Asymptotic Tracking Research Articles

Trajectory tracking of a bouncing ball in a triangular billiard by unfolding and folding the billiard table

In this paper, a feedback control law is proposed to solve the asymptotic tracking problem for a ball bouncing in a triangular billiard. This controller is designed by firstly transforming the billiard table into a surface on which the ball moves without experiencing any impact, i.e. by reflecting the billiard table rather than the ball trajectory and designing a position feedback controller based on such an unfolded billiard table. Furthermore, it is shown how such an infinite-billiard table can be mapped to the surface of a torus, thus leading to bounded trajectories of the ball.

Nonlinear spacing policies for vehicle platoons: A geometric approach to decentralized control

In this paper a decentralized approach to the platooning problem with nonlinear spacing policies is considered. A predecessor–follower control structure is presented in which a vehicle is responsible for tracking of a desired spacing policy with respect to its predecessor, regardless of the control action of the latter. From the perspective of geometric control theory, we state necessary and sufficient conditions for the existence of decentralized controllers that guarantee tracking and asymptotic stabilization of a general nonlinear spacing policy. Moreover, all nonlinear spacing policies for which there exists a decentralized state feedback controller that achieves asymptotic tracking are characterized. It is shown that string stability is a consequence of the choice of spacing policy and sufficient conditions for a spacing policy to imply string stability are given. As an example, we fully characterize all state feedback controllers that achieve the control goals for a given nonlinear spacing policy, guaranteeing asymptotic tracking for a heterogeneous platoon. The results are illustrated through simulations.

Asymptotic tracking position control with active oscillation damping of a multibody Mars vehicle using two artificial augmentation approaches

The Valles Marineris Explorer Cooperative Swarm navigation, Mission and Control research project aims to explore the Valles Marineris canyon system on Mars with, among others, multibody rotary-wing unmanned aerial vehicles (UAVs) comprising of a hexrotor system and a helium-filled balloon being attached to it by means of a rope. In this paper, we develop a high-fidelity closed-loop control system in MATLAB® and Simulink™ to present the application of an adequate flight controller guaranteeing (1) asymptotic tracking position control of the multibody flight system, (2) suppression of the balloon’s swinging motion in forward flight case, and (3) stabilization of the rope angle around its equilibrium for steady-state conditions. Applying feedback linearization for the outer loop and analytical backstepping for the inner loop of a nonlinear cascaded control design model of the hexrotor system, we propose an extension of the baseline flight controller by two artificial augmentation approaches to cope with the balloon dynamics. Basically, by utilizing oscillation damping feedbacks of the uncertain plant which are applied as additional commands to either the inner or the outer loop’s reference model. Simulation results are presented for an eight-shaped flight maneuver at the bottom of Valles Marineris proving that the augmentation units increase the flight controller capabilities to suppress modeling errors artificially—without changing the baseline control laws. The augmentation units actively damp the balloon motion in the forward flight case for non-steady-state conditions to counteract the rope oscillations and finally stabilize the rope angle around its equilibrium, so that the Mars vehicle is able to reach a steady-state in position when its extraterrestrial mission profile is successfully completed.

Adaptive NN-based finite-time trajectory tracking control of wheeled robotic systems

The trajectory tracking and finite-time control problems of wheeled robotic systems with nonlinear dynamics and uncertainties are investigated in this paper. An adaptive neural network (NN)-based control technique is developed to deal with the nonlinearities and uncertainties. Then, finite-time dynamic control and kinematic control schemes are constructed on the basis of the adaptive estimations to remedy the negative influence of uncertainties and nonlinearities. Specific forward and azimuthal angular velocities are developed by using NN-based kinematic control schemes to obtain the finite-time tracking of the desired position trajectory for the wheeled robotic system. Furthermore, the asymptotic tracking of the specific forward and azimuthal angular velocities is further achieved based on the finite-time dynamic control schemes with uncertainties and nonlinear dynamics. The efficacy of the developed finite-time tracking control approach is substantiated by a robotic system.

Adaptive asymptotic tracking control of uncertain fractional-order nonlinear systems with unknown quantized input and control directions subject to actuator failures

This article addresses an adaptive backstepping control design for uncertain fractional-order nonlinear systems in the strict-feedback form subject to unknown input quantization, unknown state-dependent control directions, and unknown actuator failure. The system order can be commensurate or noncommensurate. The total number of failures is allowed to be infinite. The Nussbaum function is used to deal with the problem of unknown control directions. Compared with the existing results, the control gains can be functions of states and the knowledge of quantization parameters and characteristics of the actuator failure are unknown. By applying the backstepping control approach based on the frequency-distributed model, it is proved that all the closed-loop signals remain bounded and the output tracking error converges to the origin asymptotically. Finally, the effectiveness of the proposed controller is demonstrated by two simulation examples.

The G -Asymptotic Tracking Property and G -Asymptotic Average Tracking Property in the Inverse Limit Spaces under Group Action

Firstly, we introduce the definitions of G -asymptotic tracking property, G -asymptotic average tracking property, and G -quasi-weak almost-periodic point. Secondly, we study their dynamical properties and characteristics. The results obtained improve the conclusions of asymptotic tracking property, asymptotic average tracking property, and quasi-weak almost-periodic point in the inverse limit space and provide the theoretical basis and scientific foundation for the application of tracking property in computational mathematics, biological mathematics, and computer science.

A Moving Target Tracking Control of Quadrotor UAV Based on Passive Control and Super-Twisting Sliding Mode Control

A novel asymptotic tracking controller for an underactuated quadrotor unmanned aerial vehicle (UAV) is proposed to solve a moving target tracking problem. Firstly, the control system is decoupled into the position control system and the attitude control system. Secondly, a method combined artificial potential field with passivity control (APF&PC) is introduced for the positioning system to achieve high-precision tracking of moving target at a fixed distance. Thirdly, a super-twisting sliding mode (STSM) method with an improved reaching law for the attitude system is applied to ensure that the attitude converges to the desired value. Furthermore, the stabilities of two subsystems are proved, and sufficient stability conditions are derived based on the passive method and Lyapunov method, respectively. Finally, simulation results of the moving target tracking verify the superiority and robustness of the proposed control method in the presence of parameter uncertainties and external disturbances.

Robust control of uncertain robotic systems: An adaptive friction compensation approach

This paper solves the robust control problem of robotic manipulator systems with uncertain dynamics by friction compensation approach. A weighting factor is introduced to distinguish the role of friction in control process by comparing the directions of sliding vector and friction. Utilizing the weighting factor, model-based and model-free adaptive friction compensation controllers are designed to achieve asymptotical tracking of the desired joint-space trajectory according to the knowledge of friction. The damping property of friction is fully used to improve the control performance by compensating the friction harmful for the stability, and on the other hand, utilizing the beneficial friction. Numerical simulations are given to demonstrate the control performance of the proposed approach.

Output Regulation of Linear Stochastic Systems

We address the output regulation problem for a general class of linear stochastic systems. Specifically, we formulate and solve the ideal full-information and output-feedback problems, obtaining perfect, but non-causal, asymptotic regulation. A characterisation of the problem solvability is deduced. We point out that the ideal problems cannot be solved in practice because they unrealistically require that the Brownian motion affecting the system is available for feedback. Drawing from the ideal solution, we formulate and solve approximate versions of the full-information and output-feedback problems, which do not yield perfect asymptotic tracking but can be solved in a realistic scenario. These solutions rely on two key ideas: first we introduce a discrete-time a-posteriori estimator of the variations of the Brownian motion obtained causally by sampling the system state or output; second we introduce a hybrid state observer and a hybrid regulator scheme which employ the estimated Brownian variations. The approximate solution tends to the ideal as the sampling period tends to zero. The proposed theory is validated by the regulation of a circuit subject to electromagnetic noise.

Neural network-based asymptotic tracking control design for stochastic nonlinear systems

ABSTRACT This article is focused on the adaptive neural network (ANN) asymptotic tracking control design for stochastic nonlinear systems with state constraints. The neural networks are utilised to deal with unknown uncertainties. The existence of state constraints and unknown virtual control coefficients (UVCC) bring many difficulties for control synthesis and analysis. With the aid of barrier Lyapunov function, the predefined state constraints are guaranteed. By fusing the lower bounds of UVCC into Lyapunov function construction, a novel ANN asymptotic tracking control method is devised by employing the bound estimation approach and backstepping technique. The presented asymptotic tracking controller can guarantee that the tracking error converges to zero in probability and the state constraints are not violated. The validity of the developed scheme is elucidated by simulation example.

Minimum eigenvalue-based adaptive compensation of actuator faults for flexible spacecraft

This paper develops an adaptive compensation scheme for flexible spacecraft under actuator faults. System uncertainties caused by faults and flexible uncertainties are first parameterized. To deal with the uncertainty of the control gain matrix caused by faults, a new control gain matrix is constructed. By using the minimum eigenvalue of this new control gain matrix, a nominal control signal is designed. Then an adaptive control signal is designed by estimating the uncertain parameters in the nominal control signal. The developed adaptive control signal guarantees system stability and asymptotic tracking properties. Simulation results are given to verify the effectiveness of the proposed adaptive compensation scheme.

Adaptive robust dynamic surface asymptotic tracking for uncertain strict-feedback nonlinear systems with unknown control direction

For strict-feedback nonlinear systems (SFNSs) with unknown control direction, this paper synthesizes an asymptotic tracking controller by a combination of the dynamic surface control (DSC) technique, the Nussbaum gain technique (NGT) and fuzzy logic systems (FLSs). The SFNSs under study feature unknown nonlinear uncertainties and external disturbances. By utilizing the DSC technique with nonlinear filters, the issue of ‘differential explosion’ is obviated, in which the adaptive laws are constructed to conquer the effect of unknown functions. The FLSs are exploited to cope with uncertainties without any prior conditions of the ideal weight vectors and the approximation errors. In addition, by introducing the NGT, the unknown control direction problem is solved. Compared with the existing results, the proposed design procedure is able to simultaneously overcome the ‘differential explosion’ and the unknown control direction problems, and asymptotic tracking is accomplished. At the end, a second-order numerical system and a more realistic Norrbin nonlinear mathematical model are applied to confirm the feasibility of the design procedure.

Partial-state feedback multivariable MRAC and reduced-order designs

This paper develops a new model reference adaptive control (MRAC) framework using partial-state feedback for solving a multivariable adaptive output tracking problem. The developed MRAC scheme has full capability to deal with plant uncertainties for output tracking and has desired flexibility to combine the advantages of full-state feedback MRAC and output feedback MRAC. With such a new control scheme, the plant-model matching condition is achievable as with an output or state feedback MRAC design. A stable adaptive control scheme is developed based on LDS decomposition of the plant high-frequency gain matrix, which guarantees closed-loop stability and asymptotic output tracking. The proposed partial-state feedback MRAC scheme not only expands the existing family of MRAC, but also provides new features to the adaptive control system, including additional design flexibility and feedback capacity. Based on its additional design flexibility, a minimal-order MRAC scheme is also presented, which reduces the control adaptation complexity and relaxes the feedback information requirement, compared to the existing MRAC schemes. New results are presented for plant-model matching, error model, adaptive law and stability analysis. A simulation study of a linearized aircraft model is conducted to demonstrate the effectiveness and new features of the proposed MRAC control scheme.

Implicit function based adaptive control of non-canonical form discrete-time nonlinear systems

This paper presents a new study on adaptive state feedback output tracking control problem for uncertain discrete-time nonlinear systems in a general non-canonical form. Time-advance operations on the output of such systems result in the output dynamics being nonlinearly dependent on the control input and unknown parameters, which leads to three technical issues: implicit relative degree; nonlinearly parameterized uncertainties; and non-affine control input. To address these issues, this paper first employs feedback linearization and implicit function theory to construct a relative degree dependent normal form; then proposes an adaptive parametric reconstruction based method to simultaneously deal with linearly and nonlinearly parameterized uncertainties in the output dynamics; and finally constructs a key implicit function equation to derive a unique adaptive control law which ensures closed-loop stability and asymptotic output tracking. An explicitly iterative solution based adaptive control law is also proposed to ensure closed-loop stability and bounded output tracking within any degree of accuracy. The simulation verifies the effectiveness of the proposed adaptive control method.

Linear‐like properties arise naturally in the adaptive control setting

SummaryClassical discrete‐time adaptive controllers typically provide asymptotic stabilization and tracking; usually the affect of the noise is at best bounded‐input bounded‐output. Recently we have shown that if you design a discrete‐time adaptive controller in just the right way, then in a variety of situations you not only obtain exponential stability, but also a bounded gain on the noise in every p−norm, as well as a never‐before‐seen linear‐like convolution bound on the input–output behavior. Quite surprisingly, the approach is very natural, and relies on the use of the unmodified, original projection algorithm to carry out parameter estimation; if the set of plant uncertainty is not convex, then a multi‐estimator and switching are used. The goal of this paper is to provide an overview of the approach, discuss the results‐to‐date, and list some of the open problems.

Backstepping-Based Adaptive Control of a Class of Uncertain Incommensurate Fractional-Order Nonlinear Systems With External Disturbance

Normally, achieving asymptotic convergence in the presence of unknown arbitrarily bounded external disturbance requires noncontinuous control signals. In this article, an adaptive backstepping controller for a class of incommensurate fractional-order nonlinear systems with parametric uncertainties and external time-varying disturbance is proposed to realize asymptotic perfect output tracking. Appropriate adaptive laws are designed to deal with the uncertainties and unknown bounded disturbance. In the designed adaptive backstepping control scheme, an auxiliary function is adopted to obtain a smooth control signal. It is theoretically shown that asymptotic output tracking to a given reference signal is achieved while ensuring global system stability in the sense that all the closed-loop signals are bounded. Simulation and experimental studies demonstrate the effectiveness of the proposed backstepping control scheme.

Predefined-Time Asymptotic Tracking Control for Hypersonic Flight Vehicles With Input Quantization and Faults

This article presents a fuzzy adaptive tracking control algorithm for hypersonic flight vehicles with input quantization and faults. Different from the usual stabilization results, we present a method to achieve asymptotic tracking with predefined-time performance. An error transformation is established and a modified Lyapunov function (MLF) is constructed. The predefined-time asymptotic tracking control goal can be achieved by simply guaranteeing the boundness of the proposed MLF which is related to the transformed tracking errors. The proposed control strategy has three advantages. First, the predefined-time tracking performance can be ensured even when the initial tracking conditions are completely unknown. Second, no compensation is required for the fuzzy reconstruction errors to achieve accurate tracking control such that the control design is significantly simplified. Third, the derived controller is fault-tolerant, and since the system is controlled by quantized signals, the communication burden can be considerably reduced. Theoretical proof and numerical simulation are presented to demonstrate the effectiveness of the proposed control algorithm.

Robust adaptive radial‐basis function neural network‐based backstepping control of a class of perturbed nonlinear systems with unknown system parameters

SummaryIn this article, the robust adaptive output tracking control problem is addressed for a class of nonlinear systems with nonlinear dynamics and unknown system parameters. The nonlinear dynamics including internal parameter uncertainties and external disturbances are formulated as time‐varying state/input‐dependent perturbations. Radial‐basis function neural networks (RBFNNs) are developed to approximate the perturbations. A robust adaptive RBFNN‐based output feedback control strategy against the perturbations is developed by using backstepping technique with immeasurable states and without knowing any system parameter. Based on Lyapunov stability theorem, the asymptotic output tracking results of the closed‐loop nonlinear system are obtained in the presence of perturbations, immeasurable states, and unknown system parameters. The efficacy of the proposed adaptive RBFNN‐based output feedback control strategy is validated by simulation in a DC–DC buck converter system.

Cascade tracking control of servo motor with robust adaptive fuzzy compensation

Servo motor drive systems with high-accuracy position tracking control suffer from some uncertainties from inherent mechanical friction and varying end-load. In this paper, a novel cascade controller with feedforward robust adaptive fuzzy compensation is proposed to overcome the negative impacts of the uncertainties. An adaptive fuzzy logic system is used to estimate the friction, which is applied for designing a feedforward compensator to improve the tracking performance. Based on Lyapunov stability theory, we show that closed-loop system can ensure the semi-global asymptotic tracking performance. Our proposed compensation strategy takes advantage of utilizing the cascade P/PI controller without changing the original control system structure, which is practical and valuable to industrial applications. Simulation results indicate some merits of the proposed controll scheme in terms of the system stability, adaptivity and robustness with respect to different uncertainties.

Adaptive Asymptotic Tracking Control for a Class of Uncertain Input-Delayed Systems with Periodic Time-Varying Disturbances

In this paper, the problem of adaptive asymptotic tracking control for a class of uncertain systems with periodic time-varying disturbances and input delay is studied. By combining Fourier series expansion (FSE) with radial basis function neural network (RBFNN), a hybrid function approximator is used to learn the functions with periodic time-varying disturbances. At the same time, the dynamic surface control technique with a nonlinear filter is used to avoid the “complexity explosion” problem in the process of traditional backstepping technology. Ultimately, all closed-loop signals are guaranteed to be semiglobally uniformly bounded, and the given reference signal can be asymptotically tracked by the output signals of system. A simulation example is given to verify the effectiveness of the proposed control scheme.