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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 17
A Solution to the Fundamental Linear Complex-Order Differential Equation J.L. Adams1, T.T. Hartley1, L.I. Adams1 and C.F. Lorenzo2
1The University of Akron, Ohio, United States of America
Full Bibliographic Reference for this paper
J.L. Adams, T.T. Hartley, L.I. Adams, C.F. Lorenzo, "A Solution to the Fundamental Linear Complex-Order Differential Equation", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 17, 2006. doi:10.4203/ccp.84.17
Keywords: fractional calculus, fractional-order systems, fractional-order differential equations, complex-order derivatives, complex-order systems, complex-order differential equations.
Summary
The idea of an integer-order differintegral operator has previously been extended
to a differintegral operator of non-integer, but real, order. Further generalizations
have been made to the complex-order operator. Hartley, et al. [1] showed that when a
complex-order operator is paired with its conjugate-order operator, real
time-responses are created. Further, as long as the coefficients of the operators are
complex-conjugates, a real time-response results [1]. Systems with complex-order
differintegrals can arise from a variety of situations. They appear to arise naturally
with the Cauchy-Euler differential equation. The CRONE controller [2] makes use of
conjugated-order differintegrals in a limited manner. Such a system can result from
system identification. A system with complex-order can also be artificially
constructed and implemented using FPGA techniques.
The system whose transfer function is
This function represents the impulse response of the differential equation
The poles of the transfer function,
The Bode magnitude plot has expected characteristics. For frequencies much less
than a critical point, the magnitude is constant. For frequencies much greater than a
critical point, the magnitude response has a slope of
Several examples serve to verify the analytical results. One example shows that
these systems can display the interesting feature of dual fractional-order resonances.
Specifically, consider
References
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