Zero Phase Error Tracking Controller Research Articles

Multi-Rate Feedforward Tracking Control for Plants With Nonminimum Phase Discrete Time Models

This paper is concerned with performance enhancement of tracking control systems by multi-rate control. The feedback controller is updated at the same rate as the sampling rate of the output measurements. The feedforward controller processes the desired output signal for high accuracy tracking, and its output is updated at a rate N-times faster than the sampling rate of the output measurements. The discrete time model of the controlled plant may possess unstable zeros, and the zero phase error tracking controller (ZPETC) is used as a feedforward controller. Inter-sample behavior of the plant is included in evaluating the tracking performance of the multi-rate system. Illustrative examples are given to show advantages of the proposed multi-rate feedback/feedforward control scheme.

An optimal and adaptive design of the feedforward motion controller

The zero phase error tracking controller (ZPETC) in motion control, as proposed by Tomizuka (1987) , renders the desirable zero phase error, but with a limited gain response. Moreover, a ZPETC, which is basically in a feedforward control structure, is very sensitive to modeling error. To improve the tracking accuracy of the ZPETC, this paper presents an optimal ZPETC design with a concise polynomial digital prefilter (DPF). The parameters of this well-designed DPF are obtained through the derived L/sub 2/-norm optimization. By cascading the developed DPF to the ZPETC, the resultant optimal ZPETC greatly improves the bandwidth of the tracking control systems while maintaining the zero phase error. Compared with other optimal approaches, the present design leads to much simpler procedures and fewer computations. Furthermore, the proposed optimal ZPETC can be adequately implemented as an adaptive ZPETC by including real-time estimation technique to cope with the external load perturbation and parameter variation. Compared with the other adaptive approaches, the optimal concept is used in the present adaptive ZPETC, and it also renders more accurate results because of its improved magnitude response. Experimental results on a DC servo table with different controllers indicate that when there is no loading, the present optimal ZPETC achieves the best tracking performance. Moreover, the adaptive ZPETC achieves the most satisfactory results when an external load is applied.

Improving the Characteristics of an Electrohydraulic Servo System with a Nonsymmetrical Cylinder by ZPETC and Linearization

The DSP (Digital Signal Processor) and the feedforward compensation method known as ZPETC (Zero Phase Error Tracking Control) are applied to an electrohydraulic servo system with a nonsymmetrical cylinder in order to improve the frequency responses of the system. ZPETC is a control algorithm which renders the transfer function between an input and an actual output close to unity. This algorithm assumes the controlled system is linear. However, the relationship between the input and the flow rate at the servo valve is nonlinear. Another nonlinear factor is added when the cylinder is nonsymmetrical. Therefore, these nonlinearities should be removed. The nonlinearity is compensated using the detected load pressure, and the detected sign of the input. In this paper, the disturbance observer assists ZPETC, which needs a transfer function that accurately expresses the plant dynamics. However, a slight difference between the transfer function and the actual dynamics is unavoidable. Therefore, use of a disturbance observer is effective to reduce the difference and improves ZPETC performance.

Concurrent Design of Continuous Zero Phase Error Tracking Controller and Sinusoidal Trajectory for Improved Tracking Control

A trajectory control strategy for a nonminimum phase system is proposed. A continuous-time version of the Zero Phase Error Tracking Controller (ZPETC), which is a well-known discrete-time feedforward controller, is considered. In the continuous-time case, the overall transfer function consisting of the ZPETC and the closed-loop plant exhibits high-pass filter characteristics. This introduces serious gain errors between the desired and actual output if the desired output is made directly as the ZPETC’s input. This paper proposes the use of a specially designed sinusoidal trajectory to compensate for the gain errors. The sinusoidal trajectory imparts a synergic effect to tracking performance when combined with the continuous ZPETC. Continuous ZPETC with sinusoidal trajectory is evaluated successfully by applying to a nonminimum phase plant, single link flexible arm.

Deformation Control of Elastic Object by Robot Arm - High-Precision Deformation Control by Adaptive Feed-forward Control -

This paper derives the equations of motion of a horizontal three-link robot arm contacting an elastic object, then proposes deformation control for the elastic object by the robot arm. This deformation control involves feedback control of joint angles, feedforward control by robot arm dynamics, and adaptive feedforward control for unknown elastic object dynamics. An adaptive feedforward control law is devised using an online parameter identification algorithm based on the least squares method, zero phase error tracking control theory, and the Jacobian matrix of the robot arm. Numerical simulation and experimental results confirmed that deformation control error by the proposed adaptive control rapidly decreases with the passing of time.

Comparison of two feedforward design methods aiming at accurate trajectory tracking of the end point of a flexible robot arm

This paper discusses the design and properties of two trajectory tracking controllers for linear time-invariant systems, and compares their implementation and experimental results on a flexible one-link robot equipped with a velocity-controlled actuator. High positioning accuracy and low tracking errors within a specified bandwidth are their performance specifications. Both controllers use the same state feedback controller, but have a different feedforward design approach. Both feedforward methods design stable prefilters which approximate the unstable inverse system model. The first method designs a stable prefilter using the extended bandwidth zero phase error tracking control (EBZPETC) method. The second feedforward method adds delay to the inverse model and then uses common filter design techniques to approximate this delayed frequency response.

Deforming Control of Elastic Object by Robot Arm. High-Precision Deforming Control by Adaptive Feedforward Control.

In this paper, first, the equations of motion of the horizontal three-link robot arm in contact with an elastic object are derived. Then, the deforming control system of the elastic object by the robot arm is proposed. The deforming control system consists of the feedback control of the joint angles, the feedforward control by the robot arm dynamics and the adaptive feedforward control for the compensation control of the unknown elastic object dynamics. The adaptive feedforward control law is constructed by use of the on line parameter identification algorithm based on the least squares method, the zero phase error tracking control theory and the Jacobian matrix of the robot arm. From the numerical simulation rerults and the experimental results, it is confirmed that the control error in the deforming control by use of the present adaptive feedforward control method becomes small rapidly with the passing of time.

Adaptive Zero Phase Error Tracking Controller with Extended Gain Characteristics

This paper deals with a tracking control problem for a discrete-time non-minimum phase system in the framework of frequency domain. The contributions are as follows. First, the proposed feedforward controller can provide the overall system with the following frequency characteristics; (i) The phase shift is equal to zero for all frequencies. (ii) The gain at any given frequency is equal to unity. (iii) The maximum deviation of the gain from unity over a given frequency range is less than an arbitrarily given positive number. Especially, the frequency at which the gain is made equal to unity can be chosen arbitrarily, while in the previous schemes, the frequency is restricted to zero. Secondly, the class of desired outputs for which the deadbeat tracking can be achieved is generalized, while in the previous results the class is restricted to only steps and ramps. Thirdly, the proposed controller is given in an explicit form and the design method is simple and straightforward. Finally, an adaptive control system based on the proposed feedforward controller is presented and its effectiveness is demonstrated by simulations.

Adaptive Zero Phase Error Tracking Controller with Precision Tracking Performance

This paper deals with a discrete-time tracking control where the desired output to be tracked is partially known a priori. The problem of designing prefilter which provides the overall system with the following frequency characteristics is considered: 1) The phase is zero for all frequencies, 2) The gain at given particular frequencies is set to unity, 3) The maximum error over a given frequency range between the gain and unity is less than an arbitrary given positive number. The prefilter satisfying this problem is given in an explicit form, and is implemented in adaptive control scheme.

Zero Phase Error Tracking System With Arbitrarily Specified Gain Characteristics

This paper considers a design problem of discrete-time preview feedforward controllers such that the gain characteristics of the overall system is within an arbitrarily specified bound subject to the zero phase error condition for a plant having nonminimum phase zeros. In order to solve this problem, a feedforward controller termed Optimal-Feedforward Controller with Zero Phase Error Tracking Controller (Optimal-FCZPETC) is introduced. With this controller, the phase characteristics of the overall system is zero for all frequencies and the maximum value of the gap between the gain of the overall system and unity, which is ideal gain characteristics, is minimized under given preview steps. The choice of the preview steps is an unsolved problem. In this paper, we investigate the Optimal-FCZPETC from the viewpoint of the preview steps. The contribution is to give explicitly the minimum value of the maximum gap between the gain of the overall system and unity for given preview steps and to show that the minimum value can be made arbitrarily small as the preview steps increase. As a result, a simple algorithm is proposed to find the minimum preview steps such that the gain characteristics of the overall system is within an arbitrarily specified bound. The effectiveness is shown by simulation results.

非対称シリンダを有する電気・油圧サーボシステムのＤＳＰによるＺＰＥＴＣと線形化を用いた特性改善に関する研究

A DSP (digital signal processor) and a feedforward compensation method called ZPETC (zero phase error tracking control) are applied to an electrohydraulic servo system with a nonsymmetrical cylinder to improve on the frequency responses of the system. ZPETC is a control algorithm, rendering the transfer function between the input and actual output close to unity. This algorithm assumes the controlled system to be linear. However, the relationship between the input and flow rate at the servo valve is nonlinear. Another nonlinear factor is added when the cylinder is nonsymmetrical. Therefore, these nonlinearities should be removed. The nonlinearity of the servo valve is compensated for by using the detected load pressure, and the effect of the nonsymmetrical cylinder is compensated for by using the detected sign of the input. The DSP performs the calculations for ZPETC and for linearization. The effects of improving the frequency responses of the system by this digital signal processing technique are investigated.

Optimal Feedforward Prefilter With Frequency Domain Specification for Nonminimum Phase Systems

The paper shows the influence of the location of unstable zeros on the tracking performance of feedforward prefilters. Unstable zeros are divided into a number of classes. It is shown that existing feedforward prefilters (Zero Phase Error Tracking Control (ZPETC), E-filter, Extended Bandwidth ZPETC, ...) perform well for two classes, but fail for a particular class of unstable zeros. For this class, a characteristic frequency, fc, exists such that the induced gain error attenuates all frequencies of the reference trajectory f ≤ fc and amplifies frequencies f > fc. Hence, it is impossible to freely select the tracking bandwidth. Therefore, an optimal feedforward prefilter for discrete time nonminimum phase systems is presented to deal with this class of unstable zeros. As in the ZPETC method, the prefilter compensates for unstable zeros in the inverse system model, retains the zero phase property, and introduces small gain errors. But in addition, the design minimizes a cost function for which a least square solution is found. A frequency and time domain analysis shows the superior performance of the presented optimal prefilter design even for trajectory with high frequency components.

Control of Cutting Force for End Milling Processes Using an Extended Model Reference Adaptive Control Scheme

In this work an extended Model Reference Adaptive Control (MRAC) technique is used to control the cutting force of an end milling process. The technique incorporates Zero Phase Error Tracking Control (ZPETC) into the MRAC system. The extended MRAC controller remains stable even in the presence of marginally stable and nonminimum phase process zeros. A modified recursive least-squares estimation algorithm is used for on-line parameter identification. Simulation results are presente to compare the extended MRAC controller to the standard MRAC controller. A microprocessor system is used to implement adaptive force control of a single-input single-output milling process where the microprocessor monitors the system cutting forces and controls the desired feedrate. A constant cutting force is maintained in the presence of time-varying plant gains and a high random component of the output force. Experimental results are presented for standard MRAC and extended MRAC controllers for comparison.

High-Speed End Milling Using a Feedforward Control Architecture

Experiments are performed for end milling aluminum at 15,000 RPM spindle speed (1,508 m/min cutting speed) and up to 3 m/min table feedrate using an experimental machine tool control system. A digital feedforward controller for feed drive control incorporates the Zero Phase Error Tracking Controller (ZPETC) and feedforward friction compensation. The controller achieves near-perfect (±3 μm) tracking over a 26 mm trajectory with a maximum speed of 2 m/min. The maximum contouring error for a 26 mm diameter circle at this speed is less than 4 μm. Tracking and contouring experiments are conducted for table feedrates as high as 10 m/min. Frequency domain analysis demonstrates that the feedforward controller achieves a bandwidth of 10 Hz without phase distortion. In a direct comparison of accuracy, the machining errors in specimens produced by the experimental controller were up to 20 times smaller than the errors in specimens machined by an industrial CNC.

Robust digital tracking controller design for high-speed positioning systems

In this paper, a robust digital tracking controller for high-speed positioning is proposed.For high-speed tracking problems, the combination of a feedforward controller and a robust feedback controller is desirable, because the feedforward controller anticipates and compensates for closed-loop dynamics and the feedback controller compensates mechanical nonlinearities, parameter variations, and disturbances. A disturbance observer and PD compensation was used as a robust feedback controller and a zero-phase error-tracking controller was used as the feedforward controller.This method was applied to an X-Y table, and experimental results demonstrate that this method yields excellent tracking performance and robustness to plant parameter variations.

IMPROVING CHARACTERISTICS OF ELECTROHYDRAULIC SERVO SYSTEM WITH NONSYMMETRICAL CYLINDER USING DSP

The DSP (digital signal processor) and the feedforward compensation called ZPETC (zero phase error tracking control) are applied to an electrohydraulic servo system with a nonsymmetrical cylinder, and frequency responses of the system are greatly improved. ZPETC is a control algorithm which renders the transfer function between an input and an actual output close to unity. To use this algorithm, it is assumed that the controlled system is linear. However the relationship between load pressure and flow rate at the servo valve is highly nonlinear. Another nonlinear factor is added when the cylinder is nonsymmetrical. Therefore these nonlinearities should be removed. The nonlinearity of the servo valve is compensated using the detected load pressure, and the effect of nonsymmetrical cylinder is compensated using the detected sign of the input. DSP performs the calculations for ZPETC and for linearization of the system. The effect of improving the frequency responses of the system by this digital signal processing technique is investigated.

Robust motion controller design for high-accuracy positioning systems

This paper presents a controller structure for robust high speed and accuracy motion control systems. The overall control system consists of four elements: a friction compensator; a disturbance observer for the velocity loop; a position loop feedback controller; and a feedforward controller acting on the desired output. A parameter estimation technique coupled with friction compensation is used as the first step in the design process. The friction compensator is based on the experimental friction model and it compensates for unmodeled nonlinear friction. Stability of the closed-loop is provided by the feedback controller. The robust feedback controller based on the disturbance observer compensates for external disturbances and plant uncertainties. Precise tracking is achieved by the zero phase error tracking controller. Experimental results are presented to demonstrate performance improvement obtained by each element in the proposed robust control structure.

Extended Pole Placement Method With Noncausal Reference Model for Digital Servo-Control

The extended pole placement (EPP) method extends the well-known pole placement method to digital servo-control with known control inputs. A servo-control system designed using the EPP method achieves an accurate command response through the use of a noncausal reference model and an accurate approximation of nonminimum-phase zeros. The phase error of the command response between the desired output and the actual output may be made arbitrarily small, while its gain extends uniformly to a selectable bandwidth. A selectable feedforward bandwidth is important in controlling flexible structures where unmodeled resonances might be excited by rough feedforward inputs. Experimental results on a machine tool slide show that a controller designed by the EPP method yields better servo performance for cornering and contour tracking tasks than an E-filter-enhanced zero phase error tracking controller.

Precision tracking control of non-minimum phase systems with zero phase error

In this paper, a method is proposed for designing the command feedforward controller for zero phase error tracking control of systems having unacceptable zeros. The unacceptable zeros include both non-minimum phase zeros and lightly damped zeros. In the proposed approach, a digital preview filter along with the acceptable part of the system's inverse model is designed to form the command feedforward controller. Using preview information of the input trajectory, the designed tracking controller guarantees zero phase error and high precision tracking performance. In the proposed design, the best tracking performance can be achieved by minimizing a weighted penalty function, which controls the gain error between the desired output and the actual output. Two design cases were examined. The first is the design based on a uniform weighting function. In this study, the optimal solution of the design can be analytically obtained and three prior approaches were shown to be the special cases of the proposed solutio...

Improving Characteristics of Electrohydraulic Servo System by using DSP.

A DSP (digital signal processor) is applied to an electrohydraulic servo system, and feed-forward compensation called ZPETC (zero phase error tracking control) which is impossible by analog techniques, greatly improves the frequency responses of the system. However, the relation-ship between load pressure and flow rate for the servo valve is highly nonlinear. Because ZPETC, which is feed forward compensation, assumes the controlled system to be linear, we must make this nonlinearity linear. Therefore, the dependence of servo valve on load pressure is removed by detecting load pressure, and the system is linearized. DSP performs the calculation for ZPETC and for linearizing the system. This effect of improving frequency responses of the system by the digital signal processing technique is investigated.