Obtained Stability Criteria Research Articles

Stability analysis of linear systems with multiple time-varying delays via a region partitioning approach and reciprocally convex combination lemmas

The delays-dependent stability analysis of linear systems with multiple time-varying delays is addressed in this study. To estimate the integral term that results from the differentiation of Lyapunov–Krasovskii functional (LKF), an improved region partitioning approach and relaxed lemmas are proposed. Based on all the delayed state information, the maximum delay interval [−h,0] is separated into 2N non-overlapping subintervals. Secondly, two novel generalized reciprocally convex combination lemmas (GRCCLs) are proposed, with Bessel–Legendre-based inequality, to estimate the integral terms generated by the region partitioning approach to obtain less conservative stability criteria. Finally, the obtained stability criteria is applied to simple linear systems and load frequency control (LFC) scheme of the two-area power system for stability analysis, and the effectiveness of proposed method is verified.

Finite-time stability and applications of positive switched linear delayed impulsive systems

<abstract><p>In this paper, we study the finite-time stability and applications of positive switched linear delayed systems under synchronous impulse control, which includes two types of random switching and average dwell time switching. By constructing a type of linear time-varying co-positive Lyapunov functional, we first propose several new finite-time stability criteria. It should be emphasized that the linear term coefficient of the linear vector of the Lyapunov functional is adjusted to the difference between the weighting vector and the given vector. Then, we apply the obtained stability criteria to the linear time-varying delayed systems with impulsive effects. At last, three examples are given to demonstrate the validity of the obtained results, which includes the specific linear programming algorithm process.</p></abstract>

Stability analysis of linear systems with time-varying delay via some novel techniques

This paper concentrates on the stability analysis of linear systems with a time-varying delay. Firstly, a new Lyapunov–Krasovskii functional (LKF) is given, which involves more time delay cross-terms information. Secondly, an improved reciprocally convex inequality with higher estimation accuracy is derived. Then, on the basis of the constructed LKF and the improved reciprocally convex inequality, some new stability criteria with less conservative are proposed. Finally, the superiority of the obtained stability criteria is explained by two numerical examples.

Existence and stability of a periodic solution of a general difference equation with applications to neural networks with a delay in the leakage terms

In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a periodic solution by constructing a type of Poincaré map. The results are used to obtain stability criteria for a general discrete-time neural network model with a delay in the leakage terms. As particular cases, we obtain new stability criteria for neural network models recently studied in the literature, in particular for low-order and high-order Hopfield and Bidirectional Associative Memory (BAM).

Self-adjointness of sound-proof models for magnetic buoyancy

An ideal magneto-hydrodynamic fluid, whether fully compressible or incompressible, is a Hamiltonian system. This implies that the equations describing perturbations to any static state are self-adjoint, a fact that is useful in obtaining stability criteria. To describe weakly compressible flows, there are a number of “sound-proof” models that eliminate sound waves by making approximations to the governing equations. However, such approximations may violate the Hamiltonian structure of the system. In a recent work, we have introduced a very general sound-proof model and determined conditions under which it closely approximates the linear regime of magneto-buoyancy instability, motivated by conditions in the solar interior. In the present work, we take a complementary approach, by deriving constraints under which the linearised sound-proof system is self-adjoint. We show that there is a unique set of self-adjoint sound-proof equations that conserves the same energy as the fully compressible system.

Hybrid event-triggered impulsive flocking control for multi-agent systems via pinning mechanism

This paper studies the leader-following flocking control problem of unmanned aerial vehicle (UAV) systems by integrating a hybrid impulsive framework, where the full exchange of information only occurs at each impulsive time instant. It is also shown that the overall stability can be achieved via impulses while the continuous dynamics remains destabilizing. Meanwhile, the even-triggered and pinning mechanisms are considered to further reduce control resource usage and transmission redundancy. Moreover, topology switching and strong nonlinearity are taken into account for more practical application. Based on transmission topology structure, impulsive control theory, and Lyapunov based event-triggered strategy, some improved theorems are derived to guarantee convergence of the corresponding error dynamics without exhibiting Zeno behaviour. Compared with most existing results on continuous or impulsive systems, the obtained stability criteria are more general as they are applicable to systems involving both delayed coupling feedback and delayed impulses. In addition, a novel approach using braking and gyroscopic forces is utilized for collision avoidance purposes. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical analysis.

A novel Lyapunov stability analysis of neutral-type Cohen–Grossberg neural networks with multiple delays

The major target of this research article is to conduct a new Lyapunov stability analysis of a special model of Cohen–Grossberg neural networks that include multiple delay terms in state variables of systems neurons and multiple delay terms in time derivatives of state variables of systems neurons in the network structure. Employing some proper linear combinations of three different positive definite and positive semi-definite Lyapunov functionals, we obtain some novel sufficient criteria that guarantee global asymptotic stability of this type of multiple delayed Cohen–Grossberg type neural systems. These newly derived stability results are determined to be completely independent of the involved time delay terms and neutral delay terms, and they are totally characterized by the values of the interconnection parameters of Cohen–Grossberg neural system. Besides, the validation of the obtained stability criteria can be justified by applying some simple appropriate algebraic equations that form some particular relations among the constant system elements of the considered neutral neural systems. A useful and instructive numerical example is analysed to exhibit some major advantages and novelties of these newly proposed global stability results in this paper over some previously reported corresponding asymptotic stability conditions.

Particle swarm optimization: Stability analysis using N-informers under arbitrary coefficient distributions

This paper derives, under minimal modelling assumptions, a simple to use theorem for obtaining both order-1 and order-2 stability criteria for a common class of particle swarm optimization (PSO) variants. Specifically, PSO variants that can be rewritten as a finite sum of stochastically weighted difference vectors between a particle’s position and swarm informers are covered by the theorem. Additionally, the use of the derived theorem allows a PSO practitioner to obtain stability criteria that contains no artificial restriction on the relationship between control coefficients. The majority of previous stability results for PSO variants provided stability criteria under the restriction that certain control coefficients are equal; such restrictions are not present when using the derived theorem. Using the derived theorem, as demonstration of its ease of use, stability criteria are derived without the imposed restriction on the relation between the control coefficients for four popular PSO variants.

Impulsive Stabilization on Hyper-Chaotic Financial System under Neumann Boundary

This paper proposes a novel technique to obtain sufficient conditions for the existence and stabilization of positive solutions for a kind of hyper-chaotic financial model. Since some important economic indexes are heavily related to region, the authors consider a nonlinear chaotic financial system with diffusion, which leads to some mathematical difficulties in dealing with the infinite-dimension characteristic. In order to overcome these difficulties, novel analysis techniques have to be proposed on the basis of Laplacian semigroup and impulsive control. Sufficient conditions are provided for existence and global exponential stabilization of positive solution for the system. It is interesting to discover that the impulse strength can be larger than 1 in the newly obtained stability criterion. Numerical simulations show the effectiveness of theoretical analysis.

A relaxed binary quadratic function negative-determination lemma and its application to neutral systems with interval time-varying delays and nonlinear disturbances

This paper considers the stability problem of neutral systems with interval time-varying delays and nonlinear disturbances. Firstly, an augmented vector containing two double integral terms is introduced into the Lyapunov-Krasovskii functional (LKF). In this case, a binary quadratic function with discrete and neutral delay arises in the time derivative. To gain the negativity condition of such function, by taking full advantage of the idea of partial differential of the binary quadratic function and Taylor's formula, a relaxed binary quadratic function negative-determination lemma with two adjustable parameters is proposed, which contains the existing lemmas as its special cases and shows the great potential of reducing conservatism for the case where the tangent slope at the endpoint is far from zero. Then, based on the improved lemma, more relaxing stability criteria have been obtained via an augmented LKF. Finally, two classic numerical examples are given to attest the effectiveness and strengths of the obtained stability criteria.

Practical fixed time active control scheme for synchronization of a class of chaotic neural systems with external disturbances

The objective in this work is to consider the practical fixed-time synchronization of delayed chaotic neural networks interfered by uncertain impacts. Firstly, to derive the sufficient conditions of fixed time stability, a novel practical fixed-time stability criterion, which can be viewed as the generalization of the existed results, is set up. Then, by means of the obtained stability criterion, we develop an effective fixed time control approach to attain the practical network synchronization within a limited time. In our synchronization scheme, the error state trajectories of two chaotic neural networks move forward to the small domain of zero in a short time freeing from the initial states. Besides, different from the traditional methodologies that dealt with the external disturbances, the chatter phenomenon can be avoided since our proposed controllers don’t contain the signum functions. Finally, the effectiveness of the obtained theoretical results and synchronization scheme are testified by conducting the simulations.

Stability analysis of delayed neural network based on the convex method and the non-convex method

This paper investigates the asymptotic stability of neural network with time-varying delay. A new Lyapunov–Krasovskii functional (LKF) is constructed with some non-integral delay-product functional terms. Some parameter matrices in the LKF are not required to be positive definite. The convex method and the non-convex method are proposed to deal with the square of the time-varying delay appearing in the derivative of the LKF, respectively. Two less conservative delay-dependent stability criteria in the form of linear matrix inequalities (LMIs) are established. Finally, two widely used numerical examples are given to illustrate the validity and superiority of the obtained stability criteria.

Improved stability criterion for distributed time-delay systems via a generalized delay partitioning approach

<abstract><p>This paper researches the problem of stability analysis for distributed time-delay systems. A newly augmented Lyapunov-Krasovskii functional (LKF) is first introduced via a generalized delay partitioning approach. Then, a less conservative stability criterion is derived by introducing a novel Jensen inequality to estimate the integral terms in the derivative of LKF. The stability condition is given in terms of linear matrix inequality. Finally, the merits of the obtained stability criterion is shown by a well-known example.</p></abstract>

A binary quadratic function negative‐determination lemma and its application to stability analysis of systems with two additive time‐varying delay components

This paper concentrates on the stability problem of systems with two additive time-varying delay components. For the construction of Lyapunov–Krasovskii functional (LKF), in the case that the introduced augmented vector contains the double integral term of the state vector, a special form of the binary quadratic function with respect to two time-varying delays has often been introduced into the derivative of the LKF. In order to determine the negative definiteness of such function, by making full use of the idea of partial differential, the convex/concave property and the slope characteristic of tangent lines of the binary quadratic function, a binary quadratic function negative-determination lemma is proposed in the case of the pluses or minuses of quadratic coefficients are unknown. Then, the obtained stability criterion in the form of linear matrix inequality shows a greater advantage than the previous criteria since not only some advanced techniques are employed to treat some integral terms, but also the proposed lemma is employed to deal with quadratic terms in functional derivative. Finally, a typical example is given to verify the superiority of the derived criterion.

New stability criteria for linear impulsive systems with interval impulse-delay

The robust stability problem for linear time-delay systems with general linear delayed impulses is investigated. Different from the previous results, the impulse-delays are allowed to be larger than the impulse period. An auxiliary state variable is introduced to construct an augmented model of the impulsive system, under which the discrete dynamics introduced by impulse-delays can be highlighted. A novel piecewise Lyapunov functional is introduced to analyze the stability of the augmented model. This functional is continuous along the trajectories of the augmented model, and is not required to be positive-definite at non-impulse instants. LMI-based exponential stability conditions are derived, which depend on both the impulse-dwell-time and the impulse-delay-interval. Numerical examples show that the obtained stability criteria are able to handle the benefit/harmful impulse-delays.

Discretized quasi‐time‐dependent control for asynchronously switched systems with dynamic dwell‐time

AbstractConsidering that actuator delay is inevitable and a discretized control design method is more preferable in practice, this article is concerned with the discretized quasi‐time‐dependent (QTD) control for continuous‐time asynchronously switched systems via dynamic output‐feedback. The persistent dwell‐time (PDT) is employed to analyze the stability and ‐gain for switched systems in the framework of multiple QTD Lyapunov function approach, and the concept of dynamic dwell‐time (DDT) is used to characterize a category of switching signals for asynchronously switched systems with measurable switching lag, which means that the controlled plant can adjust its dwell‐time on some mode dynamically according to the switching lag of the corresponding controller. First, the stability and ‐gain are analyzed for switched systems with PDT. Then, by constructing controller‐dependent multiple QTD Lyapunov functions, a QTD control synthesis condition is derived based on the obtained stability criterion. The developed controller can guarantee the closed‐loop stability and performance for the underlying system under a class of identified DDT switching signals. Subsequently, a discretized QTD control scheme is proposed by choosing the discretized quadratic version of the constructed Lyapunov functions. The synthesis condition is formulated as a set of finite‐dimensional LMIs, and hence it can be easily tractable. Finally, the proposed control scheme is applied to the tracking control of a mobile robot to illustrate its effectiveness and application potential.

Global exponential stability of discrete-time higher-order Cohen–Grossberg neural networks with time-varying delays, connection weights and impulses

In this paper, an auxiliary model-based nonsingular M-matrix approach is used to establish the global exponential stability of the zero equilibrium, for a class of discrete-time high-order Cohen–Grossberg neural networks (HOCGNNs) with time-varying delays, connection weights and impulses. A new impulse-free discrete-time HOCGNN with time-varying delays and connection weights is firstly constructed, and the relationship between the solutions of original systems and new HOCGNNs is indicated by a technical lemma. From which, the global exponential stability criteria for the zero equilibrium are derived by using an inductive idea and the properties of nonsingular M-matrices. The effectiveness of the obtained stability criteria is illustrated by numerical examples. Compared with the previous results, this paper has three advantages: (i) The Lyapunov–Krasovskii functional is not required; (ii) The obtained global exponential stability criteria are applied to check whether a matrix is a nonsingular M-matrix, which can be conveniently tested; (iii) The proposed approach applies to most of discrete-time system models with impulses and delays.

Impulsive stabilization and stability analysis for Gilpin–Ayala competition model involved in harmful species via LMI approach and variational methods

Firstly, a dynamic analysis for reaction–diffusion Gilpin–Ayala competition model involved in harmful species is considered under Dirichlet boundary value condition. Existence of multiple stationary solutions is verified by way of Mountain Pass lemma, and the local stability result of the null solution is obtained by employing linear approximation principle. Secondly, the authors utilize variational methods and linear matrix inequality (LMI) technique to deduce the LMI-based global exponential stability criterion on the null solution which becomes the unique stationary solution of a Markovian jumping ecosystem with delayed feedback under a reasonable boundedness assumption on population densities. Particularly, LMI criterion is involved in free weight coefficient matrix, which reduces the conservatism of the algorithm. In addition, a new impulse control stabilization criterion is also derived, in which no differentiable assumptions on time-delayed functions are proposed. Finally, three numerical examples show the effectiveness of the proposed methods. It is worth mentioning that the obtained stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria.

Existence and global asymptotic stability criteria for nonlinear neutral-type neural networks involving multiple time delays using a quadratic-integral Lyapunov functional

In this paper we consider a standard class of the neural networks and propose an investigation of the global asymptotic stability of these neural systems. The main aim of this investigation is to define a novel Lyapunov functional having quadratic-integral form and use it to reach a stability criterion for the under study neural networks. Since some fundamental characteristics, such as nonlinearity, including time-delays and neutrality, help us design a more realistic and applicable model of neural systems, we will use all of these factors in our neural dynamical systems. At the end, some numerical simulations are presented to illustrate the obtained stability criterion and show the essential role of the time-delays in appearance of the oscillations and stability in the neural networks.

Further results on delay-dependent stability for neutral singular systems via state decomposition method

Abstract This paper studies the delay-dependent stability for neutral singular systems. In the light of state decomposition method, a novel augmented Lyapunov-Krasovskii functional including less decision variables is developed. Then by means of zero-value equations technology, some sufficient stability conditions in the form of linear matrix inequalities are acquired, which guarantees the non-impulsiveness, regularity and stability for the proposed neutral singular systems. The obtained stability criterion takes the sizes of both the discrete- and neutral- delays into account. They are less conservative than those presented by previous analytical approaches. Numerical examples are given to show the feasibility of our method and the interrelation between the discrete- and neutral- delays.