Page 142 - Mathematical Techniques of Fractional Order Systems

130 Mathematical Techniques of Fractional Order Systems

from irrationality and the roots of both can be expressed by means of
Lambert W functions in certain cases. It was also pointed out that LTI dis-
tributed order systems with exponential weight functions have infinitely
many poles just like the infinite-dimensional time-delay systems.
The methodology employed in this chapter may be extended to obtain
exact representations of the solutions of distributed order systems with more
general weight functions. Moreover, the graphical technique offered by Jiao
et al. (2012) and used in stability analysis of distributed order systems in this
chapter can also be applied to systems with more general weight functions.
Nevertheless, the characteristic functions become more complex and are no
longer able to be expressed by means of Lambert W functions. Studying dis-
tribution of system poles on the complex plane however is not possible with-
out analyzing the characteristic function of the system. Thus a rigorous study
of the characteristic functions of distributed order systems which can be
expressed in terms of Lambert W functions in the special case of exponential
weight functions is an interesting path for future research.

REFERENCES
Aslam, M.S., Raja, M.A.Z., 2015. A new adaptive strategy to improve online secondary path
modeling in active noise control systems using fractional signal processing approach. Signal
Proc. 107, 433 443.
Azar, A.T., Vaidyanathan, S., Ouannas, A., 2017. Fractional order control and synchronization
of chaotic systems, Stud. Comput. Intelligence, Vol. 688. Springer-Verlag, Germany.
Banwell, T.C., Jayakumar, A., 2000. Exact analytical solution for current flow through diode
with series resistance. Electron. Lett. 36 (4), 291 292.
Bekir, A., Guner, O., Cevikel, A., 2017. The exp-function method for some time-fractional dif-
ferential equations. IEEE/CAA J Automat Sinica 4 (2), 315 321.
Bell, W.W., 2004. Special Functions for Scientists and Engineers. Courier Corporation, London.
Cao, W., Zeng, F., Zhang, Z., Karniadakis, G.E., 2016. Implicit-explicit difference schemes for
nonlinear fractional differential equations with nonsmooth solutions. SIAM J. Sci. Comput.
38 (5), A3070 A3093.
Caputo, M., Carcione, J.M., 2013. A memory model of sedimentation in water reservoirs. J.
Hydrol. 476, 426 432.
Chechkin, A., Gorenflo, R., Sokolov, I., 2002. Retarding subdiffusion and accelerating superdiffu-
sion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66 (4), 046129.
Chechkin, A.V., Gorenflo, R., Sokolov, I.M., Gonchar, V.Y., 2003. Distributed order time frac-
tional diffusion equation. Fract. Calculus Appl. Anal. 6 (3), 259 280.
Corless, R.M., Gonnet, G.H., Hare, D.E., Jeffrey, D.J., Knuth, D.E., 1996. On the lambertw func-
tion. Adv. Computat. Math. 5 (1), 329 359.
Dumlu, A., Erenturk, K., 2014. Trajectory tracking control for a 3-dof parallel manipulator using
λ μ
fractional-order pi d control. IEEE Trans. Ind. Electron. 61 (7), 3417 3426.
Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., 1954. Tables of Integral Transforms,
Vol. 2. McGraw-Hill Book Company, Incorporated, London.
Fisher, B., Al-Sirehy, F., 2015. Convolutions involving the exponential function and the expo-
nential integral. Math. Moravica 19 (2), 65 73.