Delay-independent Stability Conditions Research Articles

Stability Analysis of Monotone Systems via Max-Separable Lyapunov Functions

We analyze stability properties of monotone nonlinear systems via max-separable Lyapunov functions, motivated by the following observations: first, recent results have shown that asymptotic stability of a monotone nonlinear system implies the existence of a max-separable Lyapunov function on a compact set; second, for monotone linear systems, asymptotic stability implies the stronger properties of D-stability and insensitivity to time delays. This paper establishes that for monotone nonlinear systems, equivalence holds between asymptotic stability, the existence of a max-separable Lyapunov function, D-stability, and insensitivity to bounded and unbounded time-varying delays. In particular, a new and general notion of D-stability for monotone nonlinear systems is discussed, and a set of necessary and sufficient conditions for delay-independent stability are derived. Examples show how the results extend the state of the art.

A simple necessary and sufficient LMI condition for the strong delay-independent stability of LTI systems with single delay

This paper is concerned with strong delay-independent stability of linear time-invariant (LTI) systems with a single time-delay. Stability analysis of linear delay-systems is complicated by the need to locate the roots of a transcendental characteristic equation. In this paper we propose a convex necessary and sufficient condition for strong delay-independent stability. This result mainly follows from the Kronecker sum properties and the Kalman–Yakubovich–Popov lemma, which allows us to present the main result in terms of a single linear matrix inequality (LMI) feasibility test. The result is illustrated by simple numerical examples.

Discrete and Continuous Distributed Delays in Replicator Dynamics

In this paper, we study evolutionary games and we examine the stability of the evolutionarily stable strategy (ESS) in the continuous-time replicator dynamics with distributed time delays. In many examples, the interactions between individuals take place instantaneously, but their impacts are not immediate and take a certain amount of time which is usually random. In this paper, we study the consequences of distributed delays on the stability of the replicator dynamics. The main results are that (i) under the exponential delay distribution, the ESS is asymptotically stable for any value of the rate parameter; (ii) the necessary and sufficient conditions for the asymptotic stability of the ESS under uniform and Erlang distributions; and (iii) for the discrete distributed delays, we derive a necessary and sufficient delay-independent stability conditions. We illustrate our results with numerical examples from the Hawk–Dove game.

Internally Positive Representations and Stability Analysis of Linear Difference Systems with Multiple Delays

This work introduces the Internally Positive Representation of linear continuous-time difference systems with multiple delays. The technique, originally developed for delay-free linear systems, and recently extended to differential systems with multiple delays, proved to be a useful tool to exploit results available for positive systems also for systems that are not positive. As a result, a sufficient delay-independent stability condition for difference systems with constant delays is given, which is shown to be less conservative than similar existing results.

Stability of Linear Fractional Differential Equations with Delays: a coupled Parabolic-Hyperbolic PDEs formulation.

Fractional differential equations with delays are ubiquitous in physical systems, a recent example being time-domain impedance boundary conditions in aeroacoustics. This work focuses on the derivation of delay-independent stability conditions by relying on infinite-dimensional realisations of both the delay (transport equation, hyperbolic) and the fractional derivative (diffusive representation, parabolic). The stability of the coupled parabolic-hyperbolic PDE is studied using straightforward energy methods. The main result applies to the vector-valued case. As a numerical illustration, an eigenvalue approach to the stability of fractional delay systems is presented.

Partial stability analysis of some classes of nonlinear systems

A nonlinear differential equation system with nonlinearities of a sector type is studied. Using the Lyapunov direct method and the comparison method, conditions are derived under which the zero solution of the system is stable with respect to all variables and asymptotically stable with respect to a part of variables. Moreover, the impact of nonstationary perturbations with zero mean values on the stability of the zero solution is investigated. In addition, the corresponding time-delay system is considered for which delay-independent partial asymptotic stability conditions are found. Three examples are presented to demonstrate effectiveness of the obtained results.

On the stability analysis of switched nonlinear systems with time varying delay under arbitrary switching

This paper addresses the stability problem for a class of switched nonlinear time varying delay systems modeled by delay differential equations. By transforming the system representation under the arrow form and using a new constructed Lyapunov function, the aggregation techniques, the Borne-Gentina practical stability criterion associated with the properties, new delay-independent stability conditions of the considered systems are established. Compared with the existing results in this area, the obtained result is explicit, simple to use and allows us to avoid the problem of searching a common Lyapunov function. Finally, an example is provided, with numerical simulations, to demonstrate the effectiveness of the proposed method.

Delay-independent sliding mode control of time-delay linear fractional order systems

This paper considers the problem of delay-independent stabilization of linear fractional order (FO) systems with state delay. As in most practical systems in which the value of delay is not exactly known (or is time varying), a new approach is proposed in this paper, which results in asymptotic delay-independent stability of the closed-loop time-delay FO system. For this purpose, a novel FO sliding mode control law is proposed in which its main advantage is its independence to delay. Furthermore, a novel appropriate delay-independent sliding manifold is suggested. Additionally, two theorems are given and proved, which guarantee the occurrence of the reaching phase in finite time and the asymptotic delay-independent stability conditions of the dynamic equations in the sliding phase. Finally, in order to verify the theoretical results, two examples are given and simulation results confirm the performance of the proposed controller.

Asymptotic Stability Conditions and Estimates of Solutions for Nonlinear Multiconnected Time-Delay Systems

This paper examines certain classes of multiconnected (complex) systems with time-varying delay. Delay-independent stability conditions and estimates of the convergence rate of solutions to the origin for those systems are derived. It is shown that the exponents in the obtained estimates depend on the parameters of Lyapunov functions constructed for the corresponding isolated subsystems. The problem of computing parameter values that provide the most precise estimates is investigated. Some examples are presented to demonstrate the effectiveness of the proposed approaches.

Stability analysis of time-delay systems by nonlinear approximation

We consider a nonlinear complex (large-scale) system with time delay. It is assumed that the corresponding isolated subsystems are homogeneous, and the zero solutions of the subsystems are asymptotically stable when delay is equal to zero. By the Lyapunov direct method, and the Razumikhin approach, delay-independent stability conditions for the complex system are obtained. These conditions are formulated in terms of solvability of auxiliary systems of algebraic inequalities. An example is given to demonstrate effectiveness of the presented results.

On stability analysis of discrete-time uncertain switched nonlinear time-delay systems

This paper addresses the stability analysis problem for a class of discrete-time switched nonlinear time-delay systems with polytopic uncertainties. These considered systems are characterized by delayed difference nonlinear equations which are given in the state form representation. Then, a transformation under the arrow form is employed. Indeed, by constructing an appropriated common Lyapunov function, and also by resorting to the Kotelyanski lemma and the M-matrix proprieties, new delay-independent stability conditions under arbitrary switching law are deduced. Compared with the existing results of switched systems, those obtained results are formulated in terms of the unknown polytopic uncertain parameters, explicit and easy to apply. Moreover, this method allows us to avoid the search for a common Lyapunov function which is a difficult matter. Finally, a numerical example is presented to illustrate the effectiveness of the proposed approach.

Delay-Independent Stability Conditions for Some Classes of Nonlinear Systems

Some classes of nonlinear time-delay systems are studied. It is assumed that the zero solution of a system is asymptotically stable when delay is equal to zero. By the Lyapunov direct method, and the Razumikhin approach, it is shown that in the case when the system is essentially nonlinear, i.e., the right-hand side of the system does not contain linear terms, the asymptotic stability of the trivial solution is preserved for an arbitrary positive value of the delay. Based on homogeneous approximation of a time-delay system some stability conditions are found. We treat large-scale systems with nonlinear subsystems. New stability conditions in certain cases, critical in the Lyapunov sense, are obtained. Three examples are given to demonstrate effectiveness of the presented results.

On Delay-Independent Stability of a Class of Nonlinear Positive Time-Delay Systems

We present a condition for delay-independent stability of a class of nonlinear positive systems. This result applies to systems that are not necessarily monotone and extends recent work on cooperative nonlinear systems.

Delay-independent stability of homogeneous systems

Abstract A class of nonlinear systems with homogeneous right-hand sides and time-varying delay is studied. It is assumed that the trivial solution of a system is asymptotically stable when delay is equal to zero. By the usage of the Lyapunov direct method and the Razumikhin approach, it is proved that the asymptotic stability of the zero solution of the system is preserved for an arbitrary continuous nonnegative and bounded delay. The conditions of stability of time-delay systems by homogeneous approximation are obtained. Furthermore, it is shown that the presented approaches permit to derive delay-independent stability conditions for some types of nonlinear systems with distributed delay. Two examples of nonlinear oscillatory systems are given to demonstrate the effectiveness of our results.

Stability analysis for a class of switched nonlinear time-delay systems

This paper investigates the stability analysis for a class of discrete-time switched nonlinear time-delay systems. These systems are modelled by a set of delay difference equations, which are represented in the state form. Then, another transformation is made towards an arrow form. Therefore, by applying the Kotelyanski conditions combined to the M−matrix properties, new delay-independent sufficient stability conditions under arbitrary switching which correspond to a Lyapunov function vector are established. The obtained results are explicit and easy to use. A numerical example is provided to show the effectiveness of the developed results.

Asymptotic stability of cellular neural networks with multiple proportional delays

Proportional delay is a kind of unbounded time-varying delay, which is different from unbounded distributed delays. In this paper, asymptotic stability of the equilibrium point of cellular neural networks with multiple proportional delays is presented. Sufficient conditions for delay-dependent global asymptotic stability and delay-independent uniform asymptotic stability of the system are obtained by employing matrix theory and constructing Lyapunov functional. Two examples are given to illustrate the effectiveness of the obtained results and less conservative than previously existing results.

Asymptotic Stability and Decay Rates of Homogeneous Positive Systems With Bounded and Unbounded Delays

There are several results on the stability of nonlinear positive systems in the presence of time delays. However, most of them assume that the delays are constant. This paper considers time-varying, possibly unbounded, delays and establishes asymptotic stability and bounds the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case. Specifically, we present a necessary and sufficient condition for delay-independent stability of continuous-time positive systems whose vector fields are cooperative and homogeneous. We show that global asymptotic stability of such systems is independent of the magnitude and variation of the time delays. For various classes of time delays, we are able to derive explicit expressions that quantify the decay rates of positive systems. We also provide the corresponding counterparts for discrete-time positive systems whose vector fields are nondecreasing and homogeneous.

A necessary and sufficient condition for delay-independent stability of linear time-varying neutral delay systems

In this paper, stability analysis of linear time-varying neutral delay systems is considered. A necessary and sufficient condition for delay-independent global asymptotic stability of such systems is derived. Eventually, two examples are given in order to show the results established.

On Lyapunov stability of scalar stochastic time-delayed systems

In this paper, we obtain an analytical Lyapunov- based stability conditions for scalar linear and nonlinear stochastic systems with discrete time-delay. The Lyapunov–Krasovskii and Lyapunov–Razumikhin methods are applied with techniques from stochastic calculus to obtain the regions of mean square asymptotic stability in the parameter space. Both delay-independent and delay-dependent stability conditions are analyzed corresponding to both additive and multiplicative stochastic Brownian motion excitation in the Ito form. It is also shown that the derived sufficient conditions are less conservative in comparison with other numerical LMI-based Lyapunov approaches. A range of different stability charts are obtained based on the derived Lyapunov-based stability criteria, which are also compared with numerical first and second moment stability boundaries computed using the stochastic semidiscretization method. A Lipschitz condition is used to treat nonlinear functions of the current and delayed states.

On Stability of Discrete-Time Delay-Difference Equations for Arbitrary Delay Variations

This paper focuses on the concept of delay-independent stability for dynamical systems described by continuous-time linear delay-difference equations and the corresponding stability notion in discrete-time domain. The problem will be formulated with respect to delay-parameter space. Our intention is to summarize delay-independent stability condition and to provide, in a compact formulation, an appropriate numerical method for its computation, at least for two dimensional delay case. Obtained results are applied in stability analysis of the discrete-time delay-difference equations. Such a strong stability condition appears to be necessary for the existence of specific invariant regions in the state-space. Some illustrative examples complete the paper.