Stability and stabilization of linear positive systems on time scales

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Positivity

Stability and stabilization of linear positive systems on time scales Zbigniew Bartosiewicz1 Received: 17 May 2019 / Accepted: 10 January 2020 © The Author(s) 2020

Abstract It is shown that a positive linear system on a time scale with a bounded graininess is uniformly exponentially stable if and only if the characteristic polynomial of the matrix defining the system has all its coefficients positive. Then this fact is used to find necessary and sufficient conditions of positive stabilizability of a positive control system on a time scale. Keywords Positive linear system · Positive stability · Positive stabilization Mathematics Subject Classification 93D05 · 93D15

1 Introduction In positive systems the state variables take nonnegative values. Such systems appear in biology, medicine and economics [1,2]. We study here linear positive systems on time scales. A time scale is a model of time. Time may be continuous, discrete or mixed—partly continuous and partly discrete. The delta derivative, which is used in delta differential equations that model positive systems on time scales, may be equal to ordinary derivative or may be equal to a difference quotient, depending on the time scale and a particular point (see Appendix for the precise definitions). It is known that in the continuous-time case the positive system x˙ = Ax is exponentially stable (at 0) if and only if the characteristic polynomial of the matrix A, χ A (λ) = det(λI − A), has all its coefficients positive (see [1,2]). We show that this is true for a system on an arbitrary time scale as long as the graininess function of

This work has been supported by the Bialystok University of Technology Grant No. S/WI/00/2020.

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Zbigniew Bartosiewicz [email protected] Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15-351 Białystok, Poland

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Z. Bartosiewicz

the time scale (which measures the distance between a particular time instant and the next instant) is bounded and uniform exponential stability is considered. Thus any discretization (in particular nonuniform) of a positive continuous-time uniformly exponentially stable system that preserves positivity gives an uniformly exponentially stable system. We rely here on [3,4], where uniform exponential stability of linear systems was studied. Stability of linear systems on time scales was also considered in [5–14]. Various authors considered different concepts of stability and different variants of the same concept. For example, in the definition of exponential stability, to estimate the solutions, either the standard exponential function was used or the exponential function on the time scale. Relations between asymptotic stability and exponential stability for continuous-time systems were studied in [15]. If a control system is not uniformly exponentially stable, we can try to use feedback to stabilize the system. If our system is positive, it is natural to require that the feedback preserves positivity. This procedure is called positive stabilization. We g