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Direct tracking control using time-optimal trajectories
Control Eng. Practice, Vol. 4, No. 9, pp. 1231-1240, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/96 $15.00 + 0.00
Pergamon PII:S0967-0661 (96)00129-3
DIRECT TRACKING CONTROL USING TIME-OPTIMAL TRAJECTORIES Kyihwan Park*, Jung-Hyun Lee**, Soo-Hyun Kim** and Yoon Keun Kwak** *Departmentof Mechatronics, Kwang-JuInstitute of Science and Technology, 502, Sangam-dong, Kwangsan-gu, Kwangju 506-303, Korea **Departmentof Mechanical Engineering, Korea AdvancedInstitute of Science and Technology(KAIST), 373-1, Kusong-dong, Yusong-gu, Taejon305-701, Korea
(Received April 1995; in final form July 1996)
Abstract: Direct tracking control which is capable of high positioning speed and accuracy is developed using adaptive model-following control (AMFC). Direct tracking control relieves the difficulty and inconvenience of selecting the switching points in conventional hybrid control composed of bang-bang, velocity, and position control. AMFC can make the controlled plant follow, as closely as possible, a desired reference model where the time optimized positioning signal is generated. Direct tracking control using AMFC is applied to a tracking system of a magneto-optical disk drive to improve its slow tracking performance. Compared with the hybrid control system, the tracking time is much decreased in the direct tracking control system. According to the simulation and experimental results, an average access time as small as 19 ms is achieved. Keywords: Direct tracking control, time-optimal, adaptive model-following control, hybrid control, linear tracking actuators, magneto-optical disk drives.
1. INTRODUCTION The most important thing to be considered in servocontrol for a tracking system is how fast it moves to the target track, with the accuracy required. A conventional method of meeting these requirements is to use hybrid control, composed of bang-bang control, velocity control, and position control. For the minimum time requirement, a bang-bang controller is used because it is the theoretical solution for timeoptimal control (Leitmann, 1966; Ananthanarayanan, 1982). For the high positional accuracy requirement, a high-precision position controller is also used. For a smooth connection between those control modes, a velocity controller can be used. These control modes can be combined to establish the desired performance by switching the system from one control mode to another. However, it is difficult to calculate the switching
points where bang-bang control is connected to velocity control and where velocity control is connected to position control. Indeed, when the switching point is not well selected, the hybrid control method may provide worse speed performance than when only position control is employed. The difficulty of selecting a correct switching point is due to the fact that it is calculated based on the system model. In other words, since bang-bang control is open-loop, its performance characteristics are dependent on the system components and the plant characteristics. This dependency makes it hard to determine the switching point correctly when system model uncertainties exist. To compensate for the system uncertainties, closedloop velocity control is commonly used, together with position control (Koumura, et al., 1989; Inada, et al., 1986; Eguchi, et al., 1990). In this method, when a target track is determined, a velocity trajectory is generated and stored in advance in a 1231
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look-up table, because it takes time to calculate the velocity trajectory in a real-time control process. The velocity control can increase the system bandwidth compared to the bang-bang control. However, a substantial effort is still required in correctly anticipating the switching point in order to calculate the velocity trajectory in advance. As an experimental way of approaching the timeoptimal control problem, the switching points in bang-bang control can be obtained by applying maximum and minimum voltages to the real system, and measuring the phase-plane trajectories for moves of many different lengths. This collection of data is then mapped directly to the best choices for the available free parameters used for generating the trajectory, in a single-step process (Oswald, 1974; Brown and Ma, 1968). This method, however, requires several iterations of experiments to increase the reliability of the switching point. Therefore, the above methods impose difficulties and inconvenience in theoretically and experimentally selecting the switching point. To solve these problems, the approach taken in this work is to use the position trajectory directly without switching the control modes, while maintaining the concept of timeoptimal control. This control method is called "direct tracking" control to distinguish it from hybrid control. If a time optimized position trajectory is generated as a reference, and if it is ensured that the error between the reference position trajectory and the real position trajectory approaches zero, it is possible to achieve the desired performance of high speed and accurate tracking motion. In order to do this, an adaptive model-following control (AMFC) loop has been built. The time-optimized position trajectory is simply generated by integrating, twice, a bang-bang signal which is easily constructed using simple physical laws according to the traveling distance. Direct tracking control using the AMFC scheme is applied to a magneto-optical disk drive (MODD) to improve its slow tracking performance, which is a significant disadvantage of the current magnetooptical system. A performance comparison between direct tracking control and conventional hybrid control is experimentally presented in this paper.
2. AMFC SYSTEM DESIGN In this section, the development of adaptive modelfollowing control is reviewed (Landau, 1979; Goodwin and Sin, 1984). The real plant to be controlled is described by = Apx+npup,
(1)
where Ap and Bp are the real plant matrices, and
--~Rof~r~nce[~--{~ _N
-]MechanismtK~(e,t)~ Fig. 1. A parallel AMF Cconfiguration with signalsynthesis adaptation.
x = [x~ x2] r. x~ and x2 are the tracking actuator position and the velocity of the real plant, and Up is the input voltage to the plant. Similarly, the reference model is described by ~t, = AmXm+ Braun,
(2)
where Am and B m a r e the model plant matrices, and x,, =[x~,l x~,21r. x,,l and Xm~ are the desired tracking actuator position and velocity of the reference model, and u~ is the input voltage to the reference model. Consider a parallel AMFC configuration with signalsynthesis adaptation, as shown in Fig. 1. If up = up1 + u : ,
(3)
where up1 is the linear control signal, and Up2 is the adaptation signal, one can then choose up1 = - K p y+K mx~ +K,, Urn, Up2 = A K p ( e , t ) x + A K , ( e , t ) u m ,
(4) (5)
where Kp, K,,, and K, are linear control gains. AKp(e,t) and AK~(e,t) are adaptation control gains having matrices of appropriate dimensions. Rewriting eq.(1) using eqs. (3), (4), and (5), gives = IA v - B v K v +BpKm +BpAKp(e,t)lx + BpIK., + AK. (e,t)lu,. + B?Kme. (6) Recall that the error e = xm- x. Subtracting eq. (6) from eq. (2), and adding and then subtracting the term (Am - BpKm )x, the following vector differential equation is obtained: - BpKr~)e + {Bp[Kp - K m - AKp(e,t)l + Am - Ap }x + {Bin - Bp IK, + AKp (e, t)l}Um. (7)
= (Am
To have perfect model following, it is necessary to ensure that for any u,~, piecewise continuous, and initial state e. = 0, e(t) = ifft) = 0. This can be achieved if the last two terms in the right-hand side of
Direct Tracking Control using Time-Optimal Trajectories eq.(7) equal zero for any x and urn. The following results are derived from the above statement. A~, - A p = B p ( K , , - K ° ) ,
(8)
B,. = BpK°u,
(9)
where K~ and K°u are the unknown values of Kp and K , that would ensure the perfect model following. In addition to this, the remaining unforced system = (A,, - BpK~)e
(10)
must be asymptotically stable, which implies that the matrix (Ar~ - BpK~, ) must be a Hurwitz matrix. The pair ( A ~ , B p ) must be completely controllable. This controllability can beproved using the values of (Am, Bp ) in Section 4. From eq. (2), perfect model following can be achieved if rank [ Bp ] = rank [Be, An - Ap ] = rank [Bp, Bm].
(ll) To ensure the global stability of an adaptive system, Lyapunov's direct method and hyperstability theory are commonly used in the adaptive literature (Narendra and Annawamy, 1989). To achieve global stability in an adaptive system, the hyperstability and positivity theory is used, which shows that an adaptive feedback system is asymptotically hyperstable, as described in detail in the appendix.
3. DIRECT TRACKING CONTROL SYSTEM 3.1 Tracking system for a magneto-optical disk drive A magneto-optical disk is a memory device which reads and writes data optically, using a magnetic field. Because magneto-optical disk systems are erasable they are being increasingly used as external computer memory devices. The MODD has some advantages over the traditional magnetic floppy disk drive; namely, a higher capacity due to a high track density, and less susceptibility to mechanical wear because only the laser beam affects the surface of the magneto-optical disk. On the other hand, it has one significant disadvantage compared to a magnetic system; namely, its slow access performance, which results in a low data-transfer rate. The relatively sluggish performance of the MODD derives from a few factors (Marchant, 1990; Sierra, 1990): the reading/writing mechanisms need two degrees of actuation, tracking and focusing, leading to a need for heavy actuators; many of the optical components required for transmitting and receiving a laser beam also make the system heavy; also a small track pitch causes tracking difficulties.
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Two separate tracking servo mechanisms have traditionally been used in tandem in the the tracking operation, for coarse and fine tracking (Koumura, et al., 1989; Inada, et al., 1986). In the tracking operation, the moving part or tracking actuator, which has an objective lens and a focusing actuator, is driven radially towards the target track. It is first coarsely positioned in the vicinity of the target track by a coarse tracking actuator. Next, it is fmely positioned by a fine tracking actuator such as a galvano-mirror. In this tracking operation, most of the access time is spent on coarse positioning, since the fine tracking actuator has a higher bandwidth than the coarse one. Therefore, it can be said that the access time is mostly dependent on the tracking time, which is determined by coarse tracking. In the MODD system, position and velocity information are embedded in the magneto-optical disk for tracking, to enable an optical head to approach a target track quickly and accurately. Since position information, for example, is written on the disk, no external position sensing system is required. However, optical components, such as a laser beam and optical pick up devices, are required for an optical head to read the position and velocity information directly from an optical disk. An external measurement device is used in the work described here, to replace the role of the optical disk. A laser interferometer is used as a position and velocity sensor. A newly developed moving magnet-type tracking actuator has been implemented for a MODD. The system to be used as a coarse tracking actuator is shown in Fig. 2 and described in (Park and BuschVishniac, 1994). It consists of an air core solenoid, four permanent magnets attached to aluminum rods, a focusing actuator which is mounted on the tracking actuator, 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.
Fig. 2. View of the developed linear coarse tracking actuator of a magneto-optical disk drive.
Kyihwan Park et al.
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Since the mass center of the tracking actuator is near the centerline of the two aluminum rods, the driving force acts on the mass center of the total system, and is assumed to be proportional to the current in the conductor. A detailed configuration of the tracking actuator, which shows the geometry of the permanent magnet and air core solenoid, is shown in Fig. 3. The definitions of the design specifications are listed in Table 1. To obtain the dynamic characteristics of the tracking system, a frequency response test was performed experimentally, and is shown in Fig. 4. 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 Gp can be uniquely determined by employing a curve-fitting technique. However, since the parameters of the plant tend to vary from time to time because of nonlinear friction, stiction, or temperature variation, Gp is considered to be bounded between Gp# and Gp2; the maximum and minimum of Gp. They are expressed respectively as 22,000 s2 +80s+100' 10,000 Gp2 = sZ O.O01s 3+ +20s+100"
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-
(12) (13)
only a mass and a viscous damping coefficient, Since
Table 1 The specification of the tracking actuator Description Width of permanent magnet(a ) Length of permanent magnet(b ) Height of permanent magnet(c ) Inner radius of solenoid(rs ) Outer radius of solenoid(r2) Length of solenoid(/ ) Interval between permanent magnets(lp) Air gap(lg) Coil resistance Number of turns Mass of tracking actuator Force constant
Value 6 mm 12 mm 4 mm 2 mm 24 mm 10 mm 26 mm 1 mm 8.5 1000 0.025 kg 1.0 N/A
Fig. 4. The open-loop characteristics of the tracking system. the proposed tracking actuator is composed of ideally it should show the first-order delay plus integral characteristics. The difference between the transfer functions can be attributed to modeling errors associated with the system's static and dynamic characteristics.
3.2 Reference model
The reference model is simply chosen to be a double integrator because it provides the position trajectory easily and accurately. Therefore, the open-loop transfer function of the reference model, Gin, is represented as
11,000
(14)
Gm- - -
S2
The constant in the numerator is determined such that the maximum voltage and current of the air core solenoid do not exceed the electrical power availability. From eq. (14), the model plant matrices Am and Bm are
A':E: "°=[11,000} 0
Figures 5(a) and (b) show how a position trajectory is generated from the bang-bang signal through the velocity trajectory, v,, and am are the maximum velocity and maximum acceleration of the tracking system respectively, and they are determined from the electrical power availability and plant characteristics. ts is the switching point (time) where the maximum acceleration is changed to the maximum deceleration. t.~s and ts2 are the switching points(times) where the maximum acceleration is changed to zero acceleration, and zero acceleration is changed to the maximum deceleration, respectively. Then, tsl is obtained as vm/am. Since v,, and am are determined when mechanical and electrical specifications are known, the only thing to be selected for producing the reference position
Direct Tracking Control using Time-Optimal Trajectories x
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when a target point is given. Now that the real tracking system model and reference model have been obtained, it remains only to build an adaptive control loop to ensure that the plant behaves as specified by means of the reference model that produces a time-optimized position trajectory.
X
v
x. 4. SIMULATION RESULTS
tsl ts2
t ]
tsl ts2
t
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Fig. 5 Procedures of how time-optimized position trajectories are generated by double integration of a bang-bang signal, (a) through a triangular velocity profile and (b) through a trapezoidal velocity profile.
trajectory is the switching point(time), which can be is calculated according to the distance traveled. If the actuator moves with its maximum acceleration am, the velocity v is obtained as a ~ L . Then, the distance traveled, x , is x = vts = a j 2 , .
(IV)
When the target distance is smaller than xm, the reference position trajectory is generated from the bang-bang signal shaped as shown in Fig. 5(a). When x is given, the switching point is determined as from eq. (16). If the tracking actuator follows the trajectory well, the tracking time is expected to be ~rf/am
2xf-f/a,. .
P and D are obtained by solving eqs (A-5) and (A6) respectively. In one of the solutions of the Popov integral inequality introduced in Section 2, it was assumed that, AKp(e,0) = AK,,(e,0) = 0, and that F~(t)=Fa > 0, F ' ( t ) = F ' > O ,
Fb(t)=Fb,
F~'(t)= Fb'
(16)
The longest distance x~ to be traveled by an actuator with a triangular velociW profile is
x= : v . t , , : v ~ / a . .
In order to make the error dynamics described by eq. (10) decay at a fast rate, Km is selected properly, using the information on Am and Bp. The positive definite matrix Q is selected so that H(s) has a strictly positive real transfer function. When Q is
Gh(t) = G~,(t) = 1.
(21)
Several simulation results give a few design guidelines; all linear control gains, Ku and Kp, have some freedom in their determination because they have little influence on the tracking performance. tx, 13, Fa,F,', K,,, and Kp are chosen as 1, 10, 1, 1, 10, and [ 1 1], respectively. The AMFC system designed in Section 2 is represented by the detailed configuration in Fig. 6. All the block diagrams were constructed in SIMULINK, which is a commercial system dynamics
(18)
When the target distance is larger than xm, the reference position trajectory is generated from the bang-bang signal shaped as shown in Fig. 5(b). The distance traveled is X -X m = Vmts2.
(19)
If the tracking actuator follows the trajectory well, the tracking time is expected to be tsl + t.,.2. Through the above procedure, the position trajectory is generated from integrating the bang-bang signal twice, without switching from one control mode to another, while maintaining the time-optimal principle. The position trajectory can be generated in real time
Fig. 6. The block diagram of the AMFC control scheme.
Kyihwan Park et al.
1236 Position (ram)
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Fig. 7. Simulation results Case I : Control performance with sensor noise when the plant model is described by ~,, =22,00o/t:+80s÷10o).
Fig. 8. Simulation results - Case II • Control performance with sensor noise when the plant model is described by c,,~=lO.OOO/(O.OO~s'..,'~+2o.,+1oo).
and control program that includes MATLAB (The MathWorks Inc., 1992). To simulate real system environments as closely as possible, a maximum current constraint for the solenoid was imposed. The maximum and minimum currents were set to +2.0 A and -2.0 A, respectively. Since the sampling time currently used for servo control systems of MODD is decreasing due to the rapid development of digital signal processors, a sampling time of 0. lms was used for the digital simulation. Since the actual actuator has stiction or nonlinear friction and since these cause a finite gain at low frequency, integrators in the feedback gain loops were introduced to suppress the steady-state error (Semba, 1993).
model located between Gpz and Gp2. For this simulation, the modeled dynamics are of relatively low frequency, and are not excited by the sensor noise transmitted through control. When the system dynamics has a large coefficient of third order, the steady-state tracking error increases. This result seems to be reasonable because only two states (position and velocity) are used in the feedback loop for this simulation. In the simulation two plant model were applied and the experimental tests used control gains obtained from the simulation results.
Investigations into how the AMFC system performs for the two different plant models will give a few design guidelines to determine the control gains in the AMFC. The first simulation corresponds to trajectory tracking scheduled to travel 6 mm with sensor noise when the plant model is described by Gpz, eq. (12). 6 mm was chosen because it is the average tracking distance of a 3.5 in disk drive. Figure 7 shows the reference and plant positions, reference and plant velocities, and the control input and applied noise. There are almost no tracking errors observed in the position and velocity trajectories. The second simulation corresponds to the same situation, with the plant model described by Gp2, eq. 13. Figure 8 shows the reference and plant positions, the reference and plant velocities, and the control input and applied noise.
Figure 9 shows an experimental setup for the timeoptimal tracking system applied to a 3.5 in MODD. The data acquisition and digital control module were built using a PC/C31 data acquisition board from Loughborough and an TMS320C31 digital signal processor. A IBM PC 486 serves as a host computer. An OFV501 laser interferometer and an OFV3000 vibrometer controller from Polytec were used as position and velocity sensors. The experimental results using hybrid control and direct tracking control are presented for comparison.
Referring to Figs feedback noise, tracking position performance can
7 and 8, the AMFC controller, with provides accurate control of the trajectory. From these results, good also be anticipated as to the plant
5. EXPERIMENTAL RESULTS
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Fig. 10 A control scheme used for the conventional hybrid control system.
Figure 10 shows a control scheme used for the conventional hybrid control system. The bang-bang control loop is cut off a few tracks ahead o f 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 a PID controller are used for velocity and position control, respectively. Figure 11 shows the experimental result o f the speed performance for a 3.5 in M O D D whose average
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tracking distance is 6 mm. The measured velocity, position, and position error are presented, with the control inputs. The average tracking time is around 25ms including the settling time, 7 or 8 ms, required for velocity and position control. Figures 12 to 14 show the experimental results of the speed performance when adaptive model-following control was used: the measured position signals, position errors, velocity signals, and control inputs are presented, with the reference positions and reference velocities for the different tracking distances, 3 mm, 6 mm, and 10 mm. The saturation voltage is set to 6 V which corresponds to 2 A. However, when the reference velocity reaches zero, it was set to 2 V to reduce the noise in the steady state.
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Kyihwan Park et al.
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Fig. 13. The experimental results when the AMFC is used - Case II: for a 6 mm tracking distance.
The maximum velocity was restricted to 0.68 m/s. The maximum acceleration was 75 m/s 2 Then, the largest distance traveled in the triangular mode is 6.2 mm from eq. (17). Therefore, bang-bang inputs which generate triangular velocity profiles are used for the 3 mm and 6 mm tracking distances as shown in Figs 12 and 13 respectively. A bang-bang input which generates a trapezoidal velocity profile was used for the 10 mm tracking distance, as shown in Fig. 14. However, the bang-bang inputs are not shown in the figures. Referring to the simulation data presented in Figs 7 and 8 , the experimental results are similar to the simulation results. There are almost no steady-state
Fig. 14 The experimental results when the AMFC is used - Case III: for a 10 mm tracking distance.
tracking errors observed in the position and velocity trajectory. Especially, the position error in the steady state is within the sensor noise level. It can be estimated from Fig. 13 that the average tracking time to move 6 mm is around 19 ms. This result is similar to the theoretical result which is obtainable from eq. (18). This represents an improvement over the 25 ms tracking time achieved by using the conventional controller. It is also a marked improvement over the roughly 50 ms tracking times reported for commercial magneto-optical systems of comparable size. Therefore, it can be concluded that direct tracking control using the adaptive following control scheme provides high tracking accuracy, as well as high tracking speed.
Direct Tracking Control using Time-Optimal Trajectories 6. CONCLUSION Direct tracking control using an adaptive modelfollowing control scheme is implemented here for the time-optimal control of a high-speed tracking system. The real plant is experimentally obtained from the frequency response. The reference model is chosen so that it is simply described, and the switching point is easily and accurately determined. Direct tracking control relieved the difficulty and inconvenience of theoretically and experimentally selecting the switching point. Direct tracking control was applied to a prototype magneto-optical disk drive to improve the system's speed performance. The simulation and experimental results demonstrate an excellent model-following capability. Compared with the hybrid control system, the tracking time is considerabley decreased in the direct tracking control system. According to the experimental results, an average access time as low as 19 ms can be obtained for a 3.5 in MODD. This result indicates that magneto-optical disk systems can be competitive with magnetic disk systems in terms of access times.
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Company, Menlo Park, California. Oswald, R.K. (1974). Design of a disk file headpositioning servo, IBM d. Res. Develop., Vol. 18 pp.506-512. Park, K. H. and Busch-Vishniac, I. (1994). A New Tracking System for Magneto-optical Disk Drives. Part I KSME Vol. 9, pp 295 -302. Semba, T. (1993). "'Model-following Digital Servo Using Multirate Sampling for an Optical Disk Drive, Japan d. Applied Physics Vol. 32 Pt. 1 No. 11B, pp5385-5391. Sierra, H. M. (1990). An Introduction to Direct Access Storage Devices, Academic Press, San Diego, Califomia. The MathWorks, I n c . (1993). User's guide, SIMULINK, Natick, Mass. 01760.
APPENDIX Consider the adaptive feedback system, depicted in Fig. A1, which is formed by a linear time-invariant feedforward block and feedback block which can be nonlinear time-varying. The feedback block belongs to the family of linear, nonlinear and time-varying systems which satisfy an input/output relation of the form
REFERENCES Ananthanarayanan, K.S. (1982). Third-order theory and bang-bang control of voice coil actuator, IEEE Trans. on Magnetics, Vol MAG-18, No. 3, pp.888-892. Brown, C.J. and Ma, J.T. (1968). Time-optimal Control of a Moving-Coil Linear Actuator, IBMJ. Res. Devlop., Vol. 12 pp.372-379. Eguchi, N., Tobita, M. and Ogawa, M. (1990). An 86 mm Magneto-optical disk drive with a Compact and Fast-seek-time Optical Head, Optical Data Storage, Proc. SPIE Vol. 1316, pp 2-10. Goodwin, G. C. and Sin, K. S. (1984). Adaptive Filtering Prediction and Control, Prentice-Halt, Englewood Cliffs. Inada, H., Nomura, T., Iwanaga, T. and Koumura, K. (1986). Tracking Servo for Small Size Optical Disk Systems," SPIE, Optical Mass Data Storage 11, Vol. 695, pp. 130-137. Koumu ra, K., Takizawa, F., Iwanaga, T. and Inada, H. (1989).High Speed Accessing using Split Optical Head, SPIE, Data Storage Topical Meeting, Vol. 1078, pp. 239-243. Landau, I. D. (1979). Adaptive Control (The Model Reference Approach), Marcel Dekker, New York. Leitmann, G. (1966). An Introduction to Optimal Control, McGraw-Hill Book Company, New York. Narendra, K. S. and Annawamy A. M. (1989). Stable Adaptive Systems, Prentice-Hall, Englewood Cliffs. Marchant, Alan B. (1990). Optical Recording, A Technical Overview, Addison-Wesley Publishing
rl(O,t,)= I~' vrwdt >--y~, tl >0,
(A-I)
where v is the input vector, w is the output of the feedback block, and y2o is a finite positive constant. This integral inequality will be referred to as the "Popov integral inequality". Hyperstability and positivity theory say that the feedback system is asymptotically hyperstable if two conditions are satisfied (Landau, 1979): the equivalent feed-forward block is a strictly positive real transfer function or matrix, and the feedback block must satisfy the Popov integral inequality of eq. (A-l). To transform the AMFC system into the form of an equivalent feedback system composed of two blocks, one in the feed-forward path and one in the feedback path, eq. (7) in Section 2 is rewritten by the following equations: e=(A,.-BpKm)e+Bpwj,
(A-2)
w~,.J Linear v "]Time-invariant~Block Non-linear w ITime-varying Block Fig. A1. The standard feedback system.
nonlinear
time-varying
Kyihwan Park et al.
1240 v = De,
+ AK ~(0),
(A-3)
(A-8)
where wl represents the input of the feedforward block using the eqs (8) and (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 ensured, v is the input of the feedback block.
The feedback equations are obtained in the feedback block using eqs. (A-7) and (A-8) as follows:
First, examine the adaptation mechanism in the feedforward block. The feed-forward transfer function, H(s), represented as
=[~qh(v,t,x)dx +q)2(v,t)+AKu(O)]u,. (A-9)
H(s)=D(sI-Am +BpKm)-lBp,
(A-4)
should be a strictly positive real transfer matrix if and only if there must exist a P and a positive definite matrix Q such that
(A,~-BpKm)rP+P(Am-BpKm) D=BprP.
=-Q,
(A-5) (A-6)
Next, for the adaptation mechanism in the feedback block, one of the solutions of the Popov integral inequality introduced by Landau (1979) is used. The adaptation control gains, AKp and AKu are described using an integral plus proportional adaptation law as
AKp(e,t)= AKp(v,t) = ~O,(v,t,x)dt + 02(v,t) +AKp(0),
(A-7)
AKu(e,t)= A K p ( v , t ) = ~(pl(v,t,x)dt +qo2(v,t)
w = -wl = [AKp (v,t) + K ° - Kp lx +[AK.(v,t)+ K ° -K~Iu. : [j'~¢,(v,t, x)d'c +02 (v,t)+AKp (0)]y
where AK ° =AKp - K p +K °,
(A-t0)
AK°u= A K u - K . + K ° . .
(A-11)
Here, it is necessary find solutions in the feedback eq (A-9), such that the Popov integral inequality is satisfied. One of the solutions is listed below: 01(v, t, x) = Fo(t - x)v(x)lGo (t)y(x)] r, x >__t (A- 12) 02 (v, t) = F" (t)v(x)lG" (t)y(t)] r,
(A- 13)
q)~(v,t,x) = Fb(t -x)v(x)IGb(t)u,,(x)lr,x >-t (A-14) ~P2(v, t) = F~(t)v(x)IG~ (t)u~ (t)l r
(A- 15)
where F o ( t - x ) and F~ ( t - x ) 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'(t), Fg(t), G'(t) and G~,(t) are time-varying positive definite matrices for all t>0.
Pergamon PII:S0967-0661 (96)00129-3
DIRECT TRACKING CONTROL USING TIME-OPTIMAL TRAJECTORIES Kyihwan Park*, Jung-Hyun Lee**, Soo-Hyun Kim** and Yoon Keun Kwak** *Departmentof Mechatronics, Kwang-JuInstitute of Science and Technology, 502, Sangam-dong, Kwangsan-gu, Kwangju 506-303, Korea **Departmentof Mechanical Engineering, Korea AdvancedInstitute of Science and Technology(KAIST), 373-1, Kusong-dong, Yusong-gu, Taejon305-701, Korea
(Received April 1995; in final form July 1996)
Abstract: Direct tracking control which is capable of high positioning speed and accuracy is developed using adaptive model-following control (AMFC). Direct tracking control relieves the difficulty and inconvenience of selecting the switching points in conventional hybrid control composed of bang-bang, velocity, and position control. AMFC can make the controlled plant follow, as closely as possible, a desired reference model where the time optimized positioning signal is generated. Direct tracking control using AMFC is applied to a tracking system of a magneto-optical disk drive to improve its slow tracking performance. Compared with the hybrid control system, the tracking time is much decreased in the direct tracking control system. According to the simulation and experimental results, an average access time as small as 19 ms is achieved. Keywords: Direct tracking control, time-optimal, adaptive model-following control, hybrid control, linear tracking actuators, magneto-optical disk drives.
1. INTRODUCTION The most important thing to be considered in servocontrol for a tracking system is how fast it moves to the target track, with the accuracy required. A conventional method of meeting these requirements is to use hybrid control, composed of bang-bang control, velocity control, and position control. For the minimum time requirement, a bang-bang controller is used because it is the theoretical solution for timeoptimal control (Leitmann, 1966; Ananthanarayanan, 1982). For the high positional accuracy requirement, a high-precision position controller is also used. For a smooth connection between those control modes, a velocity controller can be used. These control modes can be combined to establish the desired performance by switching the system from one control mode to another. However, it is difficult to calculate the switching
points where bang-bang control is connected to velocity control and where velocity control is connected to position control. Indeed, when the switching point is not well selected, the hybrid control method may provide worse speed performance than when only position control is employed. The difficulty of selecting a correct switching point is due to the fact that it is calculated based on the system model. In other words, since bang-bang control is open-loop, its performance characteristics are dependent on the system components and the plant characteristics. This dependency makes it hard to determine the switching point correctly when system model uncertainties exist. To compensate for the system uncertainties, closedloop velocity control is commonly used, together with position control (Koumura, et al., 1989; Inada, et al., 1986; Eguchi, et al., 1990). In this method, when a target track is determined, a velocity trajectory is generated and stored in advance in a 1231
Kyihwan Park et al.
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look-up table, because it takes time to calculate the velocity trajectory in a real-time control process. The velocity control can increase the system bandwidth compared to the bang-bang control. However, a substantial effort is still required in correctly anticipating the switching point in order to calculate the velocity trajectory in advance. As an experimental way of approaching the timeoptimal control problem, the switching points in bang-bang control can be obtained by applying maximum and minimum voltages to the real system, and measuring the phase-plane trajectories for moves of many different lengths. This collection of data is then mapped directly to the best choices for the available free parameters used for generating the trajectory, in a single-step process (Oswald, 1974; Brown and Ma, 1968). This method, however, requires several iterations of experiments to increase the reliability of the switching point. Therefore, the above methods impose difficulties and inconvenience in theoretically and experimentally selecting the switching point. To solve these problems, the approach taken in this work is to use the position trajectory directly without switching the control modes, while maintaining the concept of timeoptimal control. This control method is called "direct tracking" control to distinguish it from hybrid control. If a time optimized position trajectory is generated as a reference, and if it is ensured that the error between the reference position trajectory and the real position trajectory approaches zero, it is possible to achieve the desired performance of high speed and accurate tracking motion. In order to do this, an adaptive model-following control (AMFC) loop has been built. The time-optimized position trajectory is simply generated by integrating, twice, a bang-bang signal which is easily constructed using simple physical laws according to the traveling distance. Direct tracking control using the AMFC scheme is applied to a magneto-optical disk drive (MODD) to improve its slow tracking performance, which is a significant disadvantage of the current magnetooptical system. A performance comparison between direct tracking control and conventional hybrid control is experimentally presented in this paper.
2. AMFC SYSTEM DESIGN In this section, the development of adaptive modelfollowing control is reviewed (Landau, 1979; Goodwin and Sin, 1984). The real plant to be controlled is described by = Apx+npup,
(1)
where Ap and Bp are the real plant matrices, and
--~Rof~r~nce[~--{~ _N
-]MechanismtK~(e,t)~ Fig. 1. A parallel AMF Cconfiguration with signalsynthesis adaptation.
x = [x~ x2] r. x~ and x2 are the tracking actuator position and the velocity of the real plant, and Up is the input voltage to the plant. Similarly, the reference model is described by ~t, = AmXm+ Braun,
(2)
where Am and B m a r e the model plant matrices, and x,, =[x~,l x~,21r. x,,l and Xm~ are the desired tracking actuator position and velocity of the reference model, and u~ is the input voltage to the reference model. Consider a parallel AMFC configuration with signalsynthesis adaptation, as shown in Fig. 1. If up = up1 + u : ,
(3)
where up1 is the linear control signal, and Up2 is the adaptation signal, one can then choose up1 = - K p y+K mx~ +K,, Urn, Up2 = A K p ( e , t ) x + A K , ( e , t ) u m ,
(4) (5)
where Kp, K,,, and K, are linear control gains. AKp(e,t) and AK~(e,t) are adaptation control gains having matrices of appropriate dimensions. Rewriting eq.(1) using eqs. (3), (4), and (5), gives = IA v - B v K v +BpKm +BpAKp(e,t)lx + BpIK., + AK. (e,t)lu,. + B?Kme. (6) Recall that the error e = xm- x. Subtracting eq. (6) from eq. (2), and adding and then subtracting the term (Am - BpKm )x, the following vector differential equation is obtained: - BpKr~)e + {Bp[Kp - K m - AKp(e,t)l + Am - Ap }x + {Bin - Bp IK, + AKp (e, t)l}Um. (7)
= (Am
To have perfect model following, it is necessary to ensure that for any u,~, piecewise continuous, and initial state e. = 0, e(t) = ifft) = 0. This can be achieved if the last two terms in the right-hand side of
Direct Tracking Control using Time-Optimal Trajectories eq.(7) equal zero for any x and urn. The following results are derived from the above statement. A~, - A p = B p ( K , , - K ° ) ,
(8)
B,. = BpK°u,
(9)
where K~ and K°u are the unknown values of Kp and K , that would ensure the perfect model following. In addition to this, the remaining unforced system = (A,, - BpK~)e
(10)
must be asymptotically stable, which implies that the matrix (Ar~ - BpK~, ) must be a Hurwitz matrix. The pair ( A ~ , B p ) must be completely controllable. This controllability can beproved using the values of (Am, Bp ) in Section 4. From eq. (2), perfect model following can be achieved if rank [ Bp ] = rank [Be, An - Ap ] = rank [Bp, Bm].
(ll) To ensure the global stability of an adaptive system, Lyapunov's direct method and hyperstability theory are commonly used in the adaptive literature (Narendra and Annawamy, 1989). To achieve global stability in an adaptive system, the hyperstability and positivity theory is used, which shows that an adaptive feedback system is asymptotically hyperstable, as described in detail in the appendix.
3. DIRECT TRACKING CONTROL SYSTEM 3.1 Tracking system for a magneto-optical disk drive A magneto-optical disk is a memory device which reads and writes data optically, using a magnetic field. Because magneto-optical disk systems are erasable they are being increasingly used as external computer memory devices. The MODD has some advantages over the traditional magnetic floppy disk drive; namely, a higher capacity due to a high track density, and less susceptibility to mechanical wear because only the laser beam affects the surface of the magneto-optical disk. On the other hand, it has one significant disadvantage compared to a magnetic system; namely, its slow access performance, which results in a low data-transfer rate. The relatively sluggish performance of the MODD derives from a few factors (Marchant, 1990; Sierra, 1990): the reading/writing mechanisms need two degrees of actuation, tracking and focusing, leading to a need for heavy actuators; many of the optical components required for transmitting and receiving a laser beam also make the system heavy; also a small track pitch causes tracking difficulties.
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Two separate tracking servo mechanisms have traditionally been used in tandem in the the tracking operation, for coarse and fine tracking (Koumura, et al., 1989; Inada, et al., 1986). In the tracking operation, the moving part or tracking actuator, which has an objective lens and a focusing actuator, is driven radially towards the target track. It is first coarsely positioned in the vicinity of the target track by a coarse tracking actuator. Next, it is fmely positioned by a fine tracking actuator such as a galvano-mirror. In this tracking operation, most of the access time is spent on coarse positioning, since the fine tracking actuator has a higher bandwidth than the coarse one. Therefore, it can be said that the access time is mostly dependent on the tracking time, which is determined by coarse tracking. In the MODD system, position and velocity information are embedded in the magneto-optical disk for tracking, to enable an optical head to approach a target track quickly and accurately. Since position information, for example, is written on the disk, no external position sensing system is required. However, optical components, such as a laser beam and optical pick up devices, are required for an optical head to read the position and velocity information directly from an optical disk. An external measurement device is used in the work described here, to replace the role of the optical disk. A laser interferometer is used as a position and velocity sensor. A newly developed moving magnet-type tracking actuator has been implemented for a MODD. The system to be used as a coarse tracking actuator is shown in Fig. 2 and described in (Park and BuschVishniac, 1994). It consists of an air core solenoid, four permanent magnets attached to aluminum rods, a focusing actuator which is mounted on the tracking actuator, 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.
Fig. 2. View of the developed linear coarse tracking actuator of a magneto-optical disk drive.
Kyihwan Park et al.
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)~%: t1111111,IIIIIIll, 0., oo IIIt1111/°:°~11,o, ,o: ,o, Frequency (Hz)
Since the mass center of the tracking actuator is near the centerline of the two aluminum rods, the driving force acts on the mass center of the total system, and is assumed to be proportional to the current in the conductor. A detailed configuration of the tracking actuator, which shows the geometry of the permanent magnet and air core solenoid, is shown in Fig. 3. The definitions of the design specifications are listed in Table 1. To obtain the dynamic characteristics of the tracking system, a frequency response test was performed experimentally, and is shown in Fig. 4. 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 Gp can be uniquely determined by employing a curve-fitting technique. However, since the parameters of the plant tend to vary from time to time because of nonlinear friction, stiction, or temperature variation, Gp is considered to be bounded between Gp# and Gp2; the maximum and minimum of Gp. They are expressed respectively as 22,000 s2 +80s+100' 10,000 Gp2 = sZ O.O01s 3+ +20s+100"
Gpt
-
(12) (13)
only a mass and a viscous damping coefficient, Since
Table 1 The specification of the tracking actuator Description Width of permanent magnet(a ) Length of permanent magnet(b ) Height of permanent magnet(c ) Inner radius of solenoid(rs ) Outer radius of solenoid(r2) Length of solenoid(/ ) Interval between permanent magnets(lp) Air gap(lg) Coil resistance Number of turns Mass of tracking actuator Force constant
Value 6 mm 12 mm 4 mm 2 mm 24 mm 10 mm 26 mm 1 mm 8.5 1000 0.025 kg 1.0 N/A
Fig. 4. The open-loop characteristics of the tracking system. the proposed tracking actuator is composed of ideally it should show the first-order delay plus integral characteristics. The difference between the transfer functions can be attributed to modeling errors associated with the system's static and dynamic characteristics.
3.2 Reference model
The reference model is simply chosen to be a double integrator because it provides the position trajectory easily and accurately. Therefore, the open-loop transfer function of the reference model, Gin, is represented as
11,000
(14)
Gm- - -
S2
The constant in the numerator is determined such that the maximum voltage and current of the air core solenoid do not exceed the electrical power availability. From eq. (14), the model plant matrices Am and Bm are
A':E: "°=[11,000} 0
Figures 5(a) and (b) show how a position trajectory is generated from the bang-bang signal through the velocity trajectory, v,, and am are the maximum velocity and maximum acceleration of the tracking system respectively, and they are determined from the electrical power availability and plant characteristics. ts is the switching point (time) where the maximum acceleration is changed to the maximum deceleration. t.~s and ts2 are the switching points(times) where the maximum acceleration is changed to zero acceleration, and zero acceleration is changed to the maximum deceleration, respectively. Then, tsl is obtained as vm/am. Since v,, and am are determined when mechanical and electrical specifications are known, the only thing to be selected for producing the reference position
Direct Tracking Control using Time-Optimal Trajectories x
i
am
__kVm
(a) ts I
t
ts l
t
1235
when a target point is given. Now that the real tracking system model and reference model have been obtained, it remains only to build an adaptive control loop to ensure that the plant behaves as specified by means of the reference model that produces a time-optimized position trajectory.
X
v
x. 4. SIMULATION RESULTS
tsl ts2
t ]
tsl ts2
t
tsl ts2 f
Fig. 5 Procedures of how time-optimized position trajectories are generated by double integration of a bang-bang signal, (a) through a triangular velocity profile and (b) through a trapezoidal velocity profile.
trajectory is the switching point(time), which can be is calculated according to the distance traveled. If the actuator moves with its maximum acceleration am, the velocity v is obtained as a ~ L . Then, the distance traveled, x , is x = vts = a j 2 , .
(IV)
When the target distance is smaller than xm, the reference position trajectory is generated from the bang-bang signal shaped as shown in Fig. 5(a). When x is given, the switching point is determined as from eq. (16). If the tracking actuator follows the trajectory well, the tracking time is expected to be ~rf/am
2xf-f/a,. .
P and D are obtained by solving eqs (A-5) and (A6) respectively. In one of the solutions of the Popov integral inequality introduced in Section 2, it was assumed that, AKp(e,0) = AK,,(e,0) = 0, and that F~(t)=Fa > 0, F ' ( t ) = F ' > O ,
Fb(t)=Fb,
F~'(t)= Fb'
(16)
The longest distance x~ to be traveled by an actuator with a triangular velociW profile is
x= : v . t , , : v ~ / a . .
In order to make the error dynamics described by eq. (10) decay at a fast rate, Km is selected properly, using the information on Am and Bp. The positive definite matrix Q is selected so that H(s) has a strictly positive real transfer function. When Q is
Gh(t) = G~,(t) = 1.
(21)
Several simulation results give a few design guidelines; all linear control gains, Ku and Kp, have some freedom in their determination because they have little influence on the tracking performance. tx, 13, Fa,F,', K,,, and Kp are chosen as 1, 10, 1, 1, 10, and [ 1 1], respectively. The AMFC system designed in Section 2 is represented by the detailed configuration in Fig. 6. All the block diagrams were constructed in SIMULINK, which is a commercial system dynamics
(18)
When the target distance is larger than xm, the reference position trajectory is generated from the bang-bang signal shaped as shown in Fig. 5(b). The distance traveled is X -X m = Vmts2.
(19)
If the tracking actuator follows the trajectory well, the tracking time is expected to be tsl + t.,.2. Through the above procedure, the position trajectory is generated from integrating the bang-bang signal twice, without switching from one control mode to another, while maintaining the time-optimal principle. The position trajectory can be generated in real time
Fig. 6. The block diagram of the AMFC control scheme.
Kyihwan Park et al.
1236 Position (ram)
Position (mm) Reference
! I
p o s ~ t i ~ 1i
0
0.005
0.01
0.015
0.02
0.025 Time (s)
Velocity(m/s)
I oI 0 I
0
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~,~,.-,
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0.01
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t ]
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0.005
0.01
0.015
0.02
0.025 Time is)
-t0
0
0.005
0.01
0.015
0.02
0.025 Time (s)
Fig. 7. Simulation results Case I : Control performance with sensor noise when the plant model is described by ~,, =22,00o/t:+80s÷10o).
Fig. 8. Simulation results - Case II • Control performance with sensor noise when the plant model is described by c,,~=lO.OOO/(O.OO~s'..,'~+2o.,+1oo).
and control program that includes MATLAB (The MathWorks Inc., 1992). To simulate real system environments as closely as possible, a maximum current constraint for the solenoid was imposed. The maximum and minimum currents were set to +2.0 A and -2.0 A, respectively. Since the sampling time currently used for servo control systems of MODD is decreasing due to the rapid development of digital signal processors, a sampling time of 0. lms was used for the digital simulation. Since the actual actuator has stiction or nonlinear friction and since these cause a finite gain at low frequency, integrators in the feedback gain loops were introduced to suppress the steady-state error (Semba, 1993).
model located between Gpz and Gp2. For this simulation, the modeled dynamics are of relatively low frequency, and are not excited by the sensor noise transmitted through control. When the system dynamics has a large coefficient of third order, the steady-state tracking error increases. This result seems to be reasonable because only two states (position and velocity) are used in the feedback loop for this simulation. In the simulation two plant model were applied and the experimental tests used control gains obtained from the simulation results.
Investigations into how the AMFC system performs for the two different plant models will give a few design guidelines to determine the control gains in the AMFC. The first simulation corresponds to trajectory tracking scheduled to travel 6 mm with sensor noise when the plant model is described by Gpz, eq. (12). 6 mm was chosen because it is the average tracking distance of a 3.5 in disk drive. Figure 7 shows the reference and plant positions, reference and plant velocities, and the control input and applied noise. There are almost no tracking errors observed in the position and velocity trajectories. The second simulation corresponds to the same situation, with the plant model described by Gp2, eq. 13. Figure 8 shows the reference and plant positions, the reference and plant velocities, and the control input and applied noise.
Figure 9 shows an experimental setup for the timeoptimal tracking system applied to a 3.5 in MODD. The data acquisition and digital control module were built using a PC/C31 data acquisition board from Loughborough and an TMS320C31 digital signal processor. A IBM PC 486 serves as a host computer. An OFV501 laser interferometer and an OFV3000 vibrometer controller from Polytec were used as position and velocity sensors. The experimental results using hybrid control and direct tracking control are presented for comparison.
Referring to Figs feedback noise, tracking position performance can
7 and 8, the AMFC controller, with provides accurate control of the trajectory. From these results, good also be anticipated as to the plant
5. EXPERIMENTAL RESULTS
i Current amplifier!
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Time(5ms/div)
Fig. 10 A control scheme used for the conventional hybrid control system.
Figure 10 shows a control scheme used for the conventional hybrid control system. The bang-bang control loop is cut off a few tracks ahead o f 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 a PID controller are used for velocity and position control, respectively. Figure 11 shows the experimental result o f the speed performance for a 3.5 in M O D D whose average
Velocity (m/s) s.oo;~ Sn~l~Xre'm* 1.0 0.75 " Reference ivieasu~a - ...... •. . . . . . . . 0.50.. veolcity~..velocity ..... :~..... ..... . . . . ..... 0.25 0.0 Time(5ms/div Position error (~m) 24001 ::
s.oo~
1 8 0 0 [ . . . . i ......... . . . . .
', ........ -
1200 I. . ......... ~:....
i ....... ~
6oo ......
....
....... i
..... .....
~9~fxsr0P .... . . . . . . .
Time(5ms/div Control input(V)
5.oo~/
Sn@l~l STOP
2
Velocity (m/s) Control input(V) 1.01 : ~ ~ ~ 8 + : % : 0.75 ..................................... +i....i...i ......... i ..... 6
+
050[ . . . . . .
"
0 . 0 ~
-6 -8
, i Bang-bang input ~:.
-4
Velocity..\.--....~.... i ......... i ..... 2
. . . . . . . . i . . . . . . :,.i . . . . . . . . ~
....
-2
: ...... --i..-.i...i ± ;---'~......... - - " 'i -..... - 4
'
0 25 [........ ! ~ i
0
:
.
....... +"+'-i
.......
1121.....
Fig. 12 The experimental results when the AMFC is used - Case I: for a 3 mm tracking distance.
-i Position input -~
,
....... T ......... i + " : : + i
Time(5ms/div)
0
~-~
...................
:
:
. . . . . . i .................. i ...... - - i Velocity input. Time (5.0ms/div) Position error(~m)
Position (mm) |0
..... i..
7.5
~
:
~
i
1000
5.0 [
....... i ......... i ......... +........ + ..+......... 750 Position i ; _i ......... ! li ...... ::~ ' ~ Settling time 500
2.5
+. . . . . .
[ i~ ~
+..... E:rror
~25ms;
...
+......... 250
~-~! Time (5.0ms/div
Fig. 11 The experimental result when the hybrid controller is used for a 3.5" in magneto-optical disk drive whose average tracking distance is 6 mm.
tracking distance is 6 mm. The measured velocity, position, and position error are presented, with the control inputs. The average tracking time is around 25ms including the settling time, 7 or 8 ms, required for velocity and position control. Figures 12 to 14 show the experimental results of the speed performance when adaptive model-following control was used: the measured position signals, position errors, velocity signals, and control inputs are presented, with the reference positions and reference velocities for the different tracking distances, 3 mm, 6 mm, and 10 mm. The saturation voltage is set to 6 V which corresponds to 2 A. However, when the reference velocity reaches zero, it was set to 2 V to reduce the noise in the steady state.
1238
Kyihwan Park et al.
Position (mm) 10
5 .oo~/
S r e 1~:1. STOI
7.5
Reference ~ i:: i ~. . . . . 5.0 ....... p?sition : '~';::I:i:: ....... i ..... ::i ...... : 2.5 ~ position / i i :: 0.0 ~ ' .i . ; . - ~- 1 9 \ m- s !- - " i -" ----, . . . . ' . . . . ' . . . . Time(5ms/div) Velocity (m/s) 5 .ooW/ Sngl~l STOP 1.0 !Reference ~ i ~ ! i 0.75 ...... ! vel0ci! .::.~...i..M~asured......... i......... i......... i 0.50
Position (mm) 10
5.0~s/
7.5 ......... Refer( ......... i~e(ere= 5.0 ......... positi(
~........~ i ~.~ i ...... i .........
...... ~::/~-~: ~ -
~~Me~ured 2.5 .......... )... ....... ~ ~ ! y o ~ 0.0
..... i!......... i ......... i ......... 1
....... i ....
i ....
i ....
/
Time(5ms/div) Velocity (m/s)
5.oo~/
Snsl~1 STOP
0.75 0.50
0.25 0.0
.... ! .... i ..... i
0.25 0.0
Time(5ms/div) Position error (gin) 400l i ~
:
Y-
s .OO~s/
snml~tl
STOP
800it ................................... ~ ........ ~......... ~......... i ......... 2ooi ......... i ......... i ......... ) ........ ~ ........ i
6°°I .........
5nsl~l STOP
m
~
"
...........................
..... ; .......... i ......... i .........
Time(5ms/div) Position error (rm) 2400
~
SnS1~1 STOP
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;
1800 .............. i!.................. i ....... ~. . . . . . . . . . . . . . . . . . . . i.. 1200 .............. ~:.................. ~ .................................... i 600
o
o
)
i
Time(5ms/div) Control input(V)
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-
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i ........
~.
i
"
.7..
-- .................. i., 2..,:2:.,.,~..,; ~.,;-......
-6 Time(5ms/div)
-8 Time(5ms/div)
Fig. 13. The experimental results when the AMFC is used - Case II: for a 6 mm tracking distance.
The maximum velocity was restricted to 0.68 m/s. The maximum acceleration was 75 m/s 2 Then, the largest distance traveled in the triangular mode is 6.2 mm from eq. (17). Therefore, bang-bang inputs which generate triangular velocity profiles are used for the 3 mm and 6 mm tracking distances as shown in Figs 12 and 13 respectively. A bang-bang input which generates a trapezoidal velocity profile was used for the 10 mm tracking distance, as shown in Fig. 14. However, the bang-bang inputs are not shown in the figures. Referring to the simulation data presented in Figs 7 and 8 , the experimental results are similar to the simulation results. There are almost no steady-state
Fig. 14 The experimental results when the AMFC is used - Case III: for a 10 mm tracking distance.
tracking errors observed in the position and velocity trajectory. Especially, the position error in the steady state is within the sensor noise level. It can be estimated from Fig. 13 that the average tracking time to move 6 mm is around 19 ms. This result is similar to the theoretical result which is obtainable from eq. (18). This represents an improvement over the 25 ms tracking time achieved by using the conventional controller. It is also a marked improvement over the roughly 50 ms tracking times reported for commercial magneto-optical systems of comparable size. Therefore, it can be concluded that direct tracking control using the adaptive following control scheme provides high tracking accuracy, as well as high tracking speed.
Direct Tracking Control using Time-Optimal Trajectories 6. CONCLUSION Direct tracking control using an adaptive modelfollowing control scheme is implemented here for the time-optimal control of a high-speed tracking system. The real plant is experimentally obtained from the frequency response. The reference model is chosen so that it is simply described, and the switching point is easily and accurately determined. Direct tracking control relieved the difficulty and inconvenience of theoretically and experimentally selecting the switching point. Direct tracking control was applied to a prototype magneto-optical disk drive to improve the system's speed performance. The simulation and experimental results demonstrate an excellent model-following capability. Compared with the hybrid control system, the tracking time is considerabley decreased in the direct tracking control system. According to the experimental results, an average access time as low as 19 ms can be obtained for a 3.5 in MODD. This result indicates that magneto-optical disk systems can be competitive with magnetic disk systems in terms of access times.
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Company, Menlo Park, California. Oswald, R.K. (1974). Design of a disk file headpositioning servo, IBM d. Res. Develop., Vol. 18 pp.506-512. Park, K. H. and Busch-Vishniac, I. (1994). A New Tracking System for Magneto-optical Disk Drives. Part I KSME Vol. 9, pp 295 -302. Semba, T. (1993). "'Model-following Digital Servo Using Multirate Sampling for an Optical Disk Drive, Japan d. Applied Physics Vol. 32 Pt. 1 No. 11B, pp5385-5391. Sierra, H. M. (1990). An Introduction to Direct Access Storage Devices, Academic Press, San Diego, Califomia. The MathWorks, I n c . (1993). User's guide, SIMULINK, Natick, Mass. 01760.
APPENDIX Consider the adaptive feedback system, depicted in Fig. A1, which is formed by a linear time-invariant feedforward block and feedback block which can be nonlinear time-varying. The feedback block belongs to the family of linear, nonlinear and time-varying systems which satisfy an input/output relation of the form
REFERENCES Ananthanarayanan, K.S. (1982). Third-order theory and bang-bang control of voice coil actuator, IEEE Trans. on Magnetics, Vol MAG-18, No. 3, pp.888-892. Brown, C.J. and Ma, J.T. (1968). Time-optimal Control of a Moving-Coil Linear Actuator, IBMJ. Res. Devlop., Vol. 12 pp.372-379. Eguchi, N., Tobita, M. and Ogawa, M. (1990). An 86 mm Magneto-optical disk drive with a Compact and Fast-seek-time Optical Head, Optical Data Storage, Proc. SPIE Vol. 1316, pp 2-10. Goodwin, G. C. and Sin, K. S. (1984). Adaptive Filtering Prediction and Control, Prentice-Halt, Englewood Cliffs. Inada, H., Nomura, T., Iwanaga, T. and Koumura, K. (1986). Tracking Servo for Small Size Optical Disk Systems," SPIE, Optical Mass Data Storage 11, Vol. 695, pp. 130-137. Koumu ra, K., Takizawa, F., Iwanaga, T. and Inada, H. (1989).High Speed Accessing using Split Optical Head, SPIE, Data Storage Topical Meeting, Vol. 1078, pp. 239-243. Landau, I. D. (1979). Adaptive Control (The Model Reference Approach), Marcel Dekker, New York. Leitmann, G. (1966). An Introduction to Optimal Control, McGraw-Hill Book Company, New York. Narendra, K. S. and Annawamy A. M. (1989). Stable Adaptive Systems, Prentice-Hall, Englewood Cliffs. Marchant, Alan B. (1990). Optical Recording, A Technical Overview, Addison-Wesley Publishing
rl(O,t,)= I~' vrwdt >--y~, tl >0,
(A-I)
where v is the input vector, w is the output of the feedback block, and y2o is a finite positive constant. This integral inequality will be referred to as the "Popov integral inequality". Hyperstability and positivity theory say that the feedback system is asymptotically hyperstable if two conditions are satisfied (Landau, 1979): the equivalent feed-forward block is a strictly positive real transfer function or matrix, and the feedback block must satisfy the Popov integral inequality of eq. (A-l). To transform the AMFC system into the form of an equivalent feedback system composed of two blocks, one in the feed-forward path and one in the feedback path, eq. (7) in Section 2 is rewritten by the following equations: e=(A,.-BpKm)e+Bpwj,
(A-2)
w~,.J Linear v "]Time-invariant~Block Non-linear w ITime-varying Block Fig. A1. The standard feedback system.
nonlinear
time-varying
Kyihwan Park et al.
1240 v = De,
+ AK ~(0),
(A-3)
(A-8)
where wl represents the input of the feedforward block using the eqs (8) and (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 ensured, v is the input of the feedback block.
The feedback equations are obtained in the feedback block using eqs. (A-7) and (A-8) as follows:
First, examine the adaptation mechanism in the feedforward block. The feed-forward transfer function, H(s), represented as
=[~qh(v,t,x)dx +q)2(v,t)+AKu(O)]u,. (A-9)
H(s)=D(sI-Am +BpKm)-lBp,
(A-4)
should be a strictly positive real transfer matrix if and only if there must exist a P and a positive definite matrix Q such that
(A,~-BpKm)rP+P(Am-BpKm) D=BprP.
=-Q,
(A-5) (A-6)
Next, for the adaptation mechanism in the feedback block, one of the solutions of the Popov integral inequality introduced by Landau (1979) is used. The adaptation control gains, AKp and AKu are described using an integral plus proportional adaptation law as
AKp(e,t)= AKp(v,t) = ~O,(v,t,x)dt + 02(v,t) +AKp(0),
(A-7)
AKu(e,t)= A K p ( v , t ) = ~(pl(v,t,x)dt +qo2(v,t)
w = -wl = [AKp (v,t) + K ° - Kp lx +[AK.(v,t)+ K ° -K~Iu. : [j'~¢,(v,t, x)d'c +02 (v,t)+AKp (0)]y
where AK ° =AKp - K p +K °,
(A-t0)
AK°u= A K u - K . + K ° . .
(A-11)
Here, it is necessary find solutions in the feedback eq (A-9), such that the Popov integral inequality is satisfied. One of the solutions is listed below: 01(v, t, x) = Fo(t - x)v(x)lGo (t)y(x)] r, x >__t (A- 12) 02 (v, t) = F" (t)v(x)lG" (t)y(t)] r,
(A- 13)
q)~(v,t,x) = Fb(t -x)v(x)IGb(t)u,,(x)lr,x >-t (A-14) ~P2(v, t) = F~(t)v(x)IG~ (t)u~ (t)l r
(A- 15)
where F o ( t - x ) and F~ ( t - x ) 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'(t), Fg(t), G'(t) and G~,(t) are time-varying positive definite matrices for all t>0.