Fractional-order system

A general dynamical system of fractional order can be written in the form[3]

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

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

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

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

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

and when they are also rational:

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

When ${\displaystyle q=1}$, 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.

Exponential laws are a 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.[4]

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

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

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.[5]

In nuroscience, it has been found that single rat neocortical pyramidal neurons adapt with a time scale that depends on the time scale of changes in stimulus statistics. This multiple time scale adaptation is consistent with fractional order differentiation, such that the neuron's firing rate is a fractional derivative of slowly varying stimulus parameters.[6]

Consider a fractional-order initial value problem:

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

• Acoustic attenuation
• Differintegral
• Fractional calculus
• Fractional order control
• Fractional order integrator
• Fractional Schrödinger equation
• Fractional quantum mechanics

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