Stability Analysis of Impulsive Positive Systems

Zhang, Jishi; Wang, Yan‐Wu; Xiao, Jiang‐Wen; Guan, Zhi‐Hong

doi:10.3182/20140824-6-za-1003.00021

Cited by 37 publications

(34 citation statements)

References 14 publications

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“…The following result can be seen as a "positive systems" counterpart of the results in [56,57,72] and provides a stability condition in terms of the discrete-time system (2.3). It can also be connected to [39]: Theorem 3.7 (Minimum dwell-time). Let A ∈ R n×n and J ∈ R n×n be a Metzler and a nonnegative matrix, respectively.…”

Section: Stability Under Minimum Dwell-timementioning

confidence: 99%

“…We consider here the case of linear positive impulsive systems, a class of systems that seems to have been quite overlooked until now as only very few results can be found; see e.g. [37][38][39]. Such systems can be used to represent certain classes biochemical, population or epidemiological models having deterministic jumps in their dynamics.…”

Section: Introductionmentioning

confidence: 99%

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Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Briat

2017

Nonlinear Analysis: Hybrid Systems

139

126

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Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinitedimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.

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Section: Stability Under Minimum Dwell-timementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Briat

2017

Nonlinear Analysis: Hybrid Systems

139

126

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“…Impulsive systems often arise in population dynamics, economics, electrical, etc. Therefore, impulsive systems also attract a lot of attention in the past few decades; we refer the reader to related works . Among them, continuous‐time impulsive positive systems have received some attentions .…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, impulsive systems also attract a lot of attention in the past few decades; we refer the reader to related works . Among them, continuous‐time impulsive positive systems have received some attentions . This is partly due to the fact that impulsive positive systems can be used to represent certain classes of population models, epidemiology, ecosystems, etc, which having deterministic jumps in their dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…This is partly due to the fact that impulsive positive systems can be used to represent certain classes of population models, epidemiology, ecosystems, etc, which having deterministic jumps in their dynamics. Zhang et al first investigated stability of impulsive positive linear systems; Briat further considered the dwell‐time stability of impulsive positive linear systems. Due to the importance of time delays in physics analysis, Wang et al, Hu et al, and Yang and Zhang further studied the stability of impulsive positive delay systems.…”

Section: Introductionmentioning

confidence: 99%

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Impulsive control of continuous‐time homogeneous positive delay systems of degree one

Yang

Zhang

2019

Intl J Robust & Nonlinear

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Summary This paper investigates the impulsive control of continuous‐time homogeneous positive delay systems of degree one. By using max‐separable Lyapunov functions and Razumikhin technique, a number of stability criteria for continuous‐time homogeneous impulsive positive delay systems of degree one are obtained. It should be noted that it is the first time that impulsive stabilization results for the addressed systems are given. Three numerical examples are presented to demonstrate the effectiveness of the control strategy.

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Stability analysis of positive switched impulsive systems with delay on time scales

Lü

Wang

et al. 2020

Intl J Robust & Nonlinear

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In this paper, stability analysis of positive switched impulsive systems with delay on time scales is investigated. Firstly, a necessary and sufficient condition is provided for positivity of the considered systems. Secondly, by respectively considering the situations that the switching behavior and the impulsive phenomenon happen simultaneously and asynchronously, the uniformly asymptotical stability problems are addressed. Then, based on the linear copositive Lyapunov-Krasovskii functional technique, two less conservative stability criteria are proposed. It are shown that these two criteria allow the timescale derivative of related Lyapunov functionals to be non-negative at some intervals. Moreover, these two criteria are effective for not only continuous cases and discrete cases, but also some other positive switched impulsive systems on general uniform or hybrid time domains. Finally, a non-continuous and non-discrete example is given to illustrate the effectiveness of the theoretic results. K E Y W O R D S positive switched impulsive systems, stability, time scale 1 INTRODUCTION Positive systems play an important role in many practical areas, such as economics, populations dynamics, ecology and Boolean network, etc. 1-7 The distinctive characteristics of this kind of systems are that their state variables, inputs and outputs take only nonnegative values. The common approaches for stability analysis of linear time-invariant positive systems are the Hurwitz criterion, the Routh criterion, and the Lyapunov technique (including the linear copositive Lyapunov function theorem and the diagonal quadratic Lyapunov function method). 2 And the Lyapunov function approach has been successfully introduced to stability analysis of nonlinear or delayed positive systems. 8,9 Recently, several other kinds of positive systems have also received continued attention, such as switched positive systems, 10-14 impulsive positive systems, 15-19 and switched impulsive positive systems, 20-23 etc. When studying switched impulsive positive systems, a natural question is that do the switching behavior and the impulsive phenomenon happen simultaneously or asynchronously? Unfortunately, the literatures 21-23 only consider the former. Actually, from the viewpoint of reality, it is more reasonable to consider the latter. Moreover, in, 21-23 the necessary conditions for analyzing stability of switched impulsive positive systems are to construct Lyapunov functionals with time-derivative or time-shift being strictly negative. However, this requirement can be surely weakened by the ideas in. 24,25 Generally speaking, continuous positive systems and discrete positive systems are always investigated separately. As far back as 1937, the work 26 stated that a major task of mathematics was to harmonize the continuous and the discrete, to

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Stability Analysis of Impulsive Positive Systems

Cited by 37 publications

References 14 publications

Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Impulsive control of continuous‐time homogeneous positive delay systems of degree one

Stability analysis of positive switched impulsive systems with delay on time scales

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