Delay-independent Stability Criteria Research Articles

A norm stability condition of neutral-type Cohen-Grossberg neural networks with multiple time delays

By constructing an appropriate Lyapunov functional, this paper obtains a novel delay-independent stability criterion of neutral-type Cohen-Grossberg neural networks possessing multiple time delays. Although this type of system cannot be represented as vector–matrix form due to the presence of multiple delays, our stability conclusion is fully defined by the infinite norm of parametric matrices and the network parameters first time. Due to the feasibility and simplicity of method proposed, our stability conclusion reducing the computational complexity while also reducing the conservatism compared with several onetime literature. Two concrete neural network models are applied to confirm the effectiveness and superiority of our conclusion.

On the Existence of the Exact Solution of Quaternion-Valued Neural Networks Based on a Sequence of Approximate Solutions.

In many practical applications, it is difficult or impossible to obtain the exact solution of the mathematical model due to the limitations of solving methods and the complexity of the neural network itself. A natural problem is given as follows: does the exact solution of quaternion-valued neural networks (QVNNs) exist when successively improved approximate solutions can be obtained? Fortunately, the Hyers-Ulam stability happens to be one of the important means to deal with this problem. In this article, the issue of Hyers-Ulam stability of QVNNs with time-varying delays is addressed. First, inspired by the Hyers-Ulam stability of general functional equations, the concept of the Hyers-Ulam stability of QVNNs is proposed along with the QVNNs model. Then, by utilizing the successive approximation method, both delay-dependent and delay-independent Hyers-Ulam stability criteria are obtained to ensure the Hyers-Ulam stability of the QVNNs considered. Finally, a simulation example is given to verify the effectiveness of the derived results.

Stability Criteria for Time-Delay Systems From an Insightful Perspective on the Characteristic Equation

This article provides a delay-dependent and a necessary and sufficient delay-independent stability criterion for linear autonomous continuous-time systems with a discrete delay. We take a simple perspective on the two-variable formulation of the characteristic equation, which leads to the following advantageous but not widespread delay-independent criterion: The sum of the coefficient matrix of the delay-free term and the elementwise unitarily rotated matrix of the delay term must remain Hurwitz for all rotation angles. A graphical test for the latter is shown to require no more than three lines of code. Concerning delay-independent stability, our main contribution is to extend this sufficient criterion to a necessary and sufficient one. Concerning delay-dependent stability, the focus is on the critical delay that bounds the initial delay interval of stability. We formulate a constrained minimization problem that gives the exact value of this critical delay. The taken perspective is especially insightful in terms of how the coefficient matrices may look like, and, for scalar systems, the delay-dependent stability chart becomes obvious at a first glance. The presented criteria are complementary to the well-known frequency-sweeping test, which results from another of three possible perspectives on the two-variable criterion. Besides of a unified treatment of these different perspectives, the article also discusses corollaries of the one taken, which include Mori’s famous criterion.

Stability Analysis of Linear Neutral Delay Systems With Two Delays via Augmented Lyapunov–Krasovskii Functionals

Stability analysis is considered in this paper for linear neutral delay systems subject to two different delays in both the state variables and the retarded derivatives of state variables. By choosing a suitable state vector indexed by an integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , a new augmented Lyapunov-Krasovskii functional (LKF) is constructed, and a stability criterion based on linear matrix inequalities is developed accordingly. It is shown that the proposed condition is less conservative than the existing methods due to the introduction of the delay-product-type integral terms in the LKF. The resulting stability criterion is then applied to the robust stability analysis of neutral delay systems with norm-bounded uncertainty. Moreover, a delay-independent stability criterion is developed based on the proposed LKF, and its frequency-domain interpretation is also given. These developed stability criteria indexed by an integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> exhibit a hierarchical character: the larger the integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , the less conservatism of the resulting stability criterion. Finally, two numerical examples are carried out to illustrate the effectiveness of the proposed method.

DYNAMICS IN A FRACTIONAL ORDER PREDATOR–PREY MODEL INVOLVING MICHAELIS–MENTEN-TYPE FUNCTIONAL RESPONSE AND BOTH UNEQUAL DELAYS

The interrelationship between predator populations and prey populations is a central problem in biology and mathematics. Setting up appropriate predator–prey models to portray the development law of predator populations and prey populations has aroused widespread interest in many scholars. In this work, we propose a new fractional order predator–prey system involving Michaelis–Menten-type functional response and both unequal delays. Utilizing the contraction mapping theorem, we prove the existence and uniqueness of the solution to the considered fractional order predator–prey system. By virtue of some mathematical analysis techniques, nonnegativeness of the solution to the involved fractional order predator–prey system is analyzed. By constructing a suitable function, the boundedness of the solution to the considered fractional order predator–prey system is explored. Making use of Laplace transform, we derive the characteristic equation of the involved fractional order predator–prey system, then by means of the stability principle and the bifurcation theory of fractional order dynamical system, a series of novel delay-independent stability criteria and bifurcation conditions ensuring the stability of the equilibrium point and the creation of Hopf bifurcation of the considered fractional order predator–prey system, are built. The global stability of the involved fractional order predator–prey system is analyzed in detail. The role of time delay in controlling the stability and the creation of Hopf bifurcation is revealed. To check the legitimacy of the derived key results, software simulation results are effectively presented. The obtained results in this work are completely novel and play a significant role in maintaining ecological balance.

EXPLORING BIFURCATION IN A FRACTIONAL-ORDER PREDATOR-PREY SYSTEM WITH MIXED DELAYS

This work chiefly develops and discusses a fractional-order predatorprey model with distributed delay and discrete delay. Applying skilly an appropriate variable substitution, a novel equivalent form of the fractional-order predator-prey model with distributed delay and discrete delay is derived. By virtue of the stability theorem and bifurcation principle of fractional-order dynamical system, we establish a delay-independent stability and bifurcation criterion ensuring the stability and the onset of Hopf bifurcation for the involved predator-prey system. The role of the time delay in stabilizing system and controlling Hopf bifurcation of the considered fractional-order predatorprey model is displayed. Software simulation results are presented to support the key theoretical fruits.

A novel approach to exponential stability in mean square of stochastic difference systems with delays

By a novel approach, we present in this paper for the first time, some explicit criteria for the exponential stability in mean square of solutions of general stochastic difference systems with time-varying delays. Both delay-independent and delay-dependent stability criteria are provided. A discussion and illustrative examples are given.

Delay-independent stability criteria for fractional order time delayed gene regulatory networks in terms of Mittag-Leffler function

This paper investigates the finite-time delay-free stability criteria for fractional-order gene regulatory networks (FOGRNs) with feedback regulation time delays. The existence of unique equilibrium points of considered systems is established based on the Banach fixed point theorem and Cauchy–Schwarz inequality. Further, when the order γ satisfies 0<γ<1 and 1<γ<2, some novel delay-free sufficient conditions are derived to ensure the finite-time stability for addressing fractional order systems by using the ideas of generalized Gronwall–Bellman inequality, equivalent norm techniques, and Laplace transform. In the end, the article comes up with two numerical cases to manifest the applicability and superiority of the obtained theoretical results.

Further Results on Bifurcation for a Fractional-Order Predator-Prey System concerning Mixed Time Delays

In the present work, we mainly focus on a new established fractional-order predator-prey system concerning both types of time delays. Exploiting an advisable change of variable, we set up an isovalent fractional-order predator-prey model concerning a single delay. Taking advantage of the stability criterion and bifurcation theory of fractional-order dynamical system and regarding time delay as bifurcation parameter, we establish a new delay-independent stability and bifurcation criterion for the involved fractional-order predator-prey system. The numerical simulation figures and bifurcation plots successfully support the correctness of the established key conclusions.

Delay-independent stability criteria for RiemannLiouville s nonlinear fractional neutral systems: Adescriptor system approach

Delay-independent stability criteria for RiemannLiouville s nonlinear fractional neutral systems: Adescriptor system approach

A Novel Approach to Mean Square Exponential Stability of Stochastic Delay Differential Equations

We present a novel approach to the mean square exponential stability of stochastic delay differential equations. Consequently, some new explicit criteria for the mean square exponential stability of general nonlinear stochastic delay differential equations are derived. Both delay-independent and delay-dependent stability criteria are given, and illustrative examples are also provided.

Stability criteria of quaternion-valued neutral-type delayed neural networks

Stability problem of quaternion-valued neutral-type delayed neural networks (QVNTDNNs) is researched in this paper. By utilizing homeomorphism principle, matrix inequality technique and Lyapunov method, both delay-independent and delay-dependent stability criteria are established in the form of linear matrix inequalities to pledge the existence, uniqueness and global stability of the equilibrium point of the considered QVNTDNNs. An example with simulations is given to manifest the validity of the exported results.

New criteria for mean square exponential stability of stochastic delay differential equations

By a novel approach, we present several new criteria for the mean square exponential stability of general non-linear stochastic delay differential equations. Both delay-independent and delay-dependent stability criteria are provided. A discussion of the obtained results is given.

A Novel Lyapunov functional with application to stability analysis of neutral systems with nonlinear disturbances

It is well-known that the global asymptotic stability analysis of neutral systems is an important concept in designing the appropriate controllers or filters for this class of systems. This paper carries out a delay-independent stability analysis of neutral systems possessing discrete time delays in the states and discrete neutral delays in the time derivative of the states in the presence of nonlinear disturbances. Some new global asymptotic stability criteria are proposed by introducing a novel Lyapunov functional. The obtained delay-independent stability criteria establish some simple and easily verifiable mathematical expressions involving the elements of the system matrices and the disturbance parameters of the neutral system. Different from the most of the previously reported stability results for neutral systems, the conditions obtained in this paper are not expressed in terms the Linear Matrix Inequalities (LMIs). Therefore, the criteria presented in this paper can be considered as the alternative results to previously published stability results stated in the LMI forms. A comparison between the results of this paper and some of previously published corresponding stability results is made to substantiate the significant improvement of the proposed results. A constructive numerical example is also presented to show applicability and the effectiveness of the proposed stability condition.

Scalar criteria for exponential stability of functional differential equations

Using a novel approach, we present some scalar criteria for the exponential stability of general functional differential equations. Both delay-independent and delay-dependent stability criteria are provided. A discussion of the obtained results is given.

Stability Analysis of a Novel Distributed Secondary Control Considering Communication Delay in DC Microgrids

With the wide application of distributed dc microgrids and the increasing of dc loads, the distributed control method has become a research hotspot. However, the distributed control method based on distributed communication network usually involves communication delay, which may lead to the failure of the controller in the practical application and even affect the stability of the system. This paper introduces a novel distributed secondary control algorithm without droop control, which can be used to realize voltage recovery and load sharing simultaneously. Based on this, two stability criteria under different conditions are derived for the dc microgrid system considering communication time delay, which are delay-dependent stability criterion under constant delay and time-varying delay-dependent stability criterion by using linear matrix inequality (LMI). These stability criteria are helpful to guide the selection of control parameters. Simulation results verify the effectiveness and feasibility of the proposed control strategy, and show the correctness of the delay-independent stability criterion and delay-dependent stability criterion.

On Exponential Stability of Neural Networks with Proportional Delays and Periodic Distribution Impulsive Effects

This paper is concerned with the problem of generalized exponential stability of impulsive neural networks with a proportional delay. More specifically, the considered network models are subject to both time-varying impulses, whose strengths are in a type of periodic distributions, and a special kind of unbounded time-varying delays called proportional delays. Based on the comparison principle, a unified delay-independent stability criterion is first derived. As an application of the derived stability conditions, the problem of designing a local state feedback control law with bounded controller gains is addressed. Finally, three examples with numerical simulations are given to demonstrate the effectiveness and advantages of the results.

On the Stability Analysis of Systems of Neutral Delay Differential Equations

This paper focuses on the stability analysis of systems modeled as neutral delay differential equations (NDDEs). These systems include delays in both the state variables and their time derivatives. The proposed approach consists of a descriptor model transformation that constructs an equivalent set of delay differential algebraic equations (DDAEs) of the original NDDEs. We first rigorously prove the equivalency between the original set of NDDEs and the transformed set of DDAEs. Then, the effect on stability analysis is evaluated numerically through a delay-independent stability criterion and the Chebyshev discretization of the characteristic equations.

An improved stability result for delayed Takagi–Sugeno fuzzy Cohen–Grossberg neural networks

This work proposes a novel and improved delay independent global asymptotic stability criterion for delayed Takagi–Sugeno (T–S) fuzzy Cohen–Grossberg neural networks exploiting a suitable fuzzy-type Lyapunov functional in the presence of the nondecreasing activation functions having bounded slopes. The proposed stability criterion can be easily validated as it is completely expressed in terms of the system matrices of the fuzzy neural network model considered. It will be shown that the stability criterion obtained in this work for this type of fuzzy neural networks improves and generalizes some of the previously published stability results. A constructive numerical example is also given to support the proposed theoretical results.

Robust asymptotic stability of interval fractional-order nonlinear systems with time-delay

This paper studies the global asymptotic stability of a class of interval fractional-order (FO) nonlinear systems with time-delay. First, a new lemma for the Caputo fractional derivative is presented. It extends the FO Lyapunov direct method allowing the stability analysis and synthesis of FO nonlinear systems with time-delay. Second, by employing FO Razumikhin theorem, a new delay-independent stability criterion, in the form of linear matrix inequality is established for ensuring that a system is globally asymptotically stable. It is shown that the new criterion is simple, easy to use and valid for the FO or integer-order interval neural networks with time-delay. Finally, the feasibility and effectiveness of the proposed scheme are tested with a numerical example.