ISA Transactions xxx (xxxx) xxx
Contents lists available at ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
Research article
Volterra filter modelling of non-linear system using Artificial Electric
Field algorithm assisted Kalman filter and its experimental evaluation
∗
L. Janjanam a , , S.K. Saha a , R. Kar b , D. Mandal b
a
b
Department of Electronics and Communication Engineering, NIT Raipur, Raipur, Chhattisgarh, 492010, India
Department of Electronics and Communication Engineering, NIT Durgapur, Durgapur, West Bengal, 713209, India
article
info
Article history:
Received 12 February 2020
Received in revised form 29 August 2020
Accepted 24 September 2020
Available online xxxx
Keywords:
Non-linear system
Benchmark systems
Discrete-time system identification
Volterra model
Kalman filter
AEF algorithm
a b s t r a c t
The main objective of this paper is to improve the identification efficiency of non-linear systems
using the Kalman filter (KF), which is optimised with the Artificial Electric Field (AEF) algorithm. The
conventional KF suffers from the proper tuning of its parameters, which leads to a divergence problem.
This issue has been solved to a great extent by the meta-heuristic AEF algorithm assisted Kalman
filter (AEF-KF). This paper proposes three steps for the identification of the systems while solving the
problem as mentioned above. Firstly, it converts the identification model to a measurement problem.
Next, the AEF algorithm optimises the KF parameters by considering the fitness function with the KF
equations. The third step is to identify the model using conventional KF algorithm with the optimised
KF parameters. To evaluate the performance of the proposed method, parameter estimation error, mean
squared error (MSE), fitness (FIT) percentage, statistical information and percentage improvement are
considered as the performance metrics. To validate the performance of the proposed method, five
distinct non-linear models are identified with the Volterra model using KF and the AEF-KF techniques
under various noisy input conditions. Besides, the practical applicability of the proposed approach
is also tested on two non-linear benchmark systems using experimental data sets. The obtained
simulation results confirm the efficacy and robustness of the proposed identification method in terms
of the convergence speed, computational time and various performance metrics as compared to KF,
Kalman smoother (KS) which is optimised using different state-of-the-art evolutionary algorithms and
also other existing recently reported similar types of stochastic algorithms based approaches.
© 2020 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
System identification plays an imperative role in various fields
of engineering [1]. Identification of system can proceed through
linear, non-linear and combination of both. Finite impulse response (FIR) and infinite impulse response (IIR) models are frequently used for linear system identification. In [2–4], researchers
have focussed on the FIR and the IIR system identification using different evolutionary algorithms like breeder genetic algorithm [2], firefly algorithm (FA) [3] and modified-interior search
algorithm (M-ISA) [4]. It has been established that M-ISA algorithm results in accurate identification of IIR systems compared
to the other recently introduced state-of-the-art evolutionary
optimisation algorithms in terms of mean squared error (MSE),
percentage improvement in MSE and computational time. However, most of the real-time systems exhibit non-linear behaviour
owing to which the estimation using linear systems can often lead
to spurious results [5]. If the output of the system is a differential
∗ Corresponding author.
E-mail address: ljanjanam.phd2018.etc@nitrr.ac.in (L. Janjanam).
or exponential or logarithmic function of input, then the system
is called a non-linear system [6].
Non-linear system identification is the primary challenging
task because most of the engineering fields, such as biomedical
engineering, chemical industry and communication engineering,
are successfully using the non-linear systems. For the identification of non-linear systems, different non-linear models such
as bilinear, Volterra, and polynomial autoregressive (PAR) are
employed [6–8]. The advantage of the bilinear system is that it
provides a possibility for modelling time series with an occasional
sharp spike. Because of this attractive feature, these systems are
used in economic modelling and control problems. In [7], authors
have used filtering based maximum likelihood recursive least
squares (F-ML-RLS) algorithm for accurate identification of nonlinear controlled autoregressive ARMA systems and have shown
that F-ML-RLS outperforms ML-RLS and recursive generalised
extended least squares (RGELS) in terms of parameter estimation
error. Hafezi et al. in [8], have proposed RGELS and recursive
maximum likelihood (RML) algorithms for parameter identification of bilinear systems with high convergence speed, low
output error, low parameter estimation error and lower computational complexity. However, these ARMA and bilinear systems
https://doi.org/10.1016/j.isatra.2020.09.010
0019-0578/© 2020 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: L. Janjanam, S.K. Saha, R. Kar et al., Volterra filter modelling of non-linear system using Artificial Electric Field algorithm assisted Kalman filter
and its experimental evaluation. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.09.010.
L. Janjanam, S.K. Saha, R. Kar et al.
ISA Transactions xxx (xxxx) xxx
In addition, among all the KF parameters, the estimation of
Q and R is practically challenging due to the change in the
noise statistics when the filter starts working. It results in poor
performance. Thus, noise statistics have significant importance
on KF performance, which motivates the research of developing
adaptive Kalman filter (AKF). The AKF uses adaptive filter along
with the KF. The function of AKF is to tune the KF parameters
depending on the estimations of the identification model. Many
researchers have developed various AKFs for tuning of Q and
R [23–30].
To get the best estimates of Q and R for different state estimation problems, various AKF techniques, such as maximum
likelihood, Bayesian, covariance, and correlation methods, are
used in [23]. However, these techniques are offline and also valid
for stationary noise or noise with slowly varying statistics. In [24],
the authors have proposed the multiple interacting models (IMM)
algorithm to solve the problem as mentioned above. However,
this IMM method is not valid for rapidly varying noise statistics.
Mohamed et al. used Innovation-based adaptive estimation (IAE)
and multiple model adaptive estimation (MMAE) methods to get
the estimated values of Q and R [25]. However, the IAE method
does not guarantee to converge to the exact values of Q and R for
rapidly changing systems, and MMAE method is suitable only if
the prior knowledge of system dynamics is known [26]. Karasalo
et al. in [26], proposed Optimisation-based Adaptive Estimation
(OAE) method to estimate the value of Q for a small window
of data. Hence, the OAE method is suitable for all the rapidly
changing systems in various real-time online applications.
Moreover, the OAE method requires no information about
system dynamics. Based on the maximum, a posterior criterion,
Sage-Husa AKF (SHAKF) can recursively estimate the unknown
noise statistics Q and R. However, it is not guaranteed to get
the right convergence noise statistics with SHAKF [27]. Besides,
in [28], Variational Bayesian (VB) based Rauch–Tung–Striebel
smoother method can only estimate the values of unknown and
constant noise statistics offline. Variational Bayesian-based AKF
(VBAKF) method is used in the literature [29] to estimate the
values of inaccurate, slowly varying noise statistic R. However,
the performance of the VBAKF will degrade for an inaccurate Q ;
since it assumes accurate Q. In [30], a novel AKF is designed based
on variational Bayesian (VB) approach to estimate both noise
statistics Q and R simultaneously for linear Gaussian state–space
models, which results in a better estimation compared to other
state-of-the-art AKFs.
The initial error covariance matrix P0− is one of the essential
tuning parameters in KF operation because it can affect the final
error covariance P. However, much attention was not given earlier. Generally, a guess of P0− tends to become meagre value after
some data samples. Hence, the tuning of P0− is very crucial [31–
33], especially in specific state estimation problems like target
tracking and design of control systems. In [31], Xu et al. proposed
an efficient AKF, which adaptively estimates P0− and R based
on the expectation–maximisation approach. However, these AKF
techniques [23–33] suffer from reduced convergence speed and
are not guaranteed to get right estimates P0− , Q and R for various
state estimation problems.
In recent days, evolutionary optimisation algorithms play a
significant role in solving various engineering problems due to
their robustness, convergence speed and near-global optimal solution finding capability. The classical optimisation algorithms are
not suitable to solve the complex real-world problems where
many counteracting objective functions are to be solved simultaneously, which make the problem much involved and more
complicated for the classical techniques. To optimise many objectives simultaneously, various multi-objective evolutionary algorithms (MOEAs) are developed, which result in the best performance when the problem has two or three objectives. However,
are not suitable for memory-based applications. Hence, Wiener–
Hammerstein systems are getting attention recently, which is a
combination of linear and non-linear models [9].
Volterra is one of the most popularly used models among
all the non-linear models due to its conceptual simplicity and
modelling accuracy. Volterra series is a linear combination of the
non-linear function of the input signal and depends linearly on
the model coefficients, known as the kernel. The Volterra model
suffers from the computational complexity because the number
of Volterra kernels increases at an exponential rate when the
number of polynomials increases. This makes the system identification using the Volterra model a very challenging task when
there is a lack of knowledge about the structure of the system to
be identified. So, researchers adopt the truncated Volterra series
for identification [10–13]. In [11], authors have identified the
higher-order Volterra kernels using real coded genetic algorithm
(RCGA) with excellent precision and satisfactory computational
complexity. However, RCGA suffers from low convergence speed
(approximately 300 iterations) at different SNRs. In [12], authors
have successfully identified the higher-order Volterra systems
using genetic algorithm (GA) with random constructive heuristic (RCH), and constructive heuristic oriented by benefit (CHOB)
approaches which result in high convergence speed. It is also
reported that CHOB results in superior performance compared to
RCH in terms of mean squared distortion and cumulative density
function (CDF) with distortion at various levels of SNR, mutation
rate (MR) and crossover rates (CR). Improved variable forgetting
factor recursive logarithmic least mean p th (IVFF-RLLMP) algorithm accurately identifies the Volterra system under impulsive
noisy environments in [13].
The KF is a famous optimal state estimator that estimates
the instantaneous states of a dynamic system corrupted by noise
using certain assumptions of observation noise. It was first proposed by Kalman and has been widely used in different realtime applications [14]. KF and its improved versions, such as
extended Kalman filter (EKF), unscented Kalman filter (UKF),
particle filter (PF) lead to a better optimal solution for the estimation problems in various applications [15–17]. When the
system is highly non-linear, PF gives more accurate estimations
compared to EKF and UKF [16]. In [18–20], KF algorithm has
been applied to identify the Volterra model. In [19,20], Batselier et al. have identified the multiple inputs and multiple
outputs (MIMO) Volterra systems using tensor network Kalman
filter. However, in [19], authors have achieved two times of the
convergence speed of the estimated Volterra kernel coefficients
with no additional cost and also better performance is reported
compared to the work in [20] due to extension of output model
matrix. In [21,22], the authors have identified the Hammerstein,
Hammerstein–Wiener and Hammerstein–Wiener ARMAX models
by using different versions of KF.
Identification of the system using KF depends on both the
system dynamics (includes initial state X̂0− and state transition
matrix φ ) which describe the behaviour of the system with time
and the stochastic model parameters namely, initial error covariance matrix (P0− ), process noise covariance matrix (Q ), and the
measurement noise covariance matrix (R) [14]. The KF results in
accurate identification if these parameters are known precisely.
However, in most of the practical applications, these stochastic
models are unknown or partially known. The use of uncertain or
biased noise statistics in KF leads to performance degradation or
even divergence. The KF algorithm uses noise statistics (Q and
R) to influence the Kalman gain, Ki which is applied to the error
between the available process information and the most recent
obtained measurement. The filter gain projects this error to the
process information to get the best estimation. So, the choices of
Q and R are the primary factors which determine the value of Ki ,
which in turn determines the operation of the KF.
2
L. Janjanam, S.K. Saha, R. Kar et al.
ISA Transactions xxx (xxxx) xxx
the existing MEO’s have encountered difficulties when resolving problems with spaces and high-dimensional objectives. The
significant difficulties encountered by MOEAs for resolving multiobjective optimisation problems are summarised in [34]. Mnasri
et al. have solved one of the complex real-world problems in [34,
35]. In [34,35], authors have solved three-dimensional indoor
deployment problem while optimising a set of objectives, such
as coverage, localisation, routing, energy consumption etc., using
Improved Many-Objective Optimisation Algorithms and accentbased multi-objective particle swarm optimisation algorithms,
respectively.
Due to multiple features of evolutionary algorithms, many researchers have been motivated to incorporate the optimisation algorithms to tune the KF parameters for various applications [36–
43]. In [39], multi-objective problems are solved by using both
KF and differential evolutionary algorithm (DEA). In [40], the
EKF is optimised by the DEA and then used for sensorless speed
control of induction motors. Ahmed et al. have introduced self
tuned KF to estimate the discrete model of a synchronous DC–
DC buck converter [41]. Kim et al. in [42], have used particle
swarm optimisation (PSO) assisted KF in estimating the state of
Lithium-Ion batteries.
In [36–42], researchers tuned only three parameters of KF,
namely, P0− , Q and R using different evolutionary algorithms
for various state estimation problems. However, in [43], Yazid
et al. have tuned all the KS parameters such as X̂0− , P0− , φ , Q
and R using PSO assisted KS (PSO-KS), GA assisted KS (GA-KS),
and artificial bee colony (ABC) algorithm assisted KS (ABC-KS)
methods for identification of Volterra systems. The primary role
of these evolutionary algorithms is to obtain the optimal set of
KS parameters, and it has been established that ABC-KS results
in better estimations compared to GA-KS and PSO-KS techniques.
The reported values of the fitness percentage (FIT %) between
the actual and the estimated coefficients is 98.64% for system
input corrupted with 20 dB and 99.92% for cleaned input. From
the literature [36–43], authors have motivated to incorporate the
evolutionary algorithms in KF to avoid the tuning problem of
KF by developing an efficient objective function. In this paper,
authors have identified the second and third-order Volterra type
non-linear systems using the basic KF with the optimal set of
all the KF parameters (X̂0− , P0− , φ , Q and R) obtained by the AEF
algorithm. The proposed AEF-KF method results in FIT values of
99.9973% and 99.9978% for 20dB additive noise input and cleaned
input, respectively.
To get the best estimates using KF, in this paper, a new method
called AEF algorithm assisted KF (AEF-KF) is proposed to solve the
non-linear system identification problem.
The major contributions made in this paper are as follows:
(6) The noteworthy point of this paper is that till date, only
one article [43] has been reported where the Volterra type
non-linear system is identified using KS, which is optimised
with a meta-heuristic evolutionary algorithm. The simulation results for MSE (dB) are presented which comfortably
outperform all the reported results.
(7) The proposed approach has identified two industryoriented benchmark systems based on real data.
The orientation of the paper is as follows: Section 2 describes
the measurement model of Volterra system and analyses the
problem statement. Section 3 elaborates the steps of the proposed
method for the parameter estimation and also the flow chart of
AEF-KF. In Section 4, discussion on simulation results for four
numbers of second-order, one number of third-order Volterra
modelling examples and two benchmark systems are presented.
Finally, the conclusion of this paper is given in Section 5.
2. Problem statement
Volterra series [11,43] is one of the most popular identification
models for non-linear systems. The causal discrete-time form of
qth order Volterra series is given in (1), and its matrix form is
given in (2).
d[n] = h0 +
N −1
∑
h1 [k1 ]u[n − k1 ]
k1 =0
+
N −1 N −1
∑
∑
h2 [k1 , k2 ]u[n − k1 ]u[n − k2 ] + · · ·
k1 =0 k2 =0
··· +
N −1 N −1
∑
∑
k1 =0 k2 =0
···
N −1
∑
hq [k1 , k2 , . . . , kq ]u
kq =0
× [n − k1 ]u[n − k2 ], . . . , u[n − kq ]
(1)
where the kernels h0 , h1 [k1 ], h2 [k1 , k2 ],. . . , hq [k1 , k2 , . . . , kq ] for
indexes k1 , k2 , . . . , kq ∈(0 to N-1) are constants; N represents the
memory size; u[n] represents the input signal, and d[n] denotes
the output of the model.
d[n] = LH T
(2)
where L denotes the Volterra kernel vector as given in (3); H is
the Volterra input vector and is given in (4). Subscript T denotes
the transpose of a vector.
L = [h0 , h1 [0], . . . , h1 [N − 1], h2 [0, 0], h2 [0, 1], . . . , h2
× [0, N − 1], . . . , hq [N − 1, N − 1, . . . , N − 1]]
(1) In this paper, the proposed AEF-KF method is used to
minimise the error between the responses of the unknown
non-linear system and the Volterra system.
(2) The function of the AEF algorithm is to obtain the optimised
set of all the KF parameters so that the proper tuning
problem of KF can be resolved.
(3) The simulation is carried out by using KF and AEF-KF techniques, with various additive noise input levels.
(4) The authors have also justified the results in terms of
the parameter estimation, fitness percentage, coefficient
convergence plots, optimum MSE values along with the
corresponding convergence plots and the percentage improvement in MSE for the proposed technique with respect
to the KF algorithm.
(5) The proposed approach accurately identifies the second
and third-order Volterra systems with high convergence
speed (only very few iterations required to converge) and
without any excessive computational complexity.
(3)
H = [1, u[n], u[n − 1], . . . , u[n − (N − 1)], u [n], u[n]u
2
× [n − 1], . . . , u[n]u[n − (N − 1)], . . . ,
u2 [n − 1], . . . , u2 [n − (N − 1)], . . . , uq
× [n − 1], . . . , uq [n − (N − 1)]]
(4)
Fig. 1 shows the block diagram of the non-linear system identification using AEF-KF technique with the help of the adaptive
filter. In this paper, the discrete-time unknown systems are considered, which are identified by the Volterra model with the
help of the proposed AEF-KF technique. Eq. (5) shows that the
output of the discrete-time unknown system is corrupted with
measurement noise, v[n].
y[n] = d[n] + v[n]
(5)
The estimated output of the Volterra model is given in (6).
ŷ[n] = L̂Ĥ T
3
(6)
L. Janjanam, S.K. Saha, R. Kar et al.
ISA Transactions xxx (xxxx) xxx
NP number of positions are randomly initialised with the KF
parameters. SA is given in (8).
−
SA = X̂0 (n̂ × 1)
[
P0− (n̂ × n̂)
φ (n̂ × n̂)
Q (n̂ × n̂)
R(m̂ × m̂)
]
(8)
The dimension of each SA is (1 × nv ar ), where nv ar is the summation of matrix sizes of all the KF parameters and is given in
(9).
n var = (n̂ × 1) + (n̂ × n̂) + (n̂ × n̂) + (n̂ × n̂) + (m̂ × m̂)
where n̂ denotes the number of coefficients in the model and m̂
is the size of the measurement vector.
Initialise algorithm-specific control parameters, such as γ0 , ε
and α . Also, initialise other control parameters, namely ‘nvar’
dimensional randomly generated velocity (Vi ) and the maximum
number of iterations (max _itr).
Fig. 1. Block diagram of the parametric system identification using AEF-KF
technique.
The mean squared error (MSE) is considered as the objective
function of this minimisation problem and is presented in (7).
Step 2: For a finite duration, random input, evaluate the fitness
value of each SA by considering the fitness function as given in
(7). In this process, each SA updates the time as per (10)–(11)
followed by the measurement as per (12)–(14) in the KF loop runs
for 50 iterations to find out estimated output ŷ(kk).
SP
MSE =
⏐
1 ∑⏐
⏐y(kk) − ŷ(kk)⏐2
SP
(9)
(7)
kk=1
where SP denotes the number of input sampling points.
This paper employs KF and the proposed AEF-KF algorithms
individually to update the parameters of the Volterra model until
both the known and the unknown system responses are equal so
that the error signal, e[n] is minimised and hence, MSE, given in
(7), is optimised to its best possible minimum value. However,
the accuracy of the results of the KF algorithm depends on the
proper choices of its parameters X̂0− , P0− , φ , Q , and R.
To solve this problem, many pieces of literature [36–43] have
proved that if the evolutionary algorithm is used to optimise the
KF parameters, then the best estimation results can be obtained.
Hence, in this paper, a recently introduced AEF algorithm is used
to optimise the KF parameters. The results show that AEF-KF
method gives the best estimation compared to conventional KF
algorithm.
X̂i−
+1 = φ X̂i
Pi−
+1
(10)
= φ Pi φ + Q
T
(11)
Ki = Pi− H T (HPi− H T + R)−1
−
(12)
−
X̂i = X̂i + Ki (yi − H X̂i )
(13)
−
(14)
Pi = (I − Ki H)Pi
In the above equations, Ki is the Kalman gain matrix; X̂i is the
estimated value of the system state Xi ; error covariance matrix Pi
is defined by E [(Xi − X̂i )(Xi − X̂i )T ]; and I represents the identity
matrix of size (n̂ × n̂). Here, E denotes the expected value.
Step 3: Among all the fitness values, choose the best (minimum)
value best(t) and the worst (maximum) value w orst(t).
Step 4: Calculate the charge of the particle Ci (t) and Coulomb’s
constantγ (t)using (15) and (17), respectively.
3. Parameter estimation using AEF-KF technique
In this study, significant steps of the KF algorithm, such as
time update and measurement update for solving the estimation
problems, as described in [14], are followed. Also, a robust metaheuristic optimisation algorithm, named as AEF algorithm [44], is
used in this paper to optimise the KF parameters, such as X̂0− , P0− ,
φ , Q , and R due to its high convergence speed, less computational
complexity, less number of parameters used, highly efficient to
solve single-objective non-linear problems and capable of achieving the near-global optimal solution. The fundamental theoretical
concepts and steps for updating the position of each candidate
solution are applied on different standard benchmark functions
to establish the performance efficiency of the AEF algorithm compared to other state-of-the-art optimisation algorithms [44]. In
the proposed method, initially, the AEF algorithm optimises all KF
parameters. Finally, the basic KF technique identifies the Volterra
kernels using the optimal set of KF parameters obtained earlier.
To get the best optimal set of KF parameters using AEF algorithm,
the positions of each search agent is randomly initialised with all
the KF parameters, where each KF parameter is initialised with
random values, and also efficient objective function is formulated.
The following steps describe the procedure for parameter estimation using the AEF-KF technique, and the corresponding flowchart
is shown in Fig. 2.
ci (t)
Ci (t) = ∑N
i=1 ci (t)
Step 1: Based on the number of particles (NP) in the AEF algorithm, generate ‘NP’ number of search agents (SAs), where
where Ci (t) and Cj (t) are the charges of ith and jth particle,
respectively; plj (t) is the position of the best fitness value obtained
(15)
where
ci (t) = exp(
fitpi (t) − w orst(t)
best(t) − w orst(t)
t
γ (t) = γ0 ∗ exp(−α
)
max _itr
)
(16)
(17)
where γ (t) is Coulomb’s constant; α is a constant parameter; γ0
denotes initial value, and t is the current iteration.
Step 5: Evaluate the total electric force acting on the ith particle,
Fil (t) using (18).
Fil (t) =
NP
∑
rand()Fijl (t)
(18)
j=1,j̸ =i
where rand() is a uniform random number in the interval [0,1],
and Fijl (t) is the force acting on the charged particle i from j at any
time t in l-dimensional search space,
Fijl (t) = γ (t)
4
Ci (t) ∗ Cj (t)(plj (t) − Bli (t))
Dij (t) + ε
(19)
L. Janjanam, S.K. Saha, R. Kar et al.
ISA Transactions xxx (xxxx) xxx
Fig. 2. The flowchart of the proposed AEF-KF technique.
by jth particle; ε is small positive constant, and Dij (t) is the
Euclidian distance between two particles i and j and is given by
(20).
Dij (t) = Bi (t), Bj (t)2


examples taken from [10,11,13,43,45,46], and [12,47], respectively. Each example is verified by using the KF and the proposed
AEF-KF techniques individually. In addition, using Monte Carlo
simulation, each of the methods is repeated for 30 independent
runs with different random population, and the averaged results
are recorded for comparison purpose. The control parameters of
the KF and AEF-KF techniques are summarised in Table 1. The
selection of initial states of the basic KF parameters is usually
adjusted manually to obtain the acceptable results [33]. Based
on the literature [33,43,48,49] and dimensions of KF matrices,
the authors have performed the trial and error method and have
reported the best KF parameters’ initial state values in Table 1
with which a good accuracy in identification results is obtained.
In the proposed AEF-KF technique, the selection of the number of
particles or the number of population (NP) can influence the final
solution of the evolutionary algorithm (EA). From this research,
it is observed that NP should not be chosen either large or
small. If the selected NP is small, then the suboptimal solution
(inferior result) is obtained because the search landscape is not
fully explored, and if NP is large, then computational complexity
is increased. In this paper, authors have achieved the best optimal
set of KF parameters by choosing the NP value of 20.
The value of ‘nvar’, reported using (9), varies based on number
of estimated Volterra kernels in the model (n̂) and the initial
states of KF parameters (X̂0− , P0− , φ , Q and R) in AEF-KF technique
are selected as random values, which are chosen between a(=
−1) and b(= +1). To set the lower (a) and upper (b) bounds of
KF parameters, authors have followed the same strategy as mentioned in [50]. Three different sets of a and b values (a = −10,
b = 10), (a = −1, b = 1) and (a = −100, b = 100) are chosen
and exhaustive experiments have been performed for Example
1. The average MSE(dB) values obtained after 30 individual runs
for these three different sets of a and b values are −151.32 dB,
−184.85 dB and −132.36 dB, respectively. From the MSE (dB)
values, it is observed that the performance of the proposed AEFKF approach is poor for the limits (a = −10, b = 10) and (a =
−100, b = 100). Therefore, authors have chosen a and b values
(20)
where B(t) represents the position of any charged particle at time
‘t’.
Step 6: Determine the acceleration of ith particle at any time t,
ali (t) using (21).
ali (t) =
Ci (t)Eil (t)
(21)
Mi (t)
where Mi (t) is the unit mass of ith particle, and the electric field
of the ith particle is given in (22).
Eil (t) =
Fil (t)
(22)
Ci (t)
Step 7: Update the velocity and position of each particle using (23)
and (24), respectively.
Vi (t + 1) = rand()Vi (t) + ai (t)
(23)
Bli (t
(24)
+ 1) =
Bli (t)
+
Vil (t
+ 1)
Step 8: Repeat Steps (2)–(7) until the termination criteria are
satisfied.
Step 9: Return the best position, which contains the optimised KF
parameters.
Step 10: Run the basic KF Eqs. (10)–(14) using the optimised KF
parameters (obtained in Step 9) until max _itr is reached.
Step 11: Return the updated estimated state X̂i .
4. Simulation results and discussions
An extensive MATLAB simulation has been performed for identification of the Volterra type non-linear system of five different
5
L. Janjanam, S.K. Saha, R. Kar et al.
ISA Transactions xxx (xxxx) xxx
Table 1
Control parameters for filter design.
and the corresponding outputs ŷ[n] of the identification model
are given in (31)–(35), respectively.
Parameters
Basic -KF
AEF-KF
NP
nv ar
–
–
–
–
1000
eye(n̂)
eye(n̂)
0.0003*eye(n̂)
1
a+(b-a).*rand(n̂,1)
–
–
–
20
31[for n̂=3]
81[for n̂=5]
115[for n̂=6]
1000
a+(b-a).*rand(n̂, n̂)
a+(b-a).*rand(n̂, n̂)
a+(b-a).*rand(n̂, n̂)
a+(b-a).*rand(m̂, m̂)
a+(b-a).*rand(n̂,1)
500
0.001
30
max _itr
φ
−
P0
Q
R
X̂0−
γ0
ε
α
y[n] = u[n] − 0.8u[n − 1] + 1.9u[n − 3] + 0.95u2 [n]
+ 1.1u[n]u[n − 2] − 0.63u[n − 2]u[n − 3] + v[n]
(26)
y[n] = −0.12 − 0.22u[n] − 0.33u[n − 1] − 0.21u [n]
2
+ 0.59u[n]u[n − 1] − 0.31u2 [n − 1] + v[n]
(27)
y[n] = 0.8u[n − 1] − 0.5u[n − 2] + 0.7u2 [n − 1]
+ 0.1u2 [n − 2] − 0.4u[n − 1]u[n − 2] + v[n]
(28)
y[n] = 0.21u [n] − 1.62u [n − 1] + 1.41u [n − 2]
2
2
2
+ 3.72u[n]u[n − 1] + 1.86u[n]u[n − 2]...
+ 0.76u[n − 1]u[n − 2] + v[n]
(29)
y[n] = u[n − 2] + 0.08u [n − 2] − 0.04u [n − 1] + v[n]
2
where ‘n̂’ is number of Volterra kernels, m̂ = 1, b = 1 and a = -1
3
(30)
ŷ[n] = h1 [0]u[n] + h1 [1]u[n − 1] + h1 [3]u[n − 3]
+ h2 [0, 0]u2 [n] + h2 [0, 2]u[n]u[n − 2]...
as -1 and +1, respectively, for all the examples. The Coulomb’s
constant, γ0 and constant α are usually chosen as higher values
to increase the search accuracy of the algorithm [44]. Hence,
the values of γ0 and α are chosen as 500 and 30, respectively,
which lead to the best results. The constant ε is chosen as a small
positive value of 0.001 to avoid force defined in (19) becomes
infinite. Authors have used maximum 1000 iterations to obtain
the best SA (which contains an optimised set of KF parameters)
due to the large dimension of search agent (SA) and examples
considered in this paper are higher-order with large memory size
systems.
In this paper, various performance metrics are considered
from [43,51–54] to show the identification efficiency of the
proposed method. The percentage error between the actual and
the estimated Volterra kernels, the statistical (mean and standard deviation) analysis of estimated Volterra kernels, MSE (dB),
percentage improvement in MSE (dB) of AEF-KF technique with
respect to KF algorithm, and fitness percentage (FIT %) between
the actual and estimated outputs for Examples 1–5 under different additive noise levels are reported in Tables 2, 3, 4, 5
and 6, respectively. The minimum (δmin ) and maximum (δmax )
percentage of errors are calculated, respectively, using (25).
δmin (%) =
⏐
⏐
⏐
⏐
⏐θi − θ̂i ⏐
min
θi
×100 and δmax (%) =
⏐
⏐
⏐
⏐
⏐θi − θ̂i ⏐
θi
max
+ h2 [2, 3]u[n − 2]u[n − 3]
ŷ[n] = h0 + h1 [0]u[n] + h1 [1]u[n − 1]
(31)
+ h2 [0, 0]u2 [n] + h2 [0, 1]u[n]u[n − 1]
+ h2 [1, 1]u2 [n − 1]
ŷ[n] = h1 [1]u[n − 1] + h1 [2]u[n − 2]
(32)
+ h2 [1, 1]u2 [n − 1] + h2 [2, 2]u2 [n − 2]
+ h2 [1, 2]u[n − 1]u[n − 2]
(33)
ŷ[n] = h2 [0, 0]u [n] + h2 [1, 1]u [n − 1]
2
2
+ h2 [2, 2]u2 [n − 2] + h2 [0, 1]u[n]u[n − 1]...
+ h2 [0, 2]u[n]u[n − 2] + h2 [1, 2]u[n − 1]u[n − 2]
ŷ[n] = h1 [2]u[n − 2] + h2 [2, 2]u [n − 2]
+ h3 [1, 1, 1]u3 [n − 1]
(35)
Figs. 3–7 show the identified outputs of non-linear system
parameters under various challenging conditions using KF and
AEF-KF techniques. It is observed that AEF-KF technique results
in an outstanding performance compared to conventional KF
algorithm.
×100 (25)
4.1. Comparison of actual and estimated outputs of the Volterra type
non-linear systems
where θi and θ̂i are the actual and the estimated Volterra kernels,
respectively. The noisy outputs y[n] of the non-linear discretetime unknown system of each example are given in (26)–(30),
The actual and the estimated outputs of Examples 1–5 using KF and AEF-KF techniques with cleaned input are shown in
Table 2
Percentage error comparison between the actual and the estimated coefficients of each model.
Example
1
2
3
4
5
Reference
–
Proposed work
Yazid et al. [43]
Yazid et al. [43]
Yazid et al. [43]
Yazid et al. [43]
–
Proposed work
–
Proposed work
–
Proposed work
–
Proposed work
Reported algorithm
KF
AEF-KF
KS
PSO-KS
GA-KS
ABC-KS
KF
AEF-KF
KF
AEF-KF
KF
AEF-KF
KF
AEF-KF
(34)
2
Minimum and maximum percentage of error
SNR(10 dB)
SNR(20 dB)
SNR(30 dB)
Without SNR
0.31%, 0.8%
0%, 0.01%
10.9%, 11.6%
3.22%, 5.15%
8.06%, 9.52%
3.22%, 5.15%
0.36%,3.48%
0%, 0%
0.3%, 6%
0%,0%
0.49%,1.4%
0%,0%
0.37%, 7%
0%, 0%
0.06%, 1.58%
0%, 0%
10%, 10%
1.93%, 2.6%
4.06%, 4.28%
1.93%, 2.5%
0.21%,3.41%
0%, 0%
0.02%, 6.7%
0%, 0%
0.81%,1.97%
0%,0%
0.25%, 13.2%
0%, 0%
0.03%, 0.74%
0%, 0%
N/R
N/R
N/R
N/R
0.31%, 1.74%
0%, 0%
0.32%, 3.7%
0%, 0%
0.49%,1.4%
0%,0%
0.32%, 4.5%
0%, 0%
0.02%, 0.6%
0%, 0%
4.76%, 9.67%
0%, 0%
3.03%, 3.33%
0%, 0%
0.23%, 1.58%
0%, 0%
0.04%,1.5%
0%, 0%
0.09%,1.11%
0%,0%
0.14%, 2.87%
0%, 0%
‘N/R’: Not Reported.
6
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ISA Transactions xxx (xxxx) xxx
Table 3
Statistical analysis of identified Volterra kernels with respect to true kernels of the proposed AEF-KF method.
Example
1
2
3
4
5
Kernels
h1 [0]
h1 [1]
h1 [3]
h2 [0, 0]
h2 [0, 2]
h2 [2, 3]
True values
SNR(10 dB)
SNR(30 dB)
Without SNR
STD
Mean
STD
Mean
STD
Mean
STD
1.9
0.95
1.1
−0.63
0.9753
−0.7756
1.8779
0.9411
1.0049
−0.6243
0.1790
0.1826
0.2008
0.1377
0.1309
0.1514
0.9826
−0.7872
1.8812
0.9441
1.0927
−0.6247
0.1654
0.1754
0.1950
0.1021
0.1117
0.1343
0.9852
−0.7912
1.8877
0.9465
1.0943
−0.6281
0.1589
0.1649
0.1902
0.1010
0.1003
0.1219
0.9990
−0.7991
1.8991
0.9494
1.0985
−0.6288
0.1453
0.1549
0.1704
0.1003
0.089
0.1115
0.0900
0.0578
0.0981
0.0845
0.0653
0.0872
−0.1167
−0.2023
−0.3256
−0.2067
0.0719
0.0545
0.0981
0.0710
0.0622
0.0642
−0.1171
−0.2187
−0.3272
−0.2081
0.0645
0.0532
0.0961
0.0667
0.0592
0.0582
−0.1198
−0.2201
−0.3295
−0.2096
0.0619
0.0492
0.0877
0.0552
0.0571
0.0557
1
−0.8
h0
h1 [0]
h1 [ 1 ]
h2 [0,0]
h2 [0,1]
h2 [1,1]
−0.12
−0.22
−0.33
−0.21
−0.1151
−0.1979
−0.3245
−0.2052
0.59
−0.31
−0.3042
h1
h1
h2
h2
h2
[1]
[2]
[1,1]
[2,2]
[1,2]
0.8
−0.5
0.7
0.1
−0.4
h2
h2
h2
h2
h2
h2
[0,0]
[1,1]
[2,2]
[0,1]
[0,2]
[1,2]
0.21
−1.62
1.41
3.72
1.86
0.76
h1 [2]
h2 [2,2]
h3 [1,1,1]
SNR(20 dB)
Mean
0.5851
0.7921
−0.4943
0.6947
0.0952
−0.3972
0.1971
1.3781
3.577
1.841
0.734
0.0571
0.0289
0.1024
0.0456
0.1521
0.0399
0.9952
0.0771
−0.0381
0.0986
0.0767
0.0646
−1.6115
1
0.08
−0.04
0.1287
0.1162
0.1298
0.0877
0.0988
0.5877
−0.3052
0.7943
−0.4956
0.6957
0.0967
−0.3991
0.1985
0.1156
0.1158
0.1254
0.0823
0.0942
1.3921
3.656
1.849
0.1746
0.0542
0.0271
0.1010
0.0431
0.1512
0.0378
0.9961
0.0781
−0.0389
0.0954
0.0751
0.0631
−1.6152
0.5891
−0.3086
0.7962
−0.4977
0.6961
0.0977
−0.3995
0.2077
0.1127
0.1142
0.1237
0.0723
0.0877
1.405
3.699
1.855
0.755
0.0532
0.0256
0.0098
0.0423
0.1501
0.0367
0.9977
0.0789
−0.0395
0.0939
0.0739
0.0625
−1.6171
0.5998
−0.3097
0.7988
−0.4995
0.6969
0.0993
−0.400
0.2101
0.1111
0.1117
0.1208
0.0712
0.0851
1.410
3.701
1.859
0.759
0.0512
0.0251
0.0091
0.0411
0.1499
0.0359
0.9998
0.0797
−0.04
0.0913
0.0711
0.0592
−1.1998
Table 4
Best output MSE (dB) of each model under different noise levels.
Example
Reference
Reported algorithm
1
–
Proposed work
–
Proposed work
Mete et al. [10]
Mete et al. [10]
–
Proposed work
–
Proposed work
–
Proposed work
KF
AEF-KF
KF
AEF-KF
RLS
DEA
KF
AEF-KF
KF
AEF-KF
KF
AEF-KF
2
3
4
5
MSE(dB)
NS (dB)
SNR(10 dB)
SNR(20 dB)
SNR(30 dB)
Without SNR
−84.4604
−179.4580
−96.6173
−201.4300
−84.6375
−182.0210
−97.2126
−201.5319
−91.5637
−184.0366
−99.8635
−201.7510
N/R
N/R
−89.4935
−239.9222
−75.9704
−168.1067
−100.6199
−546.4338
N/R
N/R
−96.1199
−243.1164
−76.6275
−168.6380
−101.3307
−546.6815
N/R
N/R
−98.5144
−243.6188
−77.4938
−169.8627
−102.9220
−546.6851
−98.4951
−184.8525
−106.3970
−202.9010
−9.6878
−9.6878
−112.5335
−243.7809
−78.8396
−171.8830
−114.6574
−546.6930
14.03
5.4
9.78
1.47
N/A
N/A
23.04
3.86
2.86
3.78
14.04
0.011
‘N/R’: Not Reported, ‘N/A’: Not Applicable, NS: Noise sensitivity.
respectively. Even though the KF and AEF-KF techniques run for
1000 iterations for each example, however, it can be observed
from Fig. 3 that the estimated outputs using KF and AEF-KF
techniques very closely merge with the actual output within 100
iterations. From Fig. 4, it is evident that the estimated outputs
slightly shift away from the actual outputs achieved using KF
algorithm. However, for AEF-KF technique, the deviation between
the actual and the estimated outputs for both noisy and noise-free
inputs are almost negligible as compared to the previous case.
From Table 2, it is observed that the minimum and maximum
percentage error between the actual and the estimated coefficients using AEF-KF technique is zero for both the cases for all the
examples under consideration except for the Example-1 under
10dB SNR case. It means that the SNR effect is almost eliminated
by using the proposed method. On the other hand, using KF algorithm, percentage errors lie in the range of 0.02%–2.87%, 0.3%–7%,
Table 5
Percentage improvement in output MSE (dB) of AEF-KF with respect to KF.
Example MSE percentage improvement of AEF-KF with respect to KF
1
2
3
4
5
SNR(10 dB)
SNR(20 dB)
SNR(30 dB)
Without SNR
112.47%
108.48%
168.08%
121.27%
443.06%
115.05%
107.31%
152.93%
120.07%
439.50%
100.99%
102.02%
147.29%
119.19%
431.16%
87.67%
90.70%
116.62%
118.01%
376.80%
Fig. 3(a)-(e), respectively. An arbitrarily chosen zoomed portion
(X-axis scale: 67.6545–67.655) of the actual and the estimated
outputs for the cleaned input and different additive noise input
levels (10 dB, 20 dB, and 30 dB) are shown in Fig. 4(a)-(e),
7
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ISA Transactions xxx (xxxx) xxx
Fig. 3. Actual and estimated outputs using KF and AEF-KF techniques: (a) Example 1, (b) Example 2, (c) Example 3,(d) Example 4, (e) Example 5.
0.02%–13.2%, and 0.03%–4.5% for the cleaned input, 10dB, 20dB
and 30dB, respectively. From Table 3, it is observed that the mean
values of estimated parameters for Examples 1–5 are very close to
the actual values and the STD values are nearly zero as achieved
by using the proposed AEF-KF technique.
510, 600, and 570 iterations are required, respectively, to meet
the optimum values. Fig. 6(a)-(e) depict the variation of one arbitrarily chosen Volterra kernel with respect to the other Volterra
kernel using KF and AEF-KF techniques under cleaned input for
Examples 1–5, respectively. It is further re-established from a
different perspective that AEF-KF technique converges faster than
the KF technique as presented in Fig. 6.
4.2. Comparison of convergence profiles of Volterra kernels
4.3. Analysis of MSE (dB) convergence plot under noisy conditions
To show how closely the estimated Volterra kernels match
with the actual Volterra kernels, the convergence profiles of the
estimated Volterra kernels of Examples 1–5 using KF and AEFKF techniques for noise-free input are shown in Fig. 5(a)-(e),
respectively.
It is observed from Fig. 5 that the estimated Volterra kernels
of all the examples converge to their optimum values almost
within 25 iterations using AEF-KF technique. On the other hand,
using KF algorithm for Examples 1–5 approximately 600, 300,
Fig. 7(a)-(e) show the variation of the output MSE (dB) values
with iteration for Examples 1–5, respectively, using the KF and
the AEF-KF techniques for different levels of additive input noise.
From Fig. 7, it can be inferred that the MSE (dB) plot for each example marginally varies using KF algorithm under different levels
of additive noise and noise-free inputs and converges slowly to
the optimum MSE value. However, using the AEF-KF technique,
8
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ISA Transactions xxx (xxxx) xxx
Fig. 4. Zoomed in plots of the actual and the estimated outputs under different levels of noise using KF and AEF-KF techniques: (a) Example 1, (b) Example 2, (c)
Example 3, (d) Example 4, (e) Example 5.
these variations are too small, and MSE (dB) values rapidly converge to much lower optimum value compared to KF algorithm.
From Table 4, it is observed that the output MSE (dB) values for
AEF-KF technique with SNRs are much lesser susceptible to the
deterioration (−0.26 dB to −5 dB) than the KF algorithm-based
approach under similar conditions (−3dB to −23 dB). In addition,
from Table 4, it is also observed that the proposed AEF method
is much more insensitive to the additive noise compared to KF,
which has been quantified in terms of noise sensitivity (NS). The
noise sensitivity is defined as the maximum difference between
the best and worst MSE (dB) values obtained under different
noisy conditions. The proposed approach makes a minimal range
(0.01 dB to 5.4 dB) of noise sensitivity compared to the KF
approach (2.86 dB to 23.04 dB).
9
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ISA Transactions xxx (xxxx) xxx
Fig. 5. The convergence of Volterra Kernels using KF and AEF-KF techniques. (a) Example 1, (b) Example 2, (c) Example 3, (d) Example 4, (e) Example 5.
output (ŷ) merges with the actual output (y) and is calculated for
Examples 1–5 under various noise levels using the proposed AEFKF method. Based on the reported results in Table 6, the fitness
percentages of Examples 1–4 lie between 99.9821% and 99.9998%
and 100% is achieved for Example 5.
From Table 6, it is also observed that for Example-2, Yazid
et al. [43] have reported a FIT of 98.64% and 99.92% for additive noise 20 dB and cleaned input cases, respectively. However,
From Table 5, it is evident that with respect to KF algorithm, AEF-KF technique results in MSE(dB) improvement of 87%
to 121%, 116% to 168%, and 376% to 443% for the unknown
systems which are modelled by using second-order with six
kernels (Examples-1, 2, and 4), with five kernels (Example-3), and
third-order with three kernels (Example-5),
respectively.
(
)
Fitness percentage, FIT (%) =
1−
∥y−ŷ∥
∥y−mean(ŷ)∥
× 100 per-
formance metric is used to show how closely the estimated
10
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ISA Transactions xxx (xxxx) xxx
Fig. 6. Variation of two Volterra kernels using KF and AEF-KF techniques: (a) Example 1, (b) Example 2, (c) Example 3, (d) Example 4, (e) Example 5.
Table 6
Comparison of fitness percentages of each model under different noise levels for proposed AEF-KF method with the other reported
approaches.
Example
Reference
Reported algorithm
FIT (%)
SNR(10 dB)
SNR(20 dB)
SNR(30 dB)
Without SNR
1
2
Proposed work
Yazid et al. [43]
Yazid et al. [43]
Yazid et al. [43]
Yazid et al. [43]
Proposed work
Proposed work
Proposed work
Proposed work
AEF-KF
KS
PSO-KS
GA-KS
ABC-KS
AEF-KF
AEF-KF
AEF-KF
AEF-KF
99.9961%
N/R
N/R
N/R
N/R
99.9972%
99.9996%
99.9821%
100%
99.9963%
90.96%
98.12%
96.14%
98.64%
99.9973%
99.9997%
99.9826%
100%
99.9967%
N/R
N/R
N/R
N/R
99.9975%
99.9997%
99.9831%
100%
99.9969%
91.32%
99.87%
97.21%
99.92%
99.9978%
99.9998%
99.9834%
100%
3
4
5
‘N/R’: Not Reported.
with the proposed method, the authors have achieved 99.9972%,
99.9973%, 99.9975%, and 99.9978% for additive noise levels of 10
dB, 20 dB, 30 dB, and cleaned input, respectively.
4.4. The optimal set of KF parameters of the proposed method
In the proposed technique, the dominant role of the AEF
algorithm is to obtain the optimal set of KF parameters. The
11
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Fig. 7. Convergence profiles of output MSE(dB) values under different noise levels using KF and AEF-KF techniques: (a) Example 1, (b) Example 2, (c) Example 3,
(d) Example 4, (e) Example 5.
optimal solution (contains the optimal set of KF parameters)
obtained after 1000 iterations using AEF algorithm for Example-5
is reported in Table 7. Authors have not reported other optimal
solution sets for the rest of the examples due to its large ‘nvar’.
this research work, each example is identified by two techniques,
such as basic KF and the proposed AEF-KF method. The proposed
AEF-KF technique identifies the model in two phases: initially, it
obtains the optimal set of KF parameters using AEF algorithm.
Finally, it identifies the system (model) using conventional KF
with the optimal set of KF parameters obtained in phase 1. The
following step by step procedure describes the computational
complexity of the adopted methods.
4.5. The computational complexity of the proposed method
In this paper, the procedure described in [44,55] is followed
for the calculation of time complexity of the proposed method
for the identification of Volterra type non-linear Examples 1–5. In
Step 1: Consider the following test program [44,55] and evaluate
for max_itr number of iterations. The value of max_itr is 1000
12
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ISA Transactions xxx (xxxx) xxx
Table 7
Optimal KF parameter matrices obtained using AEF algorithm for Example-5.
X̂0−
φ
P0−
2
−1.82*10
−335.18
5.7
1.07*102
9.2
−34.42
14.72
−38.88
159.5
−14.61
75.18
−2.9
Q
1.04
9.2*10−2
−3.6*10−2
1.2*10−2
−2.9*10−2
−4.2*10−2
1.2
0.25
1.2*10−2
6.81
8.6
11.8
R
−1.9*103
−44.3
25.20
1.4*102
−14.94
−23.47
−7.0
N/A
N/A
‘N/A’: Not Applicable
Table 8
Computational complexity for Examples 1-5 for using KF and proposed AEF-KF
techniques.
Example
1
2
3
4
5
T 1(s)
MT 2(s)
saturated heat exchanger (LSHE) as presented in [56]. The structure and water heating process of a heat exchanger is described
both theoretically and mathematically in [56]. In LSHE, the input
variable liquid flow rate is chosen as a control variable, and the
rest of the input variables such as inlet liquid temperature and
steam temperature are considered as disturbances (kept constant
to their nominal values). The outlet liquid temperature of LSHE is
considered as an output variable [56].
In this paper, Volterra series given in (1) with fixed order
(q = 2) and different memory sizes (N = 5 and 8) is applied
individually to identify the LSHE based on its available input
and output (I/O) data set which is downloaded from the DAISY
database for system identification [58] for performance comparison of Kalman filter (KF), Particle filter (PF), genetic algorithmbased KF (GA-KF), particle swarm optimisation based KF (PSOKF), Artificial bee colony optimisation algorithm based KF (ABCKF) and the proposed AEF-KF methods. Based on the knowledge
shared in [43], the authors have performed rigorous experiments
to select the control parameters of these techniques. The control
parameters selected to obtain the best result are as follows:
parameters, namely mutation rate of 0.2 and a crossover rate of
0.5 are used in GA-KF method. For PSO-KF, authors have used
constriction factor, cognitive and social parameters values as 0.8,
1.5 and 1.2, respectively. For ABC-KF method, the number of
food sources is set at 10. The population size for GA-KF, PSOKF and ABC-KF is set at 20. In [58], total 4000 I/O samples are
measured with a sampling period of 1 s is shown in Fig. 8(a) and
their normalised I/O data are shown in Fig. 8(b). In this paper, to
estimate the coefficients of the model, 3000 normalised I/O data
samples are utilised, and the remaining 1000 normalised I/O data
are used for validation purpose [58,59].
Fig. 8(c)-(d) show the comparison of the actual and estimated
output responses of six algorithms mentioned above for the validation input when N = 5 (21 kernels) and N = 8 (45 kernels),
respectively. From these figures, it is noticed that the estimated
outputs using these six algorithms appear to merge intimately
with the actual output. However, the corresponding optimal fitness percentage (FIT % ) values reported in Table 9 clearly describe
that the deviation between the outputs is more when N = 5 for
all the six algorithms. It is also observed that the proposed AEF-KF
results in the best fitness values (73.45% when N = 5 and 92.28%
when N = 8) compared to the other five algorithms. Moreover,
when N = 8, the authors have achieved better fitness values
compared to N = 5 for all the employed algorithms. Hence, the
general perception is that the higher the order of the Volterra
series, the better is the identification task.
CC
KF
AEF-KF
KF
AEF-KF
KF
AEF-KF
1.214
1.271
1.174
1.283
1.244
117.12
116.45
114.15
119.24
112.78
1.324
1.454
1.321
1.467
1.378
122.21
121.57
119.23
125.98
116.56
46.61
77.54
62.28
77.96
56.77
17.33*102
17.43*102
17.30*102
22.95*102
12.87*102
and 2000 for basic KF and AEF-KF techniques, respectively. Also,
assume computational time (CT) taken to run the test program is
T 0.
for i=1:max_itr
x = 0.55 + double ((i);
x=x + x; x=x/2; x=x*x; x=sqrt(x); x=log(x); x=exp(x); x=x/(x+2);
end
Step 2: Evaluate the computational time T 1 for the problem
(Example).
Step 3: Determine complete CT for the algorithm with the same
example and consider it T 2.
Step 4: Execute Step 3 for five times and get five T 2 values. Also,
determine the mean of these five values MT 2.
Step 5: Evaluate computational complexity (CC) using CC
MT 2−T 1
.
T0
=
The achieved CT (T 0) for the basic KF and proposed AEF-KF
technique is 0.002362 s and 0.002936 s, respectively. The computational complexity of the Examples 1–5 using KF and proposed
AEF-KF techniques are reported in Table 8. From Table 8, it is
observed that the values of T 1 and MT 2 are too high for the
proposed AEF-KF technique compared to basic KF technique. This
is because the AEF algorithm takes more time to get the optimal
set of KF parameters. Therefore, CC for AEF-KF technique is higher
for each example compared to KF. However, AEF-KF results in an
accurate identification with faster convergence (takes less than
20 iterations). Moreover, the presented simulation results have
outperformed all other reported results.
Example 6: Benchmark system: Liquid saturated heat
exchanger
The practical applicability of the proposed method is tested
on a more significant non-linear benchmark plant, the liquid
Table 9
Various performance metrics for Example 6 for validation input using different techniques.
Algorithm
KF
PF
GA-KF
PSO-KF
ABC-KF
AEF-KF
FIT (%)
MSE(%) in dB
CT(s)
q = 2 and N = 5
q = 2 and N = 8
q = 2 and N = 5
q = 2 and N = 8
q = 2 and N = 5
q = 2 and N = 8
62.46
66.44
68.12
70.52
71.86
73.45
72.23
80.21
83.32
85.68
90.15
92.28
−20.6251
−27.7788
−33.5390
−37.0157
−40.1862
−48.2984
−22.2915
−31.6883
−34.4862
−43.1994
−46.8386
−54.8848
2.73
2.75
461.12
446.54
459.65
457.28
4.54
4.61
734.69
721.31
732.15
729.56
13
L. Janjanam, S.K. Saha, R. Kar et al.
ISA Transactions xxx (xxxx) xxx
Fig. 8. LSHE identification with q = 2, N = 5 and q = 2, N = 8 using KF, PF, GA-KF, PSO-KF, ABC-KF and the proposed AEF-KF techniques (a) input and output data
sets for LSHE, (b) normalised input and output data sets for LSHE, (c) actual and estimated outputs with q = 2, N = 5, (d) actual and estimated outputs with q = 2,
N = 8, (e) convergence of output MSE(dB) values with q = 2, N = 5, (f) convergence of output MSE(dB) values with q = 2, N = 8.
The convergence of MSE (dB) values with iteration using six
algorithms for N = 5 and N = 8 are shown in Fig. 8(e)(f), respectively. The corresponding optimum MSE (dB) values
are reported in Table 9, where it is shown that the proposed
algorithm results in the best MSE (dB) value for both the cases.
The CT is taken to identify the LSHE using six techniques for
N = 5 and N = 8 are reported in Table 9. From Table 9, it is
observed that the CT is almost double for N = 8 compared to
N = 5 because the evolutionary algorithms need to optimise 45
kernels. However, the identification accuracy is more for N = 8.
Example 7: Benchmark system: Fluid level control system consisting of cascaded tanks
A detailed discussion on the fluid level control system consisting of cascaded tanks has been mentioned in [57]. The same
practical model has been considered here for the identification
14
L. Janjanam, S.K. Saha, R. Kar et al.
ISA Transactions xxx (xxxx) xxx
Fig. 9. Cascaded tanks identification with q = 2, N = 8 using KF, PF, GA-KF, PSO-KF, ABC-KF, and proposed AEF-KF techniques (a) Cascaded tanks experimental
setup [57], (b) Input and output data set for cascaded tanks (upper), actual and estimated outputs with q = 2, N = 8 (lower).
purpose as well as for the comparison of results obtained by
different techniques, namely KF, PF, GA-KF, PSO-KF, ABC-KF and
the proposed AEF-KF. The cascaded tank consists of two tanks
arranged vertically with free outlets, where the outlet of the
higher tank goes into the lower tank. The water is pumped from
the reservoir to feed into the higher tank and is controlled by
the input signal voltage, (u(n)) from where water flows towards
the lower tank. The water level is measured as y(n) (i.e., output),
and finally, the water is fed back to the reservoir through a small
opening of the lower tank. When too high input signal voltage is
applied for a specified duration of time, an overflow (saturation)
occurs in the higher tank. Then a part of water goes into the lower
tank and rest directly flows into the reservoir. The mathematical
expressions describing the state–space modelling of the cascaded
tank under no saturation condition is given in [57].
In order to model the cascaded water tanks shown in Fig. 9(a),
many researchers have utilised the available input/output data
sets provided in [57]. The experimental setup is reported in [57]
where Schoukens et al. have generated two data sets of length
1024 samples each for model estimation and validation, respectively. The excitation signal is a random phase multisine [60]
with a length of 1024 samples, and the sampling frequency is
0.25 Hz. In [61], Birpoutsoukis et al., have identified the Volterra
series coefficients for modelling of the cascaded tanks using those
data sets. In this paper, the authors have also used the same
input/output data sets [57] for modelling of the cascaded system,
as shown in Fig. 9(b) (upper). Since, Volterra series with q = 2,
N = 8 results in better identification compared to q = 2, N = 5,
so for this example, simulation results are verified only for q = 2,
N = 8.
The comparison between the actual and the estimated output
responses of KF, PF, GA-KF, PSO-KF, ABC-KF and AEF-KF with
Volterra model (q = 2, N = 8) is shown in Fig. 9(b) (lower).
It is observed from Fig. 9(b) that the estimated output using KF
deviates from the actual output. However, the estimated outputs
for the rest of the algorithms appear very close to the actual
output.
The optimal fitness percentage (FIT % ) and MSE (dB) values
obtained by using all the algorithms are reported in Table 10.
From Table 10, it is evident that the proposed AEF-KF method
results in superior performance in terms of FIT (84.27%) and
Table 10
Performance comparison of different techniques based on FIT (%), MSE (dB) and
CT for Example 7.
Algorithm
KF
PF
GA-KF
PSO-KF
ABC-KF
Proposed AEF-KF
FIT (%)
MSE(dB)
CT(s)
q = 2 and N = 8
q = 2 and N = 8
q = 2 and N = 8
71.52
76.24
78.24
81.78
83.65
84.27
10.13
4.027
−2.615
−10.00
−13.47
−13.91
4.29
4.32
729.21
711.57
725.16
723.86
MSE (−13.91 dB) compared to KF, PF, GA-KF, PSO-KF and ABC-KF
methods, respectively. In addition, the CT values for measurement
of lower tank water level for 1024 samples input by using the six
algorithms are reported in Table 10. From Table 10, it is observed
that CT required for the proposed AEF-KF approach is marginally
lower than GA-KF, ABC-KF and is slightly more compared to KF,
PF and PSO-KF. However, from the achieved FIT (%) and MSE
(dB) reported in Table 10, the proposed AEF-KF results in better
identification compared to other algorithms.
5. Conclusions
In this paper, an evolutionary optimisation algorithm based
Kalman filter (KF) has been implemented to estimate the kernels
of Volterra system which exactly match with the actual system.
The significant role of the evolutionary algorithm is to obtain the
optimal set of KF parameters so that proper tuning problem of
KF is solved. To evaluate the performance and robustness of the
proposed artificial electric field (AEF) algorithm assisted KF (AEFKF), five distinct numerical Volterra models have been accurately
identified under different additive noise input levels (10 dB, 20 dB
and 30 dB) in terms of convergence speed, computational time,
parameter estimation error, MSE (dB), fitness percentage and percentage improvement. The results are found to outperform those
of KF, KS, GA-KS, PSO-KS and ABC-KS. The AEF algorithm with
the proper balance of exploration and exploitation phases with
the help of iteration dependent Coulomb’s constant serves the
significant role to optimise the KF parameters. To validate the fact
15
L. Janjanam, S.K. Saha, R. Kar et al.
ISA Transactions xxx (xxxx) xxx
that the proposed approach meets the real-time characteristics
of practical systems, extensive simulations have been carried out
for modelling of non-linear benchmark systems (Liquid saturated
heat exchanger and cascaded water tanks) based on real data sets.
The benchmark systems identified with second-order Volterra
model containing different memory sizes N = 5 and N = 8 using
KF, PF, GA-KF, PSO-KF, ABC-KF and AEF-KF methods. It has been
established from the results that the proposed AEF-KF method
is efficient enough for solving unknown system identification
problem as compared to PF and GA, PSO, ABC algorithms assisted
KF techniques in terms of MSE (dB), fitness percentage and computational time. The identification efficiency is more when fixed
order Volterra model memory size is increased. However, the
computational complexity is also increased. This paper applies
the proposed approach to the Volterra model only; however, it
can be easily extended to other model structures such as FIR, IIR
and bilinear polynomial filters.
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Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.
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