1 Introduction

Fractional-order systems are gathering a huge interest by the scientific research community of various engineering domains (Diethelm and Ford 2001; Koeller 1984; Ladaci and Charef 2006; Reyes-Melo et al. 2004; Neçaibia et al. 2015; Rabah et al. 2016). Good reviews devoted specifically to the subject are available in the literature (Miller and Ross 1993; Oldham and Spanier 1974; Oustaloup 1995).

Fractional-order mathematical models have proved to be more accurate for description of many physical phenomena such as electrochemical processes (Ichise et al. 1971), long distributed lines (Heaviside 1922), dielectric polarization (Sun et al. 1984), viscoelastic materials (Bagley and Calico 1991), colored noise (Mandelbrot 1967) and chaos (Khettab et al. 2017).

Indeed, using an integer model instead of the fractional one to characterize these processes requires a high-order models or the neglecting of some physical phenomena like the diffusion phenomena (Yakoub et al. 2015). Unfortunately, identifying a fractional-order system is not as easy as for the integer-order case because it requires estimation of both model coefficients and fractional orders. Different approaches have been proposed for their modelization both in time and frequency domains: Based on the ability to define systems using continuous-order distributions, Hartley and Lorenzo (2003) showed that frequency domain fractional-order system identification can be performed. Least-square techniques are applied to discretized-order distributions.

Djamah et al. (2008) have investigated the identification and model reduction in non-integer systems in time domain based on a fractional integrator operating on a limited spectral range. In Poinot and Trigeassou (2004), Poinot and Trigeassou proposed to model the fractional system by a state-space representation, where conventional integration is replaced by a fractional one with the help of a non-integer integrator. This operator is itself approximated by a combination of an integrator and of a phase-lead filter. Nazarian et al. (2010) addressed the problem of identifiability of a fractional-order system having its input and output frequency contents. They studied analytically the effects of the commensurate order of the model structure and model parameters. Whereas, Dzieliński et al. (2011) proposed a comparison of identification methods based on least squares and total least squares in the frequency domain.

In Yakoub et al. (2015), a bias correction method is developed by Yakoub et al. to deal with the bias problem in the continuous-time fractional closed-loop system identification. This method is based on the least squares estimator combined with the state variable filter approach. Recently, Charef et al. (2013) and Djouambi et al. (2013) introduced also an identification method based on the recursive least squares algorithm applied to a linear regression equation derived from the linear fractional-order differential equation using adjustable fractional-order differentiators. Other researchers proposed a time-domain identification algorithm based on a genetic algorithm (GA), where the multivariable parameter identification is converted into a parameter optimization by applying GA to the identification of fractional-order systems (Zhou et al. 2013; Othman and Al-Sabawi 2013; Bouyedda and Ladaci 2016). A comparative study between heuristic methods performance has been proposed recently by the authors (Bourouba and Ladaci 2016; Bourouba et al. 2017) for the adjustment of a fractional-order PID controller. The main motivation of this work is not only to focus on the approximation of the fractional process by an integer-order system, but also on the reduction in the order of this model.

Since the last two decades, an evolutionary optimization technique called Differential Evolution (DE) (Storn and Price 1997; Chakraborty 2008) has found much interest for its efficiency and easy implementation for parameter identification and control (Martín et al. 2015). DE is an evolutionary algorithm that solves an optimization problem by iteratively trying to improve a candidate solution according to a fitness value. In other words, the DE technique can be viewed as a particle-based method that evolves in time to the solution that yields the lowest value of the fitness function.

In this work, we investigate the approximation of linear fractional systems using the DE technique. An original approach is developed in the case of linear systems. The aim of this project is to provide a simple and reduced-order model for identification, simulation and control purposes. By different numerical simulation examples we show that very competitive results to the existing identification approaches are obtained with a reduced order of the identified model.

The reminder of this paper is organized as follows. Sect. 2 presents some basics of fractional-order calculus whereas the two most important approximation methods in the frequency domain are defined in Sect. 3. The DE optimization technique is explained in Sect. 4. Section 5 presents the proposed fractional-order system identification strategy using the DE approach. Simulation examples are given in Sect. 6 to illustrate the effectiveness of the proposed identification method and concluding remarks are given in Sect. 7.

2 Fractional-Order Systems

There are several definitions of fractional-order derivative. Two important and widely applied definitions are Riemann–Liouville definition and Grünwald–Letnikov definition which is perhaps the best known due to its most suitability for the realization of discrete control algorithms (Kilbas et al. 2006).

The Riemann–Liouville definition of the fractional-order derivative is expressed as:

$$\begin{aligned} {}_{a}D^{\mu }_t f(t) =\frac{1}{\varGamma (1-\mu )}\frac{\hbox {d}}{\hbox {d}t}\left[ \int ^{t}_{a}\frac{f(\tau )}{(t-\tau )^{\mu }}\mathrm{d}\tau \right] \end{aligned}$$
(1)

where \(\varGamma (.)\) is the Euler’s Gamma function defined as,

$$\begin{aligned} \varGamma (x) =\int ^{a}_{t}\tau ^{x-1}e^{-\tau }\mathrm{d}\tau , \ \ \ \ \ x>0 \end{aligned}$$
(2)

\(a\in \mathfrak {R}\) is the time initial value with \(a < t\) and \(\mu \in \mathfrak {R}\) is the order of the derivative operation with \(0<\mu <1\).

The Grünwald–Letnikov definition is expressed as:

$$\begin{aligned} {}_{a}D^{\mu }_t f(t) = \lim _{h \rightarrow 0}h^{-\mu }\sum _{r=0}^{\left[ \frac{t-a}{h}\right] }(-1)^r\frac{\varGamma (\mu +1)}{r!\varGamma (\mu -r+1)}f(t-rh) \end{aligned}$$
(3)

where h is the time increment, \([\frac{t-a}{h}]\) is a flooring operator and the binomial coefficients (for positive integer r) are given by:

$$\begin{aligned} \left( \begin{array}{c} \mu \\ 0 \\ \end{array} \right) =1, \left( \begin{array}{c} \mu \\ r \\ \end{array} \right) =\frac{\mu (\mu -1)\cdots (\mu -r+1)}{r!} \end{aligned}$$
(4)

For many engineering applications, the Laplace transform methods are often used. The Laplace transform of the fractional derivative/integral under zero initial conditions for order is given by:

$$\begin{aligned} \mathscr {L}\left( _{a}D^{\mu }_t f(t)\right) = s^{\mu } F(s) \end{aligned}$$
(5)

A general fractional-order single input single output system can be described by a fractional differential equation of the form (Tang et al. 2015)

$$\begin{aligned}&a_{n}D^{\alpha _{n}}_t y(t) + a_{n-1}D^{\alpha _{n-1}}_t y(t) + \cdots + a_{0}D^{\alpha _{0}}_t y(t) \nonumber \\&\quad = b_{m}D^{\beta _{m}}_t u(t) + b_{m-1}D^{\beta _{m-1}}_t u(t) + \cdots + b_{0}D^{\beta _{0}}_t u(t) \end{aligned}$$
(6)

where \(D^{\gamma } = D^{\gamma }_t\) denotes the fractional derivative.

So, a fractional-order differential equation, provided both the signals u(t) and y(t) are relaxed at \(t = 0\), can be expressed in a transfer function form (Hartley and Lorenzo 2003)

$$\begin{aligned} F(s) = \frac{b_{1}s^{\beta _{1}} + b_{2}s^{\beta _{2}} + b_{3}s^{\beta _{3}} + \cdots + b_{m}s^{\beta _{m}}}{a_{1}s^{\alpha _{1}} + a_{2}s^{\alpha _{2}} + a_{3}s^{\alpha _{3}} + \cdots + a_{n}s^{\alpha _{n}}} \end{aligned}$$
(7)

where \((\beta _{m},a_{n}) \in \mathscr {L}\left( R\right) ^{2}\), and \((m,n) \in \mathscr {L}\left( N\right) ^{2}\) and \(\alpha _{n}> \alpha _{n-1}> \cdots> \alpha _{0} > 0\) and \(\beta _{m}> \beta _{m-1}> \cdots> \beta _{0} > 0\).

To implement the fractional-order transfer functions in simulation or practical studies, one way is to approximate them with integer-order transfer functions. For an exact approximation of a fractional-order transfer function with an integer-order one, the integer-order transfer function has to include an infinite number of zeros and poles.

3 Approximation of Fractional-Order Systems in the Frequency Domain

In general, any available method for frequency domain identification can be applied in order to obtain a rational function, whose frequency response fits that corresponding to the original transfer function (Monje et al. 2010). Many approximation techniques in the frequency domain have been developed in the literature like (Matsuda and Fuji 1993) and (Carlson and Halijak 1964) approaches. In this work we will focus on the most famous and used ones which are the approximation method of Oustaloup et al. (2000) and the singularity function method of Charef et al. (1992).

3.1 Oustaloup Approximation Method

Oustaloup filter approximation to a fractional-order differentiator is widely used in fractional calculus. It is based on a recursive distribution of an infinite number of zeros and negative real poles (to ensure a minimum phase behavior) Oustaloup et al. (2000).

The method is based on the approximation of a function of the from:

$$\begin{aligned} H(s) = s^{\alpha } \ \ \ \ \ \ \alpha \in \mathfrak {R}^{+} \end{aligned}$$
(8)

by a rational function (see Sabatier et al. 2003):

$$\begin{aligned} H_\mathrm{app}(s) = C \prod ^{k=N}_{k=1}\frac{s + \omega ^{'}_{k}}{s + \omega _{k}} \end{aligned}$$
(9)

where C is a gain adjusted so that both sides of (9) shall have unit gain at 1 rad/s and N is the number of poles and zeros related to the desired bandwidth and error criteria. These are obtained using the following set of synthesis formulas for zeros, poles and the gain:

$$\begin{aligned}&\omega '_{k} = \omega _{b} \omega _{u}^{(2k-1-\alpha )/N} \nonumber \\&\omega _{k} = \omega _{b} \omega _{u}^{(2k-1+\alpha )/N} \nonumber \\&C = \omega ^{\alpha }_{h} \end{aligned}$$
(10)

where \(\omega _{u}\) is the unity frequencies gain and the central frequency of a band of frequencies distributed geometrically. Let \(\omega _{u} = \sqrt{\omega _{b}\omega _{h}}\), where \(\omega _{h}\) and \(\omega _{b}\) are, respectively, the upper and lower frequencies.

3.2 Singularity Function Approximation Method

The so-called singularity function proposed by Charef et al. (1992) is very close to Oustaloup’s method, allowing the approximation of fractional-order transfer function with rational ones. This method is very easy to implement and is based on the approximation of a fractional-order function of the form (8),

$$\begin{aligned} H(s) = s^{\alpha } \end{aligned}$$

by a quotient of polynomials in s in a factorized form:

$$\begin{aligned} \hat{H}(s)=\frac{\prod _{i=0}^{n-1}\left( 1+\frac{s}{z_{i}}\right) }{\prod _{i=0}^{n}\left( 1+\frac{s}{p_{i}}\right) } \end{aligned}$$
(11)

where the coefficients are computed for obtaining a maximum deviation from the original magnitude response in the frequency domain of \(y \; dB\).

Defining :

$$\begin{aligned} a=10^{y/10(1-\alpha )}, \qquad b=10^{y/10\alpha }, \qquad ab=10^{y/10(1-\alpha )} \end{aligned}$$

The poles and zeros of the approximated rational function are obtained by applying the following formulas:

$$\begin{aligned} p_0=PT\sqrt{b},\qquad p_i=p_0(ab)^i,\qquad z_i=ap_0(ab)^i \end{aligned}$$

The number of poles and zeros is related to the desired bandwidth and the error criteria used by the expression:

$$\begin{aligned} N=\left[ \frac{\mathrm{log}\left( \frac{w_\mathrm{max}}{p_0}\right) }{\mathrm{log}(a)}\right] +1 \end{aligned}$$
(12)

4 Differential Evolution Algorithm

Differential Evolution, or DE, was developed by Storn and Price (1997). It is an evolutionary algorithm developed for solving continuous space functions based on a stochastic optimization method. DE is a vector-based metaheuristic algorithm, which has some similarity to pattern search and genetic algorithms due to its use of crossover and mutation. In fact, DE can be considered as a further development to genetic algorithms with explicit updating equations, which make it possible to do some theoretical analysis.

DE is a stochastic search algorithm with self-organizing tendency and does not use the information of derivatives. Thus, it is a population-based, derivative-free method. In addition, DE uses real numbers as solution strings, so no encoding and decoding is needed. It is implemented for solving several engineering problems due to simplicity and high efficiency (Martín et al. 2015; Moreno et al. 2006).

The mathematical formulation of Differential Evolution algorithm is introduced below. Suppose that the problem to be solved is to minimize an objective function (f), an optimization problem can be stated as follows:

$$\begin{aligned} \min f(X) \end{aligned}$$
(13)

with \(X = \left[ X_{1}, \; X_{2}, \; \ldots X_{D}, \;\right] ^{T}\) the decision vector and

$$\begin{aligned} X_{j} \in \left[ X_{\mathrm{min}, j}, \; X_{\mathrm{max}, j}\right] \ \ \ \ \ \ j= 1, 2, \ldots D \end{aligned}$$
(14)

where \(X_{\mathrm{min}, j}\) and \(X_{\mathrm{max}, j}\) are the lower and upper bounds for each decision variable, respectively.

The initial position of individuals is randomly generated from the uniform distribution,

$$\begin{aligned} X^{i}_{j}= & {} X_{\mathrm{min}, j}\,+\,\mathrm{rand}^{i}_{j}(0,1)\nonumber \\&\times \,\left( X_{\mathrm{max}, j}-X_{\mathrm{min}, j}\right) \ \ \ \ \ \ i= 1, 2, \ldots N_{P} \end{aligned}$$
(15)

where \(N_\mathrm{P}\) is the number of members in a population, \(\mathrm{rand}^{i}_{j}\) generates a random value within [0, 1] interval for each element of every individual.

This population is successfully improved by applying mutation, crossover, and selection operators. The procedure is described as follows:

4.1 Mutation

Many different techniques have been proposed in the literature to tune the mutation strategy and parameters such as population size \(N_\mathrm{P}\) (Teo 2006), the scale factor F (Brest et al. 2006), and the crossover rate \(C_\mathrm{R}\) (Pan et al. 2011) and to avoid the population convergence to a local optima. The mutation operator can not only increase the diversity of solution vectors, but also enhance the exploration capability of solution space for the DE algorithm. At generation G, for each target vector \(X^{(G+1)}_{i}\), a mutant vector \(v^{(G+1)}_{i}\), \(i={1,2, \ldots , N_\mathrm{P}}\) is generated according to the following equation (Moreno et al. 2006):

$$\begin{aligned} v^{G+1}_{i} = X^{G}_{r_{1}} + F.\left( X^{G}_{r_{2}}-X^{G}_{r_{3}}\right) \end{aligned}$$
(16)

The indexes \(r_1\), \(r_2\) and \(r_3\) are distinct randomly chosen between 1 and \(N_{P}\), and \(F \in (0, 1)\) is a user supplied real constant to control the amplification of the difference vector \((X^{G}_{r_{2}}-X^{G}_{r_{3}})\).

4.2 The Crossover

The most common crossover in DE is the uniform crossover which can be defined as follows,

$$\begin{aligned} u_{ij}^{(G+1)}=\left\{ \begin{array}{l} v_{ij}^{(G+1)},\quad \mathrm{rand}(j) \le C_{r} \ \mathrm{or} \ j = r(i) \\ x_{ij}^{(G+1)},\quad \mathrm{rand}(j) > C_{r} \ \mathrm{or} \ j \ne r(i) \end{array} \right. \end{aligned}$$
(17)

where \(j={1, 2,\ldots , D}\), \(\mathrm{rand}(j) \in [0 \; 1]\) is the \(j\mathrm{th}\) evaluation of a uniform random generator number. \(C_{r} \in [0 \; 1]\) is the crossover probability constant, which has to be determined previously by the user. \(r(i) \in {1, 2, \ldots , D}\) is a random integer.

4.3 Selection

The trial vector \(v_i^{(G+1)}\) is compared with the target vector \(X_i^G\) and the one with the better fitness value is admitted to the next generation. The selection scheme in DE can be represented by the following equation (for a minimization problem):

$$\begin{aligned} X_{i}^{(G+1)}=\left\{ \begin{array}{l} u_{i}^{(G+1)},\quad f(u_{i}^{(G+1)}) \le f(X_{i}^{(G)}) \\ X_{i}^{(G)},\quad \mathrm{otherwise} \end{array} \right. \end{aligned}$$
(18)

where \(f(u_i^{(G+1)})\) and \(f(X_i^G)\) are the objectives of \(u_i^{(G+1)}\) and \(X_i^G\); \(i\in {1, 2, \ldots , N_{P}}\).

4.4 DE Algorithm

The previous process (mutation, crossover, and selection) is applied to the whole population, obtaining the next generation population \((G+1)\). The algorithm usually continues the search until the maximum number of iterations is reached. Algorithm 1 illustrates a standard DE algorithm.

figure a

In our case, the fitness function (Objective function) \(f(-)\) has to be designed according to the requirements that must minimize the error between Bode diagram of the fractional-order system and our approximated model.

4.5 Parameters’ Setting

The choice of parameters is important. Both empirical observations and parametric studies suggest that parameter values should be fine-tuned (Yang 2014).

The scale factor F is the most sensitive one. \(F \in [0.4, 0.95]\) is a good range as demonstrated in Storn and Price (1997), Zaharie (2002).

Zaharie (2002) showed also that the crossover parameter \(C_{r} \in [0.1, 0.8]\) is a good range, and the first choice can be \(C_{r} = 0.5\).

It is suggested that the population size n should depend on the dimensionality D of the problem, also usually referred to as the number of chromosomes. That is, \(n = 5D\) to 10D. This may be a disadvantage for higher-dimensional problems because the population size can be large. However, we can first try a fixed value (say, \(n = 40\) or 100).

5 Fractional Model Approximation Using DE Algorithm

We consider the general fractional-order LTI systems represented by their transfer function (7) as follows:

$$\begin{aligned} F(s) = \frac{b_{1}s^{\beta _{1}} + b_{2}s^{\beta _{2}} + b_{3}s^{\beta _{3}} + \cdots + b_{m}s^{\beta _{m}}}{a_{1}s^{\alpha _{1}} + a_{2}s^{\alpha _{2}} + a_{3}s^{\alpha _{3}} + \cdots + a_{n}s^{\alpha _{n}}} \end{aligned}$$

Our target now is to find an approximate integer-order model with a relatively low order \(n = 5\), in the following form:

$$\begin{aligned} T(s) = \frac{b_{4}s^{4} + b_{3}s^{3} + b_{2}s^{2} + b_{1} s + b_{0}}{a_{5}s^{5} + a_{4}s^{4} + a_{3}s^{3} + a_{2}s^{2} + a_{1}s + a_{0}} \end{aligned}$$
(19)

The choice of Eq. (19) form for the configuration of the approximating rational function is the result of a careful study based on our knowledge about fractional approximation methods such as Oustaloup’s or Charef’s ones. A fifth-order integer model with eleven real coefficients is a simple configuration that can provide a good approximation of the original fractional-order system dynamics.

One should notice that the coefficient \(a_{5}\) could be simplified with the numerator, but the optimization algorithm was too sensitive to its value, and resulting model was much better in this form. The problem then is to obtain for the model (19) the optimal value of the parameters’ vector \(\theta \in \mathfrak {R}^{11}\) defined as,

$$\begin{aligned} \theta = [a_{5}\; a_{4}\; a_{3}\; a_{2}\; a_{1}\; a_{0}\; b_{4}\; b_{3}\; b_{2}\; b_{1}\; b_{0}]^{T} \end{aligned}$$
(20)

Using the DE method allows us to obtain an adequate solution for the given approximation problem if the fitness function is modeled in an adequate way. In this case, our aim is to estimate the parameters of the approximate model (19). One possibility consists of measuring the error between Bode diagrams of the original high-order fractional model and the reduced model, that is the difference between their magnitudes \(\epsilon _{M}(\omega )\) and their phases \(\epsilon _{P}(\omega )\) at each frequency \(\omega \). These errors are computed via the following equations:

$$\begin{aligned} \epsilon _{M}(\omega )= & {} \left| \left| F(s)\right| _{\mathrm{d}B} - \left| T(s)\right| _{\mathrm{d}B}\right| \nonumber \\ \epsilon _{P}(\omega )= & {} \left| \mathrm{arg}(F(s)) - \mathrm{arg}(T(s))\right| \end{aligned}$$
(21)

where \(s = j \omega \).

Let us define the combined error function f as,

$$\begin{aligned} f(\omega ) = \gamma .\epsilon _{M}(\omega ) + \beta .\epsilon _{P}(\omega ) \end{aligned}$$
(22)

where \(\gamma \in [0, 1]\) and \(\beta = (1-\gamma )/10\).

Then, a performance criterion in the frequency domain J is proposed for evaluating the reduced integer-order model using the integral of squared error (ISE) as follows:

$$\begin{aligned} J(\theta ) = \int ^{\infty }_{0} f(\omega )\; \mathrm{d}\omega \end{aligned}$$
(23)

We can now define a set of constraints for the parameters \(\theta \) as \(U \subset \mathfrak {R}^{11}\), where \(\theta _{i} \in U_{i} \ (i = 1:11)\).

The problem of finding the reduced-order model approximating the fractional-order system (7) can be written as the following optimization problem,

Find \(\theta ^{\star }\in U\) such that,

$$\begin{aligned} J(\theta ^{\star }) = \min _{\begin{array}{c} \theta \in U \end{array}} J(\theta ) \end{aligned}$$
(24)

Application of DE approximation technique to obtain a reduced integer-order model for the original high-order fractional-order model can be performed using the following procedure:

DE approximation procedure

1: Initialize models with random value.

2: For each model evaluate an error between Bode diagram of original high-order fractional-order model and reduced model.

3: Update parameters models.

4: Choose the best model and go to step 2,

5: End the max iteration, giving optimal model.

After running the DE, we obtain the best values, this optimized solution set. In this case, the resulting low integral-order model with these parameters give the same performance as the original fractional-order system.

It is worth to notice that in the context of identification, we suppose that we have no a priori information about the approximating model parameters’ values, the only index of success or failure is the objective function that quantifies the identification precision. Thus, the only constraint for the choice of the model initial parameters concerns the limitation of the search area in order to limit the search procedure. But of course such limits could be more accurate if we have by some way a priori information on these parameters, for example after precedent trials.

6 Simulation Results

In this section, we illustrate the effectiveness of the proposed method, by numerical comparative examples. We propose to use the proposed DE numerical algorithm to achieve a good approximation of fractional transfer functions using a finite dimensional integer-order rational transfer functions of relatively low order. Both Oustaloup’s and Charef’s approximation methods performance are compared to that of obtained results, to show the benefit of the proposed reduced-order modelization strategy.

For these numerical simulations, the setting parameters of DE algorithm are given in Table 1.

Table 1 DE parameters’ setting
Full size table

6.1 DE versus Oustaloup’s Method

Consider a linear fractional-order system whose transfer function is given as Tang et al. (2015):

$$\begin{aligned} F_{1}(s) = \frac{1}{0.8 s^{2.2} + 0.5 s^{0.9} + 1} \end{aligned}$$
(25)

which can be approximated using Oustaloup’s method by the following transfer function,

$$\begin{aligned}&{}_\mathrm{Ous}F^\mathrm{ap}_{1}(s)\nonumber \\&\quad = \frac{s^8 + 1128 s^7 + 2.506 \;10^{5} s^6 + 7.814 \;10^{6} s^5 + 5.322 \; 10^{7} s^4 + 5.223 \; 10^{7} s^3+ 1.119 \; 10^{7} s^2 + 3.368 \; 10^{5} s + 1995}{3.185 s^{10} + 3181 s^9 + 4.354 \; 10^{5} s^8 + 1.115 \; 10^{7} s^7 + 4.972 \; 10^{7} s^6 + 6.811\; 10^{7} s^{5} + 8.711\; 10^{7} s^{4} + 5.927\; 10^{7} s^{3} + 1.147\; 10^{7} s^{2} + 3.388\; 10^{5} s^{1} + 1997}\nonumber \\ \end{aligned}$$
(26)

Searching by DE algorithm for all the optimum parameters enables us to obtain the following fifth-order rational approximating transfer function of the form (19) with the parameters’ vector,

$$\begin{aligned} \theta ^{\star } = \left( \begin{array}{rcl} -\,40&{}.&{}727072552315327 \\ -\,14&{}.&{}983730125040781 \\ -\,2&{}.&{}253893874575709 \\ -\,0&{}.&{}336919088806179 \\ 18&{}.&{}214831716746669 \\ 1&{}.&{}000000000000000 \\ -\,40&{}.&{}739526743048707 \\ 6&{}.&{}318332816151482 \\ 15&{}.&{}992590057945506 \\ 0&{}.&{}477945195746248 \\ -\,0&{}.&{}000994017313606 \\ \end{array}\right) \end{aligned}$$
(27)

A comparative bode response of the fractional-order system (25) and the two approximating functions using Oustaloup (26) and the proposed DE technique (19) with the parameters (27) is given in Fig. 1.

A comparative step response of the three models is also provided in Fig. 2.

Fig. 1
figure 1

Bode diagrams of the true fractional model, Oustaloup’s and DE approximating models

Full size image
Fig. 2
figure 2

Comparative step response of the true fractional model, Oustaloup’s and DE approximating models

Full size image

Whereas, Table 2 gives a quantitative performance comparison of Oustaloup method results and the proposed technique using the ITAE, IAE, ISE and ITSE indexes defined as,

$$\begin{aligned} \nonumber \mathrm{IAE}= & {} \int ^{t_{f}}_{0} \left| e(t)\right| \mathrm{d}t \nonumber \\ \mathrm{ITAE}= & {} \int ^{t_{f}}_{0} t \left| e(t)\right| \mathrm{d}t \nonumber \\ \mathrm{ISE}= & {} \int ^{t_{f}}_{0} e^{2}(t) \mathrm{d}t \nonumber \\ \mathrm{ITSE}= & {} \int ^{t_{f}}_{0} t e^{2}(t) \mathrm{d}t \end{aligned}$$
(28)

where \(e(t) = y_\mathrm{fractional}(t) - y_\mathrm{Oustaloup}\) for Oustaloup approach, and \(e(t) = y_\mathrm{fractional}(t) - y_\mathrm{DE}\) for the proposed approach. y(t) is the step response.

As shown in Figs. 1 and 2, and confirmed by the performance indexes in Table 2 the DE approximation has better performance than that of Oustaloup technique despite the important order reduction proposed in our approach from a model of tenth-order to fifth-order one.

Table 2 Performance comparison between DE and Oustaloup’s approximations
Full size table

6.2 DE versus First-Order Charef’s Method

Consider a linear fractional-order system whose transfer function is given as Charef et al. (1992):

$$\begin{aligned} F_{2}(s) = \frac{1}{(s + 1)^{0.75}} \end{aligned}$$
(29)

which can be approximated using the singularity function method by the following transfer function,

$$\begin{aligned}&{}_\mathrm{Ch}F^\mathrm{ap}_{2}(s) \nonumber \\&\quad = \frac{4.642 \; 10^{2} s^9 +3.552 \; 10^{29} s^8 +6.158 \; 10^{33} s^7 +2.931 \; 10^{37} s^6 + 4.012 \; 10^{40} s^5 + 1.597 \; 10^{43} s^4 + 1.849 \; 10^{45} s^3 + 6.158 \; 10^{46}s^2 + 5.63 \; 10^{47}s + 1.166\; 10^{48}}{2.512 \; 10^{28} s^10 +2.613 \; 10^{28} s^9 +6.158 \; 10^{32} s^8 +3.985 \; 10^{36} s^7 + 7.417 \; 10^{39} s^6 + 4.02 \; 10^{42} s^5 + 6.362 \; 10^{44} s^4 + 2.932\; 10^{46} s^3 + 3.885\; 10^{47} s^2 + 1.414\; 10^{48} s + 1.166\; 10^{48}} \end{aligned}$$
(30)

where the approximation specifications are:

  • Tolerated magnitude error: \(1.5 \; \mathrm{dB}\),

  • frequency domain: \([10^{-2}, 10^{2}]\; \mathrm{rad}/\mathrm{s}\),

  • cutting frequency \(\omega _\mathrm{c}\,=\,1 \; \mathrm{rad}/\mathrm{s}\).

Searching by DE algorithm for the optimum parameters enables us to obtain the following fifth-order rational approximating transfer function of the form (19) with the parameters’ vector,

$$\begin{aligned} \theta ^{\star } = 10^{4} \times \left( \begin{array}{rcl} -\,4&{}.&{}552474479271549 \\ -\,4&{}.&{}517933307085452 \\ 0&{}.&{}315120213223303 \\ 0&{}.&{}425678830186406 \\ 0&{}.&{}020032141470495 \\ 0&{}.&{}000100000000000 \\ -\,4&{}.&{}549367841090670 \\ -\,1&{}.&{}009522336660871 \\ 0&{}.&{}596167689422095 \\ 0&{}.&{}051974053867064 \\ 0&{}.&{}000479601766028 \\ \end{array}\right) \end{aligned}$$
(31)

A comparative bode response of the fractional-order system (29), the two approximating functions using Charef (30) and the proposed DE technique (31) is given in Fig. 3.

A comparative step response of the three models is also provided in Fig. 4.

Fig. 3
figure 3

Bode diagrams of the one fractional-order pole model, Charef’s and DE approximating models

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In fact the performance of the proposed DE approach with model reduction allows a very competitive level of performance when compared to Charef’s method.

It is obvious that the order reduction implies a loss of performance, but the accuracy of the DE technique permits to maintain a good approximation as illustrated in Table 3 where the different performance indexes are lower for the proposed DE technique except for the ITAE index.

6.3 DE versus Second-Order Charef’s Method

Consider a second-order-like linear fractional-order system whose transfer function is given as Ladaci and Charef (2006):

$$\begin{aligned} F_{3}(s) = \frac{1}{(0.01\; s^2 + 0.19\; s + 1)^{0.55}} \end{aligned}$$
(32)

which can be approximated using the singularity function method by the following transfer function,

$$\begin{aligned}&{}_\mathrm{Ch}F^\mathrm{ap}_{3}(s) \nonumber \\&\quad = 4.19\; 10^{38}s^{14} + 6.43\; 10^{42} s^{13}+ 3.697\; 10^{46} s^{12} + 1.04\; 10^{50} s^{11} + 1.577\; 10^{53} s^{10} + 1.351\; 10^{56} s^{9} + 6.691\; 10^{58} s^{8} + 1.938\; 10^{61} s^{7}\nonumber \\&\qquad +\frac{ 3.294\; 10^{63} s^{6} + 3.281\; 10^{65} s^{5}+ 0.897\; 10^{67} s^{4}+ 6.246\; 10^{68} s^{3}+ 1.124\; 10^{70} s^{2} +1.01\; 10^{71} s + 3.503\; 10^{71}}{8.36\; 10^{33} s^{16} + 2.035\; 10^{38} s^{15}+ 1.856\; 10^{42} s^{14} + 8.288\; 10^{45} s^{13}+1.999\; 10^{49} s^{12} + 2.728\; 10^{52} s^{11} + 2.16\; 10^{55} s^{10}+ 1.006\; 10^{58} s^{9}}\nonumber \\&\qquad + 2.775\; 10^{60} s^{8} + 4.562\; 10^{62} s^{7} + 4.477\; 10^{64} s^{6} + 2.624\; 10^{66} s^{5} + 9.158\; 10^{67} s^{4} + 1.883\; 10^{69} s^{3}+2.215\; 10^{70} s^{2} + 1.375\; 10^{71} s + 3.503\; 10^{71} \end{aligned}$$
(33)

where the approximation specifications are:

  • Tolerated magnitude error: \(1 \; \mathrm{dB}\),

  • frequency domain: \([10^{-2}, 10^{2}]\; \mathrm{rad}/\mathrm{s}\),

  • cutting frequency \(\omega _\mathrm{c}\,=\,1 \; \mathrm{rad}/\mathrm{s}\).

Fig. 4
figure 4

Comparative step response of the one fractional-order pole model, Charef’s and DE approximating models

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Table 3 Performance comparison between DE and first-order Charef’s approximations
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By using the DE algorithm to obtain the optimum parameters we get the following fifth-order rational approximating transfer function of the form (19) with the parameters’ vector,

$$\begin{aligned} \theta ^{\star } = 10^{11} \times \left( \begin{array}{rcl} 1&{}.&{}714364329183685 \\ 0&{}.&{}209215147615226 \\ 0&{}.&{}001815278647107 \\ -\,0&{}.&{}000290403658218 \\ -\,0&{}.&{}000002269513543 \\ 1&{}.&{}000000000000000 \\ 1&{}.&{}714150025483689 \\ 0&{}.&{}026879336504952 \\ -\,0&{}.&{}002381130987253 \\ -\,0&{}.&{}000015179332574 \\ 0&{}.&{}0000000000000000 \\ \end{array}\right) \end{aligned}$$
(34)

A comparative bode response of the fractional-order system (29) and the two approximating functions using Charef (33) and the proposed DE technique (34) is given in Fig. 5. A comparative step response of the three models is also provided in Fig. 6.

Table 4 shows a quantification of the performance comparison between DE and Charef’s methods using a second-order fractional-order model. For all the performance indexes the proposed DE-based approximation and model reduction method presents lower error values. This demonstrates again the competitiveness of this technique with classical methods with the advantage of limiting the resulting model order.

6.4 DE versus Charef’s and Oustaloup’s Methods for the Fundamental Fractional System

Now, let us consider the fundamental fractional-order system whose transfer function is given as Charef (2006):

$$\begin{aligned} F_{4}(s) = \frac{1}{ 1 + (0.1\; s)^{1.7}} \end{aligned}$$
(35)

which can be approximated using the singularity function method by the following transfer function,

$$\begin{aligned}&{}_\mathrm{Ch}F^\mathrm{ap}_{4}(s) \nonumber \\&\quad =\frac{6.415 \; 10^{23} s^8 +3.042 \; 10^{28} s^7 +3.611 \; 10^{32} s^6 + 1.321 \; 10^{36} s^5 + 1.568 \; 10^{39} s^4 + 6.121 \; 10^{41} s^3 + 7.759 \; 10^{43}s^2 + 3.029 \; 10^{45}s + 2.961\; 10^{46}}{4.617 \; 10^{20} s^10 +3.042 \; 10^{25} s^9 +5.019 \; 10^{29} s^8 +2.552 \; 10^{33} s^7 + 4.221 \; 10^{36} s^6 + 2.304 \; 10^{39} s^5 + 4.142 \; 10^{41} s^4 + 2.412\; 10^{43} s^3 + 4.497\; 10^{44} s^2 + 3.72\; 10^{45} s + 2.961\; 10^{46}} \end{aligned}$$
(36)
Fig. 5
figure 5

Bode diagrams of the second fractional-order pole model, Charef’s and DE approximating models

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Fig. 6
figure 6

Comparative step response of the second fractional-order pole model, Charef’s and DE approximating models

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We can also approximate the transfer function \(F_{4}(s)\) of the fundamental fractional-order system (35) using Oustaloup’s methods which gives the following approximating function,

$$\begin{aligned}&{}_\mathrm{Ous}F^\mathrm{ap}_{4}(s)\nonumber \\&\quad =\frac{s^9 + 1012 s^8 +1.817 \; 10^{5} s^7 +6.767 \; 10^{6} s^6 + 5.386 \; 10^{7} s^5 + 9.217 \; 10^{7} s^4 + 3.391 \; 10^{7} s^3 + 2.667 \; 10^{6}s^2 + 4.352 \; 10^{4}s + 125.9}{2.505 s^10 + 867.1 s^9 +5.408 \; 10^{4} s^8 +8.566 \; 10^{5} s^7 + 8.601 \; 10^{6} s^6 + 5.493 \; 10^{7} s^5 + 9.231 \; 10^{7} s^4 + 3.392\; 10^{7} s^3 + 2.667\; 10^{6} s^2 + 4.352\; 10^{4} s + 125.9} \end{aligned}$$
(37)

And by using the proposed DE algorithm to obtain the optimum parameters we get a fifth-order rational transfer function of the form (19) with the parameters’ vector,

$$\begin{aligned} \theta ^{\star } = 10^{5} \times \left( \begin{array}{rcl} -\,2&{}.&{}407267478563445 \\ -\,0&{}.&{}008094044418957 \\ 0&{}.&{}055339212934548 \\ 0&{}.&{}006422861018111 \\ 0&{}.&{}000777097319824 \\ 0&{}.&{}000010000000000 \\ -\,2&{}.&{}408044991467207 \\ 0&{}.&{}003841695911245 \\ 0&{}.&{}090850746027396 \\ 0&{}.&{}002372343998927 \\ 0&{}.&{}000003900392555 \\ \end{array}\right) \end{aligned}$$
(38)

A comparative bode response of the fractional-order system (35) and the three approximating functions using Charef’s singularity function technique (36), Oustaloup’s technique (37) and the proposed DE method (38) is given in Fig. 7. Figure 8 gives also a comparative step response of the four models.

Table 4 Performance comparison between DE and second-order Charef’s approximations
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Table 5 shows a quantification of the performance comparison between the proposed DE-based approach, Charef’s singularity function and Oustaloup’s methods for the fundamental linear fractional-order system. For all the performance indexes the proposed DE-based approximation and model reduction method presents lower error values. This demonstrates again the competitiveness of this technique versus classical methods with the advantage of limiting the resulting integer model order to five.

7 Conclusions

In this paper a new method for approximation of the non-integer system is presented. It uses fifth-order rational models with eleven parameters to represent large-order fractional systems based on the Differential Evolution algorithm.

Fig. 7
figure 7

Bode diagrams of the fundamental fractional-order model, Charef’s, Oustaloup’s and DE approximating models

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Fig. 8
figure 8

Comparative step response of the fundamental fractional-order model, Charef’s, Oustaloup’s and DE approximating models

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Four numerical examples illustrate the efficiency of the proposed reduced-order approximation algorithm. The results obtained from the DE approach were compared with those of Oustaloup and Charef approximation techniques for fractional-order transfer functions. They showed clearly that the proposed approach provides a very competitive level of performance with a reduced model order and much less parameters.

This proposed technique will have an important positive impact in fractional systems engineering, for control, identification and simulation purposes.

Table 5 Performance comparison between DE and second-order Charef’s approximations
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Further research will concern the enhancement of DE method in order to perform the procedure of optimum seeking in a deterministic way, particularly in the context of fractional adaptive control.