Stability of fractional positive continuous-time linear systems with state matrices in integer and rational powers

Abstract

Abstract. The stability of fractional standard and positive continuous-time linear systems with state matrices in integer and rational powers is addressed. It is shown that the fractional systems are asymptotically stable if and only if the eigenvalues of the state matrices satisfy some conditions imposed on the phases of the eigenvalues. The fractional standard systems are unstable if the state matrices have at least one positive eigenvalue.Key words: stability, fractional, positive, linear, continuous-time, system, integer, rational, order. Stability of fractional positive continuous-time linear systemswith state matrices in integer and rational powers The following notation will be used: ℜ -the set of real numbers, ℜ n×m -the set of n×m real matrices, ℜ + n×m -the set of n×m real matrices with nonnegative entries and ℜ + n = ℜ + n×1 , M n -the set of n×n Metzler matrices (real matrices with nonnegative off-diagonal entries), I n -the n×n identity matrix. PreliminariesConsider the fractional autonomous continuous-time linear system BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. XX, No. Y, 2016 DOI: 10.1515/bpasts-2016 Stability of fractional positive continuous-time linear systems with state matrices in integer and rational powers

Bulletin of the Polish Academy of Sciences: Technical Sciences

“…The above considerations can be extended to linear systems with state matrices in integer and rational powers [13].…”

“…The stability of interval positive linear systems with state matrices in integer and rational powers has been analyzed in [14]. The linear systems have been intensively investigated in [11,[12][13][14][15][18][19][20] and the descriptor fractional linear systems with different fractional orders in [21].…”

Bulletin of the Polish Academy of Sciences: Technical Sciences

“…A central and automatic mechanism for controlling the prototyping process based on a program arbitrator may be an effective solution in this area. This architecture also allows the implementation of the advanced control mechanisms, such as those described in the papers [7][8][9].…”

Bulletin of the Polish Academy of Sciences: Technical Sciences

“…During the past decades, fractional order systems, including fractional order difference systems and fractional order differential systems, have been paid much attention due to their significant applications in the fields such as biology, physics, aerodynamics, electrical circuits, nonlinear oscillation of earthquake. Many significant results on the theory and application of fractional order systems have been obtained, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The basic theory of the fractional calculus are given in [1,2].…”

“…The existence of solutions for fractional differential systems has been investigated in [6][7][8][9]. The stability of fractional differential systems has been considered in [5,19]. The applications of fractional order differential systems in HIV model, SIR model, multi-agent systems and chaotic systems have been discussed in [3,4,10] and [11], respectively.…”