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Adaptive robust motion control for precise trajectory tracking applications
ISA Transactions 40 (2001) 57±71
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Adaptive robust motion control for precise trajectory tracking applications K.K. Tan a,*, S.N. Huang a, H.F. Dou a, T.H. Lee a, S.J. Chin a, S.Y. Lim b a Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Automation Technology Division, Gintic Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore 638075, Singapore
b
Received 00 Month 0000; accepted 0 Month 0000
Abstract This paper presents a robust servo control method for high precision motion control using linear actuators. The controller consists of three components : a simple feedforward compensator, a PID feedback controller, and a RBF (radial-basis function) adaptive compensator, each to ful®ll a speci®c objective. The ®rst two control components can be directly tuned based on only an estimated dominant second-order linear model. The RBF compensator is self-tuning, and it will compensate for remaining uncertainties in the system, residual of the linear model. Rigid proofs are provided, guaranteeing the robust stability of the proposed controller. Experimental results con®rm the much superior performance of the 3-tier composite control over a standard motion controller. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Robust control; Adaptive control; Radial basis functions; Linear motors
1. Introduction In spite of the advances in mathematical control theory over the last 50 years, industrial servo control loops are still essentially based on the threeterm PID controller. The main reason is due to the widespread ®eld acceptance of this simple controller which has been eective and reliable in most situations if adequately tuned. More complex advanced controllers have fared less favourably under practical conditions, despite the higher costs associated with implementation and the higher demands in control tuning. It is very dicult for
* Corresponding author. Tel.: +65-874-2110; fax: +65-7791103. E-mail address: [email protected] (K.K. Tan).
operators unfamiliar with advanced control to adjust the control parameters. Given these uncertainties, there is little surprise that PID controllers continue to be manufactured by the hundred thousands yearly and still increasing [1]. On the other hand, increasing servo-controllers for mechanical systems, such as machine tools and robots, are required to achieve high accuracy at high speed on speci®ed trajectories. Modern silicon processes and assembly of micro electromechanical systems (MEMS) all require submicron machine accuracy and in the immediate years to come, nanometer and sub-nanometer accuracy is set to become a necessity. For obvious economic factors, high accuracy must be achieved without sacri®cing high production speed. Given stringent requirements, the dilemma is that the performance from conventional feedback control
0019-0578/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0019-0578(00)00037-9
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
is often not sucient, although the simple control structure is highly desirable. In this paper, we consider speci®cally mechanical servo systems which are based on permanent magnet linear motors (PMLM). Linear motors are becoming very popular for applications requiring high speed, high accuracy operations due to their mechanical simplicity. Even then, to achieve high accuracy positioning, extraneous forces resident in the system must also be taken into account in the control design [2±4]. These include: . frictional force which is highly nonlinear in nature, and in many cases, time varying; . force ripples arising from imperfections in the underlying components.
These issues are dicult to handle within a framework of conventional feedback control. The main objective of the paper is to address the abovementioned dilemma which is widely evident in many real industrial control problems. In this paper, we view a general nonlinear system as a predominantly linear system, with a nonlinear and uncertain deviation from the linear model. For high speed response, a full-state (position, velocity and acceleration) feedforward compensator is applied based on the nominal linear model. Clearly, the performance achievable with feedforward control depends on the adequacy of the model used. To increase the closed-loop bandwidth and concurrently minimise the eects of imperfect modelling, a PID feedback controller is included in the control structure. While the simplicity in a PID structure is appealing, it is also often proclaimed as the reason for poor control performance whenever it occurs. In this paper, we demonstrate how advanced optimum control theory can be applied to tune PID control gains. The PID feedback controller is designed using the Linear Quadratic Regulator (LQR) technique for optimal and robust performance of the nominal system. The proposed feedforward plus feedback con®guration may also be referred to as a twodegree-of-freedom control [5], and it should suce for most practical requirements. However, if further performance enhancement is necessary, the third control component may be enabled. A
Radial Basis Function (RBF) is applied to model the nonlinear remnant, and this is subsequently used to linearise the closed-loop system by neutralising the nonlinear portion of the system. Therefore, the proposed controller can compensate the eects of strong nonlnearities in the system. This diers from the previous results. An ecient tuning method for the 3-tier composite controller is proposed and developed in the paper to yield a good initial set of control parameters based on only a second-order linear model, and subsequent ®ne self-tuning will enhance the performance. The proposed control system is able to guarantee the boundness of the tracking errors even in the presence of strong and unforseen disturbances. Note that the theoretical analysis is not involved in Otten [3]. Both simulation and experimental results are provided to highlight the principles of the method and to demonstrate its applicability and eectiveness in true industrial applications. 2. Problem formulation In this section, the problem relating to highprecision motion control for linear motors will be formulated in a systematic manner. 2.1. Dominant linear model with uncertainty Similar to general voltage controllable dc motors, the mechanical and electrical dynamics of a PMLM can be expressed as follows: J D Tl Tm
1
: dIa Ra Ia u KE La dt
2
T m K T Ia
3
where denotes position; J, D, Tm , Tl denote the mechanical parameters: inertia, viscosity constant, generated force and load force respectively; u, Ia , Ra , La denote the electrical parameters: input dc voltage, armature current, armature resistance and armature inductance respectively; KT denotes the
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
electrical-mechanical energy conversion constant. Because the electrical time constant RLaa is small (compared to the mechanical time constant DJ ), electrical transients decay very rapidly and a La dL dt 0 (see [6], Chapter 5). Thus, the following simpli®ed equation is given: K1 : K2 u J J
1 Tt J
4
where K1
KE KT Ra D KT ; K2 Ra Ra
5
Clearly, this is a second-order dynamical model. The model does not include extraneous nonlinear eects which may be present in the physical structure. When control performance requirements are more stringent, the above model alone may thus become inadequate. A prominent nonlinear eect associated with PMLM is due to frictional force and it may be written as [7]: : : : : = s ffric fc fs fc e fv sgn ; 6
K2 : f ; J
fripple 8:5sin 900
7
: Thus, a nonlinear function f1 ; may be used to model these nonlinear dynamical eects approximately. The servo system now becomes: Let
: K1 : K2 1 u Tl f1 ; J J J
8
9
It follows that K1 : K2 K2 : u f ; : J J J
10
: We will assume that f ; is a smooth nonlinear function which may be unknown. To this end, it may be pointed out that many discontinuous nonlinear functions may be adequately approximated by a continuous one. Thus, the model represents a large and rich class of nonlinear servo-systems. 2.2. Tracking problem The control objective can be described as follows: given a desired trajectory, the primary goal is to track a desired trajectory d as close as possible, while ensuring the states and control remain bounded for stability. In other words, the tracking error e de®ned as: e d
where fs is the level of static friction, : fc is the minimum level of Coulomb friction, s and fv are lubricant and load parameters which may be determined by empirical experiments. is an additional empirical parameter. In addition, one of the known disturbance generated in linear motors is the force ripple due to cogging and reluctance forces present in its structure. In [7], the force ripple is described as a periodic sinusoidal type signal:
: 1 TL f1 ; : J
59
11
should be as small as possible. Based on this de®nition, we may express (10) as: e
K1 : K2 K2 : e u f ; J J J K2 J K1 : d d J K2 K2
Since d t e tdt e dt 0
12
13
we t assign the system state : variables as x1 0 e tdt, x2 e and x3 e such that x x1 x2 x3 T . (12) can then be put into the equivalent state space form: : J K1 : : x Ax Bu Bf ; B d d K2 K2 14
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
2
0 1 A 40 0 0 0
3
0 1 5; K1 =J
2
3 0 B4 0 5 K2 =J
15
3. Controller Structure In this paper, we adopt a 3-tier composite control structure, as shown in Fig. 1, comprising of the following components. 3.1. Feedforward control
PID controller is appealing to practitioners, yet it is often proclaimed as the cause whenever poor control performance occurs. In this section, we will illustrate, how even with a simple control structure, the PID controller may be tuned to achieve the same level of optimum performance as advanced LQR control. The nominal portion of the system (without uncertainty) is given by: : x t Ax t Bu t; 17 where upid Kx kx1 kd1 x2 kd2 x3 :
The design of the feedforward control law is straightforward. From (14), since we :would like to 1 neutralise the term of B KJ2 d K K2 d with feedforward control, we design the feedforward control as: J K1 : d d 16 uff t K2 K2 Clearly, the reference position trajectory must be continuous and twice dierentiable, otherwise a pre compensator to ®lter the reference signal will be necessary. The only parameters required for the design of the feedforward control are the parameters of the second-order linear model. 3.2. PID feedback control The feedback control component utilises the PID control structure. The simple structure of the
18
This is a PID control structure which utilises a full-state feedback. We would like to obtain the optimal control parameters using the Linear Quadratic Regulator (LQR) technique that is well known in modern optimal control theory and has been widely used in many applications [8]. It has a very nice robustness property, i.e. if the process is of singleinput and single-output, then the control system has at least a phase margin of 60 degree and a gain margin of in®nity. This attractive property appeals to the practitioners. Thus, the LQR theory has received considerable attention since 1950s. The LQR problem requires a mimisation of an index of control accuracy of the form 1 J xT Qx lu2 19 0
Fig. 1. The block diagram of the control scheme.
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
where Q50 and l > 0. It is well-known that the LQR control with (19) is u
l 1 BT Px t
20
where P is the positive de®nite solution of the Riccati equation AT P PA
l 1 PBBT P Q 0
21
and Q HT H I. One practically useful feature associated with LQR design is that under mild assumptions, the resultant closed-loop system is always stable. We may summarise this feature in the following theorem. Theorem 1. (see [9], Theorem 31). For the system (17), if A; B is controllable and H; A is observable, then the control law given in (20) stabilizes the system (17). In this paper, the PID control uses a slightlymodi®ed version of the LQR controller, that is upid
r0 1BT Px
22
where l 1 and r0 is independent of P and it is introduced to weigh the relative importance between control eort and control errors. Note for this proposed feedback control, the only parameters required are the parameters of the second-order model and a user-speci®ed error weight r0 .
where vect
h : iT .
61
Assumption 1. The ideal weights are bounded by known positive values so that jwi j4wM ; i 1; 2; . . . m. 2. There exists an M > 0 such that Assumption : ; 4M , 8vect 2 I on a compact region I R2 . : Let the RBF functional estimates for f ; be given as: m : X : f^ ; i ; w^ i
25
i0
where w^ i are estimates of the ideal RBF weights that are provided by the following weight-tuning algorithm : w^ i r1 xT PB i
r2 w^ i
26
where r1 ; r2 > 0, and P is the solution of (21). Thus, we have : 27 ucomp f^ ; In adaptive law (26), -modi®cation [10] is used to improve the robustness of the adaptive controller in the presence of the system uncertainty and the external disturbances.
3.3. RBF adaptive control
4. Parameter estimation of nominal system
Next, we consider the design of the adaptive compensator :forthe nonlinear part of the system (10). Since f ; is a nonlinear smooth function (unknown), it can be represented as:
Under the proposed control structure, a secondorder model is necessary for computing the control parameters. Many approaches are available to yield the required linear model [11]. In this paper, we will use the parameter estimation approach [11]. Consider an ARX model given by:
m : X : f ; i ; wi i0
23
: with jj4M , where i ; is the RBF, that is : i ;
! m 24
vect cj 2 kvect ci k2 X exp = exp 2 2 2i 2j j0
y t a1 y t 1 . . . ana y t na b1 u t 1 . . . bnb u t nb e t
28
where na is the number of poles, nb 1 is the number of zeros, and e t represents the error term of the system. Based on this model, the linear predictor is given by:
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
y^ t= T t e t
29
where t
y t y t
1 na u t
y t
2 . . . 1 . . . u t
T c a1 . . . ana b1 . . . bnb
31
With (29), the prediction error becomes: T nb ;
30
" t; y t
T t :
Fig. 2. Comparison between linear model (dashed line) and actual nonlinear system (solid line).
Fig. 3. Desired position and velocity trajectories.
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
De®ne the model ®tting criteria function as: VN
N 1X y t N t1
T t
2
;
33
which is the least-square criterion. The criterion can be minimized analytically, giving the least squares estimates of the model parameters as: "
^ LS N
N 1X T t t N t1
#
1
N 1X ty t N t1
34
The persistently exciting input signals should be used. This may come in the form of explicit input sequences (e.g. PRBS) or it may arise naturally from control signal generated in the closed-loop, in which case no explicit experiment needs be conducted The parameter estimation described above, using explicit user-de®ned input signals, will yield
63
an initial set of parameters for the linear model. Thereafter, the model may be re®ned using incremental measurements of the input and output signals of the system under closed-loop control with a recursive version of the least square estimation algorithm [11], i.e. the re®nement may occur online with the motor under normal motion operations. 5. Stability analysis of proposed controller It is required to demonstrate that the state x and weights will remain bounded. The following de®nition will be used to illustrate the stability. Theorem 2. Assume that the desired references yd , : and yd are bounded. Consider the case where the controller given by (16),(18) and (27) is applied to the system (14). Then the state and the RBF estimation errors are uniformly ultimately bounded
Fig. 4. Control results: (a) proposed controller; (b) pure PID controller; (c) PID+feedforward controller.
64
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
Proof. The dierential Eq. (14) can be written as [upon applying the control (16),(18) and (27)]: : : x Ax B uff upid ucomp Bf x; x; x J K1 : d d 35 B K2 K2 m X A r0 1BBT P x B i w^ i " B
m X i wi i0
A
T
BB P
i0
#
36
" # m X r0 BB P x B i w~ i T
i0
37
Now consider the following Lyapunov function candidate: m 1X V x; w~ xT Px w~ 2 38 r1 i0
Taking the time derivative of v along the solution of (37), it can be shown that : V xT AT P PA PBBT P x xT 2ro m m X : 2X w~ i w~ i 2xT PB i w~ i 2xT PB r 1 i0 i0 4
PBBT P x
m X lmin Q 2r0 1PBBT P kxk2 2xT PB i w~ i m X
i0
r2 2x PB 2 x PB i w^ i w~ i r1 i0 T lmin Q 2r0 1PBB P kxk2 2xT PB m r2 X 2 w^ i w~ i r1 i0 lmin Q 2r0 1PBBT P kxk2 m m r2 X r2 X 2xT PB 2 w~ 2i 2 wi w~ i r1 i0 r1 i0 T
From
Fig. 5. Input±output signals used in identi®cation experiment.
T
39
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
1 2xT PB4xT PBBT Px 2 1 4xT PBBT Px 2M
40
r2 r2 2 r2 2 2 wi w~ i 4 w w~ r1 r1 i r1 i
41
we have n : V4 lmin Q 2r0 1PBBT P m o r2 X T 2 lmax PBB P kxk 2 1 r1 i0
where
1 w~ 2i 2
1 r2 2 2M w r1 i where , are small positive T constants. Let xT w~ 0 w~ 1 . . . w~ m and we have : V4
2 kk2 l1
65
42
43
n 1
min lmin Q 2r0 1PBBT P 2 r2 1 o T lmax PBB P ; 2 1 2 r1 1 r2 2 l1 2M w r1 i
44 45
Clearly, > 0 for suciently small , . In order to show the boundedness of the state and weights, note from kk2 4V4v kk2 46 where min lmin P; 1=r1 , v max lmax P; 1=r1 g, that :
47 V 4 2 V l1 v Hence, following [13], we have
Fig. 6. Comparison between estimated model (solid line) and linear motor (dashed line).
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
l1 v V 0 V t4 2
l1 v e 2
2 t v
48
From (48), we can conclude that the state and weights are bounded. This completes the proof. &
Remark 3. For the RBF basis, there are two ways to choose the centers and variances: the local training techniques (see [12]); the centers evenly spread along the desired trajectories with i=const(15) (see our simulation).
Remark 1. For the adaptive scheme (26), the ®rst term grows rapidly as xT PB i increases (which re¯ects the ``poor'' system performance). This will result in a high increase rate of w^ i and, therefore, strong feedback. The parameters r1 , r2 oer a design trade-o between the relative eventual magnitudes of kk and w~ i
6. Simulation
Remark 2. If the load TL is available, we can incorporate it into the feedforward control, that is
: u v
uff t
J K1 : TL d d K2 K2 K2
49
The linear motor model considered here for simulation is a brushed permanent magnet DC linear motor manufactured by Anorad Corp. The nonlinear model is given according to [13] as: : xv 50 ffriction m
fripple
w t
51
where ffriction is the friction force, fripple is the ripple force, u is the developed force, m is the combined
Fig. 7. Desired trajectories (I).
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
67
mass of translator and load, and w represents any other residual disturbances. The friction and ripple forces are modeled as follows: : : : : ffriction fc fs fc e x=xs fv x sgn x 52
the ®nal position. In this simulation study, we choose x0 0, xf 0:5m, tf 1s. Fig. 3 shows the desired trajectories. To apply the proposed controller, the plant is ®rst identi®ed using the standard least squares identi®cation algorithm as:
fripple b1 sin w0 x
x
53
The above model allows us to evaluate the friction force during both sticking and slipping motions. The model parameters can be found in [13]. The desired trajectories are planned as follows: xd x0 x0 xf 15 4 6 5 10 3 54 vd x0
xf 60 3
30 4
30 2
55
where the gain of 106 is used to normalize the units of the system to 1 mm, xd is the desired position trajectory, vd is the desired velocity trajectory, t= tf t0 , x0 is the initial position and xf is
: 117:6471x 8:44987u
56
where x represents the position. But the approximation by this model is not ideal as shown in Fig. 2 where the dynamics due to the friction and ripple forces are not represented in the model. The PID control is obtained by using (18), giving: t upid t r0 1 56:5463e t 31:6228 e d 0 : 20:8223e t 57 for given Q0 103 I, where I is the unit matrix. According to Theorem 2, we obtain the RBF compensator of the controller, where we choose
Fig. 8. Control performance from proposed controller with r0 2:5.
68
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
m 19, and the centers ci of the RBF basis are [{0.00430.01} 0.120.01], [0.02900.01 0.380.01], [0.08150.01 0.660.01], [0.15870.01 0.860.01], [0.2500.01 0.930.01], [0.34130.01 0.860.01], [0.41850.01 0.660.01], [0.47100.01 0.380.01], [0.49570.01 0.120.01], [0.50.01 0.000.01]. These are chosen as the mesh nodes along the desired position and velocity trajectories, respectively. We also choose 5 106 , r0 3, r1 10, r2 1. Fig. 4(a) shows the control results. The simulation results for the pure PID control and the PID+feedforward control are also shown in Fig. 4(b) and (c) respectively, where it is clear that the proposed controller has achieved a much improved performance. 7. Experiment In this section, experimental results are provided to illustrate the eectiveness of the proposed method. The linear motor used is a direct thrust
tubular servo motor manufactured by Linear Drives Ltd (LDL)(LD 3810), which has a travel length of 500 mm and it is equipped with a Renishaw optical encoder with an eective resolution of 1 mm. The dSPACE control development and rapid prototyping system based on DS1102 board is used. dSPACE integrated the entire development cycle seamlessly into a single environment, so that individual development stages between simulation and test can be run and rerun, without any frequent re-adjustment. To apply the proposed controller, the system (from control voltage to velocity) is ®rst identi®ed. The input-output signals from the identi®cation experiment are as shown in Fig. 5. Cross validation of the estimated model and actual linear motor is given in Fig. 6. The dominant linear model is computed to be: : x 24:1066x 1:26992 107 u: 58 The feedforward control is thus designed accordingly as:
Fig. 9. Control performance from PID control.
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
uff t
: 1 24:1066 d d 1:26992 107 1:26992 107
59
Based on the identi®ed model, the LQR technique is used to obtain the feedback control parameters. The solution to (21) is: 2
2:3218 P4 0:0029 1:7605 10
6
0:0029 1:1722 10 8:1781 10
5 9
1:7605 10 8:1781 10 1:0073 10
6 9
3 5
11
60
for 2
1 Q 5004 0 0
0 10 0
5
3 0 0 5 10 11
61
Thus, we have the PID control upid
r0 122:3565 0:1038 0:0001T x t
62
69
The centres of the RBF function are chosen with regularly spaced mesh as the simulation example. Note that the inputs of RBF should be normulized. The other parameters are chosen as 1, r1 0:001, r2 0:1. The sampling time is 0:0005 s. The desired trajectories for the position and velocity are shown in Fig. 7. Fig. 8 shows the performance of the proposed controller. The maximum error is about 7 mm. If a PID is used, the result is shown in Fig. 9. The maximum error for the PID is about 12.5 mm. It is clearly observed from the above ®gures that the tracking error is signi®cantly improved. To further improve the tracking error, we can tune the value of r0 with the other parameters kept constant. With r0 6:5 and the desired trajectories shown in Fig. 10, the control results are shown in Fig. 11. Notice that the nonlinearities such as cogging and reluctance forces inevitably present in linear motor are eliminated by the RBF compensator. Since the resolution of the encoder
Fig. 10. Desired trajectories (II).
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
Fig. 11. Control performance from proposed controller with r0 6:5 and other parameters to be the same as the above experiment.
used is 1 mm, it is clear from the ®gure that a positioning accuracy of 4 mm is achieved. As a matter of fact, a higher positioning accuracy can be achieved if special high precision encoders are used. 8. Conclusions This paper has considered the development of a new 3-tier composite feedforward+feedback+ adaptive controller for a class of linear motor systems. The second-order model is used as the basic foundation for the design of the feedforwardfeedback design, and an adaptive component designed based on a RBF provides for the further possibility of performance enhancement. Detailed stability analysis is provided. Simulation and realtime experiment further verify the eectiveness of the proposed scheme for the purpose of high-precision motion trajectory tracking.
References [1] K.J. Astrom, T. Hagglund, PID controllers: theory, design, and tuning, 2nd Edition, Instrument Society of America, 1995. [2] U. Shiro, K. Takuya, S. Takayuki, H. Hironobu, High performance positioning system with linear DC motor under self-tuning fuzzy control, Proc.of LDIA (1995) 295±298. [3] G. Otten et al., Linear motor motion control using a learning forward controller, IEEE/ASME Trans.on Mechatronics 2 (3) (1997) 161±170. [4] S.B. Chang, S.H. Wu, Y.C. Hu, Submicrometer overshoot control of rapid and precise positioning systems, J of the American Society for Precision Engineering 20 (1997) 161±170. [5] T. Sugie, T. Yoshikawa, General solution of robust tracking problem in two-degree-of-freedom control systems, IEEE Transaction on Automatic Control 31 (1986) 552±554. [6] S.A. Nasar, I. Boldea, Linear electric motors: theory, design and practical applications, Prentice-Hall, Englewood Clis, NJ, 1987. [7] K.K. Tan, T.H. Lee, S.Y. Lim, H.F. Dou, Learning enhanced motion control of permanent magnet linear
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71 motor, Proc. of the third IFAC International Workshop on Motion Control, Grenoble, France (1998) 397±402. [8] P. Dorato, C. Abdallah, V. Cerone, Linear-quadratic control: an introduction, Prentice Hall, Englewood Clis, NJ, 1995. [9] E.D. Sontag, Mathematical control theory: deterministic ®nite dimensional systems, Springer-Verlag, New York, 1990. [10] P.A. Ioannou, J. Sun, Robust adaptive control, PrenticeHall, Englewood Clis, NJ, 1995.
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[11] L. Ljung, System identi®cation theory for the user, Prentice-Hall, Englewood Clis, NJ, 1987. [12] J. Moody, C.J. Darken, Fast learning in networks of locally-tuned processing units, Neural Computation 1 (1989) 281±294. [13] K.K. Tan, T.H. Lee, S.N. Huang, S.Y. Lim, Adaptive control of DC Permanent Magnet linear motor for UltraPrecision Applications, International Conference on Mechatronic Technology, Taiwan, 1998, pp. 243±246
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Adaptive robust motion control for precise trajectory tracking applications K.K. Tan a,*, S.N. Huang a, H.F. Dou a, T.H. Lee a, S.J. Chin a, S.Y. Lim b a Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Automation Technology Division, Gintic Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore 638075, Singapore
b
Received 00 Month 0000; accepted 0 Month 0000
Abstract This paper presents a robust servo control method for high precision motion control using linear actuators. The controller consists of three components : a simple feedforward compensator, a PID feedback controller, and a RBF (radial-basis function) adaptive compensator, each to ful®ll a speci®c objective. The ®rst two control components can be directly tuned based on only an estimated dominant second-order linear model. The RBF compensator is self-tuning, and it will compensate for remaining uncertainties in the system, residual of the linear model. Rigid proofs are provided, guaranteeing the robust stability of the proposed controller. Experimental results con®rm the much superior performance of the 3-tier composite control over a standard motion controller. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Robust control; Adaptive control; Radial basis functions; Linear motors
1. Introduction In spite of the advances in mathematical control theory over the last 50 years, industrial servo control loops are still essentially based on the threeterm PID controller. The main reason is due to the widespread ®eld acceptance of this simple controller which has been eective and reliable in most situations if adequately tuned. More complex advanced controllers have fared less favourably under practical conditions, despite the higher costs associated with implementation and the higher demands in control tuning. It is very dicult for
* Corresponding author. Tel.: +65-874-2110; fax: +65-7791103. E-mail address: [email protected] (K.K. Tan).
operators unfamiliar with advanced control to adjust the control parameters. Given these uncertainties, there is little surprise that PID controllers continue to be manufactured by the hundred thousands yearly and still increasing [1]. On the other hand, increasing servo-controllers for mechanical systems, such as machine tools and robots, are required to achieve high accuracy at high speed on speci®ed trajectories. Modern silicon processes and assembly of micro electromechanical systems (MEMS) all require submicron machine accuracy and in the immediate years to come, nanometer and sub-nanometer accuracy is set to become a necessity. For obvious economic factors, high accuracy must be achieved without sacri®cing high production speed. Given stringent requirements, the dilemma is that the performance from conventional feedback control
0019-0578/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0019-0578(00)00037-9
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
is often not sucient, although the simple control structure is highly desirable. In this paper, we consider speci®cally mechanical servo systems which are based on permanent magnet linear motors (PMLM). Linear motors are becoming very popular for applications requiring high speed, high accuracy operations due to their mechanical simplicity. Even then, to achieve high accuracy positioning, extraneous forces resident in the system must also be taken into account in the control design [2±4]. These include: . frictional force which is highly nonlinear in nature, and in many cases, time varying; . force ripples arising from imperfections in the underlying components.
These issues are dicult to handle within a framework of conventional feedback control. The main objective of the paper is to address the abovementioned dilemma which is widely evident in many real industrial control problems. In this paper, we view a general nonlinear system as a predominantly linear system, with a nonlinear and uncertain deviation from the linear model. For high speed response, a full-state (position, velocity and acceleration) feedforward compensator is applied based on the nominal linear model. Clearly, the performance achievable with feedforward control depends on the adequacy of the model used. To increase the closed-loop bandwidth and concurrently minimise the eects of imperfect modelling, a PID feedback controller is included in the control structure. While the simplicity in a PID structure is appealing, it is also often proclaimed as the reason for poor control performance whenever it occurs. In this paper, we demonstrate how advanced optimum control theory can be applied to tune PID control gains. The PID feedback controller is designed using the Linear Quadratic Regulator (LQR) technique for optimal and robust performance of the nominal system. The proposed feedforward plus feedback con®guration may also be referred to as a twodegree-of-freedom control [5], and it should suce for most practical requirements. However, if further performance enhancement is necessary, the third control component may be enabled. A
Radial Basis Function (RBF) is applied to model the nonlinear remnant, and this is subsequently used to linearise the closed-loop system by neutralising the nonlinear portion of the system. Therefore, the proposed controller can compensate the eects of strong nonlnearities in the system. This diers from the previous results. An ecient tuning method for the 3-tier composite controller is proposed and developed in the paper to yield a good initial set of control parameters based on only a second-order linear model, and subsequent ®ne self-tuning will enhance the performance. The proposed control system is able to guarantee the boundness of the tracking errors even in the presence of strong and unforseen disturbances. Note that the theoretical analysis is not involved in Otten [3]. Both simulation and experimental results are provided to highlight the principles of the method and to demonstrate its applicability and eectiveness in true industrial applications. 2. Problem formulation In this section, the problem relating to highprecision motion control for linear motors will be formulated in a systematic manner. 2.1. Dominant linear model with uncertainty Similar to general voltage controllable dc motors, the mechanical and electrical dynamics of a PMLM can be expressed as follows: J D Tl Tm
1
: dIa Ra Ia u KE La dt
2
T m K T Ia
3
where denotes position; J, D, Tm , Tl denote the mechanical parameters: inertia, viscosity constant, generated force and load force respectively; u, Ia , Ra , La denote the electrical parameters: input dc voltage, armature current, armature resistance and armature inductance respectively; KT denotes the
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
electrical-mechanical energy conversion constant. Because the electrical time constant RLaa is small (compared to the mechanical time constant DJ ), electrical transients decay very rapidly and a La dL dt 0 (see [6], Chapter 5). Thus, the following simpli®ed equation is given: K1 : K2 u J J
1 Tt J
4
where K1
KE KT Ra D KT ; K2 Ra Ra
5
Clearly, this is a second-order dynamical model. The model does not include extraneous nonlinear eects which may be present in the physical structure. When control performance requirements are more stringent, the above model alone may thus become inadequate. A prominent nonlinear eect associated with PMLM is due to frictional force and it may be written as [7]: : : : : = s ffric fc fs fc e fv sgn ; 6
K2 : f ; J
fripple 8:5sin 900
7
: Thus, a nonlinear function f1 ; may be used to model these nonlinear dynamical eects approximately. The servo system now becomes: Let
: K1 : K2 1 u Tl f1 ; J J J
8
9
It follows that K1 : K2 K2 : u f ; : J J J
10
: We will assume that f ; is a smooth nonlinear function which may be unknown. To this end, it may be pointed out that many discontinuous nonlinear functions may be adequately approximated by a continuous one. Thus, the model represents a large and rich class of nonlinear servo-systems. 2.2. Tracking problem The control objective can be described as follows: given a desired trajectory, the primary goal is to track a desired trajectory d as close as possible, while ensuring the states and control remain bounded for stability. In other words, the tracking error e de®ned as: e d
where fs is the level of static friction, : fc is the minimum level of Coulomb friction, s and fv are lubricant and load parameters which may be determined by empirical experiments. is an additional empirical parameter. In addition, one of the known disturbance generated in linear motors is the force ripple due to cogging and reluctance forces present in its structure. In [7], the force ripple is described as a periodic sinusoidal type signal:
: 1 TL f1 ; : J
59
11
should be as small as possible. Based on this de®nition, we may express (10) as: e
K1 : K2 K2 : e u f ; J J J K2 J K1 : d d J K2 K2
Since d t e tdt e dt 0
12
13
we t assign the system state : variables as x1 0 e tdt, x2 e and x3 e such that x x1 x2 x3 T . (12) can then be put into the equivalent state space form: : J K1 : : x Ax Bu Bf ; B d d K2 K2 14
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
2
0 1 A 40 0 0 0
3
0 1 5; K1 =J
2
3 0 B4 0 5 K2 =J
15
3. Controller Structure In this paper, we adopt a 3-tier composite control structure, as shown in Fig. 1, comprising of the following components. 3.1. Feedforward control
PID controller is appealing to practitioners, yet it is often proclaimed as the cause whenever poor control performance occurs. In this section, we will illustrate, how even with a simple control structure, the PID controller may be tuned to achieve the same level of optimum performance as advanced LQR control. The nominal portion of the system (without uncertainty) is given by: : x t Ax t Bu t; 17 where upid Kx kx1 kd1 x2 kd2 x3 :
The design of the feedforward control law is straightforward. From (14), since we :would like to 1 neutralise the term of B KJ2 d K K2 d with feedforward control, we design the feedforward control as: J K1 : d d 16 uff t K2 K2 Clearly, the reference position trajectory must be continuous and twice dierentiable, otherwise a pre compensator to ®lter the reference signal will be necessary. The only parameters required for the design of the feedforward control are the parameters of the second-order linear model. 3.2. PID feedback control The feedback control component utilises the PID control structure. The simple structure of the
18
This is a PID control structure which utilises a full-state feedback. We would like to obtain the optimal control parameters using the Linear Quadratic Regulator (LQR) technique that is well known in modern optimal control theory and has been widely used in many applications [8]. It has a very nice robustness property, i.e. if the process is of singleinput and single-output, then the control system has at least a phase margin of 60 degree and a gain margin of in®nity. This attractive property appeals to the practitioners. Thus, the LQR theory has received considerable attention since 1950s. The LQR problem requires a mimisation of an index of control accuracy of the form 1 J xT Qx lu2 19 0
Fig. 1. The block diagram of the control scheme.
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
where Q50 and l > 0. It is well-known that the LQR control with (19) is u
l 1 BT Px t
20
where P is the positive de®nite solution of the Riccati equation AT P PA
l 1 PBBT P Q 0
21
and Q HT H I. One practically useful feature associated with LQR design is that under mild assumptions, the resultant closed-loop system is always stable. We may summarise this feature in the following theorem. Theorem 1. (see [9], Theorem 31). For the system (17), if A; B is controllable and H; A is observable, then the control law given in (20) stabilizes the system (17). In this paper, the PID control uses a slightlymodi®ed version of the LQR controller, that is upid
r0 1BT Px
22
where l 1 and r0 is independent of P and it is introduced to weigh the relative importance between control eort and control errors. Note for this proposed feedback control, the only parameters required are the parameters of the second-order model and a user-speci®ed error weight r0 .
where vect
h : iT .
61
Assumption 1. The ideal weights are bounded by known positive values so that jwi j4wM ; i 1; 2; . . . m. 2. There exists an M > 0 such that Assumption : ; 4M , 8vect 2 I on a compact region I R2 . : Let the RBF functional estimates for f ; be given as: m : X : f^ ; i ; w^ i
25
i0
where w^ i are estimates of the ideal RBF weights that are provided by the following weight-tuning algorithm : w^ i r1 xT PB i
r2 w^ i
26
where r1 ; r2 > 0, and P is the solution of (21). Thus, we have : 27 ucomp f^ ; In adaptive law (26), -modi®cation [10] is used to improve the robustness of the adaptive controller in the presence of the system uncertainty and the external disturbances.
3.3. RBF adaptive control
4. Parameter estimation of nominal system
Next, we consider the design of the adaptive compensator :forthe nonlinear part of the system (10). Since f ; is a nonlinear smooth function (unknown), it can be represented as:
Under the proposed control structure, a secondorder model is necessary for computing the control parameters. Many approaches are available to yield the required linear model [11]. In this paper, we will use the parameter estimation approach [11]. Consider an ARX model given by:
m : X : f ; i ; wi i0
23
: with jj4M , where i ; is the RBF, that is : i ;
! m 24
vect cj 2 kvect ci k2 X exp = exp 2 2 2i 2j j0
y t a1 y t 1 . . . ana y t na b1 u t 1 . . . bnb u t nb e t
28
where na is the number of poles, nb 1 is the number of zeros, and e t represents the error term of the system. Based on this model, the linear predictor is given by:
62
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
y^ t= T t e t
29
where t
y t y t
1 na u t
y t
2 . . . 1 . . . u t
T c a1 . . . ana b1 . . . bnb
31
With (29), the prediction error becomes: T nb ;
30
" t; y t
T t :
Fig. 2. Comparison between linear model (dashed line) and actual nonlinear system (solid line).
Fig. 3. Desired position and velocity trajectories.
32
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
De®ne the model ®tting criteria function as: VN
N 1X y t N t1
T t
2
;
33
which is the least-square criterion. The criterion can be minimized analytically, giving the least squares estimates of the model parameters as: "
^ LS N
N 1X T t t N t1
#
1
N 1X ty t N t1
34
The persistently exciting input signals should be used. This may come in the form of explicit input sequences (e.g. PRBS) or it may arise naturally from control signal generated in the closed-loop, in which case no explicit experiment needs be conducted The parameter estimation described above, using explicit user-de®ned input signals, will yield
63
an initial set of parameters for the linear model. Thereafter, the model may be re®ned using incremental measurements of the input and output signals of the system under closed-loop control with a recursive version of the least square estimation algorithm [11], i.e. the re®nement may occur online with the motor under normal motion operations. 5. Stability analysis of proposed controller It is required to demonstrate that the state x and weights will remain bounded. The following de®nition will be used to illustrate the stability. Theorem 2. Assume that the desired references yd , : and yd are bounded. Consider the case where the controller given by (16),(18) and (27) is applied to the system (14). Then the state and the RBF estimation errors are uniformly ultimately bounded
Fig. 4. Control results: (a) proposed controller; (b) pure PID controller; (c) PID+feedforward controller.
64
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
Proof. The dierential Eq. (14) can be written as [upon applying the control (16),(18) and (27)]: : : x Ax B uff upid ucomp Bf x; x; x J K1 : d d 35 B K2 K2 m X A r0 1BBT P x B i w^ i " B
m X i wi i0
A
T
BB P
i0
#
36
" # m X r0 BB P x B i w~ i T
i0
37
Now consider the following Lyapunov function candidate: m 1X V x; w~ xT Px w~ 2 38 r1 i0
Taking the time derivative of v along the solution of (37), it can be shown that : V xT AT P PA PBBT P x xT 2ro m m X : 2X w~ i w~ i 2xT PB i w~ i 2xT PB r 1 i0 i0 4
PBBT P x
m X lmin Q 2r0 1PBBT P kxk2 2xT PB i w~ i m X
i0
r2 2x PB 2 x PB i w^ i w~ i r1 i0 T lmin Q 2r0 1PBB P kxk2 2xT PB m r2 X 2 w^ i w~ i r1 i0 lmin Q 2r0 1PBBT P kxk2 m m r2 X r2 X 2xT PB 2 w~ 2i 2 wi w~ i r1 i0 r1 i0 T
From
Fig. 5. Input±output signals used in identi®cation experiment.
T
39
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
1 2xT PB4xT PBBT Px 2 1 4xT PBBT Px 2M
40
r2 r2 2 r2 2 2 wi w~ i 4 w w~ r1 r1 i r1 i
41
we have n : V4 lmin Q 2r0 1PBBT P m o r2 X T 2 lmax PBB P kxk 2 1 r1 i0
where
1 w~ 2i 2
1 r2 2 2M w r1 i where , are small positive T constants. Let xT w~ 0 w~ 1 . . . w~ m and we have : V4
2 kk2 l1
65
42
43
n 1
min lmin Q 2r0 1PBBT P 2 r2 1 o T lmax PBB P ; 2 1 2 r1 1 r2 2 l1 2M w r1 i
44 45
Clearly, > 0 for suciently small , . In order to show the boundedness of the state and weights, note from kk2 4V4v kk2 46 where min lmin P; 1=r1 , v max lmax P; 1=r1 g, that :
47 V 4 2 V l1 v Hence, following [13], we have
Fig. 6. Comparison between estimated model (solid line) and linear motor (dashed line).
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
l1 v V 0 V t4 2
l1 v e 2
2 t v
48
From (48), we can conclude that the state and weights are bounded. This completes the proof. &
Remark 3. For the RBF basis, there are two ways to choose the centers and variances: the local training techniques (see [12]); the centers evenly spread along the desired trajectories with i=const(15) (see our simulation).
Remark 1. For the adaptive scheme (26), the ®rst term grows rapidly as xT PB i increases (which re¯ects the ``poor'' system performance). This will result in a high increase rate of w^ i and, therefore, strong feedback. The parameters r1 , r2 oer a design trade-o between the relative eventual magnitudes of kk and w~ i
6. Simulation
Remark 2. If the load TL is available, we can incorporate it into the feedforward control, that is
: u v
uff t
J K1 : TL d d K2 K2 K2
49
The linear motor model considered here for simulation is a brushed permanent magnet DC linear motor manufactured by Anorad Corp. The nonlinear model is given according to [13] as: : xv 50 ffriction m
fripple
w t
51
where ffriction is the friction force, fripple is the ripple force, u is the developed force, m is the combined
Fig. 7. Desired trajectories (I).
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
67
mass of translator and load, and w represents any other residual disturbances. The friction and ripple forces are modeled as follows: : : : : ffriction fc fs fc e x=xs fv x sgn x 52
the ®nal position. In this simulation study, we choose x0 0, xf 0:5m, tf 1s. Fig. 3 shows the desired trajectories. To apply the proposed controller, the plant is ®rst identi®ed using the standard least squares identi®cation algorithm as:
fripple b1 sin w0 x
x
53
The above model allows us to evaluate the friction force during both sticking and slipping motions. The model parameters can be found in [13]. The desired trajectories are planned as follows: xd x0 x0 xf 15 4 6 5 10 3 54 vd x0
xf 60 3
30 4
30 2
55
where the gain of 106 is used to normalize the units of the system to 1 mm, xd is the desired position trajectory, vd is the desired velocity trajectory, t= tf t0 , x0 is the initial position and xf is
: 117:6471x 8:44987u
56
where x represents the position. But the approximation by this model is not ideal as shown in Fig. 2 where the dynamics due to the friction and ripple forces are not represented in the model. The PID control is obtained by using (18), giving: t upid t r0 1 56:5463e t 31:6228 e d 0 : 20:8223e t 57 for given Q0 103 I, where I is the unit matrix. According to Theorem 2, we obtain the RBF compensator of the controller, where we choose
Fig. 8. Control performance from proposed controller with r0 2:5.
68
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
m 19, and the centers ci of the RBF basis are [{0.00430.01} 0.120.01], [0.02900.01 0.380.01], [0.08150.01 0.660.01], [0.15870.01 0.860.01], [0.2500.01 0.930.01], [0.34130.01 0.860.01], [0.41850.01 0.660.01], [0.47100.01 0.380.01], [0.49570.01 0.120.01], [0.50.01 0.000.01]. These are chosen as the mesh nodes along the desired position and velocity trajectories, respectively. We also choose 5 106 , r0 3, r1 10, r2 1. Fig. 4(a) shows the control results. The simulation results for the pure PID control and the PID+feedforward control are also shown in Fig. 4(b) and (c) respectively, where it is clear that the proposed controller has achieved a much improved performance. 7. Experiment In this section, experimental results are provided to illustrate the eectiveness of the proposed method. The linear motor used is a direct thrust
tubular servo motor manufactured by Linear Drives Ltd (LDL)(LD 3810), which has a travel length of 500 mm and it is equipped with a Renishaw optical encoder with an eective resolution of 1 mm. The dSPACE control development and rapid prototyping system based on DS1102 board is used. dSPACE integrated the entire development cycle seamlessly into a single environment, so that individual development stages between simulation and test can be run and rerun, without any frequent re-adjustment. To apply the proposed controller, the system (from control voltage to velocity) is ®rst identi®ed. The input-output signals from the identi®cation experiment are as shown in Fig. 5. Cross validation of the estimated model and actual linear motor is given in Fig. 6. The dominant linear model is computed to be: : x 24:1066x 1:26992 107 u: 58 The feedforward control is thus designed accordingly as:
Fig. 9. Control performance from PID control.
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
uff t
: 1 24:1066 d d 1:26992 107 1:26992 107
59
Based on the identi®ed model, the LQR technique is used to obtain the feedback control parameters. The solution to (21) is: 2
2:3218 P4 0:0029 1:7605 10
6
0:0029 1:1722 10 8:1781 10
5 9
1:7605 10 8:1781 10 1:0073 10
6 9
3 5
11
60
for 2
1 Q 5004 0 0
0 10 0
5
3 0 0 5 10 11
61
Thus, we have the PID control upid
r0 122:3565 0:1038 0:0001T x t
62
69
The centres of the RBF function are chosen with regularly spaced mesh as the simulation example. Note that the inputs of RBF should be normulized. The other parameters are chosen as 1, r1 0:001, r2 0:1. The sampling time is 0:0005 s. The desired trajectories for the position and velocity are shown in Fig. 7. Fig. 8 shows the performance of the proposed controller. The maximum error is about 7 mm. If a PID is used, the result is shown in Fig. 9. The maximum error for the PID is about 12.5 mm. It is clearly observed from the above ®gures that the tracking error is signi®cantly improved. To further improve the tracking error, we can tune the value of r0 with the other parameters kept constant. With r0 6:5 and the desired trajectories shown in Fig. 10, the control results are shown in Fig. 11. Notice that the nonlinearities such as cogging and reluctance forces inevitably present in linear motor are eliminated by the RBF compensator. Since the resolution of the encoder
Fig. 10. Desired trajectories (II).
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K.K. Tan et al. / ISA Transactions 40 (2001) 57±71
Fig. 11. Control performance from proposed controller with r0 6:5 and other parameters to be the same as the above experiment.
used is 1 mm, it is clear from the ®gure that a positioning accuracy of 4 mm is achieved. As a matter of fact, a higher positioning accuracy can be achieved if special high precision encoders are used. 8. Conclusions This paper has considered the development of a new 3-tier composite feedforward+feedback+ adaptive controller for a class of linear motor systems. The second-order model is used as the basic foundation for the design of the feedforwardfeedback design, and an adaptive component designed based on a RBF provides for the further possibility of performance enhancement. Detailed stability analysis is provided. Simulation and realtime experiment further verify the eectiveness of the proposed scheme for the purpose of high-precision motion trajectory tracking.
References [1] K.J. Astrom, T. Hagglund, PID controllers: theory, design, and tuning, 2nd Edition, Instrument Society of America, 1995. [2] U. Shiro, K. Takuya, S. Takayuki, H. Hironobu, High performance positioning system with linear DC motor under self-tuning fuzzy control, Proc.of LDIA (1995) 295±298. [3] G. Otten et al., Linear motor motion control using a learning forward controller, IEEE/ASME Trans.on Mechatronics 2 (3) (1997) 161±170. [4] S.B. Chang, S.H. Wu, Y.C. Hu, Submicrometer overshoot control of rapid and precise positioning systems, J of the American Society for Precision Engineering 20 (1997) 161±170. [5] T. Sugie, T. Yoshikawa, General solution of robust tracking problem in two-degree-of-freedom control systems, IEEE Transaction on Automatic Control 31 (1986) 552±554. [6] S.A. Nasar, I. Boldea, Linear electric motors: theory, design and practical applications, Prentice-Hall, Englewood Clis, NJ, 1987. [7] K.K. Tan, T.H. Lee, S.Y. Lim, H.F. Dou, Learning enhanced motion control of permanent magnet linear
K.K. Tan et al. / ISA Transactions 40 (2001) 57±71 motor, Proc. of the third IFAC International Workshop on Motion Control, Grenoble, France (1998) 397±402. [8] P. Dorato, C. Abdallah, V. Cerone, Linear-quadratic control: an introduction, Prentice Hall, Englewood Clis, NJ, 1995. [9] E.D. Sontag, Mathematical control theory: deterministic ®nite dimensional systems, Springer-Verlag, New York, 1990. [10] P.A. Ioannou, J. Sun, Robust adaptive control, PrenticeHall, Englewood Clis, NJ, 1995.
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[11] L. Ljung, System identi®cation theory for the user, Prentice-Hall, Englewood Clis, NJ, 1987. [12] J. Moody, C.J. Darken, Fast learning in networks of locally-tuned processing units, Neural Computation 1 (1989) 281±294. [13] K.K. Tan, T.H. Lee, S.N. Huang, S.Y. Lim, Adaptive control of DC Permanent Magnet linear motor for UltraPrecision Applications, International Conference on Mechatronic Technology, Taiwan, 1998, pp. 243±246