Modeling

1  Laplace transform

The Laplace transform is an essential tool in linear dynamic system modeling and control system engineering. A function $F(s)$ of the complex variable $s=\sigma+j\omega$ is called the Laplace transform of the original function $f(t)$ and is defined in the following way:

\begin{equation} F(s)=\mathscr{L}\left[f(t)\right] = \int^{\infty}_{0}\mathbb{e}^{-st}f(t)\dif t. \tag{1}\end{equation}

The original function $f(t)$ can be recovered from the Laplace transform by applying the reverse Laplace transform defined as

\begin{equation} f(t)=\mathscr{L}^{-1}\left[F(s)\right]=\frac{1}{j2\pi}\int^{c+j\infty}_{c-j\infty}\mathbb{e}^{-st}F(s)\dif s, \tag{2}\end{equation}

where $c$ is greater than the real part of all the poles of function $F(s)$ [1].

Assuming zero initial conditions, the laplace transform of a generalized fractional-order operator is given by

\begin{equation} \mathscr{L}\left[\mathscr{D}^\alpha f(t)\right] = s^\alpha F(s). \tag{3}\end{equation}

It should be noted, that if initial conditions are not zero, different definitions apply for the Riemann-Liouville, Caputo and Grünwald-Letnikov fractional-order operators.

2  Fractional-order models

A fractional-order continuous-time dynamic system can be expressed by a fractional differential equation of the following form

\begin{eqnarray} a_{n}\mathscr{D}^{\alpha_{n}}y(t)+a_{n-1}\mathscr{D}^{\alpha_{n-1}}y(t)+\cdots+a_{0}\mathscr{D}^{\alpha_{0}}y(t)&=&\\b_{m}\mathscr{D}^{\beta_{m}}u(t)+b_{m-1}\mathscr{D}^{\beta_{m-1}}u(t)+\cdots+b_{0}\mathscr{D}^{\beta_{0}}u(t)&&. \tag{4}\end{eqnarray}
The system is said to be of commensurate-order if in (4) all the orders of derivation are integer multiples of a base order $\gamma$ such that $\alpha_{k},\,\beta_{k}=k\gamma,\,\gamma\in\mathbb{R}^{+}$. The system can then be expressed as
\begin{eqnarray} \sum_{k=0}^{n}a_{k}\mathscr{D}^{k\gamma}y(t)=\sum_{k=0}^{m}b_{k}\mathscr{D}^{k\gamma}u(t). \tag{5}\end{eqnarray}

Applying the Laplace transform to (4) with zero initial conditions the input-output representation of the fractional-order system can be obtained in the form of a transfer function:

\begin{eqnarray} G(s)=\frac{Y(s)}{U(s)}=\frac{b_{m}s^{\beta_{m}}+b_{m-1}s^{\beta_{m-1}}+\cdots+b_{0}s^{\beta_{0}}}{a_{n}s^{\alpha_{n}}+a_{n-1}s^{\alpha_{n-1}}+\cdots+a_{0}s^{\alpha_{0}}}. \tag{6}\end{eqnarray}

In the case of a system with commensurate order $\gamma$, and taking $\lambda = s^{\gamma}$, the continuous-time transfer function can be represented as a pseudo-rational function $H(\lambda)$:

\begin{equation} H(\lambda)=\frac{\sum\limits _{k=0}^{m}b_{k}\lambda^{k}}{\sum\limits _{k=0}^{n}a_{k}\lambda^{k}}. \tag{7}\end{equation}

Based on the concept of the pseudo-rational function, a state-space representation can be established in the form:

\begin{eqnarray} \mathscr{D}^{\gamma}\, x(t)&=&Ax(t)+Bu(t)\\ y(t)&=&Cx(t)+Du(t) \tag{8}\end{eqnarray}

3  Basic fractional-order system analysis

One of the most important aspects of dynamic system analysis is the stability assessment. In case of fractional systems given in form (4) or (8), one can use the following criterion [2].

(Matignon’s stability theorem) The fractional transfer function $G(s)=Z(s)/P(s)$ is stable if and only if the following condition is satisfied in $\sigma$-plane:

\begin{equation} \left|\arg(\sigma)\right|>q\frac{\pi}{2},\,\forall\sigma\in C,\, P(\sigma)=0, \tag{9}\end{equation}

where $\sigma:=s^{q}$. When $\sigma=0$ is a single root of $P(s)$, the system cannot be stable. For $q=1$, this is the classical theorem of pole location in the complex plane: no pole is in the closed right plane of the first Riemann sheet.

In general, for a commensurate-order fractional-order system in the form

\begin{equation} \mathscr{D}^{q}w=f(w), \tag{10}\end{equation}

where $0<q<1$ and $w\in R^{n}$ the equilibrium points are calculated by solving

\begin{equation} f(w)=0. \tag{11}\end{equation}

The equilibrium points are asymptotically stable if all the eigenvalues $\lambda_{k}$ of the Jacobian matrix $J=\frac{\partial f}{\partial w}$, evaluated at the equilibrium, satisfy the condition

\begin{equation} \left|\arg(\mathrm{eig}(J))\right|=\left|\arg(\lambda_{k})\right|>q\frac{\pi}{2},\, k=1,\,2,\,…,\, n. \tag{12}\end{equation}

Alternatively, the stability condition can also be evaluated from the state-space representation of the system (8)

\begin{equation} \left|\arg(\mathrm{eig}(A))\right|>q\frac{\pi}{2}, \tag{13}\end{equation}

where $0<q<1$ and $\mathrm{eig}(A)$ represents the eigenvalues of the state-space matrix $A$.

Stability regions of a fractional-order system are shown in Figure 1.

Figure 1. Stability regions

Time-domain analysis of fractional-order models can be conducted by using the definitions presented in the introduction. Specifically, the numerical approach can be used through applying the Grünwald-Letnikov definition.

In case of the frequency domain, the direct substitution $s=j\omega$ can be applied and the corresponding characteristics can be obtained directly. Practically all frequency-domain analysis methods are applicable.

4  Integer-order approximation of fractional operators

Using fractional-order operator approximations can be beneficial due to the abundance of tools available for regular transfer function analysis and modeling. We suggest using a very flexible approximation technique, proposed in [3], called the Oustaloup recursive filter method. It is summarized below.

To approximate a fractional-order operator $s^\gamma$ for $0<\gamma<1$, one can use the following set of formulae:

\begin{equation} s^\gamma \approx K \prod_{k=-N}^{N}\frac{s+\omega^{\prime}_k}{s+\omega_k}, \tag{14}\end{equation}

where

\begin{equation} \omega^{\prime}_k=\omega_b\left(\frac{\omega_h}{\omega_b}\right)^{\frac{k+N+1/2(1-\gamma)}{2N+1}},\quad\omega_k=\omega_b\left(\frac{\omega_h}{\omega_b}\right)^{\frac{k+N+1/2(1+\gamma)}{2N+1}},\quad K=\omega_h^\gamma. \tag{15}\end{equation}

For fractional orders $\alpha$ such that $\alpha\geq 1$ it holds

\begin{equation} s^\alpha=s^n s^\gamma, \tag{16}\end{equation}

where $n=\alpha-\gamma$ denotes the integer part of $\alpha$ and $s^\gamma$ is obtained through the Oustaloup approximation.

5  Discretization

Discretization is essential in digital implementation of controllers. Several discretization methods have been developed for continuous fractional-order models. These include FIR (finite impulse response) and IIR (infinite impulse response) filter realizations. The latter is preferred to the former, because the IIR implementation will be of lower order.

The following method for obtaining a discrete-time approximation of fractional models can be proposed.

  • Approximate the continuous-time fractional model by a rational-order transfer function $G_c(s)$ using the Oustaloup recursive filter method.
  • Use a discrete transformation with a sample period $T$ and obtain a discrete approximation $G_d(z)$ of the fractional model.

Difficulties may still arise if the filter is expected to precisely reflect the desired frequency-domain specifications because to the high order of the resulting discrete model. In case of IIR filters, the direct form implementation may lead to computational instability. It is essential to improve computational stability by using a proper realization method.

References

[1]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.
[2]D. Matignon, Generalized Fractional Differential and Difference Equations: Stability Properties and Modeling Issues,” in Proc. of Math. Theory of Networks and Systems Symposium, 1998, pp. 503–506.
[3]A. Oustaloup, P. Melchior, P. Lanusse, O. Cois, and F. Dancla, The CRONE toolbox for Matlab,” in Proc. IEEE Int. Symp. Computer-Aided Control System Design CACSD 2000, 2000, pp. 190–195.

9 Comments

  1. Hamidreza
    Posted December 15, 2013 at 17:43 | Permalink

    Hi,
    Recently I found your toolbox and trying to use it. Thank you it seems nice. I have a question for being sure about ability of your work.
    I investigate the problems around stability of fractional order time delayed systems. I think the only delay systems which your code can solve are G0(s)*exp(-t*s)? Is it possible to develop your code to practice on some plants like G(s)=1/(s^2.5+exp(-t*s)*s^2+exp(-2t*s)*s^1.5+1)?

    Regards;

    • Posted December 20, 2013 at 18:25 | Permalink

      Not using the fotf object. Though it should be possible to achieve what you want using Simulink.

      • Hamidreza
        Posted December 20, 2013 at 21:33 | Permalink

        Thank you for your reply. I’ve done that. But I couldn’t use your toolbox in Simulink (MATLAB-2010). It didn’t work and I think you didn’t finish the usage for Simulink completely. So I used Ninteger Library and it works good, but I prefer to use one toolbox for all purpose! However Simulink has very uncertain and unaccurate results. Some approximation for delay some for fractional-order some for derivation and ….! Finally I found analytic investigations are more reliable even not complete! ;)

      • Posted December 21, 2013 at 12:43 | Permalink

        If you find any issues while using FOMCON toolbox feel free to submit a bug report.

  2. vishal kumar pandey
    Posted March 7, 2014 at 07:25 | Permalink

    Hi,
    what is the use of FFIDATA objects, which is given in every toolbox.

    • Posted March 7, 2014 at 09:21 | Permalink

      ffidata objects in FOMCON allow to store frequency-domain identification data which can be later used in one of the NINTEGER toolbox identification functions included in FOMCON.

  3. mks
    Posted December 3, 2014 at 14:04 | Permalink

    how to use fomcon tool to design PID controller in model with transfer function can u suggest any idea

  4. M Farhan
    Posted September 26, 2015 at 16:07 | Permalink

    Dear Sir,Why Oustaloup Filter is called recursive filter ?

  5. a7med8
    Posted October 24, 2015 at 16:39 | Permalink

    Pls can you help i have an integer order Model (IO) and i wanna to converted it into Fractional order (FO)
    this a model For Wood and Berry Binary Distillation column it transfer function as follow

    12.8 e^-s/(16.7 S +1) -18.9 e^-3s/(21.0 S+1)

    T.F =
    6.6 e^-7s/(10.9 S + 1) -19.4.8 e^-3s/(14.4 S+1)

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