Parameter identification of fractional order linear system based on Haar wavelet operational matrix

ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research Article

Parameter identification of fractional order linear system based on Haar wavelet operational matrix Yuanlu Li a,b,n,1, Xiao Meng b,1, Bochao Zheng a,b,1, Yaqing Ding b,1 a

B-DAT, School of Information and Control, Nanjing University of Information Science & Technology, Nanjing 210044, China Jiangsu Collaborative Innovation Center on Atmospheric Environment and Equipment Technology, Nanjing University of Information Science & Technology, Nanjing 210044, China

b

art ic l e i nf o

a b s t r a c t

Article history: Received 23 February 2015 Received in revised form 9 July 2015 Accepted 20 August 2015 This paper was recommended for publication by Mohammad Haeri.

Fractional order systems can be more adequate for the description of dynamical systems than integer order models, however, how to obtain fractional order models are still actively exploring. In this paper, an identification method for fractional order linear system was proposed. This is a method based on input– output data in time domain. The input and output signals are represented by Haar wavelet, and then fractional order systems described by fractional order differential equations are transformed into fractional order integral equations. Taking use of the Haar wavelet operational matrix of the fractional order integration, the fractional order linear system can easily be converted into a system of algebraic equation. Finally, the parameters of the fractional order system are determined by minimizing the errors between the output of the real system and that of the identified system. Numerical simulations, involving integral and fractional order systems, confirm the efficiency of the above methodology. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Fractional order system System identification Haar wavelet Operational matrix

1. Introduction A fractional order system (FOS) is a system that is modeled by a fractional differential equation containing derivatives of noninteger order. Recently, considerable attention has been paid to the FOS. The reason for this is because a growing number of physical systems can be compactly described using FOS, such as the semiinfinite lossy (RC) transmission line [1,2], diffusion of heat into semi-infinite solid [2], viscoelastic materials [3,4], electrochemical processes [5], dynamics of porous media [6], continuous time random walk [7]. In addition, theoretical and experimental results have been shown that the fractional order controller had better dynamic responses and more robustness to model uncertainties in comparison with the classical controllers [8]. However, because the geometric and physical interpretation of fractional calculus is not as distinct as integer calculus, it is difficult to model real systems as FOS directly based on mechanism analysis. Therefore, system identification is a practical way to model a FOS. For integral order system (IOS), once the maximum order of the system to be identified is determined, the parameters of the model can be optimized directly. However, for a FOS, because identification requires the choice of the number of fractional order n

Corresponding author. E-mail addresses: [email protected] (Y. Li), [email protected] (X. Meng), [email protected] (B. Zheng), [email protected] (Y. Ding). 1 Tel.: þ86 2558731276.

operators, the fractional order of the operators, and finally the coefficients of the operators, the identification process of a FOS is more complex than that of an IOS [9]. Most classical identification methods cannot directly applied to identification of a FOS. Existing identification of a FOS can be mainly divided into two categories: time-domain system identification and frequencydomain system identification. In time domain [10–12], the parameters of a system to be identified are determined by minimizing the error between the output of the actual system and that of the identified system. For instance, Poinot and Trigeassou [13,14] have used fractional models to identify thermal systems, Sabatier et al. used the identified fractional order model to estimate the crankability of lead-acid batteries [15]. Compared to integer order system, the most obvious difference lies in identification of the fractional order of the operators. Therefore nonlinear optimization method has adopted to identify the order of a FOS [16,17]. Moreover, some intelligent algorithm were also applied for identification of FOS, such as genetic algorithms [18,19], differential evolution algorithm [20], particle swarm optimization [21,22]. In frequency domain, Li et al. [23] used the least squares method to investigate the frequency response identification technique. Nazarian et al. [24] developed an identification method of FOS according to input output frequency contents. Hartley et al. [9] discussed an identification method for FOS using continuous order-distributions. Besides above mentioned methods, recently, a refined instrumental variable method for continuous-time systems was extended to identify FOS [25], subspace method was proposed

http://dx.doi.org/10.1016/j.isatra.2015.08.011 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Li Y, et al. Parameter identification of fractional order linear system based on Haar wavelet operational matrix. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.08.011i

Y. Li et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

to identify a continuous-time FOS. In the proposed method, the parameter matrices were identified using the subspace-based technique and the commensurate orders were determined by using nonlinear programming [26], fractional Laguerre basis was proposed to identify FOS [27]. So, how to identify the FOS is still an open problem. Operational matrix has been widely used to deal with FOS. The main characteristic behind the approach is that it converts these problems to those of solving a system of algebraic equation thus greatly simplifying the problem. Typical examples are the blockpulse functions [12,28], the Jacobi operational matrix of fractional integration [29–31], Legendre polynomials [32–34], Chebyshev polynomials [35,36] and Haar wavelets [37,38]. Main aim of this paper is to use the Haar wavelet operational matrix to identify the FOS. The input and output signals are represented by Haar wavelets, and then the FOS described by fractional order differential equations are converted into fractional order integral equations. Taking use of the Haar wavelet operational matrix of the fractional order integration, the FOS can easily be converted into a system of algebraic equation. The parameters of the FOS are determined by minimizing the error between the output of the real system and that of the identified system. The organization of this paper is as follows: in Section 2, the fractional calculus, FOS and problem statement is introduced. In Section 3, the identification method based on Haar wavelet operational matrix is proposed. And verification of the method is provided in Section 4. Finally, conclusions are made in Section 5.

invariant (LTI) system may be described by the following fractional order differential equation: n

m

∑ ai Dαi y (t ) = ∑ bj D βj f (t ) i=0

j=0

(4)

Under the zero initial conditions, applying the Laplace transform to Eq. (4) the input–output representation of the FOS can be written in the form of a transfer function:

G (s ) =

Y (s ) b s β m + bm − 1s β m − 1 + ⋯ + b0 s β 0 = m α F (s ) an s n + an − 1 s α n − 1 + ⋯ + a 0 s α 0

(5)

where αi and βj are arbitrary real positive, f (t ) and y (t ) are the input and output of the system, respectively.

3. Identification method based on the Haar wavelet operational matrix 3.1 Haar wavelet If n = 2 j + k , where k and j are integers and j ≥ 1 and 0 ≤ k ≤ 2 j , then hn = h1 (2 j t − kb). h0 (t ) = 1 if 0 ≤ t < b and ⎧ 0 ≤ t < b /2 ⎪ 1, and 0 otherwise. 0 otherwise, h1 (t ) = ⎨ ⎪ ⎩ −1, b /2 ≤ t < b . An arbitrary signal x (t ) ∈ L2 [0, b] can be expanded by Haar wavelet, i.e., ∞

2. Fractional order system

x (t ) =

∑ ci hi (t ), i=0

2.1. The definition of fractional calculus

where the Haar coefficients ci , i = 0, 1, 2, ⋯, are determined by

There are several definitions for the general fractional differentiation and integration, such as the Grünwald–Letnikov definition, the Riemann–Liouville definition and Caputo definition [39]. Here the Riemann–Liouville fractional integral and Caputo fractional derivative were given as following, which will be used in this paper. The Riemann–Liouville fractional integration of order α > 0 is defined as

(I αf )(t ) =

1 Γ (α )

∫0

t

(t − τ )α − 1f (τ ) dτ =

1 α−1 t *f (t ) H (t ) Γ (α )

(1)

where Γ is the Gamma function, H (t ) is a Heaviside function. When the Riemann–Liouville derivative was used to model real-world phenomena, initial conditions with fractional order derivative are difficult to obtain. So we introduce a modified fractional differential operator D α proposed by Caputo,

(Dα f )(t ) =

1 Γ (n − α )

∫0

t

f (n) (τ ) (t − τ )α + 1 − n

dτ ,

(2)

where n − 1 < α < n and n is an integer. The relation between the Riemann–Liouville integral and Caputo derivative is given by the following expressions: n− 1

(I αDα f )(t ) = f (t ) −

∑ f (k) (0+) k=0

tk , k!

(6)

ci = 2 j

∫0

b

x (t ) hi (t ) dt

(7)

In practice, only the first N terms of Eq. (6) are considered, where N is a power of 2. So we have N−1

x (t ) ≈

∑ ci hi (t ) = CN T HN (t ) = x^ (t ) i=0

(8)

where the superscript T indicates transposition, the Haar coefficient vector CN and the Haar function vector HN (t ) are defined as

CN ≜ [ c0, c1, ... , cN − 1]T , T HN (t ) ≜ ⎡⎣ h0 (t ), h1 (t ), ... , hN − 1 (t ) ⎤⎦ .

(9) (10)

Taking suitable collocation points as following

ti =

(2i − 1) b , i = 1, 2, ... , N , 2N

(11)

The N-square Haar matrix ΨN × N can be defined by

⎡ ⎛ 1 ⎞ ⎛ 3 ⎞ ⎛ 2N − 1 ⎞ ⎤ b⎟ H ⎜ b⎟ ... HN ⎜ ΨN × N ≜ ⎢ HN ⎜ b⎟⎥. ⎝ 2N ⎠ N ⎝ 2N ⎠ ⎝ 2N ⎠⎦ ⎣

(12)

(3)

where n − 1 < α < n and n is an integer. 2.2. Fractional order systems A fractional order system (FOS) is a system that is modeled by a fractional differential equation containing derivatives of noninteger order. A single input single output (SISO) linear time

3.2. Block pulse operational matrix of the fractional order integral N-term Block pulse functions are defined as following

⎧ 1 ib/N ≤ t < (i + 1) b/N , i = 0, 1, 2, ⋯, (N − 1), φi (t ) = ⎨ ⎩ 0 otherwise ⎪

⎪

(13)

Please cite this article as: Li Y, et al. Parameter identification of fractional order linear system based on Haar wavelet operational matrix. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.08.011i

Y. Li et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

( I αn Dαn −1) y (t ) = I αn − an −1 ⎡⎣ Y T HN (t ) ⎤⎦ ≈ Y T PNαn×−Nan −1HN (t )

This can be written in matrix form as

ΦN (t ) = [φ0, φ2, ⋯, φN − 1]T .

(14)

According to Refs. [28,41,42], we can know the Block pulse operational matrix of the fractional order integration F α as following

(I αΦN )(t ) ≈ F αΦN (t )

(15)

where

⎡1 ⎢ ⎢0 ⎛ b ⎞α 1 α ⎢0 F =⎜ ⎟ ⎝ N ⎠ Γ (α + 2) ⎢ ⎢0 ⎢⎣ 0

3

⋮

( I αn Dα1) y (t ) = I αn − a1 ⎡⎣ Y T HN (t ) ⎤⎦ ≈ Y T PNαn×−Na1HN (t ) ( I αn Dα0 ) y (t ) = I αn − a0 ⎡⎣ Y T HN (t ) ⎤⎦ ≈ Y T PNαn×−Na0 HN (t )

(25)

Similarly, one can obtain:

( I αn D βm ) f (t ) = I αn − βm ⎡⎣ FT HN (t ) ⎤⎦ ≈ FT PNαn×−Nβm HN (t ) ⋮

ξ1 ξ 2 ⋯ ξN − 1⎤ ⎥ 1 ξ1 ⋯ ξN − 2 ⎥ 0 1 ⋯ ξN − 3 ⎥ , ⎥ 0 0 ⋱ ⋮ ⎥ 0 0 0 1 ⎥⎦

( I αn D β1) f (t ) = I αn − β1 ⎡⎣ FT HN (t ) ⎤⎦ ≈ FT PNαn×−Nβ1HN (t ) ( I αn D β0 ) f (t ) = I αn − β0 ⎡⎣ FT HN (t ) ⎤⎦ ≈ FT PNαn×−Nβ0 HN (t ) Substituting Eqs. (23)–(26) into Eq. (4), one can obtain:

(16) an Y T HN (t ) +

and

ξk = (k + 1)α + 1 − 2k α + 1 + (k − 1)α + 1.

(17)

3.3. Haar wavelet operational matrix of the fractional order integral Let

( I αHN ) (t ) ≈ PNα ×N HN (t )

(18)

where the N-square matrix PNα × N is called the Haar wavelet operational matrix of the fractional order integration. Because the Haar wavelets are piecewise constant, then we have

HN (t ) ≈ ΨN × N ΦN (t )

(19)

(20)

From (18) and (20), we have

PNα × N HN (t ) = PNα × N ΨN × N ΦN (t ) = ΨN × N F αΦN (t )

(21)

So, the Haar wavelet operational matrix of the fractional order integration PNα × N is given by

PNα × N = ΨN × N F αΨN−×1N ,

n− 1

m

i=0

j=0

∑ ai Y T PNαn×−Nai HN (t ) = ∑ bj FT PNαn×−Nβj HN (t ),

(27)

or

⎛ Y = ⎜⎜ an EN × N + ⎝

n− 1

T

∑ ai ( PNαn×−Nai ) i=0

⎞−1⎛ m ⎟ ⎜ ∑ bj P αn − βj ⎟ ⎜ N×N ⎠ ⎝ j=0

(

T⎞

) ⎟⎟⎠F,

(28)

where E is the identity matrix. Let a˜ i , b˜j , α˜ i and β˜j be the estimation of ai , bj , αi and βj , respectively. The output of the identified system can be written as T y˜ (t ) ≈ Y˜ HN (t )

(29)

where

⎛ Y˜ = ⎜⎜ a˜ n EN × N + ⎝

n− 1

T

∑ a˜ i ( PNα˜ n×−Na˜i ) i=0

⎞−1⎛ m ˜ ⎟ ⎜ ∑ b˜j P α˜ n − βj ⎟ ⎜ N×N ⎠ ⎝ j=0

(

T⎞

) ⎟⎟⎠F

The optimal estimation of parameters can be determined by minimizing the following criterion function:

From (18) and (19), we deduce

( I αHN ) (t ) ≈ ( I αΨN ×N ΦN ) (t ) = ΨN ×N ( I αΦN ) (t ) ≈ ΨN ×N F αΦN (t )

(26)

(22)

J = ‖Y − Y˜ ‖2 ,

(30)

There are many conventional optimization methods for optimization problem Eq. (30), such as nonlinear function optimization [17], chaotic ant swarm [40], Genetic algorithm [19]. In this paper, “fsolve” function in the MATLAB is used.

4. Verification of the proposed method To verify the effectiveness of the proposed method in identification of FOS, some examples are given in this section. Example 1, let us now consider a FOS [19] described by

3.4 Haar wavelet based identification of the FOS Consider the FOS described by Eq. (4), the goal of the system identification is to determine the parameters ai , bj and the fractional differential orders αi and βj according to the measured input and output data. For convenience of expression, we require , β0 < β1 < β2 < ⋯ < βm and αn > βm . Let output y (t ) and input f (t ) be expanded in Haar wavelets, i.e.:

y (t ) ≈ Y T HN (t ),

(23)

f (t ) ≈ FT HN (t )

(24)

Under the zero initial conditions, applying the fractional order integral of order αn to both sides of Eq. (4) one can obtain:

H1 (s ) =

1 , a 0 s α + a1

(31)

with a0 = 1.0, a1 = 1.0 and α = 0.7. This FOS is taken as a benchmark model in Ref. [19] and was identified by the genetic algorithm. In order to compare our result with that presented in Ref. [19], the pseudo-random binary sequence (PRBS) was taken as the input of the FOS, and the output of this FOS can be obtained by Eqs. (28) and (24). The input and output data were shown in Fig. 1. Using Matlab routine “fsolve” and the initial guess a0,0 = 0.5, a1,0 = 0.5, α0 = 0.5 and N ¼256, the identification result is presented in Table 1. Validation is carried out with the dataset in Fig. 1. The corresponding plot is given in Fig. 2. Example 2, let us consider now a FOS [21], whose transfer function is given by

Please cite this article as: Li Y, et al. Parameter identification of fractional order linear system based on Haar wavelet operational matrix. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.08.011i

Y. Li et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 1

2

0.5

1.5

Amplitude

Input signal

4

0 -0.5

1 0.5

-1 0

1

2

3

4

5

6

7

8

9

0

10

0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

Time (s)

1

10

0.5

5

x 10

Error

Output response

Time (s)

0

0

-5

-0.5 0

1

2

3

4

5

6

7

8

9

0

10

1

2

3

4

5 Time (s)

Time (s)

Fig. 3. Step responses of the identified system and error for Eq. (32).

Fig. 1. Input signal and output signal for system (31).

Table 1 Identification results for Eq. (31).

2

True value

Ref. [19]

Our result

a0 a1 α

1.0 1.0 0.7

0.985850 0.995380 0.696537

0.999990 1.000006 0.699997

Amplitude

1.5

Parameter

1 0.5 0 0

5

10

15

20

25

30

35

40

Time (s) 1

Fig. 4. Step response of the system (33).

Initial data Identified model 0.5

0

-0.5 0 5

x 10

1

2

3

4

5

6

7

8

9

10

-7

a00 = 0.5, a10 = 0.5, a20 = 0.5, α10 = 0.8 and α20 = 1.2 and N ¼256, the step responses of the true system and the identified system are shown in Fig. 3 and corresponding identified parameters are listed in Table 2. In Ref. [12], the authors used the operational matrix of the Block pulse to identify this FOS, the result is also shown in Table 2. Example 3, consider a mass–spring–damper system, whose system function is given as

0

H3 (s ) =

-5

1 . s 2 + 0.15s + 1

(33)

-10 -15 -20 0

1

2

3

4

5

6

7

8

9

10

Fig. 2. Identified system validation for system (31).

Table 2 Comparison of identification results for Eq. (32).

The step response of the system (33) is shown in Fig.4. One can see this is an oscillating system. In fact, oscillations may be undesirable in many practical systems. So a controller is needed to reduce the oscillations around the steady-state values. Here, we use a fractional order PID to reduce the oscillations of the system. Let the fractional order PID controller be given by

C (s ) = kp + ki s −λ + k d s μ, (λ , μ > 0).

Parameter

True value

Ref. [12]

Ref. [21]

Our result

a0 a1 a2 α1 α2

1.0 0.5 0.8 0.88 2.23

1.0001 0.4998 0.7996 0.8808 2.2301

0.999941 0.499879 0.800132 0.879624 2.229834

0.999942 0.499879 0.800132 0.879625 2.229835

The simple unity fractional feedback control system is shown in Fig. 5 in which H3 (s ) is the transfer function of the mass–spring– damper system, C (s ) is the transfer function of the fractional order PID, r is the input, ε is the error and y is the system's output. Then the closed-loop transfer function is

G (s ) =

H2 (s ) =

1 , a2 s β2 + a1s β1 + a 0

(34)

C (s ) H3 (s ) , 1 + C (s ) H3 (s )

(35)

i.e.

(32)

with coefficients a0 = 1.0, a1 = 0.5, a2 = 0.8, β1 = 0.88 and β2 = 2.23. This FOS is an example of Ref. [21], they used particle swarm optimization (PSO) to identify the FOS, the result is shown in Table 2. Here, Using Matlab routine “fsolve”, under the initial guess

G (s ) =

k d s λ + μ + kp s λ + ki sλ+2

+ 0.15s λ + 1 + k d s λ + μ + ( 1 + kp ) s λ + ki

.

(36)

Using the proposed method to identify these parameters ( λ , μ , t

k p , k i and kd ), under integral square error index ISE = ∫ e2 (t ) dt , 0

Please cite this article as: Li Y, et al. Parameter identification of fractional order linear system based on Haar wavelet operational matrix. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.08.011i

Y. Li et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

5

Table 3 Identified model for the elastic torsion system. Identified model

Residual

~ HIOS (s ) =

6.78008e þ 006

IOS

~ HFOS (s ) =

FOS

56.3237 s2 + 5.8927s + 490.9214 39.0923 s1.9000 + 3.3925s0.7653 + 34.6486

1.08839e þ006

2

1.5

1.5

1

1 Initial system controlled system

0.5 0 0

5

10

15

20

25

30

35

40

Speed (degrees/sec)

Amplitude

Fig. 5. Unity fractional feedback control system. x 10

0.5 0 -0.5 Identified model Measured data

-1

Time (s)

Fig. 6. Step responses of the initial system and the controlled system.

-1.5 0

1

2

3

4

5

Time (s)

Fig. 9. Elastic torsion system identified as an IOS.

1.5

x 10

4

Speed (degrees/sec)

1 0.5 0 -0.5 Identified model Measured data

-1 -1.5 0

1

2

3

4

5

Time (s)

Fig. 7. Elastic torsion system used in experiment. Fig. 10. Elastic torsion system identified as a FOS.

the optimal parameter can be obtained. Here λ = 1.0240, μ = 1.6149, k p = 14.0991, k i = 8.1034 and kd = 15.4385. The step response of the controlled system is shown in Fig. 6. Example 4, In the following example we will identify a real elastic torsion system (Fig. 7). Two masses are connected by an elastic spring which should be sufficient to drive the mass on the right. DC motor directly connected with the mass on the left. Taking a 75% duty cycle PWM signal as input of the DC motor, the speed of the mass on the right was measured. The input and output signal was shown in Fig. 8. Now we find a model for the elastic torsion system. From previous knowledge, we know that in case of IOS, this system can be modeled by a second order system. Thus one can take H˜ (s ) = b0 /s α2 + a1s α1 + a0 as a model to be identified. Identification yields the IOS model and the FOS model in Table 3. Validation is carried out by using the same dataset as for identification. The corresponding results are given in Figs. 9 and 10. Taking the square error norm as a measure of model error, one could say that the FOS model is more accurate than the IOS model in this case.

5. Conclusion

Fig. 8. Output speed of the elastic torsion system and input signal.

Taking the Haar wavelet operational matrix of the fractional order integration as a tool, a system identification method for SISO

Please cite this article as: Li Y, et al. Parameter identification of fractional order linear system based on Haar wavelet operational matrix. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.08.011i

Y. Li et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

linear time invariant FOS was proposed. Numerical simulations, involving integral and fractional order systems, confirm the efficiency of the above methodology. The proposed method can be used to find the optimal parameters for fractional order PID control. Its advantage is very simple and easy to implement, but its disadvantage is higher computational time for large N. The reason for this is that parameters are determined by nonlinear function optimization.

Acknowledgements The work was supported by the National Natural Science Foundation of China (Grant: 61271395 and 61403207).

References [1] Das S. Observation of fractional calculus in physical system description. New York: Springer; 2011. p. 101–56. λ μ [2] Bouafoura MK, Braiek NB. PI D controller design for integer and fractional plants using piecewise orthogonal functions. Commun Nonlinear Sci Numer Simul 2010;15:1267–78. [3] Koeller R. Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics. Acta Mech 1986;58:251–64. [4] Skaar SB, Michel A, Miller R. Stability of viscoelastic control systems. IEEE Trans Automatic Control 1988;33:348–57. [5] Ichise M, Nagayanagi Y, Kojima T. An analog simulation of non-integer order transfer functions for analysis of electrode processes. J Electroanal Chem Interfacial Electrochem 1971;33:253–65. [6] Dholkawala ZF, Sarma HK, Kam SI. Application of fractional flow theory to foams in porous media. J Pet Sci Eng 2007;57:152–65. [7] Hilfer R. On fractional diffusion and continuous time random walks. Phys A: Stat Mech Appl 2003;329:35–40. λ μ [8] Podlubny I. Fractional-order systems and PI D controllers. IEEE Trans Autom Control 1999;44:208–14. [9] Hartley TT, Lorenzo CF. Fractional-order system identification based on continuous order-distributions. Signal Process 2003;83:2287–300. [10] Mathieu B, Le Lay L, Oustaloup A. Identification of non integer order systems in the time domain. In: Proceedings of IMACS Multiconference: Computational Engineering in Systems Applications (CESA'96); 1996. p. 843–847. [11] Cois O, Oustaloup A, Battaglia E, Battaglia J. Non integer model from modal decomposition for time domain system identification. In: Proceedings of symposium on System Identification (SYSID); 2000. p. 989–994. [12] Tang Y, Liu H, Wang W, Lian Q, Guan X. Parameter identification of fractional order systems using block pulse functions. Signal Process 2015;107:272–81. [13] Gabano J-D, Poinot T. Fractional modelling and identification of thermal systems. Signal Process 2011;91:531–41. [14] Gabano J-D, Poinot T. Estimation of thermal parameters using fractional modelling. Signal Process 2011;91:938–48. [15] Sabatier J, Cugnet M, Laruelle S, Grugeon S, Sahut B, Oustaloup A, Tarascon J. A fractional order model for lead-acid battery crankability estimation. Commun Nonlinear Sci Numer Simul 2010;15:1308–17. [16] Poinot T, Trigeassou J-C. Identification of fractional systems using an outputerror technique. Nonlinear Dyn 2004;38:133–54. [17] Dorcak L, Gonzalez E, Terpak J. Identification of fractional-order dynamical systems based on nonlinear function optimization. In: Proceedings of the 9th international carpathian control conference. Sinaia, Romanie; 2008. p. 25–28.

[18] Othman MZ, AL-Sabawi EA. Fractional order system identification based on gentic algorithms. J Eng Sci Technol 2013;8:713–22. [19] Zhou S, Cao J, Chen Y. Genetic algorithm-based identification of fractionalorder systems. Entropy 2013;15:1624–42. [20] Tang Y, Zhang X, Hua C, Li L, Yang Y. Parameter identification of commensurate fractional-order chaotic system via differential evolution. Phys Lett A 2012;376:457–64. [21] Maiti D, Chakraborty M, Konar A. A novel approach for complete identification of dynamic fractional order systems using stochastic optimization algorithms and fractional calculus. In: Proceedings of international conference on Electrical and Computer Engineering (ICECE 2008). IEEE; 2008. p. 867–872. [22] Liu F, Burrage K. Novel techniques in parameter estimation for fractional dynamical models arising from biological systems. Comput Math Appl 2011;62:822–33. [23] Li Y, Yu S. Identification of non-integer order systems in frequency domain. Acta Autom Sin 2007;33:882. [24] Nazarian P, Haeri M, Tavazoei MS. Identifiability of fractional order systems using input output frequency contents. ISA Trans 2010;49:207–14. [25] Malti R, Victor S, Oustaloup A, Garnier H. An optimal instrumental variable method for continuous-time fractional model identification. In: Proceedings of the 17th IFAC World Congress; 2008. [26] Thomassin M, Malti RR. Subspace method for continuous-time fractional system identification. In: Proceedings of the 15th IFAC symposium on system identification (SYSID 2009); 2009. [27] Aoun M, Malti R, Levron F, Oustaloup A. Synthesis of fractional Laguerre basis for system approximation. Automatica 2007;43:1640–8. [28] Chi-Hsu W. On the generalization of block pulse operational matrices for fractional and operational calculus. J Frankl Inst 1983;315:91–102. [29] Bhrawy AH, Doha EH, Baleanu D, Ezz-Eldien SS. A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J Comput Phys 2015;293:142–56. [30] Bhrawy AH, Zaky MA. A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J Comput Phys 2015;281:876–95. [31] Bhrawy AH, Zaky MA. Numerical simulation for two-dimensional variableorder fractional nonlinear cable equation. Nonlinear Dyn 2015;80:101–16. [32] Jafari H, Yousefi S, Firoozjaee M, Momani S, Khalique CM. Application of Legendre wavelets for solving fractional differential equations. Comput Math Appl 2011;62:1038–45. [33] Saadatmandi A, Dehghan M. A new operational matrix for solving fractionalorder differential equations. Comput Math Appl 2010;59:1326–36. [34] Bhrawy AH, Doha EH, Ezz-Eldien SS, Abdelkawy MA. A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations. Calcolo 2015;1:1–17. [35] Li Y. Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun Nonlinear Sci Numer Simul 2010;15:2284–92. [36] Bhrawy AH, Alofi A. The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl Math Lett 2013;26:25–31. [37] Li Y, Zhao W. Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl Math Comput 2010;216:2276–85. [38] Ray S Saha. On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation. Appl Math Comput 2012;218:5239–48. [39] Baleanu D, Machado JT, Luo AC. Fractional dynamics and control. New York: Springer; 2012. [40] Li LX, Yang YX, Peng HP, Wang XD. Parameters identification of chaotic systems via chaotic ant swarm. Chaos Solitons Fractals 2006;28:1204–11. [41] Li Y, Sun N. Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Comput Math Appl 2011;62:1046–54. [42] Kilicman A, Al Zhour ZAA. Kronecker operational matrices for fractional calculus and some applications. Appl Math Comput 2007;187:250–65.

Please cite this article as: Li Y, et al. Parameter identification of fractional order linear system based on Haar wavelet operational matrix. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.08.011i