IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 1, JANUARY 2002 221
Robust Adaptive Numerical Compensation forFriction and Force Ripple in Permanent-Magnet
Linear MotorsK. K. Tan, S. N. Huang, and T. H. Lee
Abstract—This paper describes a robust adaptive compensationmethod for friction and force ripple present in the dynamics of per-manent-magnet linear motors. The method is used in ultraprecisepositioning applications. The compensation algorithm consists of aPID component and an adaptive component for estimating frictionand force ripple. The adaptive component is continuously refinedon the basis of just prevailing input and output signals. Computersimulations and real-time experimental results verify the effective-ness of the proposed scheme for high-precision motion trajectorytracking.
Index Terms—Cogging force, friction, permanent magnet linearmotor, ripple force.
I. INTRODUCTION
A MONG the electric motor drives, permanent magnet linearmotors (PMLMs) are probably the most naturally akin
to applications involving high-speed or/and high-precision mo-tion control. The increasingly widespread industrial applica-tions of PMLMs in various semiconductor processes, precisionmetrology, and miniature system assembly are self-evident tes-timonies of the effectiveness of PMLMs in addressing the highrequirements associated with these application areas. The mainbenefits of a PMLM include the high force density achievable,low thermal losses, and, most importantly, the high precisionand accuracy associated with the simplicity in mechanical struc-ture. Unlike rotary machines, linear motors require no indirectcoupling mechanisms as in gear boxes, chains, and screws cou-pling. This greatly reduces the effects of contact-type nonlinear-ities and disturbances such as backlash and frictional forces, es-pecially when they are used with aerostatic bearings. However,the advantages associated with mechanical transmission are alsoconsequently lost, such as the inherent ability to reduce the ef-fects of model uncertainties and external disturbances. There-fore, a reduction of these effects, either through proper phys-ical design or via the control algorithms, is of paramount im-portance if high-speed and high-precision motion control is tobe achieved.
The more predominant nonlinear effects underlying a linearmotor system are the friction (Coulomb, viscous, and stiction)and force ripples (detent and reluctance forces) arising from im-perfections in the underlying components. Due to the typical
Manuscript received April 24, 2000; revised October 4, 2001.The authors are with the Department of Electrical and Computer En-
precision positioning requirements and low offset tolerance oftheir applications, the control of these systems is particularlychallenging since conventional PID control usually does not suf-fice in these application domains.
Several characteristic properties of friction have been ob-served, which can be decomposed into two categories: staticand dynamic. The static characteristics of friction, comprisingstiction, the kinetic and viscous force, and the Stribeck effect,are functions of steady-state velocity. The dynamic phenom-enon includes presliding displacement, varying breakawayforce, and a frictional lag. Friction is often responsible for theinability of the system to achieve small steady-state trackingerrors, and it may also limit the allowable closed-loop band-width in order to avoid limit cycling. It is difficult to use astatic feedforward controller to compensate for the effects offriction, since friction exhibits a time-varying character. Hence,an efficient method for friction compensation should entailon-line adaptive algorithm to continuously identify the frictionmodel and compensate for the friction accordingly.
The two primary components of the force ripple are the cog-ging (or detent) force and the reluctance force. The coggingforce arises as a result of the mutual attraction between the mag-nets and iron cores of the translator. This force exists even inthe absence of any winding current and it exhibits a periodicrelationship with respect to the position of the translator rela-tive to the magnets. The reluctance force is due to the variationof the self-inductance of the windings with respect to the rela-tive position between the translator and the magnets. Thus, thereluctance force also has a periodic relationship with the trans-lator-magnet position. From motion control viewpoints, forceripple is highly undesirable. They can be minimized or eveneliminated by an alternative design of the motor structure orspatial layout of the magnetic materials, such as skewing themagnet, optimizing the disposition and width of the magnets,etc. However, these mechanisms often increase the complexityof the motor structure. An adaptive algorithm to eliminate theeffects of force ripples is similarly desirable.
Much effort has been devoted to the control system to over-come these problems. In [1], adaptive schemes were developedfor friction compensation. In [2], a comparison of several adap-tive friction compensation techniques was presented. A non-linear compensation technique that has a nonlinear proportionalfeedback control force for the regulation of the one degree offreedom mechanical system was proposed in [3]. To suppressdisturbance and force ripple, [4] proposed a optimal feed-back control to provide a high dynamic stiffness to external
222 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 1, JANUARY 2002
disturbances. In [5], a ripple compensation algorithm is imple-mented in the position control loop based on an identified first-order approximation of the ripple forces. In [6], a neural-net-work feedforward controller is proposed to reduce positional in-accuracy due to friction and ripple forces.
In this paper, we consider the design and realization of arobust adaptive algorithm to simultaneously suppress the fric-tion and ripple force phenomenon associated with general servomechanisms, but with a more specific view toward applicationto a linear dc motor. The controller consists of a PID feedbackcomponent and an adaptive feedforward component for frictionand force ripple compensation. In addition, a sliding-mode con-trol term included in the adaptive control law allows for robustcompensation of remaining unaccounted dynamics. The adap-tive component is continuously refined on the basis of just pre-vailing input and output signals. The tracking error is provento be asymptotically converged to the predetermined boundary.In addition, analytical quantification is given to illustrate theimprovement of the system’s transient performance. Computersimulations and real-time experimental results verify the effec-tiveness of the proposed scheme for high-precision motion tra-jectory tracking using the PMLM.
II. M ODELING OF THEPERMANENT MAGNET LINEAR MOTOR
The mathematical model for a voltage-controllable linear dcmotor system can be approximately described by the differentialequation [8]
(1)
wheretime-varying motor terminal voltage;motor position;amount of force produced by the motor inNewton/Ampere;back EMF voltage;total resistance between any two phases;moving thrust block mass;system disturbance;
where . and denote the fric-tional and ripple force respectively, and they will be discussedbelow. is the applied load force.
Let
(2)
(3)
(4)
(5)
(6)
Thus, we have
(7)
Fig. 1. Friction model.
TABLE ILINEAR MOTOR
PARAMETER
The load force is assumed to be bounded within the unknownupper bound as follows:
(8)
The frictional force affecting the movement of the translatormay be modeled as a combination of Coulomb friction, viscousfriction, and the component due to Stribeck effect, which can beinterpreted as stiction. The mathematical model may be writtenas in [7]
whereminimum level of Coulomb friction;level of static friction;lubricant parameter which may be deter-mined by empirical experiments;viscous friction parameter.
Fig. 1 graphically illustrates a basic friction model based on theformula.
The above model allows us to evaluate the friction forceduring both sticking and slipping motions. For a mechan-ical system, , , are assumed to be constants. Since
, the Stribeck effect is referred to a boundeddisturbance, that is . An adaptivelaw can be designed to directly compensate for the Coulomb
TAN et al.: ROBUST ADAPTIVE NUMERICAL COMPENSATION 223
Fig. 2. Desired trajectory.
Fig. 3. Control responses of linear motor with the proposed controller.
and viscous frictions. An adaptive sliding mode control com-ponent can address the Stribeck effect and other unaccounteddynamics such as the load force.
Apart from friction, one of the known disturbance forces gen-erated in linear drives is the force ripple due to cogging and re-luctance forces present in the structure of linear drives. Usually,the force ripple can be described by a sinusoidal function of theload position with a period of and an amplitude of , i.e.,
(9)
where , , , , are constants. In reality, the rippleforce is more complex in shape, e.g., due to variations in themagnet dimension, but it has a comparable period and amplitudeas described in (9).
The development of the robust adaptive control algorithm forestimating the parameters of the friction and force ripple modelsand accordingly compensating for these forces will be presentedin Section III.
III. A DAPTIVE CONTROL OFLINEAR MOTOR
Define the tracking errors
(10)
(11)
where and are the desired position and velocity, respec-tively. To achieve tracking control, we define a sliding surface(see [9] and [10])
(12)
where , are chosen such that the polynomialis Hurwitz.
We define another error metric, , as in [9]
224 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 1, JANUARY 2002
Fig. 4. Friction and ripple force estimates of linear motor. Dashed line represents estimated force; solid line represents output of model based friction or rippleforce.
Fig. 5. Control responses of linear motor with PID control.
where is a saturation function defined as
The function has the following useful properties:
1) if , then ;2) if , then and ;3) .The problem to be addressed is to design a control law
which ensures that the tracking error metric lies in the pre-determined boundary for all time .
TAN et al.: ROBUST ADAPTIVE NUMERICAL COMPENSATION 225
Fig. 6. Experimental setup.
Fig. 7. Block diagram of the experimental linear drive.
The following controller is constructed for the nonlinear plant(7):
where and are the estimates ofand , respectively; rep-resents the estimate of; represents the estimate of;and represents the estimates of and , respectively;is the estimate of . is an additional control, whichis given by
(16)
appears in the control law, which implies velocity needs tobe measured or estimated from an observer. Differentiatingand using the control (15), the linear motor dynamics (7) maybe written in terms of the filtered tracking errors as
(17)
where , , , ,, and .
We now specify the parameter update laws
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Theorem 1: Convergence of Adaptive Controller:Considerthe plant (7) and the control objective of tracking the desiredtrajectories, , , . The control law given by (15) with(18)–(24) ensures that the system states and parameters are uni-formly bounded and that asymptotically converge to thepredetermined boundary.
Proof: We first define a Lyapunov function candidateas
(25)
where , . Noting that outsidethe boundary layer, while inside the boundary layer, itfollows
(26)
Substituting the expressions given by (18)–(24) yields
(27)
From , it follows that . This impliesthat , , , , , , , are uniformly bounded withrespect to . To show the boundedness of the tracking error, weneed to first prove that, are bounded.
226 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 1, JANUARY 2002
Fig. 8. Software design for Simulink model.
Fig. 9. Velocity responses of linear motor with input of 1 V (displacement unit:�m, velocity unit:�m=s.
Define
(28)
From (12), it follows
(29)
Since , are chosen such that the polynomialis Hurwitz, the free system of the above equation is asymptot-
ically stable. This together with bounded, implies that,are bounded.
By definition, is either 0 or , where is given in(17). Since , , and the system parametersare bounded, this implies that the right side of (17) is bounded.This further implies that is bounded. Equation (27) and thepositive definiteness of then imply that
(30)
TAN et al.: ROBUST ADAPTIVE NUMERICAL COMPENSATION 227
By virtue of Barbalat’s lemma, we have
(31)
Applying (27), it follows that
(32)
The proof is completed.
A. Remark 3.1
Note that is actually standard PID control. In principle,we can employ any existing PID tuning methods to determine
, , . This initial PID tuning can be a coarse one.The system performance can be improved by the adaptivecomponent.
B. Remark 3.2
To achieve high-accuracy tracking, we should choose asmall . However, a small may cause chattering in thecontrol input. Therefore, there should be a tradeoff between thedesired tracking error and the discontinuity of the control inputtolerable.
IV. SIMULATION AND EXPERIMENT
In this section, we will illustrate the effectiveness of the pro-posed control scheme using computer simulation and real-timeexperiments. The performance of the proposed control schemewill be compared to that of PID control.
A. Computer Simulation
The particular linear motor model used in this simulation is adirect thrust tubular servo motor manufactured by Linear DrivesLtd. (LDL)(LD 3810) (see [11], [12]). The physical parametersare described in Table I. The model parameters of the frictionand ripple forces used in this paper are given as
N N m/s
N N
The spatial cogging frequency is assumed to berad m. The design parameters of the controller
are selected as
(33)
and the boundary of the error metricis set to 90. The initialvalues of , , , , , , are set to 50% of the truevalues.
The desired position trajectory is shown in Fig. 2. Fig. 3shows the tracking error when the proposed control is used.Fig. 4 shows a comparison of the estimated friction and rippleforces with the actual model-based forces. To illustrate the im-provement in performance, we compare the proposed controllerwith PID control. For a fair basis in comparison, the gains of thePID controller are chosen to be similar to the PID part ( )of the proposed control. Fig. 5 shows the tracking performanceunder PID control. It is clear that the proposed controller canachieve better performance than pure PID control.
Fig. 10. Position trajectory and control result in experiment.
B. Real-Time Experiment
In this section, experimental results are provided to illustratethe effectiveness of the proposed method. Fig. 6 shows the ex-perimental setup. A 70-kg load is placed on the thrust block ofthe motor. The functional block diagram is shown in Fig. 7. Thelinear motor used is a direct thrust tubular servo motor manufac-tured by Linear Drives Ltd. (LDL)(LD 3810), which has a travellength of 500 mm and it is equipped with a Renishaw optical en-coder with an effective resolution of 1m. dSPACE control de-velopment and rapid prototyping system is used. dSPACE inte-grated the entire development cycle seamlessly into a single en-vironment, so that individual development stages between sim-ulation and test can be run and rerun, without frequent read-justment. MATLAB/Simulink can be used from within dSPACEenvironment. The proposed algorithm is written in C, and em-bedded in an S-function block. The entire control system iswritten into a Simulink object as shown in Fig. 8, where theS-function blocktr3 is used to generate the desired trajectoriesand the S-function PIDRBF is the controller itself.
From a transient analysis of a step test as shown inFig. 9, we can measure the spatial cogging frequency to be9 10 rad m. The parameters of the controller are selectedas
(34)
and the desired error tolerance is set to 10. The initial values ofthe estimates are set to be zero (no prior knowledge assumed).Since the mechanical structure and other components in the
228 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 1, JANUARY 2002
Fig. 11. Control results using PID controller.
system have inherent unmodeled high-frequency dynamics thatshould not be excited, small adaption factors, , , ,
, , are used. As only the displacement measurement isavailable in the linear drive system, the velocity is derived usinga numerical difference method.
The desired position trajectory used in the experiments isshown in Fig. 10. In the same figure, the position error and con-trol effort are shown using the proposed control. In Fig. 11, thetracking performance of PID control is shown based on the sameposition trajectory. It can be observed that, under PID control, amaximum error of about 15m is obtained. On the other hand,the proposed control can achieve a maximum tracking error ofless than 8 m.
V. CONCLUSION
A robust adaptive algorithm has been developed for compen-sation of friction and force ripple arising in permanent magnet
linear motors. The objective of the control and compensationscheme is to achieve good tracking performance of PMLM-based motion systems, in the presence of the uncertain frictionand force ripple dynamics. This is achieved using an adaptivesliding control scheme. The stability of the proposed scheme hasbeen proven in the paper. Simulation and experimental resultshave verified the effectiveness of the proposed control schemeand have shown that the proposed compensation technique canachieve a superior tracking performance compared to conven-tional PID control.
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