Stability Theory Research Articles

Intermittent hold control for exponential synchronization of fractional-order delayed complex networks with its application

The paper focuses on synchronization problems of a type of fractional-order complex systems with delays. First, a general exponential stability theorem is achieved by utilizing function of complex variable method and fractional-order calculus, which can handle a type of multiple delayed complex systems. Then, a novel intermittent-hold control is designed, which can reduce the information transmission by flexibly adjusting the control interval. Next, through the proposed theorem and controller, exponential synchronization results can be obtained by constructing simple Lyapunov functional. Meanwhile, the control gains can be calculated by linear matrix inequality. Finally, two simulations are given to show the effectiveness of the proposed results, and some comparisons with different control times and holding times verify the flexibility of control time selection.

Control Error Convergence Using Lyapunov Direct Method Approach for Mixed Fractional Order Model Reference Adaptive Control

Abstract: This paper extends Lyapunov stability theory to mixed fractional order direct model reference adaptive control (FO-DMRAC), where the adaptive control parameter is of fractional order, and the control error model is of integer order. The proposed approach can also be applied to other types of model reference adaptive controllers (MRACs), provided the form of the control error dynamics and the fractional order adaptive control law are similar. This paper demonstrates that the control error will converge to zero, even if the derivative of the classical Lyapunov function is positive during a transient period, as long as tends to zero as time approaches infinity. Finally, this paper provides application examples that illustrate both the convergence of the control error to zero and the behavior of .

Collaborative Design of Observer and Controller for Singular Systems With Transmission Delay Based on Dynamic Event‐Triggered

ABSTRACTThis article investigates the dynamic event‐triggered with communication delay fault estimation and fault‐tolerant control problem of singular systems in network environments under system faults and external disturbances. By considering both the current time and transmission delay output error, a fault diagnosis observer is constructed to estimate the system state and faults simultaneously based on dynamic event‐triggered sampling. On the basis of estimating the system state and fault signals, a fault‐tolerant controller is proposed to compensate for the impact of faults on the system and maintain its performance. Through the collaborative design of observers, controllers, and event‐triggering schemes, the required performance of the faulty system is ensured while reducing the consumption of communication resources. In addition by constructing a new enhanced Lyapunov functional and utilizing linear matrix inequalities and Lyapunov stability theory, the conditions for asymptotic stability of the system are less conservative and easily obtained. Finally, the effectiveness of the proposed method was verified through simulation results of two examples.

Non‐Oscillatory Control Based Fixed‐Time Synchronization of Fuzzy Memristive Neural Networks

ABSTRACTThis paper investigates the fixed‐time (FXT) synchronization issue of fuzzy memristive neural networks (MNNs) via using incomplete Beta functions from the view of improving the estimate accuracy of settling time (ST). First, the parameter mismatching issue brought by the switching characteristics of the memristor is handled through the convex analysis method. Then, a new FXT stability theorem that provides a more accurate ST estimation is derived by using incomplete Beta functions. Furthermore, based on this result, some new sufficient conditions are obtained to ensure the FXT synchronization of considered fuzzy MNNs via designing a class of control schemes by introducing a new saturation function as well as using some inequality techniques. Significantly, the introduced FXT controller can achieve synchronization aim at bounded ST and it is not affected by the system's initial values. Finally, a numerical example is provided to verify the affectivity of introduced results.

The Existence and Stability of a Periodic Solution of a Nonautonomous Delayed Reaction–Diffusion Predator–Prey Model

In this study, we research a nonautonomous, three-species, delayed reaction–diffusion predator–prey model (RDPPM). Firstly, we derive sufficient conditions to guarantee the existence of a strictly positive, spatially homogeneous periodic solution (SHPS) for the delayed, nonautonomous RDPPM. These conditions are obtained using the comparison theorem for delayed differential equations and the fixed point theorem. Secondly, we present sufficient conditions to ensure the global asymptotic stability of the SHPS for the delayed, nonautonomous RDPPM. These conditions are established through the application of the upper and lower solution method (UALSM) for delayed parabolic partial differential equations (PDEs), along with Lyapunov stability theory. Finally, to demonstrate the practical application of our results, we numerically validate the proposed conditions using a 2-periodic, delayed, nonautonomous RDPPM.

Instability model of water–steam–liquid metal three-phase jet flow during steam generator tube rupture accident in a fast reactor

The steam generator tube rupture (SGTR) accident in a lead-cooled fast reactor (LFR) results in an injection of high-pressure subcooled water from the secondary circuit into the high-temperature liquid lead-bismuth eutectic (LBE) pool in the primary loop. Rapid depressurization and phase-change heat transfer generate the steam between the water jet and liquid LBE. Based on jet instability theory, this study develops a temporal instability model focusing on the water–steam–liquid LBE three-phase flow in a cylindrical coordinate to investigate the breakup characteristics of water jet. By introducing velocity and pressure disturbances and adopting a linear assumption to the hydrodynamic governing equations, a dispersion equation is developed, and jet characteristic parameters are then defined. The effects of various operating parameters and key dimensionless numbers are further studied. By solving the dispersion equation, it is found that smaller steam film thickness and higher steam velocity enhance jet instability. Increasing jet velocity and reducing jet radius accelerate jet breakup and promote droplet refinement, while at higher jet velocities, the influence of jet radius diminishes. Additionally, the co-flow of LBE with the jet and the viscosity effect of LBE both stabilize the water jet. This model enables quantitative prediction of the breakup behavior of water jets in liquid LBE during the initial stage of SGTR accident, aiming to reveal the fundamental mechanisms of the physical process and provide theoretical guidance for safety analysis of LFR.

Stabilizing Decomposition of Multiparameter Persistence Modules

Abstract While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is known to be unstable in a certain precise sense. Until now, it has not been clear that there is any way to get around this and build a meaningful stability theory for multiparameter module decomposition. We introduce new tools, in particular $$\epsilon $$ ϵ -refinements and $$\epsilon $$ ϵ -erosion neighborhoods, to start building such a theory. We then define the $$\epsilon $$ ϵ -pruning of a module, which is a new invariant acting like a “refined barcode” that shows great promise to extract features from a module by approximately decomposing it. Our main theorem can be interpreted as a generalization of the algebraic stability theorem to multiparameter modules up to a factor of 2r, where r is the maximal pointwise dimension of one of the modules. Furthermore, we show that the factor 2r is close to optimal. Finally, we discuss the possibility of strengthening the stability theorem for modules that decompose into pointwise low-dimensional summands, and pose a conjecture phrased purely in terms of basic linear algebra and graph theory that seems to capture the difficulty of doing this. We also show that this conjecture is relevant for other areas of multipersistence, like the computational complexity of approximating the interleaving distance, and recent applications of relative homological algebra to multipersistence.

Quaternion-Based Robust Sliding-Mode Controller for Quadrotor Operation Under Wind Disturbance

This paper presents a quaternion-based robust sliding-mode controller for quadrotors operating under significant wind disturbances. The proposed control method improves the reliability and efficiency of quadrotor control by eliminating the singularity problem inherent in the Euler angle method. The quadrotor dynamics and wind environment are modeled, and dynamic analysis is performed via numerical simulation. A realistic wind model is used, similar to a combination of deterministic and statistical models. The Lyapunov stability theory is utilized to prove the convergence and stability of the proposed control system. The simulation results demonstrate that the quaternion-based controller enables the quadrotor to follow the desired path and remain stable, even under external wind disturbances. Specifically, both position and attitude converge to the desired values within 10 s, demonstrating stable performance despite the challenging wind disturbances in both scenarios. Scenario 1 features turbulence with an average wind speed of 12 m/s and changing wind directions, while Scenario 2 models an environment with wind speeds that change abruptly and discretely over time, coupled with temporal variations in wind direction. Additionally, a comparative analysis with the conventional PD controller highlights the superior performance of the proposed RSMC controller in terms of trajectory tracking, stability, and energy efficiency. The rotor speeds remain within a reasonable and hardware-feasible range, ensuring practical applicability.

A forming limit prediction model based on tensile instability theory for orthotropic sheet metal considering distortion hardening

A forming limit prediction model based on tensile instability theory for orthotropic sheet metal considering distortion hardening

Secure Control of Networked Control Systems Under Replay Attacks: A Switching‐Like Approach

ABSTRACTThis paper focuses on the secure control design of networked control systems (NCSs) under replay attacks. Mitigating replay attacks is a challenging task since the statistical similarity between replayed signals and real observations makes them difficult to distinguish. Given the persistence of potential attackers and the probability of individual attack failures, a novel switching‐based replay attack model is introduced. By employing this replay attack model, NCSs vulnerable to replay attacks can be transformed into a class of switching‐like systems. Within the framework of the average dwell time switching strategy and multiple Lyapunov stability theory, sufficient conditions for the mean‐square exponential stability of the underlying system are presented. Compared with the existing control schemes using the delay‐based replay attack model, the proposed secure control method effectively reduces the impact of replay attacks on the control performance. The effectiveness of the proposed approach is validated through simulation experiments carried out on two practical systems.

Synchronization control of complex networks with bit-rate constraint via dynamic event-triggered strategy

A sampling-based dynamic event-triggered strategy is proposed for the synchronization control of continuous complex networks with uncertainties and delayed coupling, taking into account a bit-rate constraint in node communication. First, a synchronization error system is established by integrating the dynamics of complex networks with those of an isolated node. To conserve limited network bandwidth resources, a sampling-based dynamic event-triggered strategy is designed. By introducing bit-rate constraints to characterize the communication limitations of the system, the available bit rate for each node is allocated using an average allocation protocol. Subsequently, a binary encoding–decoding method is developed to transmit the triggered data, and the decoded signals are utilized to design the controller. Then, a closed-loop synchronization error system is derived. Given that coding errors are inevitable, an exponentially ultimately bounded definition of synchronization error is introduced. The coupling and transmission delays within the system are addressed by constructing Lyapunov-Krasovskii (L-K) functions. Utilizing Lyapunov stability theory and other technical theorems, the exponentially ultimately bounded condition for the closed-loop system is obtained. Based on this condition, the co-design of the controller gain and triggering parameters is achieved. In addition, the explicit relationship between synchronization error and bit rate is presented. Finally, the effectiveness of the proposed method is validated through three numerical examples.

The vertical shear instability in protoplanetary discs as an outwardly travelling wave. I. Linear theory

Abstract We revisit the global linear theory of the vertical shear instability (VSI) in protoplanetary discs with an imposed radial temperature gradient. We focus on the regime in which the VSI has the form of a travelling inertial wave that grows in amplitude as it propagates outwards. Building on previous work describing travelling waves in thin astrophysical discs, we develop a quantitative theory of the wave motion, its spatial structure and the physical mechanism by which the wave is amplified. We find that this viewpoint provides a useful description of the large-scale development of the VSI in global numerical simulations, which involves corrugation and breathing motions of the disc. We contrast this behaviour with that of perturbations of smaller scale, in which the VSI grows into a nonlinear regime in place without significant radial propagation.

Dynamical system analysis for extended f(P) gravity coupled with scalar field

In the present article, we have explored the physical characteristics of extended f(P) gravity through the dynamical system analysis. We choose the function f(P) in the form of a polynomial of second order, i.e. f(P)=αP+βP2, where α and β are constant parameters. As an additional dark energy component, we take a canonical scalar field ϕ, and several types of interaction between the dark components are considered. After presenting the field equation for the corresponding cosmological setup, we introduce some dimensionless variables and formulate a nonlinear autonomous system. For the different interaction scenarios, we have investigated the phase space and their physical characteristics and the dynamics of the cosmological solution associated with each critical point. For the first interaction model, the results we obtained by the analysis of phase space reveals that the cosmological solution associated with critical points exhibits two different cosmological epochs, namely the de-sitter epoch and quintessence epoch. For the second interaction model, the solutions represent the quintessence era. We have also studied the dynamical stability properties of each critical point by linear stability theory and determined possible physical constraints to the parameters. The cosmological solutions support cosmic acceleration. Moreover, an analysis of statefinder parameter {r,s} in terms of the dynamical system variable is presented. Based on the mathematical prospects, the modified theory of gravitation based on the extension of f(P) cubic gravity has the potential to manifest an accelerated expansion during late-time evolution.

Strategies for shore power facility utilisation under government subsidies: a three-party evolutionary game and simulation analysis

ABSTRACT The global shipping industry has prioritised reducing air pollutants by implementing various policies related to shore power in recent years. Despite these initiatives, both ports and shipping companies exhibit a low willingness to use shore power, resulting in substantial resource waste. Existing multi-party game studies on shore power simulations mostly targeted a static outcomes, lacking insights into dynamic subsidy allocation based on stakeholder interactions. Moreover, studies have often treated the influencing factors of the shipping sector as broad and vague concepts, making it challenging to identify the key determinants affecting stakeholders’ willingness to use shore power comprehensively. This paper aims to addresses these gaps by incorporating specific parameters into an evolutionary game model. By applying the stability theorem of differential equations, we determine the equilibrium points and assess their stability. Simulations are conducted in Matlab to explore the evolution of strategic choices for promoting shore power and evaluate the impact of various factors on ships’ shore power choices. The findings provide insights for enhancing shore power adoption through targeted policies and optimal incentives, thereby contributing to more effective environmental strategies within the shipping industry.

A predefined-time radial basis function (RBF) neural network tracking control method considering actuator faults for a new type of spraying robot

Abstract. A small-range fine-spraying collaborative robot (SFSC) for vehicle surface repair has been designed, which has 4 degrees of freedom. Conventional control methods, such as sliding mode control (SMC) have difficulty meeting the accuracy requirements when the end of the attitude adjustment robotic arm control is spraying. Focusing on the problem of tracking control of a multi-joint robot with uncertain information, such as modeling uncertainty and random interference, a predefined-time radial basis function (RBF) neural network tracking control (PRC) method considering actuator fault is proposed for a new spraying robot. Firstly, the dynamics equations of the n-joint manipulator are derived using the Euler–Lagrange equation. Then, a new predefined-time sliding mode surface is designed based on the stability theory of PRC. Combined with the Euler–Lagrange dynamics model of the two-joint manipulator, a nonsingular PRC controller is designed according to the uncertainty in model parameters and external interference. Stability of the system is proven based on Lyapunov theory. The simulation results show that the designed controller can ensure that the state convergence of the system does not depend on the initial conditions and has a faster convergence rate, shorter convergence time and good robustness.

Safe self-adjusting trajectory tracking control method for air cushion vehicle

ABSTRACT The safety of autonomous driving systems is a critical factor in the operation of high-speed air cushion vehicle (ACV). This paper presents a safety-guaranteed trajectory tracking controller for ACV. A safe self-adjusting speed guidance law is developed based on the finite-time auxiliary system. It can autonomously regulate tracking errors in accordance with the vehicle’s motion states and control inputs. Building on this framework, a trajectory tracking controller is developed utilising Lyapunov stability theory and finite-time control techniques to ensure safe operation. The stability of the proposed controller is rigorously proven. Simulation results demonstrate that the controller effectively mitigates tracking errors, ensuring that key motion parameters – such as speed, turn rate and drift angle – remain within predefined safety limits. Moreover, the control signals generated by the controller are shown to be compatible with the physical limitations of the vehicle’s actuators during trajectory tracking.

Adaptive neural observer-based output feedback anti-actuator fault control of a nonlinear electro-hydraulic system with full state constraints

This paper proposes an adaptive output feedback full state constrain (FSC) controller based on the adaptive neural disturbance observer (ANDO) for a nonlinear electro-hydraulic system (NEHS) with unmodeled dynamics. The Barrier Lyapunov Functions (BLFs) are utilized to ensure that all states of the system are specified within the constraints, and the approximation ability of radial basis function neural networks (RBFNNs) is used to cope with the unknown nonlinear functions. An adaptive neural compensation disturbance observer is elaborated to estimate the compound disturbance and oil leakage fault, effectively addressing these negative effects. Subsequently, observer-based output feedback command filter scheme is developed to diminish the explosion of complexity in the taking derivative procedure and obtain high precise tracking performance. The convergence of tracking errors into a small region around the equilibrium is demonstrated by the Lyapunov stability theory. Ultimately, simulation, experiment, and comparative studies are provided to further validate the effectiveness of the proposed control approach.

Stability and backward bifurcation in a malaria-transmission model with sterile mosquitoes

In this paper, we establish and study a malaria-transmission dynamic model of releasing sterile mosquitoes, which focuses on the mosquito populations affected by the impact of limited resources. First, we formulate a dynamic model with mosquitoes where the releasing rate of sterile mosquitoes is constant, and analyze the existence and stability of the equilibrium. Using the stability theory and method of differential equations, the threshold value of releasing sterile mosquitoes is obtained. Furthermore, we establish a malaria-transmission dynamic model with sterile release, and derive a formula for the reproductive number of infection and investigate the existence of endemic equilibria. With the symmetry of mathematical expression, it is also shown that this model may undergo backward bifurcation, where the locally stable disease-free equilibrium coexists with an endemic equilibrium. Finally, numerical simulations to illustrate our findings and brief discussions are provided.

Dissipative constraint-based multi-area power system with time-varying delays and cyber-attacks.

This work describes the dissipative constraint-based load frequency control problem for multi-area power system under load disturbances. Particularly, a new model incorporating time-varying delays and cyber-attacks are widespread in communication networks, significantly impacting control and stability. Consequently, the state-space equations of the addressed model are formulated and analyzed under the impact of false data injection attacks, and time-varying delays. The analysis is simplified by representing cyber-attacks using nonlinear functions adhering to Lipschitz continuity, while possible cyber-attacks are characterized by stochastic parameters conforming to Bernoulli distributions. Followed by the above information, stochastic analysis and Lyapunov-Krasovskii stability theory, the convex optimization problem is formulated. As a consequence, the load frequency control gains were effectively constructed, confirming that the established power model is stochastically stable and strictly (Q,S,R)-γ-dissipative. Finally, the case studies are employed to examine the usefulness of the suggested scheme.

Iterative Learning Control Applied to Interconnected RGB Light Strip

In this paper, an iterative learning control scheme is applied to the interconnected RGB light strip modeled with so-called spatially interconnected systems. The proposed strategy starts with formulating a set of Kirchhoff equalities, which lead to the 2D state-space model. In the next step, the system model is transformed into its 1D equivalent using the so-called lifting approach. Subsequently, the ILC design strategy is proposed. Due to the fact that it eventually takes the form of a differential linear repetitive process, the well-established stability theory along the trial is applied. This enables the application of computationally efficient stability tests, which employ linear matrix inequalities. Such a strategy ensures a numerically tractable design procedure. The application of this strategy ensures the convergence of the output signal to the desired reference. It is important to emphasize that the considered system can be affected by disturbances, and hence, it reflects practical situations that are inevitable in engineering practice. The effectiveness of the proposed approach is illustrated with a comprehensive simulation study.