Global Asymptotic Convergence Research Articles

Trajectory tracking with collision avoidance for nonholonomic vehicles with acceleration constraints and limited sensing

Nowadays, autonomously operated nonholonomic vehicles are employed in a wide range of applications, ranging from relatively simple household chores (e.g. carpet vacuuming and lawn mowing) to highly sophisticated assignments (e.g. outer space exploration and combat missions). Each application may require different levels of accuracy and capabilities from the vehicles, yet, all expect the same critical outcome: to safely complete the task while avoiding collisions with obstacles and the environment. Herein, we report on a bounded control law for nonholonomic systems of unicycle-type that satisfactorily drive a vehicle along a desired trajectory while guaranteeing a minimum safe distance from another vehicle or obstacle at all times. The control law is comprised of two parts. The first is a trajectory tracking and set-point stabilization control law that accounts for the vehicle’s kinematic and dynamic constraints (i.e. restrictions on velocity and acceleration). We show that the bounded tracking control law enforces global asymptotic convergence to the desired trajectory and local exponential stability of the full state vector in the case of set-point stabilization. The second part is a real-time avoidance control law that guarantees collision-free transit for the vehicle in noncooperative and cooperative scenarios independently of bounded uncertainties and errors in the obstacles’ detection process. The avoidance control acts locally, meaning that it is only active when an obstacle is close and null when the obstacle is safely away. Moreover, the avoidance control is designed according to the vehicle’s acceleration limits to compensate for lags in the vehicle’s reaction time. The performance of the synthesized control law is then evaluated and validated via simulation and experimental tests.

Reduced-order observer-based backstepping tracking control for a class of stochastic nonlinear systems

In this paper, an output feedback tracking control scheme is put forwarded for a class of stochastic nonlinear systems, whose dynamics involve not only unknown parameters but also unmeasured states multiplied by output nonlinearities. A type of reduced-order observer is first developed. By adding some output related items in the observer, the estimation error realize global asymptotic convergence under disturbance free condition, and global bounded convergence when considering disturbance. Besides, the dimension of the closed-loop system is reduced, and the update law of this observer gain is beneficial for steady tracking. After the observer was established, the controller is constructed by employing the adaptive backstepping approach, and a smooth nonsingular robust item is proposed to handle the influence of stochastic disturbance. All the signals in the closed system is proved to be globally bounded in probability. Moreover the output tracking error converges to an arbitrary small neighborhood of the origin by proper choosing of the design parameters. The simulation results based on current control scheme and the comparison with the previous method illustrate that the proposed output feedback scheme realizes good tracking performance and strong ability on stochastic disturbance attenuation.

Input–output linearization minimum sliding-mode error feedback control for spacecraft formation with large perturbations

To improve the control precision of nonlinear spacecraft formation flying, the input–output linearization minimum sliding-mode error feedback controller is presented based on the linear-decoupled spacecraft formation model by input–output linearization method incorporating the sliding-mode control. This paper proposes a new strategy to estimate and offset the system-control errors, which include various kinds of uncertainties and disturbances. To facilitate the analysis, the linear-decoupled spacecraft formation model is first given; on which basis, the concept of equivalent control error is introduced to define the entire model error. Based on the minimum sliding-mode covariance constraint, a cost function is formulated to estimate the equivalent control error and fed back to the conventional sliding-mode control. It is shown that the sliding mode after the input–output linearization minimum sliding-mode error feedback controller will approximate to the ideal sliding mode with high-control precision. In addition, the new methodology is applied to spacecraft formation flying. It guarantees global asymptotic convergence of the relative-tracking error in the presence of the large perturbations. More exactly, the two input–output linearization minimum sliding-mode error feedback controller laws (continuous sliding-mode control and nonsingular terminal sliding-mode control) are developed for this spacecraft formation flying system. Several fault-tolerant scenarios are considered to verify that the input–output linearization minimum sliding-mode error feedback controller is still effective in the presence of faults in spacecraft thrusters. Numerical simulations are performed to demonstrate the efficacy of the proposed methodology to maintain and reconfigure the spacecraft formation with existence of initial offsets and large perturbations effects.

Delay-Adaptive Control for Nonlinear Systems

We present a systematic delay-adaptive prediction-based control design for nonlinear systems with unknown long actuator delay. Our approach is based on the representation of the constant actuator delay as a transport Partial Differential Equation (PDE) in which the convective speed is inversely proportional to the unknown delay. We study two different frameworks, assuming first that the actuator state is measured and relaxing afterward. For the full-state feedback case, we prove global asymptotic convergence of the proposed adaptive controller while, in the second case, replacing the actuator state by its adaptive estimate, we prove local regulation. The relevance of the obtained results are illustrated by simulations of a biological activator/repressor system.

Adaptive control of rigid-link electrically driven robots with parametric uncertainties in kinematics and dynamics and without acceleration measurements

SUMMARYIn this paper, the backstepping strategy is used to design an adaptive tracking controller for rigid-link electrically driven robots in the presence of parametric uncertainties in kinematics, manipulator dynamics, and actuator dynamics. To avoid acceleration measurements, two techniques are exploited. One technique adds compensation control terms to the control law signal. The other uses a linear in variable property of the Jacobian matrix. Global asymptotic convergence of the end-effector motion tracking errors is shown via Lyapunov analysis. Simulation results are presented to show the effectiveness of the proposed control scheme.

A robust adaptive tracking controller for an aircraft with uncertain dynamical terms

This work presents, the design and the corresponding analysis of a nonlinear controller for an aircraft system subject to uncertainties in the dynamics and additive state–dependent nonlinear disturbance–like terms. Specifically; dynamic inversion technique in conjunction with a robust integral of the signum of the error feedback and an adaptive term is utilized in the overall controller design. Lyapunov based stability analysis techniques are then utilized to prove global asymptotic convergence of the tracking error.

Global stabilisation of nonlinear delay systems with a compact absorbing set

Predictor-based stabilisation results are provided for nonlinear systems with input delays and a compact absorbing set. The control scheme consists of an inter-sample predictor, a global observer, an approximate predictor, and a nominal controller for the delay-free case. The control scheme is applicable even to the case where the measurement is sampled and possibly delayed. The input and measurement delays can be arbitrarily large but both of them must be constant and accurately known. The closed-loop system is shown to have the properties of global asymptotic stability and exponential convergence in the disturbance-free case, robustness with respect to perturbations of the sampling schedule, and robustness with respect to measurement errors. In contrast to existing predictor feedback laws, the proposed control scheme utilises an approximate predictor of a dynamic type that is expressed by a system described by integral delay equations. Additional results are provided for systems that can be transformed to systems with a compact absorbing set by means of a preliminary predictor feedback.

Minimum sliding mode error feedback control for fault tolerant reconfigurable satellite formations with J2 perturbations

Minimum Sliding Mode Error Feedback Control (MSMEFC) is proposed to improve the control precision of spacecraft formations based on the conventional sliding mode control theory. This paper proposes a new approach to estimate and offset the system model errors, which include various kinds of uncertainties and disturbances, as well as smoothes out the effect of nonlinear switching control terms. To facilitate the analysis, the concept of equivalent control error is introduced, which is the key to the utilization of MSMEFC. A cost function is formulated on the basis of the principle of minimum sliding mode error; then the equivalent control error is estimated and fed back to the conventional sliding mode control. It is shown that the sliding mode after the MSMEFC will approximate to the ideal sliding mode, resulting in improved control performance and quality. The new methodology is applied to spacecraft formation flying. It guarantees global asymptotic convergence of the relative tracking error in the presence of J2 perturbations. In addition, some fault tolerant situations such as thruster failure for a period of time, thruster degradation and so on, are also considered to verify the effectiveness of MSMEFC. Numerical simulations are performed to demonstrate the efficacy of the proposed methodology to maintain and reconfigure the satellite formation with the existence of initial offsets and J2 perturbation effects, even in the fault-tolerant cases.

Global asymptotic tracking of asymmetrical underactuated surface vessels with parameter uncertainties

In this work, we investigate the tracking control problem of asymmetrical underactuated surface vessels with parameter uncertainties. The tracking error model is first derived via appropriate coordinate transformations, and is considered as a cascade structure composed of two subsystems. The Lyapunov redesign approach is employed to construct the control laws separately to stabilize the two subsystems with unknown model parameters. The cascade system theory is applied to prove the global uniform asymptotic convergence of the state trajectory to the reference one provided the desired yaw velocity is not vanishing. The effectiveness of the proposed control laws is verified by simulation examples.

Study on PMSM Integral Backstepping Controller Based on RBF Neural Network

In order to eliminate steady-state speed error of PMSM backstepping control system, an integral backstepping speed control algorithm is designed in this paper. By adding speed error integral factor in the speed Lyapunov function, the speed error can finally converge to zero when PMSM operates in steady-state. On this basis, an integral backstepping speed control algorithm based on RBF neural network compensation is proposed for PMSM backstepping control system used for high-altitude electric propulsion system which is vulnerable to load torque variables. The integral backstepping speed controller based on PMSM reference model can ensure global asymptotic convergence of the whole control system. In order to achieve fast robust adaptive control, the RBF neural network is adapted to online compensate dq axis current error produced by the reference speed and load torque changes. Simulink simulation results verify the feasibility of the given algorithm.

The design of nonsingular terminal sliding-mode feedback controller based on minimum sliding-mode error

To eliminate the effect of the uncertain disturbances and improve the control accuracy of spacecraft Attitude Control System, a nonlinear control algorithm named nonsingular terminal sliding-mode feedback controller is proposed in this work, which is mainly made up of nonsingular terminal sliding-mode controller and sliding-mode feedback controller. In the first place, nonsingular terminal sliding-mode controller is designed, which guarantees global asymptotic convergence of the attitude in the presence of the uncertain perturbations from the space. Despite that, it is the influence of the uncertain disturbances that hinder the control accuracy. Then, in order to promote the control accuracy, the sliding-mode feedback controller based on the principle of minimum sliding-mode error is proposed, which is used to compensate the control errors of the nonsingular terminal sliding-mode controller caused by the uncertainties. Hence, the determination principle of the weighting matrix in sliding-mode feedback controller is discussed, and the algorithm structure of the sliding-mode feedback controller is also analyzed, which provides the theoretical basis for the sliding-mode feedback controller. By contrast, an adaptive fuzzy algorithm is designed and introduced into the nonsingular terminal sliding-mode controller to improve the control accuracy, which named the nonsingular terminal fuzzy sliding-mode controller. Last but not the least, several numerical examples are presented to demonstrate the efficacy of the proposed nonsingular terminal sliding-mode feedback controller. Simulation results confirm that the control accuracy of the nonsingular terminal sliding-mode feedback controller is higher than the nonsingular terminal sliding-mode controller and the same as nonsingular terminal fuzzy sliding-mode controller. Not only is the calculation of the nonsingular terminal fuzzy sliding-mode feedback controller smaller than nonsingular terminal fuzzy sliding-mode controller, the adjusted parameters are also fewer than nonsingular terminal fuzzy sliding-mode controller obviously. The numerical results clearly indicate that the proposed nonsingular terminal sliding-mode feedback controller based on the principle of minimum sliding-mode error can compensate control errors accurately and quickly; therefore, it can reduce the effect of the uncertainties from the space indirectly.

Tracking control of surface vessels via fault-tolerant adaptive backstepping interval type-2 fuzzy control

This paper focuses on tracking control of fully actuated surface vessels along a desired trajectory in the presence of time-varying hydrodynamic disturbances. The combination of backstepping control and approximation-based adaptive technique allows the proposed controller to accommodate certain faults in the plant and the controller itself, and to handle time-varying hydrodynamic disturbances without explicit knowledge about the disturbance model. Through backstepping and Lyapunov synthesis, a state feedback fault-tolerant adaptive backstepping interval type-2 fuzzy logic controller is introduced, with the option of high-gain observer for output feedback control. The stability of the closed-loop systems is explored where sufficient condition for guaranteeing global asymptotic convergence of the tracking errors in state feedback control is proposed, whereas semiglobal uniform boundedness of the closed-loop signals in output feedback control is guaranteed. Simulation studies with a container ship are carried out. The proposed technique is found to be effective and robust.

Bounded controllers for global path tracking control of unicycle-type mobile robots

This paper presents a design of bounded controllers with a predetermined bound for global path tracking control of unicycle-type mobile robots at the torque level. A new one-step ahead backstepping method is first introduced. The heading angle and linear velocity of the robots are then considered as immediate controls to force the position of the robots to globally and asymptotically track its reference path. These immediate controls are designed based on the one-step ahead backstepping method to yield bounded control laws. Next, the one-step ahead backstepping method is applied again to design bounded control torques of the robots with a pre-specified bound. The proposed control design ensures global asymptotical and local exponential convergence of the position and orientation tracking errors to zero, and bounded torques driving the robots. Experimental results on a Khepera mobile robot verify the proposed control controller.

Some applications of passivity‐based control and invariance principles

This study examines the control of a passive plant utilising strictly passive feedback as motivated by the passivity theorem. A state representation of the plant is assumed with very few conditions imposed. No assumptions are made on the state representation of the feedback control law. However, some mild additional input-output properties of the feedback control law are assumed. Global stability of the closed-loop system and asymptotic convergence of a subset of the states is proven using invariance principles. The theoretical results in the study are applied to a number of application examples, demonstrating that a much broader class of controllers can deliver closed-loop global stability and asymptotic convergence, unlike the previous works in the literature where the examples are taken from. In one of the examples, it is demonstrated that actuator saturation constraints can readily be handled using the theory presented in this study.

Fault tolerant small satellite attitude control using adaptive non-singular terminal sliding mode

The Attitude Control System (ACS) plays a pivotal role in the whole performance of the spacecraft on the orbit; therefore, it is vitally important to design the control system with the performance of rapid response, high control precision and insensitive to external perturbations. In the first place, this paper proposes two adaptive nonlinear control algorithms based on the sliding mode control (SMC), which are designed for small satellite attitude control system. The nonlinear dynamics describing the attitude of small satellite is considered in a circle reference orbit, and the stability of the closed-loop system in the presence of external perturbations is investigated. Then, in order to account for accidental or degradation fault in satellite actuators, the fault-tolerant control schemes are presented. Hence, two adaptive fault-tolerant control laws (continuous sliding mode control and non-singular terminal sliding mode control) are developed by adopting the nonlinear analytical model to describe the system, which can guarantee global asymptotic convergence of the attitude control error with the existence of unknown external perturbations. The nonlinear hyperplane based Terminal sliding mode is introduced into the control law design; therefore, the system convergence performance improves and the control error is convergent in “finite time”. As a result, the study on the non-singular terminal sliding mode control is the emphasis and the continuous sliding mode control is used to compare with the non-singular terminal sliding mode control. Meanwhile, an adaptive fuzzy algorithm has been proposed to suppress the chattering phenomenon. Moreover, several numerical examples are presented to demonstrate the efficacy of the proposed controllers by correcting for the external perturbations. Simulation results confirm that the suggested methodologies yield high control precision in control. In addition, actuator degradation, actuator stuck and actuator failure for a period of time are simulated to demonstrate the fault recovery capability of the fault tolerant controllers. The numerical results clearly demonstrate the good performance of the adaptive non-singular terminal control in the event of actuator fault compare with the continuous sliding mode control.

Towards Human Capture Movement: Estimation of Anatomical Movements of the Shoulder

In this paper we focus on the human arm motion capture, which is motivated by the requirements in physical rehabilitation and training of stroke patients in the same way as monitoring of elderly person activities. The proposed methodology uses a data fusion of low-cost and low-weight MEMS sensors jointly to an a priori knowledge of the arm anatomy. The main goal is to estimate the arm position, the anatomical movements of the shoulder and its accelerations. We propose a discrete optimization based-approach which aims to search the optimal attitude ambiguity directly without decorrelation of ambiguity, and to computing the baseline vector consequently. The originality of this paper is to apply the discrete optimization to track the desired trajectory of a nonlinear system such as the Human Movement in the presence of uncertainties. The global asymptotic convergence of the nonlinear observer is guaranteed. Extensive tests of the presented methodology with real world data illustrate the effectiveness of the proposed procedure.

Global Exponential Convergence Properties for the Open-Loop Induction Motor

It has been recently shown by Oteafy and Chiasson that global asymptotic convergence properties can be guaranteed under certain conditions for the open-loop induction motor, that is for the induction motor which is fed by balanced sinusoidal voltages of constant electrical frequency. The slightly different analysis of this note not only confirms those results from a different point of view (involving the natural controlled variables rotor speed and flux modulus) but also clarifies that, under similar conditions, global exponential convergence properties can be actually established.

On the asymptotic property analysis for a class of adaptive control systems with σ‐modification

SUMMARYWith the effect of σ‐modification in adaptive control systems, robustness is achieved at the potential expense of destroying some of the convergence properties. This paper proposes a qualitative analysis method for the situations where the σ‐modification may lead to perfect tracking and also, given the prior knowledge of system parameters, may allow proper modification of the adaptive algorithm. The applicability of the proposed analysis method is illustrated by two examples, where the system control gains that lead to global asymptotic convergence are given explicitly. Further, simulations are performed to verify the qualitative analysis results. Copyright © 2012 John Wiley & Sons, Ltd.

Magnetic Detumbling of a Rigid Spacecraft

DOI: 10.2514/1.53074 In this paper, a control law that detumbles a spacecraft with magnetic actuators only is developed. A rigorous mathematical proof of global asymptotic convergence from arbitrary initial tumbling conditions to zero angular velocity in a time-varying magnetic field is derived. Furthermore, a simple criterion for determining a reasonable value of the control gain is developed. The selected gain results in a quasi-minimum time detumbling for different initialconditions,inthepresenceofmagneticcoilsaturation.Performanceoftheproposeddetumblingcontrollawis demonstrated by numerical simulations on a large set of test cases using a Monte Carlo approach.

On emulated nonlinear reduced-order observers for networked control systems

We consider a general class of nonlinear reduced-order observers and show that the global asymptotic convergence of the observation error in the absence of network-induced constraints is maintained for the emulated observer semiglobally and practically (with respect to the maximum allowable transmission interval) when system measurements are sent through a communication channel. Networks governed by a Lyapunov uniformly globally asymptotically stable protocol are investigated. Our results can be used to synthesize various observers for networked control systems for a range of network configurations, as we illustrate it by considering classes of immersion and invariance observers which include the circle-criterion observers.