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ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
IDENTIFICATION OF FRACTIONAL ORDER SYSTEMS WITH NONZERO INITIAL CONDITIONS
AND CORRUPTED BY NONZERO-MEAN NOISES
Shuen Wang, Yao Lu, Yinggan Tang
ABSTRACT
Most methods in the literature for fractional order systems identification ignore the initial condition and assume the measurement noise is zero mean, which may lead to incorrect estimation. In this paper, the problem of accurate parameter estimation of fractional order system with nonzero initial conditions and corrupted by nonzero-mean Gaussian noises is investigated. The initial conditions along with the mean of noise are treated as extra parameters of the system. The parameter, the differential orders and the extra parameters are simultaneously estimated via minimizing the error between the output of actual fractional order system and that of the identified system. In order to reduce the computation complexity of fractional derivatives of input and output signals, the operation matrix of block pulse function is adopted to convert the fractional order system to an algebraic one. Experimental results demonstrates the effectiveness of the proposed method.
Keywords: identification, fractional order system, operational matrix, nonzero initial conditions, nonzero mean noise.
1. Introduction
Fractional order calculus (FOC), dealing with derivatives and integrals of arbitrary orders, was first introduced by Leibnitz in 1695 [16]. Compared with integer calculus, FOC has several appealing characteristics. First, FOC is non- local in nature, which makes it suitable to describe history-dependent systems or processes [6]. Second, FOC is infinite dimensional, providing a concise way to describe distributed parameter systems [30]. Finally, FOC enlarges engineers's freedom to design controller's parameters, making it possible to obtain better control performance [24].
Recently, more and more researchers explored to use FOC to build more accurate models for systems in physics and engineering and enhance the performance of controllers. For example, the dynamical behaviour of tumor [3], the HIV/AIDS transmission [26], the dynamics of nuclear reactor [32], the flux-pressure relation in porous media [1] are well modelled using fractional order models (FOMs). As for controllers, the CRONE controllers [24], PIXD^ controller [24], fractional sliding mode controller [25] all adopted the concept of FOC.
Compared with integer calculus, FOC lacks distinct geometry and physical meaning, making it is difficult to build a FOM for system using mechanism analysis method. Currently, fractional system identification is the prevailing way to build FOM for a system. In the literature, many approaches had been proposed for fractional order system (FOS) identification and these methods can roughly be classified into two classes, i.e., the time domain identification methods and the frequency domain identification methods. In this paper, the time domain identification is concerned. The early work related to fractional order identification in time domain was owed to Cois [7], Lay [15] and Lin [19]. They proposed equation-error method and output-error method for parameter identification of fractional order systems. In [8], the fractional order state variable filter, specifically, the fractional Poisson filters, was
introduced into fractional system identification to reduce the effect of noise, while in [22], the simplified refined instrumental variable (SRIV) method was extended to identify fractional order system. Furthermore, Ref. [31] considered the differ- ential orders estimation problem in the framework of SRIV. The subspace identification of fractional order systems with or without time delay had been extensively studied in [29, 33, 18, 12] based on fractional state space model.
In FOS identification, the calculating of fractional derivative of input and output signals is a computationally expensive due to its history-dependent property of FOC. To simplify the computation, two strategies were adopted to convert the FOS to an algebraic one. One is the modulating function method [21, 20, 11, 9], which calculates the fractional derivatives of the modulating functions instead of calculating the fractional derivatives of input and output signals. Liu et al. firstly introduced the modulating function into fractional or- der system identification [20]. Then, Dai et al. adopted Gaussian function as modulating function to estimate the parameters of FOS using recursive least square (RLS) algorithm. Gao et al. [10, 11] utilized the polynomials as modulating function to estimate the parameters of FOS with known or unknown delay. In [4], the fractional differential orders and the parameters were jointly estimated by combining the modulating function method with the first-order Newton method. Another strategy is operational matrix method [28, 27, 17, 14]. Tang et al. proposed to use the operational matrix of block pulse function to convert the FOS to an algebraic equation and performed the simultaneous identification of parameters and differential orders [28]. Additionally, the FOS with time delay is considered in [27]. The Haar wavelet operational matrix was proposed to identify FOS in [17] and FOS with time delay in [14].
Though great progress had been made in FOS identification, there still exist some limitations. One limitation is that most methods mentioned above implicitly assumed the initial conditions of the FOS is
zero. In practice, it is not always true. For example, the system may have run for a long time at the moment that one collects the experimental data for identification. In this case, the collected data is not starting from zero initial conditions. If the initial condition is ignored, the identification result may be incorrect. The aberration phenomenon of FOS pointed out in [5] may give a reasonable explain for aforementioned statement. The aberration phenomenon of FOS says that if a FOS is excited by the same input under different initial conditions the outputs of the FOS are completely different.
Unfortunately, few researchers take the initial conditions into account in FOS identification. The second limitation is the way dealing with noise. Most works assume that the output of FOS is corrupted by Gaussian white noise with zero mean. To obtain unbiased estimation, bias compensated least square algorithm [9] or instrument variable method [22] were adopted. If the mean of noise is nonzero, the methods may be invalid.
There are several definitions for FOC. Among these definitions, the Riemann- Liouville (R-L) and
where r (a) denotes the Gamma function andis the convolution operator. The Caputo fractional differential is defined as
where a £ R, n - 1 < a < n, n £ N and f (k) (0) is the initial condition.
2.2. Generalized operational matrices of block pulse functions
To overcome the above limitations, in this paper, the identification of FOSs with nonzero initial conditions and nonzero-mean noise is investigated. The initial condition and the mean of noise are treated as extra parameters of FOS. The parameters of FOS along with the extra parameters are simultaneously estimated by minimizing the error between the output of the identified sys- tem and that of the actual system. To simplify the calculating of fractional derivative of input and output signal, operational matrix of block pulse function in [28] is adopted to convert the FOS to an algebraic equation. The rest of the paper is organized as follows. In Section 2, a brief mathematical background of FOC, and the block-pulse functions and their operational matrices for Riemann-Liouville integration are given. In section 3, for the fractional or- der systems with nonzero initial conditions and nonzero-mean noise, an identification framework based on operational matrix are presented. In Section 4, simulation examples are given. The conclusion is drawn in Section 5.
2. Prelilinaries
2.1. Definitions of FOC
(1)
Caputo definition are commonly used. The R-L fractional integral of order a > 0 is defined as
(2)
where For Caputo definition, one appealing characteristic is that the initial condition is specified using the integer order derivative. It is convenient to use Caputo definition for the problems related to initial condition. The R-L integral and Caputo derivatives has the following relation [23],
(3)
A set of block pulse functions in the semi-open interval [0, T) can be defined follows [28]
iam = T^f\t-sr-if(s)ds
= rb^-i * f(t),
n— i j,
I* (Daf{i)) = f(t) - Y1
k=О
(+\ _ / 777T < t < ' ' \0, '"' otherwise"
where i = 0,
m - 1.
Any absolute integrable function defined on interval [0, T) can be expanded onto a set of block pulse functions as
When the function series (5) is truncated at finite term, it can be written as
m ™ fT*{m)(t),
(6)
where fT = [f), fi, ..., f m-i] is the coefficient vector and the ith component f of the coefficient vector is defined as
with is the block pulse basis functions m approaches to infinite [2]. However, in
functions vector (T denotes the transpose). practical use, only finite terms is used and satisfied
Remark 1. The approximation error in (6) will accuracy can be obtained. approach to zero with the number of block pulse
(8)
Performing R-L integral for the block pulse function vector ¥m(t), one can get [13]
where
■">••• i
m.
= = - 2(1 l)a+1 -(/ ■ 2)Q+M = 2
Iaf(t) ~ IafT*{m)(t) = fTIaVim){t) = fTPa*(m](t).
(10)
with Performing R-L integral operation on both sides of Eq.(6) and using (8), one has
Eq.(10) states that the R-L integral of a function is converted to an algebraic equation, which greatly reduces the computation complexity.
3. Problem formulation.
Considering a FOS described by
where at (i=0, ..., n ) are the fractional differential parameters. The initial conditions of the system are orders satisfying 0 = ao< ai ... < a„. at, b £ R are the specified as x(k) (0) = xo(k), k = 0, 1, ... , an - 1, is the
ceiling function. The initial conditions are assumed to with nonzero mean E. Decomposing the noise term as be nonzeros. Let y(t) = x(t) + e(t) be the noise v(t) = e(t) - E, then v(t) becomes a zero mean noise. measurement of x(t)+e(t) is the measurement noise Performing R-L integration of order an on both
sides of Eq.(11), one has
holds. Since the low-pass filter property of integral (v(t) is very small and can be omitted. Therefore, and the noise usually is high frequency, the term Ian-ai Eq.(12) can be written as
where
Expanding the input signal u(t) and output response y(t) onto to a set of block pulse functions of size m
y(t) : YT&{m)(t),u(t) = U4(m){t),
(16)
FT = and UT = [uo,ui,...,um-i] are 1 x m
Where row vectors. Substituting (16) into (14) and utilizing (10), Eq.(14) can be converted as an algebraic equation as
where 1T = [1,..., 1]i*m Let
D = агХтРс
n —
г=0
And
(18)
then, Eq.(i7) can be written as
YT D = N.
According to (20), the coefficient vector of output Y equals
YT = N D-1.
Since
y(t) = YT^(m)(t), the measurement output of FOS can be expressed
as
(20)
(21)
(22)
y(t) = ND-1¥(m)(t). (23)
Fig. 1 The principle of parameter identification for FOS.
4. The identification process
It can be seen from (23), the measured output of the FOS system is parameterized by the parameters to be identified. In order to obtain a reason- able
y{t) = ND~^{m)(t),
estimation of these parameters, minimizing error principle can be used. Specifically, let a\ 'b, a', c'k and e'i be the estimations of a„ b, au ck and e„ respectively, then the output of the identified system is
(24)
where are the estimations of N and D. The identification of the parameters of the FOS can be accomplished by /y anc[ j ): error between the
output of the real system and that of the identified system, i. e.,
f " * Jtf; Л Л;
(ai 1° Iaii cik >
ê*) =
mm
(âi ,b,âi,Cik,êi)er
L
У к]
(25)
fc=1
where r is searching space admitted for parameters, differential orders, initial values and the mean of noise, k = 1, 2, ... , L is the sampling time point and L is the number of data used for parameter identification. The principle of the parameter identification of the FOS using the propopsed method is shown in Fig.(1).
Problem (25) is a nonlinear optimization problem, it can be solved by traditional optimization technology. For convenience, the subroutine fmincon in MATLAB optimization toolbox is adopted. For convenience, denote the identified parameters of system (11) as 8 = [a0, . . . , an-i, b, ... , cik, ... , e, ...], which contains the model parameters, differential orders and the extra parameters,
and denote the estimation of 8 as 8'. The main steps to accomplish the identification of the FOS are summarized as following
Step 1: Exciting the original FOS with an input signal u(t) and recording the measurement output y(t).
Step 2: Let k = 0, give an initial guess of the estimated parameter vector 8~o, and calculate the output of the estimated system according to Eq.(24).
Step 3: Performing an iterative process to get the next estimation 8'k using a certain optimization method.
Step 4: k=k+1, and go to Step 3 until the termination criterion are satisfied.
Remark 2. In this paper, the parameters to be identified directly are a, b, ..., a., ck, e. and the fractional differential orders a. From (18) and (19), they become the parameters of noise output y(t). Once a, ck and ei are estimated,
the initial condition xo(k) and mean of noise E can be obtained through (15) as
x(k)(0) = cik/dia., k = 0, 1, ... , an-i, and E = e/a. for any i = 0,. . ., n.
Remark 3. In this paper, the parameter identification problem is transformed to an parameter optimization problem. Through solving the optimization problem (25), one can simultaneously get a reasonable estimation of system's parameters, differential orders, initial conditions and the mean of noise. The purpose of regarding initial conditions and noise mean values as extra parameters is not only to identify them but also eliminate their influence on identification to get accurate results. The statement will verified through simulation examples.
5. Simulation Examples \
In this section, two FOSs are given to demonstrate the effectiveness of the proposed identification algorithm. In the following examples, the number of BPFs to be used in the simulations is set to m = T/h. To objectively evaluate the accuracy of identification, the relative estimation error index is adopted,
where 0' is the estimated parameter value and 0 is the real one. To show its superiority, the identification results considering the extra parameters are compared with those neglecting them. For the convenience of statement, the identification considering the extra parameters is denoted as method 1, and the identification neglecting them is denoted as method 2.
5.1. Example 1.
Consider the fractional order system
Da x(t) + ax(t) = bu(t),
(26)
4 6
Time(s) Fig. 2 Input signal of Example 1
with initial conditions xo(k), k = 0, ... , a-1. A multi-sine function
b
1
is chosen as the input, which is plotted in Fig.2. Case 1. The system's parameters are a = 0.8, a = 2,
= 3.
In this case, the initial condition of the FOS is xo = the mean of noise e(t) is E = 1,5 and v(t) is a Gaussian white noise with variance a2 = 1. The parameters to be estimated are a, b, a, xo and E. Firstly, the system is identified using method 1, i.e., the extra parameters including the initials and the mean of noise are together identified. The identification results are a' = 2,9898, 'b = 1.9899, a' = 0.8101, x'o = 1.001 and E' = 1.4898. Then, the system is identified using method 2, i.e., only the
system's parameters are identified. The identified results are a' = 2.9663, ~b = 1.9679, a' = 0.7100. Fig.3 shows the output of the actual system, the identified systems by method 1 and method 2.
To further compare the performance of method 1 and method 2, a Monte Carlo simulation is conducted. In Monte Carlo simulation, both methods are executed 100 runs independently with different initial conditions and noises. The identification results are statically presented in Table 1. Through comparison, it is obviously shown that the identification results of method 1 are more accurate than those of method 2. This fact demonstrates
that the initial conditions and the mean of noise have effect on identification results. If they are neglected in the identification process, inaccurate identification results will obtain and the identification accuracy will reduce. Therefore, the nonzero initial conditions and mean of noise can not be neglected in the identification process.
Case 2. The system's parameters are a = 1.1,
a = 2, b = 3.
In this case, the initial conditions of the FOS are chosen as xo(k) = 1, k = 0, 1, the mean of noise is E = 1.5 and v(t) is a zero-mean noise with variance o2=1. The parameters to be estimated in this case are a, b, a, x(0), x (0) and E. Firstly, the FOS is identified using method 1
Fig. 3 Output of example 1 in case 1
Table 1 Identification results with different initial conditions and noises of Example 1 in Case 1,
жо = 1.5, E = 2,(t2v = 0.5
то = 3, E = 3,al - 1
Method 1
Method 2
Method 1
Method 2
Parameters True Mean Std. Mean Std. Mean Std. Mean Std.
a 0.8 0.8092 0.0146 0.8183 0.0487 0.8085 0.0295 0.8176 0.0716
a 2 1.9955 0.0144 1.9958 0.0275 1.9707 0.0564 1.9648 0.0680
b 3 2.9931 0.0133 2.9895 0.0207 2.9868 0.0513 2.9869 0.0500
0.33 0.38 1 1.12
Fig. 4 Output of Example 1 in Case 2.
The identification results output are Aa = 2,9911, Ab = 1,9910, A a = 1,0872, *xo = 0,9987, ^ 0 = 0,9978 and ^E = 1,4903. Then, the FOS is identied using method 2 and the identification results are Aa = 2,9663, Ab = 1,9679, Aa = 0,7100. Fig.4 shows the output of the actual system, the identified system using method 1 and method
2. The Monte Carlo simulation is also conducted, in which 100 runs independent identification is performed with different initial conditions and mean of noise. Table 2 lists the statistical results of Monte Carlo simulation. From above experimental results, similar conclusion as Case 1 can be obtained.
Table 2 Identification results with different initial conditions and noises of Example 1 in
case 2.
ГО
ж0 = a^ = 1,5, E = 2, a,2 = 1
x0 = 3,
E = 3,al = 1
Method 1
Method 2
Method 1
Method 2
Parameters True Mean Std. Mean Std. Mean Std. Mean Std.
a 1.1 1.0998 0.0156 1.0223 0.0690 1.0945 0.0263 1.0302 0.0836
a 2 1.9971 0,0147 2.9637 0.1776 1.9945 0.0211 1.8481 0.2108
b 3 2.9912 0.0127 1.8955 0.1303 2.9917 0.0176 2.9691 0.2069
S(%) 0.25 3.56 0.3 4.51
Fig. 5 Output of Example 2
5.2 Example 2
Consider the fractional order system
0.8 D22 x(t) + 0.5 D09 x(t) + x(t) = 1.5 u(t) (28)
A unit step signal is chosen as the input to excite the system.
The initial conditions are xo(k)=1, k = 0, 1, 2, the mean of noise e(t) is £=1.5, and v(t) is a zero mean noise with variance Cv2 = 1. The parameters to be estimated in
this example are a2, ai, b, ai, xOk,
k = 0, 1, 2 and E.
Then, the system is identified using method 2. The identification using method 2 are ar = 0,45470, a2~ = 0,8558, b = 0,9871, af = 0,9712, a2~ = 2,2303. The output of the actual system, the identified systems using method 1 and method 2 are shown in Fig.5. Similarly, the Monte Carlo simulation, i.e., the identification with different initial conditions and mean of noise are conducted. The identification results are statistically listed in Ta0ble 3.
Table 3 Identification results with different initial conditions and noises of Example 2.
fD [2] 7T {Tj T2T O
«o = = »o = 1-5, So = = Xy0 - 3
E = 2,a2v= 0.5 E = 3, a2v = I
Method 1 Method 2 Method 1 Method 2
Parameters True Mean Std. Mean Std. Mean Std. Mean Std.
Q2 2.2 2.1925 0.0095 2.2406 0.0334 2.1922 0.01160 2.2438 0.0478
Ü1 0.9 0.9075 0.0113 0.9623 0.0201 0.9062 0.0224 0.9605 0.0638
«2 0.8 0.8114 0.0118 0.8710 0.0223 0.5164 0.0235 0.5632 0.1876
«1 0.5 0.5077 0.0110 0.5476 0.0227 0.8100 0.0202 0.8754 0.0608
b 1 1.0012 0.0119 1.0131 0.0248 1.0038 0.0273 1.0149 0.0501
4%) 0.60 3.9 0.76 4.29
From the experimental results, it is verified once again that if taking the initial conditions and mean of noise into account in the identification process, the identification accuracy can be significantly improved; otherwise, the identification accuracy is low, or even incorrect.
6. Conclusion
This paper investigates the identification problem of FOS with nonzero initial conditions and nonzero-mean noise. The initial conditions along with the mean of noise are treated as extra parameters of the system. The operation matrices of block pulse functions is adopted to convert the FOS to an algebraic system to simplify the calculation of fractional derivative of input and output signal involved in the identification process. The parameter, differential orders and extra parameters are simultaneously estimated via solving a nonlinear optimal problem. Theoretical analysis and experimental results demonstrates that taking the initial conditions and mean of noise into ac- count in the identification process can improve the identification accuracy. Otherwise, the identification accuracy is low, or even incorrect. In the future, our efforts will be devoted to generalize the proposed method to identify FOS with time delay or nonlinear FOS.
Acknowledgements This work is partially supported by the National Natural Science Foundation of China (Nos.61771418, 61273260).
Conflict of interest
The authors declare that they have no conflict of interest.
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СРАВНИТЕЛЬНЫЙ АНАЛИЗ КВАНТОВО-ХИМИЧЕСКИХ МЕТОДОВ ДЛЯ РАСЧЕТА ЭЛЕКТРОННЫХ ХАРАКТЕРИСТИК ПОЛИТИОФЕНОВ ИСПОЛЬЗУЕМЫХ _В СОЛНЕЧНЫХ ЭЛЕМЕНТАХ_
DOI: 10.31618/ESU.2413-9335.2020.6.78.1023 Нургалиев Ильнар Накипович
Доктор физико-математических наук, заведующий лаборатории Института химии и физики полимеров АН РУз, Ташкент, Узбекистан Бурханова Нилуфар Жалолитдиновна Стажёр-исследователь, Институт химии и физики полимеров АН РУз, Ташкент, Узбекистан Рашидова Сайёра Шарафовна доктор химических наук, академик, директор Института химии и физики полимеров АН РУз,
Ташкент, Узбекистан Ашуров Нигмат Рустамович доктор технических наук, заведующий лаборатории Института химии и физики полимеров АН РУз, Ташкент, Узбекистан
ABSTRACT
In this paper, a comparative analysis of semiempirical quantum chemical methods for calculating the energy of highest occupied molecular orbital (Ehomo) for various polythiophenes used in hybrid solar cells is carried out. The energies of the highest occupied Евзмо (eV) molecular orbitals calculated by quantum-chemical methods. Models built using HyperChem 8.0. software.
АННОТАЦИЯ
В данной работе проведено сравнительный анализ полуэмпирических квантово -химических методов для расчета энергии высшей занятой молекулярной орбитали (Ehomo) для различных политиофенов, используемых в гибридных солнечных элементах. Энергии высших занятых ЕВЗМО (eV) молекулярных орбиталей рассчитанные квантово-химическими методами. Модели, построены с помощью программы HyperChem 8.0.
Keywords: polymer, analysis, spectrum, macromolecule, quantum-chemical, calculation, energy, solar cells, polythiophene.
Ключевые слова: полимер, анализ, спектр, макромолекула, кванто-химический, расчет, энергия, солнечный элемент, политиофен.
Введение. На сегодняшний день значительные успехи достигнуты в области создания солнечных батарей на основе различных комбинаций сопряжённых полимеров и производных фуллеренов путём создания так называемого
объёмного р-п-гетероперехода. Масштабы приложений проводящих полимеров постоянно расширяются из-за их несложной обработки. Проводящие полимеры быстро находят новые приложения как хорошо обрабатываемые
