Rational Function Approximation of a Fundamental Fractional Order Transfer Function

Abstract

This paper introduces a rational function approximation of the fractional order transfer function $H(s) = \frac{{(\tau_{0} s)^{\alpha } }}{{[1 + (\tau_{0} s)^{2\alpha } ]}},\quad for\,\,0\, < \,\alpha \, \le \,0.5$. This fractional order transfer function is one of the fundamental functions of the linear fractional system of commensurate order corresponding to pure complex conjugate poles or eigenvalues, in s^α. Hence, the proposed approximation will be used in the solution of the linear fractional systems of commensurate order. Illustrative examples are given to show the exactitude and the efficiency of the approximation method.

1 Introduction

The theory of fractional order systems has gained some importance during the last decades (Miller et al. 1993), (Podlubny 1999), (Kilbas et al. 2006), (Monje et al. 2010), (Caponetto et al. 2010). Therefore, active research work to find accurate and efficient methods to solve linear fractional order differential equations is still underway to establish a clear linear fractional order system theory accessible to the general engineering community. More recently, a great deal of effort has been expended in the development of analytical techniques to solve them. The goal of these methods is to derive an explicit analytical expression for the general solution of the linear fractional differential equations (Charef 2006a), (Bonilla et al. 2007), (Oturanç et al. 2008), (Hu et al. 2008), (Arikoglu et al. 2009), (Odibat 2010), (Charef et al. 2011).

A linear single input single output (SISO) fractional system of commensurate order is described by the following linear fractional order differential equation:

$$ \sum\limits_{i = 0}^{N} {a_{i} D^{i\alpha } y(t) = \sum\limits_{j = 0}^{M} {b_{j} } } D^{j\alpha } u(t) $$

(1)

where u(t) is the input, y(t) is the output, α is a real number such that 0 < α < 1, a _i (1 ≤ i ≤ N) and b _i (0 ≤ j ≤ M) are constant real numbers with M ≤ N. With zero initial conditions, the fractional order system transfer function is given as:

$$ G(s) = \frac{Y(s)}{U(s)} = \frac{{\sum\limits_{j = 0}^{M} {b_{j} \left( {s^{\alpha } } \right)^{j} } }}{{\sum\limits_{i = 0}^{N} {a_{i} \left( {s^{\alpha } } \right)^{i} } }} $$

(2)

This fractional transfer function can be decomposed into several elementary fundamental functions corresponding to different types of poles, in s ^α, as:

$$ G(s) = \sum\limits_{k = 1}^{K} {H_{k} (s)} $$

(3)

where the functions H _k (s) are given, according to the poles of the fractional system, as:

• For a simple real pole:

$$ H_{k} (s) = \frac{1}{{\left( {s^{\alpha } + p} \right)}} $$

(4)

• For a pair of complex poles with negative real part:

$$ H_{k} (s) = \frac{{\omega_{n}^{2} }}{{[s^{2\alpha } + 2\zeta \omega_{n} s^{\alpha } + \omega_{n}^{2} ]}} $$

(5)

$$ H_{k} (s) = \frac{{s^{\alpha } + \omega_{n} \zeta }}{{[s^{2\alpha } + 2\zeta \omega_{n} s^{\alpha } + \omega_{n}^{2} ]}} $$

(6)

• For a pair of complex poles with null real part:

$$ H_{k} (s) = \frac{{\omega_{n}^{2} }}{{[s^{2\alpha } + \omega_{n}^{2} ]}} $$

(7)

$$ H_{k} (s) = \frac{{\omega_{n} s^{\alpha } }}{{[s^{2\alpha } + \omega_{n}^{2} ]}} $$

(8)

In previous works (Charef 2006a) and (Charef et al. 2011), (Nezzari et al. 2011), (Boucherma et al. 2011), the elementary fundamental functions defined in (4), (5), (6) and (7) have been approximated by rational ones in order to represent them by linear time-invariant system models so as to derive their closed form impulse and step responses as well as their performance characteristics. Using these approximations, simple analog circuits have been also derived to represent the above irrational functions of the fractional order system. This paper gives a rational function approximation of the fundamental function represented by the irrational transfer function of (8) which corresponds to pure complex conjugate poles or eigenvalues, in s ^α. In (Boucherma et al. 2011), the approximation of (8) has been done for ${\kern 1pt} 0.5 < \alpha < 1$. In this work the approximation of (8) will be done for ${\kern 1pt} 0 < \alpha \le 0.5$. First, the basic ideas and the derived formulations of the approximation technique are presented. Then, the impulse and step responses of this type of fractional system are derived. Finally, illustrative examples are presented to show the exactitude and the usefulness of the approximation method.

2 Rational Function Approximation

For $\left( {\tau_{0} } \right)^{\alpha } = \frac{1}{{\omega_{n} }}$, (8) can be rewritten as:

$$ H(s) = \frac{X(s)}{E(s)} = \frac{{(\tau_{0} s)^{\alpha } }}{{[1 + (\tau_{0} s)^{2\alpha } ]}}{\kern 1pt} ,\;0 < \alpha \le 0.5 $$

(9)

The above irrational function is the transfer function of the linear fractional order system represented by the following fundamental linear fractional order differential equation:

$$ \left( {\tau_{0} } \right)^{2\alpha } \frac{{d^{2\alpha } x\left( t \right)}}{{dt^{2\alpha } }} + x\left( t \right) = \left( {\tau_{0} } \right)^{\alpha } \frac{{d^{\alpha } e\left( t \right)}}{{dt^{\alpha } }} $$

(10)

In this context, the transfer function of (9) has two pure complex conjugate poles, in s^α. To represent the linear fractional order system of (10) by a linear time-invariant system model so as to derive their closed form impulse and step responses, its irrational transfer function of (9) will be approximated by a rational function. To do so, we will consider two cases based on the fractional derivative α.

2.1 Case 1: 0 < α < 0.5

For this case, the function of (9) can be decomposed in two functions as follows:

$$ H\left( s \right) = H_{1} \left( s \right) \times H_{2} \left( s \right) = \left( {\tau_{0} s} \right)^{\alpha } \times \frac{1}{{1 + \left( {\tau_{0} s} \right)^{2\alpha } }} $$

(11)

where $H_{1} \left( s \right) = \left( {\tau_{0} s} \right)^{\alpha }$ and $H_{2} \left( s \right) = \frac{1}{{1 + \left( {\tau_{0} s} \right)^{2\alpha } }}$.

In a given frequency band of interest [ω_L, ω_H], around the frequency ω₀ = (1/τ₀), the fractional order differentiator $H_{1} \left( s \right) = \left( {\tau_{0} s} \right)^{\alpha }$ can be approximated by a rational function as follows (b, 2006):

$$ H_{1} \left( s \right) = \left( {\tau_{0} s} \right)^{\alpha } \cong \tau_{0}^{\alpha } \left[ {K_{D} \frac{{\prod\limits_{i = 0}^{{N_{1} }} {\left( {1 + \frac{s}{{z_{i} }}} \right)} }}{{\prod\limits_{i = 0}^{{N_{1} }} {\left( {1 + \frac{s}{{p_{i} }}} \right)} }}} \right] $$

(12)

where the poles pi and the zeros z_i (0 ≤ i ≤ N1), the constant K_D and the number N1 of the approximation are given by:

$$ \begin{aligned} {\text{p}}_{\text{i}} {\text{ = p}}_{ 0} {\text{ (ab)}}^{\text{i}} ,{\kern 1pt} {\text{z}}_{\text{i}} {\text{ = z}}_{ 0} {\text{ (ab)}}^{\text{i}} ,K_{D} = (\omega_{c} )^{\alpha } ,{\kern 1pt} \hfill \\ N_{1} = \left\{ {Integer\left[ {\frac{{\log \left( {{{\omega_{\hbox{max} } } \mathord{\left/ {\vphantom {{\omega_{\hbox{max} } } {z_{0} }}} \right. \kern-0pt} {z_{0} }}} \right)}}{{\log \left( {ab} \right)}}} \right] + 1} \right\} \hfill \\ \end{aligned} $$

(13)

For some given real values y (dB), δ and β, the approximation parameters a, b, p₀, z₀, ω_c and ω_max can be calculated as:

$$ \begin{aligned} {\text{a}} = 1 0^{{\left[ {\frac{\text{y}}{{ 1 0\left( { 1- \alpha } \right)}}} \right]}} , {\text{b}} = 1 0^{{\left[ {\frac{\text{y}}{ 1 0\alpha }} \right]}} ,\omega_{\text{c}} = \delta \omega_{L} , \hfill \\ \omega_{ \hbox{max} } = \beta \omega_{H} {\kern 1pt} ,\;{\text{z}}_{ 0} =\upomega_{\text{c}} \sqrt {\text{b}} \;and\;{\text{p}}_{ 0} = a{\text{z}}_{ 0} \hfill \\ \end{aligned} $$

(14)

By the decomposition of the rational function of (12), we will get:

$$ H_{1} \left( s \right) = \left( {\tau_{0} s} \right)^{\alpha } \cong \tau_{0}^{\alpha } \left( {K_{D} + \sum\limits_{i = 0}^{{N_{1} }} {\frac{{k_{i} s}}{{\left( {1 + \frac{s}{{p_{i} }}} \right)}}} } \right) $$

(15)

$$ k_{i} = - \frac{{K_{D} }}{{p_{0} (ab)^{i} }}\frac{{\prod\limits_{j = 0}^{{N_{1} }} {(1 - a(ab)^{{\left( {i - j} \right)}} )} }}{{\prod\limits_{j = 0,i \ne j}^{{N_{1} }} {(1 - (ab)^{{\left( {i - j} \right)}} )} }} , {\text{i}} = 0, 1 ,\ldots ,{\text{N1}} $$

(16)

Because 0 < α < 0.5 the number 2α is then 0 < 2α < 1; hence, in a given frequency band [0, ω_H], the fractional system $H_{2} \left( s \right) = \frac{1}{{1 + \left( {\tau_{0} s} \right)^{2\alpha } }}$ can be approximated by a rational function as follows (Charef 2006a):

$$ H_{2} (s) = \frac{1}{{1 + (\tau_{0} s)^{2\alpha } }} \cong \sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{kk_{j} }}{{\left( {1 + \frac{s}{{pp_{j} }}} \right)}}} $$

(17)

where the poles pp_j and the residues kk_j (for 1 ≤ j ≤ 2N₂−1), and the number N₂ of the approximation are given, for some given real values λ and β, by:

$$ \begin{aligned} pp_{j} & = \frac{{(\lambda )^{(j - N)} }}{{\tau_{0} }}{\kern 1pt} \\ kk_{j} & = \frac{1}{2\pi }\left[ {\frac{\sin [(1 - \alpha )\pi ]}{{\cosh [\alpha \,\log (\frac{1}{{\tau_{0} pp_{j} }})] - \cos [(1 - \alpha )\pi ]}}} \right]{\kern 1pt} {\kern 1pt} \\ N_{2} & = Integer\left[ {\frac{{\log [\tau_{0} \beta \omega_{H} ]}}{\log \left( \lambda \right)}} \right] + {\kern 1pt} 1 \\ \end{aligned} $$

(18)

Therefore, the function of (11) is approximated by a rational function as follows:

$$ H\left( s \right) \cong \tau_{0}^{\alpha } \left( {K_{D} + \sum\limits_{i = 0}^{{N_{1} }} {\frac{{k_{i} s}}{{\left( {1 + \frac{s}{{p_{i} }}} \right)}}} } \right)\left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{kk_{j} }}{{\left( {1 + \frac{s}{{pp_{j} }}} \right)}}} } \right) $$

(19)

$$ \begin{aligned} H\left( s \right) & \cong \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\left( {\tau_{0}^{\alpha } K_{D} } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {pp_{j} kk_{j} } \right)}}{{\left( {s + pp_{j} } \right)}}} } \right) \\ & \quad + \left( {\sum\limits_{i = 0}^{{N_{1} }} {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\left( {\tau_{0}^{\alpha } } \right)\left( {p_{i} k_{i} } \right)\left( {pp_{j} kk_{j} } \right)s}}{{\left( {s + p_{i} } \right)\left( {s + pp_{j} } \right)}}} } } \right) \\ \end{aligned} $$

(20)

By the decomposition of the rational function of (20), we will get:

$$ \begin{aligned} H\left( s \right) & \cong \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\left( {\tau_{0}^{\alpha } K_{D} } \right)\left( {pp_{j} kk_{j} } \right)}}{{\left( {s + pp_{j} } \right)}}} } \right) \\ & \quad + \left( {\sum\limits_{i = 0}^{{N_{1} }} {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{A_{ij} }}{{\left( {s + p_{i} } \right)}} + \frac{{B_{ij} }}{{\left( {s + pp_{j} } \right)}}} } } \right) \\ \end{aligned} $$

(21)

where the residues A_ij and B_ij (for 0 ≤ i ≤ N₁ and 1 ≤ j ≤ 2N₂−1) are given by:

$$ A_{ij} = \frac{{\left( {\tau_{0}^{\alpha } } \right)\left( {p_{{_{i} }}^{2} k_{i} } \right)\left( {pp_{j} kk_{j} } \right)}}{{p_{i} - pp_{j} }},\quad B_{ij} = \frac{{\left( {\tau_{0}^{\alpha } } \right)\left( {p_{i} k_{i} } \right)\left( {pp_{{_{j} }}^{2} kk_{j} } \right)}}{{pp_{j} - p_{i} }} $$

(22)

Hence, we can write that:

$$ \begin{aligned} H\left( s \right) & \cong \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\left( {\tau_{0}^{\alpha } K_{D} } \right)\left( {pp_{j} kk_{j} } \right)}}{{\left( {s + pp_{j} } \right)}}} } \right) \\ & \quad + {\kern 1pt} \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\sum\limits_{i = 0}^{{N_{1} }} {B_{ij} } }}{{{\kern 1pt} \left( {s + pp_{j} } \right)}}} } \right) + \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\sum\limits_{j = 1}^{{2N_{2} - 1}} {A_{ij} } }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}} } \right){\kern 1pt} \\ \end{aligned} $$

(23)

$$ H\left( s \right) \cong \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{A}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}} } \right) + {\kern 1pt} \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\overline{B}_{j} }}{{\left( {s + pp_{j} } \right)}}} } \right) $$

(24)

where the residues $\overline{A}_{i}$ (0 ≤ i ≤ N₁), and $\overline{B}_{j}$ (1 ≤ j ≤ 2N₂−1), are given by:

$$ \overline{A}_{i} = \sum\limits_{j = 1}^{{2N_{2} - 1}} {A_{ij} } ,\quad \overline{B}_{j} = \left[ {\left( {\tau_{0}^{\alpha } K_{D} } \right)\left( {pp_{j} kk_{j} } \right)} \right] + \sum\limits_{i = 0}^{{N_{1} }} {B_{ij} } $$

(25)

2.2 Case 2: α = 0.5

For this case, the function of (9) can be rewritten as:

$$ H\left( s \right) = \frac{{\left( {\tau_{0} s} \right)^{0.5} }}{{1 + \left( {\tau_{0} s} \right)}} $$

(26)

From Eq. (15), the above function is approximated by a rational function as follows:

$$ H\left( s \right) \cong \tau_{0}^{0.5} \left( {K_{D} + \sum\limits_{i = 0}^{{N_{1} }} {\frac{{k_{i} s}}{{\left( {1 + \frac{s}{{p_{i} }}} \right)}}} } \right)\left( {\frac{1}{{1 + \left( {\tau_{0} s} \right)}}} \right) $$

(27)

$$ H\left( s \right) \cong \left( {\frac{{\tau_{0}^{( - 0.5)} K_{D} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right) + \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\left( {\tau_{0}^{{\left( { - 0.5} \right)}} } \right)\left( {p_{i} k_{i} } \right)s}}{{\left( {s + p_{i} } \right)\left( {s + 1/\tau_{0} } \right)}}} } \right) $$

(28)

By the decomposition of the rational function of (28), we will get:

$$ H\left( s \right) \cong \left( {\frac{{\tau_{0}^{( - 0.5)} K_{D} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right) + \left( {\sum\limits_{i = 0}^{{N_{1} }} {\left( {\frac{{C_{i} }}{{\left( {s + p_{i} } \right)}} + \frac{{D_{i} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right)} } \right) $$

(29)

where the residues C _i and D _i (for 0 ≤ i ≤ N ₁) are given by:

$$ C_{i} = \frac{{\left( {\tau_{0}^{{\left( { - 0.5} \right)}} } \right)\left( {p_{i}^{2} k_{i} } \right)}}{{p_{i} - 1/\tau_{0} }},\quad D_{i} = \frac{{\left( {\tau_{0}^{( - 1.5)} } \right)\left( {p_{i} k_{i} } \right)}}{{1/\tau_{0} - p_{i} }} $$

(30)

Hence, we can write that:

$$ H\left( s \right) \cong \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{C}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}} } \right) + \left( {\frac{{{\kern 1pt} {\kern 1pt} \overline{D} {\kern 1pt} {\kern 1pt} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right){\kern 1pt} $$

(31)

with the residues $\overline{C}_{i} = C_{i}$ (for 0 ≤ i ≤ N ₁) and $\overline{{D{\kern 1pt} }} = \left[ {\tau_{0}^{( - 0.5)} K_{D} } \right] + \left[ {\sum\limits_{i = 0}^{{N_{1} }} {D_{i} } } \right]$.

3 Time Responses

3.1 Case 1: 0 < α < 0.5

From Eq. (24), we have that:

$$ H\left( s \right) = \frac{X(s)}{E(s)} = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{A}_{i} }}{{\left( {s + p_{i} } \right)}}} } \right){\kern 1pt} + \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\overline{B}_{j} }}{{\left( {s + pp_{j} } \right)}}} } \right){\kern 1pt} $$

(32)

then,

$$ X(s) = \left[ {\left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{A}_{i} }}{{\left( {s + p_{i} } \right)}}} } \right) + \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\overline{B}_{j} }}{{\left( {s + pp_{j} } \right)}}} } \right)} \right]E(s) $$

(33)

For e(t) = δ(t) the unit impulse E(s) = 1, we will have:

$$ X(s) = \left[ {\left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{A}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}} } \right) + \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\overline{B}_{j} }}{{{\kern 1pt} \left( {s + pp_{j} } \right)}}} } \right)} \right] $$

(34)

Hence, the impulse response of (10) is:

$$ \begin{aligned} x\left( t \right) = L^{ - 1} \left\{ {X\left( s \right)} \right\} & = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\overline{A}_{i} \exp ( - p_{i} t)} } \right) \\ & \quad + \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\overline{B}_{j} \exp ( - pp_{j} t} )} \right){\kern 1pt} \\ \end{aligned} $$

(35)

For e(t) = u(t) the unit step E(s) = 1/s, (33) will be:

$$ X(s) = \left[ {\left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{A}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}} } \right) + \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\overline{B}_{j} }}{{{\kern 1pt} \left( {s + pp_{j} } \right)}}} } \right)} \right]\left( {\frac{1}{s}} \right) $$

(36)

$$ X(s) = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{A}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}\left( {\frac{1}{s}} \right)} } \right) + \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\overline{B}_{j} }}{{{\kern 1pt} \left( {s + pp_{j} } \right)}}\left( {\frac{1}{s}} \right)} } \right) $$

(37)

$$ \begin{aligned} X(s) & = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\left( {\frac{{\overline{A}_{i} }}{{p_{i} }}} \right)\left( {\frac{1}{s} - \frac{1}{{\left( {s + p_{i} } \right)}}} \right)} } \right) \\ & \quad + {\kern 1pt} \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\left( {\frac{{\overline{B}_{i} }}{{pp_{j} }}} \right){\kern 1pt} {\kern 1pt} \left( {\frac{1}{s} - \frac{1}{{\left( {s + pp_{j} } \right)}}} \right)} } \right) \\ \end{aligned} $$

(38)

Hence, the step response of (10) is:

$$ \begin{aligned} x\left( t \right) = L^{ - 1} \left\{ {X(s)} \right\} & = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\left( {\frac{{\overline{A}_{i} }}{{p_{i} }}} \right)\left[ {1 - {\kern 1pt} \exp ( - p_{i} t)} \right]} } \right) \\ & \quad + \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\left( {\frac{{\overline{B}_{j} }}{{pp_{j} }}} \right)\left[ {1 - \exp ( - pp_{j} t)} \right]} } \right) \\ \end{aligned} $$

(39)

3.2 Case 2: α = 0.5

From Eq. (31), we have that:

$$ H\left( s \right) = \frac{X(s)}{E(s)} = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{C}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}} } \right) + \left( {\frac{{\overline{D} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right) $$

(40)

then,

$$ X(s) = \left[ {\left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{C}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}} } \right) + \left( {\frac{{\overline{D} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right)} \right]E(s) $$

(41)

For e(t) = δ(t) the unit impulse E(s) = 1, we will have:

$$ X(s) = \left[ {\left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{C}_{i} }}{{\left( {s + p_{i} } \right)}}} } \right) + \left( {\frac{{\overline{D} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right)} \right]{\kern 1pt} $$

(42)

Hence, the impulse response of (10), for α = 0.5, is:

$$ \begin{aligned} x\left( t \right) = L^{ - 1} \left\{ {X\left( s \right)} \right\} & = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\overline{C}_{i} \exp ( - p_{i} t)} } \right) \\ & \quad + \left( {\overline{D} \exp ( - t/\tau_{0} )} \right) \\ \end{aligned} $$

(43)

For e(t) = u(t) the unit step E(s) = 1/s, (41) will be:

$$ X(s) = \left[ {\left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{C}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}} } \right) + \left( {\frac{{\overline{D} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right)} \right]\left( {\frac{1}{s}} \right) $$

(44)

$$ X(s) = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{C}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}\left( {\frac{1}{s}} \right)} } \right) + \left( {\left( {\frac{{{\kern 1pt} \overline{D} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right)\left( {\frac{1}{s}} \right)} \right) $$

(45)

$$ \begin{aligned} X(s) & = {\kern 1pt} {\kern 1pt} \left( {\sum\limits_{i = 0}^{{N_{1} }} {\left( {\frac{{\overline{C}_{i} }}{{p_{i} }}} \right)\left( {\frac{1}{s} - \frac{1}{{\left( {s + p_{i} } \right)}}} \right)} } \right) \\ & \quad + {\kern 1pt} \left( {\overline{D} \tau_{0} \left( {\frac{1}{s} - \frac{1}{{\left( {s + 1/\tau_{0} } \right)}}} \right)} \right) \\ \end{aligned} $$

(46)

Hence, the step response of f (10), for α = 0.5, is:

$$ \begin{aligned} x\left( t \right) = L^{ - 1} \left\{ {X(s)} \right\} & = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\left( {\frac{{\overline{C}_{i} }}{{p_{i} }}} \right)\left[ {1 - \exp ( - p_{i} t)} \right]} } \right) \\ & \quad + \left( {\overline{D} {\kern 1pt} {\kern 1pt} \tau_{0} \left[ {1 - \exp ( - t/\tau_{0} )} \right]} \right){\kern 1pt} \\ \end{aligned} $$

(47)

4 Illustrative Example

Let us first consider the fractional system represented by the following fundamental linear fractional order differential equation with α = 0.35 and τ ₀ = 2 as:

$$ \left( 2 \right)^{0.7} \frac{{d^{0.7} x\left( t \right)}}{{dt^{0.7} }} + x\left( t \right) = \left( 2 \right)^{0.35} \frac{{d^{0.35} e\left( t \right)}}{{dt^{0.35} }} $$

(48)

its transfer function is given by:

$$ H\left( s \right) = \frac{{\left( {2s} \right)^{0.35} }}{{1 + \left( {2s} \right)^{0.7} }} $$

(49)

Its rational function approximation, in a given frequency band, is given as:

$$ H\left( s \right) = \frac{{\left( {2s} \right)^{0.35} }}{{1 + \left( {2s} \right)^{0.7} }} = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{A}_{i} }}{{\left( {s + p_{i} } \right)}}} } \right) + \left( {\sum\limits_{j = 1}^{{2N_{2} - 1}} {\frac{{\overline{B}_{j} }}{{\left( {s + pp_{j} } \right)}}} } \right) $$

(50)

For the fractional order differentiator (2s)^0.35, the frequency band of approximation is [ω _L , ω _H ] = [10⁻⁴ rad/s, 10⁴ rad/s], around ω ₀ = (1/τ ₀) = 0.5 rad/s, y = 1 dB, δ = 0.1, and β = 100. For the fractional system $\frac{1}{{1 + \left( {2s} \right)^{0.7} }}$, the frequency band of approximation [0, ω _H ] = [0, 10⁴ rad/s], λ = 4 and β = 100. Then, the approximation parameters of H(s) are:

$$ \begin{aligned} & a = 1. 4 2 5 1 ,\;b = 1. 9 3 0 7,\;\omega_{c} = 10^{ - 5} ,\;\omega_{max} = 10^{6} , \\ & p_{ 0} = 1. 9 8 0 2 { *10}^{ - 5} {\kern 1pt} ,\;K_{D} = 0. 0 1 7 8,\;N_{1} = 25,\;and\;N_{2} = 11 \\ \end{aligned} $$

Hence, poles $p_{i}$, the residues $\overline{A}_{i}$ (0 ≤ i ≤ 25), the poles $pp_{j}$ and the residues $\overline{B}_{j}$ (1 ≤ j ≤ 20), are given by:

$$ \begin{aligned} p_{i} & = \left( { 1. 9 8 0 2 * 1 0^{ - 5} } \right){ (2} . 7 5 1 4 )^{i} \\ \overline{A}_{i} & = \sum\limits_{j = 1}^{21} \begin{aligned} \frac{{\left[ \begin{aligned} \left( { 1. 1 2 3 0{\kern 1pt} * 1 0^{ - 7} } \right)\left( { ( 2. 7 5 1 4 )^{i} (4)^{(j - 11)} } \right) \hfill \\ *\frac{{\prod\limits_{j = 0}^{25} {\left[ {1 - \left( {( 1. 4 2 5 1)( 2. 7 5 1 4)^{(i - j)} } \right)} \right]} }}{{\prod\limits_{j = 0,i \ne j}^{25} {(1 - ( 2. 7 5 1 4)^{(i - j)} )} }} \hfill \\ \end{aligned} \right]}}{{\left[ {\pi \left[ {\left( {0.5} \right)(4)^{(j - 11)} - \left( { 1. 9 8 0 2 * 1 0^{ - 5} } \right){ (2} . 7 5 1 4 )^{i} } \right]} \right]}} \hfill \\ *\left[ {\frac{\sin [(0.3)\pi ]}{{\cosh [\left( {0.35} \right)\log ((4)^{(11 - j)} )] - \cos [(0.3)\pi ]}}} \right]{\kern 1pt} \hfill \\ \end{aligned} \\ \end{aligned} $$

(51)

$$ \begin{aligned} pp_{j} & = \left( {0.5} \right)(4)^{(j - 11)} \\ \overline{B}_{j} & = \frac{{\left( { 0. 0 0 5 7} \right)}}{\pi }(4)^{(j - 11)} \left[ {\frac{\sin [(0.3)\pi ]}{{\cosh [\left( {0.35} \right)\log ((4)^{(11 - j)} )] - \cos [(0.3)\pi ]}}} \right]{\kern 1pt} \\ & \quad + \sum\limits_{i = 0}^{25} {\left\{ \begin{aligned} \frac{{{ - }\left( { 0. 0 0 2 9} \right)(4)^{2(j - 11)} \frac{{\prod\limits_{j = 0}^{25} {\left[ {1 - ( 1. 4 2 5 1)( 2. 7 5 1 4)^{(i - j)} } \right]} }}{{\prod\limits_{j = 0,\,i \ne j}^{25} {(1 - ( 2. 7 5 1 4)^{(i - j)} )} }}}}{{\pi \left[ {\left( {0.5} \right)(4)^{(j - 11)} - \left( { 1. 9 8 0 2 * 1 0^{ - 5} } \right){ (2} . 7 5 1 4 )^{i} } \right]}} \hfill \\ \times \left[ {\frac{\sin [(0.3)\pi ]}{{\cosh [\left( {0.35} \right)\log ((4)^{(11 - j)} )] - \cos [(0.3)\pi ]}}} \right] \hfill \\ \end{aligned} \right\}} \\ \end{aligned} $$

(52)

Figures 1 and 2 show the bode plots of the fundamental linear fractional order system transfer function of (49) and of its proposed rational function approximation of (50). We can easily see that they are all quite overlapping in the frequency band of interest.

Fig. 1.

Magnitude bode plots of (49) and of its proposed approximation.

Fig. 2.

Phase bode plots of (49) and of its proposed approximation

Figures 3 and 4 show, respectively, the impulse and the step responses of the fractional order system of (48).

Fig. 3.

Impulse response of (48) from its proposed approximation.

Fig. 4.

Step response of (48) from its proposed approximation.

As a second example, we will consider the fractional order system represented by the following fundamental linear fractional order differential equation with α = 0.5 and τ₀ = 0.16 as:

$$ \left( {0.16} \right)\frac{d\,x\left( t \right)}{dt} + x\left( t \right) = \left( {0.16} \right)^{0.5} \frac{{d^{0.5} e\left( t \right)}}{{dt^{0.5} }} $$

(53)

its transfer function is given by:

$$ H\left( s \right) = \frac{{\left( {0.16\,s} \right)^{0.5} }}{{1 + \left( {0.16\,s} \right)}} $$

(54)

Its rational function approximation, in a given frequency band, is given as:

$$ H\left( s \right) = \frac{{\left( {0.16\,s} \right)^{0.5} }}{{1 + \left( {0.16\,s} \right)}} = \left( {\sum\limits_{i = 0}^{{N_{1} }} {\frac{{\overline{C}_{i} }}{{{\kern 1pt} \left( {s + p_{i} } \right)}}} } \right) + \left( {\frac{{\overline{D} }}{{\left( {s + 1/\tau_{0} } \right)}}} \right){\kern 1pt} $$

(55)

For the fractional differentiator (0.16 s)^0.5, the frequency band of approximation is [ω _L , ω _H ] = [10⁻⁴ rad/s, 10⁴ rad/s], around ω ₀ = (1/τ ₀) = 6.25 rad/s, y = 1 dB, δ = 0.1, and β = 100. Then, the approximation parameters of H(s) are:

$$ \begin{aligned} & a = 1. 5 8 4 9 ,\;b = 1. 5 8 4 9,\;\omega_{c} = 1 0^{ - 5} ,\;\omega_{max} = 1 0^{ 6} , \\ & p_{ 0} = 1. 9 9 5 3 * 1 0^{ - 5} ,\,K_{D} = 0. 0 0 3 2\;and\;N_{1} = 28 \\ \end{aligned} $$

Hence, poles $p_{i}$, the residues $\overline{C}_{i}$ (0 ≤ i ≤ 25), and the residue $\overline{D}$, are given by:

$$ \begin{aligned} p_{i} & = \left( { 1. 9 9 5 3 * 1 0^{ - 5} } \right)\left( { 2. 5 1 1 9} \right)^{i} \\ \overline{C}_{i} & = \left( {\frac{{ - \left( { 1. 5 9 6 3 * 1 0^{ - 7} } \right)\left( { 2. 5 1 1 9} \right)^{i} }}{{\left[ {\left( { 1. 9 9 5 3 * 1 0^{ - 5} } \right)\left( { 2. 5 1 1 9} \right)^{i} - 6.25} \right]}}} \right) \\ & \quad *\;\left( {\frac{{\prod\limits_{j = 0}^{28} {\left( {1 - ( 1. 5 8 4 9)( 2. 5 1 1 9)^{(i - j)} } \right)} }}{{\prod\limits_{j = 0,\,i \ne j}^{28} {\left( {1 - ( 2. 5 1 1 9)^{(i - j)} } \right)} }}} \right) \\ \end{aligned} $$

(56)

$$ \begin{aligned} \overline{D} &= \left( { 0. 0 0 8} \right) \\ &\quad+ \sum\limits_{i = 0}^{28} {\left[ \begin{aligned} \left( {\frac{{ - \left( { 0. 0 5} \right)}}{{\left[ {6.25 - \left( { 1. 9 9 5 3 * 1 0^{ - 5} } \right)\left( { 2. 5 1 1 9} \right)^{i} } \right]}}} \right) \hfill \\ *\;\left( {\frac{{\prod\limits_{j = 0}^{28} {\left( {1 - ( 1. 5 8 4 9)( 2. 5 1 1 9)^{(i - j)} } \right)} }}{{\prod\limits_{j = 0,\,i \ne j}^{28} {\left( {1 - ( 2. 5 1 1 9)^{(i - j)} } \right)} }}} \right) \hfill \\ \end{aligned} \right]} \\ \end{aligned} $$

(57)

Figures 5 and 6 show the bode plots of the fundamental linear fractional system transfer function of (54) and of its proposed rational function approximation of (55). We note that they are overlapping in the frequency band of interest.

Fig. 5.

Magnitude bode plot of (54) and of its proposed approximation.

Fig. 6.

Phase bode plot of (54) and of its proposed approximation.

Figures 7 and 8 show, respectively, the impulse and the step responses of the fractional order system of (53).

Fig. 7.

Impulse response of (53) from its proposed approximation.

Fig. 8.

Step response of (53) from its proposed approximation.

5 Conclusion

In this paper, we have presented a rational function approximation of the fractional order transfer function $H(s) = \frac{{(\tau_{0} s)^{\alpha } }}{{[1 + (\tau_{0} s)^{2\alpha } ]}},\quad for\;0 < \alpha \le 0.5$. This fractional order transfer function is one of the fundamental functions of the linear fractional system of commensurate order, represented by the linear fractional state-space $D^{\alpha } x\left( t \right) = A\,x\left( t \right)$. H(s) corresponds to pure complex conjugate eigenvalues of the A matrix. First, closed form of the approximation technique has been derived. Then, the impulse and step responses of this type of fractional system have been obtained. Finally, illustrative examples are presented to show the exactitude and the usefulness of the approximation method.