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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. Adams
^{1}, T.T. Hartley^{1}, L.I. Adams^{1} and C.F. Lorenzo^{2}
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
, has an impulse response that is
given by the This function represents the impulse response of the differential equation
The poles of the transfer function,
are located
at
, where
and
.
The paper demonstrates that there are a finite number of poles on the primary
Riemann sheet for systems whose order is not purely imaginary. The bounds on
for a stable system are found to be the intersection of disks with center in
the fourth quadrant whose boundary contains the origin in the
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
, with some ripple
which is dependent on the value 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 . This system has primary Riemann-sheet poles at and . The step response exhibits two frequencies that correspond to these two imaginary frequencies. The frequency response contains two resonances, one at each of these frequencies. The analysis presented in this paper serves as a foundation for the study of complex-order systems involving more conjugate-order pairs. References
- 1
- T.T. Hartley, C.F. Lorenzo, and J.L. Adams, "Conjugated-Order Differintegrals", Proc. of 2005 ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach, CA, USA, September 24-28, 2005. doi:10.1115/DETC2005-84951
- 2
- A. Oustaloup, J. Sabatier, and P. Lanusse, "From fractal robustness to CRONE control", Fractional Calculus and Applied Analysis, vol. 2, no. 1, 1-30, 1999.
- 3
- T.T. Hartley, C.F. Lorenzo, "Dynamics and Control of Initialized Fractional-Order Systems", Nonlinear Dynamics vol 29 (1-4), 201-233, 2002. doi:10.1023/A:1016534921583
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