Abstract
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.
Highlights
The mathematical modeling of FO systems and processes, based on the description of their properties in terms of Frac- tional Derivatives (FDs), leads to differential equations of in- volving fractional derivatives (FD) that must be analyzed
The mathematical modeling of FO systems and processes, based on the description of their properties in terms of Frac- tional Derivatives (FDs), leads to differential equations of in- volving FDs that must be analyzed. These are generally termed as Fractional Differential Equations (FDEs)
The advantages of FDs become apparent in applications including modeling of damping behaviour of visco-elastic materials, cell diffusion processes [8], transmission of signals through strong magnetic fields, modeling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields [25]
Summary
The mathematical modeling of FO systems and processes, based on the description of their properties in terms of Frac- tional Derivatives (FDs), leads to differential equations of in- volving FDs that must be analyzed. These are generally termed as Fractional Differential Equations (FDEs). Fractional Derivatives (FDs) provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage over the IO models, which possess limited memory. These functions play important role in the theory of FC and in the theory of fractional differential equations (FDEs)
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