Global Lyapunov Function Research Articles

Output tracking of coupled stochastic strict-feedback higher-order nonlinear systems under pinning control

This paper focuses on the tracking problem for a class of coupled stochastic strict-feedback higher-order nonlinear systems under pinning control. Compared with existing stochastic nonlinear tracking designs, our approach stands out by simultaneously considering coupling and higher-order characteristics for the first time. By utilizing the backstepping technique and designing virtual controllers, we obtain actual controllers that ensure the closed-loop system has an almost surely unique solution and is bounded in probability. Additionally, the fourth-moment of the output tracking error can be rendered arbitrarily small in the long run. Notably, to address the cost implications of imposing a controller on each node, we employ a pinning control strategy to selectively control a portion of the nodes. Moreover, a global Lyapunov function is derived combining Lyapunov method with graph theory. Eventually, a numerical simulation is shown to verify the theoretical results.

Practical stabilization of multi-links highly nonlinear Takagi-Sugeno fuzzy complex networks with Lévy noise based on aperiodically intermittent discrete-time observation control

This paper investigates practical stabilization of multi-links highly nonlinear Takagi-Sugeno fuzzy complex networks with Lévy noise (MHFNLs) via aperiodically intermittent discrete-time observation control (AIDOC). It is worth pointing out that AIDOC is considered into MHFNLs for the first time. Due to the difficulty of satisfying linear conditions in real life, polynomial growth conditions are used instead of linear growth conditions. Meanwhile, a global Lyapunov function is constructed and sufficient conditions which can remain MHFNLs practically stable are gained. Finally, an application and its numerical simulations are presented to verify the effectiveness and availability of the theoretical results.

Existence of stationary distribution for stochastic coupled nonlinear strict-feedback systems with Markovian switching

In this paper, a class of stochastic coupled nonlinear strict-feedback systems with Markovian switching (SCNSSMs) are introduced, and the existence of stationary distribution for SCNSSMs under pinning control is studied for the first time. In particular, an appropriate technique for handling the impact of Markovian switching is proposed in the process of designing virtual controllers through the back-stepping method. Regarding the global Lyapunov function for an SCNSSM, we first construct a quartic Lyapunov function for each stochastic nonlinear strict-feedback system with Markovian switching of the SCNSSM, and then combine graph theory to construct the global Lyapunov function. After that, the theoretical results are applied to stochastic coupled oscillator systems with Markovian switching. Finally, the effectiveness of proposed results is verified by some numerical simulations.

A Neural Network Approach to Finding Global Lyapunov Functions for Homogeneous Vector Fields

A Neural Network Approach to Finding Global Lyapunov Functions for Homogeneous Vector Fields

Equilibrium Problem for the Stochastic Multi-Weighted Urban Public Transportation System with Time Delay: A Graph-Theoretic Method

This paper focuses on the equilibrium problem of an urban public transportation system with time delay. Time delay, multi-weights, and stochastic disturbances are considered in the urban public transportation system. Hence, one can regard the urban public transportation system as a stochastic multi-weighted delayed complex network. By combining graph theory and the Lyapunov method, the global Lyapunov function is constructed indirectly. Moreover, the response system can realize synchronization with the drive system under the adaptive controller. In other words, the urban public transportation system is balanced in the actual running traffic network. Finally, numerical examples about the Chua system and small-world network are presented to confirm the accuracy and validity of the theoretical results.

Asymptotic synchronization and topology identification of stochastic hybrid delayed coupled systems with multiple weights

This article discusses a class of stochastic hybrid delayed coupled systems with multiple weights (SHDCSMW). Both white noise and telegraph noise are included in the coupled systems. By employing Kirchhoff’s Matrix-Tree Theorem, a global Lyapunov function is rebuilt indirectly, which is closely related to Markovian switching. Moreover, based on Lyapunov stability theory and stochastic analysis, several sufficient conditions with respect to asymptotic synchronization and topology identification of SHDCSMW are derived. Finally, the validity of theoretical results is proved by numerical examples.

Pinning control for stabilization of coupled stochastic strict‐feedback nonlinear systems

AbstractThis article introduces a novel class of coupled stochastic strict‐feedback nonlinear systems (CSSFNSs) whose topology is not strongly connected. We are devoted to investigating the stabilization issue of CSSFNSs via pinning control. First, the specific form of controllers is decided through the back‐stepping method and the design of virtual controllers. Second, we find a new method to construct the global Lyapunov function of CSSFNS by employing the Lyapunov function of each stochastic strict‐feedback nonlinear system (SSFNS). Meanwhile, combining Lyapunov method with graph theory, several sufficient conditions to guarantee exponential stability in mean square of CSSFNS are obtained in the form of coefficients‐type criterion. Ultimately, our theoretical results are applied to a class of coupled oscillator systems and a predator‐prey model. Furthermore, the validity and feasibility of the theoretical results are illustrated by some numerical simulations.

Stability analysis of the stochastic strict-feedback nonlinear systems with time-varying delays on networks under pinning control

This paper is concerned with the stability analysis of the stochastic strict-feedback nonlinear systems with time-varying delays on networks (SSNSTDN) via pinning control. Different from the previous literature, the new model we studied owns the non-strongly connected topology. The actual controllers are realised by back-stepping and designing virtual controllers. In particular, a novel method for constructing global Lyapunov functions is proposed,which exploits the Lyapunov function of each stochastic strict-feedback nonlinear systems with time-varying delays (SSNSTD) to construct the global Lyapunov function of SSNSTDN. Meanwhile, combining the Lyapunov method with graph theory, several sufficient conditions to guarantee the exponential stability in mean square of SSNSTDN are obtained in the form of coefficients-type criterion. Furthermore, we apply our main results to investigating the stabilisation of stochastic coupled oscillators with time-varying delays on networks and the effectiveness of proposed theories is illustrated by some numerical examples.

Global Versus Local Lyapunov Approach Used in Disturbance Observer-Based Wind Turbine Control

This contribution presents a Lyapunov-based controller and observer design method to achieve an effective design process for more dedicated closed-loop dynamics, i.e., a maximal flexibility in an observer-based controller design with a large consistency in desired and achieved closed-loop system dynamics is intended. The proposed, pragmatic approach enhances the scope for controller and observer design by using local instead of global Lyapunov functions, beneficial for systems with widely spaced pole locations. Within this contribution, the proposed design approach is applied to the complex control design task of wind turbine control. As the mechanical loads that affect the wind turbine components are very sensitive to the closed-loop system dynamic, a maximum flexibility in the control design is necessary for an appropriate wind turbine controller performance. Therefore, the implication of the local Lyapunov approach for an effective control design in the Takagi-Sugeno framework is discussed based on the sensitivity of the closed-loop pole locations and resulting mechanical loads to a variation of the design parameters.

Synchronization for the coupled stochastic strict-feedback nonlinear systems with delays under pinning control

In this paper, a class of new coupled stochastic strict-feedback nonlinear systems with delays (CSFND) on networks without strong connectedness (NWSC) is considered, and the issue pertaining to the synchronization of the systems is discussed by pinning control. Towards CSFND, the controllers are approached by combining the back-stepping method and the design of virtual controllers. A key novel design ingredient is that the global Lyapunov function is obtained based on each Lyapunov function of stochastic strict-feedback nonlinear systems with delays (SFND). Moreover, a sufficient criterion is presented to realize the exponential synchronization by employing the graph theory and Lyapunov method. As a subsequent result, we apply the obtained theoretical results to the second-order oscillator systems and robotic arm systems. Meanwhile, numerical simulations are provided to demonstrate the validity and feasibility of our theoretical results.

Stochastic input-to-state stability for stochastic complex dynamical control networks with impulsive perturbation

In this paper, stochastic input-to-state stability (SISS) for stochastic complex dynamical control networks with impulsive perturbation (SCDCNIP) is investigated. An appropriate global Lyapunov function is developed which is closely related to topology of SCDCNIP and dynamical behaviors of every node system of SCDCNIP. Some sufficient criteria are given to guarantee SCDCNIP to meet SISS by imposing the average dwell-time (ADT) condition and reverse average dwell-time (RADT) condition. Moreover, as an application of our theoretical results, typical stochastic coupled oscillators with impulsive perturbation (SCOIP) are studied. Finally, some numerical simulation results are provided to verify the correctness and validity of our results.

The nonequilibrium potential today: A short review

A brief review is made of the birth and evolution of the “nonequilibrium potential” (NEP) concept. As if providing a landscape for qualitative reasoning were not helpful enough, the NEP adds a quantitative dimension to the qualitative theory of differential equations and provides a global Lyapunov function for the deterministic dynamics. Here we illustrate the usefulness of the NEP to draw results on stochastic thermodynamics: the Jarzynski equality in the Wilson–Cowan model (a population-competition model of the neocortex) and a “thermodynamic uncertainty relation” (TUR) in the KPZ equation (the stochastic field theory of kinetic interface roughening). Additionally, we discuss system-size stochastic resonance in the Wilson–Cowan model and relevant aspects of KPZ phenomenology like the EW–KPZ crossover and the memory of initial conditions.

Advanced Stability Conditions for TS Fuzzy Systems via Minimum-Type Multiple Switching Lyapunov Function

This article deals with the stability analysis of TS fuzzy models using the novel minimum-type multiple switching Lyapunov function. A method is presented to investigate the stability of the model which consists of three separate steps; mesh the space of the model using the Delaunay method, considering a quadratic function for each vertex of the mesh, proposing a set of conditions based on the declared quadratic functions. The proposed conditions can be categorized into two sets; local and discontinuity conditions. The local conditions exploit the local stability in each simplex of the mesh which have minimum-type multiple forms. The discontinuity conditions force the global Lyapunov function to decrement in the switching points (boundary of the neighbor simplexes). To validate the result of the proposed approach, comparative simulation examples are given to illustrate the performance of the design methodology as compared to those of previous approaches.

Asymptotic stability problem of predator–prey system with linear diffusion

Abstract As the primary killer of health, the class of infectious diseases is the greatest threat to humanity. At present, international methods of studying the large-scale spatial transmission of sudden infectious diseases from the perspective of dynamics can be divided into two categories. On the one hand, top international biomedical and medical teams discuss the restraining effects of some prevention and control strategies on infectious diseases, such as smallpox, malaria, hand, foot and mouth disease and pandemic influenza, from the perspective of pragmatism. On the other hand, researchers in theoretical physics and network science tend to use compound population network models to explore the internal dynamic mechanism of spatial transmission of infectious diseases. This paper establishes a Lotka–Volterra dispersal predator–prey system in a patchy environment. It shows the existence of model boundary equilibria and asymptotic stability under an appropriate condition. This paper adopts the method of global Lyapunov function and the results of graph theory. We also consider a predator–prey dynamical model in a patchy environment, where the prey and predator individuals in each compartment can travel among n patches.

Strict Lyapunov functions and feedback controls for SIR models with quarantine and vaccination

<p style='text-indent:20px;'>We provide a new global strict Lyapunov function construction for a susceptible, infected, and recovered (or SIR) disease dynamics that includes quarantine of infected individuals and mass vaccination. We use the Lyapunov function to design feedback controls to asymptotically stabilize a desired endemic equilibrium, and to prove input-to-state stability for the dynamics with a suitable restriction on the disturbances. Our simulations illustrate the potential of our feedback controls to reduce peak levels of infected individuals.</p>

Compositional stability criteria based on cyclically neutral supply conditions

In this paper we consider stability of large scale interconnected nonlinear systems that satisfy a strict dissipativity property in terms of local storage and supply functions. Existing compositional stability criteria certify global stability by constructing a global Lyapunov function as the (weighted) sum of local storage functions. We generalize these results by unifying spatial composition, i.e., (weighted) sum of local supply functions is neutral, with temporal composition, i.e., (weighted) sum of supply functions over a time cycle is neutral. Two benchmark examples illustrate the benefits of the developed compositional stability criteria in terms of reducing conservatism and constrained distributed stabilization.

Investigating essential factors in the spread of lassa fever dynamics through sensitivity analysis

Lassa fever is a zoonotic acute viral illness caused by Lassa virus. Since there is no vaccine yet to protect against contracting the virus, it continues to spread in West Africa. In this paper, a mathematical model of lassa transmission that considers two classes of rats: house rat and bush rat, is proposed. Theoretically, global stability of the model disease-free and endemic equilibria are established by constructing a global Lyapunov function. Sensitivity indices of the basic reproduction number are derived using the normalised forward approach to evaluate the effectiveness of control measures. The disease-free equilibrium is globally asymptotically stable when the basic reproduction number R0 1. Results from sensitivity analysis reveals that rat biting rate for infectious house rats RFH and infectious bush rats R_FB, transmission probability per contact with infectious house and bush rats (R_FH and R_FB), human recruitment rate and transmission probability per contact with infectious human hosts are highly significant in determining the severity of lassa infection. On the other hand, natural death rate of rats, natural death rate of human hosts, recovery and hospitalization rates of human hosts are critical for lassa transmission reduction. Plans that target the contact rate between house and bush rats (i.e use of indoor residual spray, fumigation of environment with pesticide) and those that target recovery rate of human hosts (i.e treatment of infectious human hosts) are recommended to control the disease.

Global asymptotic stability in mean for stochastic complex networked control systems

In this paper, global asymptotic stability in mean (GASM) for stochastic complex networked control systems (SCNCS) is studied. A fresh and suitable global Lyapunov function for SCNCS is constructed based on Kirchhoff’s matrix tree theorem in graph theory. Some sufficient criteria guaranteeing GASM for SCNCS are given, by applying Lyapunov method and stochastic analysis skills. These criteria are closely related to topological structure and coupling form functions of SCNCS. Furthermore, an application to a class of stochastic Lurie coupled control systems on networks (SLCCSN) is provided. Ultimately, a numerical test and its simulation process are presented to explain the validity of our theories.

Global stability analysis for coupled control systems and its application: Practical aspects and novel control

In this paper, the global stability of coupled control systems (CCSs) is discussed. Assembling the energy of each vertex system with the help of graph theory, a systematic method for constructing a global Lyapunov function of CCSs is proposed. Then, two kinds of stability criteria by Lyapunov-type theorem and coefficient-type theorem with the condition of the system topology are derived. Subsequently, the theoretical results are applied to the microgrid and the criterion of global asymptotical stability of the microgrid is developed. Meanwhile, based on the actual demand of the microgrid, the secondary frequency distributed consistency sliding mode control of the microgrid is proposed using the consensus algorithm. In the presence of a time-varying load, the control can not only quickly stabilize the frequency at the equilibrium point but also dynamically achieve active power sharing. Finally, the simulation of an islanded microgrid is conducted to test the validity and feasibility of our results.

Design of global smooth implicit control Lyapunov function for multiple-integrator system with input constraint

In this study, we propose a new method to design a smooth implicit control Lyapunov function and a smooth control law for a multi-integrator system with an input constraint. The proposed controller is almost linear near the origin, and it tends to a homogeneous controller of the previous implicit Lyapunov function methods as the state goes to infinity. The proposed method can be used for the backstepping method with a restricted virtual input, which is a nonlinear function of the state, and the desired virtual input is designed as a bounded function. Furthermore, we apply our method to the control of a magnetic levitation system, and a simulation result presents the advantage of the proposed method.