Lyapunov Theorem Research Articles

Fractional-order input-to-state stability and its converse Lyapunov theorem

This paper establishes input-to-state stability (ISS) for fractional-order nonlinear systems, providing a stability constraint framework for fractional-order systems affected by external inputs. The general fractional comparison principle aids in constructing Lyapunov functions, offering criteria for input-to-state stability of fractional-order nonlinear systems. The concept of weak robustness and the converse Lyapunov theorem for fractional-order nonlinear systems further enhance the completeness of input-to-state stability in these systems. By designing systems to maintain fractional-order input-to-state stability (FOISS), the system can remain robust against the effects of bounded external inputs. Serving as a foundational ”toolbox,” this work provides insights into the stability analysis of fractional-order nonlinear systems, benefiting researchers and practitioners in the field of control theory.

A Mathematical Approach to the Stability of Electroencephalography Signals Model Using Integral Equations During Epileptic Seizures

In this study, the mathematical stability of EEG signal models represented by the integral equation during an epileptic seizure is investigated. The modelling of the EEG signals should be done with utmost accuracy to understand the actual nature of the brain dynamics and to enhance seizure prediction approaches. We investigate the stability of such models by applying Lyapunov's stability theorem and show that stability is powerfully dependent on parameters such as seizure duration and intensity, along with neural connectivity. We demonstrate that some of the parameter regimes yield stable behaviour, while others give rise to instability, making EEG interpretation complicated during epileptic seizures. By comparing integral equation-based models with current approaches, their advantages are highlighted in capturing the transient dynamics of epileptic activity. This work focuses on developing solid mathematical frameworks that will allow for real-time EEG monitoring. It contributes to the wider knowledge on the dynamics of seizures in the brain and will have practical implications for the development of neurological diagnostic and therapeutic tools.

Complex network control and stability through distributed critic‐based neuro‐fuzzy learning

AbstractInspired by advancements in swarm autonomous vehicles and intelligent control systems, this research addresses the issue of frequency synchronization and phase tracking in oscillator networks. A novel distributed consensus protocol and a reinforcement learning algorithm for a multi‐agent network with a leader–follower topology, considering stability conditions, are developed. The critic‐based neuro‐fuzzy learning (CBNFL) method aims to achieve consensus and minimize local tracking errors. Additionally, an explicit synchronization condition for the network using the Lyapunov theorem is derived. Each vehicle tracks its reference phase and frequency. Employing a fuzzy critic to evaluate the current state and generate a stress signal for the controller, the method prompts adaptive parameter adjustments to minimize this signal. The proposed design's versatility and adaptability to various networks demonstrate robustness against dynamic vehicle properties and network parameter uncertainties, ensuring consistent controller performance. This approach exhibits high scalability, accommodating numerous autonomous agents. To validate the proposed learning method's efficacy, numerical simulations are conducted on a network of five oscillators. The outcomes of implementing CBNFL compared with a conventional PI controller underscore the CBNFL method's superior performance and robustness in maintaining network stability and achieving synchronization.

DYNAMICS OF BANDITRY ACTIVITIES: A MATHEMATICAL MODELING APPROACH

Banditry represents a significant worldwide social issue; in light of this, a mathematical model for the dynamics of banditry activities was developed. The model is intended to control banditry activities in society. The model encompasses six population classes and incorporates preventive measures such as preventing susceptible individuals from being kidnapped and taking the captured bandits into jail or detention. Bandits-free and bandits persistence were established. Also, the threshold parameter (Reproduction Number (Rc)) used to measure the level of banditry activities was established, as (Rc<1), banditry activities die out from the dynamical system, and if Rc>1 the banditry persists Rc>1, in the system. Numerical results demonstrate that banditry is effectively curbed when Rc<1 and persists otherwise. Stability analyses reveal the local asymptotic stability of the bandit-free equilibrium, further supported by global stability analysis employing the Lyapunov theorem. Empirical findings suggest that incarcerating captured bandits significantly outperforms lethal measures or preventing susceptible individuals from being kidnapped.

MIMO ultra-local model-based adaptive enhanced model-free control using extremum-seeking for coupled mechatronic systems

Multiple degree-of-freedom (DOF) mechatronic systems, such as robots and robotic arms, play a crucial role in modern life and production. However, due to strong coupling, uncertain dynamics, and external disturbance, accurately modeling these systems is challenging, making traditional model-based control methods impractical. To address this, this paper proposes an extremum-seeking-based adaptive enhanced model-free control for multi-input multi-output (MIMO) mechatronic systems to realize robust trajectory tracking. Unlike previous model-free control methods that decouple and reorganize the MIMO system into several single-input single-output ultra-local models, this paper develops a MIMO ultra-local model with a non-diagonal gain matrix α to approximate the system dynamics within an ultra-short time window. Time-delay estimation (TDE), Proportional-Derivative (PD) control law and accuracy compensation compose an TDE-based enhanced intelligent PD control that ensures the closed-loop stability. Furthermore, an extremum-seeking (ES) technique is designed to optimize the gain matrix α to enhance control performance. The main contributions of this paper are the development of a model-free control framework based on the MIMO ultra-local model and the successful application of ES to optimize the non-diagonal gain matrix α. Stability analysis of the closed-loop system is conducted using Lyapunov theorem. Finally, numerical simulations on a 2-DOF robotic manipulator and co-simulation results on a 3-DOF PUMA 560 robotic manipulator validate the effectiveness and superiority of the proposed methods.

Semi-Global Stabilization of Discrete-Time Linear Systems Subject to Infinite Distributed Input Delays and Actuator Saturations.

The semi-global stabilization problem of discrete-time systems subject to infinite distributed input delays and actuator saturations is investigated in this article. This article develops two low-gain feedback control laws for two types of systems, respectively. It is shown that the resulting system is semi-globally exponentially stabilized. Our results include those existing results on systems subject to only input saturations and systems subject to bounded delays and input saturations as special cases. Compared with existing results on infinite delays and actuator saturations, this article develops a more accurate scaling utilizing a more general framework. Furthermore, a novel converse Lyapunov theorem for discrete-time linear infinite-delayed systems and a novel stability analysis theorem for perturbed discrete-time linear infinite-delayed systems are developed to handle the nonlinearity induced by saturations. Finally, this article provides two numerical examples to illustrate the effectiveness of the developed theorems.

Anti-Sway Adaptive Fast Terminal Sliding Mode Control Based on the Finite-Time State Observer for the Overhead Crane System

This work proposes an adaptive rapid terminal SMC (sliding mode control) approach based on the FFTSO (fast finite-time state observer) for overhead crane trajectory tracking and anti-swing control in the presence of external disturbances and parameter uncertainty. First, the system state observation under the constraint of unknown system parameters is accomplished by designing the FFTSO based on finite-time theory. Next, a parameter-adaptive fast terminal SMC is created for an overhead crane based on the model transformation. This technique can still monitor the intended trajectory and reduce payload swing even in cases when the payload mass and wire rope length are uncertain. Next, the Lyapunov theorem is used to demonstrate the stability of the overhead crane system’s positioning and anti-swing angle control mechanism. Lastly, the platform experiments confirm that the suggested closed-loop system control technique is successful.

Finite-time stability in probability of stochastic delay systems via generalized Halanay inequality

This paper presents novel advancements in the analysis and control of stochastic systems with time-varying delays, focusing on the concept of finite-time stability. Firstly, a new kind of generalized Halanay inequality is established. With this crucial approach, two essential stochastic Lyapunov theorems on the finite-time stability in probability of stochastic delay systems with bounded time delays and unbounded time delays, respectively, are derived. In addition, upper bound estimate for the stochastic settling time is given. Furthermore, a type of nonlinear stochastic delay systems is investigated and a delay-independent controller is proposed to achieve finite-time stability in probability. The effectiveness of the results is demonstrated through two illustrative examples.

Decentralized Robust Tracking Control of Interconnected Nonlinear‐Constrained Systems: A Dynamic Event‐Sampled Method

ABSTRACTThis paper presents a decentralized robust dynamic event‐sampled tracking (EST) control law for interconnected nonlinear‐constrained systems. The core of developing such a control law is to convert the original EST control problem into the event‐sampled decentralized stabilization problem of augmented interconnected systems. To address the transformed decentralized stabilization problem, an indirect approach relying on the optimal control methodology is proposed. Initially, a group of cost functions are constructed for the nominal subsystems related to the augmented interconnected systems. Then, the dynamic event‐sampling mechanisms are introduced for lessening the computational burden. Meanwhile, the event‐sampled Hamilton–Jacobi–Bellman equations (ES‐HJBEs) are proposed for the augmented interconnected systems. To approximately solve the ES‐HJBEs, the critic approximators are used with their parameters tuned under the reinforcement learning framework. After that, the uniform ultimate boundedness of the tracking errors and the approximators' parameter estimation errors are assured based on the Lyapunov theorem. Finally, a nonlinear plant is provided to validate the decentralized robust dynamic EST control law.

Distributed two-channel dynamic event-triggered adaptive finite-time fault-tolerant containment control for multi-leader UAV formations

This paper addresses the distributed adaptive finite-time containment control problem in multi-leader UAV formations with actuator failures, limited communication, and external disturbances. A two-channel dynamic event-triggered strategy based on adaptive and sliding mode control is proposed as a containment control scheme, which solves the contradiction between the need to include containment error in the continuous term of the sliding mode surface in the adaptive law and the discontinuity of the communication between neighboring subsystems (follower UAVs) by constructing intermediate variables. Unlike the traditional event triggering condition that can only be applied to asymptotically convergent systems, the dynamic triggering condition designed in this paper, based on auxiliary variables, hyperbolic tangent function, and adaptive techniques, realizes finite-time convergence during the adjustment of the system and sets a more reasonable lower limit for the triggering thresholds, thus further reducing the communication frequency and ensuring the finite-time convergence of the system. The stability of the closed-loop system can be proved by the Lyapunov theorem. Zeno behavior can be avoided. The simulation results verify the effectiveness of the developed control algorithm.

Novel Controller Methodology for the Cessna Citation X Under Turbulence During Cruise

The design of a new control approach for the longitudinal motion of the Cessna Citation X during cruise is performed using a combination of sliding mode control (SMC) system, type 1 fuzzy logic system, and adaptive control system. This methodology is presented for 1) controlling the aircraft pitch rate and 2) stabilizing the aircraft speed during turbulence. The nonlinear model of the aircraft was generated using a simulation platform, which was designed based on flight data obtained from the highest Federal Aviation Administration–certified Level-D Research Aircraft Flight Simulator. The type 1 adaptive fuzzy logic system was implemented to approximate unknown functions for constructing the equivalent part of the SMC system that handled the effects of uncertainties and turbulence. The adaptation laws, derived from the Lyapunov theorem, were used to update the approximated functions in the control law at each flight condition and simulation iteration. Using the control systems combination, the pitch rate could follow the given reference signal, while the aircraft speed remained at a reference value with and without turbulence across the whole flight envelope. Results have shown that the proposed controllers satisfied tracking performance while generating smooth elevator deflection, both of which are important for real-aircraft applications.

Dynamic event-triggered consensus for stochastic delay multi-agent systems under directed topology

This paper addresses the mean-square exponential consensus (MSEC) challenge in stochastic delay multi-agent systems (MASs) under directed topology. The main contributions include avoiding infinitely fast sampling and conducting a random stability analysis. First, a dynamic event-triggered mechanism (DETM) with a fixed positive lower bound for the inter-execution time is designed to prohibit Zeno behavior. The proposed control strategy does not involve the expectations of system signals. Then, sampling errors are estimated using Itô isometry, which ensures that the fixed control at each work interval is maintained with adequate feedback capacity to mitigate the effects of these errors. Furthermore, with the help of a Halanay inequality in integral form and Lyapunov theorem framework, some conditions are proposed to guarantee the consensus of stochastic delay MASs under directed topology. Finally, an example is presented to prove the effectiveness of the DETM method.

Robust adaptive event-triggered control for unmanned surface vessel–unmanned aerial vehicle: Application to sea–air synchronized lawn mowing search operation

This paper investigates a robust adaptive event-triggered control algorithm with a consideration of the synchronized lawn mowing search guidance for a heterogeneous system comprising both an unmanned surface vessel (USV) and an unmanned aerial vehicle (UAV). For ensuring a cooperative global search navigation of the USV–UAV, a synchronized lawn mowing search guidance is developed with two key elements: the L1-based virtual ship (L1VS) and L1-based virtual aerial (L1VA), where the former can generate the smooth reference path for a USV according to the search task scope, while the latter can program the reference path for a UAV on the basis of its search requirements (search speed, detected capacity and communication distance). In the control loop, a robust adaptive path following control law is designed by utilizing an event-triggered mechanism and a minimal learning parameter (MLP) technique. In particular, the event-triggered mechanism has a potential role for a reduction of the unnecessary transmission resource usage for the channel of the controller to the actuator. Besides, only one learning parameters on each channel by introducing the MLP technique, implying a lower computational burden. Through the Lyapunov theorem, the semi-global uniform ultimate bounded (SGUUB) stability properties of the proposed control system. Finally, two numerical simulations are carried out to evaluate the effectiveness of the proposed strategy and the advantages compared with existing techniques.

Adaptive Output Formation Tracking for Nonlinear Multiagent Systems With Double Semi-Markovian Switching Topologies and Time-Varying Actuator Faults.

This article investigates adaptive output formation tracking control of nonlinear multiagent systems with time-varying actuator faults and unknown nonidentical control directions under double semi-Markovian switching topologies. Considering the dynamic changes of communication connections in uncertain environments, a double semi-Markov process is first introduced into the leader-follower structure to describe the random switching of communication topologies. Then, a novel adaptive distributed fault-tolerant output formation tracking control framework is established using the backstepping and Nussbaum gain technique to address matched/mismatched uncertainties and disturbances, time-varying actuator faults, and unknown nonidentical control directions. In this control framework, the independent variable of the Nussbaum function is designed as a non-negative function that monotonically increases with respect to time, thereby overcoming the presence of the absolute value of its derivative in the integration process. Based on the distributed structure, an adaptive fault-tolerant controller is further proposed to achieve the asymptotic output formation tracking in mean-square sense. The stability of the closed-loop nonlinear multiagent systems is analysed through the contradiction argument and Lyapunov theorem. The simulation example verifies the effectiveness of the proposed control strategy.

Composite anti-disturbance predictive control of unmanned systems with time-delay using multi-dimensional Taylor network

A composite anti-disturbance predictive control strategy employing a Multi-dimensional Taylor Network (MTN) is presented for unmanned systems subject to time-delay and multi-source disturbances. First, the multi-source disturbances are addressed according to their specific characteristics as follows: (A) an MTN data-driven model, which is used for uncertainty description, is designed accompanied with the mechanism model to represent the unmanned systems; (B) an adaptive MTN filter is used to remove the influence of the internal disturbance; (C) an MTN disturbance observer is constructed to estimate and compensate for the influence of the external disturbance; (D) the Extended Kalman Filter (EKF) algorithm is utilized as the learning mechanism for MTNs. Second, to address the time-delay effect, a recursive τ−step-ahead MTN predictive model is designed utilizing recursive technology, aiming to mitigate the impact of time-delay, and the EKF algorithm is employed as its learning mechanism. Then, the MTN predictive control law is designed based on the quadratic performance index. By implementing the proposed composite controller to unmanned systems, simultaneous feedforward compensation and feedback suppression to the multi-source disturbances are conducted. Finally, the convergence of the MTN and the stability of the closed-loop system are established utilizing the Lyapunov theorem. Two exemplary applications of unmanned systems involving unmanned vehicle and rigid spacecraft are presented to validate the effectiveness of the proposed approach.

Prescribed‐time resilient current‐sensorless DC power converter control

AbstractWhile conventional control techniques for stabilizing power electronic converters with constant‐power load (CPL) often rely on complete system state information, the associated challenges, including the cost and increased size due to the installation of multiple sensors, necessitate exploration of alternative methodologies. To address these concerns, a novel observer‐based sliding‐mode control approach is proposed. This approach aims to enhance cost‐effectiveness and reliability in the context of a DC‐DC buck converter supplying power to an unknown CPL. The core concept involves designing a state observer to estimate the system's current based on voltage measurements. A non‐certainty equivalent adaptive update law is then introduced to estimate the unknown CPL, incorporating feedback from voltage measurements and current estimation. To guarantee stability in the face of uncertainties in system parameters, a thorough analysis utilizing the Lyapunov theorem is conducted on the closed‐loop system. The efficacy and feasibility of the proposed control algorithm are substantiated through experimental results on the OPAL‐RT platform, showcasing its promising performance.

Fault‐tolerant formation control of multi‐agent systems with multiple leaders based on intermediate variable estimator

AbstractThis article investigates the distributed fault‐tolerant formation control problem of a multi‐leader system with process faults and system uncertainties. This study guarantees all the followers to achieve a specified time‐varying formation and track the combination of multiple leaders. By treating faults as a special type of states, an enhanced system is constructed. Furthermore, an intermediate variable observer is designed, but the observer matching condition is not necessary. The observer is used to estimate faults and system uncertainties, for reconstructing the formation controller. The observer gains are determined by solving linear matrix inequalities. Subsequently, controller gains are designed based on Lyapunov theorem and adaptive techniques. Finally, the proposed method is verified through a simulation example.

New results on asymptotic stability of time-varying nonlinear systems with applications

Abstract. In this paper, we present a converse Lyapunov theorem for the new notion of global generalized practical uniform h-stability of nonlinear systems of differential equations. We derive some sufficient conditions which guarantee the global generalized practical uniform h-stability of time-varying perturbed systems. In addition, these results are used to study the practical h-stability of models of infectious diseases and vaccination. Mathematics Subject Classification (2010): 35B40, 37B55, 34D20, 93D15, 92D30. Keywords: Epidemic models, generalized practical uniform h-stability, Gronwall’s inequalities, Lyapunov functions.

New Type-2-Fuzzy-Logic-Based Control System for the Cessna Citation X

This paper presents a new control law based on type-2 fuzzy logic system (T2FLS) for the Cessna Citation X aircraft during cruise. This methodology combines three control systems: a T2FLS, a sliding mode control (SMC) system, and an adaptive control system. Differing from conventional controllers, this control system can deal with uncertainties and aircraft dynamics variations using the T2FLS approximator. The approximated functions were updated during the simulation iterations using adaptation laws derived from the Lyapunov theorem. The adaptability characteristic of this controller allows updating the approximated functions while the other design parameters remain unchanged. This fact simplifies the controller design process and eliminates the need to use a group of conventional controllers with different gains to control the aircraft in various flight conditions. In addition, the sliding mode control system helps to guarantee the aircraft’s stability and robustness, along with two other methodologies in the presence and absence of turbulence. The performance of this control methodology was validated with a nonlinear simulation platform developed for the Cessna Citation X business aircraft using accurate flight data derived from a Level D flight simulator at the Research Laboratory in Active Controls, Avionics and AeroServoElasticity (LARCASE).

Predefined-time sliding mode position and attitude coupling control for in-cabin robots on space stations

As space science advances, the significance of space robotics research escalates, particularly in control methodologies. Addressing the work efficiency requirements during the mission of the in-cabin robot, it is necessary to develop a fast and robust position and attitude coupling controller for in-cabin robots. This paper presents a predefined-time sliding mode position and attitude coupling control method, employing the SE(3) frame description. The first, the dynamic model of the in-cabin robot is developed within the SE(3) framework. Then, the predefined-time position and attitude coupling sliding mode controller is designed with a sliding surface based on the dynamic error model, and a hyperbolic tangent function is used to suppress system chattering. In addition, the stability of the proposed control method is rigorously confirmed using the Lyapunov theorem, ensuring that the convergence time is accurately predefined. Finally, simulation results are provided to demonstrate the effectiveness of this novel control strategy. A comparative analysis with fixed-time sliding mode control highlights its superior performance.