Positivity and stability of fractional descriptor time–varying discrete–time linear systems

Abstract

The Weierstrass-Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor timevarying discrete-time linear systems. A method for computing solutions of fractional systems is proposed. Necessary and sufficient conditions for the positivity of these systems are established.

Robust Controlled Positive Delayed Systems with Interval Parameter Uncertainties: A Delay Uniform Decomposition Approach

“…Positive systems are largely encountered in many real process (biology, statistics, thermodynamics, ecology, networking, etc.). Accordingly, many researchers are continuously interested in these systems (Luenberger, 1976;Shorten et al, 2006;Zhang and Yang, 2013;Kaczorek, 2014;2016;Shuqian et al, 2014;Junfeng et al, 2017). Starting from a nonnegative initial state, the key mathematical property of positive systems is the state evolution in the positive orthant for all nonnegative inputs.…”

Reduced–Order Perfect Nonlinear Observers of Fractional Descriptor Discrete–Time Nonlinear Systems

“…Descriptor (singular) linear systems were investigated by Cuihong (2012), Dodig and Stosic (2009), Dai (1989), Duan (2010), Fahmy and O'Reill (1989), Gantmacher (1959), Kaczorek (2012b;2004;1992;2012a), Kucera and Zagalak (1988), Lewis (1983), Luenberger (1977;1978), Sajewski (2016), Van Dooren (1979) or Virnik (2008), and the positivity and stability of fractional descriptor time-varying discrete-time linear by Kaczorek (2016c), who also addressed the eigenvalues and invariants assignment by state and input feedbacks (Kaczorek, 2004;1992;2011b). The computation of Kronecker's canonical form of a singular pencil was analyzed by Van Dooren (1979).…”

Determinants of the matrices of solutions to the standard and positive linear electrical circuits

“…The Lyapunov, Bohl and Perron exponents and stability of time-varying discrete-time linear systems have been investigated in [1][2][3][4][5][6]. The positivity and stability of timevarying linear systems have been addressed in [12,16,18,20,22,23,28] and the stability of continuous-time linear systems with delays in [26]. The fractional positive linear systems have been analyzed in [10,11,13,19,21,24,25,29].…”