Stability Theory Research Articles

Finite-Time Cluster Synchronization of Fractional-Order Complex-Valued Neural Networks Based on Memristor with Optimized Control Parameters

The finite-time cluster synchronization (FTCS) of fractional-order complex-valued (FOCV) neural network has attracted wide attention. It is inconvenient and difficult to decompose complex-valued neural networks into real parts and imaginary parts. This paper addresses the FTCS of coupled memristive neural networks (CMNNs), which are FOCV systems with a time delay. A controller is designed with a complex-valued sign function to achieve FTCS using a non-decomposition approach, which eliminates the need to separate the complex-valued system into its real and imaginary components. By applying fractional-order stability theory, some conditions are derived for FTCS based on the proposed controller. The settling time, related to the system’s initial values, can be computed using the Mittag–Leffler function. We further investigate the optimization of control parameters by formulating an optimization model, which is solved using particle swarm optimization (PSO) to determine the optimal control parameters. Finally, a numerical example and a comparative experiment are both provided to verify the theoretical results and optimization method.

Event‐Based Prescribed‐Time Containment Control for Multiple Euler–Lagrange Systems via Super‐Twisting Sliding Mode

ABSTRACTThis article designs a distributed dynamic event‐triggered prescribed‐time sliding mode controller to tackle the containment control problem of multiple Euler–Lagrange systems (MELSs) with external disturbances and strict requirements on response speed. Unlike previous super‐twisting methods that achieve finite‐time control, this paper replaces the constant term in the time scaling function with a variable term and incorporates it into the design of the controller and dynamic event‐triggered mechanism (DETM), achieving prescribed‐time control (PTC), enhancing system robustness and reducing controller updates. Based on the sliding mode variable structure theory and PTC theory, a non‐linear sliding manifold is devised for the purpose of ensuring the convergence of containment error to zero within a prescribed time. The prescribed‐time super‐twisting sliding mode control (PTSTC) protocol is designed to ensure the prescribed‐time reachability of the sliding manifold, while attenuating chattering during the control process. The prescribed time can be set arbitrarily. To minimize the frequency of controller updates and the losses of the actuator, a DETM is deployed in the controller‐to‐actuator channel. Moreover, the absence of the Zeno phenomenon within a prescribed time is derived. Error signals are proven to converge to zero within a prescribed time using the Lyapunov stability theory and prescribed‐time stability criteria. Ultimately, a simulation of a manipulator system indicates that the designed controller effectively drives the containment error to converge to zero. Meanwhile, the number of triggers is reduced by more than 50% when the controller framework involves DETM compared to static event‐triggered mechanisms.

Improved segmental active disturbance rejection control for direct-drive valve controlled hydraulic shaking table

To address the performance degradation of hydraulic shaking table under nonlinear disturbances, an improved segmental active disturbance rejection controller (IS-ADRC) based on a switching control law is designed. A self-coupling PID (SC-PID) control law is designed, and an adaptive speed factor model is proposed to enhance the system’s response speed without compromising disturbance observation ability, thus avoiding oscillations and instability caused by high-gain observer. To overcome the jitter issue in traditional nonlinear state error feedback (NLSEF) control laws, an improved control law (INLSEF) based on a smooth Fal-M function is designed to replace the traditional fal function. Moreover, considering the potential for integral saturation, a switching principle is designed to switch from the SC-PID control law to the INLSEF control law at small errors, ensuring smooth system convergence. The stability of the controller is proved using Lyapunov stability theory. Results demonstrate that the proposed IS-ADRC improves the response speed and anti-disturbance ability of the hydraulic shaking table. Furthermore, the Fal-M function effectively reduces the jitter phenomenon compared to traditional ADRC.

An Evolutionary Game Model of a Regional Logistics Service Supply Chain Complex Network in a Blockchain Environment

Blockchain technology offers a novel solution to address the issue of information silos present in the complex network system of regional logistics service supply chains. Furthermore, enhancing the willingness of participants to engage in this system has emerged as a critical issue requiring urgent resolution. This study assumes bounded rationality and constructs an evolutionary game model to analyze the adoption behaviors of logistics service providers (LSPs) and logistics service integrators (LSIs) in a blockchain environment. The stability theorem of differential equations is employed to elucidate the inherent decision-making mechanisms present in the game. Acknowledging that traditional evolutionary mechanisms often neglect individual heterogeneity and the dynamic changes in the topological structures of complex networks, this study further involves the development of an evolutionary game model tailored for the complex network of a regional logistics service supply chain. After validating the effectiveness of the model, this research investigates how market environmental factors influence the adoption behaviors of LSPs and LSIs. Finally, through case simulations, the effects of key influencing factors on participatory behaviors within the regional logistics service supply chain are analyzed. The findings indicate that, in a blockchain environment, both prior to and following the adoption of complex network technology in the regional logistics service supply chain, the equilibrium stability of the network system is jointly determined by the total revenues of LSPs and LSIs. External factors in the blockchain environment significantly influence technology adoption strategies. Notably, when both the sensitivity coefficients of market demand relative to price and logistics service costs concerning quality enhancement are high, the willingness of supply chain participants to adopt the technology declines rapidly. As the quality elasticity increases, the technology adoption rates of providers and integrators initially stabilize and subsequently increase exponentially. Changes in the revenues of logistics service providers have no significant impact on technology adoption rates within the regional logistics service supply chain.

Linear theory of visco-resistive tearing instability

A new linear theory of a tearing instability is shown, where the modified LSC (Loureiro, Schekochihin, and Cowley) theory [T. Shimizu, arXiv:2209.00149 (2022)] is extended to visco-resistive MHD. In contrast to the original LSC theories [Loureiro et al., Phys. Plasmas 14, 100703 (2007)], in the modified LSC theory, the upstream open boundary condition is implemented at a finite point ξc. At this point, the original LSC theories are solved for ξc=+∞. This paper first studies when the resistivity and viscosity are uniform in space. In addition, some variations in the non-uniformity are studied. It is shown that the non-uniformity can enhance the linear growth rate, and the tearing instability can occur even in an unlimitedly thin current sheet. Unexpectedly, it suggests that the forward cascade process of the plasmoid instability (PI) does not stop, i.e., the finite differential MHD simulations fail. To stop the forward cascade, viscosity is required not only in the inner region of the current sheet but also in the outer region. When the uniform viscosity is assumed, the critical condition is predicted to be 2Pm/(Sξc)=0.06, beyond which the tearing instability, i.e., the forward cascade, stops. Here, S is the Lundquist number, and Pm is the magnetic Prandtl number. According to the critical condition, the resistivity and viscosity employed in most high-S MHD simulations of PI are too small to stop the forward cascade. This critical condition may be also applicable for the trigger problem of the current sheet destabilization.

Cooperative-Critic Learning-Based Secure Tracking Control for Unknown Nonlinear Systems With Multisensor Faults.

This article develops a cooperative-critic learning-based secure tracking control (CLSTC) method for unknown nonlinear systems in the presence of multisensor faults. By introducing a low-pass filter, the sensor faults are transformed into "pseudo" actuator faults, and an augmented system that integrates the system state and the filter output is constructed. To reduce design costs, a joint neural network Luenberger observer (NNLO) structure is established by using neural network and input/output data of the system to identify unknown system dynamics and sensor faults online. To achieve the optimal secure tracking control, an augmented tracking system is formed by integrating the dynamics of tracking error, reference trajectory, and filter output. Then, a novel cost function is designed for the augmented tracking system, which employs the fault estimation and the discount factor. The Hamilton-Jacobi-Bellman equation is solved to obtain the CLSTC strategy through an adaptive critic structure with cooperative tuning laws. Besides, the Lyapunov stability theorem is utilized to prove that all signals of the closed-loop system converge to a small neighborhood of the equilibrium point. Simulation results demonstrate that the proposed control method has good fault tolerance performance and is suitable for solving secure control problems of nonlinear systems with various sensor faults.

Improved sliding mode control for induction motor based on twisting algorithm

<p>In this paper, the output feedback tracking issue of induction motors is resolved by applying the sliding mode approach. We designed and implemented two robust sliding mode (SM) techniques to achieve high-performance control of induction motor drive; the second-order sliding mode (SOSM) approach using the twisting algorithm was compared with the classical sliding mode control. The method of decoupling electromagnetic torque and rotor flux for the induction motor was derived from the rotor field orientation control in the synchronous reference frame. The objective of the proposed methods is to control the rotor speed and the square of the rotor flux separately, in order to obtain robust control against disturbances and parametric uncertainties, and at the same time minimize the chattering phenomenon—the most significant drawback in the actual implementation of this technique. The stability of the proposed first-order sliding mode control was confirmed using Lyapunov stability theory. The availability and effectiveness of the proposed techniques were demonstrated through experimental results. The comparison between the results of the two proposed methods shows that the second-order sliding mode control using the twisting algorithm not only guarantees the same robustness and dynamic performances of traditional first-order sliding mode control but also achieves the reduction of the chattering phenomenon.</p>

Casson fluid flow of rotating magneto-convection in a vertical porous medium

This study investigates the stability of parallel buoyant magneto-convection in a rotating vertical porous medium filled with a Casson fluid. The vertical boundaries are considered isothermal rigid and kept at different uniform temperatures. Based on Darcy's law, the Navier–Stokes equation is employed. In a linear stability theory, the stability of an eigenvalue problem is found using the normal mode approach. The original three-dimensional problem is cast in an equivalent two-dimensional form using Squire's transformations. Subsequently, the two-dimensional stability of the eigenvalue problem is solved numerically using the Chebyshev collocation method. To determine the stability of the basic flow, the problem was originally solved in Gill's classical proof of 1969 [A. E. Gill, “A proof that convection in a porous vertical slab is stable,” J. Fluid Mech. 35, 545–547 (1969)]. Various basic state flow and magnetic fields were considered by varying the magnetic Prandtl number, ranging from 1×10−5 to 5×10−5. The critical stability conditions are exhibited, and the critical Rayleigh number (Rc), critical wave number (ac), and critical wave speed (cc) for the onset of convective instability are computed for different governing parameters. For the unsteady flow model, stability occurs through the marginal state mode within a certain range of Chandrasekhar numbers; however, the base flow remains stable throughout. The Casson fluid parameter and Darcy number significantly affect the neutral stability curve of the flow. Their combined influence contributes to the onset of instability in the Brinkman region. This interaction highlights the critical role of these parameters determining flow behavior.

A Mathematical Approach to the Stability of Electroencephalography Signals Model Using Integral Equations During Epileptic Seizures

In this study, the mathematical stability of EEG signal models represented by the integral equation during an epileptic seizure is investigated. The modelling of the EEG signals should be done with utmost accuracy to understand the actual nature of the brain dynamics and to enhance seizure prediction approaches. We investigate the stability of such models by applying Lyapunov's stability theorem and show that stability is powerfully dependent on parameters such as seizure duration and intensity, along with neural connectivity. We demonstrate that some of the parameter regimes yield stable behaviour, while others give rise to instability, making EEG interpretation complicated during epileptic seizures. By comparing integral equation-based models with current approaches, their advantages are highlighted in capturing the transient dynamics of epileptic activity. This work focuses on developing solid mathematical frameworks that will allow for real-time EEG monitoring. It contributes to the wider knowledge on the dynamics of seizures in the brain and will have practical implications for the development of neurological diagnostic and therapeutic tools.

Laminar burning velocity and cellular instability of spherical propagation NH3/H2/air premixed flames under elevated temperature and pressure

ABSTRACT Due to their carbon-free emission characteristics, ammonia and hydrogen are increasingly valued as clean energy sources. Laminar burning velocity and flame stability are two key indicators for assessing the combustion characteristics of fuels. Nonetheless, research on ammonia/hydrogen/air mixture flames under high-temperature and high-pressure conditions still requires further exploration. Consequently, this study measured the specific impact of hydrogen concentration on laminar burning velocity and flame instability under high-temperature and high-pressure conditions. Experiments were conducted under a range of conditions, including a temperature of 423 K, pressures from 1 to 3 atm, and hydrogen concentrations ranging from 0.1 to 0.5. To gain a comprehensive understanding of the combustion characteristics of ammonia/hydrogen/air mixture flames, the experiments also examined different equivalence ratios, varying from 0.8 to 1.4. The results showed that as the hydrogen concentration in the mixture increases, the laminar burning velocity (LBV) of the NH3/H2/air mixture increases exponentially. The pressure dependence of laminar burning velocity showed a negative correlation across different equivalence ratios, suggesting that pressure inhibits the laminar burning velocity of ammonia/hydrogen/air mixtures. A fictitious gas approach distinguished between thermal and chemical effects on the laminar burning velocity of premixed flames, indicating that the chemical effect outweighs the thermal effect. Schlieren images of ammonia/hydrogen/air flames under varying hydrogen concentrations revealed an increase in flame surface wrinkles as hydrogen concentrations increased. The stability of the ammonia/hydrogen/air flame was further investigated using linear stability theory. The dimensionless growth rate transitioned from negative to positive with increasing hydrogen concentration, indicating enhanced flame instability. Sensitivity analysis highlighted that the promoting effect of reaction R1 is particularly significant at high hydrogen concentrations, resulting in increased production of O and OH radicals, which facilitates the combustion reaction.

Coordinated tracking control for a class of fractional multi-manipulator system

In this paper, the concept of fractional-order model is introduced to solve the problem of coordinated tracking control of multi-manipulator systems under a fixed and switching topology, respectively. The control algorithm of the leader-following consensus is adopted, and linear matrix inequality (LMI) is used to acquire the parameters to be designed of the control algorithm. The sufficient conditions for achieving consensus of the fractional-order system are obtained by using Lyapunov stability theory and algebraic graph theory. Finally, the effectiveness of the theoretical results is validated by numerical simulations.

A Study of Two-Wheel Self-Balancing Robot Dynamics for Control Systems Learning

The two-wheeled self-balancing robot (TWSBR), utilizing an inverted pendulum arrangement, exhibits dynamic stability while being statically unstable. Due to this behavior, the TWSBR system has been used to demonstrate fundamental principles of stability, nonlinear dynamics, and control theory. In this work, a computational study of the TWSBR was carried out to develop a mathematical model using Newtonian mechanics that serves as an educational tool in control education for demonstrating the impact of different control algorithms such as Proportional (P), ProportionalDerivative (PD), and Proportional-Integral-Derivative (PID) on system stability. The comparison of P, PD, and PID controllers was undertaken to demonstrate the practical advantages of each controller in improving the robot's stability and responsiveness. The PID controller was designed and optimized using the Root-locus techniques, with a damping ratio equal to 1 which exhibits a fast-settling time with minimum overshoots. When an impulse response was applied to the PID controller in the simulation environment it demonstrated system can reach a dynamic balance within 1.2 seconds demonstrating the effectiveness of the proposed PID controller. The optimal gain parameters K value and Kp, Ki, and Kd parameter values were determined using root-locus analysis, which allowed the system to operate with great stability and minimum settling time. These findings highlight the TWSBR's educational significance as a hands-on tool for teaching practical applications of control theory, as well as providing valuable insights into the design and optimization of control systems for dynamic balance tasks. This study emphasizes the relevance of control algorithms in achieving rapid and stable dynamic behaviors, making it a valuable learning resource for both students and instructors. Keywords: Two-wheeled self-balancing robot, Inverted pendulum, P, PD and PID controllers, Stability analysis, Control Education

Nonsingular Fixed‐Time Terminal Sliding Mode Control Based Emotional Neural Network for Nonlinear Systems

ABSTRACTA fixed‐time nonsingular terminal sliding mode control (TSMC) method based on emotional neural network (ENN) is proposed for nonlinear systems. First of all, a new terminal sliding surface (TSS) is proposed in this article, which can adaptively adjust when the system states are far and near the origin and achieve fixed time fast convergence. Second, the equivalent control term in the controller is approximated by the ENN, which solves the singularity problem of traditional TSMC and the problem of unknown model information simultaneously. Finally, by using Lyapunov stability theorem, it is proved that the system states are stable in a fixed time during both sliding and reaching stages, which is independent of the initial values. The proposed method is verified and compared on an inverted pendulum system, which shows a better control performance.

Fixed-time adaptive fault-tolerant output consensus control for heterogeneous multi-agent systems

Aiming at the fault-tolerant output consensus problem of heterogeneous multi-agent systems with external finite energy interference in the case of simultaneous additive and multiplicative faults of the actuators, this paper proposes a fault-tolerant output control strategy combining fixed-time feedforward control and adaptive state feedback control. First, a fully distributed adaptive fixed-time observer is designed for the follower, so that each follower can estimate the leader state and the state matrix within a fixed time. Then, combined with the output adjustment equation with fault matrix, a distributed adaptive fault-tolerant output consensus control scheme is proposed to compensate for the impact of fault on the system, and the sufficient conditions for consensus system output are deduced by using Lyapunov stability theory. Finally, the effectiveness and feasibility of the proposed method are verified by a heterogeneous multi-agent faulty system composed of a leader and four followers.

Intelligent vehicle trajectory tracking with an adaptive robust nonsingular fast terminal sliding mode control in complex scenarios

This study presents a strategy for an intelligent vehicle trajectory tracking system that employs an adaptive robust non-singular fast terminal sliding mode control (ARNFTSMC) approach to address the challenges of uncertain nonlinear dynamics. Initially, a path tracking error system based on mapping error is established, along with a speed tracking error system. Subsequently, a novel ARNFTSMC strategy is introduced to tackle the uncertainties and external perturbations encountered during actual vehicle operation. The adaptive laws established for the longitudinal demand force and the front-wheel steering angle do not require prior understanding of the upper limit of the lumped uncertainty, while successfully avoiding singularities and eliminating chattering. By applying Lyapunov’s stability theorem, it is shown that the control system for trajectory tracking can reach the equilibrium point within a finite time. Following this, a torque optimization distribution control strategy is developed. Ultimately, numerical simulations are used to validate both the effectiveness of the proposed approach and its robustness across different conditions.

Real-time reachable set control for T-S fuzzy singularly perturbed systems with time-varying delay

In this paper, the problem of real-time reachable set control for fuzzy singularly perturbed systems with time-varying delays and disturbances is investigated. First, by combining the Lyapunov stability theory with the free-weighting matrix approach and linear matrix inequality technique, the real-time reachable set control problem related to fuzzy singularly perturbed systems is given adequate conditions. Subsequently, the matrix decoupling approach is applied to calculate the controller gains. Moreover, the stability boundary η ¯ can be established using the dichotomy technique by utilizing the adequate criteria in the following theorems. Finally, two examples are offered to illustrate the effectiveness of the strategy.

Phase space analysis and cosmography of a two-fluid cosmological model

Abstract In the framework of spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time, we investigate a two-fluid cosmological model where a tachyon scalar field with self-interacting potential and a modified chaplygin gas with non-linear equation of state are taken as the background fluids. We perform phase space analysis of the autonomous system obtained from the cosmological governing equations by a suitable transformation of variables. Linear stability theory is employed to characterise the stability criteria for hyperbolic critical points. Numerical investigation is carried out for non-hyperbolic points. Our study reveals that modified chaplygin fluid dominated solutions cannot provide the late-time evolution. Late-time accelerated evolution is obtained only when the solution is dominated by tachyon fluid. This study also yields a late-time scaling attractor providing similar order of energy densities in its evolution. The adiabatic sound speed is evaluated for both the fluids and test the stability of the models independently. Further, we perform cosmographic analysis in the model independent way by evaluating all the cosmographic parameters and then Om diagnostic is also found to compare our model with ΛCDM model.

Straight-Line Trajectory Tracking Control of Unmanned Sailboat Based on NMPC Velocity and Heading Joint Control

This study proposes a trajectory tracking approach for unmanned sailboats that integrates velocity and heading control using nonlinear model predictive control (NMPC). Unlike conventional methods, which typically rely on separate control strategies for maximum velocity and heading, this study employs a joint control framework based on NMPC. This approach allows for constrained control, ensuring that the sailboat operates safely at the desired velocity, forming a foundation for trajectory tracking. The trajectory tracking strategy, built on the joint control of velocity and heading, is further categorized into upwind and non-upwind tracking based on the wind direction. For non-upwind tracking, the desired heading is determined using the Line-of-Sight (LOS) navigation method, while the desired velocity is calculated through a backstepping method grounded in the Lyapunov stability theorem. For upwind tracking, a zigzag strategy is introduced, using the maximum lateral error to switch headings and ensure that the sailboat remains close to the trajectory. The simulation results show that the proposed method can effectively control the velocity and heading and realize accurate trajectory tracking of the unmanned sailboat under different wind conditions. The average error of velocity and heading control is close to zero. During trajectory tracking, the maximum control error is less than 0.35%.

Distributed adaptive event-triggered formation fault-tolerant control for multiple quadrotors

In the present paper, addressing the challenges of model uncertainty, actuator faults, and wind disturbances in quadrotor UAV formation systems, this study proposes an adaptive super-twisting integral terminal sliding mode control (ASTITSMC) strategy. This strategy integrates the “leader-follower” formation control approach with adaptive parameter adjustment to optimize the sliding mode controller, effectively estimating the upper bound of unknown disturbances and ensuring system stability. Additionally, based on the leader-follower structure, we design a distributed adaptive event-triggered formation control protocol that does not rely on global network information, thus reducing computational load and conserving communication resources, while strictly proving the avoidance of the Zeno phenomenon. Analysis based on Lyapunov stability theory demonstrates that the proposed algorithm ensures the convergence of the multi-UAV formation tracking error to zero. Simulation results indicate that the proposed control algorithm performs superiorly in terms of fault tolerance compared to two other algorithms and exhibits stronger robustness.

Cooperative midcourse guidance law with target changing and communication topology switching

Abstract This paper proposes a cooperative midcourse guidance law with target changing and topology switching for multiple interceptors intercepting targets in the case of target loss and communication topology switching. Firstly, a three-dimensional guidance model is established and a cooperative trajectory shaping guidance law is given. Secondly, the average position consistency protocol of virtual interception points is designed for communication topology switching, and the convergence of the average position of virtual interception points under communication topology switching is proved by Lyapunov stability theory. Then, in the case of the target changing, the target handover law and the handover phase guidance law are designed to ensure the acceleration smoothing, at last, the whole cooperative midcourse guidance law is given based on the combination of the above guidance laws. Finally, numerical simulation results show the effectiveness and the superiority of the proposed cooperative midcourse guidance law.