Exponential stability analysis for discrete-time homogeneous impulsive positive delay systems of degree one
Introduction
Many real world processes in areas such as industrial engineering and communications deal with physical quantities that cannot contain negative values, such systems can be referred to as positive systems. Positive systems often arise in physics, chemistry and economics, etc. Due to their importance and broad applications, more attentions have been paid to positive systems, see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and the references therein. It should be noted that in many practical applications, impulsive control as a type of hybrid control is very effective, see [5], [6], [7], [9], [11], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and the references therein. Among them, impulsive positive systems have received some attentions [5], [6], [7], [9], [11], [13], [14], [15]. This is partly due to the fact that many realistic problems can be modeled as impulsive positive systems, like population growth [32], epidemiology [33], ecosystems [34], etc. By using a linear copositive Lyapunov function, [5], [6], [9], [11], [14] investigated the stability and stabilization for continuous-time impulsive positive linear systems. Based on the mode-dependent average dwell time approach, [7] analyzed the asynchronously finite-time control of discrete impulsive switched positive linear time-delay systems. By using the multiple linear copositive Lyapunov function together with average dwell time approach, [13] investigated the guaranteed cost finite-time boundedness of discrete-time impulsive switched positive linear systems. By using max-separable Lyapunov functions, [15] investigated the exponential stability for discrete-time homogeneous impulsive positive delay systems of degree one. It should be noted that [7] is only concerned with the asynchronously finite-time control of discrete impulsive switched positive linear time-delay systems; [13] is only concerned with the guaranteed cost finite-time boundedness of discrete-time impulsive switched positive linear systems; for discrete-time homogeneous impulsive positive delay systems of degree one, [15] is only concerned with the case where the original impulse-free system is stable and enlarging impulses potentially destroy the stability property of the original system; [5], [6], [9], [11], [14] are only concerned with continuous-time impulsive positive linear systems. However, many phenomena are described by discrete-time systems. For example, in the fields of industrial engineering especially the numerical simulation, the models are always discrete-time systems. Futhermore, the behavior of discrete-time systems is sometimes different from the corresponding continuous-time systems [35], [36]. Moreover, to the best of our knowledge, less stability analysis has been previously reported for discrete-time impulsive positive systems.
Time delays exist in many physical systems. For general systems, time delays may degrade system’s performance, induce oscillations, even make systems out of control [37]. However, for discrete-time homogeneous positive delay systems, Theorem 4 in [38] showed that the discrete-time positive system is insensitive to time-varying bounded delays, which means that a discrete-time homogeneous positive delay system is globally exponentially stable if and only if the corresponding delay-free system is globally exponentially stable; Theorem 4.3 in [39] further showed that the discrete-time positive system is insensitive to time-varying unbounded delays.
In previous works that concern discrete-time impulsive positive linear systems, it is always assumed that the states on impulsive jumping moments are only related to the nearest one and the impulsive control law is usually taking the form . However, for some impulsive systems, the designed impulsive control strategy involves the time delays. The main reasons are as follows: (1) during the transmission of information on impulsive moments, time delays are inevitable and thus input delays are often encountered [17]; (2) time delays in impulses may contribute to the stabilization of some unstable systems. A control strategy, which does not work without impulsive delay feedback, can be activated to stabilize some unstable systems when impulsive delay feedback is considered [30]. (3) undesired delay disturbance in impulses can occur due to the robustness of unstable systems [19]. Therefore, in recent years, discrete-time dynamical systems with delayed impulses have been investigated, such as stability analysis of discrete-time delay difference equations with delayed impulses [21], synchronisation analysis of discrete-time complex networks with delayed heterogeneous impulses [26]. Motivated by the above discussions, we will consider in this paper more general delayed impulses taking the form . It’s worth noting that in Theorem 4 of [38], the discrete-time homogeneous positive delay system of degree one is stable. If the original discrete-time system is unstable, then the results in [38] are not applicable. So a problem arises: can the unstable discrete-time homogeneous positive delay system of degree one be stabilized by delayed impulses? Inspired by some enlightening works [20], [21], [38], we will answer the question. To the best of the authors’ knowledge, up to now, impulsive stabilization for discrete-time homogeneous positive delay systems of degree one has not been addressed yet.
The purpose of this paper is to investigate the stability of discrete-time homogeneous impulsive positive delay systems of degree one. The main contribution of this paper is as follows: (1) by employing max-separable Lyapunov functions together with Razumikhin technique and impulsive control theory, some new stability criteria for discrete-time homogeneous impulsive positive delay systems of degree one are obtained; (2) the obtained results show that the stabilizing delayed impulses can stabilize an unstable discrete-time homogeneous positive delay system of degree one; (3) the obtained results also show that a stable discrete-time homogeneous positive delay system of degree one can admit certain destabilizing delayed impulses.
The remainder of this paper is organized as follows. In Section 2, some notations and definitions are given. The main results are presented in Section 3. Two examples are stated in Section 4 to show the effectiveness of our results. Finally, conclusions are given in Section 5.
Section snippets
Notations and preliminaries
Let denote the set of nonnegative integers (integers, positive integers, real numbers, nonnegative real numbers). Let denote the n-dimensional nonnegative (real) space. Let Rn × m and denote the set of all real matrices of (n × m)-dimension and the set of all real matrices of (n × m)-dimension whose elements are nonnegative, respectively. For where aij denotes the entry in row i and column j, A ⪰ 0 (≻0) means that all elements of the matrix A are
Positivity of the system
Consider the following discrete-time nonlinear system:where is the initial function, x(m) ∈ Rn is the state variable, f, g: Rn → Rn, τ1(m) is the time-varying state delay and satisfies τ1(m) ∈ N, 0 ≤ τ1(m) ≤ τ1 for all m ∈ N, τ1( ∈ N) is the bound of the maximum allowable time-varying state delay.
Correspondingly, the controlled nonlinear system can be described as
Numerical examples
In this section, two numerical examples will be given to verify the effectiveness of the obtained results. Example 1 Consider a discrete-time homogeneous impulsive positive delay system of degree one (4) with the following system data:where About
Conclusion
Exponential stability of discrete-time homogeneous impulsive positive delay systems of degree one has been investigated in this paper. By using max-separable Lyapunov functions together with Razumikhin technique, a number of stability criteria for discrete-time homogeneous impulsive positive delay systems of degree one have been given. These results show that to stabilize an unstable discrete-time system, the stabilizing impulses should act frequently; to keep the stability of the original
Acknowledgments
The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions which improved the quality of the paper. This work was supported by the Fundamental Research Funds for the Central Universities.
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