Fractional-order system

Fractional dynamics is a field of study in physics, mechanics, mathematics, and economics investigating the behavior of objects and systems that are described by using integrations and differentiation of fractional orders, by methods in the fractional calculus. Derivatives and integrals of fractional orders are used to describe objects that can be characterized by power-law nonlocality, power-law long-term memory or fractal properties.

In the fields of dynamical systems and control theory, a fractional-order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer order.^[1] Fractional-order systems are useful in studying the anomalous behavior of dynamical systems in electrochemistry, biology, viscoelasticity and chaotic systems.^[1]

Definition

A general dynamical system of fractional order can be written in the form^[2]

H(D^{{\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}})(y_{1},y_{2},\ldots ,y_{l})=G(D^{{\beta _{1},\beta _{2},\ldots ,\beta _{n}}})(u_{1},u_{2},\ldots ,u_{k})

where $H$ and $G$ are functions of the fractional derivative operator of orders $\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}$ and $\beta _{1},\beta _{2},\ldots ,\beta _{n}$ and $y_{i}$ and $u_{j}$ are functions of time. A common special case of this is the linear time-invariant (LTI) system in one variable:

\left(\sum _{{k=0}}^{m}a_{k}D^{{\alpha _{k}}}\right)y(t)=\left(\sum _{{k=0}}^{n}b_{k}D^{{\beta _{k}}}\right)u(t)

The orders $\alpha _{k}$ and $\beta _{k}$ are in general complex quantities, but two interesting cases are when the orders are commensurate

\alpha _{k},\beta _{k}=k\delta ,\quad \delta \in R^{+}

and when they are also rational:

\alpha _{k},\beta _{k}=k\delta ,\quad \delta ={\frac {1}{q}},q\in Z^{+}

When $q=1$ , the derivatives are of integer order and the system becomes an ordinary differential equation. Thus by increasing specialization, LTI systems can be of general order, commensurate order, rational order or integer order.

Transfer function

By applying a Laplace transform to the LTI system above, the transfer function becomes

G(s)={\frac {Y(s)}{U(s)}}={\frac {\sum _{{k=0}}^{n}b_{k}s^{{\beta _{k}}}}{\sum _{{k=0}}^{m}a_{k}s^{{\alpha _{k}}}}}

For general orders $\alpha _{k}$ and $\beta _{k}$ this is a non-rational transfer function. Non-rational transfer functions cannot be written as an expansion in a finite number of terms (e.g., a binomial expansion would have an infinite number of terms) and in this sense fractional orders systems can be said to have the potential for unlimited memory.^[2]

Motivation to study fractional-order system

Exponential laws are classical approach to study dynamics of population densities, but there are many systems where dynamics undergo faster or slower-than-exponential laws. In such case the anomalous changes in dynamics may be best described by Mittag-Leffler functions.^[3]

Anomalous diffusion is one more dynamic system where fractional-order systems play significant role to describe the anomalous flow in the diffusion process.

Viscoelasticity is the property of material in which the material exhibits its nature between purely elastic and pure fluid. In case of real materials the relationship between stress and strain given by Hooke's law and Newton's law both have obvious disadvances. So G. W. Scott Blair introduced a new relationship between stress and strain given by

\sigma (t)=E{D_{t}^{\alpha }}\varepsilon (t),\quad 0<\alpha <1.

Preliminaries

Fractional integration and several forms of fractional derivatives are defined in fractional calculus. Due to involvement of definite integration in definitions of fractional integration and derivatives, these operators are non-local concepts and hence non-local geometrical and physical interpretation for these operators has been established by Igor Podlubny.

Analysis of fractional differential equations

Consider a fractional-order initial value problem:

{_{0}^{C}D_{t}^{\alpha }}x(t)=f(t,x(t)),\quad t\in [0,T],\quad x(0)=x_{0},\quad 0<\alpha <1.

Existence and uniqueness

Here, under the continuity condition on function f, one can convert the above equation into corresponding integral equation.

x(t)=x_{0}+{_{0}^{C}D_{t}^{{-\alpha }}}f(t,x(t))=x_{0}+{\frac {1}{\Gamma (\alpha )}}\int _{0}^{t}{\frac {f(s,x(s))\,ds}{(t-s)^{{1-\alpha }}}},

One can construct a solution space and define, by that equation, a continuous self-map on the solution space, then apply a fixed-point theorem, to get a fixed-point, which is the solution of above equation.

Numerical simulation

For numerical simulation of solution of the above equations, Kai Diethelm has suggested fractional Adams–Bashforth–Moulton method.

Chaos

In chaos theory, it has been observed that chaos occurs in dynamical systems of order 3 or more. With the introduction of fractional-order systems, some researchers study chaos in the system of total order less than 3.^[4]

References

1 2 Monje, Concepción A. (2010). Fractional-Order Systems and Controls: Fundamentals and Applications. Spinger. ISBN 9781849963350.
1 2 Blas M. Vinagre; C. A. Monje; Antonio J. Calderon. "Fractional Order Systems and Fractional Order Control Actions" (PDF). 41st IEEE Conference on Decision and Control. Retrieved 18 June 2013.
↑ M. Rivero; et al. (2011). "Fractional dynamics of populations". Appl. Math. Comput. 218: 1089–1095. doi:10.1016/j.amc.2011.03.017.
↑ Petras, I.; Bednarova, D. (2009). "Fractional - order chaotic systems,". Emerging Technologies & Factory Automation, 2009. ETFA 2009. IEEE Conference on , vol., no., pp. 1, 8, 22–25 Sept. 2009: 1–8.

M. Rivero; et al. (2011). "Fractional dynamics of populations". Appl. Math. Comput. 218: 1089–1095. doi:10.1016/j.amc.2011.03.017.
I. Podlubny (2001). "Geometric and physical interpretation of fractional integration and fractional differentiation". arXiv:math/0110241 .
I. Podlubny (1999). Fractional differential equations. Elsevier. ISBN 9780125588409.

An Introduction to the Fractional Calculus and Fractional Differential Equations, by Kenneth S. Miller, Bertram Ross (Editor). Hardcover: 384 pages. Publisher: John Wiley & Sons; 1 edition (May 19, 1993). ISBN 0-471-58884-9
The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order (Mathematics in Science and Engineering, V), by Keith B. Oldham, Jerome Spanier. Hardcover. Publisher: Academic Press; (November 1974). ISBN 0-12-525550-0

External links

Fractional Calculus Applications in Automatic Control and Robotics A tutorial on fractional calculus, fractional order systems and fractional order control theory.

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