SINGULAR FRACTIONAL CONTINUOUS-TIME AND DISCRETE-TIME LINEAR SYSTEMS

Abstract New classes of singular fractional continuous-time and discrete-time linear systems are introduced. Electrical circuits are example of singular fractional continuous-time systems. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and Laplace transformation the solution to the state equation of singular fractional linear systems is derived. It is shown that every electrical circuit is a singular fractional systems if it contains at least one mesh consisting of branches with only ideal supercondensators and voltage sources or at least one node with branches with supercoils. Using the Weierstrass regular pencil decomposition the solution to the state equation of singular fractional discrete-time linear systems is derived. The considerations are illustrated by numerical examples.

[1]  M. Otter,et al.  Linear Control Systems , 2009 .

[2]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[3]  Tadeusz Kaczorek,et al.  Selected Problems of Fractional Systems Theory , 2011 .

[4]  Tadeusz Kaczorek,et al.  Fractional Positive Continuous-Time Linear Systems and Their Reachability , 2008, Int. J. Appl. Math. Comput. Sci..

[5]  Vladimír Kučera,et al.  Fundamental theorem of state feedback for singular systems , 1987, Autom..

[6]  Tadeusz Kaczorek,et al.  INFINITE EIGENVALUE ASSIGNMENT BY OUTPUT-FEEDBACKS FOR SINGULAR SYSTEMS , 2004 .

[7]  James L. Melsa,et al.  Linear Control Systems , 1992 .

[8]  Tadeusz Kaczorek.  Positive linear systems with different fractional orders , 2010 .

[9]  T. Kaczorek.  Analysis of fractional electrical circuits in transient states , 2010 .

[10]  M. M. Fahmy,et al.  Matrix pencil of closed-loop descriptor systems: infinite-eigenvalue assignment , 1989 .

[11]  T. Kaczorek.  Polynomial and Rational Matrices: Applications in Dynamical Systems Theory , 2006 .

[12]  L. Dai,et al.  Singular Control Systems , 1989, Lecture Notes in Control and Information Sciences.

[13]  Marija Dodig,et al.  Singular systems, state feedback problem☆ , 2009 .

[14]  Luca Benvenuti,et al.  A tutorial on the positive realization problem , 2004, IEEE Transactions on Automatic Control.

[15]  P. Dooren.  The Computation of Kronecker's Canonical Form of a Singular Pencil , 1979 .

[16]  I. Podlubny.  Fractional differential equations , 1998 .