Sufficient Conditions For Asymptotic Stability Research Articles

Stability and Robustness Analysis of Quasi-Periodic System Subjected to Uncertain Parametric Excitations and Nonlinear Perturbations

Abstract In this work, the asymptotic stability bounds are identified for a class of linear quasi-periodic dynamical systems with stochastic parametric excitations and nonlinear perturbations. The application of a Lyapunov–Perron (L-P) transformation converts the linear part of such systems to a linear time-invariant form. In the past, using the Infante’s approach for linear time-invariant systems, stability theorem and corollary were derived and demonstrated for time periodic systems with variation in stochastic parameters. In this study, the same approach is extended toward linear quasi-periodic with stochastic parameter variations. Furthermore, the Lyapunov’s direct approach is employed to formulate the stability conditions a for quasi-periodic system with nonlinear perturbations. If the nonlinearities satisfy a bounding condition, sufficient conditions for asymptotic stability can be derived for such systems. The applications of stability theorems are demonstrated with practical examples of commutative and noncommutative quasi-periodic systems.

Event‐triggered fuzzy output feedback fault‐tolerant control for interval type‐2 Takagi–Sugeno large‐scale systems with time delays

The event-triggered output feedback fault-tolerant control problem is studied in this paper for interval type-2 Takagi–Sugeno (IT2 T-S) fuzzy large-scale systems with time delays. First, a fuzzy state observer is designed to address the immeasurable states problem. Then, by introducing a discrete event-triggered mechanism, an observer-based event-triggered fuzzy decentralized controller with dynamic parameters is synthesized in the presence of actuator faults based on the nonparallel distribution compensation (PDC) design principle. The sufficient conditions of asymptotic stability for the closed-loop systems are derived by constructing an appropriate dynamic parameter based Lyapunov–Krasovskii functional candidate. Finally, a simulation example is provided to verify the feasibility of the developed fuzzy control method.

H[formula omitted] synchronization of semi-Markovian jump neural networks with random sensor nonlinearities via adaptive event-triggered output feedback control

In this paper, the H∞ synchronization problem of semi-Markovian jump neural networks (s-MJNNs) based on adaptive event-triggered scheme (AETS) output feedback control is studied. Also, the time-varying delay and leakage delay are considered in the system model. In order to solve the problems that the state of the system is not completely observable and limited network resources, an output feedback controller with AETS is designed. At the same time, in order to describe the performance of the sensor in the feedback link, the random nonlinear phenomenon of the sensor is described by the variable complying with Bernoulli probability distribution. A suitable Lyapunov–Krasovskii functional (LKF) is constructed and the bounds of integral terms are estimated by affine Bessel–Legendre inequality. Finally, sufficient conditions for asymptotic stability of the synchronization error system are obtained. And, two numerical examples show the feasibility of the research work.

Higher‐order derivatives of generalized Lyapunov‐like functions for switched nonlinear systems

AbstractThis paper is concerned with utilizing higher‐order derivatives of generalized Lyapunov‐like functions to infer asymptotic stability of the switched nonlinear system without requiring the first derivative of the Lyapunov function for each activated subsystem to be negative definite. In contrast to the existing switched systems literature, we allow generalized Lyapunov‐like functions to increase both during the continuous flows and the discrete jumps. Firstly, a set of sufficient conditions for asymptotic stability of switched nonlinear systems are derived under a state‐dependent switching law. Then, all the above‐mentioned conditions are transformed to a set of bilinearly diagonally dominant sum of squares constraints. Bilinearly diagonally dominant sum of squares programming is proposed as a tool to solve the conditions provided. Finally, two examples are introduced to prove the effectiveness of the above method.

Asymptotical Stability of Riemann‐Liouville Nonlinear Fractional Neutral Neural Networks with Time‐Varying Delays

In this paper, the asymptotic stability of solutions is investigated for a class of nonlinear fractional neutral neural networks with time‐dependent delays which are unbounded. By constructing the appropriate Lyapunov functional, sufficient conditions for asymptotic stability of neural networks are obtained with the help of LMI. An example is presented by using the LMI Toolbox to demonstrate the effectiveness of the obtained results.

Convergence and stability of meshfree method based on radial basis function for a hyperbolic partial differential equation with piecewise constant arguments

Meshfree method based on radial basis functions is a popular tool used to numerically solve partial differential equations. This paper deals with the convergence and stability of the proposed method for hyperbolic partial differential equations with piecewise constant arguments. Firstly, we apply central difference to the time variable and θ-weighted finite difference to the original equation simultaneously, then the numerical scheme is obtained by using the meshfree method based on radial basis functions to approximate the unknown function and its second-order spatial derivative. Secondly, the convergence is analysed in -norm and the sufficient conditions for asymptotic stability of the numerical solution are acquired under which the analytic solution is asymptotically stable. Finally, our study is supported by several numerical experiments.

Dissipativity analysis for Takagi–Sugeno fuzzy system with time-varying delays and stochastic packet dropouts

The dissipative stability analysis of Takagi–Sugeno fuzzy system (TSFS) is the focus of this paper via taking the effects of both delays and packet dropouts. The novel augmented Lyapunov–Krasovskii functional (LKF) is proposed via utilizing the delay-dependent matrix, which provides the information of the delay derivative bounds. By applying the dynamic delay partition method, the delay interval is divided into several variable subintervals, which takes full advantage of time-varying delay information. In addition, the randomly occurring asynchronous or synchronous controllers are put forward to enhance the design flexibility and control performance. On the basis of the new augmented LKF and improved boundary techniques, the sufficient asymptotic stability condition is derived in the form of linear matrix inequalities (LMIs). Subsequently, numerical examples are addressed to certify the feasibility and advantage of the put forward methodology.

Event-triggered output-based control of T-S fuzzy systems under quantisation, actuator saturation and cyber attacks

This paper studies output-based event-triggered security control of networked T-S fuzzy systems under signal quantisation, actuator saturation and deception attacks governed by a Bernoulli distribution. Firstly, using periodically sampled output of a T-S fuzzy system, a discrete event-triggered mechanism (ETM) is introduced to save network bandwidth, which excludes Zeno behaviour absolutely. Secondly, a closed-loop system model is built, which integrates parameters of the T-S fuzzy plant, the ETM, quantiser, actuator saturation, stochastic attacks, network-induced delays and fuzzy dynamic output feedback controller in a unified framework. Thirdly, sufficient conditions for asymptotic stability of the T-S fuzzy system are obtained, and further a fuzzy output-based security controller is designed. Finally, examples confirm effectiveness of the proposed methods.

Novel criteria for global robust stability of dynamical neural networks with multiple time delays

This research article considers the problem regarding global robust asymptotic stability of the general type of dynamical neural networks involving multiple constant time delays. Some new sufficient criteria are proposed for the existence, uniqueness and global asymptotic stability of the equilibrium point of this neural network model whose network parameters possess uncertainties. This paper will first address the existence and uniqueness problem for equilibrium points by utilizing the Homomorphic transformation theory. Secondly, by exploiting a novel Lyapunov functional candidate, the sufficient conditions for asymptotic stability of equilibrium points of this class of delayed neural networks will be established. The derived robust stability conditions are expressed independently of the constant time delay parameters and define some novel relationships among network parameters of the considered neural network. Thus, the applicability and validity of the obtained robust stability conditions for delayed-type neural networks can be easily tested. The comprehensive comparisons among the results of the current article and some of previously derived corresponding results will also be made by giving an illustrative numerical example.

Bumpless Transfer Control for Switched Linear Systems and its Application to Aero-Engines

To avoid the bumps of the control signal in the switching instants, this paper proposes a bumpless transfer control method which guarantees the continuous of control signal while keeping the asymptotic stability of the closed-loop system. Firstly, the presented bumpless transfer control method possesses a simple structure and meanwhile not have to redesign sub-controllers, which is easy to implement. Furthermore, the method is appropriate for either stable subsystems or unstable subsystems, and be suitable for either state feedback control or dynamic output feedback control. Moreover, we give the sufficient conditions of asymptotic stability for switched linear systems under bumpless transfer control for four cases. For the original switched system with stable subsystems under state or dynamic output feedback control, the stability conditions under mode-dependent average dwell time switching are derived. For the original switched system with unstable subsystems under state or dynamic output feedback control, which is stabilized by some certain switching laws, the stability conditions under the original switching law are given. Finally, a hardware-in-the-loop simulation of an aero-engine control system is employed to verify the effectiveness and superiority of the proposed method.

A Sufficient Condition for Asymptotic Stability of Kinks in General (1+1)-Scalar Field Models

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models $$\begin{aligned} \partial _t^2\phi -\partial _x^2\phi + W'(\phi ) = 0, \quad (t,x)\in \mathbb {R}\times \mathbb {R}. \end{aligned}$$ The orbital stability of kinks under general assumptions on the potential W is a consequence of energy arguments. Our main result is the derivation of a simple and explicit sufficient condition on the potential W for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Last, motivated by the Physics literature, we present applications of the criterion to the $$P(\phi )_2$$ theories and the double sine-Gordon theory.

Nonlinear Control for Attitude Stabilization of a Rigid Body Forced by Nonstationary Disturbances with Zero Mean Values

A rigid body forced by a nonstationary perturbing torque with zero mean value is under consideration. The control strategy for attitude stabilization of the rigid body is based on the usage of dissipative and restoring torques. It is assumed that the dissipative torque is linear, while restoring and perturbing torques are purely nonlinear. A theorem on sufficient conditions for asymptotic stability of the body angular position is proved on the basis of the decomposition method, the Lyapunov direct method and the averaging technique. Computer simulation results illustrating the theorem are presented.

Asymptotical Stability and Stabilization of Continuous-Time Probabilistic Logic Networks

Discrete-time probabilistic logic networks (DT-PLNs), of which probabilistic Boolean networks (PBNs) are a special type, are an important qualitative model for gene regulatory networks (GRNs). Although a DT-PLN can predict the long-term behavior of a GRN, using it to describe the transient kinetics at the microtimescale level remains inconvenient. In this article, we investigate the problems associated with the stability and stabilization of continuous-time probabilistic logic networks (CT-PLNs). First, we demonstrate that the concept of finite-time stability for DT-PLNs cannot be extended to CT-PLNs owing to the nonsingularity of transitional probability matrices. Thus, we introduce the concept of asymptotical stability, which is defined as the convergence in distribution of the network state. Second, by developing the theory of invariant subsets for CT-PLNs, a necessary and sufficient condition for asymptotical stability with respect to a subset is proposed, which is expressed in terms of the transition rate matrix of probability. Third, for a CT-PLN with input nodes, termed a continuous-time probabilistic logic control network, we discuss the subsets that are invariant under piece-wise constant input. Based thereupon, we propose a necessary and sufficient condition under which asymptotically stabilizing sampled-data feedback exists. A method for designing sampled-data feedback is proposed. Last, Monte Carlo simulation algorithms are proposed to efficiently simulate a CT-PLN in the time domain. Examples are provided to demonstrate the methods proposed in this article.

Stability results for a nonlinear two-species competition model with size-structure

We formulate a system of integro-differential equations to model the dynamics of competition in a two-species community, in which the mortality, fertility and growth are size-dependent. Existence and uniqueness of nonnegative solutions to the system are analyzed. The existence of the stationary size distributions is discussed, and the linear stability is investigated by means of the semigroup theory of operators and the characteristic equation technique. Some sufficient conditions for asymptotical stability / instability of steady states are obtained. The resulting conclusion extends some existing results involving age-independent and age-dependent population models.

On boundedness of solutions of quadratic differential equation sets

We obtain new sufficient conditions of asymptotic stability on solution cone of the set of usual uniform quadratic differential equations. For non-uniform quadratic sets (with non-zero linear part) we give the conditions of boundedness of solutions. We determine quadratic sets whose solution is bounded for any linear part and for any initial conditions. We provide examples of applications of the method in mathematical ecology.

Distributed event‐triggered robust automatic generation control for networked power system with wind turbines

This paper proposes an event-triggered distributed networked control scheme and its robust stability analysis strategy for the automatic generation control of multi-area interconnected power system with wind turbines. First, for the large-scale distributed networked control systems, considering the networked control systems with parameters uncertainty, effective transmission bandwidth, and time-varying networked-induced delays, a new distributed networked control systems model is proposed. Then, by constructing Lyapunov–Krasovskii functional method, the sufficient conditions for asymptotical stability of the system are obtained, a design method of event-triggered distributed networked control law based on linear matrix inequality is proposed. In order to verify the effectiveness of the proposed scheme, a multi-area power system with wind turbines is considered, and the system performance under different load disturbances is simulated. The simulation results show that the scheme has reliable robust performance for multi-area distributed networked power system and improves the anti-interference ability of the power system.

Event-triggered equivalent-input-disturbance estimation and control for disturbance attenuation

A novel three-channel event-triggered equivalent-input-disturbance (EID) estimation approach is proposed for active exogenous disturbance rejection in linear systems. The overall system governing equations consisting of the physical plant, disturbance estimator, internal model, and event-triggering mechanisms (ETMs) are modeled as an augmented closed-loop system with multiple time-varying delays using an input delay method. Sufficient conditions for asymptotic stability of the augmented closed-loop system are derived together with a procedure for co-designing the controller and state observer gains and the ETM parameters that satisfy the sufficient conditions for asymptotic stability. Numerical simulations on a DC motor system are provided to illustrate the effectiveness and potential of the proposed design approach.

Observer‐based preview repetitive control for uncertain discrete‐time systems

AbstractIn this paper, the preview repetitive control (PRC) problem is considered for uncertain discrete‐time systems subject to a previewable periodic reference signal. First, an augmented system including future information on the previewable reference signal is constructed. Second, considering the state unmeasured and periodic tracking reference signal, state observer and preview repetitive controller (PRCr) are designed in both state feedback and output feedback settings. To achieve signal tracking performance, a two‐dimensional (2D) model approach and some LMI technique are presented to describe the control and learning actions of the PRC system by using the lifting technique. Then, by Lyapunov–Krasovskii functions, sufficient conditions for asymptotic stability of the 2D system are derived in the form of a set of LMIs, and meanwhile, the design of a repetitive controller with preview actions is proposed through the LMI approach. Finally, two examples illustrate the effectiveness of the results.

Standard L1 Stability for Singular Switched Positive Systems with Time-Varying Distributed Delay Under MDRDT Constraint

This paper is concerned with the standard [Formula: see text] stability with respect to singular switched positive systems (SSPSs) with time-varying distributed delays (TVDDs) under mode-dependent ranged dwell time (MDRDT) constraint. Firstly, for SSPSs with TVDDs under MDRDT constraint, a sufficient condition of asymptotical stability is proposed. Next, to analyze disturbance attenuation performance, a sufficient condition of standard [Formula: see text] stability is presented for SSPSs with TVDDs under MDRDT constraint. Via convex optimization approach, the corresponding standard [Formula: see text]-gain performance optimum can be received. In addition, the standard [Formula: see text] stability issue is also extended to the SSPSs in nonsingular case, in delay-free case and in RDT constraint case, respectively. Finally, the validity is proved by giving three examples.

Finite-Time Lyapunov Functions and Impulsive Control Design

In this paper, we introduce finite-time Lyapunov functions for impulsive systems. The relaxed sufficient conditions for asymptotic stability of an equilibrium of an impulsive system are given via finite-time Lyapunov functions. A converse finite-time Lyapunov theorem for controlling the impulsive system is proposed. Three examples are presented to show how to analyze the stability of an equilibrium of the considered impulsive system via finite-time Lyapunov functions. Furthermore, according to the results, we design an impulsive controller for a chaotic system modified from the Lorenz system.