Problem Of Asymptotic Stability Research Articles

Robust asymptotic stability analysis for fractional-order systems with commensurate time delays: The 1 < β ≤ 2 case

The robust asymptotic stability problem for fractional-order linear systems with commensurate time delays (FOSCDs) and structured perturbations is investigated in this manuscript. Based on the properties of Kronecker products and μ-analysis, the necessary and sufficient conditions are given to guarantee the asymptotic stability of FOSCDs. Then, the necessary and sufficient conditions for the robust asymptotic stability of FOSCDs with structured perturbations are investigated. To reduce the computing complexity and deal with the problems of systems with larger dimensions, a sufficient condition on the robust asymptotic stability of FOSCDs with structured perturbations is given based on the properties of Kronecker products and 2-norm transformations. Finally, several examples are illustrated to test the effectiveness of the proposed results.

Modified Nested Saturated Control for Uncertain Multiple Integrators With High-Order Nonlinear Perturbation.

This article studies the global asymptotic stability problem of multiple integrators subject to uncertain parameters and high-order nonlinear perturbation. By introducing a modified saturation function with positive tunable parameters, three types of modified nested saturated controllers with different parameter assignments are proposed to handle the uncertain integrators with high-order nonlinear perturbation. The specific mechanism of the modified nested saturated controllers with different parameter assignments, in which control parameters appear before states, saturation functions, or saturation levels, is comprehensively analyzed. With the help of a modified saturation function, the global convergence domain of each state can be kept invariable in the saturation reduction analysis. Meanwhile, a large saturation level is obtained by adjusting positive tunable parameters. Combining with derivatives calculation and M -matrix-based comparison theory, proposed saturated control schemes are utilized to handle the considered system. Finally, an active magnetic bearing (AMB) system is revisited as an uncertain mechanical system and used to verify the feasibility of the proposed control designs.

Simple robust PD control for asymptotic stabilization of uncertain spacecraft with actuator constraints and disturbances

AbstractThis paper concerns the problem of asymptotic attitude stabilization for uncertain spacecraft subject to actuator constraints and bounded external disturbances. A simple model‐free saturated robust PD control is proposed within the framework of PD design methodology combining sliding mode control strategy. Asymptotic stabilization stability is proved in terms of Lyapunov's direct method and Barbalat's lemma. Advantages of the proposed control include the easy‐going implementation with simple and intuitive structure and explicit tuning procedure and abilities to guarantee the asymptotic attitude stabilization and bounded disturbance rejection and ensure the actuator constraint is not violated by selecting the control gains a priori. Simulation comparisons demonstrate the improved performance of the proposed approach.

Non-separation Method-Based Global Stability Criteria for Takagi–Sugeno Fuzzy Quaternion-Valued BAM Delayed Neural Networks Using Quaternion-valued Auxiliary Function-Based Integral Inequality

This paper focuses on the global asymptotic stability (GAS) problem for Takagi–Sugeno (T-S) fuzzy quaternion-valued bidirectional associative memory neural networks (QVBAMNNs) with discrete, distributed and leakage delays by using non-separation method. By applying T-S fuzzy model, we first consider a general form of T-S fuzzy QVBAMNNs with time delays. Then, by constructing appropriate Lyapunov–Krasovskii functionals and employing quaternion-valued integral inequalities and homeomorphism theory, several delay-dependent sufficient conditions are obtained to guarantee the existence and GAS of the considered neural networks (NNs). In addition, these theoretical results are presented in the form of quaternion-valued linear matrix inequalities (LMIs), which can be verified numerically using the effective YALMIP toolbox in MATLAB. Finally, two numerical illustrations are presented along with their simulations to demonstrate the validity of the theoretical analysis.

On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms (see Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820]). The main result of this work establishes decay estimates up to exponential time scales for small “codimension one type” perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820], and an implementation of a version of an adapted functional framework introduced by Germain and Pusateri [Forum Math. Pi 10 (2022), p. 172].

Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations

In this paper, the problem of asymptotic stability of a kind of nonlinear perturbed neutral differential system with variable delay is discussed. The Lyapunov-Krasovski\v{\i} functional constructed, is used to obtain conditions for asymptotic stability of the nonlinear perturbed neutral differential system in terms of linear matrix inequality (LMI). The two new results (delay-independent and delay-dependent criteria) include and extend the existing results in the literature. Finally, an example of delay-dependent criteria is supplied and the simulation result is shown to justify the effectiveness and reliability of the used techniques.

Predictor-based observer and controller for nonlinear systems with input/output delays

This letter addresses the problem of asymptotic stabilization by output feedback for nonlinear systems with delays in the input and output. Under stabilizability and detectability of the linearized time-delay system, we show that asymptotic stabilization is solvable by predictor-based nonlinear observer and dynamic output compensator. The proof is based on the linearization method, the design of local nonlinear observers and observer-based controllers, as well as the Razumikhin theorem.

A new design method to global asymptotic stabilization of strict-feedforward nonlinear systems with state and input delays

&lt;abstract&gt;&lt;p&gt;This paper studies the global asymptotic stabilization problem of strict-feedforward nonlinear systems with state and input delays. We will first transform the considered system into an equivalent system by constructing the novel parameter-dependent state feedback controller and introducing the appropriate coordinate transformation. After that, the global asymptotic stability of the closed system is proved by giving the proper Lyapunov-Krasovskii functional and using the stability criterion of time-delay system.&lt;/p&gt;&lt;/abstract&gt;

Discrete-time state delayed systems with saturation arithmetic: overflow oscillation-free realization

ABSTRACT In many smart systems, such as cloud computing, networked control, manufacturing, and telecommunication, the effects of time delays are unavoidable. Saturation nonlinearity is common in many smart systems (e.g. electric systems with limited actuator power supplies, mechanical systems with position and speed constraints, fixed-point digital filters, etc.). Therefore, the study of stability of discrete-time systems (DTSs) with time delay and saturation is of high importance in practice. The global asymptotic stability (GAS) problem of such systems is investigated in this paper. The saturation nonlinearity is constrained by a convex hull with a free matrix whose infinity norm is smaller than or equal to unity. A new Lyapunov-based GAS criterion for DTSs with time-varying delay and state saturation is established. The GAS problem for DTSs with constant delay and saturation is also discussed. The obtained results can be used to ensure the absence of overflow oscillations in the considered system. In comparison to several existing criteria, the approach yields improved stability results. An example is provided to demonstrate the importance of the presented results.

Decentralized formation of multi‐agent conformable fractional nonlinear robot systems

AbstractA novel approach to address the asymptotic stability problem of the decentralized formation control of multi‐agent robot systems considering a cyclic pursuit formation graph is derived by using the classical Lyapunov's stability method and LaSalle's invariance principle. Theoretical development of a local potential function‐based control law and the formulation of the position and formation error dynamics, from the viewpoint of the multi‐variable conformable fractional‐order calculus, is given. In addition, a comparison between the integer‐order case and conformable fractional‐order case is provided. Finally, a numerical example illustrates the performance of the developed results.

Leader–follower sliding mode formation control of fractional-order multi-agent systems: A dynamic event-triggered mechanism

This paper examines the concept of the leader–follower formation for fractional-order multi-agent systems (FO-MASs) by utilizing dynamic event-triggered sliding mode control (SMC) method. Firstly, a dynamic event-triggered mechanism (DETM) is designed relying on the combined measurement vector, in which an auxiliary variable is introduced to dynamically adjust the threshold for each fractional-order agent system. Unlike most existing event-triggered communication mechanisms, whose threshold parameters are always fixed, the threshold parameters in our designed event-triggered conditions can be dynamically adjusted according to the fractional-order dynamic rule. Numerical results show that the proposed DETM can achieve a better system performance in reducing the sampling frequency and the expected formation performance. Secondly, the leader–follower formation control problem is transformed into checking the asymptotic stability problem of the closed-loop system. In addition, due to the memorability of fractional calculus operators, a distinctive condition is established to avoid the occurrence of Zeno behaviors reported in existing dynamic event-triggered schemes. Finally, a simulation example is presented to illustrate the effectiveness and feasibility of the dynamic event-triggered SMC method proposed in this paper.

Dynamics Analysis of a Nonlinear Satellite Attitude Control System Using an Exact Linear Model

This study aims to analyze the nonlinear dynamics of a satellite attitude control system equipped with reaction wheels and a PD controller. Based on the angular momentum conservation theorem for a closed mechanical system, the nonlinear equations of the attitude control system dynamics are presented as a linear system of differential equations with time-varying parameters. The asymptotic properties of the angular momentum of a mechanical system including a satellite and reaction wheels as rigid bodies are investigated. A relation has been established between the dynamic parameters of the attitude control system and the initial value of the angular momentum of the satellite. The issue of asymptotic stability for differential equations with time-varying parameters is simplified to the asymptotic stability problem for the ultimate homogeneous system of linear differential equations with constant elements. The dependencies of the dynamic parameters of the attitude control system on the constant parameters of this ultimate system of linear differential equations, as well as the initial values of the satellite’s angular momentum, enable us to apply proven and effective engineering methods. These methods are used not only for analyzing the stability of the control system but also for synthesizing the parameters of the control law based on the quality requirements of transient processes such as the stability margin, responsiveness, oscillation, transient time, and overshoot. In this case, the calculation of the control law parameters will be grounded in exact equations, not on approximate equations of the control system dynamics obtained by linearization.

On Asymptotic Stability of Discrete-Time Hybrid Systems

In this brief, we consider the asymptotic stability problem of a novel class of discrete-time hybrid systems. First, we introduce the so-called system in detail, which consists of discrete-time continuous-valued and Boolean dynamics. Second, we give the algebraic form of the systems by using the Khatri-Rao product and semi-tensor product (i.e., Cheng product). Third, we propose the concept of asymptotic stability for the hybrid systems. Further, based on Lyapunov functions, a necessary and sufficient condition is derived for testing asymptotic stability. Finally, a numerical example is presented to illustrate the effectiveness of the theoretical results.

Periodic event‐triggered consensus based on triggered communication states for multi‐agent systems with packet losses

AbstractIn this paper, periodic event‐triggered leaderless and leader‐following consensus problems are studied for linear multi‐agent systems with deterministic packet losses under the generic directed communication topology. Specially, an event‐triggered mechanism is proposed, which only needs to communicate the event‐triggered states of the agent and verify the event‐triggered condition at the periodic sampling instants. A switched system with stable and unstable subsystems is used to describe packet dropouts in a deterministic way. The incidence matrix of a directed spanning tree in the directed communication topology is utilized to construct a linear transformation so that the consensus problems are equivalently transformed into the asymptotic stability problems of reduced‐order systems. Then, some sufficient consensus conditions are derived and the consensus protocol gains are designed. Finally, simulation examples are given to show the effectiveness of the proposed results.

Adaptive memory-based event-triggering resilient LFC for power system under DoS attack

Considering denial-of-service (DoS) attacks in the communication network, this work develops the adaptive memory-based event triggering resilient LFC approach for multi-area power systems with wind power. First of all, the impact of each DoS attack is evaluated by the amount of continual information loss it causes. Meanwhile, an adaptive memory event triggering mechanism (AMETM) is proposed, where the dynamic event triggering parameter and the triggering mechanism are all related to the historical transmitted states. Then, based on the proposed AMETM and DoS attacks, a sampling-data LFC model is established. In order to construct the LFC scheme with the desired faster state response, the exponential decay index is introduced as a performance index. Next the exponential stability problem of LFC sampling model is transformed into the asymptotic stability problem of polynomial uncertain system. Secondly, the exponential stability criterion and the memory controllers design method are obtained by bilateral closed-loop Lyapunov–Krasovskii functional (LKF) and LMI technology, which ensure the proposed memory control scheme can quickly stabilize the LFC system under a longer DoS attack. Lastly, a one-area typical LFC system is presented to verify the superiority and effectiveness of the proposed AMETM. The resilient control with fast state response and DoS attack is studied by two-areas LFC systems with wind power.

Stability Analysis for a Class of Stochastic Differential Equations with Impulses

This paper is concerned with the problem of asymptotic stability for a class of stochastic differential equations with impulsive effects. A sufficient criterion on asymptotic stability is derived for such impulsive stochastic differential equations via Lyapunov stability theory, bounded difference condition and martingale convergence theorem. The results show that the impulses can facilitate the stability of the stochastic differential equations when the original system is not stable. Finally, the feasibility of our results is confirmed by two numerical examples and their simulations.

A new design method to global asymptotic stabilization of strict-feedforward stochastic nonlinear time delay systems

The stabilization problems of stochastic nonlinear systems remain challenging because many new obstacles will be encountered and need to be cautiously handled. This paper discusses the global asymptotic stabilization problem of stochastic nonlinear systems which possess strict-feedforward structure and multiple time delays. In the first place, we are trying to transform the considered system into an equivalent one by constructing the novel parameter-dependent state feedback controller and the suitable coordinate transformation. And then, through using the stochastic time delay system stability criterion and introducing the proper Lyapunov–Krasovskii functional, the global asymptotic stability of stochastic closed-loop system is proved. Finally, simulation results show the effectiveness of the parameter-dependent state feedback controller.

Memoryless parameter-dependent control strategy of stochastic strict-feedback time delay systems

For a class of stochastic strict-feedback nonlinear systems subject to different time delay states, this paper mainly concerns the problem of global asymptotic stabilization. Two new control strategies that the memoryless parameter-dependent state feedback control and the memoryless parameter-dependent output feedback control are taken into consideration, respectively. By skillfully constructing the Lyapunov-Krasovskii (L-K) functional, taking the proper determined parameter and employing the stochastic nonlinear time delay system (SNTDS) stability theory, the global asymptotic stability of the stochastic closed-loop system can be achieved. The proposed output feedback control scheme is finally utilized for the control design of the one-link manipulator system and two-stage chemical reactor system, which can verify the availability of the control approach.

Hankel norm performance of 2-D digital filters described by Roesser model with overflow arithmetic

ABSTRACT This paper is concerned with the problem of asymptotic stability with Hankel norm performance (HNP) of two-dimensional (2-D) digital filter characterised by Roesser model, where the 2-D system includes overflow nonlinearities and external interference or excitation of finite duration. The underlying nonlinearities cover the usual types of overflow arithmetic utilised in practice such as saturation, two’s complement, zeroing and triangular. To solve this problem, new sufficient criterion is proposed by employing quadratic 2-D Lyapunov function. The proposed criterion guarantees that the addressed system has 2-D HNP bound. This criterion can examine the unwanted memory effect (ME) reduction of 2-D interfered digital filters for past excitations. In the absence of external inputs, the asymptotic convergence of 2-D digital filters is established under the proposed criterion. The developed HNP criterion is in linear matrix inequality framework and, therefore, computationally tractable. In addition, the optimisation problem is formulated to obtain the minimum HNP of the 2-D interfered digital filter. To illustrate the effectiveness of the proposed criterion, a numerical example is provided. The criteria addressed in current paper can give a whole analysis framework for the unwanted MEs of filters.

Uniform convergence for directed time-varying signed networks under generalized Laplacian-class model

This paper addresses the uniform convergence problems for signed networks under directed time-varying topologies. A generalized Laplacian-class model is proposed by admitting the time-varying relative self-effect of each agent, which can be reflected by a newly introduced notion of the diagonal dominance degree. With a state transition matrix-based approach, sufficient conditions on uniform asymptotic stability and uniform bipartite consensus subject to any initial time are obtained for signed networks, respectively. It is shown that these uniform convergence behaviors depend on not only the topology structure but also the diagonal dominance degree. The uniform convergence results for time-varying signed networks are further applied to cope with the uniform asymptotic stability problem of general linear time-varying systems.