Tracking Control (tracking + control)

Distribution by Scientific Domains

Terms modified by Tracking Control

  • tracking control problem

  • Selected Abstracts


    Nonlinear Modeling and Tracking Control of a Hydraulic Rotary Vane Actuator

    PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005
    Frank Heidtmann
    Rotary vane actuators as rotational drives provide rotational movements directly because they are constructed as a joint and actuator in one. So it is possible to pass on the disadvantageous transmission kinematics used with the so far usual differential cylinders at the arms of large manipulators. However, the use of hydraulic rotary vane actuators is associated with high internal oil leakage and/or high friction. Therefore, a nonlinear dynamic model for such an actuator, driving a rigid robot arm, as well as its nonlinear control are derived. To achieve tracking control a model based control law is set up using fundamental linear differential equations for the tracking error. The control law is implemented and tested on a testbed, the produced experimental results are presented. The same control algorithm can also be used to realize nonlinear disturbance attenuation for hydraulic rotary vane actuators via tracking control. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    Robust Tracking Control For A Wheeled Mobile Manipulator With Dual Arms Using Hybrid Sliding-Mode Neural Network

    ASIAN JOURNAL OF CONTROL, Issue 4 2007
    Ching-Chih Tsai
    ABSTRACT In this paper, a robust tracking controller is proposed for the trajectory tracking problem of a dual-arm wheeled mobile manipulator subject to some modeling uncertainties and external disturbances. Based on backstepping techniques, the design procedure is divided into two levels. In the kinematic level, the auxiliary velocity commands for each subsystem are first presented. A sliding-mode equivalent controller, composed of neural network control, robust scheme and proportional control, is constructed in the dynamic level to deal with the dynamic effect. To deal with inadequate modeling and parameter uncertainties, the neural network controller is used to mimic the sliding-mode equivalent control law; the robust controller is designed to compensate for the approximation error and to incorporate the system dynamics into the sliding manifold. The proportional controller is added to improve the system's transient performance, which may be degraded by the neural network's random initialization. All the parameter adjustment rules for the proposed controller are derived from the Lyapunov stability theory and e-modification such that uniform ultimate boundedness (UUB) can be assured. A comparative simulation study with different controllers is included to illustrate the effectiveness of the proposed method. [source]


    Tracking control for sampled-data systems with uncertain time-varying sampling intervals and delays

    INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 4 2010
    N. van de Wouw
    Abstract In this paper, a solution to the approximate tracking problem of sampled-data systems with uncertain, time-varying sampling intervals and delays is presented. Such time-varying sampling intervals and delays can typically occur in the field of networked control systems. The uncertain, time-varying sampling and network delays cause inexact feedforward, which induces a perturbation on the tracking error dynamics, for which a model is presented in this paper. Sufficient conditions for the input-to-state stability (ISS) of the tracking error dynamics with respect to this perturbation are given. Hereto, two analysis approaches are developed: a discrete-time approach and an approach in terms of delay impulsive differential equations. These ISS results provide bounds on the steady-state tracking error as a function of the plant properties, the control design and the network properties. Moreover, it is shown that feedforward preview can significantly improve the tracking performance and an online extremum seeking (nonlinear programming) algorithm is proposed to online estimate the optimal preview time. The results are illustrated on a mechanical motion control example showing the effectiveness of the proposed strategy and providing insight into the differences and commonalities between the two analysis approaches. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    Tracking control for switched linear systems with time-delay: a state-dependent switching method,

    ASIAN JOURNAL OF CONTROL, Issue 5 2009
    Qing-Kui Li
    Abstract Tracking control for switched linear systems with time-delay is investigated in this paper. Based on the state-dependent switching method, sufficient conditions for the solvability of the tracking control problem are given. We use single Lyapunov function technique and a typical hysteresis switching law to design a tracking control law such that the H, model reference tracking performance is satisfied. The controller design problem can be solved efficiently by using linear matrices inequalities. Since convex combination techniques are used to derive the delay independent criteria, some subsystems are allowed to be unstable. It is highly desirable that a non-switched time-delay system can not earn such property. Simulation example shows the feasibility and validity of the switching control law. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source]


    Tracking control of a piezo-actuated stage based on a frictional model,

    ASIAN JOURNAL OF CONTROL, Issue 3 2009
    Yi-Cheng Huang
    Abstract The tracking control accuracy of a piezoelectric actuator (PEA) is limited due to the actuator's inherent hysteretic nonlinearity. Direct drive of PEA on a positioning stage with friction force will cause control problems. An approximated dynamic model of PEA with consideration of friction force is novel synthesized for control. This model is based on a second-order transfer function with two parameterization terms. The first time delay term consists of the hysteresis of piezo effect combined with frictional force lag with varying velocity. The second term is comprised of both presliding and sliding regimes. The H-infinite tracking controller is designed to compensate for the structural uncertainty associated with time delay and the unstructured frictional force in the PEA stage. Iterative Learning Control is implemented to reduce the unmodeled repetitive error by a factor of 20. Numerical simulations and experimental tests consolidate the root mean square (RMS), positioning error close to the hardware reproducibility and accuracy level. Experimental results show the controlled stage can be potentially used for precise positioning. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source]


    Neural network-based adaptive attitude tracking control for flexible spacecraft with unknown high-frequency gain

    INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 6 2010
    Qinglei Hu
    Abstract Adaptive control design using neural networks (a) is investigated for attitude tracking and vibration stabilization of a flexible spacecraft, which is operated at highly nonlinear dynamic regimes. The spacecraft considered consists of a rigid body and two flexible appendages, and it is assumed that the system parameters are unknown and the truncated model of the spacecraft has finite but arbitrary dimension as well, for the purpose of design. Based on this nonlinear model, the derivation of an adaptive control law using neural networks (NNs) is treated, when the dynamics of unstructured and state-dependent nonlinear function are completely unknown. A radial basis function network that is used here for synthesizing the controller and adaptive mechanisms is derived for adjusting the parameters of the network and estimating the unknown parameters. In this derivation, the Nussbaum gain technique is also employed to relax the sign assumption for the high-frequency gain for the neural adaptive control. Moreover, systematic design procedure is developed for the synthesis of adaptive NN tracking control with L2 -gain performance. The resulting closed-loop system is proven to be globally stable by Lyapunov's theory and the effect of the external disturbances and elastic vibrations on the tracking error can be attenuated to the prescribed level by appropriately choosing the design parameters. Numerical simulations are performed to show that attitude tracking control and vibration suppression are accomplished in spite of the presence of disturbance torque/parameter uncertainty. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    Indirect adaptive control of a class of marine vehicles

    INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 4 2010
    Yannick Morel
    Abstract A nonlinear adaptive framework for bounded-error tracking control of a class of non-minimum phase marine vehicles is presented. The control algorithm relies on a special set of tracking errors to achieve satisfactory tracking performance while guaranteeing stable internal dynamics. First, the design of a model-based nonlinear control law, guaranteeing asymptotic stability of the error dynamics, is presented. This control algorithm solves the tracking problem for the considered class of marine vehicles, assuming full knowledge of the system model. Then, the analysis of the zero-dynamics is carried out, which illustrates the efficacy of the chosen set of tracking errors in stabilizing the internal dynamics. Finally, an indirect adaptive technique, relying on a partial state predictor, is used to address parametric uncertainties in the model. The resulting adaptive control algorithm guarantees Lyapunov stability of the errors and parameter estimates, as well as asymptotic convergence of the errors to zero. Numerical simulations illustrate the performance of the adaptive algorithm. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    Nonlinear reference tracking control of a gas turbine with load torque estimation

    INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 8 2008
    B. Pongrácz
    Abstract Input,output linearization-based adaptive reference tracking control of a low-power gas turbine model is presented in this paper. The gas turbine is described by a third-order nonlinear input-affine state-space model, where the manipulable input is the fuel mass flowrate and the controlled output is the rotational speed. The stability of the one-dimensional zero dynamics of the controlled plant is investigated via phase diagrams. The input,output linearizing feedback is extended with a load torque estimator algorithm resulting in an adaptive feedback scheme. The tuning of controller parameters is performed considering three main design goals: appropriate settling time, robustness against environmental disturbances and model parameter uncertainties, and avoiding the saturation of the actuator. Simulations show that the closed-loop system is robust with respect to the variations in uncertain model and environ-mental parameters and its performance satisfies the defined requirements. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    A nonlinear adaptive speed tracking control for sensorless permanent magnet step motors with unknown load torque

    INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 3 2008
    P. Tomei
    Abstract Assuming that only stator currents and voltages are available for feedback, a nonlinear adaptive speed tracking control algorithm is proposed for a permanent magnet step motor with unknown constant load torque. It relies on three new theoretical results: the ,s-alignment' and ,c-alignment' procedures, in which the motor is forced to reach certain known equilibrium points, and an output feedback controller which guarantees asymptotic speed tracking for every initial condition belonging to an explicitly computed domain of attraction (global exponential convergence is achieved in the case of known constant load torque). Numerical simulation results illustrate the effectiveness of the proposed solution. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    Adaptive motion/force tracking control of holonomic constrained mechanical systems: a unified viewpoint

    INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 5 2007
    Chian-Song Chiu
    Abstract This paper proposes a robust adaptive motion/force tracking controller for holonomic constrained mechanical systems with parametric uncertainties and disturbances. First, two types of well-known holonomic systems are reformulated as a unified control model. Based on the unified control model, an adaptive scheme is then developed in the presence of pure parametric uncertainty. The proposed controller guarantees asymptotic motion and force tracking without the need of extra conditions. Next, when considering external disturbances, control gains are designed by solving a linear matrix inequality (LMI) problem to achieve prescribed robust performance criterion. Indeed, arbitrary disturbance/parametric error attenuation with respect to both motion and force errors along with control input penalty are ensured in the L2 -gain sense. Finally, applications are carried out on a two-link constrained robot and two planar robots transporting a common object. Numerical simulation results show the expected performances. Copyright © 2006 John Wiley & Sons, Ltd. [source]


    Robust adaptive tracking control of uncertain discrete time systems

    INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 9 2005
    Shengping Li
    Abstract In this paper, the problem of robust adaptive tracking for uncertain discrete-time systems is considered from the slowly varying systems point of view. The class of uncertain discrete-time systems considered is subjected to both ,,, to ,,, bounded unstructured uncertainty and external additive bounded disturbances. A priori knowledge of the dynamic model of the reference signal to be tracked is not completely known. For such problem, an indirect adaptive tracking controller is obtained by frozen-time controllers that at each time optimally robustly stabilize the estimated models of the plant and minimize the worst-case steady-state absolute value of the tracking error of the estimated model over the model uncertainty. Based on ,,, to ,,, stability and performance of slowly varying system found in the literature, the proposed adaptive tracking scheme is shown to have good robust stability. Moreover, a computable upper bound on the size of the unstructured uncertainty permitted by the adaptive system and a computable tight upper bound on asymptotic robust steady-state tracking performance are provided. Copyright © 2005 John Wiley & Sons, Ltd. [source]


    Adaptive tracking control for electrically-driven robots without overparametrization

    INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 2 2002
    Yeong-Chan Chang
    Abstract This paper addresses the motion tracking control of robot systems actuated by brushed direct current motors in the presence of parametric uncertainties and external disturbances. By using the integrator backstepping technique, two kinds of adaptive control schemes are developed: one requires the measurements of link position, link velocity and armature current for feedback and the other requires only the measurements of link position and armature current for feedback. The developed adaptive controllers guarantee that the resulting closed-loop system is locally stable, all the states and signals are bounded, and the tracking error can be made as small as possible. The attraction region can be not only arbitrarily preassigned but also explicitly constructed. The main novelty of the developed adaptive control laws is that the number of parameter estimates is exactly equal to the number of unknown parameters throughout the entire electromechanical system. Consequently, the phenomenon of overparametrization, a significant drawback of employing the integrator backstepping technique to treat the control of electrically driven robots in the previous literature, is eliminated in this study. Finally, simulation examples are given to illustrate the tracking performance of electrically driven robot manipulators with the developed adaptive control schemes. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    A levenberg,marquardt learning applied for recurrent neural identification and control of a wastewater treatment bioprocess

    INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 11 2009
    Ieroham S. Baruch
    The paper proposed a new recurrent neural network (RNN) model for systems identification and states estimation of nonlinear plants. The proposed RNN identifier is implemented in direct and indirect adaptive control schemes, incorporating a noise rejecting plant output filter and recurrent neural or linear-sliding mode controllers. For sake of comparison, the RNN model is learned both by the backpropagation and by the recursive Levenberg,Marquardt (L,M) learning algorithm. The estimated states and parameters of the RNN model are used for direct and indirect adaptive trajectory tracking control. The proposed direct and indirect schemes are applied for real-time control of wastewater treatment bioprocess, where a good, convergence, noise filtering, and low mean squared error of reference tracking is achieved for both learning algorithms, with priority of the L,M one. © 2009 Wiley Periodicals, Inc. [source]


    Robust tracking control for a class of MIMO nonlinear systems with measurable output feedback

    INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 1 2008
    Ya-Jun Pan
    Abstract This paper proposes a robust output feedback controller for a class of nonlinear systems to track a desired trajectory. Our main goal is to ensure the global input-to-state stability (ISS) property of the tracking error nonlinear dynamics with respect to the unknown structural system uncertainties and external disturbances. Our approach consists of constructing a nonlinear observer to reconstruct the unavailable states, and then designing a discontinuous controller using a back-stepping like design procedure to ensure the ISS property. The observer design is realized through state transformation and there is only one parameter to be determined. Through solving a Hamilton,Jacoby inequality, the nonlinear control law for the first subsystem specifies a nonlinear switching surface. By virtue of nonlinear control for the first subsystem, the resulting sliding manifold in the sliding phase possesses the desired ISS property and to certain extent the optimality. Associated with the new switching surface, the sliding mode control is applied to the second subsystem to accomplish the tracking task. As a result, the tracking error is bounded and the ISS property of the whole system can be ensured while the internal stability is also achieved. Finally, an example is presented to show the effectiveness of the proposed scheme. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    Adaptive backstepping control for a class of nonlinear systems using neural network approximations

    INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 7 2004
    K. K. Tan
    In this paper, an adaptive neural network (NN) backstepping technique is developed for tracking control of a class of nonlinear systems. NNs are used to compensate for the unknown nonlinear functions in the system. A systematic backstepping approach is established to synthesize the adaptive NN control scheme that ensures the boundedness of all the signals in the closed-loop system, and yields a small tracking error. The issue of transient performance is also addressed under an analytical framework. The effectiveness of the proposed scheme is demonstrated by computer simulations. Copyright © 2004 John Wiley & Sons, Ltd. [source]


    Sliding mode control of boost and buck-boost power converters using the dynamic sliding manifold

    INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 14 2003
    Yuri B. Shtessel
    Abstract Non-minimum phase tracking control is studied for boost and buck-boost power converters. A sliding mode control algorithm is developed to track directly a causal voltage tracking profile given by an exogenous system. The approximate causal output non-minimum phase asymptotic tracking in non-linear boost and buck-boost power converters is addressed via sliding mode control using a dynamic sliding manifold (DSM). Use of DSM allows the stabilization of the internal dynamics when the output tracking error tends asymptotically to zero in the sliding mode. The sliding mode controller with DSM links features of conventional sliding mode control (insensitivity to matched non-linearities and disturbances) and a conventional dynamic compensator (accommodation to unmatched disturbances). Numerical examples demonstrate the effectiveness of the sliding mode controller even for a known time-varying load. Copyright © 2003 John Wiley & Sons, Ltd. [source]


    Adaptive output feedback tracking control of spacecraft formation

    INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 2-3 2002
    Hong Wong
    Abstract In this paper, an adaptive, output feedback control design methodology is presented for a spacecraft formation flying (SFF) system. A Lagrangian derivation of the SFF model is considered to produce position dynamics for follower spacecraft #n relative to follower spacecraft #(n,1), where n is an arbitrary positive integer, assuming that the leader spacecraft in the formation follows a no-thrust, natural, elliptical orbit. Next, a control law is designed to provide a filtered velocity measurement and a desired adaptive compensation with semi-global, asymptotic, relative position tracking. To show the efficacy of the control algorithm, all desired trajectories are generated online by numerically solving the unperturbed nonlinear SFF dynamics with initial conditions satisfying a no-thrust, natural orbit constraint equation. The proposed control law is simulated for the case of two and three spacecraft and is shown to yield semi-global, asymptotic tracking of the relative position in addition to the convergence of disturbance parameter estimates. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    Adaptive tracking control of flexible-joint manipulators without overparametrization

    JOURNAL OF FIELD ROBOTICS (FORMERLY JOURNAL OF ROBOTIC SYSTEMS), Issue 7 2004
    Min S. Kim
    In this paper, an adaptive controller is designed for rigid-link flexible-joint robot manipulators based on link and actuator position measurements only. It is based on the adaptive integrator backstepping method and the link and actuator velocity filters are used to estimate the unknown velocity terms. Moreover, the proposed controller exploits the estimate of the joint stiffness matrix inverse to overcome the overparametrization problem, which has been a significant drawback in adaptive partial state feedback controllers. It achieves asymptotic tracking of link positions while keeping all states and signals bounded. The tracking capability of the presented method is shown through simulation results of one- and two-link flexible joint manipulators. © 2004 Wiley Periodicals, Inc. [source]


    A New Navigation Method for an Automatic Guided Vehicle

    JOURNAL OF FIELD ROBOTICS (FORMERLY JOURNAL OF ROBOTIC SYSTEMS), Issue 3 2004
    Chen Wuwei
    This paper presents a new navigation method for an automatic guided vehicle (AGV). This method utilizes a new navigation and control scheme based on searching points on an arc. Safety measure indices are defined and are generated from the output of a fuzzy neural network which define the actions the AGV is to take when in the presence of obstacles. The proposed algorithm integrates several functions required for automatic guided vehicle navigation and tracking control and it exhibits satisfactory performance when maneuvering in complex environments. The automatic guided vehicle with this navigation control system not only can quickly process environmental information, but also can efficiently avoid dynamic or static obstacles, and reach targets safely and reliably. Extensive simulation and experimental results demonstrate the effectiveness and correct behavior of this scheme. © 2004 Wiley Periodicals, Inc. [source]


    Robust face tracking control of a mobile robot using self-tuning Kalman filter and echo state network,

    ASIAN JOURNAL OF CONTROL, Issue 4 2010
    Chi-Yi Tsai
    Abstract This paper presents a novel design of face tracking algorithm and visual state estimation for a mobile robot face tracking interaction control system. The advantage of this design is that it can track a user's face under several external uncertainties and estimate the system state without the knowledge about target's 3D motion-model information. This feature is helpful for the development of a real-time visual tracking control system. In order to overcome the change in skin color due to light variation, a real-time face tracking algorithm is proposed based on an adaptive skin color search method. Moreover, in order to increase the robustness against colored observation noise, a new visual state estimator is designed by combining a Kalman filter with an echo state network-based self-tuning algorithm. The performance of this estimator design has been evaluated using computer simulation. Several experiments on a mobile robot validate the proposed control system. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source]


    Globally decentralized adaptive backstepping neural network tracking control for unknown nonlinear interconnected systems,

    ASIAN JOURNAL OF CONTROL, Issue 1 2010
    Weisheng Chen
    Abstract A globally stable decentralized adaptive backstepping neural network tracking control scheme is designed for a class of large-scale systems with mismatched interconnections. Under the assumption that the subsystems share the reference signals from the other subsystems, neural networks are used to approximate the unknown interconnections dependent on all reference signals such that the NN approximation domain can be determined a priori based on the bounds of reference signals. The proposed control approach can guarantee that all closed-loop signals are globally uniformly ultimately bounded and that the tracking errors converge to a small residual set around the origin. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source]


    Constrained PID tracking control for output PDFs of non-gaussian stochastic system based on LMIs,

    ASIAN JOURNAL OF CONTROL, Issue 5 2009
    Yang Yi
    Abstract This paper presents a new PID tracking control strategy for general non-Gaussian stochastic systems based on a square root B-spline model for the output probability density functions (PDFs). Using the B-spline expansion with modeling errors and the nonlinear weight model with exogenous disturbances, the PDF tracking is transformed to a constrained dynamical tracking control problem for weight vectors. Instead of the non-convex design algorithms, the generalized PID controller structure and the improved convex linear matrix inequality (LMI) algorithms are proposed to fulfil the PDF tracking problem. Meanwhile, in order to enhance robustness, the robust peak-to-peak measure is applied to optimize the tracking performance. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source]


    Finite-time tracking control of a nonholonomic mobile robot,

    ASIAN JOURNAL OF CONTROL, Issue 3 2009
    Zhao Wang
    Abstract In this paper, the finite-time tracking problem is investigated for a nonholonomic wheeled mobile robot in a fifth-order dynamic model. We consider the whole tracking error system as a cascaded system. Two continuous global finite-time stabilizing controllers are designed for a second-order subsystem and a third-order subsystem respectively. Then finite-time stability results for cascaded systems are employed to prove that the closed-loop system satisfies the finite-time stability. Thus the closed-loop system can track the reference trajectory in finite-time when the desired velocities satisfy some conditions. In particular, we discuss the control gains selection for the third-order finite-time controller and give sufficient conditions by using Lyapunov and backstepping techniques. Simulation results demonstrate the effectiveness of our method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source]


    Robust tracking control design for uncertain robotic systems with persistent bounded disturbances

    ASIAN JOURNAL OF CONTROL, Issue 4 2008
    Chung-Shi Tseng
    Abstract In this study, a robust nonlinear L, - gain tracking control design for uncertain robotic systems is proposed under persistent bounded disturbances. The design objective is that the peak of the tracking error in time domain must be as small as possible under persistent bounded disturbances. Since the nonlinear L, - gain optimal tracking control cannot be solved directly, the nonlinear L, - gain optimal tracking problem is transformed into a nonlinear L, - gain tracking problem by given a prescribed disturbance attenuation level for the L, - gain tracking performance. To guarantee that the L, - gain tracking performance can be achieved for the uncertain robotic systems, a sliding-mode scheme is introduced to eliminate the effect of the parameter uncertainties. By virtue of the skew-symmetric property of the robotic systems, sufficient conditions are developed for solving the robust L, - gain tracking control problems in terms of an algebraic equation instead of a differential equation. The proposed method is simple and the algebraic equation can be solved analytically. Therefore, the proposed robust L, - gain tracking control scheme is suitable for practical control design of uncertain robotic systems. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source]