Positivity and Stability of Time-Varying Discrete-Time Linear Systems

Abstract

IntroductionA dynamical system is called positive if its trajectory starting from any nonnegative initial condition state remains forever in the positive orthant for all nonnegative inputs. An overview of state of the art in positive system theory is given in the monographs [7,10] and in the papers [11][12][13][14]. Models having positive behavior can be found in engineering, economics, social sciences, biology and medicine, etc. The Lyapunov, Perron and Bohl exponents and stability of time-varying discretetime linear systems have been investigated in [1][2][3][4][5][6][7][8][9]. The positive standard and descriptor systems, and their stability have been analyzed in [13][14][15][16]. The positive linear systems with different fractional orders have been addressed in [14,17] and the singular discrete-time linear systems in [15].In this paper the positivity and asymptotic stability of the time-varying discretetime linear systems will be investigated.The paper is organized as follows. In section 2 the solution of the state-equation describing the time-varying discrete-time linear system is derived and necessary and sufficient conditions for the positivity of the systems are established. The asymptotic stability of the positive systems is addressed in section 3, where conditions for the stability are proposed. Concluding remarks are given in section 4.The following notation will be used:-the set of real numbers, -the set of real matrices, -the set of matrices with nonnegative entries and , -the identity matrix.

Global Stability of Positive Discrete-time Time-varying Nonlinear Feedback Systems

“…The dynamical properties of linear systems with state Metzler matrices have been analyzed in [17]. The decentralized stabilization of descriptor fractional positive systems with delays have been investigated in [21,22] and the practical stability and robust stability of discrete-time fractional linear systems in [18,20]. The global stability of positive time-varying nonlinear feedback timevarying systems has been considered in [13].…”

On Positive Solutions of a Delay Equation Arising When Trading in Financial Markets

“…As mentioned previously, we work with the specific initial conditions X(0) = X(1) = X 0 > 0. Although our goal, state positivity, is the same as in existing positive system theory, for example, see [12] and [13], this body of work is not in play because the matrix A(v, k) can have a negative entry.…”

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].…”