Finite-time Trajectory Research Articles

Adaptive Finite-Time Trajectory Tracking Event-Triggered Control Scheme for Underactuated Surface Vessels Subject to Input Saturation

In this paper, the trajectory tracking problem of underactuated surface vessels (USVs) that is subject to input saturation, unmodeled dynamics, and marine environmental disturbances is investigated. Firstly, USVs underactuation problem is handled by constructing a coordinate transformation. An auxiliary dynamic system is adopted to deal with input saturation. The adaptive law is presented to online update the upper bounds of the composite disturbances. The event-triggering condition is introduced to reduce drastically control execution frequency and the transmission load. By combining the auxiliary dynamic system, an adaptive technique, a first-order command filtered technique and an event-triggered condition, a novel adaptive finite time event-triggered control (AFETC) scheme is proposed, which has the following significant characteristics compared to most existing schemes: 1) the artfully constructed coordinate transformation not only avoids computational complexity but also simplifies the design; 2) the first-order command filtering vector-backstepping control method not only circumvents “explosion of complexity” in the traditional backstepping approach, but also ensures the finite-time settling of position tracking errors of USVs; 3) with the event triggered control, it reduces the control execution frequency, thereby decreasing the communication transmission load; and 4) it ensures stable tracking without requiring a priori information of the unmodelled dynamics of USVs or external disturbances. The designed robust adaptive control has proven to enable USVs to track the desired trajectory while guaranteeing the closed-loop stability in a finite time and avoiding Zeno behavior. Simulation and comparation results demonstrate the effectiveness and superiority of the proposed control strategy.

A Super-twisting SMC Design for Bidirectional DC–DC Converters in Electric Vehicle Applications

Many industrial and electrical systems, such as electric vehicles, employ DC/DC converters. Since DC/DC converters are nonlinear and time-varying systems, it is inappropriate to regulate them using linear control techniques. In this paper, a super-twisting sliding mode control (STSMC) is applied for bidirectional DC-DC converter in electric vehicle applications. The traction chain of the electric vehicle consists of a battery-supercapacitor energy storage system (ESS) linked in parallel to the DC bus through two bidirectional DC–DC converters and a synchronous reluctance motor drive (SynRM). The benefit of sharing power in an HPS, which is primarily made up of a supercapacitor and battery, is the ability to use the strengths of both sources, with the former having high specific energy and the latter having high specific power. The on-board energy storage system's performance is enhanced by this hybridization in terms of cost, mass, and charging time. In addition to the proposed control, the hybrid power system (HPS) needs a strategy for routing energy between the two sources and the motor. This strategy is based on frequency decoupling (DFS), which protects the battery from peak current during the acceleration and deceleration phases. The objective of using the control law based on super-twisting sliding mode control (STSMC) is to follow the reference trajectories in finite time with high precision, very good robustness, and a reduction in chattering. Using the MATLAB/Simulink software, a thorough simulation of an electric vehicle was performed in order to confirm the effectiveness of the suggested control approach and assess the effectiveness of the source's energy management strategy. This control method ensures a reduction in the bus voltage stress (Δv=5V) and improves the quality of the power. The overshoot voltage for a load power of 21 kw is reduced to 15V (3.2%) compared to the linear control techniques, improving the system's robustness and reliability..

Adaptive Finite-Time Trajectory Tracking Control for Coaxial HAUVs Facing Uncertainties and Unknown Environmental Disturbances

In this paper, the problems of system design, dynamic modeling, and trajectory tracking control of coaxial hybrid aerial underwater vehicles (HAUVs) with time-varying model parameters and composite marine environment disturbances are investigated. It is clear that a stable transition between different media remains a challenge in the practical implementation of amphibious tasks. For HAUVs, accurate dynamic modeling to describe complex dynamic variations during vehicle takeoff from underwater to air is a huge challenge. Meanwhile, due to the rapid changes in model parameters and the external environment, vehicles are likely to fall into the sea during the cross-domain process. An integrated continuous dynamic model considering hydrodynamic changes is established by introducing a linear switching coefficient during the process of trans-medium motion. A nonsingular fast terminal sliding-mode control (NFTSMC) algorithm combined with adaptive technology is used to design the position and attitude of the controller. With no previous knowledge of external interferences and lumped uncertainties of the HAUV, the adaptive NFTSMC (ANFTSMC) algorithm achieves the control objectives; at the same time, the inherent chattering problems of sliding mode control (SMC) are weakened. The finite-time stability of the global system is proven strictly using a series of mathematical derivations based on Lyapunov theory. The effect of the controller applied is analyzed through a series of simulations with representative working conditions. The results show that the proposed ANFTSMC can realize a “seamless” air–water trans-medium process, which proves the superiority and robustness of the proposed control algorithm.

Observer based incremental backstepping terminal sliding-mode control with learning rate for a multi-vectored propeller airship

The problem of finite-time trajectory tracking control for a multivectored propeller airship with model parameter uncertainties and unknown disturbances is addressed in the paper. In order to obtain a fast transient response and finite time convergence without singularity, an incremental backstepping nonsingular terminal sliding mode controller is designed to track desired trajectory within finite time. To overcome the limitation of backstepping terminal sliding-mode control design being dependant on prior knowledge, a nonlinear disturbance-observer is designed to estimate the external observable disturbances. Meanwhile, to reduce system chattering when the tracking error reaches the sliding mode surface, an adaptive learning rate design is proposed for the terminal sliding mode controller such that the tracking error approaches the sliding mode surface at a low speed and so the system chattering is restrained. The closed-loop trajectory tracking control system is proved to be stable and finite time convergence by using Lyapunov theory. The results are compared with traditional backstepping sliding mode control and different learning rate based control design, and they demonstrate the averaged tracking error in altitude and pitch motion is reduced more than 30% by using the disturbance-observer based incremental backstepping terminal sliding mode controller for the multivectored propeller airship to execute a realistic trajectory tracking mission, even in the presence of aerodynamic coefficient uncertainties, unknown disturbances.

Finite-time trajectory tracking control of dynamic positioning ship based on discrete-time sliding mode with decoupled sampling period

In this paper, the finite-time trajectory tracking control problem of a dynamic positioning ship with model uncertainty and time-varying environmental disturbances is investigated based on the discrete-time sliding mode with decoupled sampling period. To overcome the influence of the sampling period variation on the system’s control effect, a finite-time discrete sliding mode controller with decoupled sampling period is designed based on the new discrete-time adaptive fast power reaching law. The sampling period condition that guarantees the finite-time convergence of the system is given. A new finite-time high-order discrete disturbance observer is proposed to estimate model uncertainty and environmental disturbance, which reduces the dependence of the observer’s estimation accuracy on the sampling period. To realize the transformation from position tracking to velocity tracking in a finite time, a new finite-time discrete virtual velocity controller is designed. The proposed control strategy decouples the sampling period and enables the finite-time trajectory tracking control of the dynamic positioning ship over a wider sampling period range. The stability analysis demonstrates that all tracking errors of the closed-loop system can converge to a small domain near zero in a finite time. Two simulation examples verify the effectiveness of the proposed control method.

Event trigger based adaptive neural trajectory tracking finite time control for underactuated unmanned marine surface vessels with asymmetric input saturation

An adaptive finite time trajectory tracking control method is presented for underactuated unmanned marine surface vessels (MSVs) by employing neural networks to approximate system uncertainties. The proposed algorithm is developed by combining event-triggered control (ETC) and finite-time convergence (FTC) techniques. The dynamic event-triggered condition is adopted to avert the frequent acting of actuators using an adjustable triggered variable to regulate the minimal inter-event times. While solving the system uncertainties and asymmetric input saturation, an adaptive neural networks based backstepping controller is designed based on FTC under bounded disturbances. In addition, via Lyapunov approach it is proved that all signals in the closed-loop system are semi-global uniformly ultimately bounded. Finally, simulations results are shown to demonstrate the effectiveness of this proposed scheme.

Event-triggered adaptive finite time trajectory tracking control for Underactuated Vessel considering unknown time-varying disturbances

Abstract This paper aims to solve the finite-time trajectory tracking problem of underactuated surface ships under the influence of dynamic uncertainty, unknown external time-varying interference and limited communication resources, an event-triggered robust adaptive finite-time trajectory tracking control method for underactuated ships is designed by combing the existing trajectory tracking control methods and engineering needs in actual navigation. It can obviously improve the tracking accuracy of the ship, and complete the tracking task safely and efficiently. This scheme transforms the composite uncertain vector synthesized by uncertain parameters and external disturbances into a linear parameterized form. Next, considering the limitation of communication resources, a relative threshold event triggering mechanism is introduced to further extend the triggering time interval on the basis of the fixed threshold triggering scheme. Then, an event-triggered robust adaptive finite-time trajectory tracking control scheme is designed for underactuated ships, and a rigorous stability proof is provided for the finite-time trajectory tracking control scheme designed above through Lyapunov stability theory. Simulation experiment is carried out in MATLAB and the results show that the finite-time trajectory tracking control scheme proposed in this paper can effectively solve the problems of dynamic uncertainty, unknown time-varying interference from the outside world, and limitation of communication resources. This paper can provide theoretical support for the autonomous navigation of intelligent ships.

Cooperative trajectory shaping guidance law for multiple anti-ship missiles

Abstract To enhance the performance of anti-ship missiles cooperative attack, this paper proposes a finite-time trajectory shaping-based cooperative guidance law (TSCGL). Firstly, the cooperative guidance model is established on segmented linearisation of the missile’s heading angle. Then, a trajectory shaping guidance law for a single missile is derived by a weighted optimal energy cost function and Schwarz inequality. On this basis, a finite-time TSCGL is proposed combined with trajectory shaping technology and finite-time theory. The desirable finite-time convergence performance can ensure a simultaneous attack. Through an improved method of time-to-go estimation, it is independent of small-angle assumption and relaxes the launching conditions of the missiles. Additionally, the proposed finite-time TSCGL can achieve better damage performance through energy management. Finally, simulation results demonstrate the effectiveness and superiority of the proposed finite-time TSCGL.

Finite-time trajectory tracking control for a two degrees of freedom pneumatic dexterous finger

Finite-time trajectory tracking control for a two degrees of freedom pneumatic dexterous finger

Minimum-learning-parameter-based adaptive finite-time trajectory tracking event-triggered control for underactuated surface vessels with parametric uncertainties

For the problem of trajectory tracking control of underactuated surface vessels (USVs) affected by input saturation, parametric uncertainties, time-varying marine environmental disturbances, and limited transmission resources, a robust adaptive finite-time event-triggered control scheme is presented in this paper. First, a coordinate transformation is constructed to tackle the underactuation problem of the USV and enables the USV control law to be devised in vector form, which simplifies the design and makes it more suitable for computer processing. A continuous auxiliary dynamic system is constructed to handle the input saturation problem. The minimum learning parameter (MLP) technique is used to approximate the parametric uncertainties of the model. An adaptive law is devised to update online the upper bound of the combination of external disturbances and the approximation error generated by the MLP. Additionally, a finite-time control scheme is presented by introducing a robustifying term in the virtual control law, which improves the control performance; an introduced event triggering mechanism in the channel from the controller to the actuator decreases both the transmission burden and the control execution frequency without affecting the control performance. The theoretical analysis reveals that the devised robust adaptive finite-time event-triggered control scheme enables the USVs to track the desired trajectory in a finite time with a small number of control executions, while ensuring the closed-loop stability of the system and avoiding Zeno behavior. The simulations verify the effectiveness and the superiority of the presented control scheme.

Finite-time trajectory tracking control of quadrotor UAV via adaptive RBF neural network with lumped uncertainties

The trajectory tracking control of the quadrotor with model uncertainty and time-varying interference is studied. The RBF neural network is combined with the global fast terminal sliding mode (GFTSM) control method to converge tracking errors in finite time. To ensure the stability of the system, an adaptive law is designed to adjust the weight of the neural network by the Lyapunov method. The overall novelty of this paper is threefold, 1) Owing to the use of a global fast sliding mode surface, the proposed controller has no problem with slow convergence near the equilibrium point inherently existing in the terminal sliding mode control. 2) Benefiting from the novel equivalent control computation mechanism, the external disturbances and the upper bound of the disturbance are estimated by the proposed controller, and the unexpected chattering phenomenon is significantly attenuated. 3) The stability and finite-time convergence of the overall closed-loop system are strictly proven. The simulation results indicated that the proposed method achieves faster response speed and smoother control effect than traditional GFTSM.

Finite-time self-structuring neural network trajectory tracking control of underactuated autonomous underwater vehicles

A finite-time self-structuring neural network (SSNN) trajectory tracking control scheme is proposed for an input-constrained underactuated autonomous underwater vehicle (AUV) with unknown external disturbances. First, a time-varying barrier Lyapunov function is proposed to constrain and prevent excessive consistency errors and reduce computational effort. Second, a finite-time controller is presented to achieve finite-time convergence of the closed-loop system, while dynamic surface control (DSC) is used to reduce the computational complexity of the system. Third, a finite-time SSNN method is developed to obtain the optimal number of neurons with simple computation and better approximation to estimate uncertain disturbances. Moreover, the Lyapunov stability proof indicates that all signals in the closed-loop system are ultimately bounded uniformly with respect to the initial constraints, and the trajectory tracking errors can converge to near zero in a finite time. Finally, simulations not only evaluate the performance of the proposed controller-controlled system but also verify the effectiveness of the methodology in this paper.

A generalised proportional-derivative type scheme with multiple saturating structure for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with bounded inputs

A generalised proportional-derivative type scheme for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with input constraints is developed. Its generalised design permits the choice among multiple saturating structures. Suitable functions are involved for the appropriate shaping of every error correction term, and complementarily of their addition, to achieve the concerned type of convergence (in addition to the input saturation avoidance). Such a shaping is carried out through control parameters that act as exponential weights on the gained configuration and generalised velocity errors and on their joint action. Compared to previous finite-time approaches, such exponential weights are required to satisfy generalised comparative conditions, avoiding a fixed equivalence relation among any of them. This permits to give rise to a wider spectrum of finite-time convergent closed-loop trajectories and gives an optional type of (unconventional) exponential convergence (in addition to the conventional one). The resulting extended conditions on the exponential weights and the generalised saturating structure enlarge the controller potential for performance improvement or adjustment. The generalised design is supported through a more general analysis based on more general strict Lyapunov functions, stating more solid analytical bases for possible design extensions on the accomplishment of other control objectives. The study is further supported through experimental tests on a 3-degree-of-freedom robot manipulator.

Finite Time Trajectory Tracking with Full-State Feedback of Underactuated Unmanned Surface Vessel Based on Nonsingular Fast Terminal Sliding Mode

Marine transportation and operations have attracted the attention of more and more countries and scholars in recent years. A full-state finite time feedback control scheme is designed for the model parameters uncertainty, unknown ocean environment disturbances, and unmeasured system states in the underactuated Unmanned Surface Vessel (USV) trajectory tracking control. The external wind, wave and current environmental disturbances and model parameters perturbation are extended by Nonlinear Extended State Observer (NESO) to the state of the system, namely complex disturbances. The complex disturbances, positions and velocities of USV can be observed by NESO and feedback to USV control system. Next, the underactuated USV error model is obtained by operating the obtained feedback information and the virtual ship model. According to the error model, a Nonsingular Fast Terminal Sliding Model surface (NFTSM) is constructed to realize finite-time control. The control law is deduced through the Lyapunov stability theory to ensure the stability of the system. The results of MATLAB numerical simulations under different disturbances show that the trajectory tracking algorithm has fast responses, and a good convergence of the errors is observed, which verifies the effectiveness of the designed scheme.

Finite time extended state observer based nonsingular fast terminal sliding mode control of flexible-joint manipulators with unknown disturbance

To solve the problems concerning with finite time trajectory tracking of the flexible-joint manipulator subject to unknown disturbance, a singular perturbation theory (SPT) based decoupling controller is proposed in this paper. SPT is employed to decouple the basic dynamic model of the flexible-joint manipulator into a fast subsystem and a slow subsystem. For the slow subsystem, a finite time extended state observer (FTESO) is proposed to estimate the unknown disturbance. In order to avoid the singularity and accelerate the convergence speed of terminal sliding mode controller (TSMC), a novel nonsingular fast terminal sliding mode controller (NFTSMC) is designed to achieve finite time trajectory tracking. Besides, to attenuate the chattering effect, a nonsingular terminal sliding mode controller (NTSMC) is introduced for the fast subsystem. Finally, the finite time stability of the closed-loop system is proven by means of the Lyapunov theory and some comparative simulations are performed to verify the effectiveness of the proposed controller.

Barrier Function-based adaptive fast non-singular terminal sliding mode tracking control for marine vessels

In this paper, finite-time trajectory tracking problem of the marine vessels (MVs) disturbed by highly coupled nonlinearities and unknown ocean current is solved by developing a barrier function-based adaptive fast non-singular terminal sliding mode control (BFAFNTSMC) method. Salient features are as follows: (1) The defined fast non-singular terminal sliding mode manifold (FNTSM) which introduces the finite-time ocean current observer (OCO) is proposed to guarantee the finite-time sliding motion for mismatched disturbance attenuation. (2) By devising a novel log-type barrier function adaptive sliding mode reaching law (BFARL) based on single switching logic, the synovial variable will converge to and maintain in a predefined neighborhood of zero in a finite time, without overestimating the control gain. (3) Finite-time convergence of trajectory tracking errors to the predefined neighborhood of zero can be guaranteed, independently of the boundaries of the lumped nonlinearities. The proposed method exhibits the satisfied properties of tracking performance as well as chattering alleviation. Simulation results and comparisons are further presented to illustrate remarkable effectiveness of the proposed approach.

Command filtered-based neuro-adaptive robust finite-time trajectory tracking control of autonomous underwater vehicles under stochastic perturbations

In this paper, the problem of finite-time trajectory tracking control is studied and addressed for a 6 degree of freedom (DOF) autonomous underwater vehicle (AUV) subjected to unknown dynamic model, stochastic perturbations, external disturbances (matched and mismatched) and saturation input nonlinearities. Based on the backstepping control approach, novel finite-time control inputs are designed and proposed. Artificial neural networks (ANNs) and finite-time adaptation laws are exploited to approximate the nonlinear dynamics of AUV, the stochastic perturbations and the upper bound of external disturbances. To handle the destructive effects of saturation input nonlinearities, finite-time auxiliary system method is utilized. To overcome the explosion of complexity problem of backstepping control strategy, compensator-based finite-time command filter approach is exploited. By utilizing the Lyapunov stability theorem, it is mathematically proven and demonstrated that the suggested nonlinear control inputs are able to guarantee the semi-global finite-time stability in probability (SGFSP) of the closed-loop AUV system. Finally, numerical simulations are carried out to illustrate and depict the effectiveness and performance of the proposed neuro-adaptive robust finite-time control scheme.

Finite-Time Neuro-Sliding-Mode Controller Design for Quadrotor UAVs Carrying Suspended Payload

Due to the quadrotor’s underactuated nature, suspended payload dynamics, parametric uncertainties, and external disturbances, designing a controller for tracking the desired trajectories for a quadrotor that carries a suspended payload is a challenging task. Furthermore, one of the most significant disadvantages of designing a controller for nonlinear systems is the infinite-time convergence to the desired trajectory. In this paper, a finite-time neuro-sliding mode controller (FTNSMC) for a quadrotor with a suspended payload that is subject to parametric uncertainties and external disturbances is designed. By constructing a finite-time sliding mode controller, the quadrotor can follow the reference trajectories in finite time. Furthermore, despite time-varying nonlinear dynamics, parametric uncertainties, and external disturbances, a neural network structure is added to the controller to effectively reduce chattering phenomena caused by high switching gains, and significantly reduce the size of the control signals. Following the completion of the controller design, the system’s stability is demonstrated using the Lyapunov stability criterion. Extensive numerical simulations with various scenarios are run to demonstrate the effectiveness of the proposed controller.

Online recorded data-based finite-time composite neural trajectory tracking control for underactuated MSVs.

This paper presents an online recorded data-based composite neural finite-time control scheme for underactuated marine surface vessels (MSVs) subject to uncertain dynamics and time-varying external disturbances. The underactuation problem of the MSVs was solved by introducing the line-of-sight (LOS) method. The uncertain dynamics of MSVs are approximated by the composite neural networks (NNs). A modified prediction error signal is designed by virtue of online recorded data. The weight updating law of NN is driven by both tracking error and prediction error, introducing additional correction information to the weights of NN, thus improving the learning ability of the NN. Furthermore, disturbance observers can be devised to estimate the compound disturbances consisting of the approximation errors of NNs and external disturbances. Moreover, the smooth function is inserted into the design of the control scheme, and the finite-time composite neural trajectory tracking control of MSVs is achieved. The stability of the MSVs trajectory tracking closed-loop control system is guaranteed rigorously by the Lyapunov approach, and the tracking error will converge to the set of residuals around zero within a finite time. The simulation tests on an MSV verify the effectiveness of the proposed control scheme.

Event-triggered finite-time adaptive trajectory tracking control for a class of nonlinear non-strict-feedback systems with input saturation and application to a single-link robot

Event-triggered finite-time adaptive trajectory tracking control for a class of nonlinear non-strict-feedback systems with input saturation and application to a single-link robot