Sufficient Conditions For Stability Research Articles

Elastic stability criteria of seven crystal systems and their application under pressure: Taking carbon as an example

Elastic stability criteria are widely employed to prove the being of the lattice. Sin'ko and Smirnov have reported the applicable criteria under isotropic pressure and given the equations between the elastic constants C ~ i j and C i j under pressure. On this basis, the closed forms of necessary and sufficient conditions for elastic stability in all crystal classes have been presented, which are popular in normal pressure. However, the forms of elastic stability criteria under pressure are still fragmented in various literature studies. Carbon is an element with a rich variety of allotropes, and because of its excellent mechanical and electronic properties, it gains enduring and intense attention, while its phase diagram is poorly known. In order to systematically study the response of various carbon stabilities to pressure and offer some valuable insights into experimental exploration, we derive the total forms of mechanical stability criteria under isotropic pressure and calculate the mechanical stability of 46 carbon allotropes involving seven crystal systems under pressure.

Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments

AbstractIn this paper, the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments are investigated. Firstly, the variation formulation is derived by applying Green's formula and Galerkin finite element method to spatial direction of the original equation. Next, semidiscrete and fully discrete schemes are obtained and the convergence is analyzed in ‐norm rigorously. Moreover, the stability analysis shows that the semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are also obtained under which the analytic solution is asymptotically stable. Finally, some numerical experiments are provided to demonstrate our theoretical results.

Stabilization and Reconstruction of Sampled-Data Boolean Control Networks Under Noisy Sampling Interval

In this paper, we consider stabilization and reconstruction of sampled-data Boolean control networks (BCNs) under noisy sampling interval. A sampled-data BCN under noisy sampling interval is first converted into a probabilistic Boolean network (PBN). We then obtain some necessary and sufficient conditions for global stochastic stability of the considered sampled-data BCN under two types of noisy sampling intervals. However, in analyzing the stochastic stability of large-scale sampled-data BCNs under noisy sampling interval, using the above necessary and sufficient conditions leads to huge computational cost. Therefore, for a large-scale sampled-data BCN, we have to transform it into a size-reduced probabilistic logical network. Then by studying the stochastic stability of the probabilistic logical network, some sufficient conditions for global stochastic stability of the large-scale sampled-data BCN are obtained. Moreover, based on the given steady-state probabilities of the transformed PBN, the reconstruction problem of sampled-data BCNs under noisy sampling interval can be well solved as a linear programming (LP) problem.

Fixed-Time and Predefined-Time Stability of Impulsive Systems

Dear Editor, This letter establishes several criteria for fixed-time stability and predefined-time stability of impulsive systems. First, sufficient conditions for fixed-time stability of impulsive systems are presented to treat the destabilizing impulses and hybrid impulses involving multiple jump maps by fixed-time control without linear feedback regulation. It determines the robustness of nonlinear systems against impulsive disturbance which has destabilizing and hybrid effect to dynamics. Then, the predefined-time stability of impulsive systems is ensured under the requirement that the hybrid impulses have stabilizing accumulative effect. Finally, the validity of theoretical results is verified by the allocation of mobile agents on a segment via fixed-time control with impulsive regulation.

Deterministic and Stochastic Fixed-Time Stability of Discrete-time Autonomous Systems

This paper studies deterministic and stochastic fixed-time stability of autonomous nonlinear discrete-time (DT) systems. Lyapunov conditions are first presented under which the fixed-time stability of deterministic DT systems is certified. Extensions to systems under deterministic perturbations as well as stochastic noise are then considered. For the former, sensitivity to perturbations for fixed-time stable DT systems is analyzed, and it is shown that fixed-time attractiveness results from the presented Lyapunov conditions. For the latter, sufficient Lyapunov conditions for fixed-time stability in probability of nonlinear stochastic DT systems are presented. The fixed upper bound of the settling-time function is derived for both fixed-time stable and fixed-time attractive systems, and a stochastic settling-time function fixed upper bound is derived for stochastic DT systems. Illustrative examples are given along with simulation results to verify the introduced results.

Event-triggered dynamic output feedback control of switched cyber-physical systems with stochastic cyber attacks

This article is concerned with the problem of event-triggered dynamic output feedback control for switched cyber-physical systems subject to stochastic cyber attacks. Two kinds of stochastic cyber attacks obeyed the Bernoulli distribution are taken into account, and a switched stochastic system is derived. An event-triggered strategy is adopted to reduce unnecessary communication and save system resources, and then an event-triggered–based dynamic output feedback controller is designed. The sufficient conditions for the mean-square exponential stability of the switched stochastic system are given by taking advantage of the mode-dependent average dwell time method and the multiple Lyapunov functional technique. Finally, the validity of the proposed theoretical results is verified by two examples.

Prescribed-time formation tracking of second-order multi-agent networks with directed graphs

The objective of this paper is to solve the prescribed-time formation tracking problem for the second-order multi-agent network (MAN) with the interaction graph containing a directed spanning tree. By introducing a time-varying function, a novel prescribed-time observer-based control algorithm is presented for driving all the followers to approach a dynamic leader within the prescribed time, which can be pre-specified as a user-designed parameter in the algorithm and is regardless of the initial condition. Especially, the prescribed-time state observer, as an important component of the algorithm, is newly designed to estimate the states of the leader in a distributed manner and also within a user-determined prescribed time. In order to establish the prescribed-time convergence of the closed-loop system in the cascade form, we design a new Lyapunov function candidate and then derive the corresponding sufficient conditions for prescribed-time stability of the error states. Finally, several numerical examples are performed on six agents to verify the effectiveness of the presented algorithm.

Global Stability and Bifurcation Analysis in a Discrete-Time Two Predator-One Prey Model with Michaelis-Menten Type Prey Harvesting

This article studies a discrete-time Leslie-Gower two predator-one prey system with Michaelis-Menten type prey harvesting. Positivity and boundedness of the model solution are investigated. Existence and stability of fixed points are examined. Using an iteration scheme and the comparison principle of difference equations, we find out the sufficient condition for global stability of the positive fixed point. It is shown that the sufficient criterion for Neimark-Sacker bifurcation can be developed. It is observed that the system behaves in a chaotic manner when a specific set of system parameters is chosen, which are regulated by a hybrid control method. Examples are provided to illustrate our conclusions.

Stability of Uncertain Equations of Volterra-Levin type and an Uncertain Delay Differential Equation Via Fixed Point Method‎

‎In this work four uncertain delay differential equations of Volterra-Levin type will be considered‎. ‎Applying suitable contraction mapping and fixed point method‎, ‎the stability of the equations will be studied‎. ‎It will be shown that the solutions are bounded and‎, ‎with additional condition‎, ‎the solutions tend to zero‎. ‎Also‎, ‎a necessary and sufficient condition for the asymptotic stability of the solutions of an uncertain differential equation will be presented‎.

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.

Interlayer antisynchronization in degree-biased duplex networks.

With synchronization being one of nature's most ubiquitous collective behaviors, the field of network synchronization has experienced tremendous growth, leading to significant theoretical developments. However, most previous studies consider uniform connection weights and undirected networks with positive coupling. In the present article, we incorporate the asymmetry in a two-layer multiplex network by assigning the ratio of the adjacent nodes' degrees as the weights to the intralayer edges. Despite the presence of degree-biased weighting mechanism and attractive-repulsive coupling strengths, we are able to find the necessary conditions for intralayer synchronization and interlayer antisynchronization and test whether these two macroscopic states can withstand demultiplexing in a network. During the occurrence of these two states, we analytically calculate the oscillator's amplitude. In addition to deriving the local stability conditions for interlayer antisynchronization via the master stability function approach, we also construct a suitable Lyapunov function to determine a sufficient condition for global stability. We provide numerical evidence to show the necessity of negative interlayer coupling strength for the occurrence of antisynchronization, and such repulsive interlayer coupling coefficients cannot destroy intralayer synchronization.

Robust stabilization for a class of complex production process systems based on multi-dimensional moving pattern

This paper studies robust stabilization problem and state feedback controller design problem for a class of complex production process systems by pattern class variable instead of state variable or output variable. Firstly, the maximum entropy clustering method is introduced and modeling method based on multi-dimensional moving pattern is given. Nonlinear state space models of systems with input-delay are established by using the trajectories of pattern class variable. Then sufficient conditions for robust stabilization of the system are obtained by Linear Matrix Inequality. In addition, a state feedback controller is designed based on this condition. Finally, the effectiveness of the methods is illustrated by taking the actual working conditions data as an example.

Asymptotic stability of non‐instantaneous impulsive systems and T‐S fuzzy non‐instantaneous impulsive control for nonlinear systems

AbstractThis paper studies the asymptotic stability for a class of non‐instantaneous impulsive systems and Takagi‐Sugeno (T‐S) fuzzy non‐instantaneous impulsive control for linear and nonlinear systems. First, a class of concrete comparison system for a more accurate and universal nonlinear non‐instantaneous impulsive model is constructed. Then, the less conservative sufficient conditions of Lyapunov asymptotic stability are studied by impulsive differential inequalities and generalized comparison principle. On the other hand, an improved non‐instantaneous impulsive control scheme with T‐S fuzzy model is also presented for the asymptotic stability of the nonlinear systems. Finally, simulation results on controlling a linear and a nonlinear system are presented to illustrate the rationality and effectiveness of the obtained results.

Distributed Consensus Kalman Filter Design with Dual Energy-Saving Strategy: Event-Triggered Schedule and Topological Transformation.

In the distributed information fusion of wireless sensor networks (WSNs), the filtering accuracy is commonly negatively correlated with energy consumption. Therefore, a class of distributed consensus Kalman filters was designed to balance the contradiction between them in this paper. Firstly, an event-triggered schedule was designed based on historical data within a timeliness window. Furthermore, considering the relationship between energy consumption and communication distance, a topological transformation schedule with energy-saving is proposed. The energy-saving distributed consensus Kalman filter with a dual event-driven (or event-triggered) strategy is proposed by combining the above two schedules. The sufficient condition of stability for the filter is given by the second Lyapunov stability theory. Finally, the effectiveness of the proposed filter was verified by a simulation.

Exponential stability of linear discrete systems with multiple delays by degenerated Lyapunov–Krasovskii functionals

The problem of exponential stability of delayed discrete systems with multiple delays x(n+1)=(I+A)x(n)+∑i=1sBix(n−i),n=0,1,…is studied, where x=(x1x2…xm)T is an unknown vector, m and s are fixed positive integers, A, Bi are square constant matrices and I is a unit matrix. A new degenerated Lyapunov–Krasovskii functional is used to derive sufficient conditions for exponential stability and to derive an exponential estimate of the norm of solutions. Though often used in the study of stability, the assumption that the spectral radius of the matrix of linear terms is less than 1 is not applied here. The criterion derived is illustrated by an example and compared with previously known results.

Cloud-based predictive formation control of networked multi-agent system and its application to air bearing spacecraft simulators

This paper studies a class of networked multi-agent systems with communication delays. A centralized cloud predictive control protocol is proposed to realize formation control of multiple agents and especially the predictive method is introduced to actively compensate for the delays in the network. The analysis of closed-loop networked multi-agent systems provides the necessary and sufficient condition of stability and consensus. Finally, the proposed cloud-based predictive formation control scheme is verified by its application to 3-degree-of-freedom air-bearing spacecraft simulators platform. The results show that the scheme can effectively compensate for the delays in the forward channel and the feedback channel and can be applied to the networked multi-agent systems well.

Intelligent control of convergence rate of impulsive dynamic systems affected by nonlinear disturbances under stabilizing impulses and its application in Chua’s circuit

This paper investigates the variable convergence rate stability and variable convergence rate stabilization of impulsive dynamic linear systems with stabilizing impulses affected by nonlinear disturbances. The eigenvalues (poles) of the system state are closely related to the convergence or divergence rate of the system. Sufficient conditions for the variable convergence rate stability are obtained by using the generalized pole placement idea and the method of system transformation. By designing a memoryless state feedback controller, sufficient conditions of variable convergence rate stabilization are obtained, ensuring the asymptotic stability of the target closed-loop system and accurately adjusting the system state’s convergence rate. An algorithm for adjusting the speed of system state convergence and its flow chart are designed by referring to the C programming language. Combined with the variable convergence rate stabilization method, the intelligent control of system state convergence speed is realized at the operation level. As a representative example of chaotic systems, Chua’s circuit affected by impulses verifies the effectiveness of the variable convergence rate stabilization method.

Induced $\mathcal {L}_\infty$ Quantized Sampled-Data Control for T–S Fuzzy System With Bandwidth Constraint

This article proposes an induced <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}_\infty$</tex-math></inline-formula> sampled-data control problem for the Takagi–Sugeno (T–S) fuzzy system with dynamic quantization, in which the communication channel is assumed to be bandwidth-constrained. First, according to the input delay approach, the dynamics of the considered closed-loop system is modeled by a delay system with constrained nonlinear term. Then, for the bandwidth-constrained communication channel, a constraint condition is introduced to ensure that the adjustable parameter of the dynamic quantization strategy is limited by a bounding region. The main purpose of the problem addressed is to design a controller, such that the closed-loop system guarantees the asymptotic stability with an induced <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}_\infty$</tex-math></inline-formula> performance index. The sufficient conditions of stability with induced <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}_\infty$</tex-math></inline-formula> control performance are given as the basis of controller design. Furthermore, the induced <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}_\infty$</tex-math></inline-formula> sampled-data control scheme and the improved updating rule of dynamic parameter are proposed. Finally, an example shows the effectiveness of the the proposed controller design scheme.

Ecological Cooperative Adaptive Cruise Control for Heterogenous Vehicle Platoons Subject to Time Delays and Input Saturations

Public concerns about energy crisis and environmental issues lead to higher fuel economy standards and more stringent limitations on greenhouse gas emissions for ground vehicles, and ecological cooperative adaptive cruise control (eco-CACC) is considered to be effective in reducing fuel consumption and greenhouse gas emissions of vehicle platoons. In this paper, an eco-CACC strategy is presented for heterogeneous platoons with time delays and input saturations to achieve platoon stability, fuel economy, riding comfort and driving efficiency. The proposed eco-CACC strategy includes distributed linear feedback control (DLFC) for following vehicles and model predictive control (MPC) for the leading one. To obtain platoon stability, the DLFC protocol is designed using state errors between an ego vehicle and its preceding one, and the frequency domain method is employed to derive the sufficient conditions of platoon stability for the DLFC protocol. Then, to achieve fuel economy, passenger comfort and driving efficiency without violating the different input saturations of vehicles, the constrained fuel consumption optimization problem of the leading vehicle (CFCO-LV) is formulated based on delay MPC, where the weighted sum of the overall fuel consumption and velocity band-stop functions of vehicle platoon is minimized. To quickly obtain the MPC controller, an improved particle swarm optimization (PSO) algorithm is used to solve the CFCO-LV problem. Simulations are implemented to validate the effectiveness of the proposed eco-CACC strategy, and the simulation results demonstrate that, compared with benchmark, the proposed strategy can save 2.19%~8.24% of fuel for the heterogeneous platoon under different velocity ranges.

Finite-Time Stabilization and Optimal Feedback Control for Nonlinear Discrete-Time Systems

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Sufficient conditions for finite-time stability have recently been developed in the literature for discrete-time dynamical systems. In this article, we build on these results to develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finite-time stability and finite-time stabilization for nonlinear discrete-time controlled dynamical systems. Finite-time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that satisfies a difference inequality involving fractional powers and a minimum operator. This Lyapunov function can clearly be seen to be the solution to a difference equation that corresponds to a steady-state form of the Bellman equation, and hence, guaranteeing both finite-time stability and optimality. Finally, a numerical example is presented to demonstrate the efficacy of the proposed finite-time discrete stabilization framework.