Tracking Error Signal Research Articles

Adaptive Position Control of Electrohydraulic Servo Systems with Parameter Uncertainty using Artificial Bee Colony Optimization Algorithm

In this paper, we present a robust adaptive backstepping-based controller for precise positioning of the spool valve in an Electro-Hydraulic Servo System (EHSS) under conditions of parameter fluctuations. Classical control strategies, such as PID and linear controllers, often struggle with the nonlinearities and parameter uncertainties inherent in EHSS, leading to poor tracking performance and instability. To overcome these limitations, we employ the Artificial Bee Colony (ABC) algorithm to optimize the controller parameters, minimizing both the tracking error and control signal. The proposed controller ensures uniform ultimate boundedness of the error and control signal by utilizing a Lyapunov-based stability criterion, which guarantees that errors do not exceed a predefined bound despite uncertainties and disturbances. Simulation results validate the robustness and effectiveness of the control scheme, even in the presence of parameter variations. Additionally, a comparative analysis with sliding mode control highlights the superior performance of the proposed method, particularly in providing smoother control signals and reducing chattering while ensuring stability.

Compensation Function Observer-Based Backstepping Sliding-Mode Control of Uncertain Electro-Hydraulic Servo System

Observer-based control is the most commonly used method in the control of electro-hydraulic servo system (EHSS) with uncertainties, but it suffers from the drawback of low accuracy under the influence of large external load forces and disturbances. To address this problem, this paper proposes a novel compensation function observer-based backstepping sliding-mode control (BSMC) approach to achieve high-accuracy tracking control. In particular, the model uncertainties, including nonlinearities, parameter perturbations and external disturbances are analyzed and treated together as a lumped disturbance. Then, a fourth-order compensation function observer (CFO) is constructed, which fully utilizes the system state information to accurately estimate the lumped disturbance. On this basis, the estimate of the lumped disturbance is incorporated into the design of a backstepping sliding-mode controller, allowing the control system to compensate for the disturbance effect. The stability of the closed-loop control system under the CFO and BSMC is rigorously proven through the use of the Lyapunov theory, which guarantees that all the tracking error signals converge exponentially to the origin. Comparative simulations are carried out to show the effectiveness and efficiency of the proposed approach, i.e., compared with PID and ESO-based BSMC methods, the tracking accuracy is respectively improved by 94.86% and 88.19% under the influence of large external load forces and disturbances.

Optimal Asymptotic Tracking Control for Nonzero-Sum Differential Game Systems with Unknown Drift Dynamics via Integral Reinforcement Learning

This paper employs an integral reinforcement learning (IRL) method to investigate the optimal tracking control problem (OTCP) for nonlinear nonzero-sum (NZS) differential game systems with unknown drift dynamics. Unlike existing methods, which can only bound the tracking error, the proposed approach ensures that the tracking error asymptotically converges to zero. This study begins by constructing an augmented system using the tracking error and reference signal, transforming the original OTCP into solving the coupled Hamilton–Jacobi (HJ) equation of the augmented system. Because the HJ equation contains unknown drift dynamics and cannot be directly solved, the IRL method is utilized to convert the HJ equation into an equivalent equation without unknown drift dynamics. To solve this equation, a critic neural network (NN) is employed to approximate the complex value function based on the tracking error and reference information data. For the unknown NN weights, the least squares (LS) method is used to design an estimation law, and the convergence of the weight estimation error is subsequently proven. The approximate solution of optimal control converges to the Nash equilibrium, and the tracking error asymptotically converges to zero in the closed system. Finally, we validate the effectiveness of the proposed method in this paper based on MATLAB using the ode45 method and least squares method to execute Algorithm 2.

Adaptive disturbance observer-based finite-time command filtered control of nonlinear systems

This article examines the finite-time trajectory tracking control problem for a category of nonlinear perturbed systems. A novel observation mechanism is constructed to mitigate the effects of complex system disturbances. Then, a new adaptive finite-time backstepping nonlinear control algorithm is designed by integrating this disturbance observer with a command filter method. The synthesized control strategy not only effectively suppresses external disturbances and avoids the differential explosion problem of traditional backstepping methods but also ensures the convergence of the trajectory tracking error and filtering error compensation signal within a finite time. Finally, the validity of the suggested approach is confirmed via comparative simulations.

Neural-Network-Based Predefined-Time Adaptive Consensus in Nonlinear Multi-Agent Systems With Switching Topologies.

A predefined-time adaptive consensus control strategy is developed for a class of multi-agent systems containing unknown nonlinearity. The unknown dynamics and switching topologies are simultaneously considered to adapt to actual scenarios. The time required for tracking error convergence can be easily adjusted using the proposed time-varying decay functions. An efficient method is proposed to determine the expected convergence time. Subsequently, the predefined time is adjustable by regulating the parameters of the time-varying functions (TVFs). The neural network (NN) approximation technique is used to address the issue of unknown nonlinear dynamics through predefined-time consensus control. The Lyapunov stability theory testifies that the predefined-time tracking error signals are bounded and convergent. The feasibility and effectiveness of the proposed predefined-time consensus control scheme are demonstrated through the simulation results.

Tracking control of robot manipulators actuated by brushless DC motors: Elimination of joint velocity measurements with a robust observer

AbstractThis work introduces an innovative robust partial state feedback controller designed for robotic manipulators actuated by brushless DC motors. The proposed controller/observer structure relies solely on actuator current and robot joint position measurements, while effectively compensating for dynamic uncertainties in both the electrical (actuator dynamics) and mechanical (robot's dynamical terms) subsystems. Specifically, a model–based robust observer, eliminating the need of joint velocity measurements, is combined with a backstepping‐type controller design. As opposed to the previous model based observers that depend on the actual model parameters, the proposed observer structure utilizes the best guest estimates of the system parameters supported with a robust compensation term. This approach eliminates the need for precise knowledge of system parameters of the observer design which is a significant improvement over most of the previous results. This approach ensures the semi‐global, uniformly ultimately bounded joint tracking error signal. The overall stability of the closed‐loop system is affirmed through Lyapunov‐based arguments. Experimental studies conducted on an in‐house‐built two‐link robotic manipulator, actuated by brushless DC motors, are presented to demonstrate the effectiveness and feasibility of the proposed method.

Fixed‐time sliding mode trajectory tracking control for marine surface vessels under mismatched conditions and input saturation

SummaryThis paper addresses a trajectory tracking control problem for marine surface vessels (MSVs) in the presence of mismatched conditions and input saturation. Firstly, an MSV model under mismatched conditions is established to solve the problem of unreasonable modeling methods in the existing literature. Secondly, fixed‐time disturbance observers (FXDOs) are designed, where the designed FXDOs not only avoid selecting complicated parameters but also has simple structures, and the disturbance estimation errors can converge to the origin within a setting time. Thirdly, based on the FXDOs and an anti–saturation auxiliary dynamic system, a trajectory tracking control scheme in the presence of mismatched conditions and input saturation is constructed by virtue of fixed‐time integral sliding mode control (SMC), and the actual tracking error signals are uniformly fixed‐time bounded rather than finite‐time one. Fourthly, in order to decrease theoretical calculation values of the original setting time in fixed‐time control, the new setting time under three different parameter conditions is derived, and thus reducing the conservatism. Finally, the simulation results are provided to show the effectiveness and superiority of the designed tracking control scheme.

Global finite-time guidance algorithm and constrained formation control using novel nonlinear mapping function for underactuated multiple unmanned surface vehicles

The formation control issue of multiple underactuated unmanned surface vehicles (USVs) with state constraints is discussed in this work. First, the leader–followers system is used to present a global finite-time guidance algorithm for followers with velocity constraints. This algorithm can handle the case where the starting formation is not the intended formation. Second, the state constraints of the USV control subsystem is then handled by a novel nonlinear mapping function (NMF), which has the following benefits: (1) The designed NMF can respond to changing conditions with flexibility since it can handle symmetric constraints. (2) The situation where constraint bounds trend to infinity can be handled by it. (3) The designed NMF has a speed factor that enables it to be modified to track speed. Third, finite-time constrained controllers are proposed to provide formation control for multiple USVs by utilizing the unconstrained system. Next, while guaranteeing that the states are contained within the bounds, the theoretical analysis verifies that the system tracking error stabilizes in finite time and establishes the boundedness of all tracking error signals. To confirm the accuracy and superiority of the theoretical results, two case studies are created.

Fixed‐time fuzzy adaptive robust H∞ control for nonstrict‐feedback nonlinear systems with input saturation and output constraints

SummaryThis paper focuses on the design of fixed‐time fuzzy adaptive robust control of nonlinear systems in a nonstrict‐feedback form that includes output constraints, input saturation, and external disturbance. Fuzzy systems have been utilized to deal with uncertainties and non‐linearity in the system dynamic model. By introducing novel control and adaptive rules, there is no need to approximate the input saturation function with the hyperbolic tangent function. A Barrier Lyapunov Function (BLF) ensures that systems outputs do not violate the defined constraints. By combining the backstepping and BLF method, asymptotic convergence of tracking error signal to zero in fixed‐time is proved. At the same time, index is also met in dealing with the disturbance effects. Finally, the simulation results of the proposed control method in two examples confirm its effectiveness.

A High Order P-type Iterative Learning Control Scheme for Unknown Multi Input Multi Output Nonlinear Systems with Unknown Input Saturation

In this paper, a new high order P-type iterative learning control scheme is presented to solve the trajectory tracking problem of Multi Input Multi Output (MIMO) nonlinear systems with unknown input saturation. It is well known that most systems can be affected by input uncertainties such as saturation. This undesirable input has the potential to destabilize the system. Thus, it is important to mitigate the effect of saturation. In this paper, we take this problem into account. In addition, the controller scheme is very simple, in which the control input in each trial is adjusted by using the tracking error signals obtained from previous and current trials. As the iterations continue, the control system eventually learns the task and follows the desired trajectory with little or no errors. The asymptotic stability of the closed loop system under unknown input saturation is guaranteed over the whole finite time by using the λ-norm method. Finally, to illustrate the effectiveness of the proposed method, simulation results are presented.

Neural Network Trajectory Tracking Control on Electromagnetic Suspension Systems

A new adaptive-like neural control strategy for motion reference trajectory tracking for a nonlinear electromagnetic suspension dynamic system is introduced. Artificial neural networks, differential flatness and sliding modes are strategically integrated in the presented adaptive neural network control design approach. The robustness and efficiency of the magnetic suspension control system on desired smooth position reference profile tracking can be improved in this fashion. A single levitation control parameter is tuned on-line from a neural adaptive perspective by using information of the reference trajectory tracking error signal only. The sliding mode discontinuous control action is approximated by a neural network-based adaptive continuous control function. Control design is firstly developed from theoretical modelling of the nonlinear physical system. Next, dependency on theoretical modelling of the nonlinear dynamic system is substantially reduced by integrating B-spline neural networks and sliding modes in the electromagnetic levitation control technique. On-line accurate estimation of uncertainty, unmeasured external disturbances and uncertain nonlinearities are conveniently evaded. The effective performance of the robust trajectory tracking levitation control approach is depicted for multiple simulation operating scenarios. The capability of active disturbance suppression is furthermore evidenced. The presented B-spline neural network trajectory tracking control design approach based on sliding modes and differential flatness can be extended to other controllable complex uncertain nonlinear dynamic systems where internal and external disturbances represent a relevant issue. Computer simulations and analytical results demonstrate the effective performance of the new adaptive neural control method.

Fuzzy Sliding Mode Control on Positioning and Anti-swing for Overhead Crane

This paper proposes a novel positioning and anti-swing controller based on fuzzy sliding mode for an overhead crane. For the underactuated characteristics of the overhead crane, the trolley displacement and load swing angle are integrated into the same sliding mode surface. The boundness and convergence of each state on the sliding mode surface are analyzed. Appropriate fuzzy rules are introduced to adjust the control quantity to ensure that the system state is always on this surface. The sliding mode function is used to replace the tracking error signal to reduce the dependence of the control method on the system model, so that the control system has better robustness to parameter changes and external disturbances. The experiment and results analysis show that this proposed method can effectively deal with the dynamic deviation of the system model and external random disturbances. The load swing has a good suppression effect and the positioning of the trolley can be accomplished quickly and accurately. The proposed positioning and anti-swing controller based on fuzzy sliding mode is able to provide the high-quality and fast transportation of overhead cranes.

Adaptive tracking control for a class of stochastic nonlinear systems with full-state constraints and dead-zone

This paper seeks to address the problem of state-feedback tracking control for a class of stochastic systems whose states are constrained and input is perturbed by dead-zone. A tan-type barrier Lyapunov function (BLF) and a finite-time adaptive law are proposed such that all states of the system are constrained in the defined bounded compact sets. Based on stochastic Lyapunov theorem, an adaptive state-feedback tracking controller is designed to ensure that the considered stochastic system is semiglobally finite-time stable in probability (SGFSP) and the tracking error signal is bounded. Different from the existing control strategies for stochastic systems, the form of BLF can be used for asymmetric constraints without changing the controller, so the analysis of symmetric or asymmetric full-state constraints is unified. Finally, illustrative simulations are provided to verify the effectiveness of the proposed control strategy.

Singularity-free adaptive control of MIMO discrete-time nonlinear systems with general vector relative degrees

This paper develops a singularity-free adaptive tracking control scheme for a general class of multi-input and multi-output uncertain discrete-time nonlinear systems with non-canonical control gain matrices. The estimation of the control gain matrices, especially in some non-canonical forms, may be singular during parameter adaptation, which leads to the singularity problems of the adaptive control laws. This paper employs the matrix decomposition technique to solve the problem under a linearly parameterized adaptive control framework. The state and output feedback cases are addressed, respectively, to ensure closed-loop stability and asymptotic output tracking. Compared with the existing results, the features of the proposed adaptive control scheme include: (i) the proposed control laws do not involve the high-gain issue commonly encountered in robust control methods; (ii) two different filtered tracking error signals are introduced for the state and output feedback cases, respectively. These filters are crucial to avoid causality contradiction of the adaptive control laws commonly encountered in adaptive control of discrete-time systems; and (iii) a future time signal estimation-based adaptive control law is developed to ensure asymptotic output tracking for the output feedback case without requiring the high-gain observer. Finally, an illustrative example is given to verify the validity of the proposed control scheme.

Model-free adaptive robust control method for high-speed trains

Abstract Aiming at the robustness issue in high-speed trains (HSTs) operation control, this article proposes a model-free adaptive control (MFAC) scheme to suppress disturbance. Firstly, the dynamic linearization data model of train system under the action of measurement disturbance is given, and the Kalman filter (KF) based on this model is derived under the minimum variance estimation criterion. Then, according to the KF, an anti-interference MFAC scheme is designed. This scheme only needs the input and output data of the controlled system to realize the MFAC of the train under strong disturbance. Finally, the simulation experiment of CRH380A HSTs is carried out and compared with the traditional MFAC and the MFAC with attenuation factor. The proposed control algorithm can effectively suppress the measurement disturbance, and obtain smaller tracking error and larger signal to noise ratio with better applicability.

Finite-Time Bounded Tracking Control for a Class of Neutral Systems

In this paper, we investigate finite-time bounded (FTB) tracking control for a class of neutral systems. Firstly, the dynamic equation of the tracking error signal is given based on the original neutral system. Then, we combine it with the equations of the state vector to construct an error system, where the reference signal and the disturbance signal are fused in a new vector. Next, about the error system, we study the input–output finite-time stability problem of the closed-loop system by utilizing the Lyapunov–Krasovskii functional. We also give a finite-time stability condition in the form of linear matrix inequalities (LMIs). Furthermore, the delay-dependent and delay-independent finite-time bounded tracking controllers are designed separately for the original system. Finally, a realistic example is given to show the effectiveness of the controller design method in the paper.

Tracking Control of Multi-Agent Systems Using a Networked Predictive PID Tracking Scheme

With the rapid development of network technology and control technology, a networked multi-agent control system is a key direction of modern industrial control systems, such as industrial Internet systems. This paper studies the tracking control problem of networked multi-agent systems with communication constraints, where each agent has no information on the dynamics of other agents except their outputs. A networked predictive proportional integral derivative (PPID) tracking scheme is proposed to achieve the desired tracking performance, compensate actively for communication delays, and simplify implementation in a distributed manner. This scheme combines the past, present and predictive information of neighbour agents to form a tracking error signal for each agent, and applies the proportional, integral, and derivative of the agent tracking error signal to control each individual agent. The criteria of the stability and output tracking consensus of multi-agent systems with the networked PPID tracking scheme are derived through detailed analysis on the closed-loop systems. The effectiveness of the networked PPID tracking scheme is illustrated via an example.

Research on Beam Waveguide Focus Phase Calibration Method of Deep-space TT&C System

The angle auto-tracking capability is needed in the deep space system, but in the sum-differential dual channels tracking model, the deep-space TT&C system must calibrate its phase before tracking a satellite. Because of the big aperture and high frequency of the deep-space TT&C system antenna, it is impossible to build a calibrating tower which satisfy the far-field condition, and the traditional phase calibration method relies on the tower is not suitable. The radio star calibration method selects a appropriate satellite as the calibrating signal source, but the signal of which is very weak and is affected easily. In this paper, the principle of phase calibration is analyzed, and the beam waveguide focus calibration method is studied, the processes of antenna tracking and angle error signal demodulation are given. The beam waveguide focus calibration method simulates the far field signal through controlling the position of horn antenna which is at the focus of the beam waveguide. The method can be realized easily in engineering, and the limit of the traditional calibration tower and the affections of the weather in using radio star calibration method are avoided.

A predictive type-3 fuzzy control for underactuated surface vehicles

In this paper, a path-following control (PFC) problem of an underactuated surface vehicle (USV) is scrutinized based on an interval type-3 fuzzy logic system (IT3FLS). The suggested method is able to tackle the main limitations of the singleton fuzzy controller in terms of the approximation ability for uncertainties and exogenous disturbances of a complex nonlinear USV. Since a novel structure is employed for the FLS, the control policy is capable of ensuring robust performance against uncertainties with a wide range of variations and higher volumes of complex disturbances. Moreover, surge-guided line-of-sight (SGLOS) and auxiliary dynamics are utilized to design surge and heading control policies, adjusting the velocity of a vehicle and its position at each moment respective to the predefined path. Predictive compensators are designed to enhance the robustness by reducing the approximation error signals while the stability analysis of both tracking and approximation error signals are investigated based on the Lyapunov approach and Barbalat’s lemma. Finally, simulation results demonstrate the effectiveness of the suggested improved fuzzy-based control strategy to solve a PFC problem.

Adaptive tracking control for stochastic nonlinear systems with time-varying delays using multi-dimensional Taylor network

For stochastic nonlinear systems (SNSs) perturbed by compound uncertainties, the conventional model-based control approaches assume that the evolution behavior of uncertain variables is known. Unfortunately, such approaches are often conservative for most practical scenarios with the slow convergence speed and unsatisfactory anti-interference performance. For this sake, an adaptive control scheme based on deep deterministic policy gradient (DDPG) and multi-dimensional Taylor network (MTN) is proposed here to address the tracking problem for a category of SNSs subject to fast time-varying uncertainties, stochastic disturbance and unknown time-varying delays. The effect of time delay is embedded in the reproducing kernel Hilbert space through the error coordinate transformation. In the framework of DDPG, the MTN-based surrogate is utilized to construct the online network and target network via the temporal-difference method, which promises more desirable real-time performance due to its concise structure than conventional NN-based surrogates. In order to enhance the robustness of the system under fast time-varying uncertainties, a novel persistent excitation (PE) mechanism is designed to ensure that the control policy is appropriately rewarded or punished. Based on the PE condition, weights of MTNs converge exponentially and animate the system to evolve towards the target persistently. The tracking error and closed-loop state signals are proved theoretically to be uniformly ultimately bounded (UUB) via Lyapunov–Krasovskii functional. A numerical simulation from the process industry verifies the effectiveness of the proposed method.