Abstract
The fundamental system theory is presented for a class of real systems whose derivative order is complex. It is demonstrated that these so-called conjugate-order systems have a scale-invariance property in both the time and frequency domains, which makes them useful for describing certain phenomena in continuous media. The conditions for which these systems are guaranteed to be causal and stable are reviewed. The compactness properties of their Hankel operators, which allow them to admit finite-order approximations, are also discussed. A procedure is developed for choosing appropriate transfer-function parameter values to design a stable conjugate-order system whose frequency response meets given bandwidth, resonance, and ripple specifications.
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References
Maskarinec G., Onaral B.: A class of rational systems with scale-invariant frequency response. In: IEEE Trans. Circuits Syst. I. Fundam. Theory Appl. 41, 75–79 (1994)
Nigmatullin R.R., Arbuzov A., Salehli F., Gis A., Bayrak I., Catalgil-Giz H.: The first experimental confirmation of the fractional kinetics containing the complex power-law exponents: dielectric measurements of polymerization reaction. Phys. B Phys. Condens. Matter 388, 418–434 (2007)
Hartley, T.T., Lorenzo, C.F., Adams, J.L.: Conjugated-order differintegrals. In: Proceedings of DETC05. ASME (2005)
Nigmatullin R., Mehaute A.L.: Is there a geometrical/physical meaning of the fractional integral with complex exponent?. J. Non-Cryst. Solids 351, 2888–2899 (2005)
Oustaloup A., Sabatier J., Lanusse P.: From fractal robustness to the CRONE control. Fract. Calc. Appl. Anal. 2, 1–30 (1999)
Sornette, D.: Discrete-scale invariance and complex dimensions. Phys. Rep. 297
Adams J.L., Hartley T.T., Adams L.I.: A solution to the fundamental linear complex-order differential equation. Adv. Eng. Softw. 41, 70–74 (2010)
Matignon, D.: Stability properties for generalized fractional differential systems. In Proceedings of ESIAM, vol. 5, pp. 145–158 (1998)
Partington J.R.: An Introduction to Hankel Operators. Cambridge University Press, Cambridge (1988)
Glover K.: All optimal Hankel-norm approximations of linear multivariable systems and their L ∞-error bounds. Int. J. Control 39, 1115–1193 (1984)
Ortigueira M.D.: On the initial conditions in continuous time fractional linear systems. Signal Processing 83, 2301–2309 (2003)
Lorenzo C.F., Hartley T.T.: Initialization of fractional-order operators and fractional differential equations. J. Comput. Nonlinear Dyn. 3(2), 021101 (2008)
Conway J.B.: A Course in Functional Analysis, 2nd edn. Springer, New York (1990)
Conway J.B.: Functions of One Complex Variable, 2nd edn. Springer, New York (1978)
Adams J.L., Veillette R.J., Hartley T.T.: Hankel-norm estimation for fractional-order systems using the Rayleigh-Ritz method. Comput. Math. Appl. 59, 1773–1781 (2010)
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Adams, J.L., Veillette, R.J. & Hartley, T.T. Conjugate-order systems for signal processing: stability, causality, boundedness, compactness. SIViP 6, 373–380 (2012). https://doi.org/10.1007/s11760-012-0327-z
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DOI: https://doi.org/10.1007/s11760-012-0327-z