Convergence Of Tracking Error Research Articles

Adaptive fixed-time consensus control for a class of non-strict feedback multi-agent systems subject to input nonlinearities, state constraints, unknown control directions, and actuator faults

In this work, an adaptive fixed-time controller has been designed for a class of uncertain non-strict feedback multi-agent systems (MASs) subject to different types of input nonlinearities, time-varying asymmetric state constraints, unknown control directions, infinite number of actuator faults and external disturbances. Compared to the existing consensus control schemes, the proposed controller can handle input nonlinearities, actuator faults and unknown control directions simultaneously, while guaranteeing fixed-time convergence of the consensus tracking error and satisfying state constraints for non-strict feedback MASs. By employing a nonlinear mapping, the constrained MAS has been transformed into an unconstrained one, hence the feasibility analysis required for barrier Lyapunov function (BLF)-based controllers for state-constrained systems has been avoided. Different input nonlinearities including backlash, hysteresis, dead-zone and saturation have been tackled by employing a unified framework, hence each agent can have a different type of input nonlinearity. By using fuzzy logic systems (FLSs), system unknown dynamics have been approximated. Utilizing Lyapunov stability theorem, it has been proved that all closed-loop signals are semi-globally practically fixed-time stable (SPFTS), state constraints are satisfied and the consensus tracking is achieved. Finally, the effectiveness of the designed control scheme has been shown via two simulation studies.

Exponential attitude tracking and disturbance rejection of rigid body systems

AbstractIn this paper, we study the exponential attitude tracking and disturbance rejection problem of rigid body systems. We design a control law for the rigid body system that ensures the exponential convergence of the attitude tracking error and the angular velocity tracking error in the presence of unknown external disturbances. Furthermore, by tuning some design parameters, the exponential convergence rate can be made arbitrarily fast. The effectiveness of our approach is validated by a numerical example.

Model-Free Adaptive Sliding Mode Control-Based Active Chatter Suppression by Spindle Speed Variation

Abstract This paper presents a novel model-free adaptive sliding mode control (MFA-SMC) algorithm, which is employed to actively adjust the amplitude and frequency of spindle speed variation (SSV) for chatter suppression in turning. The SSV technique is effective for mitigating regenerative chatter, which however is not widely applied due to its poor adaptability to time-varying characteristics of machining dynamics and cutting conditions. The online adjustment of SSV parameters has been reported in previous works, whereas the proportional integral differential (PID)-type controller used cannot compensate for the interactions when abruptly changing the reference input and the parameter tuning procedure consumes much time. The proposed method integrates the model-free adaptive control (MFAC) with the global sliding mode control (GSMC). The method is data-driven and only dependent on the measured input/output data of the cutting process in this paper, which are the normalized wavelet packet entropy and SSV parameters, respectively. The bounded-input bounded-output stability and the tracking error convergence of the proposed control algorithm are guaranteed by theoretical analysis. The effectiveness of the proposed chatter suppression method is investigated with numerical simulations. Finally, experimental results demonstrate that chatter vibration under different cutting conditions can be effectively mitigated by the proposed method.

An observer-based adaptive fault-tolerant control for hypersonic vehicle with unexpected centroid shift and input saturation

This paper makes an investigation on the fault-tolerant control (FTC) problem for a hypersonic reentry vehicle (HSV) in the coexistence of unknown movement of center-of-mass, system input constraint and failure of actuator. Firstly, the dynamics of HSV’s attitude system with the unknown factors mentioned above are developed to illustrate the particularity of the researched topic. The influences of unexpected centroid shift on an FTC design can be summarized into the following three parts: unknown system uncertainties, eccentric torque as well as changing system moment of inertia matrix, which are coupled and unknown. Secondly, due to the difficulty in decoupling and estimating these influences (embodied in the output states of the system) one by one, it is the attitude system states observer that is proposed to estimate those detrimental unknown effects. The designed observer is consisted of an adaptive fault observer and an adaptive sliding-mode observer, supporting an innovative adaptive FTC scheme free from the variation of inverse matrix that might be singular due to an unexpected centroid shift. This fault-tolerant controller established in the estimated system states is derived by utilizing the above mentioned observer and adaptive backstepping control in conjunction with adaptive auxiliary compensation systems to handle the system input saturation. Moreover, the convergence of attitude tracking error and the boundedness of all closed-loop signals are achieved in the light of Lyapunov stability theory and boundedness analysis. Ultimately, simulation results are delivered to demonstrate the effectiveness of the proposed FTC scheme.

Distributed adaptive consensus tracking control for heterogeneous nonlinear multi-agent systems

In this study, our focus is to resolve the consensus problem of heterogeneous nonlinear multi-agent systems. As only some agents access reference information directly, the proposed distributed adaptive consensus controller uses only local information, by employing the command filtered backstepping approach. The uniform, ultimate boundedness of all closed-loop signals and the convergence of consensus tracking error to a small tunable compact set can be obtained using our proposed mechanism. Finally, the formation-tracking problem of nonholonomic unmanned aerial vehicles is solved with the suggested control algorithm.

Disturbance and state observer-based adaptive finite-time control for quantized nonlinear systems with unknown control directions

This paper concentrates on proposing a novel finite-time tracking control algorithm for a kind of nonlinear systems with input quantization and unknown control directions. The nonlinear functions in the system are approximated by the means of strong approximation capability of the fuzzy logic systems. Firstly, the nonlinear system with unknown control directions is transformed into an equivalent system with known control gains by coordinate transformation. Secondly, the unknown system states are estimated by a designed fuzzy state observer, and the disturbance observer is constructed to track the external disturbances. The command filtering method is proposed to approach the problem of “explosion of complexity” existed in the conventional backstepping design process. In this system, the difficulties caused by unknown control directions are solved via the Nussbaum gain approach. Finally, based on the fuzzy state observer, the controller of the original system is obtained via using the transformed system by the backstepping method. The boundedness of all signals and the convergence of tracking and observer errors at the origin are ensured for the closed-loop system, and demonstrated by the simulation result in this paper.

Fault-tolerant adaptive fractional controller design for incommensurate fractional-order nonlinear dynamic systems subject to input and output restrictions

In this article, a fault-tolerant adaptive neural network fractional controller has been proposed for a class of uncertain multi-input single-output (MISO) incommensurate fractional-order non-strict nonlinear systems subject to five different types of unknown input nonlinearities, infinite number of actuators failures and arbitrary independent time-varying output constraints. The barrier Lyapunov function (BLF)-based backstepping technique and fractional Lyapunov direct method (FLDM) have been used to design the controller and establish system stability. To tackle the incommensurate derivatives problem, in each step of the backstepping technique, an appropriate Lyapunov function has been selected based on error variables of the considered step and their boundedness has been proved from the last step to the first one. The computational burden is reduced by decreasing the number of adaptive laws. The developed controller satisfies the output constraints and guarantees convergence of the output tracking error to a small neighborhood of the origin and boundedness of all closed-loop signals. Finally, the effectiveness of the designed controller has been demonstrated via simulating three examples.

Biplane Trajectory Tracking Using Hybrid Controller Based on Backstepping and Integral Terminal Sliding Mode Control

A biplane quadrotor is a hybrid type of Unmanned Aerial Vehicle (UAV) that has advantages of both fixed-wing and rotary-wing UAVs. In this study, we design controllers using (i) Backstepping Control (BSC), (ii) Integral Terminal Sliding Mode Control (ITSMC), and (iii) Hybrid control (ITSMC + BSC), where the ITSMC controls attitude and BSC controls the altitude subsystems as per the mathematical model of biplane quadrotor. The performance of these controllers is evaluated based on the autonomous trajectory tracking containing all possible maneuvers and operation modes that the biplane quadrotor can perform. Performance analysis reveals that the BSC-based controller is susceptible to a steady-state error in altitude tracking when mass is changed. In contrast, the ITSMC and the “hybrid” controllers achieve smooth tracking in a finite time. Furthermore, the “hybrid” controller outperforms the other designs, reducing tracking error and faster convergence time.

Observer-based self-organizing adaptive fuzzy neural network control for non-linear, non-affine systems with unknown sign of control gain and dead zone: A case study of pneumatic actuators

In this paper, an observer-based self-organizing adaptive fuzzy neural network (OSOAFNN) control for non-linear, non-affine systems with the unknown sign of control gain and dead zone is presented. First, a reverse dead-zone compensator scheme to cope with the impact of dead-zone phenomenon existing in control input is investigated. Then, an observed-based control approach to address immeasurable states of the system is proposed, utilizing this approach all states of the system are not needed to be available. A self-organizing fuzzy neural network (SOFNN) technique is presented to approximate the non-linear and unknown function of the observer error dynamics. The proposed fuzzy neural network (FNN) model benefits from two main advantages: (1) the number of rules is automatically generated or pruned and (2) the parameters of antecedent and consequent part of SOFNN are updated through the hybrid tuning, simultaneously. Furthermore, the control law contains a Nussbaum function which deals with the unknown sign of control gain. As the system states are immeasurable, the strictly positive real (SPR) Lyapunov function to guarantee the closed-loop system stability, and tracking error convergence to zero is employed, as well as the boundedness of control parameters are assured through a projection law merged with adaptive law. Finally, the controller is practically implemented on a non-linear, non-affine pneumatic system with unknown dead zone which exploits just a sensor for output measuring. Experimental results show that the proposed controller has satisfactory performance in tracking different trajectories, and tracking error for desired signal case I and case II is limited to [−1.5, 1.5] and [−1,1]mm, respectively.

Data-driven ILC algorithms using AFD in frequency domain for unknown linear discrete-time systems

In conventional PID-type iterative learning control (ILC) designs, to determine the learning control gains involved, relevant model knowledge on the controlled systems is often dependent. In this paper, two completely data-driven ILC laws, the extended PD-type ILC law and the extended P-type ILC law, are designed in frequency domain for linear discrete-time (LDT) single-input single-output (SISO) systems. The designs of the proposed ILC laws are based on the approximation/identification to unknown transfer function with a novel adaptive Fourier decomposition (AFD) technique. As a result, the strictly monotonic convergence of ILC tracking error is guaranteed in a deterministic way. A numerical example on a four-axis robot arm is performed to illustrate the effectiveness of the proposed data-driven ILC algorithms

Broad-learning recurrent Hermite neural control for unknown nonlinear systems

The broad-learning systems (BLS) with advance control theories have been studied, but found to have two disadvantages: one is that the calculations are too complicated and the other is that the convergence time cannot be guaranteed. In order to mitigate the high computational loading, this study proposes a broad-learning Hermite neural network (BHNN), which has the capability of dynamic mapping and reduces the structural complexity of neural network. Meanwhile, a broad-learning recurrent Hermite neural control (BRHNC) system is proposed while maintaining finite-time stability to speed up the tracking error convergence. The proposed BRHNC system comprises two controllers: a recurrent broad controller that utilizes a BHNN to approximate on-line an ideal finite-time controller and a robust exponential controller that ensures system stability through a Lyapunov function. Meanwhile, the BHNN’s full-tuned parameter learning laws are developed to increase the approximating capacity, learning capacity and accuracy using gradient descent method. Finally, simulation and experimental results show that the BRHNC system has good control, tracking and disturbance rejection properties, while the BRHNC system requires no prior knowledge about the system dynamics.

Barrier Function Adaptive Nonsingular Terminal Sliding Mode Control Approach for Quad-Rotor Unmanned Aerial Vehicles.

This paper proposes a barrier function adaptive non-singular terminal sliding mode controller for a six-degrees-of-freedom (6DoF) quad-rotor in the existence of matched disturbances. For this reason, a linear sliding surface according to the tracking error dynamics is proposed for the convergence of tracking errors to origin. Afterward, a novel non-singular terminal sliding surface is suggested to guarantee the finite-time reachability of the linear sliding surface to origin. Moreover, for the rejection of the matched disturbances that enter into the quad-rotor system, an adaptive control law based on barrier function is recommended to approximate the matched disturbances at any moment. The barrier function-based control technique has two valuable properties. First, this function forces the error dynamics to converge on a region near the origin in a finite time. Secondly, it can remove the increase in the adaptive gain because of the matched disturbances. Lastly, simulation results are given to demonstrate the validation of this technique.

Iterative learning control for high‐speed trains with velocity and displacement constraints

AbstractIn this article, a novel iterative learning control (ILC) scheme is presented for the operation control of high‐speed train (HST), where the velocity and displacement of HST are strictly limited to ensure safety and comfort. The model of HST constructed in the article is practical in the sense that both parametric and nonparametric uncertainties of system are addressed simultaneously. Backstepping design with the newly proposed barrier Lyapunov function is incorporated in analysis to ensure the uniform convergence of the state tracking error and that the constraint requirements on velocity and displacement would not be violated during the whole operation process. In the end, a simulation study is presented to demonstrate the efficacy of the proposed ILC law.

RISE-Based Trajectory Tracking Control of an Aerial Manipulator Under Uncertainty

This study presents a robust integral of the sign of the error (RISE)-based controller for an aerial manipulator consisting of a multi-rotor and a robotic arm which guarantees tracking error convergence to zero in the presence of uncertainties. To rigorously address underactuatedness issue, the system dynamics is decomposed into the two subsystems for which a robust controller is derived. As an intermediate result, if there exists no uncertainty, we show that the nominal closed-loop system with the proposed nominal controller is asymptotically stable without assuming that the attitude error term in the underactuated part is zero by cascaded system analysis tool. Then, a robust controller combining a nominal controller and a RISE controller is proposed and applied to both subsystems. Tracking error convergence is strictly proved through Lyapunov-based stability analysis. The performance of the controller is demonstrated in simulation with comparative studies where the proposed controller outperforms the other compared controllers in error convergence.

Intelligent optimal hybrid motion/force control of constrained robot manipulator

This paper presents a neural network-based optimal control approach for hybrid motion/force control problem of robot manipulators in the presence of structured and unstructured uncertainties. The quadratic optimisation with sliding mode control and neural network is utilised to propose an optimal hybrid motion/force control of robotic manipulators. Firstly, the dynamics of robot manipulator is reduced into state-space form describing the constrained and unconstrained motion separately. Then the optimal hybrid motion/force control scheme is derived utilising the optimisation of Hamilton Jacobi Bellman (HJB) equation and sliding mode control approach. The structured and unstructured uncertainties of the system are compensated using radial basis function neural network and adaptive compensator. The radial basis function neural network approximates the unknown dynamics and adaptive compensator is used to estimate the bounds on neural network approximation error and the unstructured uncertainties of the system. The neural networks are trained in online manner using weight update algorithms derived with Lyapunov approach to guarantee the tracking stability and error convergence with prescribed quadratic performance index. Finally, the proposed control approach is verified through simulation results with two link constrained manipulator in a comparative manner.

Quantized-State-Feedback-Based Neural Control for a Class of Switched Nonlinear Systems With Unknown Control Directions

This paper investigates the problem of unknown virtual control directions in a state-quantized adaptive recursive control design for a class of arbitrarily switched uncertain pure-feedback nonlinear systems in a band-limited network. State quantization is considered for state feedback control in a band-limited network. The primary contribution of this study is to provide a quantized state feedback adaptive control strategy to address the unknown control direction and arbitrarily switched nonaffine nonlinearities. Herein, a coupling problem between Nussbaum functions and quantization errors caused by quantized state feedback control laws is considered in the Lyapunov-based design and stability analysis. A state-quantized adaptive recursive control scheme using the function approximation is constructed without a priori knowledge of the signs of the control gain functions, where the estimated parameters and Nussbaum-type functions are adaptively updated via quantized states. Theoretical lemmas are derived to show that the adaptive parameters and quantization errors of the closed-loop signals are bounded using the proposed control scheme. The boundedness of the closed-loop signals and the convergence of tracking error to a neighborhood of the origin are proved using the common Lyapunov function approach. Two simulation examples are shown to illustrate the effectiveness of the proposed theoretical result.

Exponential Stability With RISE Controllers

A class of continuous robust controllers termed Robust Integral of the Sign of the Error (RISE) have been published over the past two decades as a means to yield asymptotic tracking error convergence and asymptotic identification of time-varying uncertainties, for classes of nonlinear systems that are subject to sufficiently smooth bounded exogenous disturbances and/or modeling uncertainties. Despite the wide application of RISE-based techniques, an open question that has eluded researchers during this time-span is whether the asymptotic tracking error convergence is also uniform or exponential. This question has remained open due to certain limitations in the traditional construction of a Lyapunov function for RISE-based error systems. In this letter, new insights on the construction of a Lyapunov function are used that result in an exponential stability result for RISE-based controllers. As an outcome of this breakthrough, the inherent learning capability of RISE-based controllers is shown to yield exponential identification of state-dependent disturbances/uncertainty.

Lyapunov-Derived Control and Adaptive Update Laws for Inner and Outer Layer Weights of a Deep Neural Network

Lyapunov-based real-time update laws are well-known for neural network (NN)-based adaptive controllers that control nonlinear dynamical systems using single-hidden-layer NNs. However, developing real-time weight update laws for deep NNs (DNNs) remains an open question. This letter presents the first result with Lyapunov-based real-time weight adaptation laws for each layer of a feedforward DNN-based control architecture, with stability guarantees. Additionally, the developed method allows nonsmooth activation functions to be used in the DNN to facilitate improved transient performance. A nonsmooth Lyapunov-based stability analysis proves global asymptotic tracking error convergence. Simulation results are provided for a nonlinear system using DNNs with leaky rectified linear unit (LReLU) and hyperbolic tangent activation functions to demonstrate the efficacy of the developed method.

Nonsingular Terminal Sliding Mode Control Based on Adaptive Barrier Function for nth-Order Perturbed Nonlinear Systems

In this study, an adaptive nonsingular finite time control technique based on a barrier function terminal sliding mode controller is proposed for the robust stability of nth-order nonlinear dynamic systems with external disturbances. The barrier function adaptive terminal sliding mode control makes the convergence of tracking errors to a region near zero in the finite time. Moreover, the suggested method does not need the information of upper bounds of perturbations which are commonly applied to the sliding mode control procedure. The Lyapunov stability analysis proves that the errors converge to the determined region. Last of all, simulations and experimental results on a complex new chaotic system with a high Kaplan–Yorke dimension are provided to confirm the efficacy of the planned method. The results demonstrate that the suggested controller has a stronger tracking than the adaptive controller and the results are satisfactory with the application of the controller based on chaotic synchronization on the chaotic system.

L2‐gain robust trajectory tracking control for quadrotor UAV with unknown disturbance

AbstractIn this paper, the trajectory tracking problem of the position and yaw angle of an underactuated quadrotor UAV is studied. Considering that the quadrotor UAV often encounters unknown disturbance resulted by model parameter perturbations and environmental changes, a new L2‐gain robust control method is presented based on dissipation theory. First, the dynamic model of the quadrotor UAV with disturbances is transformed by introducing three virtual control variables. In this way, the trajectory tracking control can be implemented by the dual loop control, that is, the position control loop and the attitude control loop. Then, based on dissipation inequality, the robust position controller and attitude controller are developed to guarantee the L2‐gain of the system to the unknown disturbance. It is proved that all the tracking errors are uniformly ultimate boundedness. In addition, the controller is further reconstructed by introducing a new class K functions to improve the transient and steady‐state performances of the quadrotor UAV system. In the end, two simulation examples are provided to check the validity of the control method proposed and the convergence of the quadrotor UAV trajectory tracking error.