Note on Commensurate-Order Characteristic Root Equivalency Class of Linear Time Invariant Systems

This study investigates characteristic rootequivalency relations between commensurate order and integerorder Linear Time Invariant (LTI) systems. Author introducessome useful properties of a special class of commensurate ordersystems, which is called characteristic root equivalency class ofLTI systems. These properties present potential to facilitate designand analysis efforts of this class of commensurate order systems.In this sense, straightforward stability checking procedures anddesign approaches for commensurate order root equivalentsystems of the first and second order LTI systems aredemonstrated. Findings of the study are validated by illustrativeexamples.

PDF

___

I. Petras, "Stability of Fractional-Order Systems with Rational Orders: A Survey", Fractional Calculus And Applied Analysis, Vol.12, No.3, 2009, pp.269-298.
B. West, M. Bologna, P. Grigolini, "Physics of Fractal Operators", Springer, New York, 2003.
A. Oustaloup, "La Derivation Non Entiere: Theorie, Synthese et Applications", Hermes, Paris,1995.
I. Podlubny, "Fractional Differential Equations", Academic Press, San Diego,1999.
Y.Q. Chen, K.L. Moore, "Analytical stability bound for a class of delayed fractional-order dynamic systems", Nonlinear Dynamics, Vol.29, 2002, pp.191-200.
F.J.V. Parada, J.A.O. Tapia, J.A. Ramirez, "Erective medium equations for fractional Ficks law in porous media", Physica, Vol.373, 2007, pp. 339-353.
J.A.O. Tapia, F.J.V. Parada, J.A. Ramirez, "A fractional-order Darcys law", Physica Vol.374, 2007, pp.1-14.
P.J. Torvik, R.L. Bagley, "On the appearance of the Fractional Derivative in the Behavior of real materials", Transactions of the ASME, Vol.51, 1984, pp.294-298.
I.S. Jesus, J.A.T. Machado, "Fractional control of heat diffusion systems", Nonlinear Dynamics, Vol.54, No.3, 2008, pp.263-282.
P. Arena, R. Caponetto, L. Fortuna, D. Porto, "Nonlinear Noninteger Order Circuits and Systems - An Introduction", World Scientific, Singapore, 2000.
A. Charef, "Modeling and analog realization of the fundamental linear fractional order differential equation", Nonlinear Dynamics, Vol.46, 2006, pp.195-210.
M. Nakagava, K. Sorimachi, "Basic characteristics of a fractance device", IEICE Trans. Fundamentals, E75 – A, 1992, pp.1814-1818.
M.F. Silva, J.A.T. Machado, A.M. Lopes," Fractional order control of a hexapod robot", Nonlinear Dynamics, Vol.38, 2004, pp.417-433.
M.S. Tavazoei, M. Haeri, "Chaotic attractors in incommensurate fractional order systems", Physica D, Vol. 237, 2008, pp.2628-2637.
I. Podlubny, "Fractional-order systems and PIλDμ-controllers", IEEE Trans. on Automatic Control, Vol.44, 1999, pp.208-213.
H.F. Raynaud, A. Zerganoh, "State-space representation for fractional order controllers", Automatica, Vol.36, No.7, 2000, pp.1017-1021
B.M. Vinagre, I. Petras, P. Merchan, L. Dorcak, "Two digital realizations of fractional controllers: Application to temperature control of a solid", In: Proc. of the ECC'01, Porto, Portugal, 2001, pp. 1764-1767.
G. Madan, "Modern Control System Theory", New Age International, 1993, pp.322-324.
H.S. Ahn, Y.Q. Chen, I. Podlubny, "Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality", Appl. Math. Comput., Vol.187, 2007, pp.27–34.
R. Caponetto, G. Dongola, L. Fortuna, I. Petras, "Fractional Order Systems Modeling and Control Applications", World Scientific Series on Nonlinear Science A series Vol. 72 , World Scientific Publishing Co. Pte. Ltd, 2010.
D. Matignon, “Stability result on fractional differential equations with applications to control processing”, in Proc. IMACS-SMC Proceedings, Lille, France, 1996, pp 963-968.
A.G. Radwan, A.S. Elwakil, A.M. Soliman, "Fractional-Order Sinusoidal Oscillators: Design Procedure and Practical Examples", IEEE Transaction On Circuits and System, Vol.5, 2008, pp.2051-2063.
A.G. Radwan, A.M. Soliman, A.S. Elwakil, A. Sedeek, "On the stability of linear systems with fractional-order elements", Chaos Solitons & Fractals, Vol.40, No.5, 2009, pp.2317-2328
B.B. Alagoz, "Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations", ISA Transactions, 2017, https://doi.org/10.1016/j.isatra.2017.06.005
M. Buslowicz, "Stability Analysis of Linear Continous-time Fractional Systems Of Commensurate Order", Journal of Automation, Mobile Robotics & Intelligent Systems, Vol.3, 2009, pp.12-17.
M., Buslowicz Frequency domain method for stability analysis of linear continuous-time fractional systems. In: K. Malinowski, L. Rutkowski (Eds.): Academic Publishing House EXIT: Warsaw 2008, pp.83-92.
W.C. Wright, T.W. Kerlin, "An efficient computer oriented method for stability analysis of large multivariable systems", Journal of Fluids Engineering, Vol.92, 1970, pp.279-286.
M.S. Tavazoei, M. Haeri, "A note on the stability of fractional order systems", Mathematics and Computers in Simulation, Vol.79, No.5, 2009, pp.1566–1576.
Y.D. Ma, J.G. Lu, W.D. Chen, Y.Q. Chen, "Robust stability bounds of uncertain fractional-order systems", Fractional Calculus and Applied, Vol.17, No.1, 2014, pp.136-153.
J.G. Lu, Y.Q. Chen, "Robust stability and stabilization of fractional order interval systems with the fractional order α: The 0 < α < 1 case", IEEE Trans. Autom. Control, Vol.55, No.1, 2010, pp.152-158.
C. Li, J.C. Wang, " Robust stability and stabilization of fractional order interval systems with coupling relationships: The 0 < α < 1 case", J. of the Franklin Institute, Vol.349, No.7, 2012, pp.2406-2419
C.A. Monje, Y.Q. Chen, B.M. Vinagre, D. Xue, V. Feliu-Batlle, "Fractional-order Systems and Controls Fundamentals and Applications", Springer, 2010.
D. Xue, Y.Q. Chen, D.P. Atherton, "Linear Feedback Control Analysis and Design with MATLAB", Society for Industrial and Applied Mathematics, Philadelphia, 2007.
R.S. Barbosa, J.A.T. Machado, M.F. Silva, "Discretization of complex-order algorithms for control applications", Journal of Vibration and Control, Vol.14, 2008, pp.1349-1361
T.T. Hartley, C.F. Lorenzo, J.L. Adams, “Conjugated-order differintegrals”, in Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, CA, 2005, DETC2005–84951.