Tracking Of Periodic Trajectories Research Articles

Notch-filter-based repetitive control of fast tool servos for high-performance tracking of periodic trajectories

The achievable performance of conventional repetitive control (CRC) is constrained by the error amplification at the non-harmonic frequencies, the intrinsic sensitivity to the signal period shift and the limited tracking frequency range under fixed sampling rate. To remedy these undesirable behaviors, a notch-filter-based repetitive control (NF-RC) is proposed based on the spectrum-selection filtering scheme in this paper with application to fast tool servos (FTSs). The loop-shaping property of this NF-RC can be flexibly designed by the cascade of a series of parameterized IIR notch filters without the aforementioned limitations of the CRC, so as to provide a unified formula for various RCs and even resonant control schemes. A general realization along with several simplified versions of the IIR notch filter is derived for the digital implementation of the NF-RC. Comparative trajectory tracking experiments on a self-developed FTS have validated the superiorities of the proposed NF-RC in terms of the performance improvement for periodic reference tracking, the robustness against the signal period variations and the trackable frequency range under fixed sampling rate. Taking advantage of an excellent comprehensive performances of the NF-RC scheme, the radial sinusoidal surface and the water-drop surface were successfully produced with high form precision and excellent surface quality after the FTS-based diamond turning.

Solving Infinite-Dimensional Harmonic Lyapunov and Riccati Equations

In this paper, we address the problem of solving infinite-dimensional harmonic algebraic Lyapunov and Riccati equations up to an arbitrary small error. This question is of major practical importance for analysis and stabilization of periodic systems including tracking of periodic trajectories. We first give a closed form of a Floquet factorization in the general setting of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}$</tex-math></inline-formula> matrix functions and study the spectral properties of infinite-dimensional harmonic matrices and their truncated version. This spectral study allows us to propose a generic and numerically efficient algorithm to solve infinite-dimensional harmonic algebraic Lyapunov equations up to an arbitrary small error. We combine this algorithm with the Kleinman algorithm to solve infinite-dimensional harmonic Riccati equations and we apply the proposed results to the design of a harmonic LQ control with periodic trajectory tracking.

Cross-Coupled Repetitive Control of a Compliant Nanomanipulator for Micro-Stereolithography

This paper proposes an easy-implemented and high precision contouring control method for a self-developed compliant nanomanipulator supporting cost-effective micro-stereolithography (MSL). The proposed contouring control method is composed of a repetitive controller (RC) to achieve periodic trajectory tracking for each axis, and a cross-coupled control (CCC) for the coordination of contour errors. It is worth noting that the cross-coupled controller design and its integration to RC are not straightforward, as an incorrect feedback position may even deteriorate the contouring and an inappropriate design of the cross-coupled controller leads to instability of the system. For the proposed control structure, a correct feedback position is designed, hence a cross-coupled controller is constructed to ensure the stability and contour error reduction. Various simulations and real-time experiments are deployed on the nanomanipulator, and the comparative results validate the enhanced contour tracking performance of the proposed control method with the contour error 86 nm.

Periodic Trajectory Tracking for Control-Affine Driftless Systems on Compact Lie Groups

We treat the periodic trajectory tracking problem: given a periodic trajectory of a control-affine, left-invariant driftless system in a compact and connected Lie group G and an initial condition in G, find another trajectory of the system satisfying the initial condition given and that asymptotically tracks the periodic trajectory. We solve this problem locally (for initial conditions in a neighborhood of some point of the periodic trajectory) when G is semisimple and the system is Lie-determined (i.e., controllable), and only for a class of periodic trajectories (which we call regular). Finally, we present a set of sufficient conditions to ensure the existence of such trajectories.

Asynchronous repetitive control of switched systems via periodic event-based dynamic output feedback

Abstract This paper develops an asynchronous mode-dependent repetitive control strategy with periodic event-based dynamic output feedback for periodic trajectory tracking of continuous-time switched systems subject to time-varying switching delays between system modes and controllers and limited communication capacity in the feedback channel. By employing the input delay approach, the overall system is modelled as an augmented closed-loop switched system with both constant and time-varying state delays. A co-design framework is proposed for simultaneously designing the controller, event-triggering mechanism and mode switching signal. Under the co-design framework, using piecewise Lyapunov functional, free-weighting matrices and average dwell time technique, sufficient conditions are derived for ensuring the augmented closed-loop system to be exponentially stable with a prescribed $H_{\infty }$ attenuation level $\gamma $ for an exogenous disturbance input. The performance of the proposed controller design scheme is verified via numerical simulation results on a switched RLC series circuit system.

Tracking of Periodic Trajectories for Switched Systems Under Time-Varying Asynchronous Switching

In this technical brief, we provide an asynchronous modified repetitive controller design to address the periodic trajectory tracking problem for switched systems with time-varying switching delays between plant modes and controllers. In the feedback channel, a dynamic output feedback mechanism is adopted. By utilizing the lifting technique, the dynamic output feedback-based switched repetitive control system is transformed into a continuous-discrete two-dimensional (2D) model to differentiate the control and learning actions involved in the repetitive controller. For the transformed 2D model, by constructing a piecewise Lyapunov functional and utilizing a matrix decomposition approach, sufficient conditions in terms of linear matrix inequalities (LMIs) and the average dwell time are developed to guarantee closed-loop exponential stability. The performance of the proposed approach is illustrated via a switched RLC series circuit example and numerical simulations are provided.

Large dynamic range tracking of an XY compliant nanomanipulator with cross-axis coupling reduction

Abstract This paper presents a novel design on a large range compliant XY nanomanipulator with spatial constraints to reduce parasitic motions. Aiming at periodic trajectory tracking of the proposed XY parallel compliant nanomanipulating system, a disturbance observer based (DOB) repetitive control structure is developed such that the cross-axis coupling effects and the stiffness nonlinearity can be treated as disturbances with harmonic and non-harmonic components for rejection purposes. The mechanical design prototype is fabricated and the proposed control algorithm is experimentally validated, where enhanced tracking performance with cross-axis coupling reduction is demonstrated by achieving tracking errors less than 80 nm and a motion stroke over 1 mm.

An acceleration-based hybrid learning-adaptive controller for robot manipulators

The robust periodic trajectory tracking problem is tackled by employing acceleration feedback in a hybrid learning-adaptive controller for n-rigid link robotic manipulators subject to parameter uncertainties and unknown periodic dynamics with a known period. Learning and adaptive feedforward terms are designed to compensate for periodic and aperiodic disturbances. The acceleration feedback is incorporated into both learning and adaptive controllers to provide higher stiffness to the system against unknown periodic disturbances and robustness to parameter uncertainties. A cascaded high gain observer is used to obtain reliable position, velocity and acceleration signals from noisy encoder measurements. A closed-loop stability proof is provided where it is shown that all system signals remain bounded and the proposed hybrid controller achieves global asymptotic position tracking. Results obtained from a high fidelity simulation model demonstrate the validity and effectiveness of the developed hybrid controller.

Adaptive Repetitive Control Design With Online Secondary Path Modeling and Application to Bit-Patterned Media Recording

This paper presents an adaptive repetitive controller for active tracking (rejecting) of unknown periodic trajectories (disturbances). The proposed control law is based on a modified filtered-x least mean squares (MFX-LMS) algorithm with a novel variable step size that improves the convergence rate and fades the steady state excess error in a stochastic environment. A novel secondary path modeling scheme is also proposed to adaptively compensate for the dynamic mismatches between the internal model of the MFX-LMS and the real dynamic system in an online fashion. We further discuss the application of this adaptive controller in servo mechanisms for hard disk drives (HDDs) that use bit patterned media recording in which full spectrum tracking of a periodic trajectory is crucial. Finally, comprehensive numerical simulations and experimental implementations are presented for an HDD servo system that is subjected to periodic disturbances known as repeatable run-out.

Robust Trajectory Tracking for a Class of Hybrid Systems: An Internal Model Principle Approach

This paper deals with the asymptotic tracking of periodic trajectories for hybrid systems having linear dynamics in each operating mode and isolated discrete switching events (switching systems). Parametric uncertainties are considered and the dimension of the state vector is allowed to vary among modes. To deal with the hybrid nature of the system, and the possible discontinuities of its solutions at switching times, a properly amended tracking control problem is defined and a feedback control law based on a discontinuous version of the classical internal model principle is proposed. The innovative design of the discrete-time dynamics of the compensator guarantees the robust existence of a steady-state response giving zero tracking error in the controlled output, and local convergence to it.

An Adaptive Controller Dominant-Type Hybrid Adaptive and Learning Controller for Trajectory Tracking of Robot Manipulators

This paper proposes a new hybrid adaptive and learning control method based on combining model-based adaptive control, repetitive learning control (RLC) and proportional–derivative control to consider the periodic trajectory tracking problem of robot manipulators. The aim of this study is to obtain a high-accuracy trajectory tracking controller by developing a simpler adaptive dominant-type hybrid controller by using only one vector for estimation of the unknown dynamical parameters in the control law. The RLC input is adopted using the original learning control law, adding a forgetting factor to achieve the convergence of the learning control input to zero. We will improve and prove that the adaptive dominant-type controller could be applied for tracking a periodic desired trajectory in which adaptive control input increases and becomes dominant of the control input, whereas the other control inputs decrease close to zero. The domination of the adaptive control input gives the advantage that the proposed controller could adjust the feed-forward control input immediately and it does not spend much time relearning the learning control input when the periodic desired trajectory is switched over from the first trajectory to another trajectory. We utilize the Lyapunovlike method to prove the stability of the proposed controller and computer simulation results to validate the effectiveness of the proposed controller in achieving the accurate tracking to the periodic desired trajectory.

Intelligent Adaptive Backstepping Control System for Magnetic Levitation Apparatus

We propose an intelligent adaptive backstepping control system using a recurrent neural network (RNN) to control the mover position of a magnetic levitation apparatus to compensate for uncertainties, including friction force. First, we derive a dynamic model of the magnetic levitation apparatus. Then, we suggest an adaptive backstepping approach to compensate disturbances, including the friction force, occurring in the motion control system. To further increase the robustness of the magnetic levitation apparatus, we propose an RNN estimator for the required lumped uncertainty in the adaptive backstepping control system. We further propose an online parameter training methodology, derived by the gradient descent method, to increase the learning capability of the RNN. The effectiveness of the proposed control scheme has been verified by experiment. With the proposed adaptive backstepping control system using RNN, the mover position of the magnetic levitation apparatus possesses the advantages of good transient control performance and robustness to uncertainties for the tracking of periodic trajectories

Variable-Structure Repetitive Control: A Discrete-Time Strategy

This paper addresses the problem of periodic disturbance rejection arising from rotational plants following periodic trajectories. Repetitive control is a method usually employed in solving such a problem. The reported analysis results give sufficient conditions for stability of the controlled systems in the frequency domain. In this paper, discrete-time variable-structure repetitive control (VSRC) is proposed, where the periodicity features are taken into account and no disturbance estimate is involved explicitly. The presented VSRC control scheme ensures robust stability of the resultant quasi-sliding-mode dynamics to parameter variations and exogenous disturbances. The complete rejection of periodic disturbances and the perfect tracking of periodic trajectories are achieved. Time-domain sufficient conditions for the robust stability are presented by which the controller parameters can be chosen. A numerical simulation is conducted to illustrate the validity of the developed method.

制約を有するシステムに対する周期目標値の生成

This paper is concerned with off-line reference shaping for closed-loop systems with state/input constraints. We propose a method to generate reference signals which achieve better tracking property for given periodic trajectories subject to system constraints. In order to take account of both transient and steady state tracking performance, the new reference signals consist of two parts. This plays a key role to deal with periodic trajectory tracking. The steady state reference signals are produced first so as to minimize tracking errors in the steady state subject to given system constraints. Then, the transient reference signals are obtained in a similar way with the additional constraint that the transient states are connected to the steady states smoothly. By combining these signals, we obtain the shaped references. Its effectiveness is demonstrated through detailed simulations. Furthermore, an experimental validation is performed.

Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts

An infinitely rigid mass is considered, which moves on a plane under the action of proper control forces, within a closed region delimited by an infinitely massive and rigid circular barrier. A suitably amended tracking problem for a class of periodic trajectories, involving an infinite number of nonsmooth impacts between the mass and the barrier, is formally stated and solved.

Adaptive Control With Asymptotic Tracking Performance and Its Application to an Electro-Hydraulic Servo System

This paper presents a discrete-time adaptive controller, which incorporates internal model principle for asymptotic tracking performance of systems with parametric uncertainties, unmodeled dynamics and disturbances. Global stability and tracking performance of the adaptive system are derived under conditions on the system’s stabilizability and bounds of noise and unmodeled dynamics. It is shown that asymptotic tracking can be achieved while the unmodeled dynamics and disturbances exist. The adaptive algorithm is applied to an electrohydraulic servo system for periodic trajectory tracking and disturbance rejection. Experimental results based on an eighth-order adaptive system updated at 2560 Hz demonstrate the adaptive system’s ability in high bandwidth tracking performance under effects of system variations and finite word length real-time computation. [S0022-0434(00)03301-3]