Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems

In the paper the problems of practical stability and asymptotic stability of fractional discrete-time linear systems are addressed. Necessary and sufficient conditions for practical stability and for asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix of the system. In particular, it is shown that (similarly as in the case of fractional continuous-time linear systems) in the complex plane exists such a region, that location of all eigenvalues of the state matrix in this region is necessary and sufficient for asymptotic stability. The parametric description of boundary of this region is given. Moreover, it is shown that Schur stability of the state matrix (all eigenvalues have absolute values less than 1) is not necessary nor sufficient for asymptotic stability of the fractional discrete-time system. The considerations are illustrated by numerical examples.

[1]  Tadeusz Kaczorek,et al.  Simple Conditions for Practical Stability of Positive Fractional Discrete-Time Linear Systems , 2009, Int. J. Appl. Math. Comput. Sci..

[2]  Tadeusz Kaczorek,et al.  Practical stability and asymptotic stability of positive fractional 2D linear systems , 2010 .

[3]  Jürgen Ackermann,et al.  Sampled-Data Control Systems , 1985 .

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

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

[6]  Mohammad Saleh Tavazoei,et al.  A note on the stability of fractional order systems , 2009, Math. Comput. Simul..

[7]  Tadeusz Kaczorek,et al.  Practical stability of positive fractional discrete-time linear systems , 2008 .

[8]  YangQuan Chen,et al.  Fractional-order Systems and Controls , 2010 .

[9]  D. Sierociuk,et al.  Stability of Discrete Fractional Order State-space Systems , 2008 .

[10]  M. Buslowicz.  Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders , 2012 .

[11]  Luigi Grippo,et al.  Practical stability of discrete-time systems , 1976 .

[13]  Guido Maione,et al.  Loop‐Shaping and Easy Tuning of Fractional‐Order Proportional Integral Controllers for Position Servo Systems , 2013 .

[14]  I. Petráš.  Stability of Fractional-Order Systems with Rational Orders , 2008, 0811.4102.

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

[16]  R Melício,et al.  Power converter topologies and fractional-order controllers: Wind energy applications , 2010, SPEEDAM 2010.

[17]  Dominik Sierociuk,et al.  Some applications of fractional order calculus , 2010 .

[18]  M. Busłowicz,et al.  Stability of linear continuous-time fractional order systems with delays of the retarded type , 2008 .

[19]  Y. Chen,et al.  Fractional Order Motion Controls , 2012 .

[20]  Mathieu Moze,et al.  LMI stability conditions for fractional order systems , 2010, Comput. Math. Appl..

[21]  M. Busłowicz,et al.  STABILITY OF STATE-SPACE MODELS OF LINEAR CONTINUOUS-TIME FRACTIONAL ORDER SYSTEMS , 2011 .

[22]  M. Buslowicz.  Simple analytic conditions for stability of fractional discrete-time linear systems with diagonal state matrix , 2012 .

[23]  R. Feynman,et al.  RECENT APPLICATIONS OF FRACTIONAL CALCULUS TO SCIENCE AND ENGINEERING , 2003 .

[24]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[25]  Piotr Ostalczyk,et al.  Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains , 2012, Int. J. Appl. Math. Comput. Sci..

[26]  Boris Polyak,et al.  D-decomposition technique state-of-the-art , 2008 .

[27]  Wojciech P. Hunek,et al.  Finite approximations of a discrete-time fractional derivative , 2011, 2011 16th International Conference on Methods & Models in Automation & Robotics.

[28]  Krzysztof J. Latawiec,et al.  Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for the asymptotic stability , 2013 .

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

[30]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .