Fixed-time Convergence Research Articles

Adaptive Fault Tolerant Non-Singular Sliding Mode Control for Robotic Manipulators Based on Fixed-Time Control Law

This paper presents a fault tolerant scheme employing adaptive non-singular fixed-time terminal sliding mode control (AFxNTSM) for the application of robotic manipulators under uncertainties, external disturbances, and actuator faults. To begin, non-singular fixed-time terminal sliding mode control (FxNTSM) is put forth. This control method uses non-singular terminal sliding mode control to quickly reach fixed-time convergence, accomplish satisfactory performance in tracking, and produce non-singular and non-chatter control inputs. Then, without knowing the upper bounds beforehand, AFxNTSM is used as a reliable fault tolerant control (FTC) to estimate actuator faults and unknown dynamics. The fixed-time stability of the closed-loop system is established by the theory of Lyapunov analysis. The computer simulation results of the position tracking, control inputs, and adaptive parameters are presented to verify and illustrate the performance of the proposed strategy.

Fixed-time observer-based control of DFIG-based wind energy conversion systems for maximum power extraction

To extract maximum power from the available wind energy, it is necessary that the doubly-fed induction generator (DFIG) based wind energy conversion system (WECS) outputs that maximum power at each instant of time for different wind speeds. This can be achieved by controlling the rotor voltage of the DFIG. To make it possible, a novel maximum power point tracking (MPPT) controller in combination with state and disturbance observers and fixed-time stability notion is proposed in this paper. The sliding mode control (SMC) method is used to design the observer-based controller and ensure the robustness. The sliding mode state observer is designed while only the measurement of the stator reactive power and rotor speed is required. The modelling uncertainties and external disturbances are estimated by the sliding mode disturbance observer without information about their upper bounds in advance. Then, the combinatorial effect of uncertainties and disturbances are fully compensated by the designed controller. The fixed-time convergence issue is addressed where the bound on the settling time is user-defined using design parameters regardless of initial conditions. The elimination of chattering issue is considered in the design of the SMC laws. The stability analysis of the closed-loop system is obtained via the Lyapunov stability theory. The validity and robustness of the proposed method is tested in Simulink/MATLAB for a DFIG under two different scenarios of wind speed modelling. The proposed fixed-time method is also compared with a classical finite-time method. Comparing the results demonstrate that the proposed method outperforms the other method.

Fuzzy Adaptive Output-Feedback Constrained Trajectory Tracking Control for HFVs With Fixed-Time Convergence

This article proposes an output-feedback fixed-time trajectory tracking control methodology for hypersonic flight vehicles subject to asymmetric output constraints. In contrast to the state of the art, the most distinguishing feature of our control design lies in avoiding using conventional recursive design methods (e.g., backstepping technique, dynamic surface control, etc.) and in not relying on full-state availability. In the velocity control loop, an asymmetric integral barrier Lyapunov function is adopted to confine velocity variable within a well-defined compact set all the time. In the altitude control loop, after utilizing its cascaded property and proposing a novel scaling function, the original constrained system is transformed to an unconstrained one, which facilitates the control design and stability analysis. Moreover, the proposed control algorithm only involves one fuzzy logic approximator as well as one fixed-time differentiator in the transformed system and guarantees that the tracking errors of velocity and altitude converge into the user-defined residual sets within fixed time. Several comparative simulations have been conducted to highlight the superiorities of the developed method.

On Biased Harmonic Signal Estimation: Application to Electric Power Grid Monitoring

Parametric estimation of a biased harmonic signal is a significant technical challenge for many engineering applications. Such a problem is particularly important for electric utility grid-connected power electronic converters. This article utilizes a linear regression model of the signal to solve this interesting practical problem. A continuous-time dynamic regressor extension and mixing (DREM) based approach is then applied for parameter estimation. For practical implementation, continuous-time estimators are discretized using implicit and explicit Euler methods. We then prove that the implicit discretization can achieve fixed-time convergence for the unknown frequencies estimation. Thanks to the estimated frequencies, another DREM-based linear regression problem is solved for the parameter estimation purpose. The overall order of the proposed technique is the same as the number of unknown parameters, making the estimator suitable for real-time implementation in embedded devices. Theoretical results are validated through extensive comparative experimental studies.

Fast Nonsingular Fixed-Time Fuzzy Fault-Tolerant Control for HFVs With Guaranteed Time-Varying Flight State Constraints

This article proposes a fuzzy adaptive fault-tolerant control strategy solving the fast fixed-time constrained tracking problem for hypersonic flight vehicles. To handle the actuator faults, a two-step design method is first proposed that only needs to estimate the upper and lower bounds of actuator parameters, under which the adverse influences of actuator faults (e.g., lock-in-place, loss of effectiveness, etc.) have been compensated via an adaptive fashion instead of conventional robust ways. In comparison with the state-of-the-art, a new fixed-time corollary that has proved a smaller upper bound of convergence time under the same condition of conventional fixed-time stability is presented. To avoid singularity issues often encountered in fixed-time designs, a piecewise but differentiable switching control laws whose continuity and differentiability are guaranteed everywhere through an appropriate design is introduced. Such a design not only preserves the continuity and differentiability of virtual and actual control laws, but also ensures the continuity of their time derivatives. Fuzzy logic systems are exploited to tackle continuous unknown dynamics and asymmetric time-varying barrier functions are utilized to confine flight states within some predefined compact sets all the time provided their initial conditions remain therein. This property is of great significance in dealing with the challenge that the operating regions of the flight state variables are asymmetric and time-varying in practice, especially when executing various flight missions. Comparative simulations have been performed to validate the effectiveness of the proposed control scheme in terms of fixed-time convergence, smoothness, and time-varying constraints satisfaction.

Barrier Lyapunov function based fixed-time control for automatic carrier landing with disturbances

This paper studies the automatic carrier landing controller design problem with output constraint and fixed-time convergence requirements. The proposed flight controller is separated into guidance, attitude control, and approach power compensation system. In each submodule, universal barrier Lyapunov function and fixed-time control scheme are applied in the controller design. In addition, adaptive estimation method is incorporated to improve the robustness to the external disturbances. The proposed controller ensures the position tracking error, attitude tracking error, and angle of attack tracking error not violate the predesigned boundaries, while the tracking errors will converge to a small set around zero in fixed-time. The output constraint and fixed-time convergence properties of the proposed controller improve the landing security to a narrow flight deck in a limited operation time. The simulation results verify the proposed controller in automatic carrier landing mission.

Low-complexity observer–based output-feedback tracking control for a class of nonlinear lower-triangular systems

The tracking control problem for a group of nonlinear lower-triangular systems with multiple uncertainties is investigated in this work. Wherein, a novel performance constraint is first constructed to guarantee the fixed-time convergence of the output tracking error. Subsequently, a linear extended high-gain observer is employed to estimate the system uncertainties including the unmeasured states and the external disturbances. Based on the observer estimations, a novel output-feedback tracking control approach is formulated via using the backstepping technique, to ensure the boundedness of the closed-loop system. Compared with the existing works, the primary advantages of the proposed design are that (1) the problem of "explosion of terms" is avoided by eliminating the need for the derivative of the virtual control signals and (2) without the use of extra auxiliary techniques, the fixed-time convergence can be guaranteed, where the convergence time is independent of the system states and initial conditions. Then, the results about system stability are proved by the theoretical analysis. Moreover, an extension of the proposed approach to multi-input multi-output systems is deduced in this paper to further present its versatility. Finally, two groups of numerical examples are performed to validate the effectiveness of the proposed controller.

Adaptive Neural Network-Based Fixed-Time Tracking Controller for Disabilities Exoskeleton Wheelchair Robotic System

In this paper, an adaptive neural network approach is developed based on the integral nonsingular terminal sliding mode control method, with the aim of fixed-time position tracking control of a wheelchair upper-limb exoskeleton robot system under external disturbance. The dynamical equation of the upper-limb exoskeleton robot system is obtained using a free and typical model of the robotic manipulator. Afterward, the position tracking error between the actual and desired values of the upper-limb exoskeleton robot system is defined. Then, the integral nonsingular terminal sliding surface based on tracking error is proposed for fixed-time convergence of the tracking error. Furthermore, the adaptive neural network procedure is proposed to compensate for the external disturbance which exists in the upper-limb exoskeleton robotic system. Finally, to demonstrate the effectiveness of the proposed method, simulation results using MATLAB/Simulink are provided.

Fuzzy adaptive fixed-time fault-tolerant attitude tracking control for tailless flying wing aircrafts

This work investigates the fault-tolerant attitude tracking control problem for the tailless flying wing aircraft in the presence of an external disturbance, uncertain parameters, and time-varying actuator failures. First, the external disturbance, uncertainty, and time-varying actuator failures are skillfully transformed into a lumped uncertainty, followed by an approximator constructed by employing an interval type-2 fuzzy logic system (IT2-FLS). On this basis, a fixed-time IT2 fuzzy fault-tolerant control scheme is proposed, which can guarantee that all the signals of the closed-loop system have fixed-time convergence. The desired control torque is allocated to each control surface through a recurrent neural network (RNN) based control allocation algorithm by considering the magnitude and rate constraints of the actuators. Finally, the two representative simulations have been conducted to validate the effectiveness and superiority of the proposed control scheme and control allocation algorithm.

A fixed-time convergent and noise-tolerant zeroing neural network for online solution of time-varying matrix inversion

As a common mathematical operation, the time-varying matrix inversion (TVMI) is frequently arisen in many complex problems. It has been proved in a large number of studies that the zeroing neural network (ZNN) is a reliable tool to solve TVMI problems. However, the previous reported conventional zeroing neural network (CZNN) models only achieve undesirable exponential convergence or finite-time convergence, and they are difficult to be popularized in practical applications. Therefore, by introducing a novel activation function (NAF), a fixed-time convergence and noise-tolerant zeroing neural network (FTCNTZNN) model for solving TVMI problems is proposed in this paper. Different from existing CZNN models, the proposed FTCNTZNN model possesses both fixed-time convergence and anti-noise properties for TVMI solving. To comprehensively show the superiority of the proposed FTCNTZNN model for solving TVMI, a lot of theoretical demonstrations and comparative simulation experiments have been provided in this work. Firstly, the fixed-time convergence and robustness of the proposed FTCNTZNN model are verified by detailed theoretical analysis in noiseless and noisy environment, respectively. Then, the proposed FTCNTZNN model is applied to solve two different TVMI problems in a variety of situations. In addition, a successful manipulator trajectory tracking application using the proposed FTCNTZNN model in noisy environment further validates its practicability. Both of theoretical analysis and experimental results convincingly verify that the proposed FTCNTZNN model can effectively solve TVMI problems in noiseless and noisy environment.

Accelerated convergence with improved robustness for discrete-time parameter estimation

The dynamic regressor extension and mixing (DREM) method provides a fixed-time converging parameter estimator for a persistently excited regressor under bounded measurement noises. This note aims to develop this approach for cases with weaker excitation and regressor constraints. Several nonlinear estimation schemes with fixed-time convergence rates and improved measurement noise robustness properties are proposed here.

Finite-Time and Fixed-Time Attractiveness for Nonlinear Impulsive Systems

In this article, the finite-time and fixed-time attractiveness problems are investigated for nonlinear impulsive systems. The aim of the proposed problems is to establish some general Lyapunov theorems and settling-time estimates for finite-time and fixed-time attractiveness of nonlinear impulsive systems by analysis technics. Moreover, some comparisons with the existing results are also given. The classical finite-time and fixed-time convergence theorems for nonlinear impulse-free systems are well extended to nonlinear impulsive systems by our results. Furthermore, the existing results of finite-time and fixed-time attractiveness for nonlinear impulsive systems are well improved. Finally, some simulations are utilized to illustrate the usefulness of the theoretical analysis.

Fixed-Time Filtered Adaptive Parameter Estimation and Attitude Control for Quadrotor UAVs

In this article, a fixed-time filtered adaptive parameter estimation and control scheme is proposed for attitude tracking of quadrotor unmanned aerial vehicles. A nonsingular fixed-time sliding mode surface is constructed to avoid the existence of the singularity issue resulted from the differentiation of the sliding mode variable. Through designing auxiliary filtered matrices, a fixed-time parameter update law is developed to achieve the fixed-time parameter convergence. Then, an adaptive neural controller is designed to ensure the fixed-time convergence of attitude tracking errors in the presence of the model uncertainties. Instead of using any piecewise continuous functions, the possible singularity issue in the conventional fixed-time controller design can be overcome by constructing auxiliary functions, and thus the fixed time stability analysis becomes more concise. The efficiency of the presented scheme is validated through experiments on a practical Quanser quadrotor platform.

Tracking control for small autonomous underwater vehicles in the Trans-Atlantic Geotraverse hydrothermal field based on the modeling trajectory

• A multi-parameter high-order extended state observer is designed to estimate lumped uncertainties. • A fixed-time fault-tolerant adaptive trajectory tracking controller is presented. • The desired trajectory is constructed by interpolating the path. • The simulation topography is built according to the TAG active mound. This paper investigates the fixed-time trajectory tracking problem of small autonomous underwater vehicles (S-AUVs) in the Trans-Atlantic Geotraverse (TAG) active mound with ocean current, unknown disturbances, model uncertainties, actuator faults, and input saturations. First, the effect of the near-bottom environment of the mound on S-AUVs is analyzed in detail from three issues. Without the upper bound and gradient of lumped uncertainties, a high-order adaptive extended state observer (ESO) is designed. Then, a continuous fixed-time sliding mode manifold is applied to obtain fast convergence performance. In order to realize the fixed-time convergence of tracking errors, an adaptive fault-tolerant trajectory tracking control law with an auxiliary dynamic system (ADS) is proposed. In addition, the simulated topography of the TAG mound is constructed and the trajectory based on path points is modeled by cubic spline interpolation. The effectiveness and superiority of the proposed algorithm are validated by comprehensive simulations.

Fixed-time neural network control of a robotic manipulator with input deadzone

In this paper, a fixed-time control method is proposed for an uncertain robotic system with actuator saturation and constraints that occur a period of time after the system operation. A model-based control and a neural network-based learning approach are proposed under the framework of fixed-time convergence, respectively. We use neural networks to handle the uncertainty, and design an adaptive law driven by approximation errors to compensate the input deadzone. In addition, a new structure of stabilizing function combining with an error shifting function is introduced to demonstrate the robotic system stability and the boundedness of all error signals. It is proved that all the tracking errors converge into the compact sets near zero in fixed-time according to the Lyapunov stability theory. Simulations on a two-joint robot manipulator and experiments on a six-joint robot manipulator verified the effectiveness of the proposed fixed-time control algorithm.

A novel activation function based recurrent neural networks and their applications on sentiment classification and dynamic problems solving.

In this paper, a nonlinear activation function (NAF) is proposed to constructed three recurrent neural network (RNN) models (Simple RNN (SRNN) model, Long Short-term Memory (LSTM) model and Gated Recurrent Unit (GRU) model) for sentiment classification. The Internet Movie Database (IMDB) sentiment classification experiment results demonstrate that the three RNN models using the NAF achieve better accuracy and lower loss values compared with other commonly used activation functions (AF), such as ReLU, SELU etc. Moreover, in terms of dynamic problems solving, a fixed-time convergent recurrent neural network (FTCRNN) model with the NAF is constructed. Additionally, the fixed-time convergence property of the FTCRNN model is strictly validated and the upper bound convergence time formula of the FTCRNN model is obtained. Furthermore, the numerical simulation results of dynamic Sylvester equation (DSE) solving using the FTCRNN model indicate that the neural state solutions of the FTCRNN model quickly converge to the theoretical solutions of DSE problems whether there are noises or not. Ultimately, the FTCRNN model is also utilized to realize trajectory tracking of robot manipulator and electric circuit currents computation for the further validation of its accurateness and robustness, and the corresponding results further validate its superior performance and widespread applicability.

Adaptive nonsingular fixed-time sliding mode control for manipulator systems’ trajectory tracking

In this paper, an adaptive fixed-time controller is raised for the manipulator system with uncertain disturbances to boost the rate and precision of its trajectory tracking and solve the convergence time dependence on the system's initial conditions. First, a nonsingular fixed-time sliding mode (SM) surface and a reaching law based on an arctangent function are constructed to enhance the control scheme performance. Second, the upper bound is difficult to obtain because of the uncertainty of the disturbance. The disturbance upper bound is estimated by adaptive techniques, which do not require a priori knowledge about the upper bound and effectively inhibit the effect of disturbance on the system. Finally, the fixed-time convergence of the states is analyzed by rigorous theoretical proof, and the validity of the presented control scheme is demonstrated by simulation.

Fuzzy Adaptive Prescribed Performance Fault-Tolerant Control for HFVs with Fixed-Time Convergence Guarantee

A new fixed-time fuzzy adaptive fault-tolerant control methodology is proposed for the longitudinal dynamics of hypersonic flight vehicles (HFVs) in the presence of actuator faults, uncertain dynamics, and external disturbances. In contrast with the conventional fixed-time control schemes that typically contain the fractional powers of errors in their designs, this work develops a low-complexity control structure in the sense of removing the dependence on the need of abovementioned fractional power terms by means of prescribed performance control (PPC) method. Different from the most existing PPC approaches where the initial conditions of tracking errors are required to be known, the newly proposed prescribed performance function (PPF) can relax such restrictions through choosing properly small initial values of PPF. Fuzzy logic systems (FLSs) are employed to handle unknown dynamics, and minimal learning parameter (MLP) technique is incorporated into the design for the purpose of alleviating computation burden. Closed-loop stability is rigorously proved via Lyapunov stability theory, and simulation results are eventually given to validate the effectiveness of the proposed control strategy.

Formation Control of Unmanned Surface Vehicles Using Fixed-Time Non-Singular Terminal Sliding Mode Strategy

Unmanned surface vehicles (USVs) have been widely applied in the fields of marine hydrological exploration, marine resource exploration, area search, target detection, and military operations. In order to meet the demand of a complex ocean environment, USVs are frequently grouped together to improve the reliability of mission accomplishment. In this paper, a fixed-time control strategy, combined with a non-singular terminal sliding mode, is proposed for the formation control of USVs under complex external disturbances and system uncertainties. The main contributions of this paper are: (1) the leader–follower formation control framework is divided into a tracking control subsystem and a formation control subsystem. A new fixed-time non-singular terminal sliding mode (FTNTSM) strategy is developed for the tracking control subsystem, which dramatically increases the convergence rate and ensures closed-loop fixed-time stability; (2) a finite-time uncertain observer (FUO) is designed to observe lumped uncertainty items, which greatly increase the stability and robustness of the formation system; (3) the FUO-based fixed-time formation control (FUOFT-FC) strategy is designed for the formation control subsystem, which ensures the fast and stable formation of USVs. Fixed-time convergence of the formation system is established by Lyapunov stability analysis. Rigorous simulation and comparative studies demonstrate that the proposed method is superior to the state-of-the-art methods.

Neuro-adaptive fixed-time control with novel command filter design for nonlinear systems with input dead-zone

This paper designs novel command filters (CFs) and a singularity-free controller with fixed-time performance for strict feedback nonlinear systems with input dead-zone. Compared with traditional filter designs, such as dynamic surface control and CFs, the novel CFs designed in this paper can guarantee fixed-time convergence of filter errors and compensating signals. Moreover, the controller design excludes the singularities of control signals, which might result in severe oscillation and usually appear in the conventional fixed-time controller design. To compensate the input dead-zone, a compensation filter is constructed, where controller design and stability analysis can be simplified. In the end, a simulation example is given to verify the effectiveness of the theoretical results.