Identification of Linear Systems Having Time Delay Connected in
Series
CHAIMAE ABDELAALI1, ALI BOUKLATA1, MOHAMED BENYASSI2, ADIL BROURI1
1AEEE Department, ENSAM,
Moulay Ismail University,
MOROCCO
2Electrical Engineering Department, ESTM,
Moulay Ismail University,
MOROCCO
Abstract: - Nonlinear system identification has been a hot research field over the past two decades. A
substantial portion of the research work has been carried out based on block-structured models. Time delay is a
problem occurring in most industrial applications. The time delay can destabilize the system. Then, the latter
should be determined to control the system. This work aims to present an approach allowing the identification
of a linear system having a time delay connected in series. In this study, an identification method is proposed to
determine the system parameters. This method is based on sine inputs / or periodic stepwise input.
Key-Words: - Systems identification, time delay, series connections of linear and time delay, stability, time
delay estimation, Least Square Method (LSM).
Received: July 9, 2023. Revised: June 6, 2024. Accepted: July 7, 2024. Published: August 13, 2024.
1 Introduction
Nonlinear system identification has been a hot
research field over the past two decades, [1], [2],
[3], [4], [5], [6]. The research in this way is still
ongoing, [7], [8], [9], [10]. Several available papers
have been focused on the identification of nonlinear
systems structured by the series connection of linear
and nonlinear blocks, [11], [12], [13], as well as the
parallel connection of linear and nonlinear blocks
[14]. The nonlinear system identification is often
addressed in the case of Wiener and Hammerstein
models, [1], [15], [16], [17], [18]. The identification
techniques have been used in several application
domains, [19], [20], [21].
Several techniques and solutions have been used
to identify the nonlinear system parameters, e.g.,
stochastic methods [22], deterministic recursive
techniques [23], and frequency methods [24], [25].
The research on nonlinear systems focuses not
only on identifying their nonlinearity but also on
control, to mitigate the negative effects of non-
linearity on an affected system's performance, e.g.,
adaptive control using the backstepping method has
been proposed, [26], [27], [28], fuzzy fixed-time
control [29], passive robust control [30].
In this work, the focus is on system
identification rather than compensating for the
effects of nonlinearity. Knowing these nonlinearities
makes other operations, including control, easier. In
this way, the most studied solutions are proposed in
the case of a series connection of linear and
nonlinear blocks, i.e., the case of Hammerstein,
Wiener, Wiener-Hammerstein, or Hammerstein-
Wiener models. To increase the complexity of
nonlinear system models and to make the model
more general, the parallel connections of linear and
nonlinear subsystems can be proposed, [31], [32],
[33], [34].
Presently, the problem of identification of linear
systems connected in series with a time delay is
addressed. The proposed approach can be applied to
a linear system. Furthermore, this system can
describe many industrial systems. The rest of the
paper is organized as follows. Section 2 is devoted
to the presentation of the identification problem. In
this section, a mathematical description of a studied
nonlinear system is also presented. The
identification method of the system (linear with time
delay) is developed in Section 3. Then, examples of
simulations are proposed in Section 4.
2 Problem Statement
Most of the studied identification problems have
been focused on linear or nonlinear blocks, e.g., the
Hammerstein system (composed of nonlinear
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2024.19.25
Chaimae Abdelaali, Ali Bouklata,
Mohamed Benyassi, Adil Brouri
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element followed by a linear subsystem) and the
Wiener system (having a linear subsystem followed
by a nonlinear block). Presently, the identification
problem is focused on a linear system followed by a
time delay system. The linear system is described by
a transfer function and the time delay is of
value . When the latter is null (i.e., ), a
constant linear system is obtained.
Accordingly, the studied system can be
mathematically described as follows:
where ‘’ denotes the convolution product, is the
time delay, is the control signal, denotes
the output signal, and is impulse response of
the linear system (i.e., is the inverse Laplace
transform).
In discrete form, the transfer function can
be written as the ratio between two polynomials
and :
where is the offset operator, i.e.,
. In this case, the transfer function
can be expressed as:
Unlike several previous works (e.g., [1], [12],
[15], [32]), this transfer function is not necessarily
of nonzero static gain. Let denotes the signal
before time delay (the output of linear system).
Then, one has immediately:
which can be expressed using (4):
The latter leads to the following recursive equation:
The output can thus be written as:
Let us suppose that the time delay is a
multiple of the sampling time. This means that
. By combining this remark
with (7), one has:
(9)
It is readily seen that the output can be
rewritten as the recursive expression:
where the data vector is defined as:
,
(11)
and the parameter vector is defined as follows:
Replacing with in (9), one
immediately gets:
, (13)
which can be also written as the recursive
expression:
where the parameter vector is given in (12) and
the data vector is given as:
, (15)
In the case where the time delay is known, the
data vector given in (15) become known for
any time . Indeed, the
expression of in (15) contains only the past
values of input and output , for
and , which can be
fully determined for any time
. Furthermore, the expression of is
affine according to the parameters, and
. Then, the parameter vector can be easily
identified using, e.g., the least square method
(LSM). Let
denoting the parameter vector
estimate. Accordingly, the LSM algorithm is
described by the following equation system (16)-
(17):
for any arbitrary initial estimate value
and the
gain matrix is defined as:
for any arbitrary initial value .
The problem that arises at this stage is related to the
fact that the time delay is not known. The data
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vector in (15) is not known. If an upper bound
of time delay , the least square method (16)-
(17) remains applicable for
.
Presently, any knowledge of time delay is
required . The estimator algorithm (16)-(17)
cannot be used directly. In order to overcome this
problem, an estimate method of time delay is
proposed. In this respect, the system is excited using
a null control input, or any other constant control,
for . Note that the time delay
is not known at this stage. To ensure that
, the null control is applied until
the system output becomes null (or constant). Then,
another value (different from zero) of control is
applied ,and observing the time value when the
output begins to change. This time value
corresponds to . Once an upper
bound of time delay delay is estimated, the
estimator algorithm (16)-(17) can be used.
Other details of this study can be given in
simulation section.
Fig. 1: Example of obtained results in the estimate
of
3 Simulation
Presently, the aim is to identify the linear system
parameters and the time delay . The linear part of
system considered in simulation is given as follows:
The latter can be characterized by their transfer
functions and , respectively. Then, the
latter have as parameters the module of gains
and , respectively, and the phases
and , respectively. Firstly, the
considered system (Figure 1) is excited by the
following signal:
where:
(19)
The time delay value is . In this work, the
time delay is not supposed to be known. To estimate
, the method described in section 2 will be used.
Then, the system is excited firstly by a control of
zero value until the output returns to zero. The
system is excited with a control different from zero.
Example of obtained results for is shown
in Figure 2.
To ensure that , a control
of zero value is applied until the system output
becomes null (or constant). Then, another value of
the control is applied and observing the time
value when the output begins to change. This time
value corresponds to . The result
shown by Figure 2 allows us to estimate the time
delay value . Specifically, the latter is .
Fig. 2: System output for
Once an upper bound of time delay delay is
estimated, the estimator algorithm (16)-(17) can be
used. In this respect, the studied system is excited
using a signal with multiple frequencies. The used
control is shown in Figure 3.
Fig. 3: The used control in the estimate of
linear parameters
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Then, it follows from (14)-(15) that the output
can be expressed as the following recursive
form:
where the true parameter vector is given as:
and the data vector is given as:
, (22)
Using the estimator algorithm (16)-(17), one
obtains the estimates of , , , and shown in
Figure 4, Figure 5, Figure 6 and Figure 7,
respectively.
Fig. 4: The estimate of parameter
Fig. 5: The estimate of parameter
Fig. 6: The estimate of parameter
Fig. 7: The estimate of parameter
The obtained results given by Figure 4, Figure
5, Figure 6 and Figure 7 show that the estimate
parameters converge to their true values.
4 Conclusion
In this paper, the identification problem of nonlinear
systems having a more general structure is
discussed. Most studied nonlinear systems has been
focused on Hammerstein and Wiener ones. It is
shown that this nonlinear structure is more general
than the Hammerstein and Wiener models. Then,
the latter can be viewed as special cases of this
nonlinear system. This approach is easy and
converges quickly. Firstly, an input of a set of step
signals is used. In the second stage, sine signal input
is used to estimate the linear block parameters.
Simulation examples show that the obtained
parameter estimates are very close to the true
nonlinear system parameters.
References:
[1] F. Giri, Y. Rochdi, A. Brouri, A. Radouane,
F.Z. Chaoui, Frequency identification of
nonparametric Wiener systems containing
backlash nonlinearities, Automatica, Vol. 49,
pp. 124-137, 2013. DOI:
10.1016/j.automatica.2012.08.043.
[2] A. Brouri, Wiener-Hammerstein nonlinear
system identification using spectral analysis,
International Journal of Robust and
Nonlinear Control, Vol. 32, No. 10, pp. 6184-
6204, 2022. DOI: 10.1002/rnc.6135.
[3] G. Bottegal, R. Castro-Garcia. J. A.K.
Suykens, A two-experiment approach to
Wiener system identification, Automatica,
Vol. 93, pp. 282-289, 2018. DOI:
10.1016/j.automatica.2018.03.069.
[4] Q. Jin, Z. Wang, W. Cai, Y. Zhang,
Parameter identification for Wiener-finite
impulse response system with output data of
missing completely at random mechanism and
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2024.19.25
Chaimae Abdelaali, Ali Bouklata,
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Volume 19, 2024
time delay, International Journal of Adaptive
Control and Signal Processing, Vol. 35 (5),
pp. 811-827, 2021. DOI: 10.1002/acs.3227.
[5] H. Oubouaddi, A. Brouri, A. Ouannou, F.E.
El Mansouri, A. Bouklata, L. Gadi, Switched
Reluctance Motor Speed and Torque Control
using ACO and PSO Algorithms, WSEAS
Transactions on Systems and Control, Vol.
18, pp. 362-369, 2023.
https://doi.org/10.37394/23203.2023.18.38.
[6] M. Schoukens and K. Tiels. Identification of
block-oriented nonlinear systems starting
from linear approximations: A survey,
Automatica, Vol. 85, pp. 272-292, 2017. DOI:
10.1016/j.automatica.2017.06.044.
[7] H. Xua, L. Xua, S. Shen, Online identification
methods for a class of Hammerstein nonlinear
systems using the adaptive particle filtering.
Chaos, Solitons and Fractals, Vol. 186,
115181, 2024. DOI:
10.1016/j.chaos.2024.115181.
[8] W. Yanjiao, L. Yitinga, C. Jiehaoa, T.
Shihuaa, D. Muqing, Online identification of
Hammerstein systems with B-spline networks,
International Journal of Adaptive Control and
Signal Processing, Vol. 38, pp. 2074-2092,
2024. DOI: 10.1002/acs.3792.
[9] Z. Tong, K. Gaëtan, Identification of
secondary resonances of nonlinear systems
using phase-locked loop testing, Journal of
Sound and Vibration, Vol. 59010, 118549,
2024. DOI: 10.1016/j.jsv.2024.118549.
[10] L. Xu-Long, W. Sha, D. Hu, C. Li-Qun, A
subspace parameter identification method for
nonlinear structures under oversampling
conditions, Journal of Sound and Vibration,
Vol. 58924, 118590, 2024. DOI:
10.1016/j.jsv.2024.118590.
[11] F. Giri, Y. Rochdi, F.Z. Chaoui, A. Brouri,
Identification of Hammerstein systems in
presence of hysteresis-backlash and
hysteresis-relay nonlinearities, Automatica,
Vol. 44 (3), 767-775, 2008. DOI:
10.1016/j.automatica.2007.07.005.
[12] A. Brouri and L. Kadi, Identification of
nonlinear systems, Proceedings of the
Conference on Control and its Applications -
SIAM CT'19, Jun 19-21, Chengdu, China, pp.
22-24,2019. DOI:
10.1137/1.9781611975758.4.
[13] A. Brouri, F.Z. Chaoui, F. Giri, Identification
of Hammerstein-Wiener Models with
Hysteresis Front Nonlinearities, International
Journal of Control, Vol. 95, No. 10, pp. 3353-
3367, 2022a. DOI:
10.1080/00207179.2021.1972160.
[14] H. Oubouaddi, F.E. El Mansouri, A. Brouri,
Parameter Estimation of Linear and Nonlinear
Systems Connected in Parallel, WSEAS
Transactions on Mathematics, Vol. 22, pp.
373-377, 2023.
https://doi.org/10.37394/23206.2023.22.44.
[15] F. Giri, A. Radouane, A. Brouri, and F.Z.
Chaoui, Combined frequency-prediction error
identification approach for Wiener systems
with backlash and backlash-inverse operators,
Automatica, Vol. 50, pp. 768-783, 2014. DOI:
10.1016/j.automatica.2013.12.030.
[16] R.S. Risuleo, F. Lindsten, H.
Hjalmarsson, Bayesian nonparametric
identification of Wiener systems, Automatica,
pp. 1-8, 2019. DOI:
10.1016/j.automatica.2019.06.032.
[17] S. Jing, T. Pan, Q. Zhu, Identification of
Wiener systems based on the variable
forgetting factor multierror stochastic gradient
and the key term separation, International
Journal of Adaptive Control and Signal
Processing, Vol. 35 (12), pp. 2537-2549,
2021. DOI: 10.1002/acs.3336.
[18] G. Mzyk, P. Wachel, Wiener system
identification by input injection method,
International Journal of Adaptive Control and
Signal Processing, Vol. 34 (8), pp. 1105-
1119, 2020. DOI: 10.1002/acs.3124.
[19] A. Ouannou, A. Brouri, L. Kadi, H.
Oubouaddi, Identification of switched
reluctance machine using fuzzy model,
International Journal of System Assurance
Engineering and Management, Vol. 13 (6),
pp. 2833-2846, 2022. DOI: 10.1007/s13198-
022-01749-4.
[20] A. Ouannou, A. Brouri, F. Giri, L. Kadi, H.
Oubouaddi, S.A. Hussien, N. Alwadai, M.I.
Mosaad, Parameter Identification of Switched
Reluctance Motor SRM Using Exponential
Swept-Sine Signal, Machines, Vol. 11(6),
625, 2023. DOI: 10.3390/machines11060625.
[21] H. Oubouaddi, F.E. El Mansouri, A. Bouklata,
R. Larhouti, A. Ouannou, A. Brouri,
Parameter Estimation of Electrical Vehicle
Motor, WSEAS Transactions on Systems and
Control, Vol. 18, pp. 430-436, 2023. DOI:
https://doi.org/10.37394/23203.2023.18.46.
[22] A.G. Wu, F.Z. Fu, R.Q. Dong, Convergence
Analysis of Weighted Stochastic Gradient
Identification Algorithms Based on Latest-
Estimation for ARX Models, Asian Journal of
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2024.19.25
Chaimae Abdelaali, Ali Bouklata,
Mohamed Benyassi, Adil Brouri
E-ISSN: 2224-2856
238
Volume 19, 2024
Control, Vol. 21, No. 1, pp. 509-519, 2019.
DOI: 10.1002/asjc.1747.
[23] S. Cheng, Y. Wei, D. Sheng, and Y. Wang,
Identification for Hammerstein nonlinear
systems based on universal spline fractional
order LMS algorithm, Communications in
Nonlinear Science and Numerical Simulation.
Vol. 79, pp. 1-13, 2019. DOI:
10.1016/j.cnsns.2019.104901.
[24] A. Brouri, F. Giri, F. Ikhouane, F.Z. Chaoui,
O. Amdouri, Identification of Hammerstein-
Wiener Systems with Backlash Input
Nonlinearity Bordered By Straight Lines, 19th
IFAC World Congress, Cape Town, South
Africa, Vol. 47, No:3, August 24-29, pp. 475-
480, 2014a.
https://doi.org/10.3182/20140824-6-ZA-
1003.00678.
[25] A. Brouri, F.Z. Chaoui, O. Amdouri, F. Giri,
Frequency Identification of Hammerstein-
Wiener Systems with Piecewise Affine Input
Nonlinearity, 19th IFAC World Congress,
Cape Town, Vol. 47, No.3, August 24-29, pp.
10030-10035, 2014b.
https://doi.org/10.3182/20140824-6-ZA-
1003.00303.
[26] H. Shumin, C. Liangyun, L. Dong,
L. Heng, Command Filtered Adaptive Fuzzy
Control of Fractional-Order Nonlinear
Systems With Unknown Dead Zones, IEEE
Transactions on Systems, Man, and
Cybernetics: Systems. Vol. 54, pp. 3705 –
37161, 2024. DOI:
10.1109/TSMC.2024.3370003.
[27] W. Huanqing, T. Miao, Fuzzy finite-time
adaptive control of switched nonlinear
systems with input nonlinearities,
International Journal of Adaptive Control and
Signal Processing. Vol. 38, pp. 2570-2587,
2024. DOI:10.1002/acs.3819.
[28] W. Yibing, Z. Manli, T. Shengnan, L.
Chengda, W. Min, S. Daiki, An Adaptive
Integral Sliding Mode Control-Based
Amplitude and Phase Compensation
Repetitive Control Method, IEEE
Transactions on Industrial Electronics. pp. 1-
10, 2024. DOI:10.1109/TIE.2024.3401183.
[29] L. Yan, Z. Xuan, Fuzzy Fixed-Time Control
for Nonaffine Nonlinear Systems with
Unknown Hysteresis Inputs, International
Journal of Fuzzy Systems. Vol. 25, pp. 3036 –
3048, 2023. DOI: 10.1007/s40815-023-
01556-4.
[30] B. Ni, Z. Yuyi, L. Xiaoyong, C. Wei, J.
Changan, Passive robust control for uncertain
Hamiltonian systems by using operator
theory, CAAI Transactions on Intelligence
Technology, Vol. 7, pp 594 – 605, 2022. DOI:
10.1049/cit2.12142.
[31] A. Brouri, L. Kadi, K. Lahdachi,
Identification of nonlinear system composed
of parallel coupling of Wiener and
Hammerstein, Asian Journal of Control, Vol.
24, No. 3, pp. 1152-1164, 2022b. DOI:
10.1002/asjc.2533.
[32] A. Brouri, F. Giri, Identification of series-
parallel systems composed of linear and
nonlinear blocks, International Journal Of
Adaptive Control And Signal Processing, Vol:
37, No: 8, pp. 2021-2040, 2023. DOI:
10.1002/acs3624.
[33] H. Oubouaddi, A. Brouri, A. Ouannou, F.E.
El Mansouri, A. Bouklata, L. Gadi, Parameter
Estimation of Linear and Nonlinear Systems
Connected in Parallel, WSEAS Transactions
on Systems and Control, Vol. 18, pp. 362-369,
2023. DOI:
https://doi.org/10.37394/23206.2023.22.44.
[34] D. Westwick, M. Ishteva, P. Dreesen, and J.
Schoukens. Tensor Factorization based
Estimates of Parallel Wiener Hammerstein
Models. IFAC Papers Online, Vol. 50 (1), pp.
9468-9473, 2017. DOI:
10.1016/j.ifacol.2017.08.1470.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Chaimae Abdelaali and Ali Bouklata carried out
the simulation, writing, software, and the
optimization.
- Adil Brouri and Mohamed Benyassi project
supervision
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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