-FRF and BLTIA estimations, while ALPM has better performance in noise power spectrum estimation. The conclusions are illustrated by simulations. Keywords: frequency response function, switching piecewise time-varying system, nonparametric identification, estimation uncertainty
1 Introduction The linear time-invariant (LTI) model has been generally accepted to be an effective approximation of many physical systems in practice. Nonparametric frequency response function (FRF) describes the steady-state response of a LTI dynamic model in the frequency domain. It provides a fast and easy approach to: (a) get the first impression of model dynamic properties such as stability; (b) evaluate the quality (Signal to noise ratio, SNR) of
input-output measurements at the early stage of the experiment, providing the opportunity to improve the experiment; (c) have possibility to detect the nonlinear behavior [1]. The major disadvantage of the FRF estimation is that the accuracy is corrupted by the measurement noise and leakage error. Leakage error is generated with the truncated time series of measurements whose past inputs and future outputs are unknown [2-5]. Traditionally the FRF estimation was calculated by the spectral analysis method (SAM) [6]. In SAM, leakage error is suppressed via time domain windowing at a cost of a loss in frequency resolution [7]. Then piecewise average of the entail time series is calculated for reducing the measurement noise. In recent years, two new methods have been developed to significantly reduce the effects of leakage and measurement noise in FRF estimations. The first method is the local polynomial method (LPM) proposed by Pintelon and Schoukens [6, 8-10]. In this approach, the smooth behavior of the FRFs of system and leakage over the frequency is utilized. Then the smooth functions are approximated with truncated Taylor series expansions at the local frequency. A local least square (LS) estimator is applied to estimate the coefficients of these expansions. This procedure is repeated until all the frequencies of interest are covered. The second approach is the transient impulse response modeling method (TIRMM) [11, 12]. In this method, the information of the leakage structure is used, the leakage term is assumed as a function of global parameters, and a global LS problem is solved as against many local LS problems in LPM. In practice, many physical systems exhibit time-varying behaviors. Based on the behavior of the time variation, linear time variant (LTV) systems can be naturally classified into two categories: (1) Slow time-varying systems whose dynamics or parameters vary smoothly as functions of time, such as magnetic fields [13], aeroelastic flutter [14, 15], and draft of the power amplifier at different temperatures [16]; (2) switching piecewise time-varying systems which shift between several LTI systems, such as regime switching of system dynamics [17],
change of econometric regime [18], and hybrid systems of power electronics [19]. This paper focus on switching piecewise time-varying systems. Some researchers have paid attention on control synthesis problems of switching piecewise systems [20-22]. These studies are all based on dynamic models which are given beforehand. How to estimate an accurate dynamic model from experiment data is an important issue. Several approaches have been introduced for parametric identification of switching piecewise systems [19, 23-25]. Compared to parametric identification methods, nonparametric identification methods can estimate nonparametric models directly from measurement data, without needing any priori information of model structures [26]. Therefore, nonparametric methods are especially suitable for the systems with unknown or complex model structures, such as morphing or damaged aircrafts [27, 28], hybrid systems of power electronics [19], and biological systems [29]. The artificial neural network method (ANN) [30] has been used for nonparametric estimation, but it needs some priori data for training the input-output mapping. In this paper, the specific aspect we are dealing with is nonparametric estimation of switching piecewise systems from experiment data without needing any other required knowledge (e.g., either system structure or priori data). For nonparametric identification, Zadeh [31, 32] first introduced the definition of timevarying frequency response function (TV-FRF) to describe the time-varying behavior of system dynamics. The TV-FRF reflects frequency responses that evolve with time, providing both the frequency and time dependent spectral. Some approaches have proposed to estimate the TV-FRF. A widely used method is the short-time Fourier transform (STFT) [33, 34]. In STFT, a short sliding time window is applied, and the system is assumed to be time invariant within the window, hence STFT is only capable of tracking slow time variant behavior. The time width of the window should be selected carefully to balance the time tracking accuracy (smaller width) and the frequency resolution (larger width). Recently, Lataire [35-38] and Pintelon [39, 40] developed a new method (hereinafter to be referred as
LPM-Legendre) to estimate the TV-FRF by parameterizing the time variation using Legendre polynomial basis functions. In this method, the LTV system is equivalent to a timedependent weighted sum of a series of LTI systems, and LPM is applied to estimate FRFs of LTI systems. LPM-Legendre can achieve a better description of time variation without reducing frequency resolution compared to STFT. However, this method is only suitable for slow time-varying systems whose dynamics or parameters vary smoothly as functions of time, and its performance degrades for switching piecewise systems. To the best of our knowledge, the nonparametric TV-FRF estimation of a switching piecewise system has not been addressed yet. The aim of this paper is to fill this gap. The paper follows the idea of Lataire [35-38] and Pintelon [39, 40], but focuses on the TVFRF estimation problem of switching piecewise LTV systems. Compared to the existing methods (STFT and LPM-Legendre), there are mainly two motivations of the two proposed approaches (ATIRMM and ALPM): (1) By using non-smooth Walsh basis functions, the TVFRF of a switching piecewise system can be estimated with the much higher accuracy compared to the existing methods; (2) The approach can be selected based on the object to be estimated, ATIRMM delivers more accurate TV-FRF and BLTIA estimations, while ALPM has better performance in noise power spectrum estimation. The paper is organized as follows: in Section 2, the estimation problem of the piecewise LTV system is formulated; the assumptions and definitions of the excitation signal, the measured output, and the BLTIA are described. Section 3 proposes two new algorithms to estimate the BLTIA, the noise spectrum and the TV-FRF from the known input and the output which is corrupted by measurement noise in one experiment, followed by the estimation error analysis and the comparison of the two algorithms in Section 4. In section 5, the selection rules of design parameters in the two methods are discussed. Finally, the algorithms are illustrated on two simulations in Section 6.
2 Problem Formulation and Assumptions
Definitions and properties of the switching piecewise TV-FRF and its BLTIA are recalled, Walsh basis functions are introduced, and some assumptions which are needed in the following of this paper are given. 2.1 Model for linear switching piecewise time-varying system Definition 1 (Switching Piecewise LTV System). The dynamic system is a SISO linear open loop causal system with switching piecewise (non-smooth) time-varying stationary behavior. The LTV system switches between a few linear time-invariant (LTI) subsystems. Definition 2 (TV-FRF). A LTV causal system can be fully characterized by the TV-FRF ( which is the Fourier transform of the shifted response ( (
)
(
∫
)
) to a Dirac impulse [31,32]
)
(1)
The system function can also be expressed as the sum of a truncated series expansion of FRFs of LTI systems multiplied with a set of basis functions [40] ( where, (
(
) for
)
∑
(
)
( )
(2)
are FRFs of LTI systems which are asymptotically stable,
) is the number of LTI systems, and
( ) for
are basis functions
satisfying the following assumption. Assumption 1 (Basis Functions). Basis functions are a set of series which are functions of time, satisfying the constraints [40] ( ) with
and
∫
( )
[
for
]
(3)
the experiment time.
Basis functions are selected to characterize the system time variation. Legendre polynomial functions have perfect performance for slowly time-varying dynamics [37-39], whereas Walsh functions are adopted here to capture the switching time-varying behavior. Walsh functions are defined as [41] ( )
∏
(
(
))
(4)
where,
,
is the order,
integer satisfying
, and
∑
,
is a
.
First 5 orders of Walsh functions are shown in Figure 1. It is easy to prove that Walsh functions satisfy the constraints in (3). 1
w0 (t )
T
t
T
t
3T/4
T
t
3T/4
T
t
7T/8 T
t
1
w1 (t )
T/2
-1 1
w2 (t )
T/4
-1 1
w3 (t )
T/4
-1
T/2
1
w4 (t )
T/8
-1
3T/8
5T/8
Figure 1. First 5 orders of Walsh functions 2.2 Excitation and output Assumption 2 (Excitation). The excitation signal ( ) satisfies following assumptions: a.
The excitation is a band-limited signal with the length , and the signal bandwidth less than ½ the sampling rate
is
of measurements, such that the aliasing error can be
eliminated. b.
DFTs of the excitation at different frequencies are uncorrelated [36].
c.
DFTs of the excitation should be rough enough at all excited frequencies (vary wildly over the frequency) in order to separate the system term from the smooth transient behavior [9].
d.
The excitation signal is known exactly without noise.
Assumption 3 (Output). The output ( ) is corrupted by the stationary filtered white noise ( ), and
( ) is independent with ( ).
( ) where,
( )
( )
(5)
( ) is the noiseless output which can be expressed as (Theorem 2 of [39])
( )
[∑
( ) ( )]
()
(6)
denotes the inverse Laplace operation, ( ) is the Laplace transform of ( ).
where,
The DFTs of input-output are given as
( ) where,
, and
,
√
∑
(
is sample period,
)
(7)
is the number of sampling data.
Remark 1: a.
Assumption 2 can be naturally satisfied with band-limited random phase multisine excitation or filtered white noise excitation, both of the signals have been widely used in the frequency identification.
b.
Assumption 3 implies that the experiment operates in the open loop.
c.
Assumptions 2 and 3 indicate that the approaches proposed in this paper are developed within the generalized output-error framework (noiseless input and noisy output). In practice, the input signal is always corrupted by noise, resulting an error-in-variable framework. However, the case with noisy input-output can be equivalent to an outputerror framework with a noiseless reference signal ( ) and two noisy outputs { ( ) ( )} satisfying Assumptions 2 and 3, shown in figure 2. The detail discussion is given in [42].
( )
( )
Generator /actuator 𝑎
Dynamics
( , ) ( ) ( )
0( )
( , )
0( )
+
( )
( )
Generator /actuator
( , )
𝑎
( ) 0(
)
( )
+
( )
+ ,
( , )
0(
)
+
( )
Figure 2. Equivalent model of the LTV system with noisy input-output. Left: the LTV system with noisy input ( ) and output ( ); Right: the LTV system with the noiseless input ( ) and two noisy outputs { ( ) ( )}. The LTV system (
)
(
)⁄
(
).
Theorem 1: By using Walsh basis functions, equation (6) is equivalent to
( )
∑ { ( )
( )
where,
{ ( ) ( )} ( )} , and
nonzero states of the systems
[∑
()
( )]
( )
∑
()
(8)
( ) are transient (leakage) terms that stem from
at the time points that basis functions
( ) change.
Proof: see Appendix A. Under Theorem 1, the DFT of the output is expressed as
( ) where, (
∑
(
)
(
( )
) is the DFT of the output noise
)
( ), (
(
)
) is the rational function of the (
transient (leakage) term, which is the sum of system leakage (
), and ∑
( (
) with
(
)
)
(
)
(9)
), noise leakage
( ), i.e., (
)
∑
(
)
2.3 BLTIA Definition 2 (BLTIA). The BLTIA of the LTV system satisfying Definition 1 over the time interval [
] is defined as [39] (
)
∫
(
)
(10)
Combining (10) with (2) and (3) gives (
)
(
)
(11)
Under assumptions 2 and 3, the output DFT ( ) can be written as
( ) where,
( ) ( )
( )
(12)
( ) is the residual which is uncorrelated (but not independent) with ( ).
By using equation (12), it can be easily known that the FRF ̂ ( ̂ (
)
[ ( ) ] [| ( )| ]
) estimated by SAM [6] (13)
equals to the BLTIA (10), i.e., ̂ (
)
(
)
(14)
Equation (14) holds once the assumption 2(b) is satisfied, i.e., [ ( )
(
)]
. If the DFT of the input is correlated over the frequency, the FRF ̂ (
for
) is given
as ̂ (
)
(
)
∑
(
)∑
[
( )
(
)
| ( )|
]
( )
(15)
3 Estimation Algorithms For switching LTV systems, two new algorithms are proposed here to estimate: (1) the BLTIA and its variance; (2) the noise power spectrum at each frequency; (3) the TV-FRF. Two algorithms are extensions of TIRMM and LPM, respectively.
3.1 Algorithm based on the TIRMM (ATIRMM) In TIRMM, Hägg [11] has proved that the FRFs of both the system and leakage term are functions of global parameters for a LTI system, and the global parameters can be estimated by a global LS estimator. The FRF of the LTI system (
) at frequency
can be
extended round the frequency ( with (𝑎)
)
(
and
(
)
∑
(𝑎) (
)
)
(16)
.
The leakage term can be written as ( with
()
(
)
√
()
(17)
).
According to equation (9), the output DFT ( (
∑
)
∑
(
)
) at frequency
(
)
(
)
is written as (
)
Inserting (16) and (17) into (18) gives (
)
∑
{[
(
)
∑
(𝑎) (
)]
(
)}
(18)
√
with
(𝑎)
∑
(
()
)
(19)
.
Under the assumption that
(
()
to infinity, and also
(𝑎) decays to 0 when 𝑎 tends
) is asymptotically stable, if
. The sums in (19) can be approximated by the
truncations ∑
(𝑎) (
∑
()
)
∑ ∑
(𝑎) (
)
(20)
()
(21)
Equation (19) is approximated to (
)
∑ √
{[
(
)
∑
∑
(𝑎) (
)]
(
()
(
)}
)
(22)
The LS problem is obtained by assembling equation (22) for {
} , and at each
(
)(
neighboring frequencies are used with
. The global LS estimator consists of )
DFT frequencies
(
(
) equations for
) unknown global parameters (23)
where,
is the [(
)(
)
( [(
estimated,
is the
(
)
regressors,
is the
(
)
(
)
)]
vector of unknown parameters to be
)(
)
(
)] matrix of known
vector consisting output DFTs, and
is the
vector of residuals. Complete expressions of the matrices are given in
Appendix B. Assumption
4
regression matrix
(Identifiability).
The
(
[(
)
)(
)
(
)]
is of full column rank.
It can be verified that with the excitation using random phase multisine or filtered white noise, assumption 4 is fulfilled if the degree of freedom (DOF) (
)
[(
)(
)
of (
satisfies )]
(24)
The BLTIA estimation can be solved ̂
(
) ̂
̂
(25)
[
(26)
]
The vector of residuals is ̂
(
The noise power spectrum ̂ ̂
( (
(
)
)
) at frequency
is estimated by |̂ (
∑
)
In practice, the matrix inversion (
)
(27)
)|
(28)
in (25) and (27) is avoided by using some
numerically stable methods, such as singular value decomposition (SVD) [6].
3.2 Algorithm based on the Local Polynomial method (ALPM) (
) and (
) are expanded at frequency (
)
(
)
∑
(
)
(
)
∑
by applying Taylor’s formula (
)
(
)
(
)
(29) (30)
Inserting (29) and (30) into (18) gives (
)
∑
(
{[
∑
)
(
∑ (
)
The frequency domain window with a width ( {
the local LS estimator with
]
(
)}
(
)
)
(31) ) around
is selected to generate
}. At each frequency
, equation (31)
can be expressed in the matrix form ( )
( )
(
)(
)
(
in Appendix C.
)
) (
is ) [(
(
( ) ( )
[(
( )
where,
)(
( ) ( ) )]
(32)
is the column vector of to-be-identified paramters;
the ) (
( )
column )]
is
the
vector matrix
of of
known
output regressors,
DFTs; and
is the column vector of residuals. Expressions of matrices in (32) are given
Under assumption 4, the identifiability condition of (32) is fulfilled if the DOF of
( )
satisfies (
)(
The BLTIA estimated by ALPM at frequency
)
(33)
is the first element of estimated parameter
vector ̂ ( ), i.e., ̂ (
̂ ( )
)
(34)
The vector of residuals is ̂( The noise power spectrum ̂ ̂
(
)
( )(
(
) (
( )
( ))
) at frequency ) [(
(
)(
( ))
( )
(35)
is estimated by ) (
)]
This procedure is repeated at each central frequency
|̂ (
∑
)|
(36)
over the frequency range of
interest.
3.3 TV-FRF estimation In both ATIRMM and ALPM, the FRFs of LTI systems estimators. The TV-FRF estimation ̂ ( ̂(
(
) in (2) can be estimated by LS
) is elaborated by )
∑
̂ (
)
( )
(37)
The time resolution of the TV-FRF is closely related with the order the experiment time . The minimum resolvable time range
of Walsh functions and
is expressed as (38)
with
satisfying
.
Equation (38) shows that the time resolution can be enlarged with larger However, the reduction of measurement time
would result in an undesired loss of
estimation accuracy and decrease the frequency resolution. The number by the estimated uncertainty which is analyzed in the next section.
4 Error analysis
and smaller .
is also limited
4.1 Bias of estimated BLTIA The bias of estimated BLTIA in ATIRMM is (̂ ( With
(
))
⁄ )
) (
(
)
(39)
the absolute value of the largest eigenvalue of the stability matrix of
.
Proof: see Appendix D. The bias of estimated BLTIA in ALPM is (̂ ( with
(
)
(
) the (
(
)) )
)
(
) ((
)
(
deviation of
)
((
)
)
(40)
).
Proof: see Appendix E.
4.2 Variance of estimated BLTIA With the LS estimator, the variance of residuals is ̂ ̂
̂ In ATIRMM, ̂
(41)
̂ is the variance of residuals, ̂
̂ which is given in (27), and
the degrees of freedom (DOF) of the regressive matrix In ALPM, ̂
is
, shown in (24). ; ̂
̂ ( ) is the spectrum of residuals at frequency
̂ (
) which is
( ) is given in (33).
shown in (35), the DOF of the regressive matrix The covariance of ̂ is ( ̂)
̂ (
)
(42)
In ATIRMM, the variance of estimated BLTIA is given as 𝑎 (̂ )
( ̂ ))}|
{ 𝑎 (
[
]
(43)
In ALPM, the variance of estimated BLTIA is 𝑎 (̂ (
))
( ̂ ( ))|
[
]
for
(44)
By expanding the variance analyses results given in [43], the variances of estimated BLTIA in the two methods can be respectively approximated as
𝑎 (̂ ) 𝑎 (̂ (
(
(
)(
) ( (
(
))
(
)
) )(
) (
)̂
(45)
)
)̂ ( )
(46)
4.3 Bias of estimated noise spectrum The bias of estimated noise spectrum in ATIRMM is (̂
(
))
(∑
)|
(
|∑
((
) )
)
(
(
(
)| )
∑
)
(
|∑
)) (
(
) (47)
The bias of estimated noise spectrum in ALPM is
(̂
(
))
(
∑ (| ∑ (
∑ (|∑ (
((
)
(
)
)
)
(
(
)
((
))| )
(
)
))| )
((
)
)
)
(48)
4.4 Discussion (1) With same values of
,
and
in both ATIRMM and ALPM, the estimated BLTIA in
ALPM has smaller bias and larger variance compared to the one in ATIRMM. (2) ATIRMM has the larger bias of the estimated noise spectrum than ALPM. If the goal is to estimate the noise spectrum, ALPM may be preferred. (3) In both ATIRMM and ALPM, with the increase of
and decreases of
and
, the
biases of the BLTIA and the noise spectrum increase, while the variance of the estimated BLTIA decreases. The larger
increases the variance of the BLTIA, while inversely
decreases the bias of the noise spectrum.
5 Selection of design parameters In both of the two algorithms, the estimation accuracy is highly related to the values of design parameters
,
,
and
.
The leakage term (
) has exactly the same poles with (
( ) decay at the same rate
). It means that ( ) and
. Hence, a reasonable choice is
Based on the analyses of pervious section, the values of
.
and
are selected based on the
purpose of the identification, shown in table 1. Table 1. Selection of
and Cases
Estimation purpose
Small noise , ,
BLTIA Noise power spectrum
large noise , , ,
,
Based on extensive simulations, ranges of design parameters are given as follows to achieve good performance in many cases: a. In ATIRMM,
;
for highly damped systems, and
for lowly damped systems. b. In ALPM,
;
for highly damped systems, and
for lowly damped systems. The regression matrix of the LS estimator should be of full column rank (assumption 4), which implies
, therefore, in ATIRMM, (
)
[(
)(
)
(
)]
(49)
and in ALPM, [( In practice, the number
)(
)
(
)]
(50)
of time-variant branches are unknown, and should be selected
based on input-output data. The estimation procedure starts with is gradually increased until the absolute value of the smaller than its standard deviation, i.e., | ̂ (
)|
, and the value
FRF estimation | ̂ ( {̂ (
)| is
)}.
6 Simulation The goals of the simulation are to: (1) illustrate the estimation accuracy of each FRF and its variance; (2) prove the relationship between the BLTIA
(
) and
(
(
)
); (3)
evaluate the selection procedure of
; (4) estimate the noise spectrum; (5) identify the TV-
FRF and the time point that the switching change of the LTV dynamic occurs; (6) compare two algorithms proposed (ATIRMM and ALPM) with other methods (STFT and LPMLegendre). Two different structures of switching piecewise systems are simulated: (1) A LTV model with parallel time-varying branches; (2) A piecewise change dynamic model of a small morphing unmanned aerial vehicle (MUAV) which changes its sharp in flight.
6.1 Model with parallel structure The LTV system consists of 4 parallel time-varying branches, each
is a second-order stable
( ) are Walsh functions satisfying equation (4). The
transfer-function, shown in figure 3.
TV-FRF of this LTV system is expressed as ( Hence, the BLTIA
(
)
) (
∑
(
)
(51)
). G0
- 200 s 2 20 s 400
y0 (t )
G1
y1 (t )
- 625 s 2 3 0 s 625
u (t )
( )
G2
y2 (t)
- 454 s 2 25 s 500 G3
y3 (t )
- 796 s 2 45 s 1000
w0 (t )
w1(t)
y (t )
w2 (t)
w3 (t )
Figure 3. LTV system with parallel structure The excitation is a random phase multisine signal, which satisfies the assumption 2. ( ) where,
(
)
is the flat amplitude at each frequency,
distributed in the range (
for
). The sampling rate
, the number of sampling excited.
∑
,
(52) .
is uniformly
, the experiment time
, the frequency band [0.01Hz,10Hz] is
The output ( ) is corrupted by filtered white noise
( ), and
( ) has the following
expression ( )
( )
( )
(53)
with is the convolution operator, the noise spectrum is given in figure 6 (black line). Values of design parameters are chosen as follows: (1) The width of window is chosen as (2) For ATIRMM: The estimated FRFs
for both ATIRMM and ALPM;
; For ALPM: (
.
) are shown in figure 4. In figure 4(e), the estimated FRF ̂ (
)
has the same order of amplitude with its standard deviation. By following the procedure of selection in Section 5, the number
is selected as
, which coincides with the
number of parallel branches in (51). Figures 4 (a)-(d) show that the standard deviation
and the estimation error in ATIRMM are smaller than the ones in ALPM.
(a)
(
)
(b)
(
)
(c)
(
)
(d)
(
)
(
(e)
)
) of the model (51). Black thin line: true value of |
Figure 4. Estimated
(
gray thin line: | ̂ (
)| estimated by ATIRMM; light gray thin line: | ̂ ( {̂ (
ALPM; black thick line:
)} by ATIRMM; gray thick line :
dark gray crosses: estimated error | ̂ ( estimated error | ̂ (
)
(
)
(
(
)|; dark
)| estimated by
{̂ (
)} by ALPM;
)| by ATIRMM; light gray squares:
)| by ALPM.
For comparison, the BLTIA is also computed via SAM in (13). 500 runs are repeated to perform a Monte Carlo simulation. The simulation result ̂ (
) is given by the light gray
crosses in figure 5. It is obvious that ATIRMM and ALPM achieve higher BLTIA estimation accuracy than SAM, moreover, the reliable estimation can be obtained only in one experiment.
Figure 5. Estimated BLTIA ̂ ( line: | ̂ (
). Black line: | ̂ (
)| estimated by ALPM; gray crosses: ̂
)| estimated by ATIRMM; gray thick
estimated by SAM.
Figure 6 shows that the noise power spectrum estimated by ALPM has the smaller bias than the one estimated by ATIRMM.
Figure 6. Estimated noise spectrum line: ̂
(
(
). Black line: true value of
) estimated by ATIRMM; dark gray line: ̂
(
(
); light gray
) estimated by ALPM.
6.2 Model of small MUAV Morphing aircrafts can change their shapes or configurations substantially to achieve high performance for a changing mission environment in flight [27]. The methods proposed in the paper can be used to estimate the nonparametric model of a small MUAV. In the simulation, the MUAV has two shapes which can be changed in flight: a) the wing is telescoped in the body for high speed cruise, and b) the wing is unfold for economic cruise and landing. The MUAV can be considered as a switching piecewise system under the assumption that the change between two sharps is instantaneous. The linearized dynamic model (
) of the
( ) and pitch rate output ( ) at the specified velocity and
MUAV with elevator input altitude is given as follows.
(
where,
)
{
(
)
(
)
is the time point that the sharp of aircraft changes,
The simulation time
, and the sampling rate
(54)
. . The excitation is a
random phase multisine signal shown in (52). The frequency range [0.1Hz,15Hz] is excited, over the excited frequencies. The output ( ) is disturbed
and the amplitude
by the measurement noise with the SNR 20dB. The designed parameters are chosen as: a) for both ATIRMM and ALPM, b) for ATIRMM:
; for ALPM:
. The BLTIA is easy to be calculated ( ) The FRFs ̂ (
∫
(
)
(55)
) estimated by ATIRMM are shown in figure 7. ̂ (
of amplitude with
{̂ (
the amplitude | ̂ (
)| gradually decreases, and it becomes close to
)}, hence
) has the same order
. It can be seen that with the increase of {̂ (
)}.
,
(a) ̂ (
)
(b) ̂ (
)
(c) ̂ (
)
(d) ̂ (
)
(e) ̂ (
)
(f) ̂ (
)
(g) ̂ (
)
(h) ̂ (
)
(i) ̂ (
)
(j) ̂ (
)
(k) ̂ (
)
(l) ̂ (
)
Figure 7. ̂ ( {̂ (
)
(n) ̂ (
)
(o) ̂ (
)
(p) ̂ (
)
) estimated by ATIRMM. Black solid line: | ̂ (
)| ; black dash line:
)}.
The FRFs ̂ ( (2) is
(m) ̂ (
) estimated by ALPM are given in figure 8. The number of significant FRFs in
. For
,|̂ (
)| falls within its uncertainty
{̂ (
)}. It shows that
the variance estimated by ALPM is slightly larger than the one estimated by ATIRMM.
(a) ̂ (
)
(b) ̂ (
)
(c) ̂ (
)
(d) ̂ (
)
(e) ̂ (
)
(f) ̂ (
)
(g) ̂ ( Figure 8. FRFs ̂ ( |
(̂ (
(h) ̂ (
)
)
) estimated by ALPM. Black solid line: | ̂ (
)|; black dash line:
))|.
Estimated ̂ (
) and ̂ (
(
) are compared with the true value of
) in (55), shown
in figure 9. Both ATIRMM and ALPM can achieve high accurate BLTIA estimations. Compared to ALPM, ̂ (
) estimated by ATIRMM has smaller error and uncertainty.
(a) Figure 9. |̂ (
(b)
Estimated BLTIA. Figure (a): | ̂ (
)| by ALPM (top light gray thin line),
thick line),
(̂ (
(dark line), | ̂ (
)| by ATIRMM (top dark gray thin line), {̂ (
)} by ATIRMM (bottom dark gray
)) by ALPM (bottom light gray thick line). Figure (b): true |
)
(
)| by ATIRMM (dark gray line), | ̂ (
)
(
(
)|
)| by ALPM
(light gray line). In ATIRMM, the number
of parallel time-varying branches is selected as 15, and the
minimum resolvable time range FRF occurs between
⁄
and
is ⁄ ⁄
. In figure 10(b), the switching change of the TV-
, which implies that the dynamic changes between
two LTI subsystems within the time range of (36.75s-42.875s). In ALPM, the time resolution is ⁄ , and the switching change occurs between in figure 10 (c).
⁄ and
⁄ (i.e., 36.75s-49s), shown
ATIRMM and ALPM are also compared to the existing methods which are STFT and LPMLegendre. In STFT, the width of the sliding time window is selected as 30s. Under the assumption that the system dynamic is time invariant within the window, STFT is not capable to capture the switching time variation, shown in figure 10(d). By using smooth basis functions, LPM-Legendre also has poor performance with larger estimation error and uncertainty, shown in figure (e).
a) True TV-FRF
c) TV-FRF estimated by ALPM
b) TV-FRF estimated by ATIRMM
d) TV-FRF estimated by STFT
e) TV-FRF estimated by LPM_Legendre Estimated TV-FRF ̂ (
Figure 10.
{ ̂(
bottom surface:
) . Upper surface: estimated TV-FRF ̂ (
)}.
It is also possible to identify two subsystems estimated TV-FRF ̂ ( equal to ̂ (
);
(
) and
(
). In this simulation, the estimated ̂
) and ̂ (
), respectively. Estimated ̂
(
) in (54) from the ) and ̂
(
) and ̂
(
(
) are
) are given
in figure 11. It shows that the two algorithms also have ability to estimate the LTI subsystems of the switching time-variant system.
a) ̂
(
b) ̂
)
(
)
Figure 11. Estimated FRFs of LTI subsystems between which the LTV system switches. Black line: true values of | and | ̂ |̂
(
(
)
)
7 Conclusions
( (
(
)| and |
(
)|; dark gray line: | ̂
)| by ATIRMM; light gray line: | ̂
)| by ALPM.
(
( )
)
( (
)|
)| and
Two new approaches which are ATIRMM and ALPM are proposed for estimating the nonparametric TV-FRF and its BLTIA of a switching piecewise LTV system. The TV-FRF is approximated by a sum of multi FRFs of LTI systems multiplied with basis functions. The Walsh functions are adopted to capture fast transients and switching piecewise dynamics. The proposed methods can estimate the TV-FRF with higher accuracy and smaller uncertainty compared to the existing methods. The noise spectrum is also estimated by the two methods. Results of error analysis and simulations show that ATIRMM has a better performance on estimating the TV-FRF and the BLTIA, whereas ALPM is more suitable to estimate the noise power spectrum.
Appendix A. Proof of Theorem 1 Theorem 1 is proved for For
, the proof of the cases for
follows the same lines.
, the equivalent model of the LTV system is given in Figure A.1. 0(
( )
0(
1(
2(
)
)
0(
)
1(
)
2(
)
) 0(
1(
)
2(
)
)
)
Figure A.1 Equivalent model of the LTV system for We have ( )
{
( ) ( )}
( )
{
( ) ( )}
( )
{
( ) ( )}
( )
(A.1)
(a) It is obvious that the first term of (A.1) is equal to
{ ( ) ( )} with
( )
.
(b) The second term of (A.1) is
()
{ ( ) { ()
( )}}
(A.2)
{ ( ) ( )} {
( )
with
( )
Inserting
{
()
()
(A.3)
( ) ( )} ⁄
{
into (A.3) gives
⁄
{ ( ) ( )} with ̅ ( )
()
()
̅ ()
()
(A.4)
⁄
( ) ⁄
We define that ̃ ( )
{
( ) { ̅( )}},
with ̅( )
{
⁄
( )
(A.5)
⁄
̅ ( ) can be expressed as ̅ ( ) where, at
̃ ( )
( )
(A.6)
( ) is the transient term that stems from the nonzero state of the system
( )
, and ( )
{
The relation of ̃ ( ) ̅ ( ), and
(
)
(A.7)
( ) is shown in figure A.2.
~ y 1(t)
y 1(t)
y 1(t) T/2
T
Figure A.2 Relation of ̃ ( ) ̅ ( ), and
( )
Insert (A.5) and (A.6) into (A.4)
{ ( ) ( )}
()
{ ( ) { ̅( )}}
()
()
( ) { ̅( )}}
{
( ) ( )}
{ {
( )(
{ ̅( )}
( ))}
( )
( )
{
( ) { ̅( )
{
( ) { ( )
( )}}
{
( ) { ( )
( )}}
( ) the transient term of LTI system
with
( )}}
( ) ( ) ( ) ( ).
( )
, and
(A.8)
(c) The third term of (A.1) is
{ ( ) ( )}
()
()
()
(A.9)
⁄ ( )
Inserting
⁄
{
into (A.9) gives ⁄
{ ( ) ( )} ( )
With ̅
{
⁄
( )
()
and ̅
⁄
̅ () ( )
̅ ()
(A.10)
⁄
( )
{
()
⁄
We define that
̅
̃
( )
{
( ) {̅
( )}}, with ̅
( )
{
̃
( )
{
( ) {̅
( )}}, with ̅
( )
{
( ) and ̅
⁄
( )
(A.11)
⁄ ⁄
( )
(A.12)
⁄
( ) can be expressed as
̅
( )
̃
( )
( )
with
( )
{
(
̅
( )
̃
( )
( )
with
( )
{
(
)
(A.13)
)
(A.14)
Inserting (A.13) and (A.14) into (A.10) gives {
( ) ( )}
( )
( )
̃
( )
( )
{
( )(
{̅
( )}
{̅
{
( ) { ̅
{
( ) { ( )
( )}}
{
( ) { ( )
( )}}
̃
( )
̅
( )
( ) ( )}
( ))}
( )}} ( )
()
( ) ( ) ( )
( ) ( )
( )
(A.15)
with
( ) the transient term of LTI system
( )
( )
, and
( ).
Combining equations (A.1), (A.2), (A.8) and (A.15) gives ( )
( ) { ( )
{
( ) { ( )
{ The cases for
{
( ) { ( )
( )
()
( )}} ( )}}
( )}} (A.16)
exactly follow the same lines. Theorem 1 is proved.
Appendix B. The expressions of the matrices in (23) ( [
(
]
)
) ( )
with
(B.1)
[ (
[(
( )
)(
) (
)]
)]
with
(
)
(
)
[ ( )]
(
[
[
]
(B.2)
( )
(
) [(
)(
(
) (
)
)] ) (
)]
(B.3)
with
[
] (
) (
( ) (
[ (
) (
[
(
(B.4)
( )
)
)
(
(
( )
( (
( )
) )
(B.5)
)] )
( )
)
)
(
( ) (
) (
)
(
( )
)
) ( )
(
)]
(B.6)
[
(
]
) (
)
(B.7)
with √
√
√
√
√
√
[√
√
√
(
) (
)
(B.8)
]
Appendix C. The expressions of the matrices in (32) ( ( )
[
)
(
) (
(
)
)(
) (
]
(C.1)
)
( ) ( ) [(
( )
( ( (
( )
( )
(
(
[
)
( [
)
(
( )]
(C.4)
) [(
(
( )
)(
) )
(
)
(
( )
)
)
(
(
(C.3)
)]
( ) (
) (
)
(
( )
)
(
( )
) ) )
(
[
(
)]
( ( (
( )
( )
) ) )
(
[
[
(C.2)
( ) ( )]
[
( )
)]
) (
( (
)]
) )
]
)
(C.6) (
(
(C.5)
)
(
) )
) (
) )
(
]
) (
)
(C.7)
Appendix D. Proof of (39) The
(
(
)
) in (16) can be expressed as (
)
(
(
⁄ )
)
(D.1)
( ⁄ ) .
with
(
Using Taylor formula to expand ( ⁄
)
⁄ )
∑
(
(
⁄ )𝑎)
(D.2)
(
(
(D.3)
Inserting (D.2) into (D.1) (
)
∑
The accurate expression of output DFT ( (
)
∑
{[
(
)
{
with
) at frequency )]
(
)}
√
is
∑
()
(
)}
(
) (D.4)
} (
The terms (
(𝑎) (
∑
⁄ )𝑎)
)
) neglected in (22) are expressed as {[
∑ ∑
(
)
∑ (𝑎) (
{[∑
)]
(𝑎) ( )]
(
)}
√
∑
()
(D.5)
)
(D.6)
Insert (D.3) into (D.5) (
)
(
)
(
)
(𝑎)
∑
(
(
where, (
(
)
∑
)
∑ ∑
( with
)
{[
√
(
)
(𝑎)
{[∑ { ∑
∑
( ⁄ )
(
()
∑
(
(𝑎 ⁄ ))
(𝑎 ⁄ ))
(
]
]
(
)} (D.7)
)}
)}
(D.8) (
) (
)
the absolute value of the largest eigenvalue of the stability matrix of
(D.9) [36].
(D.4) can be rewritten into matrix forms as (D.10)
where, )(
, )
, and
(
are the matrices
] th components are
(
),
)
of which the [(
(
) , and
(
),
respectively. The expert value of ̂ is written as {̂ }
{(
{(
)
where, {(
)
}
)
(D.11)
} ,
)
{(
} ,
and
}.
The expect value of estimation ̂ is {̂ } |[
where, Basis ( )
]
(D.12) |[
,
( ) ,
functions ( )
,
]
|[
, and
are linearly
]
independent,
hence
inputs
( ) are linearly independent for non-zero inputs ( ) (lemma4 of [39]). It
is easy to deduce that (D.13) ( ⁄ ) Based on Theorem 6.15 in [6],
(D.14)
is equal to ) (
(
)
(D.15)
Inserting (D.13), (D.14), and (D.15) into (D.12) gives {̂ }
(
)
( ⁄ )
) (
(
)
(D.16)
Equation (39) is proved. Appendix E. Proof of (40) Taylor expansions of (
) and (
) can be written as
(
)
(
)
∑
(
)
(
)
∑
( (
(
) )
√
) (
(E.1) )
(E.2)
The remainder of (E.1) is depends on the (
)th derivation of the system FRF, collect
equations (E.1) and (E.2) into (31) (
)
∑
{[
(
)
(
∑
)
∑
(
(∑
)
(
{
with
(
)
)
]
(
)}
(
)
(
)
(
)) ((
(
)
)
)
∑
(
((
√
)
)
) (E.3)
)
)
}
(E.3) can be written into matrix forms as ( )
( ) ( )
√
( )
((
)
(
)
( ))
((
(E.4) (
∑
(
)
( (
)
(
)
∑
[
)
)
∑ with
(
(∑
)
(
)
)
(E.5) ]
The expert value of ̂ is { ̂ ( )}
{(
( )
( ))
( ) ( )}
( )
( )
( )
( ) (E.6)
( ) is the true vector of parameters,
where,
( ) ( )
{(
( )
( ))
( )
{(
{(
( )
(
( ) (∑ ( )
( ))
( ))
( ) )
( )
(
√
}
(E.7)
)
( )) ((
((
)
)
)}
(E.8)
)}
(E.9)
( )
(E.10)
The expect value of BLTIA can be written as {̂ ( where,
( ),
)}
( ), and
(
)
( )
( )
( ) are the first components of
( ). ( )
( )
( ) are independent, hence
( )
( ), and
( ) ( )
(
(E.11)
)
(
) ((
)
)
(E.12)
Pintelon (appendix 7.C of [6]) has proved that ( )
{(
( )
( ))
( )
√
((
)
)}
((
)
)
(E.13)
Inserting (E.11), (E.12), and (E.13) into (E.10) gives {̂ (
)}
(
)
(
)
(
) ((
)
)
((
)
)
(E.14)
Equation (40) is proved.
Acknowledgements The authors would like to thank the reviewers for their valuable suggestions. References [1] Rik Pintelon, Johan Schoukens, Wendy Van Moer, Yves Rolain, Identification of linear systems in the presence of nonlinear distortions, IEEE Transactions on Instrument and Measurement, 50 (2001), no.4, 855-863. [2] Rabiner L., Allen J.B., On the implementation of a short time spectral analysis method for system identification, IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-28 (1980) 69-78. [3] Peter E. Wellstead, Non-parametric methods of system identification, Automatica, 17 (1981) 55-69. [4] Schoukens J., Rolain Y. and Pintelon R., Improved frequency response function measurements for random noise excitations, IEEE Transactions on Instrumentation and Measurement, 47 (1998) 322-326. [5] Widanage W.D., Douce J.L., and Godfrey K.R., Effects of overlapping and windowing on frequency response function estimates of systems with random inputs, IEEE Transactions on Instrumentation and Measurement, 58 (2009) 214-220. [6] Rik Pintelon, Johan Schoukens, System Identification: A Frequency Domain Approach, second ed., IEEE Press, 2012. [7] Daniel Belega, Dario Petri, Frequency estimation by two- or three-point interpolated Fourier algorithms based on cosine windows, Signal Processing, 117 (2015), 115-125.
[8] Rik Pintelon, Johan Schoukens, Gerd Vandersteen, Kurt Barbé, Estimation of nonparametric noise and FRF models for multivariable systems- part I: theory, Mechanical Systems and Signal Processing, 24 (2010) 573-595. [9] Johan Schoukens, Gerd Vandersteen, Kurt Barbé, Rik Pintelon, Nonparametric preprocessing in system identification: a powerful tool, European Control Conference (ECC), Budapest (2009) 1-14. [10] Johan Schoukens, Rik Pintelon. High quality frequency response function measurements without user interaction, UKACC International Conference on Control, Coventry (2010) 1-5. [11] Per Hägg, Hakan Hjalmarsson, A least squares approach to direct frequency response estimation, 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, USA (2011) 2160 – 2165. [12] Per Hägg, On structured system identification and nonparametric frequency response estimation, Doctoral Thesis, KTH School of Electrical Engineering (2014). [13] Easwar Magesan, Alecandre Cooper, Honam Yum, Paola Cappellaro, Reconstructing the profile of Time-varying magnetic fields with quantum sensors, Physical Review 88 (2013) 1-16. [14] Dimitriadis, G., Cooper, J.E., Flutter prediction from flight flutter test data, Journal of Aircraft, 38 (2001) 355-367, DOI: 10.2514/2.2770. [15] J. Ertveldt, J. Lataire, R. Pintelon, S. Vanlanduit, Frequency-domain identification of time-varying systems for analysis and prediction of aeroelastic flutter, Mechanical systems and signal processing, 47 (2014) 225-242. [16] Chen, S.Y., Yuan, J. S., Adaptive gate bias for power amplifier temperature compensation. IEEE Transactions on Device and Material Reliability, 11 (2011) 442-449. [17] Jahanshah Kabudian, Mohammad Mehdi Homayounpour, Seyed Mohammad Ahadi, Bernoulli versus Markov: Investigation of state transition regime in switching-state acoustic models, Signal Processing, 89 (2009), 662-668. [18] James D. Hamiltion, Analysis of time series subject to changes in regime, Journal of Econometrics, 45 (1990) 39-70. [19] R.P. Aguilera, B.I. Godoy, J.C. Agüero, An EM-based identification algorithm for class of hybrid systems with application to power electronics, International Journal of Control, 87 (2014) 1339-1351. [20] Peng shi, Xiaojie Su, Fanbiao Li, Dissipativity-based filtering for fuzzy switched systems with stochastic perturbation, IEEE Transactions on Automatic Control, (2015).
[21] Xiaojie Su, Peng Shi, Ligang Wu, Yong-Duan Song, Fault detection filtering for nonlinear switched stochastic system, (2015). [22] Peng Zhang, Jiangtao Cao, Guoliang Wang, Mode-independent guaranteed cost control of singular markovian delay jump systems with switching probability rate design, International Journal of Innovative Computing, Information and Control, 10(2014) 12911303. [23] Jose Ragot, Gilles Mourot, Didier Maquin, Parameter estimation of switching piecewise linear system, Proceedings of the 42nd IEEE conference on Decision and Control Maui, Hawaii USA, 6 (2003) 5783-5788, Vol.6. [24] Xiaojie Su, Ligang Wu, Peng Shi, C.L. Philip Chen, Model approximation for fuzzy switched systems with stochastic perturbation, IEEE Transactions on Fuzzy Systems, 23(2015) 1458-1473. [25] Andras Hartmann, Susana Vinga, Joao M. Lemos, Hybrid identification of time-varying parameter with particle filtering and expectation maximization, Mediterranean Conference on Control and Automation, 6 (2013) 884-889. [26] Matthew S. Holzel, Eugene A. Morelli, Real-time frequency response estimation from flight data, Journal of Guidance, Control, and Dynamics, 36 (2012) 1406-1417. [27] Terrence A. Weisshaar, Morphing aircraft technology – new sharps for aircraft design, Meeting Proceedings RTO-MP-AVT-141, (2006) 1-20. [28] Jared Grauer, Eugene Morelli, Method for real-time frequency response and uncertainty estimation, Journal of Guidance, Control, and Dynamics, 37 (2014) 336-343. [29] Kaustubh R. Joshi, Natasha Neogi, William H. Sanders, Dynamic partitioning of large discrete event biological systems for hybrid simulation and analysis, Hybrid Systems: Computation and Control, 7th international workshop, HSCC, (2004) 463-476. [30] Iman Sadeghkhani, Abbas Ketabi, Rene Feuillet, An intelligent switching overvoltages estimator for power system restoration using artificial neural network, International Journal of Innovative Computing, Information and Control, 10(2014) 1791-1808. [31] Zadeh, L. A, The determination of the impulsive response of variable networks, Journal of applied Physics, 21 (1950) 642-645. [32] Zadeh, L. A, Frequency analysis of variable networks, Proceedings of the IRE, 38 (1950) 291-299. [33] Allen, J.B., Rabiner, L., A unified approach to short-time Fourier analysis and synthesis, Proceedings of the IEEE, 65 (1977) 1558-1564.
[34] J.K. Hammond, P.R. White, The analysis of non-stationary signals using time–frequency methods, Journal of Sound and Vibration, 190 (1996) 419-447. [35] John Lataire, Rik Pintelon, Estimating a nonparametric colored-noise model for Linear slowly time-varying systems, IEEE Transactions on Instrumentation and Measurement, 58 (2009) 1535-1545. [36] John Lataire, Ebrahim Louarroudi, Rik Pintelon, Detecting a time-varying behavior in frequency response function measurements, IEEE Transactions on Instrumentation and Measurement, 61 (2012) 2132-2143. [37] John Lataire, Rik Pintelon, Ebrahim Louarroudi, Non-parametric estimate of the system function of a time-varying system, Automatica, 48 (2012) 666-672. [38] John Lataire, Ebrahim Louarroudi, and Rik Pintelon, Non-parametric best linear time invariant approximation of a linear time-varying system, IEEE Instrumentation and Measurement Technology Conference (I2MTC), (2012) 1050-1055. [39] Rik Pintelon, Ebrahim Louarroudi, John Lataire, Detection and quantification of the influence of time variation in frequency response function measurements using arbitrary excitations, IEEE Transactions on instrumentation and measurement, 61 (2012) 33873395. [40] Rik Pintelon, Ebrahim Louarroudi, John Lataire, Nonparametric time-variant frequency response function estimates using arbitrary excitations, Automatica, 51 (2015) 308-317. [41] Rui Zou, Hengliang Wang, Ki H. Chon, A robust time-varying identification algorithm using basis functions. Annals of Biomedical Engineering, 31 (2003) 840-853. [42] Rik Pintelon, Ebrahim Louarroudi, John Lataire, Detection and quantification of the influence of time variation in closed-loop frequency-response-function measurements, IEEE transactions on instrumentation and measurement, 62 (2013) 853-863. [43] Michel Gevers, Per Hägg, Hakan Hjalmarsson, Rik Pintelon, Johan Schoukens, The transient impulse response modeling method and the local polynomial method for nonparametric system identification, 16th IFAC Symposium on System Identification, Brussels, Belgium, 16 (2012) 55-60.
Highlights
Two new approaches (ATIRMM and ALPM) are proposed for estimating the nonparametric TV-FRF of a switching piecewise LTV system in one experiment; The switching time variation is modelled by Walsh functions;
Besides the TV-FRF, the best linear time invariant approximation (BLTIA) and the noise spectrum are also estimated with small uncertainty; Error analyses show that ATIRMM delivers more accurate TV-FRF and BLTIA estimates, while ALPM has better performance on noise power spectrum estimate.