New Stability Tests of Positive Standard and Fractional Linear Systems

Abstract

New tests for checking asymptotic stability of positive 1D continuous-time and discrete-time linear systems without and with delays and of positive 2D linear systems described by the general and the Roesser models are proposed. Checking of the asymptotic stability of positive 2D linear systems is reduced to checking of suitable corresponding 1D positive linear systems. It is shown that the stability tests can be also applied to checking the asymptotic stability of fractional discrete-time linear systems with delays. Effectiveness of the tests is shown on numerical examples.

Minimum energy control of positive 2D continuous-discrete linear systems with bounded inputs

“…Theorem 5 Letū(t) ∈ ℜm + for t ∈ [0,t f ] be an input satisfying (14) that steers the state of the positive system (1) from…”

“…Using (16) and (30) compute the input (16) and t f satisfying the condition (14) for given U ∈ ℜ n + and x f ∈ ℜ n + .…”

“…, q to x f = x(t f , 0) + x(t f , 1) + ... + x(t f , q) ∈ ℜ n + and minimizes the performance index (13) Proof If the conditions (17) are met thenû(t) ∈ ℜm + for t ∈ [0,t f ]. We shall show that the input satisfying (14) steers the state of the system from x(0, i) = 0, i = 0, 1, . .…”

“…From the above considerations we have the following procedure for computation the optimal inputs satisfying the condition (14) that steers the state of the system from x 0 = 0 to x f ∈ ℜ n + and minimize the performance index (13).…”

Stability Tests of Positive Fractional Continuous-time Linear Systems with Delays

“…The problems of stability and control of system with delays have been considered in [3,11,12,13,25]. The stability and the robust stability of positive discretetime linear systems without delays and with delays have been investigated in [1][2][3][4][5][6][7][8][9][10][14][15][16][17][18][19][20][21][22][23][24][25][26].…”