Positive Delay Systems Research Articles

Separation of solutions and the attractivity of fractional-order positive linear delay systems with variable coefficients

This article has two purposes. First, we study the separation of solutions to mixed-order positive linear delay systems with variable coefficients and the lower estimates of the separation of solutions to those systems. Second, we consider the global attractivity of fractional-order positive linear delay systems and describe precisely the rate of convergence of solutions to their equilibrium in some specific cases. To do this, the unified approach we use here is that the comparison principles have been modified to accommodate fractional-order delay systems. In addition, numerical simulations are introduced to illustrate the validity of the theoretical results.

Retracted: Global Exponential Stability of Both Continuous-Time and Discrete-Time Switched Positive Time-Varying Delay Systems with Interval Uncertainties and All Unstable Subsystems

Retracted: Global Exponential Stability of Both Continuous-Time and Discrete-Time Switched Positive Time-Varying Delay Systems with Interval Uncertainties and All Unstable Subsystems

Conditions of stability and guaranteed convergence rate for linear time-varying discrete-time delay systems

This work introduces novel results on the stability of linear time-varying discrete-time delay systems, addressing both internal stability and input-to-state stability. We start from the special case of positive delay systems and provide conditions of global exponential stability, both delay-dependent and delay-independent. Moreover, we formulate an explicit bound on the exponential decay rate of the state evolution, as a function of the maximum delay. We then show that the aforementioned stability conditions also prove input-to-state stability. The stability conditions consist of simple linear inequalities, albeit infinite-many due to the time-varying nature of the system. To overcome the computational burden, we also introduce a simple relaxation that allows to check the stability with a finite test, at the expense of conservatism. Finally, leveraging comparison systems and results from the theory of non-negative matrices, we generalize all the aforementioned results to systems with no sign constraint, for which we still provide stability conditions by means of linear inequalities and guaranteed convergence rate. Some consequences and a comparative example are provided for the well investigated special case of systems with constant matrices and time-varying delays.

A Switching Strategy for Stabilization of Discrete-Time Switched Positive Time-Varying Delay Systems with All Modes Being Unstable and Application to Uncertain Data

The stability problem of switched systems plays an essential role in the study of long-term behavior. In fact, systems containing both time delay and uncertainty terms may lead to performance degradation of those systems. Therefore, we are interested in the robust stability for discrete-time switched positive time-varying delay systems with interval uncertainties in the case of all modes being unstable. Based on the proposed time-scheduled multiple co-positive Lyapunov–Krasovskii functional of each mode, new sufficient conditions for the global uniform asymptotic stability of the systems are derived. An effective time-dependent switching law utilized in this work is mode-dependent dwell time. In addition, the robust stability criteria in an asymptotic sense are formulated for the systems without time-varying delay. Compared with the existing related works, our results are less conservative and more general than some previous research. Finally, two numerical examples are provided to illustrate the effectiveness and correctness of the developed theoretical results.

Global Exponential Stability of Both Continuous-Time and Discrete-Time Switched Positive Time-Varying Delay Systems with Interval Uncertainties and All Unstable Subsystems

The global stability problem for a class of linear switched positive time-varying delay systems (LSPTDSs) with interval uncertainties by means of a fast average dwell time (FADT) switching is analyzed in this paper. A distinctive feature of this research is that all subsystems are considered to be unstable. Both the continuous-time and the discrete-time cases of LSPTDSs with interval uncertainties and all unstable subsystems (AUSs) are investigated. By constructing a time-scheduled multiple copositive Lyapunov-Krasovskii functional (MCLKF), novel sufficient conditions are derived within the framework of the FADT switching to guarantee such systems in the case of continuous-time to be globally uniformly exponentially stable. Based on the above approach, the corresponding result is extended to the discrete-time LSPTDSs including both interval uncertainties and AUSs. In addition, new stability criteria in an exponential sense are formulated for the studied systems without interval uncertainties. The efficiency and validity of the theoretical results are shown through simulation examples.

Robust Finite-Time Control of Discrete-Time Switched Positive Time-Varying Delay Systems with Exogenous Disturbance and Their Application

Many practical systems can be modeled in terms of uncertainties, which refer to the differences or errors between actual data and mathematical simulations. However, systems including slight uncertainties and exogenous disturbances may lead to the instability of those systems. Besides, the behavior of systems is preferable to investigate within a prescribed bound over a fixed time interval. Therefore, in this paper, we study a robust finite-time control of discrete-time linear switched positive time-varying delay systems with interval uncertainties and exogenous disturbance. A distinctive feature of this research is that the considered systems consist of finite-time bounded subsystems and finite-time unbounded subsystems. A class of quasi-alternative switching signals is validly designed to analyze the mechanism and switching behaviors of the systems among their subsystems. By utilizing a copositive Lyapunov–Krasovskii functional method combined with the slow mode-dependent average dwell time and the fast mode-dependent average dwell time switching techniques, new sufficient conditions containing several symmetric negative-definite matrices are derived to guarantee robust finite-time control of the systems. These results are applied to a water-quality controllability model in streams to the standard level. Finally, the consistent results between the theoretical analysis and the corresponding numerical simulations are shown.

Finite-Time Boundedness of Linear Uncertain Switched Positive Time-Varying Delay Systems with Finite-Time Unbounded Subsystems and Exogenous Disturbance

The problem of finite-time boundedness for a class of linear switched positive time-varying delay systems with interval uncertainties and exogenous disturbance is addressed. This characteristic research is that the studied systems include the finite-time bounded subsystems and finite-time unbounded subsystems. Both a slow mode-dependent average dwell time and a fast mode-dependent average dwell time switching techniques are utilized reasonably. And by applying a copositive Lyapunov-Krasovskii functional, novel delay-dependent sufficient criteria are derived to guarantee such systems to be finite-time bounded concerning the given parameters and designed switching signal. Furthermore, new finite-time boundedness criteria of the systems without interval uncertainties are also obtained. Finally, the efficiency of the theoretical results is presented in two illustrative examples.

Global Uniform Asymptotic Stability Criteria for Linear Uncertain Switched Positive Time-Varying Delay Systems with All Unstable Subsystems

In this paper, the problem of robust stability for a class of linear switched positive time-varying delay systems with all unstable subsystems and interval uncertainties is investigated. By establishing suitable time-scheduled multiple copositive Lyapunov-Krasovskii functionals (MCLKF) and adopting a mode-dependent dwell time (MDDT) switching strategy, new delay-dependent sufficient conditions guaranteeing global uniform asymptotic stability of the considered systems are formulated. Apart from past studies that studied switched systems with at least one stable subsystem, in the present study, the MDDT switching technique has been applied to ensure robust stability of the considered systems with all unstable subsystems. Compared with the existing results, our results are more general and less conservative than some of the previous studies. Two numerical examples are provided to illustrate the effectiveness of the proposed methods.

Exponential stability analysis for discrete-time homogeneous impulsive positive delay systems of degree one

The purpose of this paper is to investigate the exponential stability for discrete-time homogeneous impulsive positive delay systems of degree one. By virtue of max-separable Lyapunov functions together with Razumikhin technique, both the stability results that impulses act as stabilizers and the stability results that impulses act as perturbations are obtained. It should be noted that it is the first time that impulsive exponential stabilization results and Razumikhin type exponential stability results for discrete-time homogeneous impulsive positive delay systems of degree one are given. Two numerical examples are provided to show the effectiveness of the proposed results.

Stability of switched positive linear delay systems with mixed impulses

ABSTRACTThis paper investigates the global exponential stability of both continuous-time and discrete-time switched positive linear delay systems with the coexistence of destabilising and stabilising impulses. Based on the comparison principle, with the help of discretised multiple linear copositive Lyapunov functions, some delay-independent stability results are obtained. It should be noted that it is the first time that discrete-time switched delay systems with asynchronous switching and impulses are considered, and the switching effect can be stabilising in the obtained results of this paper. Some numerical examples are provided to demonstrate the effectiveness and superiority of the derived results.

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

SummaryThis 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.

Exponential stability of homogeneous impulsive positive delay systems of degree one

This paper investigates the global exponential stability of homogeneous impulsive positive delay systems of degree one. By using the max-separable Lyapunov functions, a sufficient criterion is obtained for exponential stability of continuous-time homogeneous impulsive positive delay systems of degree one. We also provide the corresponding counterpart for discrete-time homogeneous impulsive positive delay systems of degree one. Our results show that a stable impulse-free system can keep its original stability property under certain destabilising impulsive perturbations. It should be noted that it's the first time that the exponential stability results for homogeneous impulsive positive delay systems of degree one are given. Numerical examples are provided to demonstrate the effectiveness of the derived results.

$\mathscr {L}_\infty$ Control for Positive Delay Systems With Semi-Markov Process and Application to a Communication Network Model

This paper deals with the problem of $\mathscr {L}_\infty$ control for positive delay systems with semi-Markov process. The system is subjected to a semi-Markov process that is time-varying, dependent on the sojourn time, and related to Weibull distribution. The main motivation for this paper is that the practical systems such as the communication network model (CNM) described by positive semi-Markov jump systems (S-MJSs) always need to consider the sudden change in the operating process. To deal with the corresponding problem, some criteria about stochastic stability and $\mathscr {L}_\infty$ boundedness are presented for the open-loop positive S-MJSs. Further, some necessary and sufficient conditions for state-feedback controller satisfying $\mathscr {L}_\infty$ boundedness and positivity of the resulting closed-loop system is established in standard linear programming. Finally, the practical system about the CNM is given to verify the validity of the proposed method.

Exponential Stability for Continue-Time Switched Positive Delay Systems With All Unstable Subsystems

In this paper, the stability of continue-time switched positive delay systems (SPDSs) with all unstable subsystems is investigated. Based on the time-scheduled multiple co-positive Lyapunov-Krasovskii functional (MCLKF) method combined with fast average dwell time (ADT) techniques, a sufficient condition is obtained to ensure the underlying system is exponentially stable. Compared with the results in existing literature, our work has two advantages. On the one hand, the proposed method is independent of the maximum of corresponding elements of each subsystem matrix, which overcomes the limitation of the existing method. On the other hand, our result is less conservative than that of the previous literature. In addition, the stability condition of the system in this paper is given by matrix inequality, which can be solved by nonlinear programming. Finally, two simulation examples are provided to illustrate the correctness and effectiveness of the main result.

Construction of the Lyapunov–Krasovskii Functionals for some Classes of Positive Delay Systems

A criterion is established for the diagonal stability of positive linear difference-differential systems with discrete and distributed delay. The problem is studied of existence of a common diagonal Lyapunov–Krasovskii functional for a family of these systems. The approach developed is used to obtain conditions of the absolute stability and estimates of the time of transient processes for nonlinear difference-differential systems with sector type nonlinearities.

Stability of positive delay systems with delayed impulses

This study investigates the global exponential stability of positive delay systems with delayed impulses. By using linear copositive Lyapunov functions together with Razumikhin techniques, a number of criteria for global exponential stability of positive delay systems with delayed impulses are provided. It should be noted that it is the first time that the Razumikhin type exponential stability results for positive delay systems with delayed impulses are given. Numerical examples are provided to demonstrate the effectiveness of the derived results.

Stability and L1-gain analysis for positive delay systems with large delay period

In this paper, the problems of stability and [Formula: see text]-gain performance analysis for positive delay systems with large delay period are investigated. The maximum allowed delay bound is expanded. Under the limitations of frequency and length rate of the large delay period, the sufficient conditions guaranteeing stability of the considered system are obtained and the weighted [Formula: see text]-gain performance analysis is established. Detailed proofs are presented by using co-positive Lyapunov–Krasovskii functional candidates with large delay integral terms and the switching method; where the switching method involves transforming the original system into a positive switched delay system. All the obtained conditions can be solved via linear matrix inequalities. Finally, the validity of the proposed results is illustrated by a numerical example.

Stability and Performance Analysis for Positive Fractional-Order Systems With Time-Varying Delays

This paper addresses the stability and $L_{\infty}$ -gain analysis problem for continuous-time positive fractional-order delay systems with incommensurate orders between zero and one. A necessary and sufficient condition is firstly given to characterize the positivity of continuous-time fractional-order systems with bounded time-varying delays. Moreover, by exploiting the monotonic and asymptotic property of the constant delay system by virtue of the positivity, and comparing the trajectory of the time-varying delay system with that of the constant delay system, it is proved that the asymptotic stability of positive fractional-order systems is not sensitive to the magnitude of delays. In addition, it is shown that the $L_{\infty}$ -gain of a positive fractional-order system is independent of the magnitude of delays and fully determined by the system matrices. Finally, a numerical example is given to show the validity of the theoretical results.

Containment control of multi-agent systems with unbounded communication delays

In this paper, we address the containment control problem for multi-agent systems under heterogeneous unbounded communication delays with emphasis on the convergence rate analysis. Different from most works on multi-agent systems, we resort to a viewpoint from the area of positive delay systems. We first cast the containment control problem into the stability analysis of an associated error system. In order to capture the convergence rate, we introduce a nondecreasing positive function whose reciprocal represents the decay rate of the associated error system. Under the assumption that each follower has access to at least one leader and some mild hypotheses on the communication delays, an explicit condition is given to characterise the decay rate of the associated error system in terms of linear programming. In addition, we provide several special cases when the communication delays are restricted by linear, sublinear and logarithmic growth rates, respectively. Finally, through numerical examples, it is shown that the convergence rate is dominated by the delays being the highest order infinitely large quantity.

Relationships between asymptotic stability and exponential stability of positive delay systems

This paper explores the relations between asymptotic stability and exponential stability of continuous-time and discrete-time positive systems with delay. A system is said to be positive if its state and output are non-negative whenever the initial condition and input are non-negative. Two results are obtained. First, if a positive system is asymptotically stable for all bounded (further continuous, for continuous-time systems) time-varying delays, then it is exponentially stable for all such delays. In particular, if a positive system is asymptotically stable for a given constant delay, then it is exponentially stable for all constant delays. Second, if the involved delays are unbounded, then the positive system may be not exponentially stable even if it is asymptotically stable.