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POSITIONING AND TRACKING CONTROL OF AN X-Y TABLE WITH SLIDING MODE CONTROL Jian Wang, Hendrik Van Brussel and Jan Swevers

Division PMA, Dept. Mechanical Engineering Katholieke Universiteit Leuven Celestijnenlaan 300B, B-3001 HEVERLEE, Belgium

Abstract: In this paper, de-coupled discrete-time sliding-mode tracking controllers are designed for an x-y sliding table driven directly by linear motors. The controllers are designed using an integrated reaching law method, based on a simpli?ed model of the current control loop of these motors. To achieve high bandwidth tracking performance, a feedforward controller is added. Although the sliding table is operating in the presence of friction, no friction compensation is applied. Experimental results are presented. Keywords: motion control, sliding mode control, trajectory tracking

1. INTRODUCTION

friction may further improve the tracking performance, Van den Braembussche (1998). It is a common strategy to add a feedforward controller to the tracking control system in order to increase the tracking accuracy, Tomizuka (1993). However, the robustness of the feedforward controller is seldom addressed in the literature. In fact, the feedforward controller as a part of a twodegree-of-freedom controller structure is usually designed based on the closed-loop transfer function. If the closed-loop transfer function is depending on the plant model, like in traditional controllers, no robustness property can be expected in the feedforward controller. This problem can be partly solved by applying the SMC technique in the closed-loop, Wang et al. (2003). It has been proved that the closed-loop system designed with SMC may have a characteristic equation independent of the plant model. This paper is arranged as follows: Section 2 gives a short description of the test setup of the xy slide table and presents a simpli?ed model of the current control loop. Section 3 describes the

Machine tools for high-speed machining require good tracking accuracy at high feed rates. To increase the feed rate, there is a trend toward using direct-drive ball screw slides and directdrive linear motors. Good tracking performance is necessary to achieve accurate contour pro?les as well as to ensure a good surface ?nish of the machined parts. Non-linear e?ects, such as friction, especially at low speed, backlash, and mechanical compliance in the drive-train, as well as periodic disturbances force caused by the cutting process limit the tracking accuracy. Robust and wellperformed motion control design is thus a critical factor to achieve high tracking performance in high-speed machining applications. Several robust controller design methods have been investigated by the authors. It has been shown, on a linear motor test setup, that the sliding-mode control (SMC) technique shows more robust performance than H∞ control. Some feedforward compensation of the motor ripple and

design procedure for the two-degree-of-freedom tracking controller with discrete-time SMC. The validation test of this tracking controller is then described in section 4. Finally, some conclusions are drawn in section 5.

2. X-Y SLIDE TABLE DESCRIPTION AND SIMPLIFIED MODEL The traditional design of machine tool axes, consisting of a rotary motor and a ball-screw transmission is limited in speed, acceleration, stroke and accuracy. The use of linear direct drive technology can greatly increase the performance of machine tools because of the following:

and Schoukens (2001). They are mainly in?uenced by the motor constants and the moving mass. For this purpose, the frequency response function (FRF) of each axis is measured using stepped sine excitation (See Figure 2). The gains of the double-integrator models are ?tted to these FRF’s yielding following results for the x- and y-axis respectively: Hx (s) = Hy (s) = 209828 Kx = 2 s s2 Ky 224273 = s2 s2 (1) (2)

Frequency response of the x?axis/double integrator gain Kx=209827.9338 120 test data lease squares model

- no backlash and less friction; - high closed-loop bandwidth, which is only limited the by encoder bandwidth, power electronics and the resonances of the machine frame, leads to high contouring accuracy. A directly-driven x-y cross slide is used as a demonstrator. See Fig. 1. Both axes of this machine are directly driven by linear motors, one motor for the top or y-axis, and two motors working in parallel for the x-axis.

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Frequency response of the y?axis/double integrator gain Ky=224272.7614 120 100 80 dB of ?m/volt 60 40 20 0 ?20 ?40 ?1 10

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test data lease squares model

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Fig. 2. Linear Motor Current Loop Model Identi?cation

3. SLIDING-MODE TRACKING CONTROLLER DESIGN WITH FEEDFORWARD 3.1 Closed-loop design To design a controller for the trajectory tracking task, the general idea is to realize an overall system transfer function close to 1 up to some cut-o? frequency. Usually, this kind of controller consists of a feedforward part and a feedback part, namely, a two-degree-of-freedom tracking controller structure. In our approach, a sliding-mode controller (SMC) is employed in the feedback part. There exist many di?erent design methods of SMC for both continuous-time and discrete-time systems, see Utkin (1977), Gao and Hung (1993), and Hung and Gao (1993), etc. Once the feedback has been realized, the design of the feedforward part as a direct inverse to the closed-loop system becomes trivial, except in case of non-minimum-phase zeros. For a solution to this problem, see Tomizuka (1987), Torfs et al. (1992), Gross et al. (1994) and Chen et al. (1995).

Fig. 1. X-Y cross slide for high-speed milling machine Each motor is con?gured to work in its current control mode. The motion controller will be designed based on the motor’s current loop model. If the bandwidth of the current loop is much higher than the bandwidth of the position control loop, and the nonlinear disturbances, such as the friction, are ignored, the motor can be considered as a machine which can generate a force proportional to the command signal, thus a simple double integrator model can be used as the current loop model. See Wild and Dodds (1998). The gains Kx and Ky of the double-integrator model for each axis of the x-y table are identi?ed experimentally, using the least squares frequency domain identi?cation technique, Pintelon

SMC is a kind of discontinuous control law in which the controller switches between two different structures according to the instantaneous plant state. Under SMC, a sliding mode can be established for a continuous-time control system when all the plant states are constrained on a sliding surface. It is important to know that, for a discrete-time control system, the sliding mode on a sliding surface can never be maintained, because this will require an in?nitely high switching frequency between the two di?erent controller structures. As a moderate approach, a quasi-sliding mode can be realized by maintaining all the plant states inside a small band around the sliding surface. This small band is called the quasi-sliding mode band. The reaching law method is one of the SMC design methods, and was ?rst proposed by Gao and Hung (1993) for continuous-time systems, and later on, for discrete-time systems in Gao and Wang (1995). The idea of the reaching law method is simple. Assuming that the sliding surface is de?ned as: s(k) = Csl x(k) = 0 where x(k) is the state vector of the plant. When the plant is not yet on the sliding surface, the evaluation of s(k) in Equation (3) is nonzero, and can be interpreted as a measure of the distance of the states to the sliding surface in the state space. Then the process of establishing the quasi-sliding mode band is nothing but driving the distance s(k) from any value towards the neighborhood of zero. A di?erence equation, called the reaching law, can be used to describe this dynamic process. One of the typical reaching laws is given in Gao and Wang (1995): s(k + 1) = (1 ? qT )s(k) ? T sign[s(k)] (4) (3)

with u(k) the input signal, and f (k) the disturbance signal. The DSMC control law is: ? u(k) = ?Ksl x(k) ? ?sl sign[s(k)] (6) The closed-loop system under DSMC is: ? x(k + 1) = Aw x(k) ? B ?sl sign[s(k)] (7)

With the closed-loop system matrix Aw , disturbance and uncertainty limit Ud , feedback gain Ksl , and the discrete-time controller switching gain ? ?sl determined as follows: Aw = A ? BKsl Ksl = (Csl B) [Csl A ? (1 ? qT )Csl ] ? ?sl = (Csl B)?1 ( + Ud )T

?1

(8) (9) (10)

?Ud T < Csl [?Ax(k) + P f (k)] < Ud T (11) The dynamics of the closed-loop system are mainly determined by the eigenvalues of the matrix Aw . One of its eigenvalues is (1 ? qT ), determining the reaching law in Equation (4), and Csl is its corresponding eigenvector, determining the sliding surface in Equation (3). So, the selection of the eigenvalues of the matrix Aw will determine the reaching law and the sliding surface. Moreover, Aw will be independent of the plant model system matrix A, if the plant model is written in the controller canonic form. This means that the characteristic equation of the plant model does not in?uence the characteristic equation of the closed-loop system. This is an important robustness property of the DSMC. 3.2 Feedforward controller design The average closed-loop transfer function with DSMC after the quasi-sliding mode has been established becomes: Tc (z) = C(zI ? Aw )?1 B. (12) The feedforward part can be designed as the direct inverse to the closed-loop transfer function given in Equation (12). Notice that the output matrix C in the controller canonic form is not changed by the DSMC, which indicates that the plant zeroes are not modi?ed. This may cause some problems in designing the feedforward part of the tracking controller. If the plant uncertainty and disturbance greatly in?uence the zeroes of the plant model, then the robustness of the total tracking controller will be lost, because the zeroes cannot be modi?ed by the DSMC. Furthermore, if the plant model is non-minimum-phase, or contains unstable zeroes, then the feedforward part of the tracking controller cannot be designed as the direct inverse to the closed-loop system. In this case, the method of the ZPETC by Tomizuka (1987), or its variants, Torfs et al. (1992) and Chen et al. (1995), may be applied.

where T is the sampling period, the decaying time constant q > 0, the reaching law switching gain > 0, and (1 ? qT ) > 0. The design of a sliding mode controller with reaching law method usually involves the following procedures: (1) determine the decaying time constant q in reaching law; (2) determine the switching surface speci?ed with Csl ; An integrated design procedure for selecting the reaching law and the sliding surface for a discretetime SMC (DSMC) is given in Wang et al. (2003). The realized closed-loop system with DSMC is independent of the plant model. The integrated design procedure is summarized as follows. Suppose that the disturbed plant model in the state space is: x(k + 1) = (A + ?A)x(k) + Bu(k) + P f (k) (5)

4. ROBUST PERFECT TRACKING CONTROLLER(PTC) DESIGN 4.1 Implementation algorithm We are now ready to design a DSMC and feedforward tracking controller for the x-y slide table. The tracking controller will be implemented for each axis. As the controllers for each axis are implemented on the same dSPACE DS1102 DSP controller board, there is no synchronization problem between the two axes. The double-integrator models (Equations (1) and (2)) are converted to discrete time using the forward Euler method and a sampling period of T = 1/2200s, yielding, for the x- and y-axis respectively: Hx (z) = Hy (z) = 209828T 2 z 2 ? 2z + 1 (13)

the reaching law in Equation (4) and the sliding line in Equation (3) of the DSMC for both axes, yielding: (1 ? qT ) = e?2πf0 T , Csl = [1 ? e?2πf0 T ] (21) Finally, the DSMC control law in Equation (6) becomes: u(k) = ? [2 ? 2e?2πf0 T ? 1 + e?4πf0 T ]x(k) ? ? ?sl sign[s(k)] (22) The feedforward part of the controller is obtained by inverting the closed loop transfer function. Since this transfer function is minimum phase, this inversion can be done exactly, yielding for the x- and y-axis, respectively: F Fx (z) = z 2 ? 2e?2πf0 T z + (e?2πf0 T )2 209828T 2 2 ?2πf0 T z ? 2e z + (e?2πf0 T )2 F Fy (z) = 224273T 2 (23) (24)

224273T 2 (14) z 2 ? 2z + 1 Transforming these models to a controller canonical state space form yields the following A and B matrices: A= 2 ?1 ; B= 10 1 0

T

4.2 Tracking Performance Test Results The tracking performance of the controllers is validated experimentally using circle tracking tests. The quality of the obtained circle is measured by the circular deviation. The circular deviation is de?ned as the minimum radial separation between concentric circles enveloping the actual path, see ISO234-4. Instead of being evaluated as the minimum radial separation around the least squares circle, we evaluate the circular deviation as the minimum radial separation around the reference circle. The circular trajectory has a radius of 10mm, and the desired circular speed is 12.57rad/s. The tests are performed for di?erent closed-loop system natural frequencies f0 . The discrete-time controller ? switching gain ?sl in Equation (6) is set to 10 in T order to take into account the disturbance and uncertainty limit Ud and the reaching law switching gain ε. The results are shown in Figure 3. Figure 3 shows that increasing the closed-loop system natural frequency improves the tracking accuracy. Of course, the closed-loop natural frequency cannot be increased without limit. The structural resonance and the noise in the measurement signal put limitations on the achievable performance. To analyze the robustness of the tracking performance of the controller, some experiments are performed by tracking the same circle of radius 10mm at di?erent circular speeds in the range between 0.1Hz and 2.3Hz, or tangential speeds between 0.006m/s and 0.145m/s. The closed-loop natural frequency of the DSMC is set to 45Hz. A circular speed higher than 2.3Hz results in controller saturation. The friction force is the main

;

(15)

In order to have the same tracking performance in both axes, we need the same closed-loop dynamics for x- and y-axis. This can be done by selecting identical closed-loop poles for both axes. The dynamics of a second-order continuous-time system are determined by the damping ratio ζ and the natural frequency ω0 as follows: H(s) =

2 ω0 2 s2 + 2ζω0 s + ω0

(16)

The poles of the corresponding discrete-time system are given by the characteristic equation, see ?str¨m and Wittenmark (1997): A o z 2 + a1 z + a2 = 0 where a1 = ?2e?ζω0 T cos( 1 ? ζ 2 ω0 T ) a2 = e?2ζω0 T (18) (19) (17)

Recall that in Section 3, we have concluded that (1 ? qT ) is one of the closed-loop system eigenvalues, and has to be a real number, Csl is its corresponding eigenvector. The condition for Equation (17) to have two real roots is: a2 ? 4a2 = 1 4e?2ζω0 T (cos2 ( 1 ? ζ 2 ω0 T ) ? 1) ≥ 0 (20)

The only possibility is to have the damping ratio ζ = 1, and the two roots are z1,2 = e?2πf0 T , with ω0 = 2πf0 . These two poles are used to determine

Circular deviation versus the closed?loop natural frequency

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some extra friction model or motor ripple force model are used to provide additional feedforward compensation to the tracking controller, we rely on the DSMC to assure robustness.

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ACKNOWLEDGMENT This research is sponsored by the Belgian program of Interuniversity Poles of Attraction by the Belgian State, Prime Minister’s O?ce, Science Policy Programming (IUAP). The scienti?c responsibility is assumed by its authors.

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Fig. 3. Circular Deviation versus Closed-Loop Natural Frequency source of disturbance in this tracking system. The in?uence of the friction on the tracking system is di?erent for di?erent speeds. At low circular speed, the friction force is more signi?cant than at high circular speed, compared to the motor forces required to perform the circular path. Some results are shown in Figure 4. From Figure 4, we ?nd that the tracking performance is not greatly in?uenced by the tracking speed until the controller is saturated. This may indicate that the tracking controller is robust against friction.

Circular deviation versus the tracking speed

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REFERENCES Karl J. ?str¨m and Bj¨rn. Wittenmark. A o o Computer–controlled systems: theory and design. Prentice Hall, Upper Saddle River, New Jersey, 3 edition, 1997. ISBN 0-13-3148998. Chwan-Hsen Chen, Hendrik Van Brussel, and Jan Swevers. Extended pole placement method with noncausal reference model for digital servocontrol. Trans. of the ASME J. of Dynamic Systems, Measurement, and Control, 117:641– 644, December 1995. Weibing Gao and James C. Hung. Variable structure control of nonlinear system: A new approach. IEEE Trans. On Industrial Electronics, 40:45–55, 1993. Weibing Gao and Yufu Wang. Discrete-time variable structure control systems. IEEE Trans. On Industrial Electronics, 42:117–122, 1995. Eric Gross, Masayoshi Tomizuka, and William Messner. Cancellation of discrete time unstable zeros by feedforward control. Trans. of the ASME J. of Dynamic Systems, Measurement, and Control, 116:33–38, March 1994. John Y. Hung and Weibing Gao. Variable structure control: A survey. IEEE Trans. On Industrial Electronics, 40:2–22, 1993. ISO234-4. Test code for machine tools – part 4: Circular tests for numerically controlled machine tools, 1996. ISO234-4. Rik Pintelon and Johan Schoukens. System Identi?cation: A Frequency Domain Approach. IEEE Press, Piscataway, NJ., 1 edition, 2001. ISBN 0-7803-6000-1. Masayoshi Tomizuka. Zero phase error tracking algorithm for digital control. Trans. of the ASME J. of Dynamic Systems, Measurement, and Control, 109:65–68, March 1987. Masayoshi Tomizuka. On the design of digital tracking controllers. Trans. of the ASME J. of Dynamic Systems, Measurement, and Control, 115:412–418, June 1993. Dirk Torfs, Joris De Schutter, and Jan Swevers. Extended bandwidth zero phase error tracking control of nonminimal phase system. Trans.

circular deviation (?m)

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Fig. 4. Circular Deviation versus Tracking Speed

5. CONCLUSION The discrete-time sliding-mode controller, based on the integrated reaching law design method has been successfully applied on an x-y machine tool feed table. This DSMC in combination with feedforward, Wang et al. (2003), is robust against outside disturbance because the closed-loop system characteristic equation is independent of the plant model. The DSMC design for the x-y table is based on a simpli?ed double-integrator model. Unlike most tracking controller approaches, where

of the ASME J. of Dynamic Systems, Measurement, and Control, 114:347–351, Sepember 1992. Vadim I. Utkin. Variable structure systems with sliding modes. IEEE Trans. Automatic Control, 22(2):212–222, 1977. Pieter Van den Braembussche. Robust Motion Control of High–Performance Machine Tools with Linear Motors. Phd dissertation, Katholieke Universiteit Leuven, K. U. Leuven, Belgium, 1998. Jian Wang, Hendrik Van Brussel, and Jan Swevers. Robust perfect tracking control with discrete sliding mode controller. Trans. of the ASME J. of Dynamic Systems, Measurement, and Control, 125:27–32, March 2003. Harald G. Wild and Stephen J. Dodds. Robust control of high dynamic linear motors. In Gerhard Schweitzer, editor, Proceedings of the 4th International Conference on Motion and Vibration Control, ETH Zurich, Switzerland, 1998.

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