Nonlinear Tracking Differentiator Research Articles

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.

Design of Backstepping Control Based on a Softsign Linear–Nonlinear Tracking Differentiator for an Electro-Optical Tracking System

To address the problems of a low tracking accuracy and slow error convergence in high-order single-input, single-output electro-optical tracking systems, a backstepping control method based on a Softsign linear–nonlinear tracking differentiator is proposed. First, a linear–nonlinear tracking differentiator is designed in conjunction with the Softsign excitation function, using its output as an approximate replacement for the conventional differentiation process. Then, this is combined with backstepping control to eliminate the “explosion of complexity” problem in conventional backstepping procedures due to repeated derivation of virtual control quantities. This reduces the workload of parameter tuning, takes into account the rapidity and stability of signal convergence, and improves the trajectory tracking performance. This method can ensure the boundedness of the system signal. The effectiveness and superiority of this control method are verified through simulations and experiments.

Position Identification Algorithm of PMSM Based on Improved Nonlinear Tracking Differentiator Method

Position Identification Algorithm of PMSM Based on Improved Nonlinear Tracking Differentiator Method

Frequency analysis of a discrete-time fast nonlinear tracking differentiator algorithm based on isochronic region method

<p>In fault detection, feedback control, and other fields, real-time differential estimation of a given signal in a complex noise environment is an important but challenging task. In this paper, a discrete-time fast nonlinear tracking differentiator (FNTD) based on hyperbolic tangent functions was proposed. To start, the differential signal acquisition problem was equated to the time-optimal control (TOC) law for constructing a double-integral system using a state feedback approach. Next, the FNTD algorithm based on the hyperbolic tangent function was presented by utilizing the isochronic region (IR) method in the discrete time domain. Then, the frequency-domain characteristics of the FNTD were analyzed and the rule for tuning the parameters was provided by the frequency scan test method. Finally, the simulation results demonstrated that the proposed FNTD had fast and accurate tracking performance, as well as excellent filtering and differential extraction capability compared with other differentiators.</p>

Robotic abrasive belt grinding of complex curved blades based on a novel force control architecture integrating smooth trajectories

Constant force control and smooth trajectory planning are regarded as key strategies to improve the surface quality of blades in robotic abrasive belt grinding. However, some technical challenges still remain: (1) external normal grinding force with time-consuming sensing, large fluctuations, and force overshoot, and (2) lack of global C2 continuous grinding trajectories. In this paper, a unified architecture is proposed to concurrently address the above issues. The main contribution of this paper is to provide a complete solution for robots to automatically accomplish the grinding of complex curved blades. The grinding process is arranged as follows: Firstly, cubic B-spline and C2 continuous quaternion spline curves are used to generate the global C2 continuous trajectory, thereby avoiding vibrations and unexpected slowdowns during grinding. Secondly, the actual grinding force is sensed based on a low-speed Fourier series trajectory and the least square algorithm. Finally, a novel force control scheme consisting of an adaptive hybrid impedance (AHI) control scheme and two nonlinear tracking differentiator (NTD) modules is proposed to overcome the problems of large force fluctuations and force overshoot. Experimental results demonstrate that the proposed solution delivers better performance. Furthermore, the surface roughness Ra of the machined blade is controlled within 0.8 μm, which meets the requirements of real-world industrial applications.

Dual PID Adaptive Variable Impedance Constant Force Control for Grinding Robot

High-precision and low-overshoot force control are important to guarantee the material removal rate and surface quality of robot grinding. However, traditional force control methods are subjected to positional disturbance, stiffness disturbance, contact process nonlinearity, and force-position coupling, leading to difficulties in robot constant force control. Therefore, how to achieve smooth, stable, and high-precision constant force control is an urgent problem. To address this problem, a dual PID adaptive variable impedance control is established (DPAVIC). Firstly, PD control is used to compensate for the force error, and PID is used to update the damping parameters to compensate for the disturbance. Secondly, a nonlinear tracking differentiator is used to smooth the desired force and reduce the contact force overshoot. Then, the stability, convergence, and effectiveness of the force control algorithm are verified via theoretical analysis, simulations, and experiments. The force tracking error and overshoot of a conventional impedance controller (CIC), adaptive variable impedance control (AVIC), and DPAVIC are analyzed. Finally, the algorithm is used in grinding experiments on a thin-walled workpiece. The force tracking error is controlled within ±0.2 N, and the surface roughness of the workpiece is improved to Ra 0.218 μm.

Precise and Efficient Pointing Control of a 2.5-m-Wide Field Survey Telescope Using ADRC and Nonlinear Disturbance Observer.

Linear active disturbance rejection control (LADRC) has been widely used to improve the tracking accuracy and anti-disturbance performance of telescope servo control under disturbances. However, the linear extended state observer (LESO) is sensitive to noise, and its bandwidth is limited by the resonant frequency of the telescope. To enhance the LARDC's ability to attenuate disturbances, a novel cascade anti-disturbance structure (NCADS) with LADRC on the outer speed loop and a nonlinear disturbance observer (NDOB) on the inner current loop is proposed. The NDOB compensates for the dominant disturbance through feedforwarding the q-axis current reference, and the LESO compensates for the residual disturbance on the outer speed loop. First, the NCADS is introduced in a three-closed-loop control framework of PMSM. Then, the design method of the controller for each loop and the NDOB are presented, the parameter-tuning method based on bandwidth is demonstrated, and the convergence of the NDOB is proved. Furthermore, to improve the searching and tracking efficiency of wide-field survey telescopes, the nonlinear tracking differentiator (NTD) was modified to plan the transition process of the position loop, which only needs to set the maximum speed and acceleration of the telescope. Finally, simulations and experiments were performed on a 2.5-m-wide field survey telescope. The experimental results verify that the proposed NCADS method has a better anti-disturbance performance and higher tracking precision than the conventional method, and the improved NTD method does not need to tune parameters and achieved a fast and smooth transition process of the position loop.

Robust distributed formation control with prescribed performance for nonlinear multi‐agent systems subjected to compound disturbances

SummaryThis article is concerned with the distributed leader‐following formation control for nonlinear multi‐agent systems (MASs) subject to compound disturbances. Three aspects of the control objective: formation control, collision avoidance and connectivity preservation are achieved by the designed controller composed of three control laws. The formation control law is developed based on the barrier Lyapunov function and the backstepping method, which can prescribe the performance of the close‐loop system to some extent. Within our scheme, an adaptive finite‐time disturbance observer is used to enhance the robustness of the control system against compound disturbances including parameter perturbations, actuator bias faults and external disturbances. Meanwhile, a nonlinear tracking differentiator is introduced for each follower to cancel complex differential calculations and the requirement for the leader's acceleration information. Furthermore, the auxiliary control laws for collision avoidance and connectivity preservation are designed based on the artificial potential field method, which plays an important role during the formation shaping. Finally, complete stability analysis and vivid numerical simulation are carried out.

Model-free finite-time formation containment control of underactuated UUVs subject to input delays and unknown interaction information

This paper put forwards a model-free adaptive neural finite-time formation-containment (FC) control scheme for time-delayed underactuated unmanned underwater vehicles (UUVs) with unknown dynamic models and some interaction information under switching topologies. The UUVs are steered by multiple child leaders moving along with a root leader, where all leaders are virtual and defined by a new generalized dynamic. A novel autoregressive model-based state predictor is created for each UUV, which can construct future motion states online relying on current and stored states. Considering the containment layer, a finite-time super-twisting observer is used to recover the unavailable interaction information. Moreover, observer-based finite-time kinematic guidance laws employing only 3D positions of the neighboring UUVs are devised, together with protocols for completing formation in finite time. Aiming to accurately identify unknown dynamics and control gains simultaneously, estimator-based adaptive neural networks are introduced exploiting both historical and real-time data. Finally, model-free anti-disturbance finite-time kinetic control protocols are designed based on the data-driven adaptive neural networks and nonlinear tracking differentiators. It is proven that all signals of the closed-loop FC system are bounded, the deployed FC tracking can be achieved in finite time. Simulation results verify the feasibility and superiority of the proposed FC control scheme.

Finite-Time Output Feedback Cooperative Formation Control for Marine Surface Vessels With Unknown Actuator Faults

This article studies the finite-time output-feedback cooperative formation problem of multiple marine surface vessels (MSVs) without using velocity information, where each MSV contains unknown time-varying actuator faults, ocean disturbances, and model uncertainties. Considering the performance specifications and requirements in practice, the formation errors and velocities are also required to change in a predefined compact set. Aiming at the challenge of unknown input gain caused by unknown time-varying multiplicative faults, a novel nonlinear extended state observer with an adaptive law is first constructed, which cannot only recover unmeasurable velocities from position-heading information but also simultaneously compensate for ocean disturbances, model uncertainties, and additive faults. Subsequently, combining the barrier Lyapunov function and the nonlinear tracking differentiator technique, a unified finite-time output-feedback cooperative control framework is developed, including both kinematics and kinetics, to prevent constraint deviation and avoid the computational complexity of the traditional backstepping method. Using finite-time stability theory, the designed output feedback cooperative controller can ensure that the desired cooperative performance is achieved within a finite time and tracking errors converge to a small neighborhood of the origin while ensuring that the closed-loop system is semiglobally uniformly ultimately bounded. Simulation examples are shown to demonstrate the effectiveness of the proposed control scheme.

A composite adaptive robust control for pneumatic servo systems with time-varying inertia

This article proposes a composite adaptive robust control for a pneumatic servo system with time-varying inertia to achieve a good tracking performance. The proposed design not only preserves the merits of both direct adaptive robust control and indirect adaptive robust control but also obtains a better performance of the working condition with time-varying inertias. Firstly, prescribed tracking performance is guaranteed without knowing the bounds of parametric and non-parametric uncertainties by leveraging a discontinuous projection and deterministic robust control technique. And a nonlinear tracking differentiator is used to obviate differential signal in backstepping. Secondly, the online parametric estimation algorithm is developed by intelligently integrating tracking error dynamics and physical plant dynamics. By effectively using the information of both tracking error and prediction error, a high adaptation gain can be used for achieving a faster parameter convergence and smaller tracking error. Moreover, exponential convergence of both tracking error and prediction error can be obtained under certain conditions. Lastly, comparative experiments on different conditions are carried out for a single-rod pneumatic cylinder driven by a proportional directional valve to demonstrate the superiority of the proposed method.

Design and Stability Analysis of Sliding Mode Controller for Non-Holonomic Differential Drive Mobile Robots

This paper presents a novel extended state observer (ESO) approach for a class of plants with nonlinear dynamics. The proposed observer estimates both the state variables and the total disturbance, which includes both exogenous and endogenous disturbance. The study’s changes can be summarized by developing a sliding mode higher-order extended state observer with a higher-order augmented state and a nonlinear function for the estimation error correction terms (SMHOESO). By including multiple enhanced states, the proposed observer can monitor total disturbances asymptotically, with the second derivative of the total disturbance serving as an upper constraint on the estimation error. This feature improves the observer’s ability to estimate higher-order disturbances and uncertainty. To extend the concept of the linear extended state observer (LESO), a nonlinear function can modify the estimation error in such a way that the proposed observer can provide faster and more accurate estimations of the state and total disturbance. The proposed nonlinearity also reduces the chattering issue with LESOs. This research thoroughly examines and analyzes the proposed SMHOESO’s convergence using the Lyapunov technique. According to this analysis, the SMHOESO is asymptotically stable, and the estimation error can be significantly reduced under real-world conditions. In addition to the SMHOESO, a modified Active Disturbance Rejection Control (ADRC) scheme is built, which includes a nonlinear state error feedback (NLSEF) controller and a nonlinear tracking differentiator (TD). Several nonlinear models, including the Differential Drive Mobile Robot (DDMR), are numerically simulated, and the proposed SMHOESO is compared to several alternative types, demonstrating a significant reduction in controller energy, increased control signal smoothness, and accurate tracking of the reference signal.

Adaptive neural network observer‐based filtered backstepping control for nonlinear systems with fuzzy dead‐zone and uncertainty

SummaryThis study proposes a high gain observer‐based filtered backstepping control scheme for nonlinear systems with unknown dynamics and dead‐zone constraint. Majority of the previously papers describe dead‐zone by the conventional certain models, assume that all state variables are available and invoke the backstepping or conventional dynamic surface control (CDSC) approaches for controller design. This work describes the dead‐zone by the fuzzy model, invokes radial basis function neural network (RBFNN) to model the unknown functions and proposes a RBFNN‐based high gain observer to estimate unmeasurable states. To avoid the complexity explosion in the backstepping approach and eliminate the sensitivity to the time‐constant of the first‐order filters in the CDSC, a nonlinear tracking differentiator (NTD) with finite time convergence property is used instead of the first‐order filters in the CDSC to obtain derivative of the virtual inputs. Moreover, the error compensation mechanism and auxiliary signal are proposed to eliminate the influence of the filtering error and input nonlinearity, respectively. Despite the fuzzy dead‐zone, the proposed scheme makes the closed‐loop signals uniformly ultimately bounded. In comparison with the existing results, (i) some assumptions like certain model of dead‐zone and full state measurement are removed, (ii) “explosion of complexity” and sensitivity to the time constant of the first‐order filters are eliminated, (iii) effect of the filtering error and input nonlinearity are compensated, (iv) number of adjustable parameters and online computational burden are decreased effectively. Simulation results on the two well‐known examples verify the effectiveness and applicability of the proposed new design technique.

Simulation Study of Improved DDSRF-PLL under Distorted Grid Voltage

When the grid voltage is distorted and harmonic and DC components are mixed, the phase locking accuracy of decoupled double synchronous reference frame for phase-locked loop (DDSRF-PLL) will be greatly reduced. This paper proposes an improved DDSRF-PLL(IDDSRF-PLL) structure based on Frequency Adaptive Sliding Discrete Fourier Transform (FASDFT) filtering. A cascade SDFT structure is adopted for FASDFT filtering. The SDFT with fixed sliding window width at the front stage extracts the fundamental wave signal of power grid voltage to eliminate the influence of harmonics and DC components. The latter SDFT detects the sliding window width of power grid frequency update in real time through a nonlinear tracking differentiator (TD) to suppress spectrum leakage. Then DDSRFT-PLL is used to separate the positive and negative sequence of the fundamental signal and lock the phase. Finally, the simulation results in Matlab/Simulink verify the correctness and effectiveness of the proposed method.

On convergence of nonlinear tracking differentiator based on finite-time stable system subject to stochastic noise

In this paper, we consider convergence for a nonlinear tracking differentiator constructed from a finite-time stable system, where the input signal is corrupted by bounded stochastic noise. The existence of global weak solutions is firstly proved. It is shown that the nonlinear tracking differentiator tracks the input signal in the almost surely practical sense. With the same tuning parameter, the convergence accuracy is shown to be higher than that of the linear tracking differentiator. Some numerical simulations are performed to validate the effectiveness of the proposed differentiator.

Output-Feedback Flexible Performance-Based Safe Formation for Underactuated Unmanned Surface Vehicles

This article studies a cooperative time-varying formation of underactuated unmanned surface vehicles (USVs) with position-heading measurements, following the philosophies of (i) balance operation between performance constraints and safety requirements including collision avoidance and connectivity maintenance, and (ii) concise finite-time control design based on the integrated finite-time extended state observer (IFTESO) to facilitate the practical implementation. At first, to accomplish the guidance design of networked vehicles in the complex ocean environment and guarantee the performance of cooperative tracking errors, a flexible performance-based control scheme is investigated by feeding the potential effects of safety requirements into the modification signals of the user-specified performance functions. Then, without the application of cumbersome nonlinear tracking differentiators and backstepping techniques, a output-feedback IFTESO-based method provides a concise solution with a convenient design by taking advantage of homogeneous system theories, which further reduces the traditional computation consumption. Finally, the performance of the proposed scheme is validated by numerical simulations.

Nonlinear Tracking Differentiator Based Prescribed Performance Control for Space Manipulator

Nonlinear Tracking Differentiator Based Prescribed Performance Control for Space Manipulator

Cooperative trajectory tracking control of MUSVs with periodic relative threshold event-triggered mechanism and safe distance

This paper addresses the problem of cooperative trajectory tracking control problem for multiple unmanned surface vehicles (MUSVs) with event-triggered mechanism and safe distance, proposes a novel event-triggered adaptive sliding mode control strategy via nonlinear tracking differentiators (NTD) and periodic relative threshold event-triggered mechanism (PRTETM). Firstly, the formation autonomous navigation system is constructed by multiple leader–follower cooperative structure to establish via the synchronization scheme that guarantee the safe distance each others. Next, a novel adaptive sliding mode control with the square root form of position error is introduced to prevent the oscillation or divergence of the system caused by the larger position error. Then, a concise adaptive control strategy to compensate for dynamic model uncertainties and external environmental disturbances. Meanwhile, the NTD is used to quickly track the differential signal and achieve filter performance. Besides, to obtain smaller update frequency and less energy consumption, the PRTETM without Zero behavior and input saturation are introduced to facilitate the triggering of control inputs and avoid the actuator physical constraints. Theoretical analysis proves that the proposed cooperative control system is uniformly ultimately bounded. Finally, numerical simulations and comparisons consolidate that the effectiveness of the above control scheme and superior in the fixed threshold scheme.

Nonlinear tracking differentiator and PD controller for piezoelectric actuators in a robotic hand

This article deals with the modeling and control design for piezoelectric actuators of a micro-robotic system. Piezoelectric actuators generally exhibit complex and nonlinear behavior due to their inherent asymmetric hysteresis phenomenon, creep nonlinearity, and oscillatory characteristics. Moreover, these actuators are subject to an external disturbance force caused by the interaction with the manipulated objects. It has been shown in practice that these characteristics have a negative and significant impact on the performance of piezoelectric actuators, which increases the difficulty of their control. The main objective of this paper is to account simultaneously for all undesired and complex characteristics outlined above. To this end, our main idea is to lump the hysteresis and creep phenomena, as well as the disturbing manipulation force together as one total disturbance term. The total disturbance is first estimated in real-time via a nonlinear tracking differentiator, then feedback control law is designed to compensate for the disturbance and achieve a good transient profile. To validate and demonstrate the efficacy of the proposed approach, experiments are conducted, and results obtained are discussed.

A Variable Parameter Linear Tracking Differentiator and Its Application in Large Ground-based Telescopes

This paper is concerned with the characteristics of a tracking differentiator (TD) in arranging transition process for the physical system. In order to overcome disadvantages of linear TD (LTD) in practice, a novel variable parameter linear tracking differentiator (VLTD) is proposed. By designing the speediness factor as a function of the tracking error, the VLTD can track a large range of set values with reasonable speed and acceleration. Analysis shows that VLTD can converge to its set value under certain conditions. Meanwhile, the speed and acceleration bounds are added to the VLTD, which guarantees that the proposed transition signal really plays a transitional role. The numerical simulation results emphasize necessity for adding speed and acceleration bounds to the VLTD. By comparing VLTD with the nonlinear TD (NLTD) in the simulations and experiments, VLTD can achieve almost the same performance as the NLTD while it is easier to be implemented.