Lyapunov Function Research Articles

Dynamic event‐triggered asynchronous output feedback control for discrete‐time switched systems with deception attacks

AbstractThis article addresses the problem of dynamic event‐triggered asynchronous output feedback controller design for discrete‐time switched systems under deception attacks. To conserve communication resources and restrict asynchronous time, a dynamic event‐triggered mechanism is established which permits the system to switch many times in the trigger interval. Then, stability criteria for the underlying system despite the impact of deception attacks are obtained via the Lyapunov function and average dwell time method. Meanwhile, a co‐design scheme for dynamic output feedback gains and dynamic event‐triggered parameters are provided. Finally, two simulations are used to demonstrate the effectiveness of the approach.

Improved delay-dependent stability analysis of digital filters with generalized overflow arithmetic

This paper introduces a new set of conditions that establish Input-to-State Stability (ISS) for digital filters with external disturbance input and time-varying delays, all within the context of generalized overflow arithmetic. A unique augmented Lyapunov function and a novel summation inequality are applied to formulate a new criterion for ISS. This criterion assesses the ISS of digital filters in the presence of external disturbance, enabling the identification of asymptotic stability even in the absence of such disturbance. The criteria are developed using linear matrix inequalities (LMIs). The efficacy of the results is demonstrated through three numerical examples, highlighting that the proposed criteria are less conservative than some recent results.

Estimation of the Attraction Domain for the Quantum Systems Based on the Schrödinger Equation

This paper investigates a quantum system described by the Schrödinger equation, utilizing the concept of the quantum Lyapunov function. The Lyapunov function is chosen based on the mean value of a virtual mechanical quantity, where different values of P, the mean value of the virtual mechanical quantity in the Lyapunov function, have an impact on the attractive domain of the quantum system. The selected primary optimization algorithms approximating matrix P are the particle swarm optimization (PSO) algorithm and the simulated annealing (SA) algorithm. This study examines the characteristics of the system’s attraction domain under these two distinct algorithms and establishes stability conditions for the nonlinear quantum system. We introduce a method to estimate the size of the attractive domain using the Lyapunov function approach, converting the attractive domain issue into an optimization challenge. Numerical simulations are conducted in various two-dimensional test systems and spin 1/2 particle systems.

Wavelet Collocation Method for HIV-1/HTLV-I Co-Infection Model Using Hermite Polynomial.

In this study, the dynamic behavior of fractional order co-infection model with human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) is analyzed using operational matrix of Hermite wavelet collocation method. Also, the uniqueness and existence of solutions are calculated based on the fixed point hypothesis. For the fractional order co-infection model, its positivity and boundedness are demonstrated. Furthermore, different types of Ulam-Hyres stability are also discussed. The numerical solution of the model are obtained by using the operational matrix of the Hermite wavelet approach. This scheme is used to solve the system of nonlinear equationsthat are very fruitful and easy to implement. Additionally, the stability analysis of the numerical scheme is explained. The mathematical model taken in this work incorporates the biological characteristics of both HIV-1 and HTLV-I. After that all the equilibrium points of the fractional order co-infection model are found and their existence conditions are explored with the help of the Caputo derivative. The global stability of all equilibrium points of this model are determined with the help of Lyapunov functions and the LaSalle invariance principle. Convergence analysis is also discussed. Hermite wavelet operational matrix methods are more accurate and convergent than other numerical methods. Lastly, variations in model dynamics are found when examining different fractional order values. These findings will be valuable to biologists in the treatment of HIV-1/HTLV-I.

Exponential stability of second order time delay semilinear systems with bilinear control

ABSTRACT This paper explores the stabilisation of delayed bilinear hyperbolic systems, examining scenarios with delays in bilinear control and state. Our contributions are twofold: First, we establish existence results under specific conditions for these systems. Second, we design bilinear controls ensuring exponential stability under Lipschitz nonlinear perturbations. For control delay, stability is demonstrated using Lyapunov's abstract function. In contrast, for state delay, we show an equivalence between the stabilisation of the delayed system and the observability of the corresponding undamped system. Our theoretical findings are exemplified through practical models, emphasising their applicability in real-world scenarios.

Adaptive robust safety‐critical control of switched systems via single barrier function

SummaryThis paper addresses the problem of adaptive robust safety‐critical control (ARSCC) for switched systems, where the safety of subsystems is not necessary. A novel ARSCC framework and an executable algorithm are presented to solve the ARSCC problem by finding a switching signal and controllers of subsystems. To estimate uncertainties, a novel switched piecewise‐constant adaptive law is presented, which guarantees a pre‐computable estimation error boundary. A single barrier function (SBF) method is proposed to ensure safety of switched systems with uncertainties, where the safety of subsystems is satisfied only in some subregion of a given safe set, instead of the whole safe set. Based on the SBF method, a novel state‐dependent switching law possessing different dwell times for different subsystems, is established to orchestrate the switching among potentially unsafe subsystems. As a special case of the SBF method, a common barrier function method is presented to achieve safety of switched systems with uncertainties under switching signals with given dwell times. In addition, some sufficient conditions are derived to obtain safety and asymptotic stability for switched systems with uncertainties by combining SBF and single Lyapunov function. Finally, a switched RLC circuit system is given to illustrate the effectiveness of the theoretical results.

The Lorenz system as a gradient-like system

We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for 0⩽ρ⩽12 when σ = 10 and β=8/3 , significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.

Backstepping control of hypersonic vehicles under input and output constraints with model uncertainties and disturbances

In this paper, considering the physical restriction of hypersonic vehicle systems in a practical flight environment, the altitude tracking problem is studied with input magnitude, input rate, output constraints, system parameter uncertainties and disturbances, which expands the traditional considerations of control methods in the hypersonic vehicle field. The hyperbolic tangent function is employed to address the system input constraints. Furthermore, the incorporation of barrier Lyapunov function serves to ensure that the system output is limited in the prescribed limitations. In particular, two extended state observers are integrated to eliminate errors from model uncertainties and disturbances. Based on the effective combination of these techniques and auxiliary variable designs, a novel backstepping controller is proposed to simultaneously handle problems of input and output constraints, model uncertainties and disturbances. The closed-loop system stability is demonstrated using the Lyapunov theory. Simulation results show the effectiveness and superiority of the proposed controller compared with the traditional backstepping controller.

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.

Resilient sampled-data control for fractional-order PMVG-based WTS with actuator saturation and probabilistic faults using fuzzy Lyapunov function method

The goal of this study is to present a new fuzzy Lyapunov function-based resilient sampled-data control method for a fractional-order permanent magnet vernier generator (PMVG)-based wind turbine system (WTS) that addresses actuator saturation and probabilistic faults. To achieve this, we first model the dynamical fractional model of the PMVG-based WTS as a T-S fuzzy model. Next, we introduce a fuzzy Lyapunov function within a unified framework that incorporates actuator saturation, probabilistic faults, and uncertainty information from control gain matrices. We then present an actuator fault model that represents actuator faults as stochastic variables with a specific probability distribution. Subsequently, we formulate a robust asymptotic stability criterion for closed-loop systems using linear matrix inequality (LMI), which integrates the fuzzy Lyapunov function and membership function information with fractional integral inequality techniques based on sampling time. We also propose an LMI approach to identify the domain of attraction that meets the necessary actuator saturation criteria. Finally, we apply the proposed method to a PMVG-based WTS, demonstrating its superiority and practical applicability through a comparison with existing methods using a numerical example of a nonlinear truck trailer system.

Command filtering‐based adaptive backstepping control for a class of discrete‐time uncertain nonlinear systems with mismatched disturbances

SummaryThis paper investigates the adaptive fuzzy backstepping control problem for uncertain discrete‐time nonlinear systems with mismatched disturbances. A novel adaptive fuzzy command filtered backstepping approach is proposed, which can overcome the noncausal problem by using the command filter. The filter errors generated by filter are removed by constructing error compensation mechanism. The fuzzy logic systems are used to approximate the uncertain nonlinear dynamics, and the unknown mismatched disturbances are actively attenuated by using adaptive control technique. Based on the proposed new weighed Lyapunov functions, all the signals in the close‐loop system are proved to be uniformly ultimately bounded. The viability of proposed method is demonstrated by simulation results.

Delay-dependent exponential stability of highly nonlinear stochastic functional switched systems: an average dwell time approach

The delay-dependent exponential stability of stochastic functional switched systems (SFSSs) with highly nonlinear coefficients is investigated in this paper. By virtue of the mode-dependent average dwell time (MDADT) technique, the existence of global solution and qth moment exponential stability is studied without linear growth condition (LGC). Both of drift and diffusion coefficients have functional terms which can embody many existing delay cases, e.g. constant delay, time-varying delay and distributed delay, etc. By employing common/multiple Lyapunov function method, sufficient delay-dependent stability criteria of qth moment exponential stability are proposed. The MDADT condition reveals the positive role of switching signal in the delay-dependent stability analysis. Finally, an example is given to validate our theoretical results.

Asynchronous nonfragile guaranteed performance control for singular switched positive systems: An event‐triggered mechanism

SummaryThis paper addresses the guaranteed performance control problem for a class of singular switched positive systems, where the switching signal is subject to a state‐dependent process. Firstly, the causality, regularity, positivity, and asymptotical stability of the considered systems are discussed. In order to prevent the occurrence of data collisions and reduce the consumption of communication resources, the event‐triggered (E‐T) mechanism is applied. This is one of the initial attempts to introduce the E‐T scheme for such special systems. Besides, an asynchronous nonfragile controller under the E‐T scheduling scheme is designed to make certain that the resulting closed‐loop systems are causal, regular, positive, asymptotically stable, and have a guaranteed performance value . Through the application of the co‐positive Lyapunov function method and the min‐projection strategy, the corresponding sufficient conditions are given in the form of linear programming (LP). Finally, the effectiveness of the controller proposed in this paper is verified via two simulation examples.

Admissibility Analysis and Controller Design Improvement for T-S Fuzzy Descriptor Systems

In this paper, a stability analysis and the controller improvement of T-S fuzzy Descriptor system are studied. Firstly, by making full use of the related theory of fuzzy affiliation function and combining the design method of fuzzy Lyapunov function with the method of inequality deflation, a stability condition with wider admissibility and less system conservatism is proposed. The advantage of this method is that it is not necessary to ensure that each fuzzy subsystem is progressively stable. We also maximise the boundary of the derivatives of the affiliation function mined. Secondly, a PDC controller and a Non-PDC controller are designed, and the deflation conditions for the linear matrix inequalities of the two controllers are constructed. Finally, some arithmetic simulations and practical examples are given to demonstrate the effectiveness of the method studied in this paper, and the results obtained are less conservative and have larger feasible domains than previous methods.

Event-triggered quasi-time-varying [formula omitted] filtering for switched systems via multiple trigger-dependent Lyapunov functionals

This paper investigates the event-triggered filtering problem for switched systems from the perspective of switched impulsive systems (SISs) by exploiting the characteristics of trigger and impulse behavior. By constructing multiple trigger-dependent Lyapunov functionals (MTDLFs), an event-triggered quasi-time-varying (ET-QTV) H∞ filtering scheme is developed. Since the MTDLFs properly capture the event-induced impulse property of SISs, they release less conservatism. The filtering stability in the scenario of frequent plant switching during an interevent interval is also guaranteed by incorporating the average dwell time switching. The filtering synthesis condition is formulated as a set of QTV matrix inequalities defined over a finite time horizon, based on which a sum of squares-based filter design algorithm is presented further. Additionally, the derivation of a lower bound on interevent intervals is clarified for the event-triggering mechanism (ETM) to exclude the Zeno phenomenon. Finally, the effectiveness and potential practicability of the proposed filtering scheme are validated using a morphing aircraft model.

A novel fixed-time fuzzy adaptive tracking control for nonlinear high-order systems with state constraints

A fixed time fuzzy adaptive tracking controller is presented for high-order nonlinear systems, which includes unknown nonlinear functions and satisfies full state constraints. This is a constructive control scheme based on integrator barrier Lyapunov function, unique dominant adaptive parameter related to fuzzy approximation, and complex high-order back-stepping technique, which ensures that the tracking error can be confined to a preassigned neighbourhood including the origin within a fixed time independent of initial values, and all the states remain the given constraints. A practical application to single joint manipulator tracking control is provided, where comparative simulation results are illustrated for two different preassigned tracking errors. Another numerical example is offered to further manifest the feasibility of the theoretical results and the effectiveness of the controller in this paper.

Finite-time composite learning control for flexible-link manipulator based on disturbance observer

With the development of robotic technology and theory, there is an expectation for robots to operate in complex environments. This paper addresses the joint tracking control problem of flexible-link manipulator system under unknown external disturbances, modelling uncertainties, and actuator saturation constraints. A research is conducted, proposing a composite learning control scheme based on adaptive finite-time disturbance observer (AFTDO). The designed AFTDO can estimate the upper bound of unknown disturbances by adaptive laws, and a neural network controller is developed based on the AFTDO algorithm to compensate and suppress disturbances from multiple sources, ensuring the stability of all variables within a finite time. Simultaneously, a barrier Lyapunov function (BLF) is chosen to guarantee that the system satisfies constraints under flexible vibrations. Finally, through simulation comparisons with other control strategies, the feasibility and superiority of the proposed method are convincingly demonstrated.

Adaptive event-triggered security control for networked teleoperation systems.

An adaptive event-triggered security control method is proposed for networked robotic teleoperation systems subject to time-varying delays and false data injection (FDI) attacks. An event-triggered scheme is designed via the position and velocity signals of the master and slave robots, where the triggering thresholds can change adaptively with the system states. The position and velocity triggered signals and the feedback error between the operator and the environment forces are utilized to detect whether the transmitted signals suffered from FDI attacks. Then, a switching controller is designed, which is able to adopt the corresponding control strategy according to the corresponding detection result. The stability of the system and the convergence of the tracking errors are proved through Lyapunov functions, and the proposed method is validated through practical experiments. The proposed method is able to adaptively adjust the triggering frequency and effectively reduce the transmitted data, thus saving the network resources. Meanwhile, it can ensure the stability of networked teleoperation systems under time-varying delays and FDI attacks, as well as the position and force tracking performance.

Analysis of a stochastic SEIuIrR epidemic model incorporating the Ornstein-Uhlenbeck process

This article aims to analyze a stochastic epidemic model SEIuIrR (Susceptible-exposed-undetected infected-detected infected (reported -recovered) assuming that the transmission rate at which people undetected become detected is perturbed by the Ornstein–Uhlenbeck process. Our first objective is to prove that the stochastic model has a unique positive global solution by constructing a nonnegative Lyapunov function. Afterward, we provide a sufficient criterion to prove the existence of an ergodic stationary distribution of the mode by constructing a suitable series of Lyapunov functions. Subsequently, we establish sufficient conditions for the extinction of the disease. Finally, a series of numerical simulations are carried out to illustrate the theoretical results.

Event-triggered adaptive neural prescribed performance admittance control for constrained robotic systems without velocity measurements

In this paper, an event-triggered adaptive neural prescribed performance admittance control (ETANPPAC) scheme is proposed to control the constrained robotic systems without velocity sensors. To ensure compliance during human–robot interaction, the reference trajectory is obtained by reshaping the desired trajectory for the robotic systems based on the admittance relationship, where a saturation function is used to constrain the reference trajectory, avoiding excessive contact forces that could render the trajectory inexecutable. Moreover, a barrier Lyapunov function is used to constrain the tracking errors for prescribed performance, where a velocity observer and a radial basis function neural network are designed to estimate the velocity and the uncertainty of the robotic systems, respectively, to enhance control performance. To reduce the communication burden, an event-triggered mechanism is introduced and the Zeno behavior is avoided with the event-triggered condition. The stability of the whole control scheme is analyzed by the Lyapunov function. Simulation and experimental tests demonstrate that the proposed ETANPPAC scheme can track the desired trajectory well under constraints and reduce the communication burden, thereby achieving better efficiency for controlling the robotic systems compared with similar control schemes.