Vector Lyapunov Functions Research Articles

Lyapunov Vector Functions, Rotation of Vector Fields, Guiding Functions, and the Existence of Poisson Bounded Solutions

Lyapunov Vector Functions, Rotation of Vector Fields, Guiding Functions, and the Existence of Poisson Bounded Solutions

Exponential stability of singularly perturbed systems with mixed impulses

This paper investigates the exponential stability problem for a class of singularly perturbed impulsive systems in which the flow dynamics is unstable and is affected at discrete time instants by impulses that have both destabilizing and stabilizing effects. More precisely the impulses have stabilizing effects on the slow variables but destabilizing effects on the fast ones. Thus, a first contribution of our work is related to stability analysis of singularly perturbed impulsive systems in the case when neither the flow dynamics nor the impulsive one is stable. In order to take full advantage of the jump matrix structure and its stabilizing effects on the slow dynamics, we introduce a new impulse-dependent vector Lyapunov function. This function allows us to better describe the behavior between two consecutive impulses as well as the jumps at impulse instants. Several numerically tractable criteria for stability of singularly perturbed impulsive systems are established based on vector comparison principle. Additionally, upper bounds on the singular perturbation parameter are derived. Finally, the validity of our results is verified by two numerical examples.

Vector Lyapunov Functions, Complete Sets of Guiding Functions, and the Existence of Poisson Bounded Solutions

Based on the synthesis of the method of vector Lyapunov functions and the method of complete sets of guiding functions, we obtain sufficient conditions for the existence of Poisson bounded solutions to systems of differential equations, as well as solutions which are partially bounded in the Poisson sense.

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.

Input-to-state stability of impulsive reaction–diffusion neural networks with infinite distributed delays

In this paper, we focus on the global existence–uniqueness and input-to-state stability of the mild solution of impulsive reaction–diffusion neural networks with infinite distributed delays. First, the model of the impulsive reaction–diffusion neural networks with infinite distributed delays is reformulated in terms of an abstract impulsive functional differential equation in Hilbert space and the local existence–uniqueness of the mild solution on impulsive time interval is proven by the Picard sequence and semigroup theory. Then, the diffusion–dependent conditions for the global existence–uniqueness and input-to-state stability are established by the vector Lyapunov function and M-matrix where the infinite distributed delays are handled by a novel vector inequality. It shows that the ISS properties can be retained for the destabilizing impulses if there are no too short intervals between the impulses. Finally, three numerical examples verify the effectiveness of the theoretical results and that the reaction–diffusion benefits the input-to-state stability of the neural-network system.

Вектор-функции Ляпунова, полные наборы направляющих функций и существование ограниченных по Пуассону решений

Вектор-функции Ляпунова, полные наборы направляющих функций и существование ограниченных по Пуассону решений

Structural-Identification Aspects of Decision-Making in Systems with Bouc-Wen Hysteresis

Considering the structural analysis problem of systems properties with Bouc-Wen hysteresis (BWH), various approaches are proposed for the identification of BWH parameters. The applied methods and algorithms are based on the design of parametric models and consider a priori information and the results of data analysis. Structural changes in the BWH form a priori. Methods for the Bouc-Wen model (BWM) identification and its structure estimation are not considered under uncertainty. The study’s purpose is the analysis the structural problems of the Bouc-Wen hysteresis identification. The analysis base is the application of geometric frameworks (GF) under uncertainty. Methods for adaptive estimation parameters and structural of BWM were proposed. The adaptive system stability is proved based on vector Lyapunov functions. An approach is proposed to estimate the identifiability and structure of the system with BWH. The method for estimating the identifiability degree based on the analysis of GF is considered. BWM modifications are proposed to guarantee the system’s stability and simplify its description.

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.

Stochastically Globally Exponential Stability of Stochastic Impulsive Differential Systems with Discrete and Infinite Distributed Delays Based on Vector Lyapunov Function

This paper deals with stochastically globally exponential stability (SGES) for stochastic impulsive differential systems (SIDSs) with discrete delays (DDs) and infinite distributed delays (IDDs). By using vector Lyapunov function (VLF) and average dwell-time (ADT) condition, we investigate the unstable impulsive dynamics and stable impulsive dynamics of the suggested system, and some novel stability criteria are obtained for SIDSs with DDs and IDDs. Moreover, our results allow the discrete delay term to be coupled with the nondelay term, and the infinite distributed delay term to be coupled with the nondelay term. Finally, two examples are given to verify the effectiveness of our theories.

Iterative Learning Control Design for Discrete-Time Stochastic Switched Systems

Discrete-time linear systems with switching in the repetitive mode are considered. The systems are subjected to random disturbances, and the measurements are corrupted by additive noises. Two iterative learning control design methods are proposed. Both of the methods involve an auxiliary 2D model in the form of a discrete repetitive process. The first method is based on the dissipativity conditions established for the auxiliary model with a special choice of the supply rate and storage function. This choice allows finding a control law (in the general case, nonlinear) that ensures the convergence of the learning process. The second method adopts a linear iterative learning control update law of a given form, while the convergence of the learning process is ensured by the stability conditions of the auxiliary 2D model. The structure of both control laws includes a stationary Kalman filter. The stability conditions are obtained using the divergent method of vector Lyapunov functions. An example is given to demonstrate the capabilities and features of the new method.

Razumikhin-type theorem for pth exponential stability of impulsive stochastic functional differential equations based on vector Lyapunov function

This paper is concerned with the pth moment exponential stability of stochastic functional differential equations with impulses. Based on average dwell-time method, Razumikhin-type technique and vector Lyapunov function, some novel stability criteria are obtained for impulsive stochastic functional differential systems. Two examples are given to demonstrate the validity of the proposed results.

Asymptotic stabilization of general nonlinear fractional-order systems with multiple time delays

In this paper, two new control methods based on a Lyapunov-like function and a vector Lyapunov function separately were put forward to solve the asymptotic stabilization problem of general fractional-order nonlinear systems with multiple time delays. First, we deduced a new asymptotic stabilization control criterion based on a Lyapunov-like function after discussing the asymptotic stability criterion of general fractional-order nonlinear systems. Moreover, to address the problem of multiple time delays of the nonlinear system, we derived another asymptotic stabilization control criterion based on a vector Lyapunov function and an M-matrix via the new controller. Finally, the feasibility and effectiveness of the proposed controllers were verified by several common fractional-order nonlinear systems.

Iterative Learning Control Design for Switched Systems

In this paper, discrete-time linear systems with parameter switching in the repetitive mode are considered. A new iterative learning control design method is proposed. This method is based on the construction of an auxiliary 2D model in the form of a discrete repetitive process; the stability of the auxiliary model guarantees the convergence of the learning process. Stability conditions are derived using the divergent method of Lyapunov vector functions. The concept of the average dwell time in pass direction is introduced. An example that demonstrates the capabilities and features of the new method is presented.

State Observer-Based Iterative Learning Control of an Uncertain Continuous-Time System

Linear systems with the affine model of parametric uncertainty that operate in a repetitive mode are considered. For such systems, a new iterative learning control design method is proposed. This method is based on the use of a full-order state observer and an auxiliary 2D model in the form of a differential repetitive process whose stability guarantees the convergence of the learning process. For obtaining stability conditions, the divergent method of vector Lyapunov functions is used. An example illustrating the features and advantages of the new iterative learning control design method is presented.

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.

Iterative Learning Control of a Multiagent System under Random Perturbations

A multiagent system in which each of the agents is described by a linear discrete-time model with random perturbations (external random disturbances affecting the plant and measurement noises) is considered. Networked modifications of iterative learning control laws based on minimizing the deviations from a reference model and also based on the theory of stochastic stability of repetitive processes using the divergent method of vector Lyapunov functions are proposed. These modifications are compared with each other by an illustrative example of iterative learning control for a group of gantry robots.

Management of life modelling of regional socio-economic systems

The study posed and solved the problems of theoretical and methodological substantiation of the management of modeling of the provision of vital functions of the population and the functioning of national economic facilities. An approach to modeling the level of life and stability of the region’s dynamic systems is presented. The mathematical apparatus and methodological support were used using the methods of linear matrix inequalities and non-quadratic Lyapunov vector functions, together with the method of optimizing linear series of a combination of partial and average values of integrated socio-economic indicators. Methods for evaluating managerial decisions and differentiating regions at the level of constituent entities of the Russian Federation are proposed and tested using the example of Vladimir, Ivanovo and Ryazan regions. Studies have established that the level of economic viability of the population is a priority indicator in characterizing the level and quality of public administration and determining the differentiation of a region (territory). As a result of the analysis, we came to the conclusion that the solution to the problems of increasing the sustainability of the economic development of regional economies should be associated with the following processes: strategic management as a continuous process of substantiating and selecting promising goals for the sustainable development of the region and developing specific plans and programs to achieve these goals; financial modeling as a system of forms, methods and techniques for managing cash flows, financial and material resources of the region, aimed at ensuring sustainable development of the territory; optimization of production and technical activities as a set of methods, principles and management tools for the transition to modern technologies and environmental quality standards; legal regulation as a process of targeted impact on social, economic, environmental and social relations occurring in the region, with the help of legal (legal) institutions; formation of the organizational structure and strengthening of the administrative resource (intensification of the impact of socio-political forces on socio-economic development). In these conditions, one of the guarantors of the implementation of measures to achieve balance and sustainable development should be local authorities.

Stability of Switched Differential Repetitive Processes and Iterative Learning Control Design

In this paper general stability conditions of differential nonlinear repetitive processes with switching are obtained. The approach is based on the development of a method that uses vector Lyapunov functions and the properties of the counterpart of its divergence. The obtained results are applied to iterative learning control design for switched linear system. An example that demonstrate effectiveness of the new design is given.

Iterative Learning Control for Switched Systems in the Presence of Input Saturation

The paper considers iterative learning control for differential and discrete switched linear systems with control input saturation. A new design is developed based on the use of common vector Lyapunov functions. An example demonstrating the features and advantages of the new design is given.

Dissipativity-Based Asynchronous Control for 2-D Markov Jump Systems in Continuous-Time Domain

In Markov jump systems (MJSs), it is really difficult for the controller to perfectly follow the plant's mode transitions due to a number of practical factors. Therefore, this article considers to develop an asynchronous controller for a class of two-dimensional continuous-time MJSs. The relation of the plant's mode and the controller's mode is modeled by the hidden Markov model, i.e., the plant's mode influences the controller's transitions in a conditional probabilistic style. By defining a vector Lyapunov function, a sufficient condition for mean-square exponential stability and ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> )- ε dissipativity is derived, whose feasible solutions give birth to a desired asynchronous controller. Finally, the influence of conditional probabilities on dissipativity performance is further studied through simulations.