Tracking Differentiator Research Articles

A Positioning and Tracking Performance–Enhanced Composite Control Algorithm for the Macro–Micro Precision Stage

In the macro–micro composite motion process, the macro stage achieves rapid motion in a large workspace, while the micro stage realizes precise positioning within a small displacement. The large-stroke and high-speed motion of the macro stage is influenced by nonlinear friction, overshooting, and vibrations, making it challenging to achieve rapid stabilization within the travel range of the micro stage, thereby impacting equipment operation efficiency. This paper proposes a composite controller structure designed to solve the fast point-to-point positioning of the macro stage driven by a linear motor and enhance the trajectory tracking performance of the stage. The proposed composite control algorithm (CCA) includes velocity feed-forward, gain-scheduled proportional integral differential (PID) control, and a plug-in repetitive control method. By employing a tracking differentiator, velocity feed-forward, and a gain-scheduled PID controller, the control algorithm can realize rapid stabilization and positioning for the macro stage. Through velocity feed-forward, gain-scheduled PID control, and a plug-in repetitive controller, the control algorithm can reduce the trajectory tracking error of the stage. Simulations and experimental studies of point response and sinusoidal trajectory tracking are carried out on a macro–micro stage to verify the effectiveness of the composite controller for the linear motor. The experimental results demonstrate that the proposed composite controller effectively reduces the macro stage’s settling time and overshoot, and improves the accuracy of sinusoidal trajectory tracking, which lays a foundation for submicron positioning accuracy in high-speed macro–micro motion stages.

Positioning anti-sway control method for tower crane based on improved MFAC

A model-free adaptive control (MFAC) based on partial format is proposed for the complex modeling and uncertain model parameters of the tower crane (TC) in this paper. Firstly, considering the complexity of the TC nonlinear system, the data model of the TC is obtained through the partial format dynamic linearization method. Then, a model-free adaptive controller is designed for the TC, incorporating the change rate of the tracking error and its derivative into the objective function to address the inefficiency of traditional model-free controllers in controlling TCs. The convergence and stability of the proposed controller are also proved. To further improve the dynamic performance of the system, a tracking differentiator (TD) is introduced after the input signal. Finally, MATLAB simulation and experiments are conducted to verify that the proposed method can achieve precise trolley positioning while suppressing the load swing angle.

Nolinear tracking differentiator based practical prescribed time tracking control for perturbed wheeled mobile robot

Nolinear tracking differentiator based practical prescribed time tracking control for perturbed wheeled mobile robot

Four-stage cascaded adaptive sliding mode control for automatic carrier landing with airwake disturbances and uncertainties

ObjectiveThis paper addresses the carrier landing affected by airwake, parametric uncertainties, and carrier deck motion, its main target being the design of a novel sliding mode based automatic carrier landing system to obtain accurate tracking of the reference trajectory, robustness in terms of disturbances and uncertainties, as well as excellent touchdown accuracy. ApproachFor an aircraft nonlinear dynamics, written under a four-stage cascaded strict feedback form, the design of the novel landing control architecture involves the design of a guidance subsystem, robust sliding mode controllers (for the control of the heading angle, attitude angles, and angular rates), an approach power compensation system, adaptive control laws suppressing the uncertainties and disturbances, a Kalman filter for deck motion prediction, a block computing the reference trajectory, a tracking differentiator block for deck motion compensation, and first-order command filters. Main resultsThe software validation process proves the effectiveness of the sliding mode based control scheme and the suppression of the uncertainties and disturbances. Also, the comparison between the performances of the sliding mode control based carrier landing system and the ones associated to other automatic carrier landing systems shows the superiority of the sliding mode based control scheme, as well as its better touchdown accuracy and landing success rate. SignificanceThis study innovatively transforms the general carrier landing problem into a time-varying tracking control problem for cascaded strict feedback dynamics with disturbances and uncertainties. The new designed automatic carrier landing system is the first control architecture in the literature employing the sliding mode control augmented by adaptive control laws for carrier landing, subjected to airwake, deck motion, and uncertainties.

Attitude Tracking of Rigid-Liquid-Flexible Coupling Spacecraft by Active Disturbance Rejection Control

Considering the complex attitude dynamics of Rigid-Liquid-Flexible Coupling Spacecraft, especially when the large amplitude sloshing of liquid exists in rapid attitude maneuver, the conventional control design method based on the linearized model is not suitable for the significant coupling and strong nonlinearity. By contrast, Active Disturbance Rejection Control is model-independent and easy to design and implement. In this paper, the three main components of Active Disturbance Rejection Control, Tracking Differentiator, Extended State Observer and Nonlinear State Error Feedback are comprehensively applied to the attitude tracking control of Rigid-Liquid-Flexible Coupling Spacecraft. Firstly, two cascaded Tracking Differentiators are used to arrange the transition process of the attitude command, which not only reduces overshoot and oscillation of the tracking process, but also obtains the angular acceleration information of attitude trajectory for feed-forward compensation. Secondly, the total disturbance caused by liquid sloshing and flexible vibration is observed by Extended State Observer and compensated synchronously. Finally, the Nonlinear State Error Feedback is used to further improve the transient behavior and steady-state quality of the control system. The simulation results show that the tracking accuracy of the attitude and angular velocity using the Active Disturbance Rejection Control are about 3 to 27 times and 6 to 115 times that of the PID control, respectively. The convergence time, overshooting are also significantly less than the PID control.

Tracking Differentiator-Based Identification Method for Temperature Predictive Control of Uncooled Heating Processes

The temperature control of uncooled heating processes presents challenges due to a substantial lag and the absence of active cooling mechanisms, which can lead to overshoot and oscillations. To address these issues, we propose an anti-disturbance identification method based on a tracking differentiator (TD) and an input-constrained temperature predictive control (ICTPC) strategy. Our approach specifically considers the impact of unknown disturbances on model identification within a second-order heating process. By employing a TD to differentiate the input and output signals, we effectively minimize the identification error caused by low-frequency disturbances, yielding a robust anti-disturbance identification technique. Following this, we establish input constraints to limit the amplitude and variation of the control input, ensuring a more controlled and predictable system response. Using the identified model, an ICTPC algorithm is designed to achieve stable temperature control in uncooled heating processes. Experimental results from a typical uncooled heating system demonstrate that our method not only significantly reduces overshoot but also effectively mitigates temperature fluctuations, leading to enhanced control performance and system stability. This study provides a practical solution for temperature control in systems without cooling capabilities, offering substantial improvements in the efficiency and quality of industrial production processes.

Active Disturbance Rejection Control of Hydraulic Quadruped Robots Rotary Joints for Improved Impact Resistance

Hydraulic actuated quadruped robots have bright application prospects and significant research values in unmanned area investigation, disaster rescue and other scenarios, due to the advantages of high payload and high power to weight ratio. Among these fields, inevitable collision of robots may occur when contact with unknown objects, step on empty objects, or collapse, all of which have an impact on the working hydraulic system. To overcome the unknown external disturbances, this paper proposes an active disturbance rejection control (ADRC) strategy of double vane hydraulic rotary actuators for the hip joints of the quadruped robots. Considering the order of the valve-controlled actuator model, a three-stage tracking differentiator, a four-stage extended state observer, and a state error feedback controller are designed relatively, and the extended state observer is adopted to observe and compensate the uncertainty of external load torque of the system. The effectiveness of the ADRC method is verified in simulation environment and a single joint experimental platform. Moreover, the impact experiments of the limb leg unit are carried out after introducing the proposed ADRC strategy into hip joint, the limb leg unit of quadruped robots presents better impact resistance ability.

An efficient tracking differentiator based active disturbance rejection control for flight environment simulation system

The flight environment simulation system plays a crucial role in aeroengine testing, replicating real mission pressure and temperature conditions. However, existing systems face challenges in handling strong measurement noises and total disturbances during transient tests, leading to inaccurate simulations and potential actuator damage. To address these gaps, an enhanced tracking differentiator based active disturbance rejection control scheme is proposed and implemented. Our method introduces two key innovations: a discrete-time optimal control law that dynamically adjusts control inputs based on state transition and sampling times, and a phase-advancing method that mitigates filtering phase delays by incorporating derivative signals to predict input trends. These components are integrated into a novel tracking differentiator, enhancing the estimation capability of the extended state observer in active disturbance rejection control. Frequency-domain analysis demonstrates the proposed tracking differentiator's superior filtering performance and phase quality. Simulations of intake environment pressure during aeroengine transient tests reveal that, compared to traditional tracking differentiator-based active disturbance rejection control, the proposed approach significantly reduces both control signal oscillations and setpoint deviations by 75.44% and 53.9%, respectively. These improvements effectively address environment simulation deviations and unnecessary actuator wear, thereby enhancing the accuracy and reliability of aeroengine testing.

A Real-Valued Kalman Estimation Method for Harmonic Signal Analysis in Biomedical Applications

This work presents a methodology for obtaining the harmonic estimation of biomedical signals such as electrocardiogram, cardiorespiratory and blood pressure signals. The proposed methodology is achieved using polynomial approximation and the Kalman filter. As advantage, the technique includes instant estimations of signal harmonics and its derivatives using a real-valued model. Furthermore, a comparison of the results is conducted with the Savitzky-Gola, nonlinear tracking differentiator methods, extended state observer and digital differentiator base on Taylor series. The results suggest that the proposed method has the potential to enhance the quality of signal measurements, especially in the presence of noise.

Adaptive event-triggered finite-time prescribed performance control of PMSM stochastic system considering time-varying delays

This paper investigates the finite-time prescribed performance tracking control problem of permanent magnet synchronous motor (PMSM) considering stochastic disturbances and time-varying delays under event-triggered mechanism. A quintuple polynomial finite-time prescribed performance function (FPPF) is introduced to ensure the transient and steady-state performance of the system output, and a nonlinear transformation function is employed to convert the constrained error into an unconstrained one. The Lyapunov-Krasovskii function is constructed to address time delays. And the system uncertainties are approximated by the radial basis function neural networks (RBFNN). For the “explosion of complexity” caused by backstepping method, a tracking differentiator (TD) is employed. By combining finite time control, command filtering backstepping control, and event-triggered mechanism, the effect of the filter errors is decreased, and the update frequency of the control signals are reduced. It is shown that the proposed controller can guarantee finite time convergence bounded of all signals in the closed-loop system, and the tracking error can converge in finite time. Finally, simulation results are presented to illustrate the effectiveness of the proposed controller.

On the convergence of the linear tracking differentiator for signals with KH-integrable derivatives

On the convergence of the linear tracking differentiator for signals with KH-integrable derivatives

Adaptive backstepping control using a novel dual-differentiator for multi-degree-of-freedom manipulator uncertain systems with unknown multi-disturbances

In this article, the adaptive sliding mode backstepping controller (ASMBC) using a novel dual-differentiator is proposed for multi-degree-of-freedom (MDOF) manipulator systems with uncertain states and multi-disturbances. The dual-differentiator comprises a tracking differentiator (TD) and a disturbance observer (DOB) based on the proposed TD. A novel tracking differentiator based on the inverse hyperbolic sine function and terminal attractor function (IHSTD) is introduced to reconstruct unknown states like velocity responses of the manipulator systems. Furthermore, the other differentiator, DOB based on aforementioned IHSTD (IHSTD-DOB), is involved in estimating the uncertain and stochastic multi-disturbances affecting the manipulator systems. Then, an ASMBC scheme, combining proposed dual-differentiator, is developed for achieving accurate tracking control of manipulator. Additionally, to handle the “explosion of term” issue in backstepping control, the designed IHSTD is also utilized to estimate the derivative of the virtual control law. The stability of the controller is rigorously analyzed using the Lyapunov method. Finally, numerical simulation results are presented to validate the effectiveness of the proposed scheme. Comparison experiments with the traditional DOB, the scheme except IHSTD, and the classical sliding mode control method are carried out by Simulink. The results illustrate that the present control method not only has an excellent tracking performance but also accurately estimates the unpredictable multi-disturbance and reconstructs the unknown states of the manipulator systems.

Dynamical analysis and event-triggered adaptive finite-time prescribed performance control of the FO coupled MEMS resonators

This paper explores the dynamic behaviors and control method of weakly coupled micro-electro-mechanical system (MEMS) resonators with fractional-order (FO) dynamics. In the dynamics analysis session, Lyapunov exponents, bifurcation theory, and phase diagrams are used to analyze the effects of system parameters, coupling stiffness, and FO on the complex dynamical behaviors such as periodicity, pseudo-periodicity, and chaos oscillations. To suppress chaotic oscillations, a dynamic event-triggered finite-time prescribed performance controller is proposed. The unknown nonlinear functions are approximated through the interval type-3 fuzzy system (IT3FS) with an adaptive law. A novel finite-time prescribed performance function (finite-time PPF) is constructed to establish constrained boundaries for tracking errors. Compared with the existing PPFs, the proposed approach allows for more flexible setting of the convergence boundary before reaching steady-state. Subsequently, the constrained tracking errors are mapped to an unconstrained form using a nonlinear transformation function, whether the error constraint is symmetric or asymmetric. To circumvent the “explosion of complexity” arising from backstepping design process, a tracking differentiator (TD) is utilized. Additionally, to alleviate strain on communication resources, an event-triggered mechanism is devised to update control signals. The trigger threshold of the control signals can be adaptively adjusted based on the values of the Lyapunov function and the dynamic auxiliary variables. This control method guarantees finite-time convergence for all signals of the system. Finally, extensive simulations are performed to verify the effectiveness of the proposed algorithm.

Distributed Estimator-Based Containment Control for Multi-AUV Systems Subject to Input Saturation and Unknown Disturbance

This article addresses the containment control issue for multi-AUV systems with the intervention of both external disturbance and input saturation. Firstly, a distributed estimator is established for the sake of acquiring precise estimation information of the desired position and its derivative for each follower AUV in the system. Next, on the basis of the proposed distributed estimator, a virtual control law is designed for each follower AUV. Then, due to the difficulty in obtaining accurate information about the derivative of the virtual control law, a linear tracking differentiator is introduced. Additionally, a disturbance observer is employed to tackle the composite disturbance, which mainly contains the internal model uncertainties and external bounded disturbances. Meanwhile, the issue of input saturation is handled by constructing the auxiliary system. Furthermore, a containment control law is designed with the assistance of the introduced linear tracking differentiator, the established disturbance observer, and the constructed auxiliary system. Additionally, the Lyapunov stability theory is applied to analyze the stability of the multi-AUV system. Finally, simulation results are given to confirm the feasibility of the proposed containment control scheme.

Adaptive finite time tracking control for the perturbed nonlinear systems by using backstepping technique and a fast tracking differentiator

Adaptive finite time tracking control for the perturbed nonlinear systems by using backstepping technique and a fast tracking differentiator

Design and analysis of tracking differentiator based on SO(3)

Motivated by the issue of insufficient dynamic performance and tracking accuracy in SO(3)-based attitude tracking differentiators during large-angle maneuvers and complex trajectory tracking, a novel design approach for a three-degree-of-freedom attitude tracking differentiator within the SO(3) framework is proposed by incorporating second-order system theory and Lie group theory and improving the classical tracking differentiator. The kinematics model and error dynamics model of a rigid body on SO(3) are derived, and a reasonable virtual control input on SO(3) is constructed subsequently in order to achieve better dynamic response and tracking performance. Simulation and experimental results validate that the designed tracking differentiator could realize rapid and smooth convergence during large-angle maneuvers, and the initial large tracking error rapidly drops to near zero in a short period of time; additionally, it can also track expected time-varying curves well in complex trajectory tracking, with initial errors rapidly decreasing and maintaining at normal levels, demonstrating excellent tracking and control capabilities. There are strong application prospects for this new approach in addition to its theoretical significance.

Combining finite-time control and prescribed tracking performance for uncertain PMSM driven steer-by-wire system with unknown disturbance

To solve the uncertainties in permanent magnet synchronous motor (PMSM) driven steer-by-wire (SbW) system, a prescribed-performance-control (PPC) based finite-time fuzzy controller is proposed. Specifically, the dynamics of SbW system driven by PMSM is analyzed, and combine PPC technology to transform the tracking error coordinates. Then, the velocity signal is obtained from the sigmoid function-based tracking differentiator by using the position signal. To solve the problem of mismatched disturbance, fuzzy logic system (FLS) are used as a universal approximator to estimate the lumped nonlinear friction and other unknown functions, the boundary of lumped disturbance is estimated by the designed adaptive updating law. By fusing the designed virtual controller and the intermediate control law, the controller needs no precise parameters of system, under which the effect of uncertainty of control gain can be compensated completely. Finally, rigorous theoretical analysis based on the Lyapunov stability theory is provided to demonstrate the finite-time stability of the closed-loop system under consideration. Simulation and experiment results are presented to show the effectiveness of the developed control approach, and some comparisons are given to show the rapid and accurate position tracking control performance.

Distributed synchronization method of multi-motor driving system’s accelerated backstepping tracking control

This paper proposes a distributed synchronization control method and an accelerated backstepping tracking control scheme for the multi-motor driving system (MMDS). In the first step, we create a dynamic model of the MMDS with complex nonlinear dynamics, encompassing elements such as the dead zone, frictions, and disturbances. Next, in order to tackle the challenge of load tracking, we fuse a speed function, a cosine barrier function, a second-order tracking differentiator (TD), and a disturbance compensator into the backstepping approach. Lastly, to address potential issues related to diverse torque inputs, which could result in the overload occurrences, we put forward a novel distributed synchronization control scheme. This scheme aims to achieve torque synchronization for the MMDS while simultaneously ensuring superior load tracking performance. In the distributed synchronization control, a communication network is built to achieve the local coupling and improve the synchronization efficiency, and a corresponding mean deviation coupling synchronization control scheme is designed. Lyapunov theory is utilized to demonstrate the stability of the introduced control scheme. The simulation experimental results for the MMDS show the effectiveness of the proposed scheme.

Adaptive predefined-time tracking control of uncertain permanent-magnet synchronous motor drive chaotic system

A predefined-time tracking control algorithm is formulated for the Permanent Magnet Synchronous Motor(PMSM) driver system subjected to chaotic oscillations and undeterministic parameters, which can determine the upper bound of settling-time by controller gains. Radial Basis Function Neural Networks(RBFNNs) are employed to approximate the unknown nonlinear functions as universal approximators. The innovative predefined-time tracking differentiators are constructed to estimate the derivatives of the virtual inputs, resolving “the explosion of complexity” in traditional backstepping method. Then, an adaptive predefined-time controller is proposed in the backstepping framework. Lyapunov’s theoretical analysis proves that all signals in the closed-loop systems are uniformly ultimately bounded and the output tracking errors can converge to a small neighborhood of the origin within the predefined-time. Finally, simulation studies are performed to demonstrate that the proposed control scheme can quash the chaotic behaviors of the PMSM drive system and ensure fast-tracking performance even under unknown parameters.

Identification of linear flat outputs using neural networks—Examples of two-degree-of-freedom underactuated mechanical systems

This paper proposes a neural networks-based approach of finding flat output of linearized underactuated mechanical systems (UMS). Given that differential flatness and controllability are equivalent for linear systems, the problem is equivalent to finding the Brunovsky canonical form of linearized UMSs. We use a two degree-of-freedom (2DOF) system to illustrate the theoretical development. The proposed method identifies the local flat output of nonlinear mechanical systems from the measurements only, without a detailed mathematical model. The identification allows us to combine the well-known active disturbance rejection control (ADRC) and differential flatness control. A time-domain direct identification (TDDI) algorithm and its variant based on the algebraic method are proposed for flat output identification (FOID). The neural networks for implementing the TDDI algorithm called FOID-NN are created to evaluate flat output candidates and to use the flat output to reconstruct the system states in terms of a linear mapping. The neural networks are trained with the loss functions defined by reconstruction errors. Two special layers, namely tracking differentiator (TD) and algebraic layer, are inserted in the FOID-NN to handle noisy signal differentiations. Simulations of a cart–pole system and a stable 2DOF nonlinear UMS are carried out to show the range of applications of the FOID-NN. The experimental data of an underactuated rotary crane are used to validate the identified flat output.