Lyapunov Theorem Research Articles

Distributed Cooperative Fault-Tolerant Control of High-Speed Trains With Input Saturation and Actuator Faults

In this work, a novel distributed fault-tolerant control strategy is proposed for cooperative operation of high-speed trains (HSTs) subject to unmodeled dynamics, time-varying external disturbances, input saturation and actuator partial failures. The cooperative control of HSTs is first transformed to the consensus problem of a multi-agent system. Then, the distributed control laws that are equipped with a command filter and an adaptive neural network are developed to achieve the consensus task among the HSTs with predefined displacement and speed profiles. Further, an auxiliary dynamical system is introduced to compensate for the influence of input saturation. Moreover, an adaptive failure compensation strategy is designed to deal with the unknown and time-varying actuator faults. The convergence of the proposed controller is analyzed rigorously by applying the Lyapunov theorem, and the effectiveness is demonstrated by numerical simulations.

Convergence of stochastic approximation via martingale and converse Lyapunov methods

In this paper, we study the almost sure boundedness and the convergence of the stochastic approximation (SA) algorithm. At present, most available convergence proofs are based on the ODE method, and the almost sure boundedness of the iterations is an assumption and not a conclusion. In Borkar and Meyn (SIAM J Control Optim 38:447–469, 2000), it is shown that if the ODE has only one globally attractive equilibrium, then under additional assumptions, the iterations are bounded almost surely, and the SA algorithm converges to the desired solution. Our objective in the present paper is to provide an alternate proof of the above, based on martingale methods, which are simpler and less technical than those based on the ODE method. As a prelude, we prove a new sufficient condition for the global asymptotic stability of an ODE. Next we prove a “converse” Lyapunov theorem on the existence of a suitable Lyapunov function with a globally bounded Hessian, for a globally exponentially stable system. Both theorems are of independent interest to researchers in stability theory. Then, using these results, we provide sufficient conditions for the almost sure boundedness and the convergence of the SA algorithm. We show through examples that our theory covers some situations that are not covered by currently known results, specifically Borkar and Meyn (2000).

Integrated sliding mode intelligent fractional-order backstepping control of warp tension based on fixed-time extended state observer

The purpose of this study is to develop an integral sliding mode intelligent fractional-order backstepping controller based on a fixed-time extended state observer (ISMIFOBC-ESO) to control warp tension. First, the time-varying mathematical model of the let-off system was studied and converted into the state equation. Due to the parameter uncertainty and healed motion of the let-off system, the total external disturbance of the system is unknown. We designed a fixed-time extended state observer to deal with this situation, the estimated total interference is updated into the control algorithm in real time to reduce the tracking error. The nonlinear integral sliding surface is designed and integrated into the backstepping control to ensure the finite time convergence and stability of the algorithm. The fractional-order theory with more degrees of freedom is introduced further to enhance the control accuracy and robustness of the algorithm, the stability is proved by the Lyapunov theorem and the control law is designed. The simulation experiment verifies the effectiveness of the designed scheme compared with the other two controllers in the case of sinusoidal-like signals and step signals.

An adaptive robust backstepping improved control scheme for mobile manipulators robot

In this study, an approach with the position control scheme is proposed for the mobile manipulator robot, by applying the adaptive strategies with the backstepping control technique. In the proposed backstepping controller, the fixed problem of main control parameters has been improved by the adaptive self-updating process. In addition, this control method requires no prior knowledge of the control system, such as the unknown dynamics and disturbances. And, to relax uncertainties, an adaptive robust controller is also designed as a compensator for the main backstepping controller. Moreover, all adaptive updating algorithms are designed based on the Lyapunov theorem so that the proposed control system performances are guaranteed. The advantage of the improved control system will be evaluated via simulation results for the 3 - DOF MMR system.

Normal forms, invariant manifolds and Lyapunov theorems

<abstract><p>We present an approach to Lyapunov theorems about a center for germs of analytic vector fields based on the Poincaré–Dulac and Birkhoff normal forms. Besides new proofs of three Lyapunov theorems, we prove their generalization: if the Poincaré–Dulac normal form indicates the existence of a family of periodic solutions, then such a family really exists. We also present new proofs of Weinstein and Moser theorems about lower bounds for the number of families of periodic solutions; here, besides the normal forms, some topological tools are used, i.e., the Poincaré–Hopf formula and the Lusternik–Schnirelmann category on weighted projective spaces.</p></abstract>

A Novel Neural Network-Based Robust Adaptive Formation Control for Cooperative Transport of a Payload Using Two Underactuated Quadcopters

Designing a controller for the cooperative transport of a payload using quadcopter-type unmanned aerial vehicles (UAVs) is a very challenging task in control theory because these vehicles are underactuated mechanical systems. This paper presents a novel robust adaptive formation control design for the cooperative transport of a suspended payload by ropes using two underactuated quadcopters in the presence of external disturbances and parametric uncertainties. The structure of the proposed controller is divided into two subsystems: fully actuated and underactuated. An integral sliding mode adaptive control strategy is proposed for the fully actuated subsystem, and for the underactuated subsystem, an adaptive control strategy based on the combination of Backstepping and sliding mode is proposed. Then, the control parameters of the sliding surfaces of both control subsystems are adaptively tuned by a neural network. In addition, to improve the robustness of the proposed controller, a disturbance observer is incorporated to estimate and compensate the lumped disturbances. The asymptotic stability of the cooperative transport system is verified with the Lyapunov theorem. Finally, numerical simulations are performed in MATLAB/Simulink environment, and the results show that the proposed controller successfully transports the payload safely and without oscillations. Moreover, the desired formation pattern is maintained throughout the flight task, even with external disturbances and parametric uncertainties.

Dynamics Analysis of Cannibalistic Model With Density Dependence in Mature Stage

Based on considering mature individuals with density dependence, a two-stage structure model of egg-mature individuals with cannibalism is established. The dynamic behavior of the equilibrium of the model is discussed from two aspects: when no cannibalism exists, the global asymptotic stability of the equilibria is proved by the Lyapunov theorem; when cannibalism exists, it makes the model have saddle-node bifurcation by using the center manifold theorem. By constructing Dulac function, there is no limit cycle in the two-dimensional autonomous system, therefore, the global stability of the equilibria is obtained. Finally, the theoretical results are verified by numerical simulation.

Application of Adaptive Integral Robust Controller in Laser Galvanometer Motor

High-speed motion control of the galvanometer has become a very important part of laser processing. Due to the parameter uncertainty of the galvanometer itself, non-linearity can affect the friction compensation effect. In this paper, we propose to reduce the steady-state tracking error of the galvanometer system by means of an integral control term, while estimating the friction behavior using a nonlinear modified LuGre observer. Then an adaptive integral robust control scheme is used to eliminate the effects of nonlinear friction and external disturbances. And the stability of the galvanometer system is proved by the Lyapunov theorem. Simulation verification by MATLAB shows that the control scheme proposed in this paper can accurately follow the system commands. And the controller can effectively suppress the effects of unknown disturbances and parameter uncertainties compared with traditional control strategies. The simulation results show that the proposed control scheme significantly improves the response speed and tracking accuracy of the galvanometer system.

Mild solutions, variation of constants formula, and linearized stability for delay differential equations

The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré–Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.

ISS and iISS for discontinuously time-varying time-delay systems

Novel sufficient conditions are given for input-to-state stability (ISS) to imply integral-ISS (iISS) of time-varying time-delay systems. These conditions are more general and less restrictive than existing ones. Outstanding points of generality are that neither continuity with respect to time nor Lipschitz continuity with respect to the input is required from the function defining the system dynamics. Increased generality is achieved via the analysis of properties of solutions, without requiring converse Lyapunov theorems.

On the Converse Safety Problem for Differential Inclusions: Solutions, Regularity, and Time-Varying Barrier Functions

This article presents converse theorems for safety in terms of barrier functions for unconstrained continuous-time systems modeled as differential inclusions. Via a counterexample, we show the lack of existence of autonomous and continuous barrier functions certifying safety for a nonlinear system that is not only safe but also has a smooth right-hand side. Guided by converse Lyapunov theorems for (nonasymptotic) stability, time-varying barrier functions and appropriate infinitesimal conditions are shown to be both necessary as well as sufficient under mild regularity conditions on the right-hand side of the system. More precisely, we propose a general construction of a time-varying barrier function in terms of a marginal function involving the finite-horizon reachable set. Using techniques from set-valued and nonsmooth analysis, we show that such a function guarantees safety when the system is safe. Furthermore, we show that the proposed barrier function construction inherits the regularity properties of the proposed reachable set. In addition, when the system is safe and smooth, we build upon the constructed barrier function to show the existence of a smooth barrier function guaranteeing safety. Comparisons and relationships to results in the literature are also presented.

Kontrola mašine za čišćenje ulica sa bočnom četkom za različite površine puteva koristeći PID i kontrolere kliznog režima

This paper examines the side brush control technologies for a novelty semi-autonomous road sweeper design. This study proposes a side brush structure and offers a brush control solution to improve working efficiency and reduce abrasive brush. For the mechanical system using a parallelogram mechanism, the direction of movement when raising and lowering the brush is always parallel to the road surface. The modeling of the side brush mechanism shows that this is a nonlinear system. Therefore, the Sliding Mode Control(SMC) was proposed and established from the dynamics equation. The Lyapunov theorem demonstrates its stability. Besides, we also consider the proportional-integral-derivative (PID) controller to evaluate the responsiveness of the linear controller for a nonlinear system. Finally, the parameters of the controllers are optimized by a genetic algorithm to consider the response of the sliding mode control compared to the PID controller to control the road sweeper side brush with different references.

Autonomous Leader–Follower Formation of Vehicular Robots Using the Lyapunov Method

This paper focuses on the control of two self-driven vehicles (leader–follower) in a multi-obstacle environment, while maintaining formation. The acceleration-based control input design governs the overall movement and control of the rovers. This is accomplished through the application of APF functions that support the leader robot to reach the desired target while avoiding obstacles and maintaining formation. The Lyapunov theorem was used for the control design of the leader and follower vehicles. An effective mathematical model was designed and run through the MATLAB software for simulation verification. The simulation results obtained illustrate the behavior of the leader–follower vehicles with respect to the controllers designed. Therefore, this paper looks at the efficiency of the vehicles to converge at a predefined target, from random points in a predefined workspace, while avoiding fixed and moving obstacles. The technique may be applied in transportation and defense sectors where environments are a risk prone to human health or safety.

Observer‐based single‐network incremental adaptive dynamic programming for fault‐tolerant control of nonlinear systems with actuator faults

SummaryA model‐free incremental adaptive fault‐tolerant control (FTC) scheme is proposed for a class of nonlinear systems with actuator faults. To deal with actuator faults and guarantee the approximate optimal performance of the nominal nonlinear system without any prior knowledge of system dynamics, a single‐network incremental adaptive dynamic programming (SIADP) algorithm based on incremental neural network observer is developed to design an active fault‐tolerant control (AFTC) policy. An approximate linear time‐varying system is obtained by incremental nonlinear technique, in which the relevant matrix parameters are identified by recursive least square estimation. Then, a SIADP algorithm‐based fault‐tolerant controller is developed. Based on the redundancy characteristic and function of actuators, a grouping scheme of actuators is introduced. An incremental neural network observer is designed to approximate the actuator faults. The novel SIADP scheme is constructed with a simplified single critic neural network to shorten the learning time and decrease the computational burden in the control process, in which the norm of the weight estimations of critic neural network is updated. Moreover, based on the Lyapunov theorem, the uniformly ultimately bounded stability of the closed‐loop incremental system is proved. Finally, simulations are given to verify the effectiveness of the proposed FTC scheme.

Practically Robust Fixed-Time Convergent Sliding Mode Control for Underactuated Aerial Flexible JointRobots Manipulators

The control of an aerial flexible joint robot (FJR) manipulator system with underactuation is a difficult task due to unavoidable factors, including, coupling, underactuation, nonlinearities, unmodeled uncertainties, and unpredictable external disturbances. To mitigate those issues, a new robust fixed-time sliding mode control (FxTSMC) is proposed by using a fixed-time sliding mode observer (FxTSMO) for the trajectory tracking problem of the FJR attached to the drones system. First, the underactuated FJR is comprehensively modeled and converted to a canonical model by employing two state transformations for ease of the control design. Then, based on the availability of the measured states, a cascaded FxTSMO (CFxTSMO) is constructed to estimate the unmeasurable variables and lumped disturbances simultaneously in fixed-time, and to effectively reduce the estimation noise. Finally, the FxTSMC scheme for a high-order underactuated FJR system is designed to guarantee that the system tracking error approaches to zero within a fixed-time that is independent of the initial conditions. The fixed-time stability of the closed-loop system of the FJR dynamics is mathematically proven by the Lyapunov theorem. Simulation investigations and hardware tests are performed to demonstrate the efficiency of the proposed controller scheme. Furthermore, the control technique developed in this research could be implemented to the various underactuated mechanical systems (UMSs), like drones, in a promising way.

Lyapunov stability theorems for $$\psi $$-Caputo derivative systems

We introduce the stability concepts for \(\psi \)-Caputo derivative systems in the sense of Lyapunov’s idea. Two new Lyapunov theorems are formulated for the stability analysis of such systems. Many new inequalities that involve \(\psi \)-Caputo derivative are developed for the application of the proposed Lyapunov theorems. Examples are provided and it is shown that the proposed theorems are very useful for predicting the limiting behavior of the trajectory of such systems starting from any initial condition imposed at any initial time on the real axis.

Sliding mode speed control of permanent magnet synchronous motor based on improved reaching law

PurposeThis paper aims to present a sliding mode control method based on disturbance observer (DO) for improving the reaching law of permanent magnet synchronous motor (PMSM).Design/methodology/approachAiming at the insufficiency of the traditional exponential reaching law used in sliding mode variable structure control, an exponential reaching law related to the speed error is proposed. The improved exponential reaching law can adaptively adjust the size of the constant velocity term in the reaching law according to the size of the speed error, so as to adaptively adjust the speed of the system approaching the sliding mode surface to overcome the control deviation and improve the dynamic and steady state performance. To improve the anti-interference ability of the system, a DO is proposed to observe the external disturbance of the system, and the observed value is used to compensate the system. The stability of the system is analyzed by Lyapunov theorem. The effectiveness of this method is proved by simulation and experiment.FindingsSimulation and experiment show that the proposed method has the advantages of fast response and strong anti-interference ability.Research limitations/implicationsThe proposed method cannot observe the disturbance caused by the change of internal parameters of the system.Originality/valueA sliding mode control method for PMSM is proposed, which has good control performance. The proposed method can effectively suppress chattering, ensure fast response speed and have strong anti-interference ability. The effectiveness of the algorithm is verified by simulation and experiment.

Multicluster Consensus for Large-Scale Heterogenous Manned/Unmanned Aerial Team With Random Link Failure via Pinning Control

This brief investigates the multi-cluster consensus for heterogenous manned/unmanned aerial system with random link failure via pinning control method. Each cluster is composed of first-order and second-order dynamic models. A distributed control protocol is proposed to construct the weakly connected topology with restricted communication load in large-scale heterogenous aerial team. Based on the graph theory, matrix analysis and Lyapunov theorem, sufficient consensus conditions are derived. Numerical examples validate the theoretic results.

Automatic Landing for Carrier-Based Aircraft Under the Conditions of Deck Motion and Carrier Airwake Disturbances

This article studies the automatic landing problem for the carrier-based aircraft in the presence of deck motion and carrier airwake disturbance. First, an automatic regressive model based prediction algorithm is proposed to generate the motion information of the deck, which in turn is used to generate corrections in the commands of the aircraft's heading and flight path to guarantee landing accuracy. Then, a fast fixed-time stable system (FFSS) is proposed. Based on the FFSS, the sliding-mode-based command differentiators are designed to estimate the first-order derivatives of the reference commands in finite time. Moreover, using the modified reaching law, the incremental sliding mode control (ISMC) is presented to design the landing guidance and control system to provide fast and accurate flight control to touchdown, where the airwake disturbances are compensated by the designed adaptive super twisting extended state observers (ASTESOs). Besides, to improve the stability during the final approach, an approach power compensation subsystem maintaining the angle of attack is proposed using the ISMC and ASTESO. The stability of the closed-loop system is analyzed using the Lyapunov theorem. Finally, comparative simulation results demonstrate the effectiveness of the proposed control scheme over state-of-the-art methods on the landing accuracy and robustness.

Robust Adaptive Neural Control for Wing-Sail-Assisted Vehicle via the Multiport Event-Triggered Approach.

This article presents a robust adaptive neural control algorithm for the wing-sail-assisted vehicle to track the desired waypoint-based route, where the event-triggered mechanism is with the multiport form. The main features of the proposed algorithm are three-fold: 1) the communication burden, in the channel from the sensor to the controller as well as the actuator, has been reduced for the merits of the multiport event-triggered approach. The feedback error signals and the control input will be updated only on the event-triggered time point; 2) for the wing-sail-assisted vehicle, the thrust force is provided by devices with the propeller and the sail. From this consideration, the proper sail force compensation is derived on the basis of information about the current heading angle and the wind direction. The corresponding control law can guarantee the energy-saving for the propeller; and 3) in the algorithm, the system uncertainties are remodeled by the neural-network approximator. Furthermore, by fusion of the robust neural damping and dynamic surface control (DSC) techniques, the corresponding gain-related adaptive law is developed to address constraints of the gain uncertainty and the environmental disturbances. Through the Lyapunov theorem, all signals of the closed-loop control system have been proved to be with the semiglobal uniform ultimate bounded (SGUUB) stability, including the triggered time point and the intermediate triggered interval. Finally, the numerical simulation and the practical experiment are illustrated to verify the effectiveness of the proposed strategy.