Vector Lyapunov Functions Research Articles

ASYMPTOTIC STABILITY ANALYSIS AND STABILIZATION CONTROL OF FRACTIONAL-ORDER VEHICLE SUSPENSION SYSTEM WITH TIME DELAY

Our study focuses on the analysis of asymptotic stability and the development of asymptotic stabilization control methods for fractional vehicle suspension systems (FVSS). We begin by constructing a mathematical model for FVSS using the state-space equations of Caputo fractional calculus. Initially, we utilize the fractional Routh–Hurwitz criterion to derive the necessary conditions for asymptotic stability and instability in the open-loop system of the fractional vehicle suspension. Subsequently, we propose a novel control strategy for FVSS and establish an associated asymptotic stabilization criterion by combining a new vector Lyapunov function with the [Formula: see text]-matrix method. Moreover, we extend the fractional-order vehicle suspension model to include time delay resulting from the interactions between different variables in the real system, thus creating a FVSS with time delay. Based on the vector Lyapunov function, [Formula: see text]-matrix measure, and Razumikhin interpretation, we develop a control strategy specifically tailored for FVSSs with time delay. Lastly, we compare two numerical simulations of the FVSS, one with time delay and one without, to demonstrate the accuracy, effectiveness, and applicability of the proposed method presented in our paper.

Construction of vector Lyapunov function for nonlinear large-scale system with periodic subsystems

Construction of vector Lyapunov function for nonlinear large-scale system with periodic subsystems

Networked Iterative Learning Control Under Changing Operating Mode of Agents and Configuration of Information Network

The paper considers the iterative learning control (ILC) design problem of a network system under changing operating mode of subsystems (agents) and configuration of information network. The network system consists of identical agents, which are discrete linear dynamic plants operating in a repetitive mode. The operating modes of agents depend on their parameters and the reference trajectory, which must be tracked with required accuracy at output of system. The configurations of information network define the group of functioning agents and the type of information exchange between them. The mode and the configuration change takes place in accordance with certain external rules. The control design is based on the divergent method of the vector Lyapunov function. For reducing the transient error caused by the mode change and the connection of new agents, a special rule for switching the ILC law is proposed. The results of modeling the obtained control law for a group of manipulators with flexible link are presented.

The Stability of Set-Valued Differential Equations with Different Initial Time in the Sense of Fractional-like Hukuhara Derivatives

This paper investigates set-valued differential equations with fractional-like Hukuhara derivatives. Firstly, a novel comparison principle is given by introducing the upper quasi-monotone increasing functions. Then, the stability criteria of Lipschitz stability and practical stability of such equations with different initial time are obtained via the new comparison principle and vector Lyapunov functions.

NEW RESULTS ON K- STABILITY AND BOUNDEDNESS OF SOLUTIONS OF A CERTAIN DIFFERENTIAL EQUATIONS IN BANACH SPACES INTERACTING WITH WEAKLY SUBSYSTEMS

The article gives new results of K- Stability and Boundedness of solutions of certain differential equations in Banach Spaces interacting with weakly subsystems. Some results from the theory of semigroups, and the description of the general method of solution of the posed problem were also given. The application of the matrix – valued Lyapunov function is discussed and the main theorem of the method is formulated for differential equations in a Banach Space. The vector Lyapunov function was the main tool applied for the analysis of K – Stability and Boundedness of a system in Banach Spaces. All the results of this Article for the ‘equation (2.2)’ are essentially new.

Adaptive Identification of System with Bouc-Wen Hysteresis Modifications

The adaptive identification method developed to evaluate the parameters of the Bouc-Wen hysteresis (BWH). The adaptive approach based on the use of adaptive observer. We synthesize adaptive identification algorithms using the second Lyapunov method. Requirements for the input of the system which guarantee the identification of parameters considered. We propose BWH modifications (BWHM). Adaptive algorithms for estimating BWHM parameters developed. The boundedness of adaptive system processes shown in coordinate and parametric spaces. We prove the exponential dissipativity of processes in an adaptive system by using the Lyapunov vector function method. Estimating method proposed for signaling uncertainty in the system.

Finite-Time Stabilization for Continuous Triangular Systems via Vector Lyapunov Function Approach

This article provides finite-time stabilizing control laws for the continuous triangular systems, in which the nominal dynamics is the chain of power integrators with the power of a positive odd rational number. For a class of continuous lower-triangular systems, the linear controllers are designed. By invoking a vector Lyapunov function theory, there is no need to use the usual back-stepping technique and a quite number of inequality computations are avoided. For a class of continuous upper-triangular systems, the nested-saturation controllers are constructed. In this case, the use of a vector Lyapunov function theory also helps to reduce computational burden, and allows us to summarize a same group of parameter conditions guaranteeing both the reduction of saturated terms and the finite-time stability of the reduced system.

Asymptotic and Robust Stabilization Control for the Whole Class of Fractional-Order Gene Regulation Networks with Time Delays

Throughout this article, a novel control strategy for fractional-order gene regulation networks (FOGRN) of all categories is designed by using the vector Lyapunov function in combination with the M-matrix measure. Firstly, a series of puzzles surrounding the asymptotic stability of two-dimensional FOGRN are studied, and a new asymptotic stability control strategy is formulated based on the vector Lyapunov function in combination with the M-matrix measure, ensuring that the controlled FOGRN has a strong robust stability. In addition, the corresponding asymptotic stability criterion is deduced. On this basis, the problem of asymptotic stability of a three-dimensional FOGRN is studied. Based on the new method, a stabilization control strategy is also formulated with the corresponding asymptotic stability criterion deduced, ensuring that the controlled FOGRN has a strong robust stability as well. Finally, this novel method’s effectiveness and generality are authenticated via simulation experiments.

Bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks under signed digraph via adaptive strategy

In this work, an adaptive controller, formulated as linear feedback controls plus nonlinear parts, is synthesized to achieve bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks (FMNNs), in which follower FMNNs are linearly coupled under a signed digraph. The salient features of this research lie in two aspects: (1) the assumption on time-varying delays is very weak, since it neither requires boundedness of delays nor restricts the differentiation of time delay functions; (2) the adaptive controller contains no delay term, and it is feasible for both bounded and unbounded activation functions. As the preparatory work for stability analysis of the controlled synchronization error system, the ready-made results on delayed incommensurate fractional-order linear positive systems, especially stability condition and comparison principle, are perfected by relaxing premises of time-varying delays. Besides, a group of differential inclusion inequalities are established as a powerful aid in scaling the bipartite synchronization error system. More relevantly, an algebraic synchronization criterion, formulated in terms of coupling strength, inner coupling matrix, Laplacian matrix and FMNN system parameters, is proved with the benefit of vector Lyapunov function and positive system theory. With the controller utilized, linear feedback controls can be exerted merely on partial neurons of some selected FMNNs, which is exemplified by numerical simulations.

Lag quasi-synchronization of incommensurate fractional-order memristor-based neural networks with nonidentical characteristics via quantized control: A vector fractional Halanay inequality approach

This work realizes lag quasi-synchronization of incommensurate fractional-order memristor-based neural networks (FMNNs) with nonidentical characteristics via quantized control. The motivations behind this research work are threefold: (1) quantized controllers, which generate discrete control signals, can be more easily realized in computers than non-quantized controllers, and can consume smaller communication capacity; (2) incommensurate orders in a single FMNN and nonidentical characteristics in drive-response FMNNs are inescapable due to the differences among the circuit elements used to implement FMNNs; (3) convergence analysis of delayed incommensurate fractional-order nonlinear systems, which is the basis for the derivation of synchronization criterion, has not been handled perfectly. As an effective tool for convergence analysis of delayed incommensurate fractional-order nonlinear systems, especially for estimation of ultimate state bound, a vector fractional Halanay inequality is established at first. Then, a quantized synchronization controller, in which the dead-zone is introduced into some logarithmic quantizers to avoid chattering phenomenon, is designed. By means of vector Lyapunov function together with the newly derived vector fractional Halanay inequality, the synchronization criterion is proved theoretically. Lastly, numerical simulations supplementarily illustrate the correctness of the synchronization criterion. In contrast with the hypotheses in the relevant literature, the hypotheses in this paper are weaker.

A novel asymptotic stability condition for a delayed distributed order nonlinear composite system with uncertain fractional order

This paper mainly proposes a novel stability condition for distributed order composite systems with delay based on some properties of Caputo fractional-order derivatives in distributed order form that we extend and the generalization of a new method of vector Lyapunov function combined with M-matrix. First, we extend some properties of the Caputo fractional-order derivative to its distributed order form of it effectively. Next, we use a new method to solve the stability problem of the distributed order systems via a series of class−κ functions. Then, we propose the novel stability condition of the distributed order composite systems whose interconnect part is not only linear but also nonlinear based on the generalization of a new method which is vector Lyapunov function combined with M-matrix. Moreover, we solve the stability problem of the distributed order composite systems with time-delay. Finally, we provide several numerical simulation examples for all different cases show the correctness and usefulness of the novel stability condition.

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.

Stability analysis and feedback design of a direct current–direct current power converter by vector Lyapunov function

SummaryTo prove the stability of hybrid systems, where, several subsystems are in series–parallel arrangement, Vector Lyapunov functions are appropriate substitutes for the scalar Lyapunov ones. By assigning a scalar Lyapunov function to each subsystem, the stability of the whole hybrid system can be analyzed. In this paper, having reviewed the preliminaries, a direct current (DC) –DC converter working as an incremental voltage source is selected. We show that the conditions of the vector Lyapunov stability are satisfied by deriving the variation interval for the feedback coefficients and obtaining the appropriate feedback factor given the desired output. Simplicity and less restrictedness of vector Lyapunov method is shown comparatively. All the derived mathematical relations are validated by numerical computations.

On stability with respect to the part of variables of a non-autonomous system in a cylindrical phase space

Abstract. The stability problem of a system of differential equations with a right-hand side periodic with respect to the phase (angular) coordinates is considered. It is convenient to consider such systems in a cylindrical phase space which allows a more complete qualitative analysis of their solutions. The authors propose to investigate the dynamic properties of solutions of a non-autonomous system with angular coordinates by constructing its topological dynamics in such a space. The corresponding quasi-invariance property of the positive limit set of the system’s bounded solution is derived. The stability problem with respect to part of the variables is investigated basing of the vector Lyapunov function with the comparison principle and also basing on the constructed topological dynamics. Theorem like a quasi-invariance principle is proved on the basis of a vector Lyapunov function for the class of systems under consideration. Two theorems on the asymptotic stability of the zero solution with respect to part of the variables (to be more precise, non-angular coordinates) are proved. The novelty of these theorems lies in the requirement only for the stability of the comparison system, in contrast to the classical results with the condition of the corresponding asymptotic stability property. The results obtained in this paper make it possible to expand the usage of the direct Lyapunov method in solving a number of applied problems.

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.

Guiding Functional Families, Lyapunov Vector Functions, and the Existence of Poisson Bounded Solutions

Guiding Functional Families, Lyapunov Vector Functions, and the Existence of Poisson Bounded Solutions

Novel asymptotic stabilisation conditions of nonlinear fractional-order composite systems via the decentralised control

ABSTRACT In this paper, we consider decentralised control of a nonlinear fractional-order composite system and show novel sufficient conditions for the asymptotic stabilisation of the fractional-order composite system by a set of local state feedback controllers. First, we design a local state feedback controller for each nonlinear fractional-order subsystem and a Lyapunov function that guarantees the stabilisation of the isolated nonlinear fractional-order subsystem by the controller. Then, we construct a vector Lyapunov function from the set of the Lyapunov functions, and show the stabilisation of the nonlinear fractional-order composite system. Finally, two numerical simulation examples of the nonlinear fractional-order composite systems are performed to show the correctness and usefulness of the proposed decentralised control.

Asymptotic Stabilization for a Class of Linear Fractional-Order Composite Systems

In this paper, we present the design for a decentralized control method comprising a series of local state feedback controllers for a class of linear fractional composite systems. In addition, the corresponding asymptotic stabilization criterion is derived. First, we design the local state feedback controllers for each subsystem of the linear fractional composite system. Then, based on the vector Lyapunov function, we combine these local state feedback controllers into a single decentralized controller through which the asymptotic stabilization criterion is proposed for the class of linear fractional composite system. Finally, numerical simulation of a class of linear fractional composite systems is used to verify the accuracy and effectiveness of the decentralized control method.

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.

Global stabilization of fractional-order memristor-based neural networks with incommensurate orders and multiple time-varying delays: a positive-system-based approach

This paper addresses global stabilization of fractional-order memristor-based neural networks (FMNNs) with incommensurate orders and multiple time-varying delays (MTDs), where the time delay functions are not necessarily bounded. First, without assuming that time delay functions are bounded, the asymptotical stability condition is given for fractional-order linear positive system with incommensurate orders and MTDs. Then, comparison principle for such a system is established. By virtue of two kinds of vector Lyapunov functions (absolute-value-function-based and square-function-based vector Lyapunov functions), stability condition of fractional-order linear positive system and comparison principle, two stabilization criteria are derived and the equivalence between them is illustrated. In comparison with the reported criterion, the criteria derived in this paper are less conservative, since they allow controller parameters to satisfy weaker algebraic conditions. Lastly, numerical examples are displayed to validate the availability of the controller and correctness of the stabilization criteria.