Tracking Error Estimation Research Articles

Adaptive observer-based event-triggered finite time formation tracking for multi-agent systems with a non-cooperative leader

The event-triggered time-varying formation tracking for a class of second-order multi-agent systems (MASs) subject to a non-cooperative leader is investigated in this paper. First, in the presence of the unknown input of the leader and external disturbances, a distributed observer with adaptive parameters is put forward for followers to estimate the velocity tracking error. Then, based on the estimated tracking error and an auxiliary variable, a finite time formation controller is further constructed, which is updated depending on a pre-designed event-triggered mechanism. As a result, the desired time-varying formation configuration can be realized in finite time with less communication resource consumption. It’s noted that the constructed formation strategy doesn’t rely on any global information and thus is fully distributed. The stability of the controlled multi-agent system is proved rigorously and it’s verified that event-triggered intervals are with a positive lower bound. At last, simulations are carried out to illustrate the effectiveness of the presented algorithm.

Passivity based hierarchical multi-task tracking control for redundant manipulators with uncertainties

Most research so far on hierarchical multi-task control with strict priority requires exact dynamic and kinematic models. However, when a robot picks up tools of unknown lengths, orientations and mass, its kinematics and dynamics are varying and unavailable. There are few controllers that can achieve asymptotic tracking of multi-task in the presence of kinematic and dynamic uncertainties. Thus, in this paper we present a passivity-based adaptive Jacobian hierarchical multi-task controller. New reference velocity and kinematic regressor matrix are defined for the multi-task case with strict priority to guarantee the asymptotic convergence of both tracking error and parameter estimation error. A novel natural adaption law is specially designed for this method to guarantee the correctness of estimated dynamic parameters and increase the rate of convergence. Experiment results are presented to illustrate the performance of the proposed controller.

Approximate Optimal Tracking Control for Partially Unknown Nonlinear Systems via an Adaptive Fixed-Time Observer

This paper investigates a novel adaptive fixed-time disturbance observer (AFXDO)-based approximate optimal tracking control architecture for nonlinear systems with partially unknown dynamic drift and perturbation under an adaptive dynamic programming (ADP) scheme. To attenuate the impact of disturbance, a novel AFXDO was designed based on the principle of a fixed-time stable system without prior information of disturbance, making disturbance observer errors converge to zero in a fixed time independent of initial estimation error. Additionally, approximate optimal control is conducted by incorporating the real-time estimation of AFXDO into a critic-only ADP framework to stabilize the dynamics of tracking errors and strike a balance between consumption and performance. In particular, to address the heavy calculation burden and oscillation phenomenon in the traditional actor–critic structure, an improved adaptive update law with a variable learning rate was developed to update the weight for adjusting the optimal cost function and optimal control policy simultaneously, avoiding the initial chattering phenomenon and achieving a prescribed convergence without resorting to dual networks. With the efforts of AFXDO and a weight law with a variable learning rate, the track errors were achieved with fast transient performance and low control consumptions in a fixed time. By revisiting Lyapunov stability, the tracking error and weight estimation error were proven to be uniformly ultimately bounded, and the designed control tended to optimal control. The simulations were carried out on quadrotor tracking to demonstrate the effectiveness of the developed control scheme, which achieves rapid convergence by lower control consumption in 4 s, where the cost function is reduced by 19.13%.

Improved indirect adaptive line-of-sight guidance law for path following of under-actuated AUV subject to big ocean currents

This paper presents a novel indirect adaptive disturbance observer (IADO)-based line-of-sight (LOS) guidance law for path following of under-actuated AUV in the presence of time-varying big ocean currents. The influences of ocean currents on AUV are expressed in the kinematic model. An indirect adaptive disturbance observer is designed and utilized to estimate and compensate for the ocean currents-induced uncertainties when following a curved path. In addition, an adaptive look-ahead distance is proposed for better following the curved path with sharp turnings. Then, an improved LOS guidance law is designed to follow the desired path accurately by introducing an auxiliary variable to the conventional LOS and utilizing the indirect adaptive disturbance observer to estimate the ocean current-induced disturbances in the kinematic model. Furthermore, the closed-loop cascade IADO-LOS guidance system, composed of a tracking error subsystem and estimation error system, is proven to be globally uniformly asymptotically stable according to the Lyapunov stability theory. The proposed guidance law is intended for maneuvering AUVs in the horizontal plane to follow a predefined curved path with sharp turning under time-varying big ocean currents. Extensive simulation studies demonstrate that the proposed approach has good tracking accuracy, low chattering, and anti-jamming ability.

Improved disturbance observer-based fixed-time adaptive neural network consensus tracking for nonlinear multi-agent systems

This paper is concerned with the problem of fixed-time consensus tracking for a class of nonlinear multi-agent systems subject to unknown disturbances. Firstly, a modified fixed-time disturbance observer is devised to estimate the unknown mismatched disturbance. Secondly, a distributed fixed-time neural network control protocol is designed, in which neural network is employed to approximate the uncertain nonlinear function. Simultaneously, the technique of command filter is applied to fixed-time control, which circumvents the “explosion of complexity” problem. Under the proposed control strategy, all agents are enable to track the desired trajectory in fixed-time, and the consensus tracking error and disturbance estimation error converge to an arbitrarily small neighborhood of the origin, meanwhile, all signals in the closed-loop system remain bounded. Finally, a simulation example is provided to validate the effectiveness of the presented design method.

Composite Learning Finite-Time Control of Robotic Systems With Output Constraints

In this article, for robotic systems with uncertain dynamics and time-varying asymmetric output constraints, we present a composite learning finite-time control scheme with salient features benefited from two design steps. In the first step, unlike existing composite adaptive/learning control algorithms, which result in either exponential convergence or finite-time convergence but exhibit a potential singularity issue, a modified nonsingular terminal sliding mode-based composite learning controller is adopted such that both tracking error and parameter estimation error converge to zero in finite time without singularity. The unknown parameter learning law is constructed by using online historical data and regressor extension, which gives a benefit of relaxing the typically required stringent persistent excitation with a much weaker excitation condition termed interval excitation. In the second step, a universal time-varying asymmetric barrier function (UTABF) is adopted to directly constrain the system output rather than the most commonly used barrier Lyapunov function, which indirectly converts the output constraints into the conservative tracking error constraints. Moreover, the UTABF can handle both constrained and unconstrained cases uniformly without the need for changing the control structure. Both theoretical analysis and experiments results on an industrial manipulator confirm the benefits and effectiveness of the proposed control scheme.

Asymptotic switched composite adaptive control with application to robotic interaction tasks

AbstractThis article proposes a new switched adaptive control design for uncertain switched systems with composite (time‐driven and state‐dependent) switching and shows its applicability in switched impedance control. A composite switched adaptive control design, consisting of the direct switched adaptive control and the indirect switched adaptive control counterpart, is developed to improve the control performance. Specifically, a new stability condition for composite switching is proposed by making use of differential matrix equations and Sylvester matrix equations, which are a generalization of Lyapunov matrix equations. The design results in a time‐varying multiple Lyapunov function that is decreasing at the switching instants. From the theoretical point of view, the relevance of this work is the construction of the adaptive laws that guarantee asymptotic tracking error and asymptotic estimation for the direct and indirect switched adaptive control loops, respectively. From the practical point of view, the relevance of this work is validated in a new switched impedance control for the robot interaction with uncertain and discontinuous environments.

Neural network-based event-triggered integral reinforcement learning for constrained [formula omitted] tracking control with experience replay

Since input constraints and external disturbances are unavoidable in tracking control problems, how to obtain a controller in this case to save communication and data resources at the same time is very challenging. Aiming at these challenges, this paper develops a novel neural network (NN)-based event-triggered integral reinforcement learning (IRL) algorithm for constrained H∞ tracking control problems. First, the constrained H∞ tracking control problem is transformed into a regulation problem. Second, an event-triggered optimal controller is designed to reduce network transmission burden and improve resource utilization, where a novel threshold is proposed and its non-negativity can be guaranteed. Third, for implementation purpose, a novel NN-based event-triggered IRL algorithm is developed. In order to improve data utilization, the experience replay technique with an easy-to-verify condition is employed in the learning process. Theoretical analysis proves that the tracking error and weight estimation error are uniformly ultimately bounded. Finally, simulation verification shows the effectiveness of the present method.

Adaptive high-order sliding mode control for proton exchange membrane fuel cell air-feed system based on high-gain observer

The proficient control algorithms for the air-feed system of proton exchange membrane fuel cell can significantly improve the output net power and prevent the polymeric membranes deterioration. To this end, a novel adaptive high-order sliding mode (HOSM) control method for air feed system is proposed in this paper. There are several meaningful contributions. Firstly, in order to reduce the number of redundant sensors and overall cost, the high-gain observer is implemented to estimate unmeasured tracking error derivatives. Secondly, without the priori knowledge of accurate dynamic model and parameters, the proposed adaptive HOSM controller ensures a real sliding mode established in finite time and a continuous control input signal to the motor. Thirdly, based on the estimated tracking error derivatives, the developed adaptive schemes guarantee the control gain would not be overestimated without the information about the bounds of the uncertainties/perturbations, which significantly suppresses the formidable chattering caused by high frequency switching of the control signal. Finally, the robustness and effectiveness of the adaptive HOSM controller are indicated by numerical simulations and hardware-in-loop experiments.

Continuous adaptive integral recursive terminal sliding mode control for DC motors

Parametric uncertainties and unmodeled disturbances limit the tracking performance improvement of DC motor systems. In this paper, a third-order continuous adaptive integral recursive terminal sliding mode (3-AIRTSM) control scheme is formulated for high-performance motion control of DC motors. The residual parameter uncertainties and unmodeled disturbances are synthesised as the lumped uncertainties. Then, the derivative of the lumped uncertainties is analysed whose upper bound is denoted by system state and control signal dependent terms with unknown coefficients. Meanwhile, a novel third-order sliding mode structure including integral element is constructed to obtain the chattering-free control input. In the proposed controller, the equivalent control term functions as feedforward model compensation, and the reaching control input handles the lumped uncertainties with adaptive gain coefficients, which reduces the design conservation and avoids the high-gain feedback issue. The closed-loop system stability analysis indicates that both tracking error and gain estimation errors converge to zero in finite time under the proposed controller. Comparative experimental results verify the proposed 3-AIRTSM controller can realize the best tracking performance with the modest or the lowest control effort, compared with 3-IRTSM and 3-RTSM controllers.

Event-Triggered Optimal Parallel Tracking Control for Discrete-Time Nonlinear Systems

A novel event-triggered optimal tracking control (ETOTC) method is developed for discrete-time nonlinear systems in this study. For the time-invariant desired trajectory, we prove that the tracking error is asymptotically stable, and an upper bound of the real performance index can be predetermined by a design parameter. For the time-varying desired trajectory, the developed triggering condition reduces communication costs by relaxing the restriction of the asymptotic stability of the closed-loop system, and we prove that the tracking error is uniformly ultimately bounded (UUB). The developed ETOTC method entails obtaining the next state of the real system. Therefore, a parallel control approach is proposed to predict the next state by constructing a parallel system for the real system. Neural networks (NNs) and adaptive dynamic programming (ADP) techniques are utilized in the parallel control approach. Moreover, the stability analysis of the closed-loop system is shown, and the tracking error and NN weight estimation errors are proved to be UUB using the Lyapunov approach. Finally, we validate the developed ETOTC method through two simulations.

Nonlinear Proportional-Integral Disturbance Observers for Motion Control of Permanent Magnet Synchronous Motors

In this paper, we present nonlinear proportional integrator (N-PI) disturbance observers (DOBs) for enhancing the motion tracking performance of a surface-mounted permanent magnet synchronous motor in rapidly varying speed regions. By presenting an N-PI-DOB for load torque estimation with a torque modulation technique, we show that the electro-mechanical system, including the motion controller, disturbance observers, and inverter controller, can be presented in a three-cascaded form, in which the stability analysis of the closed-loop system can be easily performed. After analyzing the disturbances of the current tracking error dynamics, we designed another N-PI-DOB and a Lyapunov-based nonlinear controller for the current loop to enhance the motion tracking performance. With these N-PI-DOBs and the motion controller, we analyzed the stability of the motion tracking error dynamics and estimation error dynamics. We experimentally performed a comparative study with and without N-PI-DOB to validate the effectiveness of the proposed method under the conditions of unknown load torque and rapid speed variation.

An enhanced UAV safety control scheme against attacks on desired trajectory

The detection and compensation of attacks are crucial for UAV safety. This paper focuses on the case where the desired trajectory sent by ground control station (GCS) is attacked. Based on the analysis of the attack transmission mechanism, an integrated scheme of attack detection and compensation is presented. The attack detection mechanism is established based on the trajectory tracking error and state estimation error. In addition, the proposed scheme can not only detect attacks but also distinguish attacks from unknown abrupt disturbances. In the proposed scheme, attack observer (AO) and learning observer (LO) are developed to address the coupling problem of additive and multiplicative attacks. Finally, the effectiveness of the proposed scheme is validated by both simulation and experimental tests.

Observer-Based Composite Adaptive Type-2 Fuzzy Control for PEMFC Air Supply Systems

Polymer electrolyte membrane fuel cell (PEMFC) air supply systems are usually affected negatively by model uncertainties, external disturbance, and unmeasured variables. In this article, we propose a composite adaptive type-2 fuzzy controller based on a high-gain observer and a disturbance observer for oxygen excess ratio (OER) of PEMFC air supply systems. First, the derivatives of system output, which are unavailable due to limited sensors, are estimated via the high-gain observer. Then, interval type-2 fuzzy logic systems (IT2 FLSs) are adopted to approximate the unknown system dynamics and the disturbance observer is designed to estimate compound disturbance including unknown external disturbance and fuzzy approximation error. Finally, in order to improve the tracking performance, two composite adaptive updating laws are constructed by utilizing the estimated tracking error and the modeling error. Theoretical analysis shows that the system tracking error is uniformly ultimately bounded by Lyapunov stability theory. Numerical simulations and hardware-in-loop experiments are presented to demonstrate the effectiveness and superiority of the proposed controller.

General first-order target registration error model considering a coordinate reference frame in an image-guided surgical system.

Point-based rigid registration (PBRR) techniques are widely used in many aspects of image-guided surgery (IGS). Accurately estimating target registration error (TRE) statistics is of essential value for medical applications such as optically surgical tool-tip tracking and image registration. For example, knowing the TRE distribution statistics of surgical tool tip can help the surgeon make right decisions during surgery. In the meantime, the pose of a surgical tool is usually reported relative to a second rigid body whose local frame is called coordinate reference frame (CRF). In an n-ocular tracking system, fiducial localization error (FLE) should be considered inhomogeneous, that means FLE is different between fiducials, and anisotropic that indicates FLE is different in all directions. In this paper, we extend the TRE estimation algorithm relative to a CRF from homogeneous and anisotropic to heterogeneous FLE cases. Arbitrary weightings can be assumed in solving the registration problems in the proposed TRE estimation algorithm. Monte Carlo simulation results demonstrate the proposed algorithm's effectiveness for both homogeneous and inhomogeneous FLE distributions. The results are further compared with those using the other two algorithms. When FLE distribution is anisotropic and homogeneous, the proposed TRE estimation algorithm's performance is comparable with that of the first one. When FLE distribution is heterogeneous, proposed TRE estimation algorithm outperforms the other two classical algorithms in all test cases when ideal weighting scheme is adopted in solving two registrations. Possible clinical applications include the online estimation of surgical tool-tip tracking error with respect to a CRF in IGS. Graphical Abstract This paper provides the target registration error model considering a coordinate reference frame in surgical navigation.

Distributed Adaptive Observer-Based Control for Output Consensus of Heterogeneous MASs With Input Saturation Constraint

Without assuming that the global connectivity information of the interaction graph is available to any agent, fully distributed saturated controller design for output consensus of heterogeneous multi-agent systems (MASs) under directed leader-follower communication topologies is addressed in this paper. The main difficulty of the problem lies in the coupling of the nonlinearities caused by the input saturation and the adaptive feedback control. To decouple these two types of nonlinearities, a distributed adaptive observer is first designed for each follower to realize asymptotical estimation of leader’s state, where the adaptive coupling gain is embedded to estimate the graph connectivity, and then an extra tracking error observer is introduced to tackle input saturation constraint. The controller is then constructed based upon these two observers, which is fully distributed with only the information of the distributed adaptive observer transmitted between neighboring agents and the local output information of each agent involved. To reduce the order of the controller, an in-depth analysis of the proposed controller is conducted, and the novel controller is designed by removing the tracking error observer and modifying the multi-level saturation control parameters for each follower. Furthermore, to remove the limitation of using absolute output information of each agent, novel controller is proposed with tracking error estimation designed in a distributed manner, which can achieve output consensus tracking for heterogeneous non-introspective MASs with undirected subgraph among followers.

Robust composite adaptive neural network control for air management system of PEM fuel cell based on high-gain observer

Polymer electrolyte membrane (PEM) fuel cell system is usually affected negatively by external disturbance, model uncertainties and unmeasured variables. In this paper, a robust composite adaptive neural network controller using high-gain observer is proposed to achieve stable oxygen excess ratio control for PEM fuel cell air management system. First, the derivatives of system output, which are unavailable due to the limited sensors, are estimated via high-gain observer. Then, a neural network is adopted to estimate the unknown system dynamics and the additional robust term is used to attenuate the compound disturbance including unknown external disturbance and neural network approximation error. Finally, a composite adaptive updating laws are constructed by utilizing estimated tracking error and modeling error to improve the tracking performance. In contrast to the existing controllers applied in PEM fuel cell air management system, this controller has a better control performance in the practical application. By means of Lyapunov stability analysis, it is theoretically proved that the system tracking error is uniformly ultimately bounded. The effectiveness and practicability of the proposed controller are validated by hardware-in-loop experiment.

Decentralized Finite-time [formula omitted] tracking control for a class of interconnected Markovian jump system with actuator saturation

An observer-based decentralized tracking control technique is developed for a saturated interconnected Markovian jump system in the sense of finite-time stability (FTS). To improve tracking performances, a L2−L∞ performance criterion which is relevant to both estimated error and estimated tracking error is proposed. A sufficient condition for finite-time bounded (FTB) is established. In view of the obtained condition, a method with the one-step procedure to seek the gains of observer and controller is investigated. By virtue of this procedure, the problem of the decentralized tracking stabilization is transformed into a convex optimization problem subject to constraints in the form of linear matrix inequalities (LMIs). Furthermore, the tracking control problem of interconnected Markovian jump system without saturation is also investigated, and the results are obtained directly to avoid taking congruent transformation. Simulation is presented to show the effectiveness and applicability of the obtained results.

Quantifying Uncertainty in Aquatic Telemetry: Using Received Signal Strength to Estimate Telemetry Error

AbstractTelemetry error is not regularly considered in aquatic studies; however, when it is considered, it is generally treated as a static value despite high variation that can occur even within a single tracking event. We describe a simple procedure for using received signal strength (RSS) to estimate telemetry error. We recorded the RSSs of ground‐ and air‐based detections of large (11 g; 11 × 59 mm) and small (8 g; 11 × 43 mm) 172‐MHz radio transmitters across a range of distances. Received signal strength was an excellent predictor of the distance to transmitter for ground tracking detections (large transmitter: r2 = 0.98; small transmitter: r2 = 0.97) and a fair predictor for aerial detections (large transmitter: r2 = 0.49; small transmitter: r2 = 0.57). We also manipulated transmitter antenna lengths and unexpectedly found that both transmitters performed better with shorter antennas relative to factory lengths. With calibrated models relating RSS to the distance to transmitter, RSSs from field‐collected data can be used to approximate telemetry error and draw spatial confidence areas around location estimates for analysis and interpretation. During a concurrent movement study, we estimated telemetry error for 2,436 detections of fish at large. Ground tracking error estimates ranged from 1 to 131 m (median = 24 m), and aerial error estimates were most often less than 300 m but were as high as 1 km. The benefits of representing telemetry data as spatial confidence areas guided by the RSS of each detection are discussed. With appropriate caution, this method will provide a more robust alternative to the assumption that error is constant, negligible, or both.

Composite Model Reference Adaptive Control with Parameter Convergence Under Finite Excitation

A new parameter estimation method is proposed in the framework of composite model reference adaptive control for improved parameter convergence without persistent excitation. The regressor filtering scheme is adopted to perform the parameter estimation with signals that can be obtained easily. A new framework for residual signal construction is proposed. The incoming data are first accumulated to build the information matrix, and then its quality is evaluated with respect to a chosen measure to select and store the best one. The information matrix is built to have full rank after sufficient but not persistent excitation. In this way, the exponential convergence of both tracking error and parameter estimation error can be guaranteed without persistent oscillation in the external command, which drives the system. Numerical simulations are performed to verify the theoretical findings and to demonstrate the advantages of the proposed adaptation law over the standard direct adaptation law.