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A high speed tracking system using adaptive model-following control
Mechatronics Vol. 5, No. 8, pp. 857-872, 1995
© 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved. 0957-4158/95 $9.50+0.00
Pergamon 0957-4158 (95) 00055-0
A HIGH SPEED TRACKING SYSTEM USING ADAPTIVE MODEL-FOLLOWING CONTROL J. H. LEE,* K. H. PARK, t S. H. KIM* and Y. K. KWAK* *Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Kusong-dong, Yusong-gu, Taejon 305-701, Korea and tDepartment of Mechatronics, Kwangju Institute of Science and Technology (K-JIST), 572 Sangam-dong, Kwangsan-ku, Kwangju 506-303, Korea
(Received 14 January 1995; revised and accepted 20 June 1995) Abstract--In the time optimal control problem, it is important to select a switching point accurately in order to improve tracking speed performance. However, the accurate selection of the switching point is not an easy task because of the damping coefficient variations and modeling uncertainties in actual systems. To relieve the difficulty and inconvenience of this task, adaptive model-following control (AMFC) is implemented. AMFC can make the controlled plant follow as closely as possible a desired reference model whose switching point can be calculated easily and accurately, assuring that the error between the reference model and the real system approaches zero. The AMFC is applied for a tracking actuator of a magneto-optical disk drive (MODD) to improve its slow tracking performance. According to the simulation and experimental results, an average tracking time as small as 20 ms is obtained for a 3.5 in. magneto-optical disk drive.
1. INTRODUCTION Time optimal control is essential in engineering applications where mechanisms must operate repetitively at high speed and with great precision. The time optimal problems can be approached in a n u m b e r of ways using theoretical and experimental approaches [1, 2]. Bang-bang control is the theoretical solution for time optimal control. For a high precision tracking system, a high precision position controller is also necessary. For smooth connection of these control modes, a velocity controller can be used. The advantages of each control m o d e are combined to establish the desired p e r f o r m a n c e by switching the system from one control m o d e to another. H e r e , bang-bang control plays the most important role of determining the tracking time. Bang-bang control is alway an extremal control and involves N - 1 drive signal polarity reversals for a system with an N t h order state equation [3]. The extremal control for the system with a second order system, which describes most real mechanical systems, has one polarity reversal from +Vs to - V s at a switching point, x~, where the m a x i m u m acceleration changes to the m a x i m u m deceleration. For the best p e r f o r m a n c e of bang-bang control, one needs to correctly calculate the switching point. The switching point is calculated based on the system model. Since bang-bang control, however, is open-loop, its p e r f o r m a n c e characteristics are dependent on the system components and the plant characteristics. This dependency m a k e it hard to determine the switching point accurately when system model uncertainties 857
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exist. The inaccuracy of the switching point increases in a high frequency operating condition, i.e. high bang-bang input frequency. To compensate for the system uncertainties, closed-loop velocity control is usually used together with bang-bang control [4, 5]. In velocity control, when a target track is determined, a velocity trajectory is generated and stored in a look-up table in advance because it take time to calculate the velocity trajectory in a real time control process. The velocity control can increase the system bandwidth compared to bang-bang control, especially if the system is hard to model, and the switching point is hard to predict, However, substantial efforts are still required to anticipate the switching point correctly because it cannot be expected that the extreme inputs +V~ and -V~ have equal magnitudes in practical cases and hence the switching point x~ is not easily calculable. In an experimental approach to the time optimal control problem, the switching points in bang-bang control can be obtained by applying +1/~ and -V~ to the real system and measuring the phase plane trajectories for various travel lengths. This collection of data then is mapped directly to the best choices for the available free parameters used for generating the trajectory in a single step process [6]. This method, however, requires several iterations of experiments to increase the reliability of the switching point. Therefore, the above methods have the difficulty and inconvience in theoretically and experimentally selecting the switching point. The approach taken to relieve these problems in this work is to make the controlled plant follow, as closely as possible, a desired reference model whose switching point can be calculated easily and accurately. We build an adaptive loop to assure that the controlled plant always behaves as specified by means of a reference model that produces the desired output for a given input. This control method is called adaptive model-following control (AMFC). If the reference model is designed so that it has a high system bandwidth, the tracking time can be reduced if we can assure that the error between the reference model and the real system approaches zero. Therefore, it can be said that the AMFC scheme is a good alternative for bang-bang control and velocity control. Position control is switched after the AMFC scheme's task is completed. The AMFC scheme is applied to a magneto-optical disk drive to improve its slow tracking performance which is the significant disadvantage of the current magnetooptical systems.
2. SWITCHING POINT IN BANG-BANG C O N T R O L
The system to be analyzed is shown in Fig. 1 and described in [7]. To reduce the average accessing time of a magneto-optical disk drive, a newly developed moving magnet type tracking actuator is implemented. It consists of an air core solenoid, four permanent magnets connected on glass rods, a focusing actuator which is mounted on the tracking actuator, lens assembly, and linear ball bearings. The tracking actuator moves horizontally on the ball bearings by using the interacting force generated between the solenoid and permanent magnets. The assembly of the tracking actuator can be treated as a rigid body whose mass and viscous friction coefficient are M and c. Since the mass center of the tracking actuator is near the centerline of the two
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Permanent Permanent magnet magnetholder Focusing actuator holder
Linear ball bearing Base
f Laser t~eam~ path
- Glass rod Air-core solenoid
Fig. 1. Configurationof the developed linear tracking actuator. glass rods, the driving force acts on the mass center of the total system, and it is assumed to be proportional to the current in the conductor. The moving mass of the tracking actuator can be measured by a scale and it is around 0.025 kg. However, the damping coefficient is hard to measure, and is subject to variations due to temperature change, moving direction, and location during stroke. Let us investigate how the switching point changes according to the difference in the damping coefficient.
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The block diagram of the tracking system is represented in Fig. 2. Here r, I, F and xl indicate the input signal, the current, the force generated from the actuator, and the position of the actuator, respectively. K,, and K, are the current amplifier gain and the force constant measured as 0.35 A/V and 1 N/A, respectively for the system tested. Therefore, the open loop transfer function of the velocity control system, G,, is theoretically G,-
xl r
K~KI
0.35
-
s ( M s + c)
.
(1)
s ( M s + c)
The dynamic equation can be represented in terms of state variables: 2i = x2 c
.G -
x~ +
M
(2) 1 --u,
(3)
M
where Xl and x2 respectively are the position and the velocity of the tracking actuator. The control input u is defined as Ku K t r . Equations (2) and (3) are conveniently shown on a phase plane with coordinates xl and x2. Figure 3 shows how the switching points are determined by the graphical representation in the phase plane ( & - x 2 ) when the tracking distance is 6 mm, and u = 1.5 V. From the figure we can see that the switching points change for the different damping coefficients. If the damping coefficient, c, is zero, the switching point occurs exactly at the middle point of the target position. As the damping
s(Ms+c) Fig. 2. Block diagram representation of the tracking system.
c-90M c=60M c=30M c=15M c=3M xl(m)
/ '/ /2":
0.006 0.004 .
0.002
c=30M
0 . ,
~ 0
c=I5M c=3M 0.2
0.4
0.6
0.8
1 x 2 (m/s)
Fig. 3. Variation of the switching point for the different damping coefficients.
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coefficient increases, the switching point tends to shift upward to the target point. This implies that a slight change of the damping coefficient or mass in the tracking system leads to the incorrect estimation of the switching point. In addition to the change of the damping coefficient, the theoretical model is subject to change because of the modeling uncertainties. Therefore, we have a limitation in achieving a fast tracking time using bang-bang control. The approach taken to relieve problems such as the change of the damping coefficient and modeling uncertainties is to make the controlled plant follow, as closely as possible, a desired reference model whose switching point can be calculated easily and accurately. In other words, if the distance to move a target track from a current position is determined, we can generate a triangular velocity profile, for example, which is obtained by integrating the bang-bang signal whose switching point is in the middle of the distance. This relation can be understood in terms of Fig. 4. Since the maximum acceleration, a,~, is fixed due to the current constraint, the switching point, ts, can be easily obtained by a simple physical law. Therefore, we can say the switching point is dependent only on the moving distance. The velocity profile obtained from the procedure is used for the reference input. If the adaptation mechanism works properly, the adapted plant will follow the reference input well, the triangular velocity profile.
3. AMFC SYSTEM DESIGN FOR A T R A C K I N G SYSTEM
The adaptive model-following control scheme is well established in theory [8-12]. In this section, we shall briefly review the AMFC design. If we take the velocity and acceleration as the state variables, the real plant to be controlled is described by = Aey
+
Bpttp,
(4)
where A p and Bp are the real plant matrices, and y = [Yl Y2] r. Yl and Y2 are the tracking actuator velocity and the acceleration of the real plant, and Up is the input voltage to the plant. Similarly, the reference model is described by Xm --- AmXm + Braun, where
Am
and
B m are
(5)
the model plant matrices, and Xm = [x,,1 Xm2]r. Xml and Xm2
acceleration
velocity
1.!_ s
I
Time, t
Fig. 4. Proceduresfor generation of the velocitytrajectories by integration of bang-bang signal.
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are the desired tracking actuator velocity and the acceleration of the reference model, and Um is the input voltage to the reference model. We consider a parallel AMFC configuration with signal-synthesis adaptation as shown in Fig. 5. If we define (6)
u 1, = u~l + up2,
where Up1 is the linear control signal and Up2 is the adaptation signal, we can then choose g/pl
=
- K p y + KuUm
(7)
up2 = AKp(e, t)y + AK.(e, t)Um,
(8)
where Kp and K . are linear control gains, and AKp and A K . are adaptation control gains having matrices of appropriate dimensions. Recall that the error e between the states of the model and those of the plant is X m - y- Then, using Eqns (7) and (8), we obtain the following vector differential equation: = Ame + [A,,, - At, + BpKp - BpAKp(e, t)]x + [B,,, - BpK, - BpAK,(e, t)]Um.
(9)
To have perfect model following, we must assure that for any Urn, piecewise continuous, and initial state e0 = 0 we shall have e(t) = e(t) = 0. We conclude that this can be achieved if the last two terms in the right hand side of Eqn (9) equal zero. In addition to this, the remaining unforced system t~ = Ame
(10)
must be asymptotically stable, which implies that the matrix A m must be a Hurwitz matrix. In short, we can summarize that the objective of the adaptation mechanism is to generate the two time-varying matrices AKp(e, t) and A K , ( e , t) to assure that the generalized state error, e, goes to zero.
_1 Reference
tim
=
+
Plant
I Adaptation Kp(e,t)) 1~ Mechanism ( Ku(e,t)
Fig. 5. A parallel AMFC configuration with signal-synthesisadaptation.
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For this purpose, we use the hyperstability and positivity theory which says that a feedback system is asymptotically hyperstable if two conditions are satisfied [8]: the equivalent feedforward block is a strictly positive real transfer function or matrix, and the feedback block must satisfy the Popov integral inequality. To transform the system into the form of an equivalent feedback system composed of two blocks, one in the feedforward path and one in the feedback path, Eqn (9) is rewritten by the following equations: = (Am)e + W
(11)
v = De,
(12)
where W represents the last two terms of Eqn (9). The matrix D is especially chosen in order to be able to meet the specification required for the linear part so that the stability of the system is assured, v is the input of the feedback block. Let us first examine the adaptation mechanism in the feedforward block. The feedforward transfer function, H(s), represented as H(s) = D(sI - A m ) -1,
(13)
should be a strictly positive real transfer matrix, or there must exist a P satisfying the Lyapunov equation such that (Am)rP + P(Am) = - Q ,
(14)
D = P,
(15)
in which Q is a positive definite matrix. Next, for the adaptation mechanism in the feedback block, we use one of the solutions of the Popov integral inequality introduced by Landau [8]. The adaptation control gains, AKp and AK,, are described using an integral plus proportional adaptation law as t
AKp(e, t)
= fo I~)1(¥' t, r ) d r + q~2(v, t) + AKp(e, 0)
(16)
t
AK.(e, t) = f ~Pl(V, t, r) + apz(V, t) + AK.(e, 0), Jo
(17)
where q,l(V, t, ~) =
Vo(t
-
~)v(r)[~;a(t)x(~)] T,
r
¢
q52(V, t) = Fa(t)v(t)[Ga(t)x(t)]
T
!
~P2(v, t) = Fb(t)v(t)[Gb(t)Um(t)]
(18)
(19)
~Pl(V, t, r) = F b ( t - r)v(~)[Gb(t)um(~)] r, P
~ >1 t
T
,
r >-t
(20)
(21)
where F a ( t - r) and F b ( t - r) are positive definite matrix kernels whose Laplace transforms are positive real transfer matrices with a pole at s = 0. Ga and Gb are positive definite constant matrices; F'a(t), F~,(t), G'a(t) and G~(t) are time-varying positive definite matrices for all t t> 0. The equivalent feedback system composed of two blocks, one in the feedforward path and one in the feedback path, is illustrated in Fig. 6.
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Linear Parl r-
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Fig. 6. Equivalent feedback representation of adaptive model-followingcontrol.
4. APPLICATION TO THE MAGNETO-OPTICAL DISK DRIVE To begin constructing the A M F C system, we first need to know the real plant matrices Ap, Up and model plant matrices A .... B m as described in Eqns (4) and (5). A frequency response test for the tracking actuator velocity versus input signal is experimentally performed to obtain dynamic characteristics of the tracking system and is shown in Fig. 7. The experimentally obtained amplitude ratio and phase angle are denoted by the legend 'o'. The open loop transfer function of the experimentally obtained real system can be most closely fitted to a second order system and is denoted by the solid line: Gp =
t
14
•
(22)
0.0005s -~ + s + 40 Then, from Eqn (22) F
t)
A, = •] - 8 0 , 0 0 0
-21000 j '
Bp = [0 228,000].
(23)
Now that we have obtained the real tracking system model, it remains to build an adaptive control loop to assure that the plant behaves as specified by means of a reference model that produces a desired output. In choosing the reference model, three facts are considered. Firstly, the reference model should be simply described so that the switching point is determined easily and accurately. Secondly, there must be a trade-off between the model-following capability and control effort. If the reference model is the same as the real system model, no control effort is needed. However, as
Tracking system using adaptive model-following control
II IIIII
I I
1
IIIIll
]~[tlt[
865
I EIE II
IIIIIII ILIL Ill I Illl tll Jl LIIh II I Iqllh LII
~-OO
0.1
10
1
100
10OO
Frequency (Hz)
0 -loo
I[I I II I I [ [ ~ l I III I Ill II LILl 11111 T+I
.~ ~ -200
~. -3oo
0.1
1
10
100
10OO
Frequency (Hz) Fig. 7. Frequency response test for va vs r.
the real system's requirement of the model-following capability increases, the control effort also increases. Finally, the highest order of the characteristic equation of the reference model should be the same as that of the plant model for an easy construction of the adaptation mechanism. With the above considerations, we define the reference model Gr, though it is not unique, as 114
Gr =
•
(24)
0.0001s 2 + s + 1 The reference model Gr can be treated almost as a integrator on the time scale of interest because of the small coefficient of s z. If a bang-bang input is applied to the G~, a trapezoidal velocity profile is generated, which means an easy selection of the switching point in the middle of the tracking distance. Then from Eqn (24) Am
=
E0
-10,000
-10,000
'
B m = [0
1,140,000].
(25)
The reference model described by the above transfer function is constructed in the digital computer. Here, (Ap, Be) and (Am, B m ) a r e stabilizable, and A m is a Hurwitz matrix, i.e. the reference model is asymptotically stable. The next step is to follow the procedures which are presented in section 3. The detailed configuration is given in Fig. 8. All block diagrams are constructed in SIMULINK, which is a commercial system dynamics and control program that includes M A T L A B [13]. To simulate real system environments as closely as possible, a maximum current constraint for the solenoid is imposed. The maximum and minimum currents are set to +4 A and - 4 A, respectively. Since the sampling time currently used for the servo
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v'y2
'alpha
Fig. g. Block diagram of the AMFC control scheme.
control system of a magneto-optical disk drive is less than 0.1 ms [5], digital simulation uses 0.1 ms sampling time. The positive definite matrix Q is selected so that H(s) has a strictly positive real transfer function. When Q is Q =
0
0.1
"
(26)
P and D are obtained by solving Eqns (14) and (15), respectively. In one of the solutions of the Popov integral inequality introduced in section 3, we assume that AKp(e, 0) = AK,,(e, 0) = 0, and we choose V',,(t) = F',, > 0.
F,(t) = V,, > 0.
,
G,,(t) = Go(t) --
~ _
0
0
a'2 l '
a~ ;> ()"
Fh(t) = Fb, 0;2 ~ O,
F;(t)
--
r'h
G~(t) = G'h(t) = 1.
(27)
Several simulation results give a few design guidelines; all linear control gains, K,, and some freedom because they have little influence on the tracking performance. Since ct,2 and F',, are associated with the tracking acceleration they should be kept small, o(1, oQ, Fa, F'a, K,, and Kp are chosen as 1, 0.01, 1, 0.1 1, and [1 1] respectively.
Kp, have
The simulation result shown in Fig. 9 corresponds to trajectory tracking scheduled to go 6 mm, which is chosen because it is the average tracking distance of a 3.5 in. disk drive. Figure 9 shows the reference velocity and plant velocity; and control input and applied white noise. There are almost no tracking errors observed in the velocity trajectory.
Tracking system using adaptive model-following control Velocity (m/s) 0.8
0 0
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0.005
0.01
867
~:~:~'~ 0.015
0.02
0.025 Time (s)
Control input & sensor noise(V)
1!
Time (s) Fig. 9. The simulation result.
5. EXPERIMENT
5.1. Tracking velocity profile in a random access device In a search for an adequate velocity profile, it may be found that most positioners have velocity versus time curves which may be closely represented by triangles or trapezoids. According to Hertrich [14], the average tracking time, tav, can be represented by using the speed factor tav = Ta~/(L/a),
(28)
where L and a are a maximum moving distance in the data storage and acceleration. ra is a dimensionless time constant and is expressed as r~
_
1
3
15~/fl(/3_(1 + 3),) - 5/32(1 + 2),) + 15/3(1 + ),) + 5).
(29)
Here, /3 is defined as the ratio of the largest distance traveled in the triangular velocity mode to the maximum distance, L. ), is introduced to modify the motion times in accordance with overshoots and sluggishness encountered in the actual system. In the practical case now under consideration, the maximum distance L as well as the acceleration a will be constant. Hence, the average tracking time, according to Eqn (28), becomes strictly proportional to the time constant, to. In this work, we choose ), = 0.3, /3 = 0.4 since r a turns out to be almost unaffected by a variation of/3 in the range of 0.4 <~/3 ~< 1. With these values, r o is around 1.22. When the maximum distance, L, in a 3.5in. magneto-optical disk is 1 8 m m , the largest distance, b, traveled in the triangular mode is calculated as 6.4 mm. Since the average tracking
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distance is around 6 mm, the triangular velocity mode should be applied to find out the average tracking time. If the real plant velocity can follow the model reference velocity by using the AMFC control scheme, we can obtain the average tracking time by using Eqn (28).
5.2. Experimental results
To begin with, the experimental result using the hybrid controller composed of bang-bang, velocity, position control is presented for comparison with that using AMFC. The bang-bang control loop is cut off a few tracks ahead of a target track and then the velocity control loop is connected to the velocity control signal. This permits the tracking actuator to slide smoothly toward the target track by using a velocity trajectory to cover the remaining distance. When the velocity reaches almost zero, the velocity control loop is cut off and the position control loop is connected to the position control signal. This allows the tracking actuator to settle down onto the target track with the required accuracy. A P and PID controller are used for velocity and position control, respectively. Figure 10 shows the experimental result of the speed performance for a 3.5 in. magneto-disk drive whose average tracking distance is 6 mm. We see that the average tracking time is around 25 ms including the settling time, 7 or 8 ms, required in both velocity and position control. The servo control system for the high speed tracking actuator of the magnetooptical disk drive is shown in Fig. 11. With the help of the tracking motion analysis discussed in section 5.1, the adaptive controller produces the velocity profiles, i.e. triangles or trapezoids, as a reference model according to the tracking distances to be traveled. When the velocity reaches almost zero, the adaptive control loop is cut off and the position control loop is connected to the position command input signal. This allows the tracking actuator to settle down onto the target track with the required accuracy. A PID controller is used for the position controller and implemented with Velocity tin/s) 1.0
Control input(V) 8
075 050
•
//
0.25 0.0
~
--"
6
2 . . . .
4
Velocity \
2
,
0 -2
--[
Bang-bang input
.....
Z/
Position mput
-vr Velocity input
-6
x'
o8 Time (5.0ms/div)
Fig. ll). The experimental result when the hybrid controller is used for a 3.5 in. magneto-opticaldisk drive whose average tracking distance is 6 ram.
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Switching signal from I. . . . . . . . t, . . . . . . . . . . . . . . . . . . . . . , DAboard : ! i . system .
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Amplifier[-] . . . . . I .
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+
signals into[ AD board
[PlDcontroller I ; [ for position I"" --" -- - [ 1 control ]
k._
k..~ Position Velocity
Fig. 11. Servocontrol scheme for the high speed tracking actuator of the magneto-optical disk drive. analog circuits. AD7512DI is used for the switcher. The data acquisition and digital control module are built using a PC/C31 data acquisition board from Loughborough and a TMS320C31 digital signal processor. A IBM PC 486 serves as a host computer. As the velocity and position sensors, a OFV 3000 laser vibrometer from Polytec is used. Actually, in the magneto-optical disk drives, the position and velocity information is obtained by counting the tracks embedded in the optical disk and the frequency of the tracking error signal, respectively. Since position information, for example, is written on the disk, no external position sensing system is required. In short, an optical disk works like an intrinsic sensor for quick and accurate tracking. To have the position and velocity information directly measured from the optical disk, however, we require optical components such as a laser beam and optical pick up devices. With the consideration of cost and scope of the work, we exclude the use of the optical disk as a measurement device, so it is necessary to develop an external measurement device to replace the role of the optical disk. In this work, a laser vibrometer which is compatible with the optical disk in terms of accuracy is used for replacing the role of the optical disk and to measure the position and velocity. Figure 12 shows the experimental results of the speed performance for different bang-bang input frequencies (55, 85, 125 and 250 Hz). The velocity responses from the reference model, and the AMFC system are shown together. The triangular model reference signals are generated from the corresponding bang-bang inputs. The bang-bang input is not shown in the figures because of the lack of a displaying channel. The velocity using the AMFC system follows the reference model well up to 250 Hz. Figure 13 shows the experimental results of the speed performance for different tracking distances (2, 6, 7.5 and 9 mm). We can see that high speed and accurate tracking performance are achieved. Note that we set the speed factor fl = 0.4, and the largest distance b traveled in the triangular mode is 6.4 mm. Therefore, a triangular velocity mode should be applied to the tracking distances of 2 and 6 mm while a trapezoidal mode should be applied to the tracking distances of 7.5 and 9 mm. The maximum acceleration is 90 ms -2. The maximum velocity is restricted to 0.73ms -1.
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Velocity (m/s) 1
ro.oos
Velocit of Velocity ref~enc
0.75 0.50
5.00"{/
A
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et al.
Velocity (m/s)
, - o .O0,a
0.50
Velocityof ~
~.
02'1 .......
0.25 -''}
sr,gl§l. STOI
0.751
'
Velocityof
/fAMFC system
I reference model ~ N , ,
0
5 .oo~/
1I
%
•
. . . . i ........ i
. . . . . ~ ' " AMFC syst
Time (5.0ms/div)
Time (5.0ms/div)
(b)
(a) Velocity (m/s) 1
~--0.00s
z
0.75
.......
0.50
Velocity of reference model
5 .oo~/
%n~l~l STOI
Velocity 0 n / s ) 1
r-o.oos
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5.~,~/
Z
0.75
Velocity of
{).50
2-
Velocity of , , 0.25 ~ r e l e r e n c moael e
:
Velocity of A_MFCsystem
I
÷ +
Time (5.0ms/div) (c)
Time (5.0ms/div) (d)
Fig. 12. Experimental results o f the speed performance for different bang-bang input frequencies. (a) 55 H z , (bt 85 H z , (c) 125 H z a n d (d) 250 H z ,
Hence, we can figure out that the average tracking time to move 6 mm is around 20 ms adding the yt~ from Eqn (28). This analytical result agrees with the experimental result, as shown in Fig. 13b. This represents a significant improvement over the 25 ms tracking time achieved by using bang-bang control with velocity and position control added. It is also a marked improvement over the roughly 50 ms tracking times reported for commercial magneto-optical systems of comparable size. The first reason for the fast tracking time is seemingly due to the AMFC's control effectiveness. As observed from the control input to the plant as shown in Fig. 9, the application of the AMFC scheme shows the effect of applying the multiple signal polarity reversal as if our system were a high order system. The other reason, viewed from a different angle, is due to the electrical power effectiveness. When the conventional controller (which uses bang-bang control with velocity and position control added) is used, as shown in Fig. 10, the maximum negative peak current in the decelerating range is larger in magnitude than the maximum positive peak current in the accelerating range due to the back emf voltage. This fact indicates that we cannot employ a full electrical power evenly in both ranges, implying that a tracking actuator does not operate in a fully bang-bang mode. On the other hand, since it can operate in a fully bang-bang mode in the AMFC scheme, a higher acceleration is obtainable from it, which contributes to a reduction in the tracking time. Therefore, we can conclude that AMFC provides not only an easy and correct selection of the switching point of the tracking system's velocity profile, but also high tracking speed.
Tracking system using adaptive model-following control Velocity (m/s)
Position (mm)
5.00~/
!
:
llo
I
0.7:
+1! i i 7.5 02 ...... Veloeity.~: ...... i......... i - P o s i t i o n ! ....... 5.0
0.25 ......... i ........ ~ / ~
........ ;F.., ............ •....
J2.5
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I
"
5.oo~/
• .,:i.,,,
0.75
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871
..... •
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Position(mm) 10
:
Pos t on .... ,17.5 !'~
........
~ ..-../s.o
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0
Time (5.0ms/div)
Time (5.0ms/div)
(a) Velocity (m/s) 1 r
(b) s .oo~/
!
~
:
:
Position (ram) 10 :
0.75
7.5
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7.5 5.0
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I
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I
(C)
Time (5.0ms/div) (d)
Fig. 13. Experimental results of the speed performance for different tracking distances. (a) 2 mm, (b) 6 ram, (c) 7.5 mm and (d) 9 ram.
6. CONCLUSION Adaptive model-following control is implemented for a high speed tracking system. It can relieve the difficulty and inconvenience of selecting the switching point in bang-bang time optimal control using a simple reference model. Adaptive modelfollowing control is applied to a prototype magneto-optical disk drive to improve the tracking speed performance. A excellent model-following capability is demonstrated in the simulation and experiment. According to the experimental results, an average tracking time as small as 20 ms can be obtained when a 3.5 in. magneto-optical disk drive is applied. This result indicates that magneto-optical disk systems can be competitive with the purely magnetic disk systems in terms of tracking time.
REFERENCES 1. Ananthanarayanan K. S., Third-order theory and bang-bang control of voice coil actuator. IEEE Trans. Magnetics MAG-18(3), 888-892 (1982). 2. Brown C. J. and Ma J. T., Time optimal control of a moving-coil linear actuator. IBM J. Res. Develop. 12,372-379 (1968). 3. Leitmann G., An Introduction to Optimal Control. McGraw-Hill, New York (1966). 4. Ishibashi H., Tanaka S., Shimizu R., Kuwamoto M. and Imura M., High •speed accessing magneto-optical disk drive. Optical Storage Technology and Applications, SPIE Vol. 899, pp. 8-15 (1989).
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5. Koumura K., Takizawa F., lwanaga T., lnada H. and Yamanaka Y., High speed accessing using split optical head. Data Storage Topical Meeting, SPIE Vol. 1078, pp. 239-243 (1989). 6. Oswald R. K., Design of a disk file head-positioning servo. IBM J. Res. Develop. 18,506-512 (1974). 7. Park K. H., Development of tracking and focusing actuators for magneto-optical disk drive systems. Ph.D. dissertation, Department of Mechanical Engineering, The University of Texas at Austin, TX (1993). 8. Landau Y. D., Adaptive Control (The Model Re]erence Approach). Marcel Dekker, New York (1979). 9. Goodwin G. C. and Sin K. S., Adaptive Filtering Prediction and Control. Prentice-Hall, Englewood Cliffs, NJ (1984). 10. Liaw C. M. and Lin F. J., A discrete adaptive induction position servo drive. 1EEE Trans. Energy Conversion 8,350-356 (1993). 11. Lin F. J. and Liaw C. M., Reference model selection and adaptive control for induction motor drives. IEEE Trans. Automat. Control 38, 1594-1600 (1993). 12. Benjelloun K., Mechlih H. and Boukas E. K., A modified reference adaptive control algorithm for DC servomotor. Second 1EEE Conf. on Control Application, pp. 941-946, Vancouver, September (1993). 13. The MathWorks, Inc., User's guide, SIMULINK Version 1.2. Natick, MA 01760 (1992). 14. Hertrich F. R., Average motion times of positioners in random access devices, l B M J. Res. Develop. 9, 124-133 (1965).
© 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved. 0957-4158/95 $9.50+0.00
Pergamon 0957-4158 (95) 00055-0
A HIGH SPEED TRACKING SYSTEM USING ADAPTIVE MODEL-FOLLOWING CONTROL J. H. LEE,* K. H. PARK, t S. H. KIM* and Y. K. KWAK* *Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Kusong-dong, Yusong-gu, Taejon 305-701, Korea and tDepartment of Mechatronics, Kwangju Institute of Science and Technology (K-JIST), 572 Sangam-dong, Kwangsan-ku, Kwangju 506-303, Korea
(Received 14 January 1995; revised and accepted 20 June 1995) Abstract--In the time optimal control problem, it is important to select a switching point accurately in order to improve tracking speed performance. However, the accurate selection of the switching point is not an easy task because of the damping coefficient variations and modeling uncertainties in actual systems. To relieve the difficulty and inconvenience of this task, adaptive model-following control (AMFC) is implemented. AMFC can make the controlled plant follow as closely as possible a desired reference model whose switching point can be calculated easily and accurately, assuring that the error between the reference model and the real system approaches zero. The AMFC is applied for a tracking actuator of a magneto-optical disk drive (MODD) to improve its slow tracking performance. According to the simulation and experimental results, an average tracking time as small as 20 ms is obtained for a 3.5 in. magneto-optical disk drive.
1. INTRODUCTION Time optimal control is essential in engineering applications where mechanisms must operate repetitively at high speed and with great precision. The time optimal problems can be approached in a n u m b e r of ways using theoretical and experimental approaches [1, 2]. Bang-bang control is the theoretical solution for time optimal control. For a high precision tracking system, a high precision position controller is also necessary. For smooth connection of these control modes, a velocity controller can be used. The advantages of each control m o d e are combined to establish the desired p e r f o r m a n c e by switching the system from one control m o d e to another. H e r e , bang-bang control plays the most important role of determining the tracking time. Bang-bang control is alway an extremal control and involves N - 1 drive signal polarity reversals for a system with an N t h order state equation [3]. The extremal control for the system with a second order system, which describes most real mechanical systems, has one polarity reversal from +Vs to - V s at a switching point, x~, where the m a x i m u m acceleration changes to the m a x i m u m deceleration. For the best p e r f o r m a n c e of bang-bang control, one needs to correctly calculate the switching point. The switching point is calculated based on the system model. Since bang-bang control, however, is open-loop, its p e r f o r m a n c e characteristics are dependent on the system components and the plant characteristics. This dependency m a k e it hard to determine the switching point accurately when system model uncertainties 857
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exist. The inaccuracy of the switching point increases in a high frequency operating condition, i.e. high bang-bang input frequency. To compensate for the system uncertainties, closed-loop velocity control is usually used together with bang-bang control [4, 5]. In velocity control, when a target track is determined, a velocity trajectory is generated and stored in a look-up table in advance because it take time to calculate the velocity trajectory in a real time control process. The velocity control can increase the system bandwidth compared to bang-bang control, especially if the system is hard to model, and the switching point is hard to predict, However, substantial efforts are still required to anticipate the switching point correctly because it cannot be expected that the extreme inputs +V~ and -V~ have equal magnitudes in practical cases and hence the switching point x~ is not easily calculable. In an experimental approach to the time optimal control problem, the switching points in bang-bang control can be obtained by applying +1/~ and -V~ to the real system and measuring the phase plane trajectories for various travel lengths. This collection of data then is mapped directly to the best choices for the available free parameters used for generating the trajectory in a single step process [6]. This method, however, requires several iterations of experiments to increase the reliability of the switching point. Therefore, the above methods have the difficulty and inconvience in theoretically and experimentally selecting the switching point. The approach taken to relieve these problems in this work is to make the controlled plant follow, as closely as possible, a desired reference model whose switching point can be calculated easily and accurately. We build an adaptive loop to assure that the controlled plant always behaves as specified by means of a reference model that produces the desired output for a given input. This control method is called adaptive model-following control (AMFC). If the reference model is designed so that it has a high system bandwidth, the tracking time can be reduced if we can assure that the error between the reference model and the real system approaches zero. Therefore, it can be said that the AMFC scheme is a good alternative for bang-bang control and velocity control. Position control is switched after the AMFC scheme's task is completed. The AMFC scheme is applied to a magneto-optical disk drive to improve its slow tracking performance which is the significant disadvantage of the current magnetooptical systems.
2. SWITCHING POINT IN BANG-BANG C O N T R O L
The system to be analyzed is shown in Fig. 1 and described in [7]. To reduce the average accessing time of a magneto-optical disk drive, a newly developed moving magnet type tracking actuator is implemented. It consists of an air core solenoid, four permanent magnets connected on glass rods, a focusing actuator which is mounted on the tracking actuator, lens assembly, and linear ball bearings. The tracking actuator moves horizontally on the ball bearings by using the interacting force generated between the solenoid and permanent magnets. The assembly of the tracking actuator can be treated as a rigid body whose mass and viscous friction coefficient are M and c. Since the mass center of the tracking actuator is near the centerline of the two
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Permanent Permanent magnet magnetholder Focusing actuator holder
Linear ball bearing Base
f Laser t~eam~ path
- Glass rod Air-core solenoid
Fig. 1. Configurationof the developed linear tracking actuator. glass rods, the driving force acts on the mass center of the total system, and it is assumed to be proportional to the current in the conductor. The moving mass of the tracking actuator can be measured by a scale and it is around 0.025 kg. However, the damping coefficient is hard to measure, and is subject to variations due to temperature change, moving direction, and location during stroke. Let us investigate how the switching point changes according to the difference in the damping coefficient.
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The block diagram of the tracking system is represented in Fig. 2. Here r, I, F and xl indicate the input signal, the current, the force generated from the actuator, and the position of the actuator, respectively. K,, and K, are the current amplifier gain and the force constant measured as 0.35 A/V and 1 N/A, respectively for the system tested. Therefore, the open loop transfer function of the velocity control system, G,, is theoretically G,-
xl r
K~KI
0.35
-
s ( M s + c)
.
(1)
s ( M s + c)
The dynamic equation can be represented in terms of state variables: 2i = x2 c
.G -
x~ +
M
(2) 1 --u,
(3)
M
where Xl and x2 respectively are the position and the velocity of the tracking actuator. The control input u is defined as Ku K t r . Equations (2) and (3) are conveniently shown on a phase plane with coordinates xl and x2. Figure 3 shows how the switching points are determined by the graphical representation in the phase plane ( & - x 2 ) when the tracking distance is 6 mm, and u = 1.5 V. From the figure we can see that the switching points change for the different damping coefficients. If the damping coefficient, c, is zero, the switching point occurs exactly at the middle point of the target position. As the damping
s(Ms+c) Fig. 2. Block diagram representation of the tracking system.
c-90M c=60M c=30M c=15M c=3M xl(m)
/ '/ /2":
0.006 0.004 .
0.002
c=30M
0 . ,
~ 0
c=I5M c=3M 0.2
0.4
0.6
0.8
1 x 2 (m/s)
Fig. 3. Variation of the switching point for the different damping coefficients.
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coefficient increases, the switching point tends to shift upward to the target point. This implies that a slight change of the damping coefficient or mass in the tracking system leads to the incorrect estimation of the switching point. In addition to the change of the damping coefficient, the theoretical model is subject to change because of the modeling uncertainties. Therefore, we have a limitation in achieving a fast tracking time using bang-bang control. The approach taken to relieve problems such as the change of the damping coefficient and modeling uncertainties is to make the controlled plant follow, as closely as possible, a desired reference model whose switching point can be calculated easily and accurately. In other words, if the distance to move a target track from a current position is determined, we can generate a triangular velocity profile, for example, which is obtained by integrating the bang-bang signal whose switching point is in the middle of the distance. This relation can be understood in terms of Fig. 4. Since the maximum acceleration, a,~, is fixed due to the current constraint, the switching point, ts, can be easily obtained by a simple physical law. Therefore, we can say the switching point is dependent only on the moving distance. The velocity profile obtained from the procedure is used for the reference input. If the adaptation mechanism works properly, the adapted plant will follow the reference input well, the triangular velocity profile.
3. AMFC SYSTEM DESIGN FOR A T R A C K I N G SYSTEM
The adaptive model-following control scheme is well established in theory [8-12]. In this section, we shall briefly review the AMFC design. If we take the velocity and acceleration as the state variables, the real plant to be controlled is described by = Aey
+
Bpttp,
(4)
where A p and Bp are the real plant matrices, and y = [Yl Y2] r. Yl and Y2 are the tracking actuator velocity and the acceleration of the real plant, and Up is the input voltage to the plant. Similarly, the reference model is described by Xm --- AmXm + Braun, where
Am
and
B m are
(5)
the model plant matrices, and Xm = [x,,1 Xm2]r. Xml and Xm2
acceleration
velocity
1.!_ s
I
Time, t
Fig. 4. Proceduresfor generation of the velocitytrajectories by integration of bang-bang signal.
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are the desired tracking actuator velocity and the acceleration of the reference model, and Um is the input voltage to the reference model. We consider a parallel AMFC configuration with signal-synthesis adaptation as shown in Fig. 5. If we define (6)
u 1, = u~l + up2,
where Up1 is the linear control signal and Up2 is the adaptation signal, we can then choose g/pl
=
- K p y + KuUm
(7)
up2 = AKp(e, t)y + AK.(e, t)Um,
(8)
where Kp and K . are linear control gains, and AKp and A K . are adaptation control gains having matrices of appropriate dimensions. Recall that the error e between the states of the model and those of the plant is X m - y- Then, using Eqns (7) and (8), we obtain the following vector differential equation: = Ame + [A,,, - At, + BpKp - BpAKp(e, t)]x + [B,,, - BpK, - BpAK,(e, t)]Um.
(9)
To have perfect model following, we must assure that for any Urn, piecewise continuous, and initial state e0 = 0 we shall have e(t) = e(t) = 0. We conclude that this can be achieved if the last two terms in the right hand side of Eqn (9) equal zero. In addition to this, the remaining unforced system t~ = Ame
(10)
must be asymptotically stable, which implies that the matrix A m must be a Hurwitz matrix. In short, we can summarize that the objective of the adaptation mechanism is to generate the two time-varying matrices AKp(e, t) and A K , ( e , t) to assure that the generalized state error, e, goes to zero.
_1 Reference
tim
=
+
Plant
I Adaptation Kp(e,t)) 1~ Mechanism ( Ku(e,t)
Fig. 5. A parallel AMFC configuration with signal-synthesisadaptation.
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For this purpose, we use the hyperstability and positivity theory which says that a feedback system is asymptotically hyperstable if two conditions are satisfied [8]: the equivalent feedforward block is a strictly positive real transfer function or matrix, and the feedback block must satisfy the Popov integral inequality. To transform the system into the form of an equivalent feedback system composed of two blocks, one in the feedforward path and one in the feedback path, Eqn (9) is rewritten by the following equations: = (Am)e + W
(11)
v = De,
(12)
where W represents the last two terms of Eqn (9). The matrix D is especially chosen in order to be able to meet the specification required for the linear part so that the stability of the system is assured, v is the input of the feedback block. Let us first examine the adaptation mechanism in the feedforward block. The feedforward transfer function, H(s), represented as H(s) = D(sI - A m ) -1,
(13)
should be a strictly positive real transfer matrix, or there must exist a P satisfying the Lyapunov equation such that (Am)rP + P(Am) = - Q ,
(14)
D = P,
(15)
in which Q is a positive definite matrix. Next, for the adaptation mechanism in the feedback block, we use one of the solutions of the Popov integral inequality introduced by Landau [8]. The adaptation control gains, AKp and AK,, are described using an integral plus proportional adaptation law as t
AKp(e, t)
= fo I~)1(¥' t, r ) d r + q~2(v, t) + AKp(e, 0)
(16)
t
AK.(e, t) = f ~Pl(V, t, r) + apz(V, t) + AK.(e, 0), Jo
(17)
where q,l(V, t, ~) =
Vo(t
-
~)v(r)[~;a(t)x(~)] T,
r
¢
q52(V, t) = Fa(t)v(t)[Ga(t)x(t)]
T
!
~P2(v, t) = Fb(t)v(t)[Gb(t)Um(t)]
(18)
(19)
~Pl(V, t, r) = F b ( t - r)v(~)[Gb(t)um(~)] r, P
~ >1 t
T
,
r >-t
(20)
(21)
where F a ( t - r) and F b ( t - r) are positive definite matrix kernels whose Laplace transforms are positive real transfer matrices with a pole at s = 0. Ga and Gb are positive definite constant matrices; F'a(t), F~,(t), G'a(t) and G~(t) are time-varying positive definite matrices for all t t> 0. The equivalent feedback system composed of two blocks, one in the feedforward path and one in the feedback path, is illustrated in Fig. 6.
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Linear Parl r-
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'~ L
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l .
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Fig. 6. Equivalent feedback representation of adaptive model-followingcontrol.
4. APPLICATION TO THE MAGNETO-OPTICAL DISK DRIVE To begin constructing the A M F C system, we first need to know the real plant matrices Ap, Up and model plant matrices A .... B m as described in Eqns (4) and (5). A frequency response test for the tracking actuator velocity versus input signal is experimentally performed to obtain dynamic characteristics of the tracking system and is shown in Fig. 7. The experimentally obtained amplitude ratio and phase angle are denoted by the legend 'o'. The open loop transfer function of the experimentally obtained real system can be most closely fitted to a second order system and is denoted by the solid line: Gp =
t
14
•
(22)
0.0005s -~ + s + 40 Then, from Eqn (22) F
t)
A, = •] - 8 0 , 0 0 0
-21000 j '
Bp = [0 228,000].
(23)
Now that we have obtained the real tracking system model, it remains to build an adaptive control loop to assure that the plant behaves as specified by means of a reference model that produces a desired output. In choosing the reference model, three facts are considered. Firstly, the reference model should be simply described so that the switching point is determined easily and accurately. Secondly, there must be a trade-off between the model-following capability and control effort. If the reference model is the same as the real system model, no control effort is needed. However, as
Tracking system using adaptive model-following control
II IIIII
I I
1
IIIIll
]~[tlt[
865
I EIE II
IIIIIII ILIL Ill I Illl tll Jl LIIh II I Iqllh LII
~-OO
0.1
10
1
100
10OO
Frequency (Hz)
0 -loo
I[I I II I I [ [ ~ l I III I Ill II LILl 11111 T+I
.~ ~ -200
~. -3oo
0.1
1
10
100
10OO
Frequency (Hz) Fig. 7. Frequency response test for va vs r.
the real system's requirement of the model-following capability increases, the control effort also increases. Finally, the highest order of the characteristic equation of the reference model should be the same as that of the plant model for an easy construction of the adaptation mechanism. With the above considerations, we define the reference model Gr, though it is not unique, as 114
Gr =
•
(24)
0.0001s 2 + s + 1 The reference model Gr can be treated almost as a integrator on the time scale of interest because of the small coefficient of s z. If a bang-bang input is applied to the G~, a trapezoidal velocity profile is generated, which means an easy selection of the switching point in the middle of the tracking distance. Then from Eqn (24) Am
=
E0
-10,000
-10,000
'
B m = [0
1,140,000].
(25)
The reference model described by the above transfer function is constructed in the digital computer. Here, (Ap, Be) and (Am, B m ) a r e stabilizable, and A m is a Hurwitz matrix, i.e. the reference model is asymptotically stable. The next step is to follow the procedures which are presented in section 3. The detailed configuration is given in Fig. 8. All block diagrams are constructed in SIMULINK, which is a commercial system dynamics and control program that includes M A T L A B [13]. To simulate real system environments as closely as possible, a maximum current constraint for the solenoid is imposed. The maximum and minimum currents are set to +4 A and - 4 A, respectively. Since the sampling time currently used for the servo
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v'y2
'alpha
Fig. g. Block diagram of the AMFC control scheme.
control system of a magneto-optical disk drive is less than 0.1 ms [5], digital simulation uses 0.1 ms sampling time. The positive definite matrix Q is selected so that H(s) has a strictly positive real transfer function. When Q is Q =
0
0.1
"
(26)
P and D are obtained by solving Eqns (14) and (15), respectively. In one of the solutions of the Popov integral inequality introduced in section 3, we assume that AKp(e, 0) = AK,,(e, 0) = 0, and we choose V',,(t) = F',, > 0.
F,(t) = V,, > 0.
,
G,,(t) = Go(t) --
~ _
0
0
a'2 l '
a~ ;> ()"
Fh(t) = Fb, 0;2 ~ O,
F;(t)
--
r'h
G~(t) = G'h(t) = 1.
(27)
Several simulation results give a few design guidelines; all linear control gains, K,, and some freedom because they have little influence on the tracking performance. Since ct,2 and F',, are associated with the tracking acceleration they should be kept small, o(1, oQ, Fa, F'a, K,, and Kp are chosen as 1, 0.01, 1, 0.1 1, and [1 1] respectively.
Kp, have
The simulation result shown in Fig. 9 corresponds to trajectory tracking scheduled to go 6 mm, which is chosen because it is the average tracking distance of a 3.5 in. disk drive. Figure 9 shows the reference velocity and plant velocity; and control input and applied white noise. There are almost no tracking errors observed in the velocity trajectory.
Tracking system using adaptive model-following control Velocity (m/s) 0.8
0 0
.
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.
.
.
.
.
.
0.005
0.01
867
~:~:~'~ 0.015
0.02
0.025 Time (s)
Control input & sensor noise(V)
1!
Time (s) Fig. 9. The simulation result.
5. EXPERIMENT
5.1. Tracking velocity profile in a random access device In a search for an adequate velocity profile, it may be found that most positioners have velocity versus time curves which may be closely represented by triangles or trapezoids. According to Hertrich [14], the average tracking time, tav, can be represented by using the speed factor tav = Ta~/(L/a),
(28)
where L and a are a maximum moving distance in the data storage and acceleration. ra is a dimensionless time constant and is expressed as r~
_
1
3
15~/fl(/3_(1 + 3),) - 5/32(1 + 2),) + 15/3(1 + ),) + 5).
(29)
Here, /3 is defined as the ratio of the largest distance traveled in the triangular velocity mode to the maximum distance, L. ), is introduced to modify the motion times in accordance with overshoots and sluggishness encountered in the actual system. In the practical case now under consideration, the maximum distance L as well as the acceleration a will be constant. Hence, the average tracking time, according to Eqn (28), becomes strictly proportional to the time constant, to. In this work, we choose ), = 0.3, /3 = 0.4 since r a turns out to be almost unaffected by a variation of/3 in the range of 0.4 <~/3 ~< 1. With these values, r o is around 1.22. When the maximum distance, L, in a 3.5in. magneto-optical disk is 1 8 m m , the largest distance, b, traveled in the triangular mode is calculated as 6.4 mm. Since the average tracking
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distance is around 6 mm, the triangular velocity mode should be applied to find out the average tracking time. If the real plant velocity can follow the model reference velocity by using the AMFC control scheme, we can obtain the average tracking time by using Eqn (28).
5.2. Experimental results
To begin with, the experimental result using the hybrid controller composed of bang-bang, velocity, position control is presented for comparison with that using AMFC. The bang-bang control loop is cut off a few tracks ahead of a target track and then the velocity control loop is connected to the velocity control signal. This permits the tracking actuator to slide smoothly toward the target track by using a velocity trajectory to cover the remaining distance. When the velocity reaches almost zero, the velocity control loop is cut off and the position control loop is connected to the position control signal. This allows the tracking actuator to settle down onto the target track with the required accuracy. A P and PID controller are used for velocity and position control, respectively. Figure 10 shows the experimental result of the speed performance for a 3.5 in. magneto-disk drive whose average tracking distance is 6 mm. We see that the average tracking time is around 25 ms including the settling time, 7 or 8 ms, required in both velocity and position control. The servo control system for the high speed tracking actuator of the magnetooptical disk drive is shown in Fig. 11. With the help of the tracking motion analysis discussed in section 5.1, the adaptive controller produces the velocity profiles, i.e. triangles or trapezoids, as a reference model according to the tracking distances to be traveled. When the velocity reaches almost zero, the adaptive control loop is cut off and the position control loop is connected to the position command input signal. This allows the tracking actuator to settle down onto the target track with the required accuracy. A PID controller is used for the position controller and implemented with Velocity tin/s) 1.0
Control input(V) 8
075 050
•
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Fig. ll). The experimental result when the hybrid controller is used for a 3.5 in. magneto-opticaldisk drive whose average tracking distance is 6 ram.
Tracking system using adaptive model-following control
869
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Fig. 11. Servocontrol scheme for the high speed tracking actuator of the magneto-optical disk drive. analog circuits. AD7512DI is used for the switcher. The data acquisition and digital control module are built using a PC/C31 data acquisition board from Loughborough and a TMS320C31 digital signal processor. A IBM PC 486 serves as a host computer. As the velocity and position sensors, a OFV 3000 laser vibrometer from Polytec is used. Actually, in the magneto-optical disk drives, the position and velocity information is obtained by counting the tracks embedded in the optical disk and the frequency of the tracking error signal, respectively. Since position information, for example, is written on the disk, no external position sensing system is required. In short, an optical disk works like an intrinsic sensor for quick and accurate tracking. To have the position and velocity information directly measured from the optical disk, however, we require optical components such as a laser beam and optical pick up devices. With the consideration of cost and scope of the work, we exclude the use of the optical disk as a measurement device, so it is necessary to develop an external measurement device to replace the role of the optical disk. In this work, a laser vibrometer which is compatible with the optical disk in terms of accuracy is used for replacing the role of the optical disk and to measure the position and velocity. Figure 12 shows the experimental results of the speed performance for different bang-bang input frequencies (55, 85, 125 and 250 Hz). The velocity responses from the reference model, and the AMFC system are shown together. The triangular model reference signals are generated from the corresponding bang-bang inputs. The bang-bang input is not shown in the figures because of the lack of a displaying channel. The velocity using the AMFC system follows the reference model well up to 250 Hz. Figure 13 shows the experimental results of the speed performance for different tracking distances (2, 6, 7.5 and 9 mm). We can see that high speed and accurate tracking performance are achieved. Note that we set the speed factor fl = 0.4, and the largest distance b traveled in the triangular mode is 6.4 mm. Therefore, a triangular velocity mode should be applied to the tracking distances of 2 and 6 mm while a trapezoidal mode should be applied to the tracking distances of 7.5 and 9 mm. The maximum acceleration is 90 ms -2. The maximum velocity is restricted to 0.73ms -1.
870
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Time (5.0ms/div) (d)
Fig. 12. Experimental results o f the speed performance for different bang-bang input frequencies. (a) 55 H z , (bt 85 H z , (c) 125 H z a n d (d) 250 H z ,
Hence, we can figure out that the average tracking time to move 6 mm is around 20 ms adding the yt~ from Eqn (28). This analytical result agrees with the experimental result, as shown in Fig. 13b. This represents a significant improvement over the 25 ms tracking time achieved by using bang-bang control with velocity and position control added. It is also a marked improvement over the roughly 50 ms tracking times reported for commercial magneto-optical systems of comparable size. The first reason for the fast tracking time is seemingly due to the AMFC's control effectiveness. As observed from the control input to the plant as shown in Fig. 9, the application of the AMFC scheme shows the effect of applying the multiple signal polarity reversal as if our system were a high order system. The other reason, viewed from a different angle, is due to the electrical power effectiveness. When the conventional controller (which uses bang-bang control with velocity and position control added) is used, as shown in Fig. 10, the maximum negative peak current in the decelerating range is larger in magnitude than the maximum positive peak current in the accelerating range due to the back emf voltage. This fact indicates that we cannot employ a full electrical power evenly in both ranges, implying that a tracking actuator does not operate in a fully bang-bang mode. On the other hand, since it can operate in a fully bang-bang mode in the AMFC scheme, a higher acceleration is obtainable from it, which contributes to a reduction in the tracking time. Therefore, we can conclude that AMFC provides not only an easy and correct selection of the switching point of the tracking system's velocity profile, but also high tracking speed.
Tracking system using adaptive model-following control Velocity (m/s)
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Fig. 13. Experimental results of the speed performance for different tracking distances. (a) 2 mm, (b) 6 ram, (c) 7.5 mm and (d) 9 ram.
6. CONCLUSION Adaptive model-following control is implemented for a high speed tracking system. It can relieve the difficulty and inconvenience of selecting the switching point in bang-bang time optimal control using a simple reference model. Adaptive modelfollowing control is applied to a prototype magneto-optical disk drive to improve the tracking speed performance. A excellent model-following capability is demonstrated in the simulation and experiment. According to the experimental results, an average tracking time as small as 20 ms can be obtained when a 3.5 in. magneto-optical disk drive is applied. This result indicates that magneto-optical disk systems can be competitive with the purely magnetic disk systems in terms of tracking time.
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5. Koumura K., Takizawa F., lwanaga T., lnada H. and Yamanaka Y., High speed accessing using split optical head. Data Storage Topical Meeting, SPIE Vol. 1078, pp. 239-243 (1989). 6. Oswald R. K., Design of a disk file head-positioning servo. IBM J. Res. Develop. 18,506-512 (1974). 7. Park K. H., Development of tracking and focusing actuators for magneto-optical disk drive systems. Ph.D. dissertation, Department of Mechanical Engineering, The University of Texas at Austin, TX (1993). 8. Landau Y. D., Adaptive Control (The Model Re]erence Approach). Marcel Dekker, New York (1979). 9. Goodwin G. C. and Sin K. S., Adaptive Filtering Prediction and Control. Prentice-Hall, Englewood Cliffs, NJ (1984). 10. Liaw C. M. and Lin F. J., A discrete adaptive induction position servo drive. 1EEE Trans. Energy Conversion 8,350-356 (1993). 11. Lin F. J. and Liaw C. M., Reference model selection and adaptive control for induction motor drives. IEEE Trans. Automat. Control 38, 1594-1600 (1993). 12. Benjelloun K., Mechlih H. and Boukas E. K., A modified reference adaptive control algorithm for DC servomotor. Second 1EEE Conf. on Control Application, pp. 941-946, Vancouver, September (1993). 13. The MathWorks, Inc., User's guide, SIMULINK Version 1.2. Natick, MA 01760 (1992). 14. Hertrich F. R., Average motion times of positioners in random access devices, l B M J. Res. Develop. 9, 124-133 (1965).