Time and Frequency Domain Analysis of the Linear Fractional-order Systems (original) (raw)
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www.ijacsa.thesai.org Time and Frequency Domain Analysis of the Linear Fractional-order Systems
2015
— Recent years have seen a tremendous upsurge in the area related to the use of Fractional-order (FO) differential equations in modeling and control. FO differential equations are found to provide a more realistic, faithful, and compact representations of many real world, natural and man-made systems. FO controllers, on the other hand, have been able to achieve a better closed-loop performance and robustness, than their integer-order counterparts. In this paper, we provide a systematic and rigorous time and frequency domain analysis of linear FO systems. Various concepts like stability, step response, frequency response are discussed in detail for a variety of linear FO systems. We also give the state space representations for these systems and comment on the controllability and observability. The exercise presented here conveys the fact that the time and frequency domain analysis of FO linear systems are very similar to that of the integer-order linear systems. Keywords- Fractional...
The modelling and analysis of fractional-order control systems in frequency domain
2000
This paper deals with fractional-order controlled systems and fractional-order controllers in the frequency domain. The mathematical description by fractional transfer functions and properties of these systems are presented. The new ways for modelling of fractional-order systems are illustrated with a numerical example and obtained results are discussed in conclusion.
Linear fractional order controllers; A survey in the frequency domain
Annual Reviews in Control, 2019
Today, there is a great tendency toward using fractional calculus to solve engineering problems. The control is one of the fields in which fractional calculus has attracted a lot of attention. On the one hand, fractional order dynamic models simulate characteristics of real dynamic systems better than integer order models. On the other hand, Fractional Order (FO) controllers outperform Integer Order (IO) controllers in many cases. FO-controllers have been studied in both time an frequency domain. The latter one is the fundamental tool for industry to design FO-controllers. The scope of this paper is to review research which has been carried out on FO-controllers in the frequency domain. In this review paper, the concept of fractional calculus and their applications in the control problems are introduced. In addition, basic definitions of the fractional order differentiation and integration are presented. Then, four common types of FO-controllers are briefly presented and after that their representative tuning methods are introduced. Furthermore, some useful continuous and discrete approximation methods of FO-controllers and their digital and analogue implementation methods are elaborated. Then, some Matlab toolboxes which facilitate utilizing FO calculus in the control field are presented. Finally, advantages and disadvantages of using FO calculus in the control area are discussed. To wrap up, this paper helps beginners to get started rapidly and learn how to select, tune, approximate, discretize, and implement FO-controllers in the frequency domain.
The modelling and analysis of fractional-order control systems in discrete domain
2000
This paper deals with fractional-order controlled systems and fractional-order controllers in the frequency domain. The mathematical description by fractional transfer functions and properties of these systems are presented. The new ways for modelling of fractional-order systems are illustrated with a numerical example and obtained results are discussed in conclusion.
International Journal of Dynamics and Control, 2016
Fractional order (FO) differentiation is a generalization of classical integer differentiation to real or complex orders. The origin of this concept dates back to the early days of classical differential calculus, although its inherent complexity postponed its use and application to the engineering world. In the last decades, the developments in computing technologies combined with the unique advantages of FO differ-integrals in modeling complex phenomena, have led to ongoing research interest towards using fractional calculus (FC) as an optimal tool to describe the dynamics of complex systems. Apart from this, FC is currently gaining more and more popularity in the engineering community. Nowadays, the adoption of FC in control engineering has been gaining more and more momentum, both in modeling, identification, and controller design. The aim of this special issue is to promote further development of FC in control engineering, stability analysis of FO systems, solutions for fractional order continuous-time linear systems, signal processing, approximations for fractional B Cristina I. Muresan
Stability Analysis of Fractional-order Systems
Ijca Proceedings on International Conference and Workshop on Emerging Trends in Technology, 2012
Fractional-order (FO) systems are a special subset of linear time-invariant (LTI) systems. The transfer functions (TFs) of these systems are rational functions with polynomials of rational powers of the Laplace variable 's'. FO systems are of interest for both controller design and modelling purpose. It has been shown that FOPID controller gives better response as compared to integerorder(IO) controllers. FO systems provide the accurate models for many real systems. The stability analysis of FO systems, which is quite different from that of integerorder(IO) systems analysis, is the main focus of this paper. Stability is defined using Riemann surface because of their multi-valued nature of the FO transfer functions (FOTFs). In this paper, various approaches viz., time domain analysis, frequency domain analysis, state space representation are discussed. Both the types of FO systems, with commensurate and incommensurate TFs, are discussed.
Modeling and Analysis of Multiple Fractional Order Systems
This paper provides a rational function approximation of the irrational transfer function of the fundamental linear fractional order differential equation, namely,) () () () (0 t e t y dt t y d m m m = + whose transfer function is given by ]) (1 [ 1) () () (0 m s s E s Y s H + = = for 0 < m < 2. Simple methods, useful in system and control theory, which consists of approximating, for a given frequency band, the irrational transfer function of this fractional order system by a rational function are presented. The impulse and step responses of this system are derived. We use a heuristic method to extract temporal and frequency characteristics performances. Illustrative examples are presented to show the exactitude and the usefulness of the approximation methods.
Stability Analysis and Fractional Order Controller Design for Control System
2017
In this paper, a new approach to stability for fractional order control system is proposed. Here a dynamic system whose behavior can be modeled by means of differential equations involving fractional derivatives. Applying Laplace transforms to such equations, and assuming zero initial conditions, causes transfer functions with no integer powers of the Laplace transform variable s to appear. In recent time, the application of fractional derivatives has become quite apparent in modeling mechanical and electrical properties of real materials. Fractional integrals and derivatives have found wide application in the control of dynamical systems when the controlled system and the controller are described by a set of fractional order differential equations. In the existing work, a fractional order system has been signified by a higher integer order system. Fractional calculus provides an excellent instrument for the description of memory and hereditary properties of various materials and pr...
State-space controller design for the fractional-order regulated system
2001
In this paper we will present a mathematical description and analysis of a fractional-order regulated system in the state space and the state-space controller design based on placing the closed-loop poles on the complex plane. Presented are the results of simulations and stability investigation of this system.
International Journal of Engineering and Technology
In this paper, a new comparative approach has been proposed for reliable controller design. Scientists and engineers are often confronted with the analysis, design, and synthesis of real-life problems. The first step in such studies is the development of a 'mathematical model' which can be considered as a substitute for the real problem. The mathematical model is used here as a plant. Fractional integrals and derivatives have found wide application in the control of dynamical systems when the controlled system and the controller are described by a set of fractional order differential equations. Here the stability of fractional order system is checked at the different level and it is found that the stability region is large in the complex plane. This large stability region provides the more flexibility for system implementation in the control engineering. Generally, an analytically or experimentally approaches are used for designing the controller. If a fractional order controller design approach used for a given plant then the controlled parameter gives the better result.