Comparison of the methods for the calculation of fractional-order differential equations

Real objects in general are fractional-order systems, although in some types of systems the order is very close to an integer order. Since major advances have been made in this area in the last decades, it is possible to consider also the real order of the dynamical systems by using fractional order of the differential equations. Such models are more adequate for the description of dynamical systems than integer-order models. Appropriate methods for the numerical calculations of fractional-order differential equations are needed. In this contribution we will compare some previous methods used for simulation purposes with the methods based on approximate formulas for numerical inversion of Laplace transforms. The verification and comparison will be based mainly on the accuracy and computing time which is very important e.g. in the tasks of simulation or identification using optimization methods where too many calculations are needed and faster methods can save time very significantly. © 2011 IEEE

Analogue realization of fractional-order dynamical systems

As it results from many research works, the majority of real dynamical objects are fractional-order systems, although in some types of systems the order is very close to integer order. Application of fractional-order models is more adequate for the description and analysis of real dynamical systems than integer-order models, because their total entropy is greater than in integer-order models with the same number of parameters. A great deal of modern methods for investigation, monitoring and control of the dynamical processes in different areas utilize approaches based upon modeling of these processes using not only mathematical models, but also physical models. This paper is devoted to the design and analogue electronic realization of the fractional-order model of a fractional-order system, e.g., of the controlled object and/or controller, whose mathematical model is a fractional-order differential equation. The electronic realization is based on fractional-order differentiator and integrator where operational amplifiers are connected with appropriate impedance, with so called Fractional Order Element or Constant Phase Element. Presented network model approximates quite well the properties of the ideal fractional-order system compared with e.g., domino ladder networks. Along with the mathematical description, circuit diagrams and design procedure, simulation and measured results are also presented

Electronic realization of the fractional-order system

A great deal of modern methods for investigation, monitoring and control of the processes in the area of mining and processing of earth resources utilize approaches based upon modeling of these processes using the mathematical or physical models. As it results from the last research works, majority of the real processes in general are fractional-order dynamical processes (arbitrary real order including integer order), however in some types of processes the order is very close to an integer order. In these cases the application of fractional-order models is more adequate than integer-order models for the investigation of these dynamical processes. Comparison of the methods for solving fractional-order mathematical models we have described in our past paper on the 10th International Multidisciplinary Scientific Geo-Conference-SGEM 2011. The solution of many practical problems can be simplified with the help of electronic analog models utilizing so called fractional-order elements or constant phase elements. This paper is devoted to the design and realization of the electronic fractional-order model of the fractional-order system whose mathematical model is fractional-order differential equation. Along with the mathematical description, circuit diagrams and design procedure presented are also simulation results. © SGEM2012 All Rights Reserved by the International Multidisciplinary Scientific GeoConference SGEM Published by STEF92 Technology Ltd

A method for incorporating fractional-order dynamics through PID control system retuning

Proportional-Integral-Derivative (PID) controllers have been the heart of control systems engineering practice for decades because of its simplicity and ability to satisfactory control different types of systems in different fields of science and engineering in general. It has receive widespread attention both in the academe and industry that made these controllers very mature and applicable in many applications. Although PID controllers (or even its family counterparts such as proportional-integral [PI] and proportional-derivative [PD] controllers) are able to satisfy many engineering applications, there are still many challenges that face control engineers and academicians in the design of such controllers especially when guaranteeing control system robustness. In this paper, we present a method in improving a given PID control system focusing on system robustness by incorporating fractional-order dynamics through a returning heuristic. The method includes the use of the existing reference and output signals as well as the parameters of the original PID controller to come up with a new controller satisfying a given set of performance characteristics. New fractional-order controllers are obtained from this heuristic such as PIλ and PIλDμ controllers, where λ, μ ∈ (0, 2) are the order of the integrator and differentiator, respectively. © 2013 Academic Publications, Ltd

Comparison of the electronic realization of the fractional-order system and its model

Real objects in general are fractional-order (FO) systems, although in some types of systems the order is very close to an integer order. Also application of FO models is more adequate for the description of dynamical systems than integer-order models. This paper is devoted to the analogue electronic realization of the FO systems, e.g. controlled objects and/or controllers whose mathematical models are FO differential equations. The electronic realization is based on FO differentiator and FO integrator where operational amplifiers are connected with appropriate impedance or in our realization with so called Fractional Order Element (FOE) or Constant Phase Element (CPE). Presented network model in spite of its simplicity approximates quite well the properties of the ideal FO system compared with e.g. domino ladder networks. Also presented, along with the mathematical description, are simulation and measurement results. © 2012 IEEE