Lyapunov Function Technique Research Articles

Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment

In this paper, we derive and analyze a reaction-diffusion cholera model in bounded spatial domain with zero-flux boundary condition and general nonlinear incidence functions. The parameters in the model are space-dependent due to the spatial heterogeneity. By applying the theory of monotone dynamical systems and uniform persistence, we prove that the model admits the global threshold dynamics in terms of the basic reproduction number ℜ0, which is defined by the spectral radius of the next generation operator. When all model parameters are strictly positive constants, we study three types of nonlinear incidence functions to achieve the global stability results on the unique positive cholera-endemic steady state (CESS) whenever it exists. For all these examples, the sharp threshold property based on the basic reproduction number was completely established by using Lyapunov functional techniques under some realistic assumptions. Our numerical results reveal that when ℜ0 > 1, the convergence speed of the solution to the CESS becomes faster as the diffusion coefficient d becomes larger in the spatially homogeneous case. While in the spatially heterogeneous case, cholera can not be controlled by limiting the movement of host individuals, and the spatial heterogeneity does not always enhance the disease persistence.

Asynchronous repetitive control of switched systems via periodic event-based dynamic output feedback

Abstract This paper develops an asynchronous mode-dependent repetitive control strategy with periodic event-based dynamic output feedback for periodic trajectory tracking of continuous-time switched systems subject to time-varying switching delays between system modes and controllers and limited communication capacity in the feedback channel. By employing the input delay approach, the overall system is modelled as an augmented closed-loop switched system with both constant and time-varying state delays. A co-design framework is proposed for simultaneously designing the controller, event-triggering mechanism and mode switching signal. Under the co-design framework, using piecewise Lyapunov functional, free-weighting matrices and average dwell time technique, sufficient conditions are derived for ensuring the augmented closed-loop system to be exponentially stable with a prescribed $H_{\infty }$ attenuation level $\gamma $ for an exogenous disturbance input. The performance of the proposed controller design scheme is verified via numerical simulation results on a switched RLC series circuit system.

Distributed tracking control for multiple Euler–Lagrange systems with communication delays and input saturation

This study mainly investigates the problem of distributed tracking control for time-varying delay existing multiple Euler–Lagrange systems considering full-state constraints and input saturation under the directed graph. Specifically, the system under consideration consists of system uncertainties and external disturbances. In the control law design, a distributed observer is first designed that the followers can obtain the leader’s time-varying information. Then the barrier Lyapunov function technique is used to make sure the system errors can converge to a certain range while the anti-windup method is utilized to overcome the influence of control input saturation. Further, in order to prevent chattering, an adaptive law is given. Numerical simulations are given to verify the proposed algorithms.

Observer-based Event-triggered Sliding Mode Control for Markov Jump Systems with Partially Unknown Transition Probabilities

This paper investigates the event-triggered sliding mode control problem for discrete-time Markov jump systems under the unavailable states and partially unknown transition probabilities. To save the limited computational source, an event-triggered scheme is implemented to determine whether the current data should be sent or not, and an observer is constructed to estimate the unmeasurable states of the system. Then, on the basis of the Lyapunov functional technique, the sufficient conditions of stochastic stability for the closed-loop system are derived. Moreover, the sliding mode controller based on event-triggered mechanism is designed to ensure that the state trajectories of the closed-loop system can be driven onto the predefined sliding manifold and maintain there for all subsequent time. Finally, a numerical example is utilized to demonstrate the effectiveness of the proposed method.

Global Stability of a Markovian Jumping Chaotic Financial System with Partially Unknown Transition Rates under Impulsive Control Involved in the Positive Interest Rate

The intrinsic instability of the financial system itself results in chaos and unpredictable economic behavior. To gain the globally asymptotic stability of the equilibrium point with a positive interest rate of the chaotic financial system, pulse control is sometimes very necessary and is employed in this paper to derive the globally exponential stability of financial system. It should be pointed out that the delayed feedback model brings an essential difficulty so that the regional control method has to be adopted. In this paper, the author firstly employs impulsive control, regional control, the Lyapunov function technique, and variational methods to derive the stochastically globally asymptotic stability criterion of the economic balance point with a positive interest rate for a delayed feedback financial system with Markovian jumping and partially unknown transition rates. Besides, the mathematical induction method and the proof by contradiction are applied synthetically to deduce the globally exponential stability of the equilibrium point with a positive interest rate for the impulsive financial system without time-delays. Moreover, numerical examples illustrate that under suitable data conditions on the two main criteria mentioned above, the interest rates are positive decimals when the financial system reaches stability, which means better economic significance.

Well-posedness and long time behavior of singular Langevin stochastic differential equations

In this paper, we study damped Langevin stochastic differential equations with singular velocity fields. We prove the strong well-posedness of such equations. Moreover, by combining the technique of Lyapunov functions with Krylov’s estimate, we also establish exponential ergodicity for the unique strong solution.

Stability analysis of switched systems on time scales with all modes unstable

In this paper, global uniform asymptotic stability problems of switched systems on time scales with all modes unstable are considered. First, a more general stability criterion is proposed for nonlinear switched systems on time scales, in which the value of Lyapunov function at some switching instants may not necessarily decrease. Second, based on the discretized Lyapunov function technique, a sufficient condition is derived for stability analysis of linear switched systems with all subsystems unstable on a special kind of time scale (the graininess function is constant). It is shown that this approach does not require the system matrix of one subsystem must commute with the one of the other subsystem, which is essential for stability analysis of linear switched systems on non-uniform time domains by the eigenvalue based approach in the existing results. Moreover, the idea of this paper not only provides a unified approach to study continuous switched systems and their discrete counterparts simultaneously, but also is effective to investigate stability problems of other systems on uniform or non-uniform time domains. Two examples are given for illustrating the effectiveness of the proposed results.

Dissipative Filtering for Switched Fuzzy Systems With Missing Measurements.

This paper investigates the dissipative filtering problem for a class of discrete-time switched fuzzy systems with missing measurements. The fuzzy plant under consideration incorporates characteristics of Takagi-Sugeno fuzzy systems and switched systems simultaneously. The occurrence of missing measurements is described by a stochastic variable that satisfies the Bernoulli binary distribution, which characterizes the effect of data loss in information transmission between the plant and the filter. Utilizing the Lyapunov function technique, sufficient conditions are developed to ensure that the resultant filtering error system is exponentially stable and strictly dissipative. Two simulation examples are presented to illustrate the validity of the proposed method.

Dynamic output feedback control for network‐based switched linear systems with performance dependent event‐triggering strategies

This study investigates the dynamic output feedback L ∞ control for network-based switched linear systems subject to communication delay and quantisation. To reasonably configure the network communication resources according to the dynamic and steady-state characteristics of the system, the performance dependent event-triggering strategies are proposed. Moreover, the quantiser and dynamic output feedback controllers are applied, and the time-delay closed-loop switched system models corresponding to the system dynamic and steady-state processes are developed using a transformation technique. Subsequently, by utilising the average dwell time and piecewise Lyapunov functional techniques, sufficient conditions are derived to render the state of the time-delay closed-loop switched system globally uniformly ultimately bounded with an L ∞ performance level. In particular, the coupling between the switching instants and the data updating instants in the actuator is analysed during the system performance analysis. Furthermore, the design conditions of the event-triggered dynamic output feedback controllers are presented. Finally, numerical simulations verify the effectiveness of the proposed approach.

Event-triggered neural adaptive failure compensation control for stochastic systems with dead-zone output

This paper investigates event-triggered adaptive compensation control in the face of uncertain stochastic nonlinear system with actuator failure and output dead-zone. It is still an arduous task and challenge to design a compensation controller for uncertain stochastic nonlinear system. In order to avoid damaging output caused by the nonlinearity of the system, blended neural network integration with Nussbaum-type function is proposed. It is established to ensure the provision of the tracking error constraints, which is based on backstepping Lyapunov function technique. Additionally, system transmission resource constraints and actuator failure problems exist simultaneously in the system, which is extremely challenging for control design. More transmission resources are demanded when the system is suffered with actuator failures, while the system transmission resource is limited. Thus, the requirements cannot be achieved. It is difficult and challenged to ensure the system tracking performance. Using the proposed event-triggered controller and combining the Lyapunov synthesis, a novel optimization algorithm is deduced to guarantee the closed-loop system stability and the convergence of the tracking error. The simulation results illustrate the effectiveness of the proposed neural networks adaptive control approach.

Composite adaptive disturbance observer‐based control for switched stochastic systems with multiple disturbances subject to mode‐dependent average dwell time switching

The problem of composite adaptive disturbance observer-based control and weighted H ∞ performance analysis for a class switched stochastic systems with multiple disturbances subject to mode-dependent average dwell time switching are studied. A disturbance observer and an adaptive updated law are constructed to further design a state feedback controller. Some sufficient criteria are established to check the stability of the switched stochastic systems with multiple disturbances in terms of the multiple Lyapunov function technique. Furthermore, a composite adaptive disturbance observer based-controller is also given. Finally, two examples are proposed to prove the validity of the obtained results.

Adaptive practical preassigned finite‐time stability for a class of pure‐feedback systems with full state constraints

SummaryThis paper focuses on an adaptive practical preassigned finite‐time control problem for a class of unknown pure‐feedback nonlinear systems with full state constraints. Two new concepts, called preassigned finite‐time function and practical preassigned finite‐time stability, are defined. In order to achieve the main result, the pure‐feedback system is first transformed into an affine strict‐feedback nonlinear system based on the mean value theorem. Then, an adaptive preassigned finite‐time controller is obtained based on a modified barrier Lyapunov function and backstepping technique. Finally, simulation examples are exhibited to demonstrate the effectiveness of the proposed scheme.

Exponential Stability of Neural Networks with Markovian Switching Parameters and General Noise

This paper investigates the problem of exponential stability of Neural Networks (NNs) with Markovian parameters and general noise. The model in this paper with general noise is more suitable for many real nervous systems than NNs with white noise. Criteria for the exponential stability of the NNs with Markovian switching parameters and general noise in both the mean square and p-th moment are derived by utilizing the random analysis method and Lyapunov functional method techniques. The exponential stability of NNs without Markovian switching is given as a special case. Finally, simulation result in two examples are discussed to illustrate the theoretical results.

Sensor fault detection and estimation for switched power electronics systems based on sliding mode observer

In this paper, the problem of sensor fault detection and estimation (FDE) for power electronics systems such as DC-DC buck-boost converter is investigated. Firstly, the power electronics system is modeled, and the fault model is given. Then based on the switched Lyapunov function technique, a sliding mode observer (SMO) is designed for the switched error systems, which can guarantee that the error systems are asymptotically stable with a prescribed H∞ performance index. Next, comparing the set threshold with the residual which is produced by the sliding mode observer, the sensor fault detection can be achieved. Meanwhile, based on the method of fault reconfiguration, fault estimation is realized. Finally, in order to show the feasibility of this sensor fault detection and estimation, the simulation of three kinds of sensor faults including open-circuit, gain deviation and noise abnormity is given in this article.

Exponential Synchronization of Stochastic Memristive Neural Networks with Time-Varying Delays

This paper pays attention to the synchronization control methodology for stochastic memristive system. On the framework of Lyapunov functional, stability theory and free-weighting matrices technique, some brand-new solvability criteria are established to ensure the exponential synchronization goal of the target model. Considering the introduce of some free-weighting matrices, the obtained synchronization verdict will be much more applicable. Finally, the living example is included to show the effectiveness of the presented methodology.

Reliable Control Against Sensor Failures for Markov Jump Systems With Unideal Measurements

The problem of state-feedback reliable control against sensor failures for Markov jump nonlinear systems taking into account the mixed time delays is investigated in this paper. To describe the phenomenon of imperfect transmission between the system and the controller, a mode-dependent stochastic measurement fading model is adopted. By introducing two mutually independent and mode-dependent fading channel coefficients, the undulation of communication workload caused by the change of system mode can be accurately described. Also, the mixed time delays including the discrete and the infinite distributed delays are considered. Our goal is to design a reliable controller, which can guarantee the passivity of the closed-loop control system even when some sensors experience faults or failures. To realize this reliable controller, we introduce an additional matrix and utilize some inequality techniques. By using Lyapunov function technique, the gain of the designed controller can be solved. The merits and effectiveness of the developed design scheme are verified by a simulation example.

Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback

This paper investigates the problem of output tracking control for a class of delayed switched linear systems via state-dependent switching and dynamic output feedback control. A state-dependent switching rule and the switching regions are proposed. Then the stability and tracking performance analysis are proposed based on single Lyapunov function technique, respectively. The design problem of dynamic output feedback controllers can be solved efficiently by using linear matrix inequalities (LMIs) toolbox. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.

UIO-based Fault Estimation and Accommodation for Nonlinear Switched Systems

This paper investigates the problems of fault estimation and fault accommodation for a class of nonlinear switched systems. First, an augmented switched system is constructed by forming an augmented state vector composed of the state vector and the fault vectors. Then, an unknown input observer (UIO) is designed for the augmented switched system to estimate the augmented state vector. With the assist of the average dwell-time (ADT) method and the switched Lyapunov function technique, sufficient conditions are obtained to guarantee that the error system is globally uniformly asymptotically stable (GUAS) with a prescribed H∞ performance index. An algorithm is provided to show the procedures on how to design the UIO. Moreover, the results are extended to the measurement disturbances case. Based on signal compensation principle, a dynamic output feedback controller is designed to ensure the asymptotical stability of the closed-loop system. Finally, simulation results are presented to demonstrate the proposed technique.

A co-design methodology for cyber-physical systems under actuator fault and cyber attack

This paper investigates the controller design problem of cyber-physical systems (CPSs) to ensure the reliability and security when actuator faults in physical layers and attacks in cyber layers occur simultaneously. The actuator faults are time-varying, which cover bias fault, outage, loss of effectiveness and stuck. Besides that, some state-dependent cyber attacks are launched in control input commands and system measurement data channels, which may lead state information to the opposite direction. A novel co-design controller scheme is constructed by adopting a new Lyapunov function, Nussbaum-type function, and direct adaptive technique, which may further relax the requirements of actuator/sensor attacks information. It is proven that the states of the closed-loop system asymptotically converge to zero even if actuator faults, actuator attacks and sensor attack are time-varying and co-existing. Finally, simulation results are presented to show the effectiveness of the proposed control method.

The stochastic extinction and stability conditions for nonlinear malaria epidemics.

The stochastic extinction and stability in the mean of a family of SEIRS malaria models with a general nonlinear incidence rate is presented. The dynamics is driven by independent white noise processes from the disease transmission and natural death rates. The basic reproduction number $R^{*}_{0}$, the expected survival probability of the plasmodium $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$, and other threshold values are calculated, where $\mu_{v}$ and $\mu$ are the natural death rates of mosquitoes and humans, respectively, and $T_{1}$ and $T_{2}$ are the incubation periods of the plasmodium inside the mosquitoes and humans, respectively. A sample Lyapunov exponential analysis for the system is utilized to obtain extinction results. Moreover, the rate of extinction of malaria is estimated, and innovative local Martingale and Lyapunov functional techniques are applied to establish the strong persistence, and asymptotic stability in the mean of the malaria-free steady population. %The extinction of malaria depends on $R^{*}_{0}$, and $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$. Moreover, for either $R^{*}_{0}>1$, or $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})>\frac{1}{R^{*}_{0}}$, whenever $R^{*}_{0}\geq 1$, respectively, extinction of malaria occurs. Furthermore, the robustness of these threshold conditions to the intensity of noise from the disease transmission rate is exhibited. Numerical simulation results are presented. %\textbf{(200 to 300 words)}.