Nonlinear Tracking Differentiator Research Articles

Image Dynamics-Based Visual Servoing for Quadrotors Tracking a Target With a Nonlinear Trajectory Observer

In this correspondence paper, an image dynamics-based visual servoing for quadrotors is proposed to realize stable hovering and tracking. Four perspective image moments are adopted as visual features to control all the independent degrees of freedom of a quadrotor. The complicated interaction matrix is simplified by projecting original image to virtual image plane. On this basis, the dynamics of the system is determined by considering the dynamics of image features and the quadrotor simultaneously. Backstepping controllers are then designed to stabilize the visual servoing system of the quadrotor. In reality, it is unrealistic to have exact prior knowledge about the trajectory parameters of an unpredictable moving target. To solve this problem, a trajectory observer based on nonlinear tracking-differentiator to estimate trajectory parameters of the target is firstly integrated into the quadrotor with image dynamics, which guarantees a satisfactory performance. The effectiveness of the proposed approach is verified by simulations.

Tracking differentiator based on improved Gaussian distribution

Concerning the problems of the existing tracking differentiator, such as large computation, complex parameter tuning and output chattering, a nonlinear tracking differentiator (GTD) based on the improved Gaussian distribution was proposed. It is a simple calculating nonlinear tracking differentiator with easy parameters tuning and strong ability of noise restraint. Firstly, the amplitude gain and exponential factor are introduced to improve the Gaussian distribution, then the acceleration function is constructed by utilizing the improved Gaussian distribution. Secondly, the global asymptotic stability of the designed TD is proved by constructing Lyapunov function. And its singularity type and phase trajectory distribution are analyzed by the phase plane analysis method, and the qualitative law of parameter adjustment is also given. Finally, simulations are performed and results are compared with linear differentiator and anti-hyperbolic sinusoidal based TD. It concludes that the differential signal of the improved Gaussian-distribution-based nonlinear tracking differentiator does not have the high frequency flutter behavior in steady state. Moreover, the GTD not only has good dynamic response but also has strong filtering ability.

A novel non-singular terminal sliding mode control-based integrated missile guidance and control with impact angle constraint

Abstract A novel impact angle constrained integrated guidance and control (IGC) scheme is proposed with non-singular terminal sliding mode control (NTSMC) technique for missile to intercept a maneuverable target. Firstly, an IGC design model is innovatively established in a strict feedback form which selects the normal overload as the system state in replace of the angle of attack so that the system states both are measurable. On this basis, an IGC law is proposed to guarantee the impact angle converge to the expected command within a finite time and to improve the robustness against the multiple disturbances which include the model parameter uncertainties and the target maneuvering by incorporating the NTSMC method and the extended state observer (ESO) technique. In order to avoid the chattering phenomenon, the smooth derivative of the intermediate control command which is obtained by the introduced nonlinear tracking differentiator (TD) is adopted into the control loop. Then, the stability of closed-loop integrated system is rigorously proved by utilizing the Lyapunov theorem. Finally, extensive contrast simulation results are presented to validate the efficiency, superiority and robustness of the suggested IGC scheme.

Robust adaptive NN tracking control for MIMO uncertain nonlinear systems with completely unknown control gains under input saturations

This paper focuses on the study of tracking control problem for a class of multiple-input-multiple-output (MIMO) uncertain nonlinear systems with completely unknown control gains under input saturations. A hyperbolic tangent function is introduced to approximate the input saturation nonlinearity. Based on the above, a novel robust adaptive neural network (NN) tracking control scheme is proposed combining finite-time nonlinear tracking differentiator (FTNTD), Nussbaum-type function and adaptive NN with backstepping design tool. Within our scheme, a constructed novel nonlinear function featured by “small-error large-gain and large-error small-gain” is inset into virtual and actual control laws to achieve a balance between the feedback control gain and the system control performance and prevent the over-learning of parameter adaptive laws caused by the input saturation. By means of a newly constructed non-quadratic Lyapunov function, it is theoretically shown that all the signals in the closed-loop control system are semi-globally uniformly ultimately bounded. Finally, the effectiveness of our proposed scheme is verified by the simulations of two simulation examples.

Modular neural dynamic surface control for position tracking of permanent magnet synchronous motor subject to unknown uncertainties

This paper is concerned with the adaptive position tracking of permanent magnet synchronous motors (PMSMs) subject to model uncertainties and unknown loads. A modular neural dynamic surface control (MNDSC) method is used to devise the position tracking controller. Specifically, a predictor module based on neural networks (NNs) is designed, which is able to fast identify the unknown nonlinearities and uncertainties of PMSMs without introducing high-frequency oscillations in the learning process. Next, a position tracking controller module is designed based on a modified dynamic surface control where a second-order nonlinear tracking differentiator (NLTD) instead of a first-order filter is used to extract the time derivatives of virtual control law. The salient features of the proposed position tracking controller for PMSMs are as follows. First, the transient performance can be improved compared with the previous control method. Second, the unknown nonlinearities of the PMSM can be approximated by the NNs. Third, the predictor module and the controller module are decoupled by using the modular design. The stability of the position tracking system cascaded by the predictor module and the controller module is proved by cascade theory and input-to-state stability theory. Finally, the performance of the proposed MNDSC strategy for PMSMs is verified through simulations.

Output-Feedback Cooperative Formation Maneuvering of Autonomous Surface Vehicles With Connectivity Preservation and Collision Avoidance.

In this paper, a cooperative time-varying formation maneuvering problem with connectivity preservation and collision avoidance is investigated for a fleet of autonomous surface vehicles (ASVs) with position-heading measurements. Each vehicle is subject to unknown kinetics induced by internal model uncertainty and external disturbances. At first, a nonlinear state observer is used to recover the unmeasured linear velocity and yaw rate as well as unknown uncertainty and disturbances. Then, observer-based cooperative time-varying formation maneuvering control laws are designed based on artificial potential functions, nonlinear tracking differentiators, and a backstepping technique. The stability of closed-loop distributed formation control system is analyzed based on input-to-state stability and cascade stability. The salient features of the proposed method are as follows. First, cooperative time-varying formation maneuvering with the capability of connectivity preservation and collision avoidance can be achieved in the absence of velocity measurements. Second, the complexity of the cooperative time-varying formation maneuvering control laws is reduced without resorting to dynamic surface control. Third, the uncertainty and disturbance are actively rejected in the presence of position-heading measurements. Simulation results are given to substantiate the proposed output feedback control method for cooperative time-varying formation maneuvering of ASVs with connectivity preservation and collision avoidance.

Nonsingular Fast Terminal Sliding Mode Tracking Control for a Class of Uncertain Nonlinear Systems

Aiming at the tracking control problem of a class of uncertain nonlinear systems, a nonsingular fast terminal sliding mode control scheme combining RBF network and disturbance observer is proposed. The sliding mode controller is designed by using nonsingular fast terminal sliding mode and second power reaching law to solve the problem of singularity and slow convergence in traditional terminal sliding mode control. By using the universal approximation of RBF network, the unknown nonlinear function of the system is approximated, and the disturbance observer is designed by using the hyperbolic tangent nonlinear tracking differentiator (TANH-NTD) to estimate the interference of the system and enhance the robustness of the system. The stability of the system is proved by the Lyapunov principle. The numerical simulation results show that the method can shorten the system arrival time, improve the tracking accuracy, and suppress the chattering phenomenon.

Estimation of angular velocity and acceleration with Kalman filter, based on position measurement only

This paper presents a method to obtain good synchronous and instantaneous estimates in position, velocity and acceleration; using a position measurement only. The proposed method shows better estimates in position, velocity and acceleration than other methods such as nonlinear tracking differentiator (TD), extended state estimation (ESO) and digital differentiator based on Taylor series (DDBTS). The proposed method allows for an increased accuracy of estimates by generating a feedback frequency obtained to the position measurement signals. The model and method proposed in this paper reduce the error when the measurement signal presents a change in frequency. The model update and adjusts simultaneously to changes in sample rates using a feedback frequency estimate. The proposed method was validated with the QNET DC Motor Control Trained (DCMCT). Since the method requires the measurement position only from an encoder, it eliminates the need for more sensors for velocity and acceleration, thus begin less costly.

A new guidance law for trajectory tracking of an underactuated unmanned surface vehicle with parameter perturbations

This paper proposes a novel trajectory tracking controller for an underactuated unmanned surface vessel (USV) with multiple uncertainties and input constraints. Unlike previous dynamic surface control (DSC) methods that utilize a first-order filter to obtain derivatives and avoid “explosion of complexity”, we introduce nonlinear tracking differentiators (NTDs) that achieve satisfactory differential performance and fast tracking response and control the vessel to converge to the pseudo-yaw angle and surge. First, a new guidance law for yaw angle and surge is constructed. Second, inner and outer disturbances caused by uncertainties can be observed by reduced-order extended state observers. With the new guidance law, the design process of the controller is simplified and easy to implement. The simulation results show the trajectory tracking error can be stabilized to a certain extent under the parameter perturbations and other uncertainties, which verified the effectiveness of the proposed algorithm.

Robust Adaptive Trajectory Linearization Control for Tracking Control of Surface Vessels With Modeling Uncertainties Under Input Saturation

This paper develops a novel adaptive trajectory tracking control strategy to enhance the tracking performance for surface vessels with unmodeled dynamics and unknown time-varying disturbances. A high robustness and precision trajectory tracking controller is presented by using trajectory linearization control (TLC) technology, neural network, extended state observer (ESO), nonlinear tracking differentiator, and auxiliary dynamic system. First, the greatest advantage of this paper is that the TLC technology is first introduced into the field of surface vessels motion control, which provides a new direction for TLC technology research. Then, to further enhance the control performance and robustness of the system, the neural network with minimum learning parameter is used to replace the classical radial basis function neural network to approximate unmodeled dynamics, which can reduce the burden of computing. A novel reduced-order ESO is constructed to estimate unknown time-varying disturbances to achieve real-time compensation. Meanwhile, nonlinear tracking differentiator is employed to realize the derivative of virtual control command, as well as to provide command filtering. In addition, an auxiliary dynamic system is designed to reduce the risk of actuator saturation. The stability of the closed-loop system is guaranteed based on the Lyapunov criteria. Lastly, the comparison results demonstrate the superior performance of the proposed approach.

A New Mechanical Resonance Suppression Method for Large Optical Telescope by Using Nonlinear Active Disturbance Rejection Control

Aiming at solving the problem of the multi-low-frequency mechanical resonances appearing in the large optical telescope control system, this paper proposes a novel control method based on nonlinear active disturbance rejection control (NADRC) and proportional-integral (PI) control. In the proposed control framework, a nonlinear tracking differentiator (NTD)-based feedforward control is designed to improve the tracking performance of the system. Then, the principle of suppression of mechanical resonance of this method is analyzed. Compared with the most commonly used acceleration feedback control (AFC) method, the theoretical analysis shows that the proposed method is more effective for suppressing the low-frequency mechanical resonance. Finally, the proposed method is applied to a large optical telescope, and the experimental results show that the proposed method is better than AFC.

Error-Driven-Based Nonlinear Feedback Recursive Design for Adaptive NN Trajectory Tracking Control of Surface Ships With Input Saturation

In this paper, we investigate the trajectory tracking control problem of surface ship subject to the dynamic uncertainties, unknown time-varying disturbances and input saturation. To handle the non-smooth input saturation nonlinearity and compensate the ship dynamic uncertainties, Gaussian error function and adaptive neural network technique are employed. In control design, to obtain the transient motion reference signal, finite-time nonlinear tracking differentiator is applied to generate virtual reference signal and to extract the derivative of virtual control law. Referring to the effects of the kinematics subsystem on the kinetics subsystem caused by the error of tracking differentiator, and the effects of the input saturation on the control accuracy and the dynamic quality of the trajectory tracking control system, we propose an error-driven-based nonlinear feedback recursive design technique to design trajectory tracking control law, and employ a new non-quadratic Lyapunov functions to analyze the trajectory tracking control system stability. The proposed control scheme fully embodies the characteristics of the lowgain and high-gain control, and overcomes the effect of tracking differentiator error on closed-loop system by recursive design method. Simulation results verify the effectiveness of our proposed control scheme.

Second-Order Hierarchical Fast Terminal Sliding Model Control for a Class of Underactuated Systems Using Disturbance Observer

A second-order hierarchical fast terminal sliding mode control method based on disturbance observer (DOSHFTSM) is proposed for a class of fourth-order underactuated systems. In the first step, the fourth-order underactuated system is divided into two subsystems, and the integral sliding surface is designed for each subsystem. Then, the first-order fast terminal sliding surface is defined by using the integral sliding surface and its derivatives of each subsystem, and the switching control items of the system are designed according to the first-order fast terminal sliding surface of the subsystem. Secondly, the second-order sliding surface is designed by using the first-order fast terminal sliding surface of each subsystem. On the premise of ensuring the stability of Lyapunov, the switching control term is designed by using the variable coefficient double power reaching law to eliminate the system jitter. Finally, based on the principle of hyperbolic tangent nonlinear tracking differentiator, a hyperbolic tangent nonlinear disturbance observer (TANH-DOC) is designed to estimate the uncertainties and external disturbances of the system and compensate them to the sliding mode controller to improve the robustness of the system. The stability of the system is proved by using Lyapunov principle. The validity of this method is verified by numerical simulation and physical simulation of inverted pendulum system.

Finite-time model-assisted active disturbance rejection control with a novel parameters optimizer for hypersonic reentry vehicle subject to multiple disturbances

In this paper, a scheme which combines finite-time model-assisted active disturbance rejection control and novel greedy criterion-based salp swarm algorithm is developed for attitude tracking problem of hypersonic reentry vehicle with multiple disturbances. To simplify control structure, the control scheme is designed within the framework of active disturbance rejection control completely. To lessen tuning parameters, a sigmoid function-based tracking differentiator is employed to generate a more realizable attitude transient profile instead of utilizing conventional nonlinear tracking differentiator. Taking known model information as model-assisted term, finite-time model-assisted extended state observers are constructed to estimate the lumped disturbance in attitude and angular rate loop with employment of function fal. To achieve rapid response and strong robustness, finite-time model-assisted control laws are derived by utilizing function fhan. Finally, a novel greedy criterion-based salp swarm algorithm is used to optimize control parameters in finite-time model-assisted control laws to achieve optimal solution that minimizes the tracking error and energy consumption. Comparative simulations are conducted to illustrate the effectiveness of the proposed scheme.

Consensus Maneuvering for a Class of Nonlinear Multivehicle Systems in Strict-Feedback Form.

In this paper, a consensus maneuvering problem for nonlinear multivehicle systems in strict-feedback form is investigated. The consensus maneuvering problem includes a geometric task and a dynamic task. The geometric task means that all trajectories of follower vehicles converge to a parameterized path. The dynamic task is to drive the system to satisfy a desired dynamic assignment. A consensus maneuvering controller is developed for each vehicle based on a modular design approach. First, an estimator module is designed based on an echo state network, which is used to estimate uncertain nonlinearities. Then, a controller module is designed based on a modified dynamic surface control method through the use of a second-order nonlinear tracking differentiator. Finally, a path update law is designed based on a distributed maneuvering error feedback and a filtering scheme. The proposed controller is distributed in the sense that the path information is accessed by a small number of follower vehicles only. The stability of the closed-loop system cascaded by the estimator module and the controller module is analyzed based on input-to-state stability theory and cascade theory. Simulation results are provided to demonstrate the efficacy of the proposed consensus maneuvering controllers for uncertain nonlinear strict-feedback systems.

Robust Sliding Mode Control of PMSM Based on a Rapid Nonlinear Tracking Differentiator and Disturbance Observer.

Torque ripples caused by cogging torque, flux harmonics, and current measurement error seriously restrict the application of a permanent magnet synchronous motor (PMSM), which has been paid more and more attention for the use in inertial stabilized platforms. Sliding mode control (SMC), in parallel with the classical proportional integral (PI) controller, has a high advantage to suppress the torque ripples as its invariance to disturbances. However, since the high switching gain tends to cause chattering and it requires derivative of signals which is not readily obtainable without an acceleration signal sensor. Therefore, this paper proposes a robust SMC scheme based on a rapid nonlinear tracking differentiator (NTD) and a disturbance observer (DOB) to further improve the performance of the SMC. The NTD is employed to providing the derivative of the signal, and the DOB is utilized to estimate the system lumped disturbances, including parameter variations and external disturbances. On the one hand, DOB can compensate the robust SMC speed controller, it can reduce the chattering of SMC on the other hand. Experiments were carried out on an ARM and DSP-based platform. The obtained experimental results demonstrate that the robust SMC scheme has an improved performance with inertia stability and it exhibits a satisfactory anti-disturbance performance compared to the traditional methods.

Nonlinear Control of Marine Surface Vessels

In the present study, a robust yaw control law design derived from nonlinear extended state observer (NESO) based nonlinear state error feedback controller (NSEFC) in conjunction with nonlinear tracking differentiator (NTD) for marine surface vessels is presented. As marine vessel operates in an environment where significant uncertainties and disturbances are present, an NESO is used to estimate the effect of the uncertainties and disturbances along with the plant states leading to a robust design through disturbance estimation and compensation. Convergence of NESO and NTD is demonstrated. The notable feature of the formulation is that to achieve robustness, accurate plant model or any characterization of the uncertainties and disturbances is not needed. Efficacy of the design is illustrated by simulation. Further, performance of the proposed design is compared with some existing controllers to showcase the effectiveness of the proposed design.

An improved tracking-differentiator filter based on Taylor's formula

Because of the ability of tracking and filtering the reference signal (RS), and the ability of obtaining the generalized derivative of the RS, nonlinear tracking differentiators (NTDs) have been widely applied in many of the engineering fields. However, there are phase delays when applying NTDs. These phase delays will affect subsequent performance. In order to improve the accuracy of NTDs, a new phase compensation method based on Taylor's formula is proposed in this paper, which could reduce the phase delays significantly. Some simulation results are provided to demonstrate the effectiveness of the method.

State estimation of a robot joint by a novel nonlinear tracking differentiator

PurposeThis paper aims to present a method for improving the state estimation of a robot in the presence of noise measurement, which can improve the performance of the robot controller.Design/methodology/approachIn this work, a novel nonlinear tracking differentiator (NTD) was formulated to solve the problems of phase lag, low stability and amplitude attenuation faced by traditional tracking differentiators, which can be used for the state estimation of a robot. Based on the user-defined function stu() with linear and nonlinear characteristics, the authors establish a new acceleration function of NTD and confirm its global asymptotic stability by using the Lyapunov method and the system equivalence method. Phase plane analysis shows that the origin is its stable nodal point or focus point and uncovers the basic constraint conditions for parameter regulation. In addition, the convergence property and robustness performance against noises are studied by describing function method.FindingsComparative simulations, robot state estimation experiments and joint trajectory tracking experiments have indicated that NTD proposed integrates tracking rapidness, accuracy and transitional stability and has high approximation and filtering effects on generalized derivatives of the signal, which contribute to an excellent performance of robot controller in stability and response speed in practice.Originality/valueThe main contribution of this paper lies in the design of a novel NTD, which successfully improves the state estimation of a robot joint in noisy surroundings, the tracking performance of robot controller and the stability of the system.

Fractional Calculus Guidance Algorithm in a Hypersonic Pursuit-Evasion Game

<p class="p1">Aiming at intercepting a hypersonic weapon in a hypersonic pursuit-evasion game, this paper presents a fractional calculus guidance algorithm based on a nonlinear proportional and differential guidance law. First, under the premise of without increasing the complexity degree of the guidance system against a hypersonic manoeuvering target, the principle that the differential signal of the line-of-sight rate is more sensitive to the target manoeuver than the line-of-sight rate is employed as the guidelines to design the guidance law. A nonlinear proportional and differential guidance law (NPDG) is designed by using the differential derivative of the line-of-sight rate from a nonlinear tracking differentiator. By using the differential definition of fractional calculus, on the basis of the NPDG, a fractional calculus guidance law (FCG) is proposed. According to relative motions between the interceptor and target, the guidance system stability condition with the FCG is given and quantitative values are also proposed for the parameters of the FCG. Under different target manoeuver conditions and noisy conditions, the interception accuracy and robustness of these two guidance laws are analysed. Numerical experimental results demonstrate that the proposed guidance algorithms effectively reduce the miss distance against target manoeuvers. Compared with the NPDG, a stronger robustness of the FCG is shown under noisy condition.</p>