Extending the Root-Locus Method to Fractional-Order Systems

The well-known root-locus method is developed for special subset of linear time-invariant systems known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variable s. Such systems are defined on a Riemann surface because of their multivalued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root-locus method such as asymptotes, roots condition on the real axis, and breakaway points are extended to fractional case. It is also shown that the proposed method can assess the closed-loop stability of fractional-order systems in the presence of a varying gain in the loop. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.

[1]  Georges Oppenheim,et al.  Continuous-time fractional ARMA processes , 1994 .

[2]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[3]  C. Byrnes.  Root-Locus and Boundary Feedback Design for a Class of Distributed Parameter Systems , 1994 .

[4]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[5]  Banu Onaral,et al.  An algorithm for determining global parameters of minimum-phase systems with fractional power spectra , 1989 .

[6]  T. Poinot,et al.  Identification of Fractional Systems Using an Output-Error Technique , 2004 .

[7]  I. Podlubny.  Fractional-Order Systems and -Controllers , 1999 .

[8]  R. Rubin,et al.  Singularities of Linear System Functions , 1961 .

[9]  Duarte Valério,et al.  Tuning of fractional PID controllers with Ziegler-Nichols-type rules , 2006, Signal Process..

[10]  D. Matignon.  Stability properties for generalized fractional differential systems , 1998 .

[11]  Le Page,et al.  Complex Variables and the Laplace Transform for Engineers , 2010 .

[12]  Igor Podlubny,et al.  On Fractional Derivatives, Fractional-Orlder Dynamic Systems and PIXD~-controllei~s , 1997 .

[13]  B.M. Vinagre,et al.  Frequency domain identification of a flexible structure with piezoelectric actuators using irrational transfer function models , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[14]  Allan M. Krall,et al.  A Closed Expression for the Root Locus Method , 1963 .

[15]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[16]  Alexis Chauchois,et al.  Use of noninteger identification models for monitoring soil water content , 2003 .

[17]  A. M. Eydgahi,et al.  Complementary root locus revisited , 2001, IEEE Trans. Educ..

[18]  Horst R. Beyer,et al.  Definition of physically consistent damping laws with fractional derivatives , 1995 .

[19]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[20]  Ezekiel Bahar,et al.  Numerical Technique to Trace the Loci of the Complex Roots of Characteristic Equations , 1981 .

[21]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[22]  Alain Oustaloup,et al.  The CRONE Control of Resonant Plants: Application to a Flexible Transmission , 1995, Eur. J. Control.

[23]  Gerald Berman,et al.  The Asymptotes of the Root Locus , 1963 .

[24]  W. Hayman,et al.  COMPLEX FUNCTIONS An Algebraic and Geometric Viewpoint , 1988 .

[25]  Xavier Moreau,et al.  The CRONE Suspension , 1996 .

[26]  Allan M. Krall,et al.  The Root Locus Method , 1970 .

[27]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[28]  Anissa Zergaïnoh-Mokraoui,et al.  State-space representation for fractional order controllers , 2000, Autom..