Fractional Order Systems

Abstract

PrefaceThis book is devoted to fractional order systems, their applications to modelling and control. It is based on derivatives and integrals of arbitrary (real) order, fractional differential equations and methods of their solution, approximations and implementation techniques.The advantages of fractional calculus have been described and pointed out in the last few decades by many authors. It has been shown that the fractional order models of real systems are regularly more adequate than usually used integer order models.Applications of these fractional order models are in many fields, as for example, rheology, mechanics, chemistry, physics, bioengineering, robotics and many others.At the same time, fractional integrals and derivatives are also applied to the theory of control of dynamical systems, when the controlled system and/or the controller is described by fractional differential equations.The main goal of the book is to present applications and implementations of fractional order systems. It provides only a brief theoretical introduction to fractional order system dedicating almost all the space to the modelling issue, fractional chaotic system control and fractional order controller theory and realization.The book is suitable for advanced undergraduates and graduate students.It is organized as follows: Chapter one is a brief introduction to the fractional order systems. Some historical notes, definitions and fundamentals are described.Chapter two is dedicated to Fractional Order PID Controller defining their stability regions when first order with time delay plant have to be controlled in closed loop. Chapter three is on fractional order chaotic systems. In this chapter, a survey of well-known chaotic systems is presented. Mathematical models of nonlinear dynamical systems contain the fractional derivatives. Total order of the system is less than three, however, the chaotical phenomena, as for example, in strange attractors can be observed in such systems.In chapter four the operator s m , where m is a real number, is approximated via the binomial expansion of the backward difference and then a hardware implementation of differintegral operator is proposed using Field Programmable Gate Array (FPGA). This building block represents the basic element to implement fractional order control systems.Chapter five is devoted to microprocessor implementation of the fractional order controllers. Fundamentals on discrete approximations of a fractional operator as well as control algorithm for implementation of the controllers are described. Also presented are three examples of the discrete fractional order controllers implemented on PIC, PC with PCL card, and PLC, respectively. A real measurement and obtained results are shown for each particular case. Some concluding remarks close this chapter.Chapter six is dedicated to the implementation of the fractional order PID controller by using the analog counter part of FPGA that is Field Programmable Analog Array (FPAA).Chapter seven presents a possible implementation ...

Application of fractional order transfer functions to modeling of high — Order systems

“…Modeling fractional order transfer functions with the use of MATLAB/SIMULINK requires us to apply models: integer order and finite-dimensional.In this paper the approximations proposed by Charef and Oustaloup, dedicated to approximate both elementary plants described by (2) and (3) will be presented. They are very close and their general idea consists in fitting the Bode magnitude plot of approximation to Bode magnitude plot of approximated trasfer function.…”

“…Rcently new, powerful computation tools allow many Authors [1], [2], [8], [9], [14], [16], [19] to apply the fractional -order calculus to describe a number of physical phenomena: heat and mass transfer ( [15], diffusion, supercapacitors [18] [25] and so on [10], [11], [22]. A transfer function applying the Laplace trasform:…”

Application of Real Ant Colony Optimization algorithm to solve space and time fractional heat conduction inverse problem

“…In case of optimization algorithms inspired by nature, another good feature of these algorithms is the fact that they do not need any requirements about minimized function, except the existence of the solution. Fractional calculus is very useful to model many various types of physical and technical phenomena [8,9,11,25,26,30,34]. Application of fractional calculus can be found, for example, in electrical engineering [26], control theory [8,11], mechanics [9].…”

“…Fractional calculus is very useful to model many various types of physical and technical phenomena [8,9,11,25,26,30,34]. Application of fractional calculus can be found, for example, in electrical engineering [26], control theory [8,11], mechanics [9]. In papers [30,46] the authors consider the model of heat conduction in ceramic and composite medium.…”

On the observability of conformable linear time-invariant control systems

“…A new interest has been developed to fractional-order systems in the area of control theory. One can find the uses and importance of fractional derivatives in the field of control theory, dynamical systems, nanotechnology, viscoelasticity, financial modeling, anomalous transport, and anomalous diffusion, see, for example, [2,4,6]. In modern control theory, controllability and observability have become the backbone, which was introduced by Kalman in 1960, see [8], and [9].…”