Levant's Differentiator Research Articles

Speed regulation system based on proportional resonance self-coupling PI control

In this paper, an improved self-coupled PI control dual-loop control strategy is proposed for the vector control of permanent magnet synchronous electric speed control system, to improve the poor motor control effect caused by the dead zone of the inverter, the presence of harmonics in the motor magnetic field distribution, and the error in the sampling process of the current. In order to reduce the tracking error of the signal, an acceleration feed-forward method is adopted, and the self-coupled PI controller with an additional Levant differentiator is introduced into the speed loop of the permanent magnet synchronous motor to ensure the accurate derivation of the input signal containing noise to improve the robustness of the system. In addition, this strategy combines the proportional term of the autotuned PI control structure with the standard resonator to form a proportional resonant autotuned PI in the current loop, so that the permanent magnet synchronous motor can better compensate the current in the current loop control to achieve the effect of current harmonic suppression. Finally, the control mechanism of the servo system analyzed, and based on the mathematical model of the controlled object PMSM, the design method of the improved self-coupling PI control structure is given. The experimental results show that the proposed control strategy can reduce the impact of the motor's own parameter changes on the system performance and has a good current harmonic suppression in the low-speed domain, which verifies the effectiveness of the designed improved self-coupled PI controller.

Output-mask-based adaptive NN control for stochastic time-delayed multi-agent systems with a unified event-triggered approach

This paper investigates a class of output-mask-based adaptive neural network (NN) tracking control for nonlinear stochastic time-delayed multi-agent systems (STMASs) based on a unified event-triggered approach. The output signal relies on an output mapping acted as a mask, defined as a privacy-protection-like method, so that the internal state of one agent cannot be identified by other distrustful eavesdroppers or attackers. Moreover, the construction of a unified event-triggered control scheme retains the advantages of the saturation threshold triggering strategy, incorporates distributed errors, and increases the flexibility of thresholds. Furthermore, for stochastic time-delay multi-agent systems, the initial value limitation of the conventional first-order filter is removed by a first-order Levant differentiator, and a new estimation term in the fuzzy observer is established to solve the nonlinear fault. The unknown function in pure-feedback form is addressed via combining Butterworth low-pass filter and radial basis function neural networks (RBF NNs). Finally, the boundedness of all signals in the closed-loop systems is demonstrated, and the effectiveness of the proposed algorithm is verified by some simulation results.

Distributed finite-time bearing-based formation control for underactuated surface vessels with Levant differentiator

In this paper, a distributed bearing-based formation control scheme with finite-time convergency is proposed for multiple underactuated surface vessels (USVs). By virtue of the guide point-based model transformation method, the dimension of underactuated tracking error dynamics can be reduced from three to two. This allows us to convert the actuator model to a fully form that matches dimension-reduced tracking error dynamics. Further, to reduce the computation complexity, a finite-time recruited filter is designed according to first-order Levant differentiator. At meanwhile, auxiliary error compensation signals are introduced to address unknowns stemming from filter process and system dynamics. Since merely relative-distances and inner bearings are utilized in the proposed bearing-based formation approaches, the proposed control scheme provides a feasible solution to achieve formation control under body-fixed frame. Finally, theoretical analysis and numerical simulations are presented to demonstrate the efficacy.

Real-Time Control of Underactuated Systems in a Mechatronics Kit Via a Novel LMI-Based Sliding Mode Approach

Real-time implementation of a sliding mode control scheme based on the unit vector approach, for a variety of underactuated configurations of a mechatronics kit (rotational, inertia, and double pendulum) is presented in this work. Details are given to perform diffeomorphisms leading to the required normal form, based on which design conditions are cast as linear matrix inequalities, thus improving numerical systematicness of the traditional methodology. Once the control law is designed, its implementation requires reliable estimates of the velocities since the different plants under consideration do not measure them; to this end, Levant's robust differentiator is employed. Results are provided that show the effectiveness of the proposal.

Finite-Time Composite Learning Control of Strict-Feedback Nonlinear System Using Historical Stack.

This article investigates the finite-time control of the strict-feedback nonlinear system using composite learning based on the historical stack. The controller design adopts the backstepping scheme while the nonlinear function is introduced to avoid the singularity problem. The first-order Levant differentiator is introduced to obtain the filtered command signal and the compensation signal is further constructed. To indicate the learning performance, the historical data over the moving time window are analyzed to construct the predictor error using the maximum-minimum singular value algorithm. Furthermore, the finite-time neural update law is proposed. The stability of the closed-loop system is analyzed via the Lyapunov approach. The performance of the proposed method is verified using simulations.

Closed-Form Stability Conditions and Differentiation Error Bounds for Levant's Arbitrary Order Robust Exact Differentiator

This paper proposes, for the first time, closed-form stability conditions and differentiation error upper bounds for arbitrary orders of Levant's robust exact Differentiator. Based on these conditions, a tuning rule is provided to select the Differentiator parameters. Numerical examples demonstrate the application of the proposed tuning rule and compare the conservativeness of the obtained conditions to existing results.

Active disturbance rejection control for a transport equation via a differentiatior

This paper deals with the stabilization of a transport equation subject to a boundary disturbance. Our feedback design relies on the so-called strategy called active disturbance rejection control (ADRC). The unknown disturbance is estimated by Levant's differentiator and one of the feature of this differentiator is that it allows to estimate in finite-time the disturbance. We prove the existence of solutions of the closed-loop system and the global asymptotic stability of the closed-loop system. A numerical example is given to illustrate the efficiency of our strategy.

Command filter‐based adaptive neural network controller design for uncertain nonsmooth nonlinear systems with output constraint

SummaryThis paper investigates the command filter‐based adaptive neural network tracking control problem for uncertain nonsmooth nonlinear systems. First, an integral barrier Lyapunov function is introduced to deal with the symmetric output constraint and make the output comply with prescribed restrictions. Second, by the Filippov's differential inclusion theory and approximation theorem, the considered nonsmooth nonlinear system is converted to an equivalent smooth nonlinear system. Third, the Levant's differentiator is used to deal with the “explosion of complexity” problem. An error compensation mechanism is established to attenuate the effect of the filtering error on control performance. Then, an adaptive neural network controller is set up by resorting to the backstepping technique. It is strictly mathematically proved that the tracking error can converge to an arbitrarily small neighborhood of the origin and all the signals in the closed‐loop system are semi‐globally uniformly ultimately bounded. Finally, a numerical example and an application example of the robotic manipulator system are provided to demonstrate the availability of the proposed control strategy.

Distributed finite-time performance-prescribed time-varying formation control of autonomous surface vehicles with saturated inputs

This paper studies a time-varying formation control problem of a swarm of autonomous surface vehicles (ASVs) with prescribed constraints. Every ASV suffers from unavailable velocities, saturated inputs, internal uncertainties, and external disturbances. A distributed finite-time performance-prescribed time-varying formation control method is proposed for ASVs. Specifically, an adaptive fuzzy state observer (AFSO) is firstly proposed to recover the unavailable velocity by using saturated inputs and estimated disturbances from the fuzzy logic system. Secondly, a tunnel prescribed performance (TPP) function is designed to limit tracking errors with a concise form and smaller overshoot. At the kinematic level, with TPP-based error transformation, a C1 finite-time time-varying guidance law is proposed by using the smooth switch functions and recovered velocities. Next, a finite-time command filter based on Levant differentiator is presented to smooth the guidance signals. At the kinetic level, a C1 finite-time saturated control law is developed with the estimated velocities and disturbances. Then, it proves that tracking errors of the proposed closed-loop system converge into a small neighborhood around the equilibrium within finite time while evolving under TPP constraints regardless of saturated inputs. Finally, comparison results are provided to demonstrate the effectiveness of the proposed distributed finite-time performance-prescribed time-varying formation control method.

Finite-time command-filtered backstepping control for nonlinear systems with input delay and time-varying full-state constraints

This paper examines a finite-time command-filtered backstepping control proposed for a class of strict-feedback nonlinear systems with input delay and time-varying asymmetric full-state constraints. The existing control method either ignores the effect of input delay or converges asymptotically in infinite time. A command-filtered backstepping design is used to decrease computational burden. A first-order Levant differentiator is employed to replace the command filter for estimating the virtual control signals in finite time. The time-varying asymmetric barrier Lyapunov function (TVABLF) and command-filtered backstepping design are applied to alleviate the significant challenges caused by the backstepping approach and full-state constraints. A novel finite-time delay compensation mechanism is proposed to remove the impact of input delay. The closed-loop signals are proved to be practical bounded. Simulation results are provided to demonstrate the effectiveness of the proposed control scheme.

Modeling and Adaptive Tension Control of Chain Transmission System With Variable Stiffness and Random Load

In the fully mechanized mining face, proper chain tension of chain transmission system is of great significance in reducing the failure rate and improving the service life of scraper conveyor. However, the disturbance caused by random load and variable stiffness makes it very difficult to control chain tension. An adaptive tension control scheme combining a neural command filtering backstepping algorithm and online identification is proposed in this article based on a novel tension model. First of all, the proposed tension model is represented by a linear regression part and a nonlinear mapping part, which effectively reflects the chain transmission characteristics and avoids the full measurement of each chain ring. Then, Levant's differentiator and state observer are used to expand input variables for identification algorithm and neural networks estimation. To avoid the adverse effect of random load on parameter adaption, stiffness online identification algorithm is employed in improving the tension tracking performance. To compensate for nonlinear mapping part and uncertain nonlinearity caused by random load, neural adaptive estimator is effectively integrated with the robust integral term in backstepping design. Moreover, Lyapunov stability of the proposed controller is guaranteed. Finally, comparative experimental results of six controllers are completed to verify the effectiveness of the proposed controller in the static and random load mode.

Output feedback control for the driving cylinder of hydraulic support with error constraint

The position control of the driving cylinder of hydraulic support affects the straightness of the underground coal mining face, so this paper focuses on improving tracking performance of the driving cylinder system. Since only position signal of the system is measurable, a Levant differentiator is designed to estimate the velocity and acceleration of the driving cylinder system, which can reduce the effect of the measurement noise. Besides, the parameter changes, modelling errors and external disturbances are treated as a lumped disturbance, an extended disturbance observer is constructed to estimate the lumped disturbance and compensate for the proposed controller. Moreover, barrier Lyapunov function is introduced to limit the tracking error into a small set around zero, and then the stability of the close-loop system is proved. Both the performance of differentiator and disturbance observer is verified in simulation experiments in MATLAB/Simulink. Finally, an experiment rig for the driving cylinder system of hydraulic support is established and comparative experiments are carried out. The experiment results show that the proposed extended disturbance observer-based output feedback controller with error constraint has a satisfied effectiveness in improving tracking performance.

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.

Unknown Dynamics Estimator-Based Output-Feedback Control for Nonlinear Pure-Feedback Systems

Most existing adaptive control designs for nonlinear pure-feedback systems have been derived based on backstepping or dynamic surface control (DSC) methods, requiring full system states to be measurable. The neural networks (NNs) or fuzzy logic systems (FLSs) used to accommodate uncertainties also impose demanding computational cost and sluggish convergence. To address these issues, this paper proposes a new output-feedback control for uncertain pure-feedback systems without using backstepping and function approximator. A coordinate transform is first used to represent the pure-feedback system in a canonical form to evade using the backstepping or DSC scheme. Then the Levant's differentiator is used to reconstruct the unknown states of the derived canonical system. Finally, a new unknown system dynamics estimator with only one tuning parameter is developed to compensate for the lumped unknown dynamics in the feedback control. This leads to an alternative, simple approximation-free control method for pure-feedback systems, where only the system output needs to be measured. The stability of the closed-loop control system, including the unknown dynamics estimator and the feedback control is proved. Comparative simulations and experiments based on a PMSM test-rig are carried out to test and validate the effectiveness of the proposed method.

Output feedback control of electro-hydraulic asymmetric cylinder system with disturbances rejection

To achieve accurate position control of electro-hydraulic asymmetric cylinder system with only available displacement signal, an output feedback controller is proposed in this paper. The dynamic model of the system is expressed as a Brunovsky form, which helps to estimate the system states and simplify the controller structure. Then Levant differentiator is introduced to estimate the position, velocity and acceleration of the asymmetric cylinder system based on the output signal, which can reduce the impact of measurement noise compared to the means of calculating the time derivative of the displacement signal directly. Besides, a high gain disturbance observer is designed to reject the lumped disturbance rejection of the system including parameter uncertainty, modelling error and external disturbance. Moreover, a sliding mode surface is introduced to the controller design and a robust item with continuous function is applied to compensate for estimation errors. According to Lyapunov theory, the developed output controller is pledged to be stable that can realize disturbance rejection control as well as backstepping-free control. Furthermore, a large-size asymmetric cylinder experimental rig is set up to simulate practical applications environment. Comparative experimental results reveal the validity and potential practical meaning of the developed control approach.

Neuroadaptive Finite-Time Control for Nonlinear MIMO Systems With Input Constraint.

This article considers the problem of finite-time (FT) tracking control for a class of uncertain multi-input-multioutput (MIMO) nonlinear systems with input backlash. A modified FT command filter is designed in each step of backstepping, which ensures the output of the filter can faster approximate the derivatives of virtual signals, suppress chattering, and relax the input signal limit of the Levant differentiator. Then, the corresponding improved FT error compensation mechanism is adopted to reduce the negative impact of filtering errors. Furthermore, a neural-network-adaptive technology is proposed for MIMO systems with input backlash via FT convergence. It is shown that desired tracking performance can be implemented in finite time. The simulation example is presented to illustrate the effectiveness and advantages of the new design method.

Observer Based Sliding Mode Control for Hydraulic Driven Barrel Servo System With Unknown Dynamics

The position tracking control of a barrel system driven by a two-stage hydraulic cylinder with long transmission lines was addressed in this paper. First, an active balancing system was designed to balance the gravitational torque which may deteriorate the performance of the barrel control system. The active balancing system is totally isolated from the barrel control system; thus the dynamics of the barrel servo system is significantly simplified. The effects of the long transmission lines were also considered during the modeling of the barrel servo system. In order to exactly estimate the unmeasured states and the mechanical disturbance, an extended state observer was designed by employing the Levant's Differentiator, thus the estimation errors would converge to zero exponentially. Then, based on the estimated signals, the barrel system was transformed into the pure integration chain form, thus the sliding mode technology can be directly utilized. After that, a sliding mode controller is designed to make the position output of the barrel to track different motion trajectories as close as possible in the presence of model uncertainty and unknown dynamics. The closed loop system was proven to be asymptotical stable in the sense of Lyapunov theory. Three different motion trajectories were tested to verify the effectiveness of the proposed controller.

Levant's Arbitrary-Order Exact Differentiator: A Lyapunov Approach

We propose a family of smooth explicit Lyapunov functions (LF) for Levant's differentiator, which is a discontinuous real-time robust and exact differentiator. These LF allow the analysis of the convergence and performance properties and the design of appropriate gains for the differentiator.

A novel adaptive nonsingular terminal sliding mode controller design and its application to active front steering system

SummaryNote that the amplitude of chattering existing in the sliding mode control method is proportional to the magnitude of the control gain. Therefore, the key issue to diminish the chattering is to decrease the value of sliding mode controller's gain to an acceptable minimal level defined by the so‐called reaching condition for the sliding mode's existence. For this reason, the nonsingular terminal sliding mode (NTSM) control method and the adaptive technique have been considered in this paper to develop a novel adaptive NTSM control method, which can be used to search the minimal value of the control gain automatically in the presence of the external disturbances. Meanwhile, the average value of a high‐frequency switching signal in the adaptive law can be provided by Arie Levant's differentiator rather than a low‐pass filter. The rigorous mathematical proof verifies that the system states can converge to the origin within a finite time under the proposed adaptive NTSM controller. Both the academic example and the practical application to an active front steering system are illustrated to show that the presented adaptive NTSM controller has better control performance than the conventional sliding mode controller.

Discrete-Time Implementation of Homogeneous Differentiators

The discrete-time version of Levant's arbitrary order robust exact differentiator, which is a forward Euler discretized version of the continuous-time algorithm enhanced by linear higher order terms, is extended by taking into account also nonlinear higher order terms. The resulting differentiator preserves the asymptotic accuracies with respect to sampling and noise known from the continuous-time algorithm. It is demonstrated in a simulation example and by differentiating a measured signal that the nonlinear higher order terms allow reducing the high-frequency switching amplitude whenever the (n+1)th derivative of the signal to be differentiated vanishes, leading to an improvement in the precision.