^{1}

^{1}

^{2}

^{1}

^{2}

The fractional order calculus (FOC) is as old as the integer one although up to recently its application was exclusively in mathematics. Many real systems are better described with FOC differential equations as it is a well-suited tool to analyze problems of fractal dimension, with long-term “memory” and chaotic behavior. Those characteristics have attracted the engineers' interest in the latter years, and now it is a tool used in almost every area of science. This paper introduces the fundamentals of the FOC and some applications in systems' identification, control, mechatronics, and robotics, where it is a promissory research field.

The fractional order calculus (FOC) was unexplored in engineering, because of its inherent complexity, the apparent self-sufficiency of the integer order calculus (IOC), and the fact that it does not have a fully acceptable geometrical or physical interpretation [

In the latter years FOC attracted engineers' attention, because it can describe the behavior of real dynamical systems in compact expressions, taking into account nonlocal characteristics like “infinite memory” [

Bearing these ideas in mind, this paper is organized as follows. Section

The intuitive idea of FOC is as old as IOC, it can be observed from a letter written by Leibniz to L'Hopital in 1695 [

Its applications in engineering were delayed because FOC has multiple definitions [

Riemann-Liouville:

Integral:

Grünwald-Letnikov:

Integral:

Derivative:

Caputo:

Cauchy:

We can choose one definition or another, depending on the application and the preference of the designer. In [

Some other tools of interest for engineers are the classical transforms of Laplace and Fourier, that are valid and used in order to simplify operations like convolution and can be applied to solve FOC differential equations. In FOC the Laplace transform is defined as [

As shown, this transform takes into account all initial conditions from the first to the

In addition to the problem for which definition must be chosen based on its properties or implementation complexity, the engineers may know the implications of using a mathematical tool. An easy way to understand it, is by plotting it in a figure and seeing what is happening when it is applied. Pitifully for FOC it is a lack, but some approaches were proposed in the last decade, as will be presented in Section

In the case of integral order calculus, there is a well-accepted geometrical explanation which clearly relates some physical quantities, for example, instant rate of change of a function completely explains the relationship between concepts like position and speed of an object. Unfortunately, until the last decade there was no geometrical interpretation of the fractional order derivatives. One of them was proposed in [

Then for

If

For values of

Therefore the values near to the evaluation time (present) have more influence over the result than those that are far from it. This interpretation is shown in Figure

Tenreiro fractional order derivative interpretation. Here values near the evaluation point have a more significant effect over “the present” than others.

A geometric interpretation based on Riemann-Louville-definition ((

With this information a tridimensional graph is drawn with axes

Podlubny fractional order derivative interpretation.

Another geometrical interpretation, this time based fractal dimension was proposed in [

For

Function kernel

In the derivative case (

Note that the kernel of (

Cantor set with

In this case the

Fractional order can represent systems with high-order dynamics and complex nonlinear phenomena using few coefficients [

Another way to obtain the response of a fractional order system is by using analogical circuits with fractional order behavior as shown in Figure

Recursive low pass RC filter.

Fractor is a parallel capacitance with fractional order behavior. It uses fractal geometry when fabricated. (a) Introducing a type of fractal tree. (b) Presenting the link diagram, and (c) The circuit equivalence.

Fractal tree

Fractal branch

Equivalent circuit of a fractal branch

Anyone of these approaches could be used in engineering applications, in this paper we introduce its use in systems' identification, control theory and robotics.

Fractional order dynamical systems can be modeled using the Laplace transform-like transfer functions [

Some high-order systems would be approximated with a compact fractional order expression, it is useful in cases where an approach between holistic and detailed description of the process is required. As an instance the model of the 5th order [

This 8-parameter system would be well approximated by

Comparison between a high-order integer system and its approximation by a fractional one.

Magnitude in dB

Phase in dB

Many real systems are better identified as fractional order equations [

Adjusting the model is accomplished by finding the parameters

Another instance of the fractional order formulation is presented in [

Another example of identification of a biological system was presented in [

Circuit used for identification of fractance in fruits and vegetables.

A nonparametric method introduced in [

Block diagram of the identification of a system by

Just as an example, we propose an experiment with synthetic data, simulating the vibration present in a gearbox. These kinds of systems are highly complex as several frequencies and their harmonics are exited by the rotation of the axes, unbalanced pieces, meshing between gears, bearing balls interaction, backslash between pieces among others. When the system has a failure, harmonics and side-bands are added to the frequency spectrum and the dynamical model of the system may change. If these models were known a predictive maintenance strategy would be proposed based on comparison between them.

Unfortunately as there are many components interacting and some have nonlinear behavior, a dynamical model of integer order is frequently difficult to obtain and involve several parameters that are hardly comparable. Notwithstanding, as shown in Figure

Magnitude of the Bode's plot of two complex systems, one represents the vibration signal of a rotational system without a failure (a) and the other is (b) a system with a teeth broken on the transmission box. Note that when approximating by a fractional order equations the order changes from a system to another.

System without failure

System with failure

Dynamic systems are typically fractional order, but often just the controller is designed as that, as the plant is modeled with integer order differintegral operators. A robust fractional order controller requires less coefficients than the integer one [

Classification of dynamical system grouping by order of the plant and the controller.

Order of system | Order of controller |
---|---|

Integer | Integer |

Integer | Fractional |

Fractional | Integer |

Fractional | Fractional |

In [

The typical fractional controller in literature are [

(i)

Block diagram of

(ii)

The function approximation to the dyke behavior was

(iii)

There is not a rigorous formula to design this type of controller, some techniques to adjust it are artificial intelligence, as swarm intelligence [

Block diagram of the

(iv)

In [

In industrial environments the robots have to execute their task quickly and precisely, minimizing production time. It requires flexible robots working in large workspaces; therefore, they are influenced by nonlinear and fractional order dynamic effects [

In [

A cooperative cell of robots achieving a desired task.

Another interesting problem in robotics which can be treated, with FOC is the control of flexible robots, as this kind of light robots use low power actuators, without self-destruction effects when high impacts occurs. Nevertheless significant vibrations over flexible links make a position control difficult to design, because it reveals a complex behavior difficult to approximate by linear differential equations [

Another case was analyzed in [

A small overshoot in fractional order controllers is an important characteristic when accuracy and speed are desired in small spaces. In [

An application in a robot with legs was presented in [

In this paper some basic concepts of FOC and some applications in engineering were presented. However, its inherent complexity, the lack of a clear geometrical interpretation and the apparent sufficiency of the integer calculus have delayed its use outside the area of mathematics. Nowadays, some applications have begun to appear but they are still at the initial stage of development. In the near future, with a deep understanding of FOC's implications, its use in systems' identification will increase, as it captures very complex behavior neglected by IOC, and in control of systems this tool open a wide range of desired behavior, where the integer one is just a special case.

The authors acknowledge support received from the Universidade Estadual de Campinas—UNICAMP (Brazil), Intituto Superior de Engenharia do Porto—I.S.E.P. (Portugal), and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—CAPES (Brazil), that made this study be possible.