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:

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

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

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

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

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:

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 $H(\lambda)$:

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

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

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

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

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

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.

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:

where

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

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.