毕业设计英语翻译Tracking-control-and-active-disturbance-rejection.pdf

International Journal of Machine Tools received in revised form 12 July 2007; accepted 13 July 2007 Available online 26 July 2007 Abstract Control systems are usually required to track reference signals while operating under the infl uence of disturbances. A fast tool servo system for noncircular machining application works under such conditions, resulting in large control efforts. This paper presents a linear active disturbance rejection controller design for a voice coil motor-driven fast tool servo system for noncircular machining application. The controller is designed through an extended state observer to estimate and compensate the variant dynamics of the system, nonlinearly variable cutting load, and other uncertainties. Then, a simple proportional derivative controller produces the control law. To improve the tracking performance of the fast tool servo, the tracking error from the trial-cutting workpiece is added to the reference input and used as feed-forward error compensation. In such a combined control arrangement, the active disturbance rejection controller provides active disturbance rejection ability for the controller, and the feed-forward error compensation controller improves the tracking precision. Both the tracking control and disturbance rejection performances are thus enhanced. In real-time control and implementation, the effects of fi nite word length, position feedback resolution, and short sampling period are analyzed and addressed. Machining experiments are conducted, and the results illustrate the control system synthesis procedures and a substantial improvement over the tracking error generated by the linear active disturbance rejection controller alone. r 2007 Elsevier Ltd. All rights reserved. Keywords: Fast tool servo; Active disturbance rejection control; Feed-forward compensation; Noncircular machining 1. Introduction Noncircular machining is a single-point cutting process that generates a workpiece with noncircular shaped cross- sections by controlling the cutting tool motion in the direction normal to the surface of the workpiece. In such a machining process, the cutting tool is driven by a fast tool servo (FTS), and the cutting tool motion must synchronize with the spindle rotation. Noncircular machining is well known for its excellent fl exibility, surface generation ability, and precision. This is due, to a great extent, to a FTS that helps make the cutting tool track a desired trajectory swiftly through programming. High rigidity and precision of the lathe are the other major factors that produce what is generally considered to be the best non- abrasive method to obtain high, even mirror fi nishes. Such noncircular surfaces are used in a wide range of products, such as engine pistons and camshafts 13. It is obvious that along with higher spindle speeds, more complicated workpiece shapes and higher profi le accuracy require better FTS performance. To improve the machin- ing productivity, adaptability, and accuracy of the non- circularmachiningprocess,theFTSmustthusbe simultaneously designed with high throughput drive, high bandwidth,longstroke,andhighaccelerationand accuracy.However,theinherentcontradictionswith respect to stroke, bandwidth, acceleration, and accuracy of the FTS lead to complexity and diffi culty in its design and implementation. Hence, FTS problems have drawn the attention of many researchers. In the design and implementation of the FTS, one problem is the design of the mechanical and electrical structure of the linear actuator. Various kinds of actuators, such as electromagnetic, piezoelectric, voice coil motor (VCM), and electro-hydraulic driven actuators, or their ARTICLE IN PRESS 0890-6955/$-see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.07.002 ?Corresponding author. Tel.: +861062773270; fax: +861062782351. E-mail address: wud (D. Wu). combinations 46, have been developed, focusing on the trade-off among stroke, bandwidth, and acceleration via the application requirements. Another problem is the design of a tracking control algorithm for generating accurate dynamic cutting tool motion, which is essential for successful implementation. In the noncircular machin- ing process, the FTS has to suffer from a nonlinearly variable cutting load, machine tool dynamics, and other uncertain disturbances. Moreover, there are the problems of inevitable, nonlinear aging and wear of actuator components, which make the plant time-variable. A high disturbance rejection capability must be thus established by the FTS to minimize the effects of the additional dynamics, disturbances, and variation. On the other hand, tracking control of FTS is needed to minimize the error between the reference and actual cutting tool positions while maintain- ing a high productivity that demands a shorter sampling period. Hence, the issue of how to control a FTS system under varying cutting load and system dynamics has become a paramount and challenging research topic. In tracking control problems, the control input must be applied so that the plant output follows a time varying desired output. This problem is closely related to system dynamic inverse. Intuitively speaking, if the dynamic inverse of a controlled plant is placed between the reference signal and the control input, then the plant output must come close to the reference signal. The feed-forward strategy thus easily meets such a tracking control require- ment. Of course, the problem is more complicated because of the stability and the realization concerns of the inverse systems 7. Tomizuka 8 proposed a zero phase error tracking control (ZPETC) algorithm to be applied as a feed-forward control law for the positioning of the tool along the desired time varying signal. He attempted to add feed-forward zeros to compensate for the plant unstable zeros. In addition, the concept of repetitive control was applied in noncircular machining because the reference input of the FTS is approximately periodic. Based on the Internal Model Principle, a periodic signal generator 1=1 ? e?ts is included in the feedback loop of the repetitive control system, which generates infi nitely large feedbackgainsattheperiodicsignalsfundamental frequency and its harmonics. Therefore the periodic signals can be tracked or rejected asymptotically provided the closed-loop system is stable 9. However, design methods for the tracking controller mentioned above assume that the plant model is accurately known. When the controlled plant is poorly known and is subject to uncertainties and variations, the tracking controller must have adequate adapting or learning capability so that an acceptable level of tracking performance is maintained. Hence, various solutions have been developed to improve the robustness of the feed-forward controller and the repetitive controller 1,2,1014. Moreover, a FTS using the fuzzy proportional, integral, derivative (PID) controller has been proposed by Liu et al. 15, in which the control parameters were continuously updated based on the position error and its varying rate to face the variations. However, it is diffi cult to tune the control parameters of the fuzzy PID, especially when the plant is variable. This research is motivated by the tracking control and disturbance rejection problems in the noncircular machin- ing process and tries to develop a new control strategy and algorithm which need no explicit plant model and allow for the easy tuning of the parameters and robust with disturbances. In this paper, a novel methodology for preciseand robust tracking control is presented. It combines the active disturbance rejection control (ADRC) concept 16 with the input revising, feed-forward error compensation strategy 17 to achieve both good robust stability and tracking performance under plant dynamics uncertainties and variations. This approach uses the active disturbance rejection controller to estimate and compen- sate the uncertainties and variations of the system, stabilize the plant, and then cascade the feedback system with the feed-forward controller to maintain the specifi ed tracking performance. As a design paradigm 18,19, ADRC maintains the kernel of the classical and effective PID control but attemptsto reject thedisturbancesactively.Inthis approach, a new and signifi cant notion of the extended state observer (ESO) is proposed. Unlike most existing observers, the ESO adds another dimension to the system instead of reducing the system order. The ESO views the nonlinearsystemdynamicsvariations,systemmodel uncertainty, and external disturbances as the extended state to be estimated equally and compensated during each sampling period. Actually, this process converts the plant into a standard linear system of series integrators. The dynamic feedback linearization of the system is thus achieved.TheESOstrategyresultsintheinherent robustness and disturbance rejection of the ADRC 20. Furthermore, a compensation block based on previous tracking errors is involved in the feed-forward loop of the system to revise the input signal and thus improve tracking accuracy. This paper is organized as follows. First, in Section 2, a second-order rough dynamic model is proposed to capture the fundamental drive principle and structure behavior of the actuator. Next, Section 3 presents the ADRC and feed- forward error compensation strategies and their respective digital controller design as well as stability analysis. Then, the experimental system, process, and results are shown in Section 4. Finally, concluding remarks are given in Section 5. 2. Modeling 2.1. Modeling of the actuator Although an explicit mathematic model is not required in the ADRC concept, much knowledge of the plant is particularly convenient for the controller design and parameter tuning. In particular, an approximate, even ARTICLE IN PRESS D. Wu et al. / International Journal of Machine Tools moreover, the different parameters and variables have physical interpretations. The drawback, however, is that it may be diffi cult and time-consuming to build the model from fi rst principles. Mathematical model building often has to be combined with experiments. Therefore, the model in Eq. (2) is measured with a software-based dynamic signal analyzer, using a swept sine method that generates fi xed amplitude sine waves of various frequencies 22. Since the frequency response of a system frequently needs to be described by algebraic expression usually comprised of the ratio of two frequency-dependent polynomials, the LEVY method of evaluating polynomial coeffi cients is used to determine the nominal frequency response, which is based on the miniaturization of the weighted sum of squares of the errors between the absolute magnitudes of the actual function and polynomial ratio for various frequencies 23. For simplifi cation, the nominal frequency response is a curve fi tted with a linear, second-order transfer function described as Gps 128850 s2 89:43s 20842 .(3) Such a simple linear second-order model is suffi cient for the forthcoming controller design and analysis. 2.2. Modeling of the reference input signal Fig. 2 schematically shows the relative position and motion between the cutting tool and the workpiece in noncircular machining. The in-and-out motion of the cutting tool must be synchronized with the workpiece rotation. When the noncircular machining process is applied to cut the engine piston with elliptical cross-section, the reference signal r of the cutting tool motion can be described as r rmax? rmaxrmin ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi r2 maxsin 2 ot r2 mincos2ot q,(4) where rmaxand rminare the maximal and minimal radius, respectively, and ot is the spindle angular position. Because the difference between rmaxand rminis generally smallenoughcomparedwithrmaxorrmin,namely, rmax?rmin5rmin, an approximate representation of Eq. (4) can be given as r ? rmax? rmin 2 1 ? cos2ot.(5) ARTICLE IN PRESS Flexure Cutting tool Moving shaft roller Coil Soft ironMagnet Fig. 1. Confi guration of VCM driven linear actuator. r workpiece cutting tool acutator rmax rmin Fig. 2. Schematic relationship between the workpiece and the cutting tool. D. Wu et al. / International Journal of Machine Tools y; _ y;w bu,(8) where y is the position output of the actuator, b is a constant as described in Eq. (2), u is the input force generated typically by the VCM. w is an extraneous unknown input force, such as the cutting load (known as the external disturbance), and ft;y; _ y;w represents the combined effect of internal dynamics and external dis- turbance on acceleration. In the model-based design, assuming that the desired closed-loop dynamics is y gy; _ y(9) the feedback control design fi nds an approximate, usually linear,time-invariant,anddisturbance-freeanalytical expression of ft;y; _ y;w, fy; _ y ? ft;y; _ y;w(10) through the modeling process. Then, the control law is designed as u ?fy; _ y gy; _ y b (11) to satisfy the design goal, approximately if not exactly. It should be noted that both the well-known pole- placement method for linear time-invariant systems and the feedback linearization method for nonlinear systems can be characterized in Eq. (11). The key assumption here is that the analytical expression fy; _ y is suffi ciently close to its corresponding part ft;y; _ y;w in physical reality. Therefore, when designing such a controller, it takes a great deal of effort to obtain an accurate dynamic model, especially if the plant is nonlinear. The tasks of modeling and prediction are also compounded by model parameters subject to changes over time and unpredictable distur- bances. However, one cannot guarantee that an accurate model is always available. It is thus worthwhile to fi nd alternatives to model-based control techniques and attempt to combine them with existing process control techniques. As a new design paradigm, ADRC represents a departure from classical as well as modern control theory. The ADRC concept was fi rst proposed by Han 16. Contrary to all existing conventions, this concept is perhaps not required to obtain the analytical expression of ft;y; _ y;w for feedback control design. Instead, all that is needed is the estimation of its value in real time. Specifi cally, let f be the estimate of ft;y; _ y;w at time t, then u ?f u0=b,(12) where u represents the control variable, corresponding to the input force generated by the VCM. Ignoring the estimation error in f, the plant in Eq. (8) is reduced to a unit gain double integrator, y ft;y; _ y;w b?f u0=b ? u0(13) which can be easily controlled. This shows that ADRC attempts to actively and directly estimateand compensatethe generalized disturbance ft;y; _ y;w. In return, such a strategy reduces the control of a complicated, perhaps nonlinear, time-varying and uncertain process in Eq. (8) to the simple problem in Eq. (13). The key difference between this approach and previous approaches is that no explicit expression of ft;y; _ y;w is assumed here. Only one thing is required to obtain the order of the system and the approximate value of b in Eq. (8). Thus, ADRC can be widely applied in industry regardless of whether the plant is linear or nonlinear, time-invariant or time-variant. A description of the unique control concept and successful industrial applications of ADRC can be found in 19,24,25. ARTICLE IN PRESS D. Wu et al. / International Journal of Machine Tools y; _ y;w in real time, it is necessary to designastateobserver.However,asmostexisting observers are based on the mathematical model of the plant, the presence of disturbances, dynamical uncertain- ties,andnonlinearitiesposechallengesinpractical applications. To this end, the high-performance robust observer design problem has been a topic of considerable interest recently, and several advanced observer designs have been proposed. As a unique observer design, a class of ESO was originally proposed by Han 16. The main idea of the ESO is to use an augmented state space model that often includes ft;y; _ y;w as an additional state. With consideration of the plant in Eq. (8), it can be augmented as _ x 1 x2; _ x 2 x3 bu; _ x 3 h; y x1; 8 : (14) where ft;y; _ y;w is treated as an extended state, x3. Here both ft;y; _ y;w and its derivative h _ ft;y; _ y;w are assumed unknown. By making ft;y; _ y;w a state, however, it is now possible to estimate ft;y; _ y;w by using a state estimator. This implies that a third-order observer should be designated for a two-order plant while implementing ADRC. Because the VCM driven actuator is approximated as a second-order system, a third-order ESO is designed to estimate and compensate the disturbances. Fig. 3 illustrates thearchitectureofthedevelopedactivedisturbance rejection controller for noncircular machining. Compared with the general ADRC confi guration 15, the controller removes the tracking differential controller (TD). There are two reasons for doing this. First, the desired trajectory of the actuator in noncircular machin- ing is often represented as differentiable functions such as Eqs. (5) and (6), making it easy to obtain their differentia- tion directly. Second, although the TD can track the desired trajectory as fast as possible, there still exists a slight delay that will inevitably lead to unnecessary tracking error. Since the controller is to be implemented by a digital signal processor (DSP), a discrete-time ESO is designed and represented as ? z1k ? yk; z1k 1 z1k hz2k ? b1?; z2k 1 z2k hz3k ? b2? bu; z3k 1 z3k ? hb3?; 8 : (15) where e is t