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

Abstract

Abstract. 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.

Stability conditions for linear continuous-time fractional-order state-delayed systems

“…We apply the D decomposition method of Nejmark (see [23] for the state of the art and [10] as an example of application) for determining the stability region of (4) in the complex…”

“…The problems of stability of linear continuous-time and discrete-time fractional order systems, standard and positive, have been investigated in the above mentioned monographs and in many papers, see [5][6][7][8][9][10][11][12] for example, and references therein.…”

On simplified forms of the fractional-order backward difference and related fractional-order linear discrete-time system description

“…One may check that the system is marginally stable [9][10][11][12]. The limit cycle is presented in Fig.…”

“…The simplified forms of the FOBDs (19b) and (19c) in the discrete system description should preserve the system stability conditions [10][11][12][13] (asymptotic, marginal stability, instability). There should be a parity between model and their simplified form.…”

Controllability, reachability and minimum energy control of fractional discrete-time linear systems with multiple delays in state

“…The problem of stability of linear fractional order discretetime systems has attracted considerable attention recently, see [20][21][22][23][24].…”