Lyapunov Function Method Research Articles

On the Global Stability of Nonlinear Hurwitz Control Systems

The problem of designing nonlinear control systems by the algebraic polynomial-matrix method using quasilinear models is considered. The quasilinear models of nonlinear plants are easily created on the basis of their nonlinear equations in the Cauchy form. To create these models only the differentiability of the plants nonlinearities is required. The solution to the design problem of the nonlinear Hurwitz control systems using the algebraic polynomial-matrix method is available if the quasilinear model of the plant is controllable. The design with application of this method consists of creating the quasilinear model of the nonlinear plant, generating several polynomials, composing and solving a system of linear algebraic equations. The theorem about the global stability of the equilibrium of the nonlinear systems, represented in the quasilinear model, is proved by the method of Lyapunov functions. The numerical examples of the design of nonlinear Hurwitz control systems are given. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article is concerned with the creation of stable nonlinear control systems with nonlinear elements since the linear systems can’t fulfill the quality requirements demanded from modern control systems. Additionally, the known design methods of nonlinear systems, such as the input-state feedback linearization, backstepping, passivity and others, require the transformation of the original nonlinear equations of the plant into some special form. These transformations are often very difficult to find and to execute. A new algebraic polynomial-matrix design method of the nonlinear control systems using the quasilinear models of nonlinear plants is proposed in the article. This method can be applied if the nonlinearities of the plant are differentiable, and the quasilinear model of the plant is controllable. The theorem about global stability of the equilibrium is proved for the systems designed with this approach. The quasilinear models are easily created on the basis of the original equations in the Cauchy form for a given nonlinear plant. The algebraic polynomial-matrix design method is very simple: the quasilinear model is created; several polynomials are calculated with the use of this model and the linear algebraic equations system is composed. The solution of this system allows us to write down the expression which defines the control law as a nonlinear function of the plant’s state variables. This system will be globally or locally stable if the conditions of the theorem or the corollaries, proved in this article, are satisfied. The numerical examples illustrate the application of the suggested approach to the design of nonlinear Hurwitz control systems for the nonlinear plants. The main advantages of this approach: the quasilinear models are created quite easily; the nonlinear control law is found as a solution to the system of linear equations. The results can be applied to the creation of nonlinear control systems of nonlinear plants in many industries: shipbuilding, aircraft construction, automobile construction, agriculture and many others.

The Stationary Distribution and Extinction of a Stochastic Five‐Dimensional COVID‐19 Model

This paper investigates a stochastic SEIAIR epidemic model with nonlinear incidence and contacting distance to describe the transmission dynamics of COVID‐19. Firstly, we show the global existence of positive solution of the system. Then, using the Lyapunov function method and theory of stochastic analysis, we set out the sufficient conditions of the existence and uniqueness of an ergodic stationary distribution to the stochastic model. Furthermore, we obtain the sufficient condition for extinction of the disease. Finally, we go through some numerical simulations to demonstrate the theoretical results. Our study extends and improves the results of previous studies.

Синтез адаптивной робастной системы управления трикоптером с поворотными винтами в условиях неопределенных аэродинамических коэффициентов, частично неопределенной матрицы входа, влияния внешних возмущений и ограничения входных воздействий

In this article, an adaptive robust control system for a tricopter equipped with three rotary propellers is developed and studied under the following conditions: a) uncertain aerodynamic coefficients; b) partially uncertain input matrix; c) influence of unknown external disturbances; and d) input constraints. When constructing a complete nonlinear mathematical model of the dynamics of a tricopter with rotary propellers in the form of Lagrange-Euler equations, the input constraints are taken into account. The synthesis of an adaptive robust control system for a tricopter with rotary propellers is carried out on the basis of the modified computed torque control (Li–Slotine method) and the function approximation technique to eliminate the influence of uncertainty. Unknown external disturbances are compensated by a smooth sliding mode control using an adaptive estimate of the upper bound of the sum of external disturbances and approximation errors of uncertain functions in the mathematical model. To compensate for the negative effects of input constraints, an auxiliary dynamic system is constructed, the state variables of which are used to synthesize the control law. The robustness of the adaptive system and the desired tracking accuracy of the reference signals are proven using the Lyapunov function method. Simulation results using MatLab/Simulink are presented to illustrate the effectiveness of the proposed control law.

Asymptotic stability of impulsive stochastic switched system with double state-dependent delays and application to neural networks and neural network-based lecture skills assessment of normal students

&lt;abstract&gt;&lt;p&gt;This article investigates the stability problem of impulsive stochastic switched systems with double state-dependent delays. In the designed system, unstable and stable impulses are taken into consideration, respectively, and they do not need to function simultaneously with switching behavior. Additionally, two new ideas, i.e., mode-dependent switching density and mode-dependent impulsive density, are developed. Based on the Lyapunov function method and comparison principle, the asymptotic stability criteria for an impulsive stochastic switched system with state-dependent delays are given. Moreover, the application of theoretical results to neural networks and the neural network-based lecture skills assessment of normal students is analyzed. Finally, two numerical examples are provided to illustrate the effectiveness and reliability of the theoretical criteria.&lt;/p&gt;&lt;/abstract&gt;

Hopf bifurcation in a predator-prey model under fuzzy parameters involving prey refuge and fear effects

&lt;p&gt;In ecology, the most significant aspect is that the interactions between predators and prey are extremely complicated. Numerous experiments have shown that both direct predation and the fear induced in prey by the presence of predators lead to a reduction in prey density in predator-prey interactions. In addition, a suitable shelter can effectively stop predators from attacking as well as support the persistence of prey population. There has been less exploration of the effects of not only fear but also refuge factors on the dynamics of predator prey interactions. In this paper, we unveil several conclusions about a predator-prey system with fuzzy parameters, considering the cost of fear in two prey species and the effect of shelter on two prey species and one predator. As the first step of the investigation, the boundedness and non-negativity of the solutions to the system are put forward. Using the Jocabian matrix and Lyapunov function methods, we further analyze the existence and stability of the available equilibria and also the existence of Hopf bifurcation, considering the fear parameter as the bifurcation parameter that has been observed by applying the normal form theory. Finally, numerical simulations help us better understand the dynamics of the model, in which some interesting chaotic phenomena are also exhibited.&lt;/p&gt;

Синтез нелинейной и адаптивно-робастной систем со скользящими режимами в управлении динамикой трикоптера с поворотными винтами при действии неизвестных внешних возмущений

The article develops a complete nonlinear mathematical model of tricopter dynamics with rotary propellers in the form of Lagrange-Euler equations and two systems of nonlinear and adaptive-robust control with sliding modes, operable under the action of unknown external disturbances. The nonlinear control system with sliding modes and constant controller parameters is synthesized based on the Lyapunov function method. The adaptive-robust system is developed on the basis of the Lyapunov function method with a smooth sliding mode control and using an adaptive estimate of the upper bound of external disturbances. The performance of the proposed control systems under the action of unknown external disturbances and variations in aerodynamic coefficients is investigated. A comparative study of the constructed control systems is conducted in the MatLab/Simulink software environment.

Stability Results for Nonlinear Fractional Differential Equations with Incommensurate Orders

This paper aims to establish some novel conditions for revealing the stability of nonlinear incommensurate fractional-order systems formulated using Caputo differential operator. This will be accomplished through introducing satisfactory proofs for these conditions with the help of using Lyapunov function method. All theoretical findings will be verified numerically by observing the wished stability is occurring on the systems at hand.

Adaptive neural control for delayed discrete-time switched systems under deception attacks

This paper focuses on the issue of adaptive neural control for discrete-time switched systems with time delay and deception attacks. Firstly, the switching signal is constrained by the dwell time. Considering that the deception attacks are unknown, the neural network technique is employed to approximate the attack signals. Then, an adaptive state feedback controller is established to compensate for the adverse effects of deception attacks for switched systems. Meanwhile, sufficient conditions for the boundedness of the switched system are given through the Lyapunov functional method, and the controller gains can be obtained by resolving the linear matrix inequality. Finally, the feasibility of the proposed method is illustrated via a numerical example.

Modeling Control of Supercoiling Dynamics and Transcription Using DNA-Binding Proteins.

Nearly all natural and synthetic gene networks rely on the fundamental process of transcription to enact biological feedback, genetic programs, and living circuitry. In this work, we investigate the efficacy of controlling transcription using a new biophysical mechanism, control of localized supercoiling near a gene of interest. We postulate a basic reaction network model for describing the general phenomenon of transcription and introduce a separate set of equations to describe the dynamics of supercoiling. We show that supercoiling and transcription introduce a shared reaction flux term in the model dynamics and illustrate how the modulation of supercoiling can be used to control transcription rates. We show the supercoiling-transcription model can be written as a nonlinear state-space model, with a radial basis function nonlinearity to capture the empirical relationship between supercoiling and transcription rates. We show the system admits a single, globally exponentially stable equilibrium point. Notably, we show that mRNA steady-state levels can be controlled directly by increasing a length-scale parameter for genetic spacing. Finally, we build a mathematical model to explore the use of a DNA binding protein, to define programmable boundary conditions on supercoiling propagation, which we show can be used to control transcriptional bursting or pulsatile transcriptional response. We show there exists a stabilizing control law for mRNA tracking, using the method of control Lyapunov functions and illustrate these results with numerical simulations.

Global Event-Triggered Funnel Control of Switched Nonlinear Systems via Switching Multiple Lyapunov Functions.

In this article, the global event-triggered (ET) funnel tracking control problem is studied for a class of switched nonlinear systems with structural uncertainties, where the solvability of the control problem for each subsystem is not needed. A switching multiple Lyapunov functions (MLFs) method is established, where MLFs are designed to handle switched inverse dynamics, and a switching barrier Lyapunov function is constructed to address switched sampled errors that may compromise system stability. This is achieved alongside a new switching dynamic event-triggering mechanism (DETM). By combining this method with backstepping, a dwell-time state-dependent switching law and an ET funnel controller of each subsystem are constructed, effectively eliminating the issue of the "explosion of complexity" encountered in traditional backstepping without using dynamic surface control or command filters. Additionally, the designed switching DETM ensures that the tracking error always evolves within a performance funnel in any consecutive triggering interval, excluding Zeno behavior, and guaranteeing positive constant lower bounds for two consecutive triggering intervals and any switching interval, respectively. Finally, an example is provided to show the validity of the theoretical results.

Fuzzy Adaptive Finite-Time Resilient Control Against Unknown False Data Injection Attacks for MIMO Nonlinear Switched Systems With Unknown Dead Zone.

This article studies the finite-time adaptive resilient control problem for MIMO nonlinear switched systems with the unknown dead zone. The sensors of the controlled systems suffer from unknown false data injection (FDI) attacks so that all states cannot be directly applied to the design process of the controller. To address this negative impact of FDI attacks, a new coordinate transformation is designed in control design. Moreover, the Nussbaum gain technique is introduced to deal with the difficulty of unknown time-varying weights caused by FDI attacks. Based on the common Lyapunov function method, a finite-time resilient control algorithm is designed by employing compromised state variables, which ensures that all signals of the closed-loop systems are bounded under arbitrary switching rules even in the presence of unknown FDI attacks. Compared with the existing results, the proposed control algorithm not only enables the controlled systems to reach an equilibrium state in a finite time but also removes the assumption that the sign of the attack weights is positive. In the end, a practical simulation example proves that the designed control method is valid.

Adaptive neural DSC for nonlinear switched systems with prescribed performance and input saturation

This paper solves the problem of an adaptive neural dynamic surface control for a class of uncertain strict-feedback nonlinear systems with guaranteed transient and steady-state performance under arbitrary switchings. First, by utilizing the prescribed performance control, the prescribed tracking control performance can be ensured, while the requirement for the initial error is removed. Second, radial basis function (RBF) neural networks (NNs) are used to handle unknown nonlinear functions, the Gaussian error function is employed to represent a continuous differentiable asymmetric saturation model and the dynamic surface control (DSC) technique is used to overcome the problem of 'explosion of complexity' inherent in the control design. At last, by using the common Lyapunov function method in combination with the backstepping technology, a common adaptive neural controller is constructed. The designed controller overcomes the problem of the over-parameterization, and further alleviates the computational burden. Under the proposed common adaptive controller, all the signals in the closed-loop system are bounded, and the prescribed transient and steady tracking control performance are guaranteed under arbitrary switchings. Simulation studies demonstrate the effectiveness of the proposed method.

A mathematical model of meningitis with antibiotic effects

The mathematical model in this study is a SCIR-type meningitis disease spread model, namely susceptible (S), carrier (C), infected (I), and recovery (R). In the model used, there are two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point. The conditions and stability of the equilibrium point are determined by the basic reproduction number, which is the value that determines whether or not the spread of meningitis infection in a population. The results of this study show that the stability of the disease-free equilibrium point and the endemic equilibrium point are locally asymptotically stable and by using the Lyapunov Function method it is found that the disease-free equilibrium point will be globally stable when, while the endemic equilibrium point will be globally stable when numerical simulations perform to support the theoretical results.

Synchronization for fractional-order neural networks with unbounded delays

This paper deals with synchronization analysis problem for a class of fractional-order neural networks with unbounded delays. Using the Lyapunov function method combined with fractional Halanay inequality, we derive a novel sufficient condition for asymptotic stability of the error system resulting in two neural networks are synchronized. The obtained conditions are given in terms of linear matrix inequalities, which therefore can be efficiently checked. A numerical example is proposed to illustrate the effectiveness of the obtained results.

Finite‐time bipartite synchronization of T–S fuzzy coupled neural networks via quantized adaptive control

AbstractBased on the framework of signed graphs, this paper investigates the finite‐time (FET) bipartite synchronization of Takagi‐Sugeno (T–S) fuzzy coupled neural networks with output coupling and state‐dependent parameters. Adaptive control schemes with quantizer are designed, and generalized norm is considered. Based on the framework of bipartite synchronization and FET synchronization, several sufficient conditions are established to realize the goal of synchronization by using Lyapunov functional method. Moreover, some numerical simulations are provided to illustrate the theoretical results.

Predefined Time Fuzzy Adaptive Control of Switched Fractional-Order Nonlinear Systems with Input Saturation

Article Predefined Time Fuzzy Adaptive Control of Switched Fractional-Order Nonlinear Systems with Input Saturation Lusong Ding, and Weiwei Sun * 1 Institute of Automation, Qufu Normal University, Qufu 273165, China * Correspondence: wwsun@hotmail.com Received: 31 August 2023 Accepted: 31 October 2023 Published: 21 December 2023 Abstract: This article investigates the predefined-time fuzzy adaptive tracking control problem for a class of nonlinear switched fractional-order systems with input saturation and external disturbances under a nonstrict feedback structure. By combining the backstepping technique and the common Lyapunov function method, a predefined-time switching control method is constructed based on a novel fractional-order auxiliary function. The fuzzy logic system and the adaptive method are introduced to identify unknown compounded continuous functions. Moreover, the issue of calculating explosion and the problem of singularity are tackled through the newly proposed predefined-time and filter-based dynamic surface control. Especially, the construction of a continuous term in the controller eliminates possible chattering. The developed control strategy achieves that the closed-loop system is practically predefined-time stable under arbitrary switchings, where the upper bound of the settling-time can be defined by users in advance. Finally, two simulation examples are illustrated to prove the feasibility and effectiveness of the presented scheme.

On linear stabilization of a class of nonlinear systems in a critical case

In this paper, we address the stabilization problem for nonlinear systems in a critical case. Namely, we study the class of canonical nonlinear systems. Canonical nonlinear systems or chain of power integrators is an important subject of research. Studying such systems is complicated by the fact that they cannot be mapped onto linear systems. Moreover, they have the uncontrollable first approximation. Previous results on smooth stabilization of such systems were obtained under the assumption that the powers in the right-hand side are strictly decreasing. In this work, we consider a case of non-increasing powers in the right-hand side for a three-dimensional system. A popular approach for studying such systems is the backstepping method, which is a method of step-wise stabilization. This method requires a sequential investigation of lower-dimensional subsystems. Backstepping enables the study of a wide range of nonlinear triangular systems but requires technically complex and cumbersome computations. Therefore, a natural question arises about constructing stabilizing controls of a simple form. Polynomial controls can serve as an example of such controls. In the paper, we demonstrate that linear controls can be considered as stabilizing controls. We derive sufficient conditions for the coefficients of the linear control that ensure the asymptotic stability of the zero equilibrium point of the corresponding closed-loop system. The asymptotic stability is proven using the Lyapunov function method, which is found as the sum of squares. The negative definiteness of the Lyapunov function derivative in a neighborhood of the origin guarantees asymptotic stability. In contrast to the case of strictly decreasing powers, additional conditions on the control coefficients, apart from their negativity, emerge. The obtained result extends to a broader class of nonlinear systems through stabilization by nonlinear approximation. This allows the consideration of systems with higher-order terms in the right-hand side. The effectiveness of the applied approach is illustrated by several model examples. The method used in this work to investigate the case of non-increasing powers can be applied to systems of higher dimensions.

Reachable set control for nonlinear Markov jump cyber–physical systems with false data injection attacks

This paper presents the problem of reachable set controller design for a class of nonlinear Markov jump cyber–physical systems (MJCPSs). The adverse effects of bounded peak external disturbances, uncertainties and false data injection (FDI) attacks with certain probability on MJCPSs are considered. The aim is to design an appropriate controller to guarantee that the actual state of the closed-loop MJCPSs is defined by a specified set. Firstly, the Lyapunov functional method is introduced because the time-varying delays are carried by FDI attacks. By using the Lyapunov functional method, an ellipsoid description of the reachable set containing all the states of the system originating from the origin is obtained. Then, the controller gain can be obtained via a set of feasible linear matrix inequalities (LMIs). Finally, the effectiveness of the designed reachable set controller is proved by simulation results.

Neuroadaptive Performance Guaranteed Control for Multiagent Systems With Power Integrators and Unknown Measurement Sensitivity.

This article investigates the adaptive performance guaranteed tracking control problem for multiagent systems (MASs) with power integrators and measurement sensitivity. Different from the structural characteristics of existing results, the dynamic of each agent is a power exponential function. A method called adding a power integrator technique is introduced to guarantee that the consensus is achieved of the MASs with power integrators. Different from existing prescribed performance tracking control results for MASs, a new performance guaranteed control approach is proposed in this article, which can guarantee that the relative position error between neighboring agents can converge into the prescribed boundary within preassigned finite time. By utilizing the Nussbaum gain technique and neural networks, a novel control scheme is proposed to solve the unknown measurement sensitivity on the sensor, which successfully relaxes the restrictive condition that the unknown measurement sensitivity must be within a specific range. Based on the Lyapunov functional method, it is proven that the relative position error between neighboring agents can converge into the prescribed boundary within preassigned finite time. Finally, a simulation example is proposed to verify the availability of the control strategy.

Identifying the switching topology of dynamical networks based on adaptive synchronization.

This paper proposes an approach for identifying unknown switching topology in a complex dynamical network. The setup is divided into two components: a primary drive network and a specialized response network equipped with switched topology observers. Each class of observers is dedicated to tracking a specific topology structure. The updating law for these observers is dynamically adjusted based on the operational status of the corresponding topology in the drive network-active if engaged and dormant if not. The sufficient conditions for successful identification are obtained by employing adaptive synchronization control and the Lyapunov function method. In particular, this paper abandons the generally used assumption of linear independence and adopts an easily verifiable condition for accurate identification. The result shows that the proposed identification method is applicable for any finite switching periods. By employing the chaotic Lü system and the Lorenz system as the local dynamics of the networks, numerical examples demonstrate the effectiveness of the proposed topology identification method.