Finite-time Convergence Of Tracking Errors Research Articles

Adaptive Multi-Input Super Twisting Control for a Quadrotor: Singular Perturbation Approach

Singular perturbations easily result in the problem of high gain. Instead of high gain feedbacks, this work proposes an adaptive two-time-scale (TTS) super twisting control (STWC) algorithm for the attitude tracking of a singularly perturbed quadrotor system (SPQS). The bounds of uncertainties are assumed to be perturbed around their nominal values. First, a nominal STWC with slow and fast control gains is designed to achieve the finite-time attitude tracking for the nominal SPQS. The finite-time stability of the close-loop system is proven by utilizing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -dependent Lyapunov method. Then, dynamically adapted slow and fast control gains initialized at their nominal values are used for the accurate finite-time convergence of tracking errors to the region around the origin, which avoids overestimating the values of control gains. Experimental results demonstrate the performance of the proposed controller in terms of attitude stabilization and tracking, and robustness to external disturbances.

Finite-time sliding-mode tracking control of uncertain chaotic systems in the presence of an external disturbance and an input saturation

ABSTRACT The study presents a Finite-time Sliding-mode Tracking Control (FSTC) technique to achieve the trajectory tracking of an uncertain chaotic system. The feature of the proposed control strategy possesses a high level of robustness even when the uncertain chaotic system is in the presence of external disturbances and input saturation. First, a nonlinear sliding surface with a fractional-order term is constructed using tracking errors so that the finite-time convergence of tracking errors can be attained. The design of the nonlinear sliding surface, i.e. the selection of control parameters, can shape the system response. Since chattering will result from such control design, a boundary layer is adopted in the controller design to mitigate the high-frequency switching of control signal inputs. The overall system stability and finite-time convergence of the tracking error are verified using Lyapunov stability criteria and the relevant lemma. Finally, the proposed FSTC is applied to the Genesio chaotic system in the presence of uncertainty, external disturbance, and input saturation. The simulation results reveal that the proposed control scheme is indeed feasible and effective for chaotic systems control.

Integral terminal sliding-mode-based singularity-free finite-time tracking control for entry vehicle with input saturation

This paper investigates the finite-time tracking control problem for entry vehicle under input saturation and uncertainty. First, a new integral terminal sliding mode surface is devised, which not only renders fast finite-time convergence, but also presents an explicit estimate of settling time. Subsequently, a singularity-free adaptive integral terminal sliding mode control approach is exploited with application of the designed integral terminal sliding mode surface, anti-windup compensator and adaptive technique, which ensures the superior control performance. By introducing linear and fractional power feedback functions into the adaptation mechanism and anti-windup compensator, the designed controller achieves finite-time convergence of tracking errors and estimation errors, while circumventing the overestimation of control gain. Moreover, the finite-time stability is proved by employing Lyapunov theory and bi-limit homogeneity technique. The results of a Monte Carlo simulation show that the final tracking precision within 3 km of the target is improved about 40% under the developed controller.

Composite trajectory tracking of a ship-borne manipulator system based on full-order terminal sliding mode control under external disturbances and model uncertainties

This paper studies the trajectory tracking control problem of a ship-borne manipulator system (SBMS) which comprises of a robotic manipulator installed on an unmanned surface vehicle (USV). In order to deal with external disturbances and model uncertainties in the SBMS, the composite trajectory tracking control method based on full-order terminal sliding mode (FOTSM) control is proposed. The main contributions of this paper are: (1) A dynamics model of the SBMS taking into consideration of the dynamic coupling between the manipulator and the USV as well as external disturbances is developed. (2) A new type of composite trajectory tracking control strategy based on FOTSM control is designed and applied for the first time to the SBMS. (3) It is proved mathematically that the proposed control strategy can guarantee stability of the closed-loop system while achieving finite-time convergence of tracking errors. Simulation studies demonstrate the effectiveness and efficiency of the proposed control strategy.

Adaptive Finite-Time Command Filtered Controller Design for Nonlinear Systems With Output Constraints and Input Nonlinearities.

This work addresses a finite-time tracking control issue for a class of nonlinear systems with asymmetric time-varying output constraints and input nonlinearities. To guarantee the finite-time convergence of tracking errors, a novel finite-time command filtered backstepping approach is presented by using the command filtered backstepping technique, finite-time theory, and barrier Lyapunov functions. The newly proposed method can not only reduce the complexity of computation of the conventional backstepping control and compensate filtered errors caused by dynamic surface control but also can ensure that the output variables are restricted in compact bounding sets. Moreover, the proposed controller is applied to robot manipulator systems, which guarantees the practical boundedness of all the signals in the closed-loop system. Finally, the effectiveness and practicability of the developed control strategy are validated by a simulation example.

Robust trajectory tracking control for a quadrotor using recursive sliding mode control and nonlinear extended state observer

The trajectory tracking performance of a quadrotor is affected by the load variations, wind gusts, inaccurate model parameters, etc. This study proposes a novel robust trajectory tracking controller for a quadrotor via a nonlinear extended state observer and recursive sliding mode control (NESO-RSMC). First, a novel NESO is applied to observe the lumped disturbance in quadrotor dynamics, and the parameter design of the NESO does not rely on the information of the unknown lumped disturbance. Subsequently, based on NESO, the RSMC scheme is constructed in detail, which reduces chattering and ensures a fast response. In addition, the finite-time convergence of tracking errors based on the NESO-RSMC scheme was rigorously proved. The numerical simulation results verify the robustness of the proposed NESO-RSMC scheme in the case of external disturbances and model parameter uncertainties. Ultimately, the NESO-RSMC scheme was implemented on a real quadrotor, and its effectiveness was further proven by outdoor flight experiments.

Adaptive fuzzy finite-time control with prescribed performance for waverider vehicles

This paper proposes an adaptive fuzzy control method with prescribed performance for Waverider Vehicles (WVs), being able to guarantee finite-time convergence and small overshoot for tracking errors. Firstly, we design a new type of performance function that is independent of the initial error, and possess finite-time convergence and small overshoot. Then, we transform the inequality constraints on tracking errors into an unconstrained equation by introducing a transformed error. On this basis, we design a prescribed performance control (PPC) approach to limit the tracking errors within prescribed funnels utilizing the transformed error and fuzzy approximation, which ensures that satisfactory transient performance and steady-state accuracy can be guaranteed for tracking errors. Compared with the existing PPC, the improvement is to assure finite-time convergence of tracking errors with almost zero overshoot. Finally, compared simulations are given to verify the advantage.

Adaptive distributed finite-time formation control for multi-UAVs under input saturation without collisions

This paper devotes to addressing the adaptive distributed finite-time formation control problem for multi-UAVs with collision avoidance and input saturation. By combining the formation reconstruction algorithm designed in this paper, the novel adaptive finite-time control law that utilizes the nonlinear mapping technology and prescribed performance control can limit the UAVs in the specified region to avoid collisions between each other. Moreover, an improved auxiliary dynamic system governed by hyperbolic functions is constructed to address the input saturation problem, while promoting to achieve the finite-time convergence of tracking errors. Finally, the convergence of the error systems and the practicability of the control law are verified by Lyapunov stability analysis and numerical simulations.

Observer-based finite-time control for trajectory tracking of lower extremity exoskeleton

In this article, an advanced observer-based finite-time trajectory tracking controller is investigated for lower extremity exoskeleton without available joint angular velocities to improve the movement ability of dependent persons, which is robust against uncertain dynamics, human active joint torque and external disturbances. First, the Lagrange principle is applied to analyze the dynamic properties of lower extremity exoskeleton driven by artificial pneumatic muscles, and its swing phase model is established. After that, a novel finite-time extended state observer is proposed to observe the lumped disturbances and unavailable angular velocities of the lower limb exoskeleton simultaneously. Furthermore, a finite-time sliding mode controller of exoskeleton is designed based on the extended state observer, and the finite-time convergence of tracking error is rigorously demonstrated based on the Lyapunov theory. Finally, the control system simulation is established and experimental tests are conducted with a voluntary subject during flexion of wearer’s knee and hip joints, the obtained results demonstrate fast and high-precision tracking performance of the proposed approach.

Finite-time tracking control for a variable stiffness pneumatic soft bionic caudal fin

This paper introduces a finite-time tracking control method for a pneumatic soft bionic caudal fin with driving fins and variable stiffness fins. Designs, fabrications and kinematic modelling are shown for the pneumatic soft bionic caudal fin. Based on a kinematic model, a dynamic nonlinear system is established for the driving fins in the pneumatic soft bionic caudal fin. A nonlinear error feedback controller is designed for the pneumatic soft bionic caudal fin via an extended state observer based on a second-order finite-time tracking differentiator. Finite-time convergence of tracking error is proved by Lyapunov approach for the dynamic nonlinear system. Finally, experimental and simulation results are shown to verify the effectiveness for the design of the pneumatic soft bionic caudal fin and the proposed control method of the dynamic nonlinear system, respectively.

Command filtered finite-time control for nonlinear systems with state constraints and its application to TCP network

This work studies the issue of finite-time tracking control for a class of nonlinear systems with state constraints. By means of command filtered backstepping technique, finite-time theory and Barrier Lyapunov Functions (BLF), a novel finite-time command filtered backstepping approach is presented to guarantee the finite-time convergence of tracking errors. The new proposed method can not only reduce the complexity of computation of the conventional backstepping control and compensate filtered errors by Dynamic Surface Control (DSC), but also can guarantee that the state variables are restricted in compact bounding sets. Moreover, the proposed controller is designed and applied to Transmission Control Protocol/Active Queue Management (TCP/AQM) network systems, which guarantees the practical boundedness of all the signals in the closed-loop system. Finally, the effectiveness and practicability of the developed control strategy are validated by a TCP network simulation example.

Finite‐time super twisting sliding mode controller based on higher‐order sliding mode observer for real‐time trajectory tracking of a quadrotor

The main focus of this study is to develop a finite-time super-twisting sliding mode control strategy for the quadrotor based on a higher-order sliding mode observer (HOSMO). 12 state variables are required to describe the motion of the quadrotor, of which six state variables, namely the position, altitude, and orientation, are assumed to be obtained from the sensors. The remaining state variables, i.e. the linear and angular velocities, are determined using the HOSMO. Besides, the HOSMO aids in determining the unknown bounded lumped disturbances acting on the quadrotor. The output of the HOSMO is utilised for implementing the finite-time super-twisting sliding mode controller (FTSTSMC). The proposed FTSTSMC ensures finite-time convergence of tracking error with chattering attenuation. The chattering analysis for a super-twisting algorithm is presented in this work. Moreover, the overall system stability is investigated using the Lyapunov theory, and an expression for the time of convergence of the tracking and estimation error is presented. The effectiveness of the proposed methodology is established using numerical simulations and its performance is compared to a finite-time sliding mode observer coupled with a combination of proportional–integral–derivative and continuous sliding-mode controller. This is then validated in real-time using the DJI Matrice 100.

Robust Finite-Time Convergence Control Mechanism for High-Precision Tracking in a Hybrid Fluid Power Actuator

In this study, a novel multiphase converged control structure with a switching mechanism is established based on the fundamentals of the fast-terminal sliding mode. This control structure achieves an excellent performance in a nonlinear dynamic system, and its primary objective is to control the piston trajectory in an innovative electrohydraulic, pneumatic, and mechanical hybrid system. This work focuses on abrupt gain-scheduled acceleration, which has been rarely studied in the literature but has increasingly diverse high-technology applications across various industries. The greatest challenge lies in the sensitivity and instability of the system during a sudden actuator acceleration. Moreover, the system parameter uncertainties and external disturbances strongly influence the degree of control of the system. By inheriting the robustness and fast convergence properties of sliding-mode algorithms, the proposed control law not only guarantees a finite-time convergence of tracking errors to their origin but also reduces the impact of composite disturbances in an extremely rapid experimental process. The effectiveness of the proposed structure is analyzed via numerical simulations and industrial implementation.

Finite‐time trajectory tracking control for autonomous airships with uncertainties and external disturbances

This study addresses the finite-time tracking control problem for autonomous airships with uncertainties and external disturbances. Six-degree-of-freedom kinematics and dynamics equations are established by considering the model uncertainties and external disturbances. To handle the model uncertainties and external disturbances, a finite-time sliding mode disturbance observer (DOB) is designed. To have a good tracking performance, a finite-time command-filtered backstepping-supertwisting controller is proposed with DOB. By using Lyapunov theory, the finite-time convergence of tracking errors and the stability of the control method is proved. Finally, the simulation results show that the controller can track the desired trajectory well in spite of model uncertainties and external disturbances.

Attitude Tracking Control of Gravity Gradient Microsatellite in Maneuvering for Space Exploration

The problem of the finite-time tracking control strategy is studied for a gravity gradient microsatellite BUAA-SAT in maneuvering. The contributions of the work are mainly shown in two aspects: 1) the analysis of the combined forcing action of the gravity and the constant thrust; 2) a method of the adaptive finite-time sliding mode tracking control. In orbital maneuvering, the gravity gradient microsatellite under a constant thrust shows complicated dynamics. The stability and pendular motion of the gravity gradient microsatellite are determined by the combined action of the gravity and the constant thrust. The dynamics characteristics revealed in this paper are conducive to derive an expected attitude trajectory for the attitude tracking control, which not only ensures the flight safety but also reduces the energy consumption of the controller. In order to avoid the vibration and the structure deformation of the coilable mast, the microsatellite is required to track the expected attitude trajectory in a finite time. Compared with existing finite-time sliding mode control methods, a desired tracking error function with a simple polynomial form is designed to eliminate the reaching phase problem and to enable the tracking error to converge to zero in a finite time; besides, an adaptive law is presented to solve the problem of the boundary of uncertainty. The Lyapunov stability analysis is performed to show the global finite-time convergence of tracking errors. A numerical case of the gravity gradient microsatellite is studied to demonstrate the effectiveness of the proposed controller.

A two-time scale control scheme for on-orbit manipulation of large flexible module

The key issue in on-orbit manipulation is to transport large flexible modules to pre-assembly position accurately. However, the cooperative manipulation by a team of space robots can induce large vibration in the robot-beam system. This paper presents a two-time scale control scheme for vibration minimization in trajectory tracking of the flexible module. The control scheme is composed of a slow controller ensuring precise tracking of the beam's rigid motion and a fast controller to minimize the systemic vibration. Our approach to the slow controller design is based on non-singular terminal sliding mode control, with which fast and finite-time convergence of tracking errors against disturbances is obtained. The fast controller guarantees minimum vibration induced during the manipulation by applying optimal control technique. Numerical results by comparison show that the two-time scale control scheme for cooperative manipulation demonstrates higher precision and finite-time convergence in tracking the rigid motion with robustness and minimizing the vibration simultaneously.

A New Fuzzy Sliding Mode Controller with a Disturbance Estimator for Robust Vibration Control of a Semi-Active Vehicle Suspension System

This paper presents a new fuzzy sliding mode controller (FSMC) to improve control performances in the presence of uncertainties related to model errors and external disturbance (UAD). As a first step, an adaptive control law is designed using Lyapunov stability analysis. The control law can update control parameters of the FSMC with a disturbance estimator (DE) in which the closed-loop stability and finite-time convergence of tracking error are guaranteed. A solution for estimating the compensative quantity of the impact of UAD on a control system and a set of solutions are then presented in order to avoid the singular cases of the fuzzy-based function approximation, increase convergence ability, and reduce the calculating cost. Subsequently, the effectiveness of the proposed controller is verified through the investigation of vibration control performances of a semi-active vehicle suspension system featuring a magnetorheological damper (MRD). It is shown that the proposed controller can provide better control ability of vibration control with lower consumed power compared with two existing fuzzy sliding mode controllers.

Finite‐Time Sliding Mode Trajectory Tracking Control of Uncertain Mechanical Systems

AbstractThe problem of finite‐time tracking control is studied for uncertain nonlinear mechanical systems. To achieve finite‐time convergence of tracking errors, a simple linear sliding surface based on polynomial reference trajectory is proposed to enable the trajectory tracking errors to converge to zero in a finite time, which is assigned arbitrarily in advance. The sliding mode control technique is employed in the development of the finite‐time controller to guarantee the excellent robustness of the closed‐loop system. The proposed sliding mode scheme eliminates the reaching phase problem, so that the closed‐loop system always holds the invariance property to parametric uncertainties and external disturbances. Lyapunov stability analysis is performed to show the global finite‐time convergence of the tracking errors. A numerical example of a rigid spacecraft attitude tracking problem demonstrates the effectiveness of the proposed controller.

Fast trajectory tracking of electromagnetic satellite formation with actuator saturation

Electromagnetic formation flight leverages electromagnetic force to control the relative position of satellites. This new propulsion technique offers a promising alternative to traditional propellant-based spacecraft formation flying since it does not consume fuel. Due to the restriction of maximum current in coils, the available inter-satellite electromagnetic force is small, and its efficient use is an important issue. In this paper, a modified far-field model is proposed to gain better accuracy of electromagnetic force approximation. Based on this model, an adaptive terminal sliding mode control is proposed to achieve fast trajectory tracking. The given method can guarantee the finite-time convergence of tracking error in the presence of bounded disturbance, input uncertainty, and saturation. Numerical simulation results demonstrate the performance and robustness of the proposed control.

A new robust control for minirotorcraft unmanned aerial vehicles

This paper presents a new robust control based on finite-time Lyapunov stability controller and proved with backstepping method for the position and the attitude of a small rotorcraft unmanned aerial vehicle subjected to bounded uncertainties and disturbances. The dynamical motion equations are obtained by the Newton–Euler formalism. The proposed controller combines the advantage of the backstepping approach with finite-time convergence techniques to generate a control laws to guarantee the faster convergence of the state variables to their desired values in short time and compensate for the bounded disturbances. A formal proof of the closed-loop stability and finite-time convergence of tracking errors is derived using the Lyapunov function technique. Simulation results are presented to corroborate the effectiveness and the robustness of the proposed control method.