Lyapunov Function Method Research Articles

Anti-bump control for switched systems with decentralised adaptive event-triggering

A control system may exist bumps during mode transitions and discontinuous control input would seriously affect performance. In this paper, the anti-bump (or bumpless) switching control for triggered and switched systems under injection attacks is studied. This control method combines decentralised adaptive event-triggering mechanism (DAETM) and anti-bump condition. The main work is as follows: First, a DAETM is considered to allocate network resources properly and resist data injection attacks. Then, a data buffer is adopted to consolidate data from decentralised channels for ensuring timely utilisation. In addition, a controller gain condition is presented to achieve the bumpless switching between subsystems. Moreover, by adopting piecewise Lyapunov function method, sufficient conditions are obtained for triggered switched systems to be exponentially stable and anti-bump under injection attacks. Finally, the design method effectiveness is verified by actual circuit simulation.

Partial state-constrained integrated guidance and control for an STT missile with self-adjusting prescribed performance via non-barrier Lyapunov function method

A state-constrained guidance and control law is vital to the flight safety of an aerial interceptor against a maneuvering target. Taking the flight angle limitations, manipulation saturation, and terminal constraints into consideration, a novel integrated guidance and control (IGC) schema with a self-adjusting prescribed performance is proposed. To avoid the feasibility condition via the conventional barrier Lyapunov function-based method, a new state-dependent nonlinear mapping function is designed, based on which the state-constrained IGC system is converted into a constrained-free model. A novel prescribed performance with a self-tuning mechanism is proposed to guarantee satisfied transit and steady guidance performances. The backstepping technique combined with super-twisting integral sliding modes and newly defined predefined-time disturbance observers is implemented to generate state-constrained virtual command signals. All the signals in the closed-loop system are strictly proven to be bounded by Lyapunov stability theory. The simulation results show the effectiveness of the proposed IGC schema.

EFFECTS OF PUBLIC AWARENESS AND VACCINATION IN MODELING THE DYNAMICS OF MONKEY-POX TRANSMISSION DISEASE IN NIGERIA

Monkey-pox disease is recognized as pathogens, disturbing animals and humans, it is among the family of orthopox virus and the disease causes lymph nodes to swell. In this paper, we developed a deterministic model system for Monkey pox infection by incorporating public awareness parameter and vaccination individual. The study verified the feasible region of the system equations and non-negativity of the solutions is achieved. The disease free and endemic equilibrium states have been obtained. The study computed and analyzed the reproduction number, of the system equations. The study presented and analysed, the global stability of disease free equilibrium and endemic equilibrium state, it has been found that when there is no transmission between human and non-human (), then meaning it is globally asymptotically stable (GAS) at DFE and using nonlinear lyapunov function, the study shows that the endemic equilibrium state is GAS if and unstable if by comparison method of lyapunov functions. Numerical Simulations were done, it was found that the effective reproduction number decreases as vaccination of individual increases, varying the public awareness, the effective reproduction number reduces to zero and becomes stable as public awareness increases. It was discovered that effective reproduction number decreases as public awareness increases.

Stability of stochastic difference equations with multiplicative and additive fading noise

In this paper, the effects of multiplicative and additive fading stochastic perturbations on the solutions of stochastic difference equations are studied, in which the strength of fading stochastic perturbations is square summable. Different from the existing research, this paper mainly studies the influence of two kinds of stochastic perturbations on the stability of the system, and applies the Lyapunov functional method, the asymptotic mean square trivial sufficiency conditions are obtained for the solutions of stochastic difference equations which based on the influence of multiplicative and additive fading noises. At the same time, the conditions of stochastic perturbation fading rate are obtained. Finally, the above conditions are verified by numerical simulation examples, and the results of numerical simulation show that: (1) the additive perturbations have less influence on the system than the multiplicative perturbations; (2) the fading stochastic perturbations can make the system reach a stable state in a faster time.

Exponential stability of stochastic inertial neural networks with generalised piecewise constant argument

This paper investigates the exponential stability of stochastic inertial neural networks (SINNs) with generalised piecewise constant argument. By applying variable substitution, the second-order system is transformed into a first-order system. Subsequently, several sufficient conditions are introduced to ensure the existence of a unique solution. Due to the presence of generalised piecewise constant argument, calculus theory is applied to estimate the dynamic effects of network states in current time and deviation function. The global mean-square exponential stability of SINNs with generalised piecewise constant argument is established using the Lyapunov function method. Finally, two numerical examples are provided to verify the validity of derived theoretical results.

Distributed Consensus Filtering Over Sensor Networks With Asynchronous Measurements

ABSTRACTIn practical sensor networks, sensor nodes may operate with different sampling periods and initial sampling time instants, and their observations may also be nonuniform. Unfortunately, the research on distributed state estimation problems over such asynchronous sensor networks is very limited. Thus, this article focuses on the distributed consensus filtering problem over sensor networks with asynchronous measurements. First, the asynchronous measurement from each sensor is synchronized by the continuous‐time state evolution equation to a unified filtering fusion time instant within a given filtering period. After measurement synchronization, the statistical characteristics of measurement noise change. The cross‐correlations between the converted measurement noises are analyzed, as well as one‐step correlations between the converted measurement noises and the process noise. Second, an optimal asynchronous distributed consensus filter is designed based on synchronization measurements under the criterion of minimum mean‐square error with the above correlations between various types of noises taken into account. Meanwhile, a suboptimal distributed consensus filtering algorithm is further proposed to reduce computational complexity. Finally, based on the Lyapunov function method, the stability of the estimation error is theoretically demonstrated with an appropriate selection of consensus filtering gain and validated through simulations.

Finite‐time lag bipartite synchronization of double‐layer networks with non‐linear coupling strength and random coupling delays via T‐S fuzzy logic theory

AbstractThis article discusses finite‐time lag bipartite (FETLB) synchronization of double‐layer networks with non‐linear coupling strength and multiple time delays. The cooperative and competitive interactions between nodes are considered based on signed graphs. To address the non‐linear couplings strength, the T‐S fuzzy logic theory is used. An intermittent control approach is introduced to achieve FETLB synchronization, effectively minimizing control costs. Moreover, by the Lyapunov functional method, we derive criteria for achieving FETLB synchronization and provide estimations for the synchronization settling time. In addition, numerical simulation affirms the validity of the theoretical findings, showcasing the practical application of the synchronization results in secure communication.

Event-triggered and quantised feedback stabilisation of switched networked control systems

In order to conserve network resources, this paper investigates the event-triggered control and quantised feedback stabilisation of switched networked control systems by introducing the vector quantizer technique. The state information undergoes quantisation and is transmitted to the controller after being sampled by the event-triggering technology. The designed event-triggered mechanism relies on quantised state values, and the generation of each quantised value also depends on the event-triggered condition. Addressing the asynchronous issue between the controller and the subsystem, we provide a sufficient condition for ensuring exponential stability of the closed-loop system and outline the controller gain design method by employing the multiple Lyapunov function and average dwell time methods. Besides, we demonstrate the explicit exclusion of the Zeno phenomenon in the event-triggered mechanism. Finally, the viability of our approach is demonstrated through simulation examples.

Resilient event-triggered [formula omitted] control for autonomous vehicle under novel important-data-based DoS attack

This paper presents the method of a resilient event-triggered H∞ controller for autonomous vehicles (AVs) to address the emerging threat of important-data-based (IDB) Denial-of-Service (DoS) attack. Specifically, a unique IDB DoS attack strategy from an adversarial perspective is proposed to enhance the attack’s destructiveness. Unlike many mature DoS attack methods, the proposed IDB DoS attack could discern the importance level for the data and selectively launch attacks on crucial information, resulting in more significant damage. Furthermore, a novel resilient event-triggered H∞ control technique is employed to mitigate the impact of the attack and counteract the negative effects produced by the IDB DoS attack. Utilizing the Lyapunov functional method, sufficient conditions are successfully derived to ensure the asymptotical stability of the resulted closed-loop system and H∞ interference suppression performance. Subsequently, simulation results demonstrate that: (1) the addressed IDB DoS attack method leads to a more severe system degradation, highlighting the potential risks associated with this novel attack approach; and (2) the proposed event-triggered H∞ controller effectively mitigates the potential impact of the IDB DoS attack, showcasing its efficiency in preserving system performance under adversarial conditions.

Distributed Adaptive Optimization Algorithm for High-Order Nonlinear Multi-Agent Stochastic Systems with Lévy Noise.

An adaptive neural network output-feedback control strategy is proposed in this paper for the distributed optimization problem (DOP) of high-order nonlinear stochastic multi-agent systems (MASs) driven by Lévy noise. On the basis of the penalty-function method, the consensus constraint is removed and the global objective function (GOF) is reconstructed. The stability of the system is analyzed by combining the generalized Itô's formula with the Lyapunov function method. Moreover, the command filtering mechanism is introduced to solve the "complexity explosion" problem in the process of designing virtual controller, and the filter errors are compensated by introducing compensating signals. The proposed algorithm has been proved that the outputs of all agents converge to the optimal solution of the DOP with bounded errors. The simulation results demonstrate the effectiveness of the proposed approach.

Distributed Consensus Tracking of Incommensurate Heterogeneous Fractional-Order Multi-Agent Systems Based on Vector Lyapunov Function Method

This paper investigates the tracking problem of fractional-order multi-agent systems. Both the order and parameters of the leader are unknown. Firstly, based on the positive system approach, the asymptotically stable criteria for incommensurate linear fractional-order systems are derived. Secondly, the models of incommensurate heterogeneous multi-agent systems are introduced. To cope with incommensurate and heterogeneous situations among followers and the leader, radial basis function neural networks (RBFNNs) and a discontinuous control method are used. Thirdly, the consensus criteria are derived by using the Vector Lyapunov Function method. Finally, a numerical example is presented to illustrate the effectiveness of the proposed theoretical method.

Designing a switching law for Mittag-Leffler stability in nonlinear singular fractional-order systems and its applications in synchronization

In this work, the stability and synchronization issue of switched singular continuous-time fractional-order systems with nonlinear perturbation are examined. Using the fixed-point principle and S-procedure lemma, a sufficient condition for the existence and uniqueness of the solution to the switched singular fractional-order system is first stated. Next, using the Lyapunov functional method in combination with some techniques related to singular systems and fractional calculus, a switching rule for regularity, impulse-free, and Mittag-Leffler stability is developed based on the formation of a partition of the stability state regions in convex cones. For synchronizing switched fractional singular dynamical systems, we propose a state feedback controller that ensures regularity, impulse-free, and Mittag-Leffler stable in the error closed-loop system. Finally, the ease of use and computational convenience of our proposed methods are illustrated by two numerical examples and a practical example about DC motor controlling an inverted pendulum accompanied by simulation results.

Stabilization of fractional nonlinear systems with disturbances via sliding mode control

AbstractIn this article, the sliding mode control (SMC) of fractional nonlinear systems (FNSs) with disturbances is studied. First, a useful fractional power‐rate inequality for is extended to a more general form . Based on the generalized inequality, the method of a modified fractional integral SMC is completed, in which the parameter of the symbolic function is extended to and . This increases the degree of freedom of the sliding mode surface (SMS). Then, the SMC of FNSs is studied for the cases of known and unknown disturbances. In the case of known disturbance, it is proved by the generalized inequality and quadratic Lyapunov function method that the state of the FNSs can converge asymptotically to zero on the SMS. For the case of unknown disturbance, a fractional disturbance observer is used to estimate the disturbance by introducing auxiliary variables . The disturbance estimation error of the proposed FNSs is proved to be bounded. An equivalent global control law is obtained and asymptotic convergence of the FNSs under the action of the controller is proved. Finally, two numerical simulation examples are given to verify the feasibility of the method proposed in this article.

Adaptive neural inverse optimal control for uncertain switched nonlinear systems: an improved predefined-time method

For the first time, this paper introduces an improved predefined-time inverse optimal method to address the adaptive tracking control problem of a class of uncertain switched nonlinear systems. The proposed approach enables the system to achieve optimal performance without resorting to the HJB equation. Firstly, we design two-time constants to improve the traditional predefined-time lemma. The improved predefined-time lemma exhibits faster convergence. Subsequently, to realise the design of a predefined-time inverse optimal controller, a specific form of the auxiliary controller is designed using Sontag functions and nonsingular functions. Furthermore, the stability of the system and the optimality of the inverse are proven through the use of a common Lyapunov function method, ensuring that tracking errors converge to the neighbourhood of zero within the predefined time. Finally, the rationality of the proposed control structure is demonstrated through the example results.

Stability of a stochastic brucellosis model with semi-Markovian switching and diffusion.

To explore the influence of state changes on brucellosis, a stochastic brucellosis model with semi-Markovian switchings and diffusion is proposed in this paper. When there is no switching, we introduce a critical value and obtain the exponential stability in mean square when by using the stochastic Lyapunov function method. Sudden climate changes can drive changes in transmission rate of brucellosis, which can be modelled by a semi-Markov process. We study the influence of stationary distribution of semi-Markov process on extinction of brucellosis in switching environment including both stable states, during which brucellosis dies out, and unstable states, during which brucellosis persists. The results show that increasing the frequencies and average dwell times in stable states to certain extent can ensure the extinction of brucellosis. Finally, numerical simulations are given to illustrate the analytical results. We also suggest that herdsmen should reduce the densities of animal habitation to decrease the contact rate, increase slaughter rate of animals and apply disinfection measures to kill brucella.

A Power Circulating Suppression Method for Parallel Transient Inverters with Instantaneous Phase Angle Compensation

A unidirectional link is typically incorporated into the DC input side of an inverter to ensure the reliability and stability of the microgrid power supply. Due to impedance and control variations among inverters, circulating currents may rapidly arise during the operation of parallel transient inverters. Nonetheless, the unidirectional power circulating in a DC unidirectional link results in energy accumulation on the DC side, causing DC bus voltage pumping and affecting MGs’ operation safety and stability. This paper proposes a non-communication-based circulation suppression strategy to suppress power circulation in parallel transients based on the local information of inverters. First, a parallel transient is modeled, and the power circulation phenomenon and its influencing factors are analyzed. Then, the instantaneous power and absorbed active energy are calculated to adjust the phase of the inverter output voltage and suppress power circulation. Moreover, the output frequency is adjusted to balance the DC-side voltage of each inverter. Then, the stability of the parallel system after adding the circulation suppression control strategy is verified using the Lyapunov function method. Finally, simulation and experimental results verify the effectiveness of the proposed power circulation suppression strategy.

Finite-time synchronization of proportional delay memristive competitive neural networks

The finite-time synchronization (FTS) is considered for proportional delay memristive competitive neural networks (PDMCNNs). By utilizing Lyapunov functional method and differential inclusion theory, two new criteria ensuring the FTS of PDMCNNs are established. These criteria with algebraic inequality forms are less complicated and easier to verify than the matrix inequality forms. In addition, the corresponding settling times have been estimated. Eventually, the effectiveness of the presented criteria and controllers is confirmed through two numerical examples, and one application about image encryption is provided.

Global Analysis of a Fractional‐Order Hepatitis B Virus Model Under Immune Response in the Presence of Cytokines

AbstractThis research proposes and investigates an epidemiological model to study the dynamic behaviors of the Hepatitis B virus (HBV) under immune response and cytokine influence. The model's stability, positivity, boundedness, and equilibria are analyzed using Lyapunov functional methods and the Routh–Hurwitz criterion under Caputo fractional derivative. The study evaluates nucleoside analogues and interferon treatments, determining critical drug efficiencies. Equilibria, including infection‐free and endemic states, are analyzed using the fundamental reproduction number, , to predict disease elimination. Numerical simulations utilize the fractional Adams method and the L1 scheme, capturing memory traces as the fractional order changes. Results show the L1 scheme effectively captures memory traces, providing empirical support for the theoretical findings. Furthermore, Ulam–Hyers stability is treated according to the equilibrium point, which describes relationships between functions. Notably, the findings of the study yielded profound insights. They revealed that the HBV system remains locally asymptotic stable at disease‐free and the endemic point when . At the same time, the simulations illustrated a correlation between the rate of infection and the rise in infected individuals, indicating the feasibility of eradicating and effectively managing HBV infections through a multifaceted approach and various measures such as vaccination and effective drug administration protocols. The proposed framework can guide medical professionals and decision‐makers in developing effective strategies to limit and eliminate the spread of HBV in the population.

Aperiodic intermittent dynamic event-triggered synchronization control for stochastic delayed multi-links complex networks

In this work, the exponential synchronization issue of stochastic complex networks with time delays and time-varying multi-links (SCNTM) is discussed via a novel aperiodic intermittent dynamic event-triggered control (AIDE-TC). The AIDE-TC is designed by combining intermittent control with an exponential function and dynamic event-triggered control, aiming to minimize the number of the required triggers. Then, based on the proposed control strategy, the sufficient conditions for exponential synchronization in mean square of SCNTM are obtained by adopting graph theoretic approach and Lyapunov function method. In the meanwhile, it is proven that the Zeno behavior can be excluded under the AIDE-TC, which ensures the feasibility of the control mechanism to realize the synchronization of SCNTM. Finally, we provide a numerical simulation on islanded microgrid systems to validate the effectiveness of main results and the simulation comparison results show that the AIDE-TC can reduce the number of event triggers.

Zero gradient sum algorithm of arbitrary initial value with constraints and communication delay based on directed graph

The distributed convex optimisation issue with inequality constraints and box constraints on the directed communication topology is examined in this paper. In the case of the communication delay between agents, we propose a piecewise zero-gradient-sum triggered control algorithm that allows arbitrary initial values. Using the log-barrier penalty method, we only need to study an unconstrained approximation problem of the original problem. Firstly, each agent's state converges to the optimal value of the related local objective function in a fixed time. Simultaneously, the upper bound of fixed time in this paper is smaller than some existing results. Then, in the case of communication delay, time-triggered and event-triggered methods with non-uniform sampling interval are proposed in this paper, which make the communication cost lower and the application broader. Using the Lyapunov function method, the sufficient conditions for each agent's state to converge to the optimal point are given, and the Zeno behaviour is excluded. Finally, to confirm the algorithm's effectiveness, we provide two numerical examples and one machine learning example as demonstrations.