Stability Analysis of 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...
Time and Frequency Domain Analysis of the Linear Fractional-order Systems
International Journal of Advanced Computer Science and Applications, 2012
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 manmade 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.
Stability Analysis and Fractional Order Controller Design for Control System
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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...
International Journal of Bifurcation and Chaos, 2012
Fractional complex order integrator has been used since 1991 for the design of robust control-systems. In the CRONE control methodology, it permits the parameterization of open loop transfer function which is optimized in a robustness context. Sets of fractional order integrators that lead to a given damping factor have also been used to build iso-damping contours on the Nichols plane. These iso-damping contours can also be used to optimize the third CRONE generation open loop transfer function. However, these contours have been built using nonband-limited integrators, even if such integrators reveal to lead to unstable closed loop systems. One objective of this paper is to show how the band-limitation modifies the left half-plane dominant poles of the closed loop system and removes the right half-plane ones. Also presented are how to obtain a fractional order open loop transfer function with a high phase slope and a useful frequency response, and how the damping contours can be use...
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.
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Bulletin of the Polish Academy of Sciences Technical Sciences, 2013
This paper presents a series of new results on the asymptotic stability of discrete-time fractional difference (FD) state space systems and their finite-memory approximations called finite FD (FFD) and normalized FFD (NFFD) systems. In Part I of the paper, new necessary and sufficient stability conditions have been given in a unified form for FD, FFD and NFFD-based systems. Part II offers a new, simple, ultimate stability criterion for FD-based systems. This gives rise to the introduction of new definitions of the so-called f -poles and f -zeros for FD-based systems, which are used in the closed-loop stability analysis for FD-based systems and, approximately, for FFD/NFFD-based ones.