Introduction

1  Introduction

The concept of the differentiation operator $\mathscr{D}=\dif/\dif x$ is a well-known fundamental tool of modern calculus. For a suitable function $f$ the $n$-th derivative is well defined as $\mathscr{D}^n f(x)=\dif f(x)/\dif x^n$, where $n$ is a positive integer. However, what would happen if we extended this concept to a situation, when $n$ is arbitrary, e.g. fractional? This was the very same question L’Hôpital addressed to Leibniz in a letter in 1695. Since then the concept of fractional calculus has drawn the attention of many famous mathematicians, including Euler, Laplace, Fourier, Liouville, Riemann, Abel and Laurent. But it was not until 1884 that the theory of generalized operators reached a satisfactory level of development for the point of departure for the modern mathematician [1].

However, fractional-order calculus was not particularly popular until recent years when benefits stemming from using its concepts became evident in various scientific fields, including system modeling and automatic control. The rise of interest to the topic of fractional differentiation is also related to accessibility of more efficient and powerful computational tools. The introduction of computer algebra systems, such as MATLAB and Mathematica, led to new possibilities for evaluating the theoretical aspects of fractional calculus in specific applications.

Recent findings support the notion that fractional-order calculus should be employed where more accurate modeling and robust control are concerned. Specifically, fractional-order calculus found its way into complex mathematical and physical problems [2, 3]. In general, fractional-order calculus may be useful when modeling any system which has memory and/or hereditary properties [4]. In the field of automatic control fractional calculus is used to obtain more accurate models, develop new control strategies and enhance the characteristics of control systems.

2  Definitions of the fractional operator

Fractional calculus is a generalization of integration and differentiation to non-integer order operator $_a\mathscr{D}_t^\alpha$, where $a$ and $t$ denote the limits of the operation and $\alpha$ denotes the fractional order such that

\begin{equation} _{a}\mathscr{D}_{t}^{\alpha}=\begin{cases} \frac{\dif\,^{\alpha}}{\dif\, t^{\alpha}} & \Re(\alpha)\gt0,\\ \\ 1 & \Re(\alpha)=0,\\ \\ \int_{a}^{t}\left(\dif\, \tau\right)^{-\alpha} & \Re(\alpha)\lt0, \end{cases} \tag{1}\end{equation}

where generally it is assumed that $\alpha\in\mathbb{R}$, but it may also be a complex number [5].

One of the reasons why fractional calculus is not yet found in elementary texts is a certain degree of controversy found in the theory [1]. This is why there exist several definitions for the fractional-order differintegral operator. Next, several popular definitions are given.

Definition 1. (Riemann-Liouville definition)

\begin{equation} _{a}\mathscr{D}_{t}^{\alpha}f(t)=\frac{1}{\Gamma(m-\alpha)}\,\left(\frac{\dif}{\dif\, t}\right)^{m}\int_{a}^{t}\frac{f(\tau)}{(t-\tau)^{\alpha-m+1}}\,\dif\tau\,, \tag{2}\end{equation}

where $m-1\lt\alpha\lt m$, $m\in\mathbb{N}$, $\alpha\in\mathbb{R}^{+}$ and $\Gamma\left(\cdot\right)$ is Euler’s gamma function.

Definition 2. (Caputo definition)

\begin{equation} _{0}\mathscr{D}_{t}^{\alpha}f(t)=\frac{1}{\Gamma(m-\alpha)}\int_{0}^{t}\frac{f^{(m)}(\tau)}{(t-\tau)^{\alpha-m+1}}\dif\,\tau, \tag{3}\end{equation}

where $m-1\lt\alpha\lt m$, $m\in\mathbb{N}$.

Definition 3. (Grünwald-Letnikov definition)

\begin{equation} _{a}\mathscr{D}_{t}^{\alpha}\, f\left(t\right)=\lim_{h\rightarrow0}\frac{1}{h^{\alpha}}\sum_{j=0}^{\left[\frac{t-a}{h}\right]}\left(-1\right)^{j}\binom{\alpha}{j}\, f\left(t-jh\right), \tag{4}\end{equation}

where $\left[\cdot\right]$ means the integer part.

3  Fractional operator properties

Fractional-order differentiation has the following properties [6, 7, 4]:

  1. If $f(t)$ is an analytic function, then the fractional-order differentiation $_{0}\mathscr{D}_{t}^{\alpha}f(t)$ is also analytic with respect to $t$.
  2. If $\alpha=n$ and $n\in\mathbb{Z}^{+}$, then the operator $_{0}\mathscr{D}_{t}^{\alpha}$ can be understood as the usual operator $\dif\,^{n}/\dif\, t^{n}$.
  3. Operator of order $\alpha=0$ is the identity operator: $_{0}\mathscr{D}_{t}^{0}f(t)=f(t)$.
  4. Fractional-order differentiation is linear; if $a,\, b$ are constants, then
    \begin{equation} _{0}\mathscr{D}_{t}^{\alpha}\left[af(t)+bg(t)\right]=a\,_{0}\mathscr{D}_{t}^{\alpha}f(t)+b\,{}_{0}\mathscr{D}_{t}^{\alpha}g(t). \tag{5}\end{equation}
  5. For the fractional-order operators with $\Re(\alpha)>0,\,\Re(\beta)>0$, and under reasonable constraints on the function $f(t)$ it holds the additive law of exponents:
    \begin{equation} _{0}\mathscr{D}_{t}^{\alpha}\left[_{0}\mathscr{D}_{t}^{\beta}f(t)\right]={}_{0}\mathscr{D}_{t}^{\beta}\left[_{0}\mathscr{D}_{t}^{\alpha}f(t)\right]={}_{0}\mathscr{D}_{t}^{\alpha+\beta}f(t) \tag{6}\end{equation}
  6. The fractional-order derivative commutes with integer-order derivative
    \begin{equation} \frac{\dif\,^{n}}{\dif\, t^{n}}\,\left(_{a}\mathscr{D}_{t}^{\alpha}f(t)\right)={}_{a}\mathscr{D}_{t}^{\alpha}\left(\frac{\dif\,^{n}f(t)}{\dif\, t^{n}}\right)={}_{a}\mathscr{D}_{t}^{\alpha+n}f(t), \tag{7}\end{equation}
    under the condition $t=a$ we have $f^{(k)}(a)=0,\,(k=0,\,1,\,2,\,…,\, n-1)$.

4  Computation examples

Example 1. Let us compute1 the Riemann-Liouville fractional derivative of order $\alpha=\frac{1}{2}$ for an elementary function $f(t)=t^2$ taking $a=0$:

\begin{multline*} _{0}\mathscr{D}_{t}^{\frac{1}{2}}\left(t^{2}\right)\stackrel{RL}{=}\frac{1}{\Gamma(1-\frac{1}{2})}\frac{\dif}{\dif\, t}\left(\int_{0}^{t}\frac{\tau^{2}}{(t-\tau)^{\frac{1}{2}-1+1}}\dif\,\tau\right)=\\ =\frac{1}{\sqrt{\pi}}\,\frac{\dif}{\dif\, t}\left(\frac{16\cdot t^{\frac{5}{2}}}{15}\right)=\frac{8\, t^{\frac{3}{2}}}{3\,\sqrt{\pi}}. \end{multline*}

Let us show that using Caputo’s definition yields the same result in this case:

\begin{equation*} _{0}\mathscr{D}_{t}^{\frac{1}{2}}\left(t^{2}\right)\stackrel{C}{=}\frac{1}{\Gamma(1-\frac{1}{2})}\int_{0}^{t}\frac{\left(\tau^{2}\right)^{(1)}}{(t-\tau)^{\frac{1}{2}-1+1}}\dif\,\tau=\frac{1}{\sqrt{\pi}}\int_{0}^{t}\frac{2\tau}{(t-\tau)^{\frac{1}{2}}}\dif\,\tau=\frac{8\, t^{\frac{3}{2}}}{3\,\sqrt{\pi}}. \end{equation*}

Example 2. Compute the fractional derivative of order $\alpha=\frac{1}{3}$ for function $f_{1}(t)=\mathrm{e}^{5t}$ and the fractional derivative of order $\alpha=\frac{1}{2}$ for function $f_{2}(t)=\sin(3t)$. In this case, in order to obtain the derivative, we use the Riemann-Liouville definition taking $a=-\infty$:

\begin{equation*} _{-\infty}\mathscr{D}_{t}^{\frac{1}{3}}\left(e^{5t}\right)=5^{\frac{1}{3}}\mathrm{e}^{5t}. \end{equation*}

We can compute the fractional derivative for the function $f_2(t)$ in the same way:

\begin{equation*} _{-\infty}\mathscr{D}_{t}^{\frac{1}{2}}\left(\sin(3t)\right)=\sqrt{3}\sin\left(3t+\frac{\pi}{4}\right). \end{equation*}

5  Thoughts on the meaning of fractional operator

The reader might be wondering, why the title of this section begins with “thoughts”. Could there be no proper explanation for the physical and geometrical meaning of fractional differentiation? Unfortunately, there is no clear, intuitive interpretation so far. Now, there exist several papers that shed some light on this matter, e.g. [8, 9].

Additionally, one may look for interpretation of fractional operators in other directions as well:

  • Fractal theory;
  • Сorrespondence to integer-order derivatives, which may be considered as a particular case of fractional derivatives.

It is also important to note the apparent reason for the difficulties with understanding the fractional derivative. The integer-order derivatives were developed with a clear application in mind. That is, they were the primary object of development, while the applications of using fractional derivatives, while considered by e.g. Leibniz and L’Hôpital, were not clear at the time.

Since the field of applications of fractional calculus is rapidly growing, it is perhaps safe to say that a clear geometric and physical interpretation of the fractional-order derivative will eventually arise.

References

[1]K. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations. Wiley, 1993.
[2]R. Hilfer, Applications of fractional calculus in physics, ser. Applications of Fractional Calculus in Physics. World Scientific, 2000.
[3]A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies). New York, NY, USA: Elsevier Science Inc., 2006.
[4]I. Podlubny, Fractional differential equations, ser. Mathematics in science and engineering. Academic Press, 1999.
[5]Y. Q. Chen, I. Petráš, and D. Xue, Fractional order control – A tutorial,” in Proc. ACC ’09. American Control Conference, 2009, pp. 1397–1411.
[6]D. Xue, Y. Chen, and D. P. Atherton, Linear Feedback Control: Analysis and Design with MATLAB (Advances in Design and Control), 1st ed.. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, 2008.
[7]C. A. Monje, Y. Chen, B. Vinagre, D. Xue, and V. Feliu, Fractional-order Systems and Controls: Fundamentals and Applications, ser. Advances in Industrial Control. Springer Verlag, 2010.
[8]I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002.
[9]J. A. T. Machado, A probabilistic interpretation of the fractional order differentiation,” Fractional Calculus & Applied Analysis, vol. 6, no. 1, pp. 73–80, 2003.

  • 1 A free CAS Maxima may be used for carrying out the computations. You can download the corresponding notebook FracCalcDefs. After evaluating the cells with the definitions, you may use the commands RLDif(t^2,t,1/2,0) or CapDif(t^2,t,1/2) to evaluate this particular fractional order derivative.

9 Comments

  1. Yanfang Liu
    Posted September 28, 2012 at 09:04 | Permalink

    The Eq. (3) seems to be missprinted.

  2. mak
    Posted May 19, 2016 at 14:23 | Permalink

    Hi,
    Is fractional PID controller in Simulink, you can change the type of derivative?
    For example – Grünwald-Letnikov, Caputo etc. ?
    What is the difference between the result for the fractional derivatives?

    Best Regards

  3. mak
    Posted May 23, 2016 at 07:40 | Permalink

    Dear Aleksei,
    What type of derivative fractional you have used to block in Simulink called – Fractional PID controller?
    Is it a derivative of fractional one known as the Riemann-Liouville, Grünwald-Letnikov, Caputo?
    Have you tried to do in Simulink derivative fractional formula Riemann-Liouville, Grünwald-Letnikov, Caputo?
    Do you have such a solution in Simulink?

  4. jairoviola
    Posted February 16, 2017 at 06:49 | Permalink

    Dear Aleksei

    I have a question about the optimization algorithm, how do you do to set a specific settling time???

    I mean, the toolbox allows to set phase margin and gain crossover frequency but how can I relate these parameters for the frequency domain in the time domain.

    I tried to do it using the Astrom equations for a second order system but it is not accurate in most cases for fractional transfer functions.

    best regards

    Jairo Viola

    • Posted February 20, 2017 at 10:34 | Permalink

      Currently, you can only select a performance index for the time domain. You cannot explicitly set the settling time. However, note that settling time is linked to the phase margin value. The greater this value, the longer the transient, and vice versa.

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