Methods for computing the time response of fractional-order systems

www.ietdl.org Published in IET Control Theory Received on 2 April 2014 Revised on 3 September 2014 Accepted on 23 September 2014 doi: 10.1049/iet-cta.2014.0354

Applications

ISSN 1751-8644

Methods for computing the time response of fractional-order systems Derek P. Atherton1 , NusretTan2 , AliYüce2 1 University

of Sussex, Richmond Building, Falmer, Brighton BN1 9QT, UK Faculty, Department of Electrical and Electronics Engineering, Inonu University, 44280, Malatya,Turkey E-mail: [email protected] 2 Engineering

Abstract: There is considerable interest in the study of fractional-order systems but obtaining accurate time domain responses is a difficult problem. This is because all methods reported on to date use approximations for the fractional derivative both for analytical-based computations and more relevantly in simulation studies. This means unlike in integer systems exact simulations are not available and thus for considering non-linear problems and comparisons with measured data no exact solution reference exists. In this study, the authors provide a major breakthrough for this situation by developing methods which allow the exact computation of the time responses of fractional-order systems.

1

Introduction

In recent years, there has been a major interest in systems described by fractional-order differential equations. These equations when transformed using the Laplace transformation lead to fractional-order transfer functions (FOTFs). There are major difficulties in finding the time domain responses to these equations because the standard procedure is to replace the fractional derivative term sα , α ∈ R by an approximation, which is the ratio of polynomials in integer powers of s. In the control field a major driving force in this respect was the need to simulate systems with fractional order, typically proportional-integral-derivative (PID) controllers. Thus, by definition, any solution must be approximate and what is a further cause for concern is that accurate results cannot be found by simulation, as similar approximations are used. To our knowledge the only known analytical solution for the step response of a FOTF is for the transfer function 1/s0.5 , which can be found in some extended tables of Laplace transforms. It was made use of in [1] to compare the step responses of various approximations but the author’s were less than forthcoming in not acknowledging its uniqueness. However, when s is replaced by jω the frequency representation is exact so that the plot of (jω)α on a Nyquist diagram is a radial line at an angle of απ/2. It is therefore clear that if frequency domain data can be used exact answers can be obtained for time domain responses. This paper looks at this problem for linear systems described by FOTFs and presents two exact methods which are compared with the approximate approaches which are currently used. One of the major difficulties with fractional order representation is the computation of time responses. The analytical solution of the output is not possible and there is not a general method for estimating it. Many studies have IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

been done in order to simulate fractional control systems over the last decade. Some of these studies are based on integer approximation models and others based on numerical approximation of the non-integer order operator [2–4]. The methods developing integer order approximations are attractive since they convert the problems related with the FOTFs into classical transfer functions. Therefore a number of methods to evaluate rational approximations have been developed. The most popular of these are the continued fractional expansion (CFE) method, Oustaloup’s method, Carlson’s method, Matsuda’s method, Chareff’s method, least-square methods and so on [5–10]. Approximation methods generally suffer from inaccuracy in the high and low frequency regions. A modified approximation method based on Oustaloup’s method has been given in [11] to improve the approximations in the low and high frequency regions. Another problem with approximation methods is that when a low order approximate model, such as first order is used, then the results will not be accurate, reliable or satisfactory and when a high order model is used, its complexity may preclude understanding, although computer solutions are still straightforward. Further the magnitude and location of errors, although possibly small, in time domain responses is not known. Some work on model reduction of high order transfer functions obtained from approximations has been done in [12] to overcome the problems of high order. However, since the model reduction approach is also an approximation further uncertainty is introduced into the results. There are also some methods based on Mittag– Leffler functions and Gamma functions for computation of the impulse and step responses of commensurate-order systems [13, 14]. However, the solution methods using Mittag–Leffler functions and Gamma functions are time consuming and tedious, and the numerical solutions used for time domain analysis suffer from a validation problem 817 © The Institution of Engineering and Technology 2015

www.ietdl.org because of the lack of analytical results. For example, the accuracy of the numerical computation given in [2] for time response evaluation depends on the step size used. Different step sizes have been used and results obtained and compared in order to validate their correctness. Some of the methods mentioned above for fractional-order systems have applications outside the realm of transient response investigations, which is the objective for the computational techniques to be presented. The availability of the exact methods allows comparison with some of the common approximations used. In this paper, two exact methods are developed for the time domain computation of FOTFs responses. Both methods use frequency domain data of the FOTF. The first method uses the Fourier series of a low frequency square wave as the input of an FOTF to compute its step and impulse responses. We shall refer to this method as the Fourier series method (FSM). The second method is based on the inverse Fourier transform of the frequency domain data and referred to as the inverse Fourier transform method (IFTM). The paper is organised as follows: in Section 2, an introduction to FOTF is given. An example is given to show that the exact frequency response of a FOTF such as Bode, Nyquist and Nichols plots can be computed similar to the integer-order transfer functions. Oustaloup’s integer-order approximation method is briefly explained. A table including integer approximations of 1/s0.5 using different methods is provided. Bode plots and step responses of 1/s0.5 using the different methods are given. Errors in approximations are demonstrated. The exact method called FSM (Fourier series method) for the time response computation of a FOTF is introduced in Section 3. Various examples are provided to show the importance of the method presented. In Section 4, the IFTM is introduced and examples are again given. Analytical formulas for the step and impulse responses of 1/sα when 0 ≤ α ≤ 1 are derived. In Section 5, the steady-state error and the influence of fractional-order integrators on convergence to zero error are studied. Concluding remarks are given in Section 6.

2

Fractional-order transfer functions

A physical system represented by a differential equation where the orders of derivatives can take any real number not necessarily an integer number can be called a fractionalorder system. The idea of non-integer order (or fractional order) differentiation/integration emerged in 1695 [15]. At the beginning, mathematicians studied it only as a theoretical subject because of its complexity and therefore other science disciplines could not use it effectively owing to the absence of exact solution methods of non-integer-order differential equations [16]. However, in recent years, facilitated by today’s computational facilities considerable attention has been given to fractional-order systems, including fractionalorder control systems. The significance of fractional order representation is that fractional-order differential equations are claimed to be more adequate to describe some real world systems than those of integer-order models [17, 18]. Thus, fractional calculus has been an important tool to be used in engineering, chemistry, physical, mechanical and other sciences [10, 19]. G(jω) =

(0.309ω3.2

−

2.8533ω1.8

+

818 © The Institution of Engineering and Technology 2015

1.362ω0.7

Different definitions of fractional-order operators such as Grünwald-Letnikov (GL), Riemann-Liouville and Caputo [13] have been proposed over the years. The Caputo definition is the most frequently used one in engineering applications [12]. This definition can be stated as L{Dα y(t)} = sα L{y(t)} −

[α]−1 

sα−i−1

i=0

di y (0) dt i

(1)

where Dα y(t) = d α y(t)/dt α indicates the Caputo derivative of y(t), α ∈ R+ is the rational number, [α] is the integer part of α and L denotes the Laplace transform. A fractional-order control system with input r(t) and output y(t) can be described by a fractional differential equation of the form [1] an Dαn y(t) + an−1 Dαn−1 y(t) + · · · + a0 Dα0 y(t) = bm Dβm r(t) + bm−1 Dβm−1 r(t) + · · · + b0 Dβ0 r(t)

(2)

or by a FOTF of the form G(s) =

Y (s) bm sβm + bm−1 sβm−1 + · · · + b0 sβ0 = R(s) an sαn + an−1 sαn−1 + · · · + a0 sα0

(3)

where ai , bj (i = 0, 1, 2, . . . , n and j = 0, 1, 2, . . . , m) are real parameters and αi , βj are real positive numbers with α0 < α1 < · · · < αn and β0 < β1 < · · · < βm . Thus, a transfer function including fractional powered s terms can be called a fractional-order transfer function, FOTF. For example, with the FOTF G(s) =

s3.2

+

3s1.8

2 + 3s0.7 + 1

(4)

replacing s by jω and using (jω)α = ωα [cos(απ/2) + j sin(απ/2)], one obtains (see (5)) Bode, Nyquist and Nichols diagrams of this equation can then be obtained. As mentioned in the introduction, there are many approximation methods and Matlab codes exist [2, 19, 20] for computation of them. For example, one of the most popular rational approximation methods is Oustaloup’s method [5] which gives a rational approximation of the fractional differentiator operator sα , 0 ≤ α ≤ 1 by a rational transfer function in other words by an integer-order transfer √ function over a frequency band [ωm , ωM ] centred at ωu = ωm ωM . The resultant transfer function is the cascade of 2N + 1 first-order transfer functions which can be expressed as sα ∼ = Ka

N  1 + (s/ωk ) 1 + (s/ωk ) k=−N

(6)

where ωk and ωk can be computed recursively and the gain Ka is chosen to make the gain of the approximated transfer function equal to the gain of sα at ωu [12]. There are many studies using Oustaloup’s recursive approximation algorithm in the literature and therefore we have generally used Oustaloup’s approximation method in this paper. The high and low transitional frequencies ωm and ωM are chosen as ωm = 10−2 rad/s and ωM = 102 rad/s, respectively. 2 + 1) + j(−0.9511ω3.2 + 0.927ω1.8 + 2.673ω0.7 )

(5)

IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

www.ietdl.org with frequency ωs = 2π/T can be written as

Obviously making N larger gives a more accurate integer approximation but doing this increases the order of the integer-order transfer function significantly. Using (6), for N = 2 and applying it to s0.2 , s0.7 and s0.8 in (4), the order of the integer approximating transfer function of (4) will be 18 and for N = 10 it will be 66. Thus, high-order integer approximations may give acceptable results for some cases but they have the disadvantage of producing high dimensional models. Rational approximations for G(s) = 1/s0.5 obtained using different methods [1] are shown in Table 1. The exact Bode plots of G(s) = 1/s0.5 and its approximations are shown in Fig. 1a where one can see that the approximations are good for a range of frequency but fail to give an exact matching. It is known that the exact step response of G(s) = 1/s0.5 is equal to √ ys (t) = 2(t/π )0.5 and its impulse response equals yi (t) = 1/ tπ [10]. The exact step response and the approximations by different methods are shown in Fig. 1b, where it can be seen that the approximations are worse as time increases. Fig. 1c shows the errors in the approximations.

x(t) =

∞ 4  1 sin(kωs t) π k k=1(2)

(7)

where T is the period of the square wave. If x(t) passes through the transfer function G(s) then the output which reveals the unit step response can be written as ys (t) ∼ =

∞ 4  1 Re[G(jkωs )] sin(kωs t) k π k=1(2)

(8)

The proof of this can be done using convolution. Let g(t) = L−1 (G(s)) and using the convolution integral, the output can be written as ∞ g(τ ) g(τ )x(t − τ ) dτ = 0 0   ∞ 4  1 sin(kωs (t − τ )) dτ × π k=1(2) k  ∞ 4 g(τ ) sin(ωs (t − τ )) dτ = π 0  1 ∞ g(τ ) sin(3ωs (t − τ )) dτ + 3 0  ∞ 1 g(τ ) sin(5ωs (t − τ )) dτ + · · · + 5 0 4 = (A1 + A2 + A3 + · · · ) π

y(t) =

3 Step and impulse responses of fractional-order systems using FSM One technique for obtaining the step response of a control system, which was used in early experimental and analogue computer work, was to observe the response to a low frequency square wave. This concept can be done analytically using frequency domain information as was done in [21], where it was made use of in computer aided design software written in Manchester autocode, as at the time the programming approach seemed easier than inverting Laplace transforms. The analytical concept, however, according to [22] was first described by Tustin in 1948. The principle is simply one of writing the low frequency square wave as a Fourier series, calculating the transmission through the transfer function for each individual frequency and then summing them at the output. A program has been written in MATLAB to implement this since the transmission of the individual frequencies is known exactly even when the transfer function is fractional. The Fourier series representation of a periodic signal x(t) can be written in either exponential or trigonometric form and we have decided to use the latter for a periodic square wave. The Fourier series for the square wave of −1 to 1

∞

(9)

If we consider the first integral A1 =

∞

g(τ ) sin(ωs (t − τ )) dτ ∞ 1 = g(τ )(ejωs (t−τ ) − e−jωs (t−τ ) ) dτ 2j 0   ejωs t ∞ e−jωs t ∞ = g(τ )e−jωs τ dτ − g(τ )ejωs τ dτ 2j 0 2j 0 0

(10)

Table 1 Integer approximations of G(s) = 1/s 0.5 using different methods [1] Method CFE high frequency method

Integer approximations of G(s) = 1/s 0.5 Hcfe (s) =

0.3513s 4 + 1.405s 3 + 0.8433s 2 + 0.1574s + 0.008995 s 4 + 1.333s 3 + 0.478s 2 + 0.064s + 0.002844 Hcar (s) =

Carlson’s method Matsuda’s method

Hmat (s) =

Least-squares method

Hls (s) =

Charref’s method Oustaloups’s method

IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

s 4 + 36s 3 + 126s 2 + 84s + 9 9s 4 + 84s 3 + 126s 2 + 36s + 1

0.08549s 4 + 4.877s 3 + 20.84s 2 + 12.995s + 1 s 4 + 13s 3 + 20.84s 2 + 4.876s + 0.08551

0.1002s 4 + 4.011s 3 + 11.26s 2 + 5.076s + 0.3694 s 4 + 8.654s 3 + 9.364s 2 + 1.771s + 0.03744

6.3s 4 + 74.84s 3 + 121.1s 2 + 29.79s + 0.9986 + 29.85s 4 + 121.8s 3 + 76.85s 2 + 7.497s + 0.1 s 5 + 74.97s 4 + 768.5s 3 + 1218s 2 + 298.5s + 10 Hous (s) = 10s 5 + 298.5s 4 + 1218s 3 + 768.5s 2 + 74.97s + 1 Hcha (s) =

s5

819 © The Institution of Engineering and Technology 2015

www.ietdl.org Magnitude(dB)

50

0

-50 -4 10

-2

10

Phase(deg)

0

Hcfe Hcar 0 Hmat 10 Hls Frequency(rad/sec) Hcha Hous Exact

10

-20 -40 -60 -4 10

-2

0

10

10 Frequency(rad/sec) a

2

4

10

10

14

16 Analytical Function

CFE Method

CFE Method

14

Carlson's Method

12

Carlson's Method

Matsuda's Method

Matsuda's Method

Least-Squares Method

Least-Squares Method

12

10

Oustaloups's Method

Oustaloups's Method Charref's Method

Charref's Method

8

10 Error

Output(Step Response)

4

2

10

8

6

6

4

4

2

2

0

0

20

0

40

60

80

100 120 Time(sec) b

140

160

180

200

-2

0

20

40

60

80

100 120 Time(sec) c

140

160

180

200

Fig. 1 Approximations by different methods a Exact Bode plots of G(s) = 1/s0.5 and its approximations b Step responses of G(s) = 1/s0.5 c Errors in the approximations

As T → ∞ and ωs → 0 the numerator of the imaginary part of G(jωs ) is multiplied by ωs so that limωs →0 ImG(jωs ) = 0 and (13) becomes

from which jωs t

−jωs t

e e G(jωs ) − G(−jωs ) 2j 2j   1 −j = cos(ωs t) + sin(ωs t) G(jωs ) 2 2   j 1 + cos(ωs t) + sin(ωs t) G(−jωs ) 2 2

A1 =

y(t) ∼ = (11)

Since ReG(−jωs ) = ReG(jωs ) and ImG(−jωs ) = −ImG (jωs ), we can write A1 = ReG(jωs ) sin(ωs t) + ImG(jωs ) cos(ωs t) Thus, the step response can be written as ∞  4  1 Re[G(jkωs )] sin(kωs t) y(t) = π k k=1(2)  1 + Im[G(jkωs )] cos(kωs t) k 820 © The Institution of Engineering and Technology 2015

(12)

∞ 4  1 Re[G(jkωs )] sin(kωs t) π k k=1(2)

(14)

which is the unit step response of G(s). Similarly, the impulse response, which is the derivative of the step response is given by ∞ dys (t) 4  (ωs Re[G(jkωs )] cos(kωs t) = yi (t) = dt π k=1(2)

− ωs Im[G(jkωs )] sin(kωs t)) ∞ 4  ∼ ωs Re[G(jkωs )] cos(kωs t) = π k=1(2)

(13)

(15)

A Matlab program has been written to compute the step and impulse responses of a transfer function G(s), using IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

www.ietdl.org the FSM. Apart from the transfer function data, the other required input parameters for the program are the frequency ωs of the square wave; the high frequency value, ωh ; the largest odd harmonic, n, to be taken in the square wave; and the simulation time parameters t and tm . The output is computed from t = 0 at equal time intervals spaced by t up to tm . These can all be input by the user but default values are available in which case the program proceeds as follows: 1. Enter G(s). 2. The frequency ωs is computed as ωs = 0.01ω3 dB , where ω3 dB is the first 3 dB point on G(jω) below G(0), which is assumed to be finite. 3. The frequency ωh is computed as ωh = 100ω3 dB and n taken equal to (ωh /ωs ) + 1 = 10001. 4. The simulation time default is to take 200 points in the interval T /4 so that t = [0:T /800:T /4]. 5. The step and impulse responses are computed from (14) and (15) where k is 1, 3, 5, . . . , 10001. Example 1: The aim of this example is to show the validity of the program by considering the integer-order transfer function G(s) =

1.5s2 + 3s + 1.5 s4 + 3s3 + 3.5s2 + 3s + 1.5

(16)

The computed frequency values are ω3 dB = 1.601 rad/s, ωs = 0.01601 rad/s and ωh = 160.1 rad/s. The simulation time was taken as t = [0 : 0.5 : 40]. The difference between the step and impulse responses obtained by Matlab and the FSM program are not discernable on graphs as seen in Figs. 2a and b so the error values are shown in Figs. 2c and d and found to be less than 2.31 × 10−7 for the step responses and less than 7.42 × 10−5 for the impulse responses. The computation time for FSM depends on the number of harmonics, n, and the simulation time. The computation time and maximum error values for this example for different values of n and default value (n = 10 001) are given in Table 2 where it can be seen that in approximately 3.5–5 s an output with an error value smaller than 10−3 can be obtained.

Table 2 Computation time of FSM for different values of n Number of harmonics (n)

Computation time, s

Maximum error

1.30 2.38 4.50 19.60 38.36

2.10 × 10−2 2.71 × 10−3 2.22 × 10−4 2.37 × 10−6 2.31 × 10−7

201 501 1001 5001 10 001

Example 2: In this example, a comparison with Oustaloup’s method and Matsuda’s method is given. Consider the following FOTF G(s) =

4s + 4 + 5s + 4

2s2.5

(17)

The parameters for this transfer function are ω3 dB = 2.908 rad/s, ωs = 0.02908 rad/s, ωh = 290.8 rad/s. Oustaloup’s third-order approximations of s0.5 is given as follows 10s3 + 104.9s2 + 48.67s + 1 (18) s0.5 ∼ = 3 s + 48.67s2 + 104.9s + 10 and the fifth-order approximation is given in Table 1. Thus replacing Oustaloup’s third and fifth-order approximations of s0.5 in (17), one obtains (see (19 and 20)) The step responses of Gous3 (s), Gous5 (s) and the step response of the actual system using the method presented are shown in Fig. 3a. The error plots between Oustaloup’s method and the FSM are given in Fig. 3b. Matsuda’s third-order approximations of s0.5 is computed from [20] as follows 18.58s3 + 254.8s2 + 108.8s + 1 s0.5 ∼ = 3 s + 108.8s2 + 254.8s + 18.58

(21)

and the fourth-order approximation is given in Table 1. Thus (see (22))

4s + 4 2s2 (s0.5 ) + 5s + 4 4s4 + 198.68s3 + 614.28s2 + 459.6s + 40 = 20s5 + 214.8s4 + 344.69s3 + 721.18s2 + 469.6s + 40 4s + 4 Gous5 (s) = 2 0.5 2s (s ) + 5s + 4 4s6 + 303.9s5 + 3373.9s4 + 7946s3 + 6066s2 + 1234s + 40 = 7 20s + 602s6 + 2814.85s5 + 5679.38s4 + 9313.94s3 + 6366.5s2 + 1244s + 40 Gous3 (s) =

4s + 4 + 5s + 4 4s4 + 439.2s3 + 1454.4s2 + 1093.5s + 74.3 = 37.16s5 + 514.6s4 + 765.6s3 + 1711.2s2 + 1112.1s + 74.3

Gmat3 (s) =

(19)

(20)

2s2 (s0.5 )

IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

(22)

821 © The Institution of Engineering and Technology 2015

2 1.5 1 0.5 0

Matlab FSM 0

10

20 30 Time(sec)

40

Output(impulse response)

Output(step response)

www.ietdl.org 1 0.5 0 -0.5 -1

Matlab FSM 0

10

a

0

0

10

30 20 Time(sec)

40

Error(in impulse responses)

Error(in step responses)

-5

x 10

2

-2

40

b

-7

4

20 30 Time(sec)

5

x 10

0

-5

-10

10

0

c

30 20 Time(sec)

40

d

Fig. 2 Difference between the step and impulse responses obtained by Matlab and the FSM program are not discernable a b c d

Step responses of (16) Impulse responses of (16) Error values in step responses Error values in impulse responses

and (see (23)) The step responses of Gmat3 (s), Gmat4 (s) and the FSM are given in Fig. 3c and the error plots in Fig. 3d. The above results using Oustaloup’s and Matsuda’s methods are typical of the approximate methods, and we have considered others, in that: 1. The errors are normally smaller the higher order the approximation of a specific method used. 2. The larger errors occur near overshoots and undershoots. 3. At any specific time in the response it is difficult to say which of two methods of the same order will give the lowest error. For this example, the maximum error over the response time is smaller for Matsuda’s fourth-order approximation than Oustaloup’s fifth. However, at time t = 0.8 s when Oustaloup’s method gives an error of zero Matsuda’s method gives an error of 1.04 × 10−3 . In control system investigations, the values of steady-state errors are important and one wishes to obtain exact results, which unfortunately as shown by the next example is not necessarily the case when approximations are used. Example 3: Consider G(s) =

1 5s0.9 + 1

(24)

It can be seen that since lims→0 G(s) = 1, the unit step response converges to 1. Oustaloup’s third and fifth-order approximations for s0.9 are 63.1s3 + 358s2 + 89.94s + 1 s0.9 ∼ = 3 s + 89.94s2 + 358s + 63.1

(25)

63.1s5 + 1303s4 + 3679s3 + 1606s2 + 108.4s + 1 s0.9 ∼ = 5 s + 108.4s4 + 1606s3 + 3679s2 + 1303s + 63.1 (26) so that the approximations for (24) are Gous3 (s) =

s3 + 89.94s2 + 358s + 63.1 316.6s3 + 1879.9s2 + 807.7s + 68.1

Gous5 (s) =

s5 + 108.4s4 + 1606s3 + 3679s2 + 1303s + 63.1 316s5 + 6623s4 + 20001s3 + 11709s2 + 1845s + 68 (28)

(27)

The exact step response of the system obtained from FSM and the step responses of Gous3 (s) and Gous5 (s) are shown in Fig. 4a. In all the cases, Oustaloup’s approximations give a steady-state error equal to 0.073 (7.3%) and increasing the approximation order further does not eliminate it. On the other hand, FSM gives a steady-state error of 0.0048 at t = 200 s and reduces further to 0.0015 at t = 900 s.

4s + 4 2s2 (s0.5 ) + 5s + 4 0.0342s5 + 19.85s4 + 102.868s3 + 135.34s2 + 55.98s + 4 = 6 2s + 26.4275s5 + 66.407s4 + 133.46s3 + 148.506s2 + 56.98s + 4

Gmat4 (s) =

822 © The Institution of Engineering and Technology 2015

(23)

IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

www.ietdl.org 0.1

1.5

Output(step response)

Oustaloup 3rd order Oustaloup 5th order FSM

Oustaloup 3rd order

Error

0.05 0 -0.05

1

-0.1

1.46

15 Time(sec)

20

25

30

25

30

25

30

25

30

Oustaloup 5th order

Error

1.4

0 -0.01

1.38 1.2 0

10

0.01

1.42

0

5

0.02

1.44

0.5

0

1.4 5

1.6 10

1.8 2 15 Time(sec)

-0.02 20

25

30

0

5

10

a

15 Time(sec)

20

b 0.2

1.5 Matsuda 3rd order Matsuda 4th order FSM

Matsuda 3rd order

Error

0.1

1.48

-0.1

0

5

10

15 Time(sec)

20

1.46 0.01 Matsuda 4th order

1.44

0.5

0.005

Error

Output(step response)

0 1

1.42

-0.005

1.4 1.4 0

0

5

10

0

15 Time(sec)

1.6 20

1.8 25

-0.01 30

0

5

10

c

15 Time(sec)

20

d

Fig. 3 Step responses of Gous3 (s), Gous5 (s), Gmat3 (s), Gmat4 (s) and the step response of the actual system and the error plots a b c d

Step responses of Gous3 (s), Gous5 (s) and actual system using FSM Error plots between Oustaloup’s methods and the FSM Step responses of Gmat3 (s), Gmat4 (s) and the FSM Error plots between the Matsuda’s method and the FSM

Example 4: In this example, the step response of the following FOTF is studied and comparison with Oustaloup’s third-order approximation method is given G(s) =

0.8s2.2

1 + 0.5s0.8 + 1

(29)

Oustaloup’s third-order approximations for s0.2 and s0.8 are 2.512s3 + 41.74s2 + 30.71s + 1 s0.2 ∼ = 3 s + 30.71s2 + 41.74s + 2.512 39.81s3 + 261.4s2 + 77.14s + 1 s0.8 ∼ = 3 s + 77.14s2 + 261.4s + 39.81

(30) (31)

Thus replacing these approximations in (29) one obtains (see (32))

Gous3 (s) =

The step responses are shown in Fig. 4b and the error between the plots in Fig. 4c. Furthermore the step response obtained from the FSM enters the 2% tolerance band at t = 46 s but the approximate step response enters at t = 73 s. The steady-state error at t = 100 s is 0.0032 for the FSM and 0.0153 for the approximate step response. For large time, the approximate step response will have a constant steady-state error of 1–100/101.3 = 0.0129. Example 5: The closed-loop transfer function of a control system can be extremely complicated. In this example the transfer function is taken as G(s) =

s5.25

+

4s3.77

0.4s1.12 + 0.3 (33) + 6s2.92 + 4s1.68 + 1.3s1.43 + 0.3

s6 + 108s5 + 2674s4 + 11351s3 + 12411s2 + 2325s + 100 2s8 + 188s7 + 3150.9s6 + 11622s5 + 15450s4 + 19272s3 + 14400s2 + 2443.1s + 101.3

IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

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823 © The Institution of Engineering and Technology 2015

www.ietdl.org 1.6 1

1.2

Output(step response)

0.8

Output(step response)

Oustaloup 3rd order FSM

1.4

0.6

0.4 FSM Oustaloup 3rd order Oustaloup 5th order

0.2

1 0.8 1.02 0.6 1 0.4

0.98

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0.96 60

0

0

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100 120 Time(sec)

140

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a Output(step response)

0.08 0.06 0.04

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1

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Error

0.02

0

50 60 Time(sec)

80

b

0.1

-0.1

40

70

70

80

90

100

0.3 0.2 0.1 0 -0.1

c

d

Fig. 4 Exact step response of the system obtained from FSM a b c d

Step response of (24) obtained from FSM and the step responses of Gous3 (s) and Gous5 (s) Step responses of Gous3 (s) of (32) and actual system using FSM Error between step responses of Gous3 (s) of (32) and actual system using FSM Step and impulse responses of G(s) of (33)

which includes the fractional power terms s0.12 , s0.25 , s0.77 , s0.92 , s0.68 and s0.43 . If we approximate each of these fractional powered terms then the order of the approximation increases considerably, it is very time consuming to evaluate and there is a high probability of making an error in the algebra. For example, if we use third-order approximations for each term then the order of the approximate transfer function will be 23 and for fifth-order approximations it will be 35. On the other hand, the FSM gives the step and impulse responses of the transfer function, shown in Fig. 4d, in a few seconds and the result is exact. Any concerns about accuracy can always be investigated by not accepting the default parameters and choosing a lower frequency square wave and possibly more harmonics. Example 6: In this example, the step responses of the FOTFs given in (29) and (33) are studied and comparison results with the GL numerical approximation method are given. The GL numerical approximation method [2, 19] can be used for step response computation of FOTFs without estimating integer-order rational transfer functions. It should be 824 © The Institution of Engineering and Technology 2015

noted that the integer-order approximation methods considered in the previous examples have been developed not only for computing transient responses but also for further analysis, and digital and analogue domain implementation. Thus, in some ways, comparing the presented approach with the GL method is more appropriate. The step response results obtained from the GL method are generally very accurate as long as the step size, t, of the simulation time is small enough. The accuracy of the GL method strongly depends on t. For an all pole FOTF, such as (29), the numerical solution to the corresponding fractional-order differential equation can be obtained on substitution of the fractionalorder derivative operator of GL [19, p. 293]. However, for the general form of FOTF with zeros, such as (33), the righthand side of the corresponding fractional-order differential equation, involving derivatives of the input u(t), must first be evaluated numerically and then the final solution can be found as for the all pole case. Fig. 5a shows step responses of the all pole FOTF given in (29) using the FSM and GL methods for different step IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

www.ietdl.org 0.5

1.6

t=0.5

FSM ( t=0.5)

Error

GL ( t=0.5)

1.4

GL ( t=0.01)

Output(Step Response)

1.2

0

1 -0.5

0

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30

40

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50 60 Time(sec)

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t=0.01 0.02

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0 -0.02 -0.04

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a

60 50 Time(sec)

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b -3

1.5

8

x 10

FSM( t=1) GL( t=1)

6

GL( t=0.5) GL( t=0.1)

4

2

Error

Output(step response)

GL( t=0.05) 1

1.45 0.5

0

1.4 -2 1.35 -4

1.3 12 0

0

20

40

14

60 Time(sec)

16

18 80

20

22

100

120

-6

0

20

c

40

60 Time(sec)

80

100

120

d

Fig. 5 Step responses of the FOTFs given in (29) and (33) using the FSM and GL methods a b c d

Step responses of (29) using FSM and GL method for different t Error between FSM and GL for t = 0.5 and t = 0.01 Step responses of (33) using FSM and GL method for different t Error between FSM and GL for t = 0.05

sizes. The errors between the FSM and GL for t = 0.5 and t = 0.01 are shown in Fig. 5b. Similarly, Fig. 5c shows step responses of the general FOTF given in (33) using the FSM and GL methods for different step sizes. The error between the FSM and GL for t = 0.05 is shown in Fig. 5d. The computation time of GL method depends on the step size t. For example, the step responses shown in Fig. 5a were obtained in 7.02 and 6.92 s using FSM method for n = 1001 and GL method for t = 0.01, respectively. As shown in Example 1, the computation time can be reduced for FSM by taking a smaller number of harmonics. We found that the computation time changed from 7.02 to 3.8 s for n = 501. On the other hand, it can be seen from Fig. 5b that the GL method gave a maximum error in the response of 2.5 × 10−2 for t = 0.01. This error can be further reduced by reducing the value of t. However, this will increase the computation time. For t = 0.005, the computation time of the GL method was 25.44 s and the maximum error was 1.3 × 10−2 . IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

4 Time responses of fractional-order systems using IFTM Another perhaps more obvious way to compute time information from known frequency data is to make use of the Fourier transform. The impulse response, g(t), corresponding to the transfer function G(s) of (3) is given by g(t) = L−1 (G(s)) where L−1 denotes the inverse Laplace transform. Assuming the impulse response is that of a stable system so that limt→∞ g(t) = 0 then the Fourier transform can be evaluated. The impulse response, g(t), only exists for t > 0 so it will be denoted by g+ (t) but as the range of t is from −∞ to ∞ for the Fourier transform one may consider the double sided function g(t) = g− (t) + g+ (t), where g− (t) exists for t < 0 and g+ (t) for t > 0. g− (t) may be taken as g+ (−t) or −g+ (−t), making the total time functioneither even or odd. Assuming the even ∞ 0 case then G(jω) = −∞ g− (t)e−jωt dt + 0 g+ (t)e−jωt dt = G− (jω) + G+ (jω), which can be shown to give G(jω) = ∞ 2 0 g+ (t) cos ωt dt. However, our concern is with the 825 © The Institution of Engineering and Technology 2015

Alternatively, if one assumes g(t) to be odd then G− (jω) = −ReG(jω) + jImG(jω), G+ (jω) + G− (jω) = 2jImG(jω) and  1 ∞ 2jImG(jω)ejωt dω g(t) = 2π −∞  2 ∞ =− ImG(jω) sin ωt dω (35) π 0 The Matlab program IFTM has been written to compute g(t) by numerical integration by both the above equations, (34) and (35). Matlab has several functions available for integration and trapz has been used. The outline of the program is given below: 1. Choice of frequency band. Lower and upper values for the frequency are defined by the vector ω = [ωl :ω:ωu ], where ω is the step value. The default value for frequency is selected as ω = [0.0001ω3 dB :0.001:100ω3 dB ] where ω3 dB is the first 3 dB point on G(jω) below G(0). 2. The frequency response data of G(s) over the chosen frequency band is computed. 3. The time for the simulation is entered in the program as t = [tl :t:tu ], where t is the step value. The default value for time is t = [0:0.1:50/ω3 dB ]. 4. For each t value within the given interval, the value of the integral given in (34) or (35) is determined and the time response is plotted. Example 7: As an example to validate the IFTM program G(s) was taken as 5 G(s) = 3 s + 3s2 + 7s + 5

(36)

The impulse response is g(t) = L−1 (G(s)) = 1.25e−t (1 − cos 2t). The impulse responses of G(s) using IFTM and exact analytical function g(t) are shown in Fig. 6a and the error plots between the impulse responses is given in Fig. 6b, respectively. From the error plots, one can conclude that the proposed method gives very accurate results. The error can be further decreased by increasing the frequency interval and decreasing the step value. Example 8: The aim of this example, discussed previously where approximations for the step response were compared with the known analytical solution for α = 0.5, is to show that the IFTM gives very accurate results for G(s) =

1 sα

(37)

when α = 0.5, and also to investigate possible analytical formulas for the step response of 1/sα . Fig. 7a shows 826 © The Institution of Engineering and Technology 2015

1 IFTM Exact

0.5

0

-0.5

0

5

10

15

10

15

Time(sec)

a -5

3

x 10

2

Error

∞ inversion integral g(t) = (1/2π) −∞ [G+ (jω) + G− (jω)]ejωt dω. It can be shown that G+ (jω) = ReG(jω) + jImG(jω) where G(jω) is the Laplace transform of g+ (t) or g(t), with s = jω, since the transform is defined for t > 0. Further from the definition, it can be seen that G− (jω) = ReG(jω) − jImG(jω), so that the integral gives  1 ∞ 2ReG(jω)ejωt dω g(t) = 2π −∞  2 ∞ = ReG(jω) cos ωt dω (34) π 0

Output(impulse response)

www.ietdl.org

1 0 -1

0

5 Time(sec)

b

Fig. 6 Impulse responses of G(s) using IFTM and exact analytical function g(t) a Impulse responses of (36) obtained from IFTM and exact analytical function b Error between impulse responses obtained from IFTM and exact analytical function

the step response using the default parameters in IFTM and the error from the analytical result. The error slowly increases with time and it is smaller than 0.015 after 200 s, although this can be reduced by using parameters other than the default ones or better Matlab functions for integration. For example, the quadl function of Matlab gave an error of less than 10−3 . The step response obtained using FSM for ωs = 0.000001 rad/s and ωh = 1 rad/s is also shown in Fig. 7a and it can be seen that the error is less than 0.1 as shown in Fig. 7a. This error becomes less than 0.08 when ωh is increased to ωh = 5 rad/s. This can be further decreased by a further increase in ωh and the inclusion of more harmonics. This example demonstrates the power of the method, as although it is intended for use in situations where G(0) is finite, good results can be obtained when this is not the case with a suitable choice of program parameters. The step response results of 1/sα for different α are shown in Fig. 7b. It can be seen that the analytical solutions for α = 0, 0.5 and 1 can be fit by a step response of the form  α t ys (t) = [Aα 2 + Bα + C] (38) π as the exact step response of 1/s1 is ys (t) = t, 1/s0.5 is ys (t) = 2(t/π )0.5 and 1/s0 is ys (t) = 1(t). Using these results in the above formula gives A + B + C = π , A/4 + B/2 + C = 2, C = 1 and the step response of 1/sα for 0 ≤ α ≤ 1 is  α t 2 ys (t) = [(2π − 6)α + (5 − π )α + 1] (39) π and the impulse response is yi (t) =

α[(2π − 6)α 2 + (5 − π )α + 1]t α dys (t) = dt tπ α

(40)

The step response of G(s) of (37) using IFTM and the analytical solutions given by (39) for α = 0.2, α = 0.4, α = 0.6 IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

www.ietdl.org

10 5

=0.1

FSM Exact

15

=0.2

10 5 0

0

50

100 150 Time(sec)

200

0

0.015

50

100 150 Time(sec)

200

0.15 0.1 Error

0.01

Output( Step Response)

15

0

Error

120

20 IFTM Exact

Step response

Step response

20

100

=0.3 =0.4

80

=0.6 =0.7

60

=0.9

=0.5

=0.8

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0.05 20

0.005

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100 150 Time(sec)

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100 120 Time(sec)

a

140

=0.8 - IFTM =0.2 - Analytic Function =0.4 - Analytic Function =0.6 - Analytic Function

Error

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0

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=0.2

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=0.8 - Analytic Function

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1 =0.6

Error

Output( Step Response)

=0.4

=0.4 - IFTM =0.6 - IFTM

50

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0.05 =0.2 - IFTM

60

180

b

80 70

160

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100 150 Time(sec)

200

-0.5

0

50

100 150 Time(sec)

200

d

c

Fig. 7 Step response using the default parameters in IFTM and the error from the analytical result a b c d

Step responses of (37) for α = 0.5 using IFTM and FSM Step responses of (37) for different values of α Step responses of G(s) of (37) using IFTM and the analytical solutions Errors between step responses obtained from IFTM and the analytical solutions for α = 0.2, α = 0.4, α = 0.6 and α = 0.8

and α = 0.8 are shown in Fig. 7c and the errors given in Fig. 7d. Thus, the analytical formulas based on iteration between the three mathematically derived exact solutions are very good.

by using the IFTM and Oustaloup’s approximation. The Oustaloup first, third, fifth and seventh-order approximations for transitional frequencies ωm = 10−3 rad/s and ωM = 103 rad/s are (see (42–45))

Example 9: In this example, we compute the impulse response of 1 G(s) = 1.8 (41) s +1

The impulse responses are shown in Fig. 8a and the error plots are given in Fig. 8b. The impulse response of the seventh-order Oustaloup’s approximate transfer function matches with the IFTM results with an error less than 0.01.

s + 251.2 251.2s2 + 2s + 251.2 s3 + 637.3s2 + 4021s + 251.2 = 251.2s4 + 4022s3 + 1275s2 + 4022s + 251.2 s5 + 809.7s4 + 38910s3 + 117500s2 + 22300s + 251.2 = 251.2s6 + 22300s5 + 118300s4 + 77810s3 + 118300s2 + 22300s + 251.2 s7 + 953.4s6 + 110900s5 + 1762000s4 + 3880000s3 + 1184000s2 + 49380s + 251.2 = 251.2s8 + 49380s7 + 1185000s6 + 3991000s5 + 3524000s4 + 3991000s3 + 1185000s2 + 49380s + 251.2

Gous1 =

(42)

Gous3

(43)

Gous5 Gous7

IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

(44) (45)

827 © The Institution of Engineering and Technology 2015

www.ietdl.org 0.8

0.5 0

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0.005

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Error

IFTM -0.6

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Output(Impulse Response)

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Error

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1.4 =1.5

=1 1.2

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1

1

Output(step response)

Output(step response)

0

b

1.4

0.8 =0.5 0.6

0.8 =0.8 0.6

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d

Fig. 8 Impulse responses and the error plots a b c d

Impulse responses of (41) obtained from IFTM and the impulse responses of Gous1 (s), Gous3 (s), Gous5 (s) and Gous7 (s) Error plots between impulse responses obtained from IFTM and impulse responses of Gous1 (s), Gous3 (s), Gous5 (s) and Gous7 (s) Step responses of (47) for different α values Step responses of (49) for different α values

The impulse response of (41) was also obtained using FSM. Both results were in agreement and the errors between the two results were smaller than 12 × 10−3 for the initial time until approximately t = 0.1 s and then they became smaller than 10−4 .

5

effect the fractional order has on the settling time for a step response as considered in the following examples. Example 10: This example investigates the effect of α on the settling time by considering a unity feedback control system that includes a plant transfer function and a fractional-order PI controller as

Steady-state error-time to converge

It has been seen from the above examples that, if high order approximations are used for a FOTF the errors in the resultant step responses are often quite small. However, one area of major interest in feedback control for step responses is the determination of any steady-state error and the time of convergence to the steady state. It has been seen in Examples 3 and 4 that approximations can give incorrect steady-state errors and obviously therefore inaccurate predictions of settling times. Since PID controllers with fractional-order integrators have been suggested for use in control systems it is very important to know exactly what 828 © The Institution of Engineering and Technology 2015

G(s) =

1 2 s + 4s + 1

and C(s) = 1 +

sα + 1 1 = (46) sα sα

Thus the closed-loop transfer function is T (s) =

sα + 1 s2+α + 4s1+α + 2sα + 1

(47)

Using the FSM and IFTM the settling times for different tolerance bands and α values are given in Table 3. When α reduces the settling time increases significantly. The step responses for different α values are given in Fig. 8c. IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354

www.ietdl.org Table 3 Time to converge for α

α

Settling time, s. For 5% (0.95–1.05) tolerance band

Settling time, s. For 3% (0.97–1.03) tolerance band

Settling time, s. For 2% (0.98–1.02) tolerance band

Settling time, s. For 1% (0.99–1.01) tolerance band

9.5 8.5 7 15 39 121

14.5 9 14.5 33 94 367

15 14.5 19.5 59 188 1070

16 16.5 49 158 655 3770

1 0.9 0.8 0.7 0.6 0.5

Table 4 Time to converge for α

α

Settling time, s. For 5% (0.95–1.05) tolerance band

Settling time, s. For 2% (0.98–1.02) tolerance band

9 6 5.5 4.5 13 35 160

13.5 7 6 15 58 230 5800

1.5 1.3 1.2 1 0.9 0.8 0.7

Example 11: Consider a unity feedback control system that includes a fractional-order plant and a fractional-order controller such as [2] 0.8s1.2 + 2 + 0.8s1.3 + 1.9s0.5 + 0.4 1.2s0.72 + 1.5s0.33 C(s) = 3sα

G(s) =

1.1s1.8

and

Fourier series of a low frequency square wave. The second method called the IFTM is based on the inverse Fourier transform of frequency domain data of a FOTF. The frequency domain data for a FOTF can be obtained exactly. Thus, the time responses obtained using both the FSM and IFTM methods with the program default parameters are very accurate as they use frequency domain data. The accuracy can be increased in most cases by a more sophisticated choice of program parameters. Many examples have been provided to show the value of the methods. These indicate the largest errors typically occur in responses for low values of time, which are normally of less concern in feedback control systems.

7 1 2

(48)

3 4

and we want to determine the settling time for varying α. The closed-loop transfer function is (see (49)) Settling times for different tolerance bands and α values are given in Table 4. The step responses for different α values are given in Fig. 8d. These last two examples clearly show the effect of variations in the fractional power α of the integrator on closedloop response. Since less phase lag is introduced the smaller the value of α, and typically less oscillatory the response, this may seem a desirable procedure. However, the ‘trade off’ is that this produces a major increase in settling time so that values below even 0.9 do not seem appropriate.

5

6 7 8 9

6

Conclusions

10

In this paper, two exact methods have been given for the time response computation of control systems with FOTFs. The first method called the FSM is based on using the

T (s) = =

11

References Vinagre B.M., Podlubny I., Herñandez A., Feliu V.: ‘Some approximations of fractional order operators used in control theory and applications’, Fractional Calc. Appl. Anal., 2000, 3, (3), pp. 231–248 Chen Y.Q., Petráš I., Xue D.: ‘Fractional order control – a tutorial’. 2009 American Control Conf., Hyatt Regency Riverfront, St. Louis, MO, USA, 10–12 June 2009, pp. 1397–1411 Krishna B.T.: ‘Studies on fractional order differentiators and integrators: a survey’, Signal Process., 2011, 91, (3), pp. 386–426 Djouambi A., Charef A., Voda A.: ‘Numerical simulation and identification of fractional systems using digital adjustable fractional order integrator’. 2013 European Control Conf. (ECC), Zürich, Switzerland, 17–19 July 2013, pp. 2615–2620 Oustaloup A., Levron F., Mathieu B., Nanot F.M.: ‘Frequency band complex noninteger differentiator: characterization and synthesis’, IEEE Trans. Circuit Syst. I, Fundam. Theory Appl., 2000, 47, (1), pp. 25–39 Carlson G.E., Halijak C.A.: ‘Approximation of fractional capacitors (1/s)1/n by a regular Newton process’, IEEE Trans. Circuit Theory, 1964, 11, (2), pp. 210–213 Matsuda K., Fujii H.: ‘H∞ -optimized wave-absorbing control: analytical and experimental results’, J. Guid. Control Dyn., 1993, 16, (6), pp. 1146–1153 Charef A., Sun H.H., Tsao Y.Y., Onaral B.: ‘Fractal system as represented by singularity function’, IEEE Trans. Autom. Control, 1992, 37, (9), pp. 1465–1470 Duarte V., Costa J.S.: ‘Time-domain implementations of non-integer order controllers’. Proc. of Control 2002, Portugal, 5–7 September 2002, pp. 353–358 Das S.: ‘Functional fractional calculus for system identification and control’ (Springer-Verlag, Berlin, Heidelberg, New York, 2008) Xue D., Zhao C., Chen Y.Q.: ‘A modified approximation method of fractional order system’. Proc. of the 2006 IEEE Int. Conf. on Mechatronics and Automation, Luoyang, China, 25–28 June 2006

C(s)G(s) 1 + C(s)G(s) 0.96s1.92 + 1.2s1.53 + 2.4s0.72 + 3s0.33 3.3s(α+1.8) + 2.4s(α+1.3) + 0.96s1.92 + 1.2s1.53 + 5.7s(α+0.5) + 1.2sα + 2.4s0.72 + 3s0.33

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Krajewski W., Viaro U.: ‘A method for the integer-order approximation of fractional-order systems’, J. Franklin Inst., 2014, 351, pp. 555–564 13 Podlubny I.: ‘Fractional differential equations’ (Academic Press, San Diego, 1999) 14 Podlubny I.: ‘Fractional-order systems and PIλ Dμ -controllers’, IEEE Trans. Autom. Control, 1999, 44, (1), pp. 208–214 15 Sun H.G., Song X., Chen Y.Q.: ‘A class of fractional order dynamic systems with fuzzy order’. Proc. of the Eighth World Cong. on Intelligent Cont. and Auto., Jinan, China, 6–9 July 2010, pp. 197–201 16 Xue D., Chen Y.Q.: ‘A comparative introduction of four fractional order controllers’. Proc. of the Fourth World Cong. on Intelligent Cont. and Autom., Shangai, P.R.C., 10–14 June 2002, pp. 3228– 3235

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IET Control Theory Appl., 2015, Vol. 9, Iss. 6, pp. 817–830 doi: 10.1049/iet-cta.2014.0354