Vector Lyapunov Function Method Research Articles

Distributed Consensus Tracking of Incommensurate Heterogeneous Fractional-Order Multi-Agent Systems Based on Vector Lyapunov Function Method

This paper investigates the tracking problem of fractional-order multi-agent systems. Both the order and parameters of the leader are unknown. Firstly, based on the positive system approach, the asymptotically stable criteria for incommensurate linear fractional-order systems are derived. Secondly, the models of incommensurate heterogeneous multi-agent systems are introduced. To cope with incommensurate and heterogeneous situations among followers and the leader, radial basis function neural networks (RBFNNs) and a discontinuous control method are used. Thirdly, the consensus criteria are derived by using the Vector Lyapunov Function method. Finally, a numerical example is presented to illustrate the effectiveness of the proposed theoretical method.

State Observer-Based Iterative Learning Control Design for Discrete Systems Using the Heavy Ball Method

The paper considers a state observer-based iterative learning control design problem for discrete linear systems. To accelerate the convergence of the learning error, a combination of the heavy ball method from optimization theory and the vector Lyapunov function method for a class of two-dimensional systems known as repetitive processes is used to develop a new design. A supporting numerical example is given, including a comparison with an existing design.

Stability and Synchronization of Delayed Quaternion-Valued Neural Networks under Multi-Disturbances

This paper discusses a type of mixed-delay quaternion-valued neural networks (QVNNs) under impulsive and stochastic disturbances. The considered QVNNs model are treated as a whole, rather than as complex-valued neural networks (NNs) or four real-valued NNs. Using the vector Lyapunov function method, some criteria are provided for securing the mean-square exponential stability of the mixed-delay QVNNs under impulsive and stochastic disturbances. Furthermore, a type of chaotic QVNNs under stochastic and impulsive disturbances is considered using a previously established stability analysis method. After the completion of designing the linear feedback control law, some sufficient conditions are obtained using the vector Lyapunov function method for determining the mean-square exponential synchronization of drive–response systems. Finally, two examples are provided to demonstrate the correctness and feasibility of the main findings and one example is provided to validate the use of QVNNs for image associative memory.

Stability of stochastic nonlinear delay systems with delayed impulses

In this paper, we study the input-to-state stability of a class of stochastic delay differential equations with delay impulses. We consider the two cases where the continuous stochastic delay system is stable and unstable, and establish different types of stability conditions by using the average dwell time and the vector Lyapunov function method. Finally, two examples are used to verify the validity of our results.

Novel asymptotic stability criterion for fractional‐order gene regulation system with time delay

AbstractAt present, Routh–Herwitz criterion is the mostly applied characteristic equation of the time‐lag fractional‐order gene regulatory network systems (FGRNs) for stability analysis. However, it is limited by linearization, which makes the result have a certain error. In addition, due to the processes of obtaining the characteristic roots are too complicated, there are too many constraints on stability criterion. To solve these problems, the vector Lyapunov function method combined with the M‐matrix is used in the present study. Based on this method, the corresponding asymptotic stability theorem is derived. Our results show that this novel method not only had fewer limitations compared with conventional methods, but also is suitable to all fractional‐order operators between 0 and 1 with a large time delay.

Stability analysis of neutral stochastic delay differential equations via the vector Lyapunov function method

In this paper, we study the pth moment exponential stability (p-ES) and the almost sure exponential stability (a-ES) of neutral stochastic delay differential equations (NSDDEs). By using the vector Lyapunov function (VLF) method, we can prove that the global solution of NSDDEs exists when the linear growth condition is removed, and we also get some stability criteria for NSDDEs. In the presented stability conditions, the established L-operator differential inequality is based on the VLF and allows cross item to exist. An example is given to verify the correctness of the proposed results.

Exponential stability for delayed complex-valued neural networks with reaction-diffusion terms

In this study, we investigate reaction-diffusion complex-valued neural networks with mixed delays. The mixed delays include both time-varying and infinite distributed delays. Criteria are derived to ensure the existence, uniqueness, and exponential stability of the equilibrium state of the addressed system on the basis of the M-matrix properties and homeomorphism mapping theories as well as the vector Lyapunov function method. The results demonstrate the positive effect of reaction-diffusion on the stability, which further improves the existing conditions. Finally, the analysis of several examples is compared to the present results to verify the correctness and reduced conservatism of the primary results.

具有反应扩散项的变时滞复数域神经网络的指数稳定性

The exponential stability of complex-valued neural networks with time-varying delays and reaction-diffusion terms was studied. Firstly, the addressed systems were separated into their real parts with the complex-valued activation functions assumed to be divided into the real parts and imaginary parts. Secondly, some sufficient conditions for ensuring the exponential stability of the equilibrium states of the systems were established based on the vector Lyapunov function method and the M-matrix theory. The obtained criteria have no free variables and reduced conservatism compared with the existing results. A numerical example proves the correctness of the obtained results.

Further research on exponential stability for quaternion-valued neural networks with mixed delays

This paper addresses the global exponential stability of a class of quaternion-valued neural networks (QVNNs) with mixed delays including time-varying delays and infinite distributed delays. Because of the noncommutativity of quaternion multiplication, the concerned quaternion-valued models separated into four real-valued parts to form the equivalent real-valued systems. Based on M-matrix properties and homomorphism mapping theories, some sufficient conditions are derived to guarantee the existence and uniqueness of the equilibrium point of the system. Conditions for ensuring the global exponential stability of the equilibrium point of the system are obtained on the basis of the vector Lyapunov function method instead of the linear matrix inequality method. Using a similar method, the mixed-delay QVNNs with parameter uncertainties are also studied, and the conditions for ensuring the global robust exponential stability of the system are established directly. The adopted approach and the obtained results in this paper complement already the existing ones. Finally, three numerical examples are provided to illustrate the feasibility and the less level conservatism of the main results.

Liouville Formula for Some Classes of Differential Equations with the Hukuhara Derivative

We use the vector Lyapunov function method and the comparison method in conjunction with methods of convex geometry and the theory of mixed volumes to obtain analogs of the Liouville formula for some classes of linear differential equations with the Hukuhara derivative defined on the space of convex compact subsets of the plane. A continual generalization of the construction of a vector Lyapunov function is proposed.

Razumikhin-type theorem for stochastic functional differential systems via vector Lyapunov function

This paper is concerned with input-to-state stability of SFDSs. By using stochastic analysis techniques, Razumikhin techniques and vector Lyapunov function method, vector Razumikhin-type theorem has been established on input-to-state stability for SFDSs. Novel sufficient criteria on the pth moment exponential input-to-state stability are obtained by the established vector Razumikhin-type theorem. When input is zero, an improved criterion on exponential stability is obtained. Two examples are provided to demonstrate validity of the obtained results.

Mean Square Exponential Stability of Stochastic Complex‐Valued Neural Networks with Mixed Delays

This paper investigates the mean square exponential stability problem of a class of complex‐valued neural networks with stochastic disturbance and mixed delays including both time‐varying delays and continuously distributed delays. Under different assumption conditions concerning stochastic disturbance term from the existing ones, some sufficient conditions are derived for assuring the mean square exponential stability of the equilibrium point of the system based on the vector Lyapunov function method and differential‐integral theorem. The obtained results not only generalize the existing ones, but also reduce the conservatism of the previous stability results about complex‐valued neural networks with stochastic disturbances. Two numerical examples with simulation results are given to verify the feasibility of the proposed results.

Iterative Learning Control Design for Multiagent Systems Based on 2D Models

This paper considers a group of systems (agents) described by linear continuous or discrete models. All systems operate in the repetitive mode with a constant pass length, with resetting to the initial state after each pass is complete. Information exchange among the systems is described by a directed graph. The problem of reaching a consensus is formulated as designing an iterative learning control law (protocol) under which the output variable of each agent converges to a reference trajectory (pass profile) as the number of passes grows infinitely. This problem is solved using an original approach based on 2D models and a 2D modification of the vector Lyapunov function method. The ultimate results are written in form of linear matrix inequalities. 2D counterparts of the Fax–Murray theorem are established. An illustrative example is given.

Hybrid Control Approach to Multi-AUV System in a Surveillance Mission

Surveillance missions for multiple autonomous underwater vehicle (AUV) system suggest the use of different modes of operation including organizing and keeping a predefined formation, avoiding obstacles, reaching static and tracking dynamic targets. While exploiting a leader-follower strategy to formation control and the vector Lyapunov function method to controller design, we use discrete-event approach and supervisory control theory to switch between operational modes.

Stochastic Exponential Robust Stability of Delayed Complex-Valued Neural Networks With Markova Jumping Parameters

This paper deals with the problem on stochastic exponential robust stability for a class of complex-valued interval neural networks with Markova jumping parameters and mixed delays, including both time-varying delays and continuously distributed delays. By applying the M-matrix theory and coupling with the vector Lyapunov function method, some sufficient conditions are derived to guarantee the existence, uniqueness, and stochastic exponential robust stability of the equilibrium point of the addressed system. The obtained results not only are easy to judge the dynamical behavior of the addressed system, but also are with lower level conservatism in comparison with some existing results. Finally, two numerical examples with simulation results are given to illustrate the effectiveness of the proposed results.

Stochastic Stability of Some Classes of Nonlinear 2D Systems

The paper considers nonlinear discrete and differential stochastic repetitive processes using the state-space model setting. These processes are a special case of 2D systems that originate from the modeling of physical processes. Using the vector Lyapunov function method, sufficient conditions for stability in the mean square are obtained in the stochastic setting, where the vast majority of the currently known results are for deterministic dynamics. Based on these results, the property of stochastic exponential passivity in the second moment is used, together with the vector storage function, to develop a new method for output feedback control law design. An example of a system with nonlinear actuator dynamics and state-dependent noise is given to demonstrate the effectiveness of the new results.

Impulsive Disturbances on the Dynamical Behavior of Complex-Valued Cohen-Grossberg Neural Networks with Both Time-Varying Delays and Continuously Distributed Delays

This paper studies the global exponential stability for a class of impulsive disturbance complex-valued Cohen-Grossberg neural networks with both time-varying delays and continuously distributed delays. Firstly, the existence and uniqueness of the equilibrium point of the system are analyzed by using the corresponding property of M-matrix and the theorem of homeomorphism mapping. Secondly, the global exponential stability of the equilibrium point of the system is studied by applying the vector Lyapunov function method and the mathematical induction method. The established sufficient conditions show the effects of both delays and impulsive strength on the exponential convergence rate. The obtained results in this paper are with a lower level of conservatism in comparison with some existing ones. Finally, three numerical examples with simulation results are given to illustrate the correctness of the proposed results.

Dynamical behaviour analysis of delayed complex-valued neural networks with impulsive effect

ABSTRACTThis paper investigates the problem of the dynamical behaviours of a class of complex-valued neural networks with mixed time delays and impulsive effect. By separating the complex-valued neural networks into the real and the imaginary parts, the corresponding equivalent real-valued systems are obtained. Some sufficient conditions are derived for assuring the exponential stability of the equilibrium point of the system based on the vector Lyapunov function method and mathematical induction method. The obtained results generalise the existing ones. Finally, two numerical examples with simulations are given to demonstrate the feasibility of the proposed results.

Impulsive effects on stability of discrete-time complex-valued neural networks with both discrete and distributed time-varying delays

In this paper, the impulsive effects on the stability of discrete-time complex-valued neural networks with both discrete and distributed time-varying delays are investigated. Several sufficient conditions are obtained ensuring the existence, uniqueness and global exponential stability of equilibrium point for the considered neural networks by the idea of vector Lyapunov function method, M-matrix theory and analytic technique. Moreover, the estimation for exponential convergence rate index is proposed. An example with simulations is provided to verify the effectiveness of the obtained results.

Stability Analysis of Stochastic Interconnected Systems by Vector Lyapunov Function Method

AbstractIn this paper, the interconnecting structure between a certain system and its lower order subsystems within the infinite‐dimension stochastic interconnected systems is analyzed. Assuming that the excitations are parametric white noises, the exponential string stability for a few classes of nonlinear stochastic interconnected systems is discussed. By using the vector Lyapunov function method, the sufficient conditions of exponential string stability are derived, expanding the scope of the parameters for systems stability. Moreover, some cases of exponential string stability for vehicle‐following systems in automated highway systems are specified. Finally, an example is shown to illustrate the proposed method.