Robust Differentiator Research Articles

Anti-disturbance leader–follower synchronization control of marine vessels for underway replenishment based on robust exact differentiators

This paper considers the leader–follower synchronization control of under-actuated marine vessels for underway replenishment. An integrated guidance and control method is presented to achieve the underway replenishment operations despite of unknown model uncertainties, external disturbances, and unknown velocities of the leader vessel. Specifically, a velocity observer based on a robust exact differentiator (RED) is firstly designed to estimate the unknown velocity of the leader vessel. Next, a finite-time light-of-sight guidance law based on the velocity observer is developed to synchronize the follower vessels with the leader vessel. Then, two disturbance observers based on REDs are designed to estimate the total disturbances composed of model uncertainties and external disturbances at the kinetic level. With the aid of the RED-based disturbance observers, an anti-disturbance nonlinear control law is presented to track the desired guidance signals in finite time. The tracking errors of the closed-loop system are proved to be ultimately uniformly bounded via Lyapunov stability theory. Finally, an example is utilized to illustrate the effectiveness of the proposed anti-disturbance synchronization control method for multiple under-actuated marine vessels.

Adaptive FIT-SMC Approach for an Anthropomorphic Manipulator With Robust Exact Differentiator and Neural Network-Based Friction Compensation

In robotic manipulators, feedback control of nonlinear systems with fast finite-time convergence is desirable. However, because of the parametric and model uncertainties, the robust control and tuning of the robotic manipulators pose many challenges related to the trajectory tracking of the robotic system. This research proposes a state-of-the-art control algorithm, which is the combination of fast integral terminal sliding mode control (FIT-SMC), robust exact differentiator (RED) observer, and feedforward neural network (FFNN) based estimator. Firstly, the dynamic model of the robotic manipulator is established for the n-degrees of freedom (DoFs) system by taking into account the dynamic LuGre friction model. Then, a FIT-SMC with friction compensation-based nonlinear control has been proposed for the robotic manipulator. In addition, a RED observer is developed to get the estimates of robotic manipulator joints’ velocities. Since the dynamic friction state of the LuGre friction model is unmeasurable, FFNN is established for training and estimating the friction torque. The Lyapunov method is presented to demonstrate the finite-time sliding mode enforcement and state convergence for a robotic manipulator. The proposed control approach has been simulated in the MATLAB/Simulink environment and compared with the system with no observer to characterize the control performance. Simulation results obtained with the proposed control strategy affirm its effectiveness for a multi-DoF robotic system with model-based friction compensation having an overshoot and a settling time less than 1.5% and 0.2950 seconds, respectively, for all the joints of the robotic manipulator.

Practical tracking control of piezoelectric actuators with time-delay estimation and nonsingular terminal sliding mode

A novel type of nonlinear robust control strategy is proposed in view of uncertain nonlinear factors, such as hysteresis, creep, and high-frequency vibration, of piezoelectric actuators (PEAs). This strategy can be used for the precise trajectory tracking of PEAs. The Bouc–Wen dynamic model is reasonably simplified to facilitate engineering application. The hysteresis term is summarized as an unknown term to avoid its nonlinear parameter identification. The controller robustness is achieved due to the nonsingular terminal sliding mode control, and the online estimation of unknown disturbances is realized because of the delay estimation technology; thus, no prior knowledge of the unknown boundary of the system is required. The precision robust differentiator is used to estimate the speed and acceleration signals in real time on the basis of the obtained displacement signals. The closed-loop stability of the system is proved by the Lyapunov criterion. Experimental results show that the proposed control strategy performs better than the traditional time-delay estimation control in terms of control accuracy and energy conservation. Therefore, the proposed control strategy can play an important role in the micro/nanofield driven by PEAs.

Non-asymptotic estimation for fractional integrals of noisy accelerations for fractional order vibration systems

In this paper, a class of fractional order vibration systems is considered, where the fractional differentiation orders are arbitrary real numbers between 0 and 2. The objective is to fast and robustly estimate the fractional integrals and derivatives of positions from noisy accelerations. In particular, the velocities and positions can be estimated by the proposed method. Since the proper fractional integrals of accelerations can be estimated using a numerical method, the main task of this work is to estimate unknown initial conditions. For this purpose, the generalized modulating functions method is adopted, which is recently developed to design non-asymptotic and robust fractional order differentiators. By distinguishing the cases where the stiffness matrix is invertible or not, algebraic integral formulas are provided for the unknown initial conditions in different situations. Hence, the unknowns can also be estimated even if the stiffness matrix is not invertible. Finally, the efficiency and robustness of the proposed estimators are verified by taking several numerical examples.

A Robust Observer-Based Control Strategy for n-DOF Uncertain Robot Manipulators with Fixed-Time Stability.

In this paper, a robust observer-based control strategy for n-DOF uncertain robot manipulators with fixed-time stability was developed. The novel fixed-time nonsingular sliding mode surface enables control errors to converge to the equilibrium point quickly within fixed time without singularity. The development of the novel fixed-time disturbance observer based on a uniform robust exact differentiator also allows uncertain terms and exterior disturbances to be proactively addressed. The designed observer can accurately approximate uncertain terms within a fixed time and contribute to significant chattering reduction in the traditional sliding mode control. A robust observer-based control strategy was formulated, according to a combination of the fixed-time nonsingular terminal sliding mode control method and the designed observer, to yield global fixed time stability for n-DOF uncertain robot manipulators. The proposed controller proved definitively that it was able to obtain global stabilization in fixed time. The approximation capability of the proposed observer, the convergence of the proposed sliding surface, and the effectiveness of the proposed control strategy in fixed time were fully confirmed by simulation performance on an industrial robot manipulator.

Second‐order adaptive integral terminal sliding mode approach to tracking control of robotic manipulators

A second-order adaptive integral terminal sliding mode controller is proposed for the trajectory tracking control of robotic manipulators with uncertainties. A second-order integral terminal sliding mode surface is designed for which an integral sliding mode (ISM) surface and a fast nonsingular integral terminal sliding mode surface are combined. By using the ISM surface, the reaching phase is removed, which enhances system robustness. The steady-state error is reduced because of the presence of an error integral term. A fast second-order nonsingular integral terminal sliding mode surface is employed to ensure that the ISM surface is able to converge to zero rapidly within a finite period of time without leading to a singularity problem. The control input of the proposed controller is continuous. Thus, the chattering phenomenon is removed. An adaptation technique is employed to estimate the upper bound of unknown lumped disturbance. The second-order derivative of position is calculated using a robust differentiator, making it practical. Simulations and experiments show that the proposed scheme improves the tracking performance and eliminates chattering.

Robust exact differentiators with predefined convergence time

The problem of exactly differentiating a signal with bounded second derivative is considered. A class of differentiators is proposed, which converge to the derivative of such a signal within a fixed, i.e., a finite and uniformly bounded convergence time. A tuning procedure is derived that allows to assign an arbitrary, predefined upper bound for this convergence time. It is furthermore shown that this bound can be made arbitrarily tight by appropriate tuning. The usefulness of the procedure is demonstrated by applying it to the well-known uniform robust exact differentiator, which is included in the considered class of differentiators as a special case.

Higher order sliding mode inspired nonlinear discrete-time observer

This work proposes a design scheme for arbitrary order discrete-time sliding mode observers for input-affine nonlinear systems. The dynamics of the estimation errors are represented in a pseudo-linear form, where the coefficients of the characteristic polynomial comprise the nonlinearities of the algorithm. The design process is reduced to a state-dependent eigenvalue placement procedure. Moreover, two different discrete-time eigenvalue mappings are proposed. As basis for the eigenvalue mappings serves a modified version of the continuous-time uniform robust exact differentiator. Due on the chosen eigenvalue mapping the proposed algorithm does not suffer from discretization chattering. Global asymptotic stability of the estimation errors for observers of order 2 and 3 is proven and the method to prove stability for higher order observers is demonstrated. The performance of a 3-rd order observer is illustrated in simulation. Simulation studies indicate that proposed discrete-time observer might possess an upper bound of its convergence time independent of the initial conditions.

Adaptive sliding-mode trajectory tracking control for state constraint master–slave manipulator systems

This study aims to propose an adaptive state-dependent gain finite-time convergent controller (using the fundamentals of the sliding mode theory) that solves the trajectory tracking for a class of state constraint master–slave robotic system (M-SRS) formed by two manipulators with the same number of articulations. The control design considers the effect of state constraints by implementing a state dependent adaptive gain. A Lyapunov-stability analysis leads to design the gain variation laws yielding proving the finite-time convergence of the sliding surface as well as the asymptotic convergence of the tracking error. The state constraints of the slave system motivate the characterization of the convergence-time as a function of the bounded uncertainties affecting the M-SRS dynamics. The forward-complete setting of the M-SRS justified the application of a robust and exact differentiator which estimated the articulation velocities for the slave robot. The estimated velocities are used as part of the realization of the output feedback controller. Numerical simulations demonstrate that the proposed control scheme provides a smaller quadratic norm of the tracking error compared with the obtained with other controllers (proportional–derivative and conventional sliding modes). The proposed control approach satisfies the state constraints while the sliding manifold converges to the origin in finite-time as justified by the theoretical stability analysis.

Observer-based robust integral of the sign of the error control of class I of underactuated mechanical systems: Theory and real-time experiments

In this paper, the control problem of a class I of underactuated mechanical systems (UMSs) is addressed. The considered class includes nonlinear UMSs with two degrees of freedom and one control input. Firstly, we propose the design of a robust integral of the sign of the error (RISE) control law, adequate for this special class. Based on a change of coordinates, the dynamics is transformed into a strict-feedback (SF) form. A Lyapunov-based technique is then employed to prove the asymptotic stability of the resulting closed-loop system. Numerical simulation results show the robustness and performance of the original RISE toward parametric uncertainties and disturbance rejection. A comparative study with a conventional sliding mode control reveals a significant robustness improvement with the proposed original RISE controller. However, in real-time experiments, the amplification of the measurement noise is a major problem. It has an impact on the behaviour of the motor and reduces the performance of the system. To deal with this issue, we propose to estimate the velocity using the robust Levant differentiator instead of the numerical derivative. Real-time experiments were performed on the testbed of the inertia wheel inverted pendulum to demonstrate the relevance of the proposed observer-based RISE control scheme. The obtained real-time experimental results and the obtained evaluation indices show clearly a better performance of the proposed observer-based RISE approach compared to the sliding mode and the original RISE controllers.

Software Sensor to Enhance Online Parametric Identification for Nonlinear Closed-Loop Systems for Robotic Applications.

This paper proposes an online direct closed-loop identification method based on a new dynamic sliding mode technique for robotic applications. The estimated parameters are obtained by minimizing the prediction error with respect to the vector of unknown parameters. The estimation step requires knowledge of the actual input and output of the system, as well as the successive estimate of the output derivatives. Therefore, a special robust differentiator based on higher-order sliding modes with a dynamic gain is defined. A proof of convergence is given for the robust differentiator. The dynamic parameters are estimated using the recursive least squares algorithm by the solution of a system model that is obtained from sampled positions along the closed-loop trajectory. An experimental validation is given for a 2 Degrees Of Freedom (2-DOF) robot manipulator, where direct and cross-validations are carried out. A comparative analysis is detailed to evaluate the algorithm’s effectiveness and reliability. Its performance is demonstrated by a better-quality torque prediction compared to other differentiators recently proposed in the literature. The experimental results highlight that the differentiator design strongly influences the online parametric identification and, thus, the prediction of system input variables.

Neuro-adaptive sliding mode control for underground coal gasification energy conversion process

ABSTRACT Due to the non-availability of model parameters, the model-base control of a nonlinear and infinite-dimensional underground coal gasification (UCG) process is a challenging task. In this paper, a robust neuro-adaptive sliding mode control (NASMC) is designed for the UCG process to maintain a desired heating value level. The unknown model parameters used in NASMC are estimated using the feed-forward neural network. Moreover, the controller also requires time derivatives of some model parameters, which are estimated by uniform robust exact differentiator. As the relative degree of the output with respect to the input is zero, therefore, to apply NASMC, the relative degree is increased to one. This approach maintains the desired heating value and provides insensitivity to input disturbance and model uncertainties. A comparison is also made between NASMC and an already designed conventional SMC. The simulation results show that NASMC exhibits better performance as compared to the conservative SMC design.

Anti-saturation adaptive finite-time neural network based fault-tolerant tracking control for a quadrotor UAV with external disturbances

This brief is focused on the finite-time tracking control of the quadrotor unmanned aerial vehicle (UAV) subject to external disturbances, parametric uncertainties, actuator faults, and input saturation. According to the principle of the Euler-Lagrangian methodology, a comprehensive dynamics is first decomposed into the position and attitude subsystems to accommodate the controller design. Different from the most of previous studies with an ideal assumption that velocity information is available, a robust exact differentiator is employed to gain the precise information of unavailable velocity in finite time. Then, by utilizing the strong approximation of the radial basis function neural network (RBFNN), the NN-based fault-tolerant control scheme is proposed to compensate for parametric uncertainties, external disturbances and actuator faults. More importantly, a novel adaptive mechanism is responsible for automatically adjusting the NN's parameters, which not only can effectively avoid the selection of large adaptive gains, but can also greatly decrease the number of online-updated leaning parameters. To solve the input saturation problem, an auxiliary dynamics system is constructed. Based on the Lyapunov theoretical framework, it is proved that all the closed-loop signals are uniformly ultimately bounded and the tracking errors can converge into small neighborhoods around the origin in finite time. Finally, simulation results are verified to intuitively reveal the good tracking performance of the introduced composite controller in terms of the finite-time error convergence, strong robustness, fault tolerance, and saturation elimination.

Non-singular terminal sliding-mode control for a manipulator robot using a barrier Lyapunov function

This study introduces a design of robust finite-time controllers that aims to solve the trajectory tracking of robot manipulators with full-state constraints. The control design is based on the construction of a distributed state constraint non-singular terminal sliding mode (CNTSM). The CNTSM design includes the gain self-adapting tuning method, which can ensure finite-time convergence to the sliding surface aside from the states to its corresponding reference trajectories. The implementation of the time-varying gain ensures the fulfillment of the accurate tracking for the references while the position and velocity constraints are satisfied permanently. A barrier Lyapunov function is proposed to develop the finite-time stability analysis of the designed controllers. The CNTSM realization uses the tracking error as well as its estimated derivative, which is calculated using a variant of adaptive super-twisting algorithm operating as robust differentiator. The proposed CNTSM is numerically evaluated on a two-link RM with uncertain inertia and Coriolis matrices. Simulation and experimental results evidence the efficiency of the CNTSM controller demonstrating a better tracking performance while the full-state constraints are satisfied in counterpart with the classical non-singular terminal sliding mode which is not able to keep such restrictions.

Neuro-adaptive fast integral terminal sliding mode control design with variable gain robust exact differentiator for under-actuated quadcopter UAV

In this paper, a robust global fast terminal attractor based full flight trajectory tracking control law has been developed for the available regular form which is operated under matched uncertainties. Based on the hierarchical control principle, the aforesaid model is first subdivided into two subsystems, i.e., a fully-actuated subsystem and an under-actuated subsystem. In other words, the under-actuated subsystem is further transformed into a regular form whereby the under-actuated characteristics are decoupled in terms of control inputs. In the proposed design, the nonlinear drift terms, which certainly varies in full flight, are estimated via functional link neural networks to improve the performance of the controller in full flight. Besides, a variable gain robust exact differentiator (VG-RED) is designed to provide us with estimated flight velocities. It has consequently reduced the noise in system’s velocities and has mapped this controller as a practical one. The finite-time sliding mode enforcement and the states’ convergence are shown, for all flight loops, i.e., forward flight and backward flight, via the Lyapunov approach. All these claims are verified via numerical simulations and experimental implementation of the quadcopter system in a Matlab environment. For a more impressive presentation, the developed simulation results are compared with standard literature.

Constrained sliding mode control of MIMO nonlinear systems

In this paper, the problems of state estimation and saturated control based on a high-order sliding mode observer (HOSMO) are evaluated for a class of multi-input and multi-output (MIMO) nonlinear systems under actuator and state constraints.For the state estimation of nonlinear systems, an estimation error system is established based on HOSMO and the convergence of the error system is analyzed using a robust exact differentiator. Then, a control scheme of the second-order sliding mode (SOSM) using the information from HOSMO is designed for uncertain nonlinear affine systems with arbitrary relative degrees. The finite-time stability of the estimated system is analyzed using the designed SOSM control law under both unconstrained and constrained actuators and states. Accordingly, the geometric analysis of the resulting sliding motions is performed to depict the maximal attraction domain of the sliding motions. Finally, two simulation examples are provided to validate the effectiveness of the presented methods.

Multivariable Binary MRAC with Guaranteed Transient Performance Using Non-homogeneous Robust Exact Differentiators

In this paper, we propose an extension of the binary model reference adaptive control (BMRAC) for uncertain multivariable linear plants with non-uniform arbitrary relative degree. The BMRAC is a robust adaptive strategy which has good transient properties and robustness of sliding mode control with the important advantage of having a continuous control signal free of chattering. The relative degree obstacle is circumvented using a multivariable version of a hybrid estimation scheme, named global robust exact differentiator (GRED). The hybrid estimator switches between robust exact differentiators (RED) based on higher-order sliding modes and lead filters in a way that the exact derivatives are globally obtained in finite time. To improve the robustness and the transient performance of the GRED, we propose a modification of the switching scheme replacing the conventional RED with a non-homogeneous one. Global exact output tracking is obtained with robustness and guaranteed transient performance without requiring stringent symmetry assumptions on the plant high-frequency-gain matrix.

Real-Time Sensorless Robust Velocity Controller Applied to a DC-Motor for Emulating a Wind Turbine

The wind power systems of variable velocity using a doubly-fed induction generator dominate large-scale electrical generation within renewable energy sources. The usual control goal of the wind systems consists of maximizing the wind energy capture and streamlining the energy conversion process. In addition, these systems are an intermittent energy source due to the variation of the wind velocity. Consequently, the control system designed to establish a reliable operation of the wind system represents the main challenge. Therefore, emulating the operation of the wind turbine by means of an electric motor is a common strategy so that the controller design is focused on the induction generator and its connection to the utility grid. Thus, we propose to emulate the dynamical operation of a wind turbine through a separately excited DC motor driving by a sensor-less velocity controller. This controller is synthesized based on the state-feedback linearization technique combined with the super-twisting algorithm to set a robust closed-loop system in the presence of external disturbances. A robust velocity observer is designed to estimate the rotor velocity based on the armature current measuring. Furthermore, a robust differentiator is designed for estimating the time derivative of the velocity error variable, achieving a reduction in the computational calculus. Experimental tests were carried using a separately excited DC motor coupled with a dynamometer to validate the proposed wind turbine emulator.

Trajectory Tracking Control of Autonomous Underwater Vehicle With Unknown Parameters and External Disturbances

Most studies so far on trajectory tracking control of autonomous underwater vehicle (AUV) have assumed that the Euler angles are exactly known. However, the AUV inevitably suffers from external environmental disturbances which are driven by wind, density, and temperature gradients. The attitude transducers cannot derive accurate attitude information of the AUV. Additionally, the uncertain hydrodynamic parameters affect the stability of the system. Consequently, it is unknown whether tracking performance of the AUV can be guaranteed. In order to overcome these drawbacks, in this paper, a finite-time controller is developed by using the nonsingular fast terminal sliding mode control technique. A robust differentiator is proposed to estimate the external disturbances and uncertain parts. Simulations are performed to show that with the proposed control laws, the AUV converges to the desired trajectory even in the presence of external disturbances and system uncertainty.

Robust Fixed-Time Inverse Dynamic Control for Uncertain Robot Manipulator System

This paper proposes a novel robust fixed-time control for the robot manipulator system with uncertainties. Based on the uniform robust exact differentiator (URED) algorithm, a robust control term is constructed. Then, a robust fixed-time inverse dynamics control (IDC) is proposed. For the proposed control method, the fixed-time stability of a closed-loop system with uncertainties is strictly proved. The newly proposed method exhibits the following two attractive features. First, the proposed control scheme extends the existing fixed-time IDC for the robot manipulator system to the robust control scheme. Second, the proposed method is strictly nonsingular rather than the commonly used approximate approach. Simulation result demonstrates the effectiveness of the proposed control scheme.