- Email: [email protected]
Cost effective repetitive controllers for data storage devices
14th World Congress oflFAC
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
B-le-Ol-3
Copyright © 1999 IF AC 14th Triennial World Congress, Beijing, P.R. China
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STORAGE DEVICES Craig Smith *
Kenji Takeuchi
Masayoshi Tomizuka *
0/ Mechanical Engineering University of California at Berkeley Berkeley, Cali/o'rnia 94720-1140, USA
* Department
[email protected], [email protected],
[email protected]
Abstract: Repetitive control for data storage devices is revisited. Attempts to make repetitive control more cost effective through reductions in memory use and/or computational complexity are presented. Frequency domain interpretations of stability and performance are used for design, and simulation and experimental studies confirm the validity of the proposed structures. Copyright © 1999 IFAC . Keywords: Data storage; discrete time systems; digital control; interpolation; inverse transfer function; periodic motion; mechanical systems
1. INTRODUCTION
The computer disk drive industry is constantly striving for increased storage densities. This requires advances in both hardware and software technologies, making the motion control of the read/write head a representative case of mechatronics. The servo system found in newer disk drives is based on the idea of sector servo, in which track information is distributed along circular tracks at equally spaced angles; equally divided areas of the disk are called sectors. With the disk rotating at a fixed angular frequency and position error information available only as the head passes from one sector to another (servo information is on the edges of sectors), the sector servo system is inherently a discrete time control system. A challenge in the design of these systems is to minimize the number of sectors (equivalently the sampling rate), while continuing to meet tracking performance requirements. In this paper, the use of repetitive control to meet this challenge is revisited with special emphasis on implementation costs. Design alternatives which attain similar levels of performance with different memory and
computation costs are evaluated by simulation and experiment. 1.1 Background For disk drives, keeping the read/write head on a particular data track is referred to as track following. Moving the head from one track to another is called track seeking. This paper considers only the track following servo loop. During track following, the plant for which servo control is being designed behaves approximately like a double integrat or. For most sector servo systems, the controller only has access to the sampled position err or of the read/write head. Stabilization of the double integrator plant can be achieved with a p r oportional plus derivative (PD) controller (implemented in discrete time). Typically, the bandwidth of the dosed loop system is constrained by the possibility of high frequency resonant modes in the actuator assembly. This bandwidth constraint will limit the gain of the feedback loop at lower frequencies if no control other than PD control is applied. Any digital controller is also subject to computation delay, measurement noise, and control noise.
891
Copyright 1999 IFAC
ISBN: 0 08 043248 4
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
14th World Congress ofTFAC
Fig. 2. Small gain theorem structure Fig. 1. Basic Repetitive Controller Structure The necessity of additional control is a result of the various disturbances which are typically present in disk drive servo systems. The combined effect of windage and the flexible printed circuit tension is a constant torque disturbance to the actuator during track following. Integral action is usually incorporated into the feedback controller to eliminate steady state errors. Another source of disturbances is the shape of the data tracks. Ideally, they are concentric circles centered at the axis of rotation of the disk, but in reality they will not be perfect circles, nor will they be perfectly centered. Deviations from the ideal shape are referred to as repetitive runout because they repeat as the disk rotates. Often, but especially for removable media disk drives, the centering error will cause most of the repetitive Iunout with the fundamental frequency matching the spindle speed. Other components of repetitive runout can be written in terms of harmonics of this fundamental frequency. If the disk is made of an anisotropic material (e.g. flexible media), contraction or expansion will make circular tracks elliptic. The resulting disturbance will be a sinusoid at the first harmonic frequency. Repetitive disturbances at the sixth harmonic and higher can appear due to mechanical faults in the spindle assembly. Some additional control is needed to reduce potentially large tracking errors due to repetitive runouts. From a frequency response point of view, the loop gain needs to be increased at the repetitive frequencies. Repetitive .control does exactly that. 1.2 Repetitive Control
The objective of repetitive control is to eliminate or reduce the effect of repeating disturbances. Thought of this way, the internal model principle is used to design repetitive controllers. In the sequel, the discrete time repetitive controller presented by Tomizuka et al. (1989) is what will be referred to as the basic repetitive controller. The structure of this controller (for a disturbance which repeats every N samples) is shown in Fig. 1. This repetitive controller structure is used for an asymptotically stable plant, so when applying this controller to the disk drive servo problem, the "plant" includes stabilizing feedback control (e.g. PD). This basic repetitive controller structure is easily separated into two components: the inverse dynamics and the internal model.
1.2.1. Basic Internal Model The internal model portion of the repetitive controller is composed partly of a delay chain. A delay chain can be easily implemented as a table with moving index. If the period of the repetitive disturbance is large (N is large), the table will be large as well. If the filter q, in Fig. 1, is set equal to 1 and the input to the internal model is set to zero, the control applied will be a repeating signal (cycle through the table). If control inputs which cancel the disturbance are stored in the table, the output of the plant will remain zero and the table values will not change. There are practical reasons, however, to introduce Iql < 1 at higher frequencies. In doing this, perfect tracking of a repetitive disturbance is sacrificed in order to ensure the stability of the closed loop system in the presence of model uncertainties. 1.2.2. Stability and Performance Analysis When analyzing the stability of this repetitive control structure, the small gain theorem is quite useful. By closing the feedback loop inside the box labeled "Plant", and calling the resulting system Gp, the block diagram in Fig. 1 can easily be regrouped as shown in Fig. 2. This grouping ignores the input, but puts the system in a form where stability is easy to check. A block d has been added to the block diagram to represent dynamiC uncertainty in the closed loop system. Applying the small gain theorem, the closed loop system will be asymptotically stable if the system inside the dashed lines, (z-cl - (1 + d) GpG inv )q , has infinity norm less than 1. An infinity nOrm smaller than 1 indicates some measure of stability robustness. Consider the case where G inv is selected such that GpGiTlfJ = z-d. The resulting sufficient condition for stability provided by the small gain theorem is IILlqlloo < 1. Some plant uncertainty is expected at higher frequencies (there may be unmodeled dynamics), so it is prudent to pick the frequency response gain of q to be 1 at low frequency and rolling off slightly at higher frequency. The choice for q described by Chiu et al. (1993) is a zerophase error FIR filter. Conveniently, such filters are easily implemented. In order to evaluate the closed loop performance of the system, however, we must return to the block diagram in Fig. 1. The closed loop transfer function from repetitive runout to the tracking error is
892
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO .. .
G er =
( 1) l+G n G c
1-
qz-N
-l-_-QZ---N""+""C"U-("":z:""_"":"d-_-G-,,-G-'n-o-)
(1)
If q = 1 at a particular repetitive frequency, it is clear that the gain of the closed loop at that frequency is zero. Choosing a q with zero phase means that the response of 1 - qz-N at harmonic frequencies ",ill be a positive number less than 1. If 11 (z-d-GpGinv)qlloo = 'Y < I, then the closed loop poles coming from the second factor of GeT will all have magnitude less than ')'. Larger ")'S mean longer settling times for the closed loop system.
1.2.3. Basic Inverse Dynamics The above stability development provides motivation (Tomizuka et al., 1989) for using the inverse dynamics or approximate inverse based on the zero-phase-error (ZPE) compensator (Tomizuka, 1987). Briefly, if the plant transfer function is of the form -1
Gp(Z
)
=
z-dpB+(Z-1)B-Cz- 1) A(z-l)
(2)
where B+ has roots all outside the unit circle and both B- and A have all roots inside the unit circle, the ZPE compensator will be: -1
Gzpe(Z
)
==
zd pA(z-l)B+(z) B-(z 1)B+(1)2
(3)
The ZPE style plant inversion requires advance knowledge of the input (which is not available in this case), so d delays are added to the ZPE inverse, G inv = z-dG zpe , such that G inv is realizable (d = dp+dB +, where d B + is the order of B+(z)). Due to the zero-phase properties designed for, z - d and GpGinv will have the same phase at all frequencies. Using Eqs. (2) and (3) along with the block diagram in Fig. 2, a sufficient condition for asymptotic stability of the closed loop system is
11(1 - a - Lla)qlloo < 1 -1 a(z
)
=
B+(z-1 )B+(z) B+(1)2
(4) (5)
The transfer function a in this case takes on positive real values less than 1. Notice that if II~qlloo < 1, and I1qlloo < 1, the condition given in Eq. (4) will be satisfied. This simplification relies heavily on the ZPE design of the inverse dynamics, and makes the frequency response of q the most significant factor in determining the nominal steady-state performance.
2. COST EFFECTIVE REPETITIVE CONTROL Now, the focus shifts to the costs of implementing a repetitive control system. Looking at the frequency domain developments of stability and performance criteria in the preceding sections,
Fig. 3. Multirate repetitive control with a reduced order internal model one might be tempted to apply an optimal loopshaping control to design the filters q and G inv · An attempt to minimize the amount of computation prohibits this approach. In fact, all modifications considered either reduce or maintain the cost relative to the "basic" repetitive controller described earlier. The delay chain internal model with FIR filter is a low computational burden, but does use a significant amount of memory. Therefore, methods of applying repetitive control with reduced memory usage are investigated. In contrast, the ZPE inverse requires significant computation. In some cases, applying this computation effort to a higher order feedback compensator may obviate the need for repetitive control.
2.1 Modifications to the Internal Model The internal model in large part determines the steady-state performance of the repetitive control scheme (see section 1.2.2). The primary cost associated with implementing the internal model in the basic repetitive controller is memory. As previously mentioned, the delay chain can be implemented as table with moving index. This table has as many elements as sectors on the disk. The filter q is selected to be a lowpass filter with a peak gain of 1 to provide a tradeoff' between tracking performance and stability robustness . In the case where the fundamental repetitive frequency and its lower order harmonics are the primary components of the repetitive disturbance, the gain of q can be quite low at higher frequencies - less attenuation of those repetitive frequencies is necessary. Alternatively, the table can be decimated. If there are N table elements and N is divisible by r, decimating by a factor of r means using a table of length M = Nlr instead. The information in the shorter table is still used to characterize the entire repetitive disturbance. It can be thought of as using only every rth element from a table of length N, and hoping that by looking at every rth element, enough information about the disturbance is present to cancel it. The high frequency components of the disturbance will not be visible (Nyquist frequency) unless aliased . If this reduced order repetitive control is applied as though the sampling rate of the system was a factor of r slower, each table value would be the control for r measurements.
It is not necessary to hold the same control for r samples. Information about the next control
893
Copyright 1999 IF AC
ISBN: 0 08 043248 4
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
is accesible. This allows for linear interpol",,"~~u of the repetitive controller at the measurement sampling instants. A block diagram analagous to that shown in Fig. 2 can be found in Fig. 3. Interpolation (I) and decimation (V) blocks have been incorporated into the diagram. The small gain theorem is again used to obtain a stability criterion (11(z-d - DGinvGpI)qlloc < 1). To evaluate performance, the r x r transfer matrix for the closed loop track runout to error (there are r runouts per sample at the slow sampling rate) is evaluated at frequencies up to 1/2r of the measurement frequency. Frequency domain analysis cannot be extended beyond this frequency, because the repetitive controller is a periodically time varying system when considered at the measurement sample rate. A significant amount of complexity is added to the design process as well; in addition to q and G inv , the interpolation and decimation schemes also affect the robust stability criterion. Specifically, they can be used to match phases associated with delays shorter than r measurement samples ( the d' in Fig. 3 is not the same as the d in Fig. 1). It should be noted that the order of V and Ginv can be switched, that is, Ginv can be implemented as a discrete system at the slower sampling rate, with the decimated plant output as its input. That option, although it has potential to save computation, is not addressed in this paper.
2.2 Approximate Inverses For disk drive track following servos, the "plant" to which repetitive control is to be applied can typically be modeled as a fifth order discrete time system. Two states come from the double integrator characteristics aforementioned, two come from the PD controller which stabilizes the double integrator system, and one state models the computation delay. This leads to a fifth order model of the inverse dynamics if the ZPE inverse is used. There are memory and more importantly, computational costs associated with a fifth order inverse. These costs may not be considered large for some applications, but for disk drives, reductions in this cost are likely to be welcomed. For any proposed inverse, the stability condition must be satisfied for a q which reflects the tradeoff between desired steady-state rejection of repetitive disturbances and robustness to uncertainty.
3. EXPERIMENTAL VALIDATION In this section, times are measured in samples, and frequencies in cycles per sample. Various repetitive controllers are designed for a sector servo disk drive system with 80 sectors.
14th World Congress ofTFAC
~2or~ll ~o~·~. 1
f~~~~t c. -360
j
.•.•
's;] I
---'----~-~-----"...----'
t..._
0.01
0.05 0.1 0.5 freq(cycles/samp) Fig. 4. Bode plot of nommal feedoack loop gain The disk drive model used is obtained from the experimental setup. It consists, as is typical, of a double integrator system with delay and a stabilizing feedback controller. The frequency response characteristics of the nominal closed loop system (corresponding to Gp in Fig. 2) are shown in Fig. 4. The nominal closed loop bandwidth is limited by the possibility of resonant modes at frequencies above 0.15 cycles/samp. The tracking performance goal is to maintain less than 0.1 tracks of error, preventing corruption of data on adjacent tracks. The experimental system exhibits track runouts primarily at the fundamental frequency. This provided a nice starting point for experimental evaluation of cost reduction ideas, because it is not a particularly challenging repetitive signal. The amplitude of this runout was approximately 17 tracks. Measurement and control noise are included in time domain simulations. For the time being, the effect of these noises on tracking performance is not considered from an analytical point of view.
3.1 Analysis
In Sections 1.2.2 and 2.1, the stability and performance of repetitive control sytems was discussed based on the small gain theorem. Plots of (Z-d VGin",GpL)q are used as a graphical indicator of stability robustness. he closed loop will be stable if the gain of this quantity is less than 1. Comparing how close different structures come to this stability limit is a way to assess relative robustness of the closed loop system. Another comparison tactic is to look at the performance at the frequencies which make up a repetitive signal. The closed loop transfer function from track runout to position error is formed and evaluated at the fundamental repetitive frequency and its harmonics. The resulting attenuations (att) are given in (dB). In addition, since this is a nominal performance, the minimum magnitude uncertainty which leads to a violation of the stability criterion is also computed at the harmonics. The idea is that these values will reflect the likelihood that the nominal performance will be acheived. If there is little tolerance for uncertainty at a particular harmonic, small uncertainties may have a large effect on the performance at that harmonic.
894
Copyright 1999 IF AC
ISBN: 008 0432484
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
14th World Congress ofTFAC
:: :::: :::··:····J····i·····'~·I···l :::: ::::: :; .. ::::: : "'S 101··r··· . . . ...... : ~-lg
~ ~ ~.!
-20
!I
•• , p'
0
....<... .
l"I' .
"
~o:H;;;;;;;;;;:,;:.,
..
. .ll
0.01
0.05 0.1 0.5 freq( cycles/samp) Fig. 5. Comparison of three different inverse dynamics used with an 80 memory internal model.
s~ 0.5?II:..:: _.
roc
zpe
att
97
une 1.0
att 97 73 58 31
. " ' : ' . . . . ';':
'.-.-.~'
.... . •
.
0.1 0.025 0.05 freq (cycles/samp) Fig. 6. Stability comparisons for r = 4
i
~rA~~{f~tLAaJ ~-5~
1- ~~I
-5='---_-'
~ g~~.,.~~~j ~-~
une 1.0
The final method in which designs will be compared is more standard. Experimental tracking errors are averaged to capture the repeating part of position error. Discrete fourier transforms are used to compute the repetitive frequency content of time domain signals.
L......._....J
0.01
gn
att 97 0.96 1.0 73 73 0.91 58 1.0 59 31 1.0 0.67 35 0.61 11 15 1.1 14 11 4.6 0.25 1.5 8.4 1.5 8.5 23 'unc' : magnitude of destabilizing uncertainty 'att' : attenuation of closed loop (dB) 0 1 2 5
une 1.0 1.0 1.0 1.0 1.1
:
I' 'E5~"" '1
hold lIlterp .' ':.' ...
o
Table 1. Comparison of inverses harmonic
I,*,,"t.liA ~....,. ~'.l~\
~ o! t.. "i"" ",±~i?til~ ::l.~
~>~ j.~.:..
.. , .. : . ~'
'g'!;n
E 0.5 .... , ......._. . .
! ! ~ ~ ~ , . € Ai ,Jf! ...... , .. , ... .
~ 100 . ;;-; I ~ ~ 50 . En A
~ 1~".'~, ' '''~~~;~ . .... +'\.,.... ..~
');"~~:
I I
10 20 30 40 50 60 70 80 time(samples) Fig. 7. Experimental averaged PES time traces
.6., which can cause instability of the closed loop system is shown for selected harmonics. Note that since feedback control is included in wha.t is considered the plant, ~ is not likely to be very large at frequencies within the bandwidth of the closed loop, so the nominal closed loop gains are close to the actual gains. When the inverse (gn) is used, the closed loop is less robust to uncertainty than the (zpe) and (sec) inverses.
3.2 Estimated Inverses
3.3 Inverses with Reduced Order Repetitive Control
To start, three different choices for G inv are compared when used in conjunction with the basic repetitive control internal model (80 memory delay chain and q = (z + 6 + z-1)/8). The bode plots of the three inverses, and magnitude plots of (z-d - GpGinv)q in each case are shown in Fig. 5. Notice that the fifth order ZPE based inverse (zpe) is nearly indistingushable from its second order approximation (sec) in the frequency domain. If the approximate inverse is set to a fixed gain (gn) , the difference in phase is significant at higher frequencies. The DC gaiml of all of these inverses are set such that the DC gain of GpG inv = O.S. This results in a slightly less aggressive repetitive controller (known as a result of practical experience). The lower plot in Fig. 5 illustrates the nomina.l robust stability of the system. Notice that the gain inverse (gn) brings the system closer to instability than the ZPE and its approximate. Choosing d so that the phases of z-d and GpGinv are close helps the closed loop system meet the stability requirements . Table 1 further compares the three proposed inverses. The nominal attenuation (in dB) of the closed loop system, and the size of the smallest uncertainty,
Similar comparisons to those above are made when the delay chain is decimated by a factor of 4. Frequency response data is therefore valid only up to one fourth of the Nyquist frequency associated with the measurement rate. Every 4th output of G inv is used to drive the internal model (decimation). Unless otherwise noted, the output of the repetitive controller is linearly interpolated between successive elements in the table. In Fig. 6, the upper plot is of the same robust stability indicator we have used before, now applied to a reduced order repetitive control system with decimation and interpolation. It is qualitatively very similar to the lower plot of Fig. 5. This is expected. The inverses are being implemented at the measurement rate of the plant: they are not affected by the memory reduction of the internal model. The upper plot in Fig. 7 shows a comparison of the averaged experimental time traces for repetitive control reduced by a factor of 4 with Ginv being set to both the (sec) and (gn) inverses described above. Both control structures achieve the tracking goal. Errors at the 9th harmonic persist when (gn) is used. This persistence is related to the fact that this harmonic is the Nyquist frequency for the reduced sample rate.
895
Copyright 1999 IF AC
ISBN: 008 0432484
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
14th World Congress ofTFAC
Table 2. Comparison between q's harmonic
q4
q128
qlS
une 0.91
att
unc
74
0.90
0.84
att 61 37
0.73
50
0.71
att 101 72
2
0.81
23
0.56
35
0.51
54
3
0.95
13
0.44
25
0.33
41
0
une 0.93
1
3.4 The Effect of q The development in section 1.2.2 suggested that the filter q can play a large role in both the stability robustness and the performance of repetitive controllers. The center plot in Fig. 6 shows how the stability of a reduced order repetitive controller (r = 4) is affected by different choices of the filter q. The inverse dynamics used for these comparisons is the cheap one, (gn). The filter q is implemented at 1/4 the measurement rate. The three q's compared are all easily implemented on a digital CQInputer because the divisions are all by powers of 2. They are, with mildest roHoff listed last: q4 = (z + 2 + z~1)/4
+ 14 + z-1)/16 q128 = (z + 126 + z-l )/128 q16
=
(z
The effect of q on the stability robustness is seen in the figure. Notice that the magnitude moves closer to 1 as q's with less roUoff are selected. The classic tradeoff of feedback control, sensitivity vs. complementary sensitivity, appears. This tradeoff is apparent in the calculated performance and uncertainty tolerances analagous to Table 1 shown in Table 2. The lower plot in Fig. 7 displays experimental corroboration of the data in the table. The traces shown are the average periodic error over 12 revolutions. Notice that although the fundamental component is completely gone from the trace for q = Q128, again the 9th harmonic is very visible.
3.5 Interpolation and A!iasing The most likely reason for the appearance of higher harmonic components in the errors is aliasing. By picking every 4th output of the inverse plant model, higher frequency components are aliased. By simulating the response to a 7th harmonic repetitive disturbance with amplitude 10 tracks, this aliasing effect is readily visible in the upper plot of Fig. 8: both 7th and 11th harmonics are present in the error signal. Holding the repetitive signal instead of interpolating increases the magnitude of the errors. The lower plot in Fig. 6 shows the effect of interpolation on the stability robustness of the reduced by 4 repetitive controller with inverse (gn) and q = (z+30+z- 1 )/32. Using the same repetitive control parameters, the effect of not interpolating the repetitive control
~
0
.
Fourier coeff ot simulated PES.
10 5 :;;; 0 ?A .f:I 1 ~ 0.5
•
~
hClld
inle,,;
I
--.J. x 0
Fourier coeff of experimental
•o!IJllljI_ : w5¥ •• 5R!I __~.~.
'ES..
1
. ." J hold interp
O--"-'~.--"--'LJL..!
25 30 20 15 coefficient Fig. 8. Experimental and simulated evidence of aliasing
0
5
10
output is examined experimentally. The Fourier coefficients are plotted in the lower half of Fig. 8. More fundamental frequency error is present when the repetitive control output is not interpolated. Furthermore, without interpolation, errors at the aliased equivalents of the fundamental frequency, the 18th and 20th harmonics, are also large. This effect is much smaller when interpolation is used. A direction for future work is to incorporate antialiasing into the decimation.
4. CONCLUSION The work presented in this paper has been motivated by the goal of making repetitive control more accessible to applications by reducing the computational and/or memory costs ofimplementation. The simulation and experimental results demonstrate that this goal can be achieved without sacrificing the objectives of repetitive controL In particular, by using a simple gain instead of a fifth order inverse and one quarter the memory of a standard repetitive controller, almost no performance at the fundamental repetitive frequency was lost. The frequency domain analysis of the reduced order repetitive controllers was restricted by the Nyquist frequency associated with the new sampling rate. Moving away from frequency domain ideas, and into operator based performance characterizations will be the next step toward making fair comparisons of the proposed designs. It may also allow for less ad-hoc choices of interpolation and decimation filters. 5. REFERENCES Chiu, Tsu-Chih et al. (1993). Compensation for repeatable and non-repeatable tracking errors in disk file systems. Proceedings of the JSME International Conference on Advanced Mechatronics pp. 710-717. Tomizuka., M. (1987). Zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems, Measurement, and Control109, 65~68. Tomizuka, M., T-C. Tsao and K-K. Chew (1989). Analysis and synthesis of discrete-time repetitive controllers. ASME Journal of Dynamic System, Measurement, and Control III , 353358.
896
Copyright 1999 IF AC
ISBN: 008 0432484
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
B-le-Ol-3
Copyright © 1999 IF AC 14th Triennial World Congress, Beijing, P.R. China
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STORAGE DEVICES Craig Smith *
Kenji Takeuchi
Masayoshi Tomizuka *
0/ Mechanical Engineering University of California at Berkeley Berkeley, Cali/o'rnia 94720-1140, USA
* Department
[email protected], [email protected],
[email protected]
Abstract: Repetitive control for data storage devices is revisited. Attempts to make repetitive control more cost effective through reductions in memory use and/or computational complexity are presented. Frequency domain interpretations of stability and performance are used for design, and simulation and experimental studies confirm the validity of the proposed structures. Copyright © 1999 IFAC . Keywords: Data storage; discrete time systems; digital control; interpolation; inverse transfer function; periodic motion; mechanical systems
1. INTRODUCTION
The computer disk drive industry is constantly striving for increased storage densities. This requires advances in both hardware and software technologies, making the motion control of the read/write head a representative case of mechatronics. The servo system found in newer disk drives is based on the idea of sector servo, in which track information is distributed along circular tracks at equally spaced angles; equally divided areas of the disk are called sectors. With the disk rotating at a fixed angular frequency and position error information available only as the head passes from one sector to another (servo information is on the edges of sectors), the sector servo system is inherently a discrete time control system. A challenge in the design of these systems is to minimize the number of sectors (equivalently the sampling rate), while continuing to meet tracking performance requirements. In this paper, the use of repetitive control to meet this challenge is revisited with special emphasis on implementation costs. Design alternatives which attain similar levels of performance with different memory and
computation costs are evaluated by simulation and experiment. 1.1 Background For disk drives, keeping the read/write head on a particular data track is referred to as track following. Moving the head from one track to another is called track seeking. This paper considers only the track following servo loop. During track following, the plant for which servo control is being designed behaves approximately like a double integrat or. For most sector servo systems, the controller only has access to the sampled position err or of the read/write head. Stabilization of the double integrator plant can be achieved with a p r oportional plus derivative (PD) controller (implemented in discrete time). Typically, the bandwidth of the dosed loop system is constrained by the possibility of high frequency resonant modes in the actuator assembly. This bandwidth constraint will limit the gain of the feedback loop at lower frequencies if no control other than PD control is applied. Any digital controller is also subject to computation delay, measurement noise, and control noise.
891
Copyright 1999 IFAC
ISBN: 0 08 043248 4
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
14th World Congress ofTFAC
Fig. 2. Small gain theorem structure Fig. 1. Basic Repetitive Controller Structure The necessity of additional control is a result of the various disturbances which are typically present in disk drive servo systems. The combined effect of windage and the flexible printed circuit tension is a constant torque disturbance to the actuator during track following. Integral action is usually incorporated into the feedback controller to eliminate steady state errors. Another source of disturbances is the shape of the data tracks. Ideally, they are concentric circles centered at the axis of rotation of the disk, but in reality they will not be perfect circles, nor will they be perfectly centered. Deviations from the ideal shape are referred to as repetitive runout because they repeat as the disk rotates. Often, but especially for removable media disk drives, the centering error will cause most of the repetitive Iunout with the fundamental frequency matching the spindle speed. Other components of repetitive runout can be written in terms of harmonics of this fundamental frequency. If the disk is made of an anisotropic material (e.g. flexible media), contraction or expansion will make circular tracks elliptic. The resulting disturbance will be a sinusoid at the first harmonic frequency. Repetitive disturbances at the sixth harmonic and higher can appear due to mechanical faults in the spindle assembly. Some additional control is needed to reduce potentially large tracking errors due to repetitive runouts. From a frequency response point of view, the loop gain needs to be increased at the repetitive frequencies. Repetitive .control does exactly that. 1.2 Repetitive Control
The objective of repetitive control is to eliminate or reduce the effect of repeating disturbances. Thought of this way, the internal model principle is used to design repetitive controllers. In the sequel, the discrete time repetitive controller presented by Tomizuka et al. (1989) is what will be referred to as the basic repetitive controller. The structure of this controller (for a disturbance which repeats every N samples) is shown in Fig. 1. This repetitive controller structure is used for an asymptotically stable plant, so when applying this controller to the disk drive servo problem, the "plant" includes stabilizing feedback control (e.g. PD). This basic repetitive controller structure is easily separated into two components: the inverse dynamics and the internal model.
1.2.1. Basic Internal Model The internal model portion of the repetitive controller is composed partly of a delay chain. A delay chain can be easily implemented as a table with moving index. If the period of the repetitive disturbance is large (N is large), the table will be large as well. If the filter q, in Fig. 1, is set equal to 1 and the input to the internal model is set to zero, the control applied will be a repeating signal (cycle through the table). If control inputs which cancel the disturbance are stored in the table, the output of the plant will remain zero and the table values will not change. There are practical reasons, however, to introduce Iql < 1 at higher frequencies. In doing this, perfect tracking of a repetitive disturbance is sacrificed in order to ensure the stability of the closed loop system in the presence of model uncertainties. 1.2.2. Stability and Performance Analysis When analyzing the stability of this repetitive control structure, the small gain theorem is quite useful. By closing the feedback loop inside the box labeled "Plant", and calling the resulting system Gp, the block diagram in Fig. 1 can easily be regrouped as shown in Fig. 2. This grouping ignores the input, but puts the system in a form where stability is easy to check. A block d has been added to the block diagram to represent dynamiC uncertainty in the closed loop system. Applying the small gain theorem, the closed loop system will be asymptotically stable if the system inside the dashed lines, (z-cl - (1 + d) GpG inv )q , has infinity norm less than 1. An infinity nOrm smaller than 1 indicates some measure of stability robustness. Consider the case where G inv is selected such that GpGiTlfJ = z-d. The resulting sufficient condition for stability provided by the small gain theorem is IILlqlloo < 1. Some plant uncertainty is expected at higher frequencies (there may be unmodeled dynamics), so it is prudent to pick the frequency response gain of q to be 1 at low frequency and rolling off slightly at higher frequency. The choice for q described by Chiu et al. (1993) is a zerophase error FIR filter. Conveniently, such filters are easily implemented. In order to evaluate the closed loop performance of the system, however, we must return to the block diagram in Fig. 1. The closed loop transfer function from repetitive runout to the tracking error is
892
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO .. .
G er =
( 1) l+G n G c
1-
qz-N
-l-_-QZ---N""+""C"U-("":z:""_"":"d-_-G-,,-G-'n-o-)
(1)
If q = 1 at a particular repetitive frequency, it is clear that the gain of the closed loop at that frequency is zero. Choosing a q with zero phase means that the response of 1 - qz-N at harmonic frequencies ",ill be a positive number less than 1. If 11 (z-d-GpGinv)qlloo = 'Y < I, then the closed loop poles coming from the second factor of GeT will all have magnitude less than ')'. Larger ")'S mean longer settling times for the closed loop system.
1.2.3. Basic Inverse Dynamics The above stability development provides motivation (Tomizuka et al., 1989) for using the inverse dynamics or approximate inverse based on the zero-phase-error (ZPE) compensator (Tomizuka, 1987). Briefly, if the plant transfer function is of the form -1
Gp(Z
)
=
z-dpB+(Z-1)B-Cz- 1) A(z-l)
(2)
where B+ has roots all outside the unit circle and both B- and A have all roots inside the unit circle, the ZPE compensator will be: -1
Gzpe(Z
)
==
zd pA(z-l)B+(z) B-(z 1)B+(1)2
(3)
The ZPE style plant inversion requires advance knowledge of the input (which is not available in this case), so d delays are added to the ZPE inverse, G inv = z-dG zpe , such that G inv is realizable (d = dp+dB +, where d B + is the order of B+(z)). Due to the zero-phase properties designed for, z - d and GpGinv will have the same phase at all frequencies. Using Eqs. (2) and (3) along with the block diagram in Fig. 2, a sufficient condition for asymptotic stability of the closed loop system is
11(1 - a - Lla)qlloo < 1 -1 a(z
)
=
B+(z-1 )B+(z) B+(1)2
(4) (5)
The transfer function a in this case takes on positive real values less than 1. Notice that if II~qlloo < 1, and I1qlloo < 1, the condition given in Eq. (4) will be satisfied. This simplification relies heavily on the ZPE design of the inverse dynamics, and makes the frequency response of q the most significant factor in determining the nominal steady-state performance.
2. COST EFFECTIVE REPETITIVE CONTROL Now, the focus shifts to the costs of implementing a repetitive control system. Looking at the frequency domain developments of stability and performance criteria in the preceding sections,
Fig. 3. Multirate repetitive control with a reduced order internal model one might be tempted to apply an optimal loopshaping control to design the filters q and G inv · An attempt to minimize the amount of computation prohibits this approach. In fact, all modifications considered either reduce or maintain the cost relative to the "basic" repetitive controller described earlier. The delay chain internal model with FIR filter is a low computational burden, but does use a significant amount of memory. Therefore, methods of applying repetitive control with reduced memory usage are investigated. In contrast, the ZPE inverse requires significant computation. In some cases, applying this computation effort to a higher order feedback compensator may obviate the need for repetitive control.
2.1 Modifications to the Internal Model The internal model in large part determines the steady-state performance of the repetitive control scheme (see section 1.2.2). The primary cost associated with implementing the internal model in the basic repetitive controller is memory. As previously mentioned, the delay chain can be implemented as table with moving index. This table has as many elements as sectors on the disk. The filter q is selected to be a lowpass filter with a peak gain of 1 to provide a tradeoff' between tracking performance and stability robustness . In the case where the fundamental repetitive frequency and its lower order harmonics are the primary components of the repetitive disturbance, the gain of q can be quite low at higher frequencies - less attenuation of those repetitive frequencies is necessary. Alternatively, the table can be decimated. If there are N table elements and N is divisible by r, decimating by a factor of r means using a table of length M = Nlr instead. The information in the shorter table is still used to characterize the entire repetitive disturbance. It can be thought of as using only every rth element from a table of length N, and hoping that by looking at every rth element, enough information about the disturbance is present to cancel it. The high frequency components of the disturbance will not be visible (Nyquist frequency) unless aliased . If this reduced order repetitive control is applied as though the sampling rate of the system was a factor of r slower, each table value would be the control for r measurements.
It is not necessary to hold the same control for r samples. Information about the next control
893
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ISBN: 0 08 043248 4
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
is accesible. This allows for linear interpol",,"~~u of the repetitive controller at the measurement sampling instants. A block diagram analagous to that shown in Fig. 2 can be found in Fig. 3. Interpolation (I) and decimation (V) blocks have been incorporated into the diagram. The small gain theorem is again used to obtain a stability criterion (11(z-d - DGinvGpI)qlloc < 1). To evaluate performance, the r x r transfer matrix for the closed loop track runout to error (there are r runouts per sample at the slow sampling rate) is evaluated at frequencies up to 1/2r of the measurement frequency. Frequency domain analysis cannot be extended beyond this frequency, because the repetitive controller is a periodically time varying system when considered at the measurement sample rate. A significant amount of complexity is added to the design process as well; in addition to q and G inv , the interpolation and decimation schemes also affect the robust stability criterion. Specifically, they can be used to match phases associated with delays shorter than r measurement samples ( the d' in Fig. 3 is not the same as the d in Fig. 1). It should be noted that the order of V and Ginv can be switched, that is, Ginv can be implemented as a discrete system at the slower sampling rate, with the decimated plant output as its input. That option, although it has potential to save computation, is not addressed in this paper.
2.2 Approximate Inverses For disk drive track following servos, the "plant" to which repetitive control is to be applied can typically be modeled as a fifth order discrete time system. Two states come from the double integrator characteristics aforementioned, two come from the PD controller which stabilizes the double integrator system, and one state models the computation delay. This leads to a fifth order model of the inverse dynamics if the ZPE inverse is used. There are memory and more importantly, computational costs associated with a fifth order inverse. These costs may not be considered large for some applications, but for disk drives, reductions in this cost are likely to be welcomed. For any proposed inverse, the stability condition must be satisfied for a q which reflects the tradeoff between desired steady-state rejection of repetitive disturbances and robustness to uncertainty.
3. EXPERIMENTAL VALIDATION In this section, times are measured in samples, and frequencies in cycles per sample. Various repetitive controllers are designed for a sector servo disk drive system with 80 sectors.
14th World Congress ofTFAC
~2or~ll ~o~·~. 1
f~~~~t c. -360
j
.•.•
's;] I
---'----~-~-----"...----'
t..._
0.01
0.05 0.1 0.5 freq(cycles/samp) Fig. 4. Bode plot of nommal feedoack loop gain The disk drive model used is obtained from the experimental setup. It consists, as is typical, of a double integrator system with delay and a stabilizing feedback controller. The frequency response characteristics of the nominal closed loop system (corresponding to Gp in Fig. 2) are shown in Fig. 4. The nominal closed loop bandwidth is limited by the possibility of resonant modes at frequencies above 0.15 cycles/samp. The tracking performance goal is to maintain less than 0.1 tracks of error, preventing corruption of data on adjacent tracks. The experimental system exhibits track runouts primarily at the fundamental frequency. This provided a nice starting point for experimental evaluation of cost reduction ideas, because it is not a particularly challenging repetitive signal. The amplitude of this runout was approximately 17 tracks. Measurement and control noise are included in time domain simulations. For the time being, the effect of these noises on tracking performance is not considered from an analytical point of view.
3.1 Analysis
In Sections 1.2.2 and 2.1, the stability and performance of repetitive control sytems was discussed based on the small gain theorem. Plots of (Z-d VGin",GpL)q are used as a graphical indicator of stability robustness. he closed loop will be stable if the gain of this quantity is less than 1. Comparing how close different structures come to this stability limit is a way to assess relative robustness of the closed loop system. Another comparison tactic is to look at the performance at the frequencies which make up a repetitive signal. The closed loop transfer function from track runout to position error is formed and evaluated at the fundamental repetitive frequency and its harmonics. The resulting attenuations (att) are given in (dB). In addition, since this is a nominal performance, the minimum magnitude uncertainty which leads to a violation of the stability criterion is also computed at the harmonics. The idea is that these values will reflect the likelihood that the nominal performance will be acheived. If there is little tolerance for uncertainty at a particular harmonic, small uncertainties may have a large effect on the performance at that harmonic.
894
Copyright 1999 IF AC
ISBN: 008 0432484
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
14th World Congress ofTFAC
:: :::: :::··:····J····i·····'~·I···l :::: ::::: :; .. ::::: : "'S 101··r··· . . . ...... : ~-lg
~ ~ ~.!
-20
!I
•• , p'
0
....<... .
l"I' .
"
~o:H;;;;;;;;;;:,;:.,
..
. .ll
0.01
0.05 0.1 0.5 freq( cycles/samp) Fig. 5. Comparison of three different inverse dynamics used with an 80 memory internal model.
s~ 0.5?II:..:: _.
roc
zpe
att
97
une 1.0
att 97 73 58 31
. " ' : ' . . . . ';':
'.-.-.~'
.... . •
.
0.1 0.025 0.05 freq (cycles/samp) Fig. 6. Stability comparisons for r = 4
i
~rA~~{f~tLAaJ ~-5~
1- ~~I
-5='---_-'
~ g~~.,.~~~j ~-~
une 1.0
The final method in which designs will be compared is more standard. Experimental tracking errors are averaged to capture the repeating part of position error. Discrete fourier transforms are used to compute the repetitive frequency content of time domain signals.
L......._....J
0.01
gn
att 97 0.96 1.0 73 73 0.91 58 1.0 59 31 1.0 0.67 35 0.61 11 15 1.1 14 11 4.6 0.25 1.5 8.4 1.5 8.5 23 'unc' : magnitude of destabilizing uncertainty 'att' : attenuation of closed loop (dB) 0 1 2 5
une 1.0 1.0 1.0 1.0 1.1
:
I' 'E5~"" '1
hold lIlterp .' ':.' ...
o
Table 1. Comparison of inverses harmonic
I,*,,"t.liA ~....,. ~'.l~\
~ o! t.. "i"" ",±~i?til~ ::l.~
~>~ j.~.:..
.. , .. : . ~'
'g'!;n
E 0.5 .... , ......._. . .
! ! ~ ~ ~ , . € Ai ,Jf! ...... , .. , ... .
~ 100 . ;;-; I ~ ~ 50 . En A
~ 1~".'~, ' '''~~~;~ . .... +'\.,.... ..~
');"~~:
I I
10 20 30 40 50 60 70 80 time(samples) Fig. 7. Experimental averaged PES time traces
.6., which can cause instability of the closed loop system is shown for selected harmonics. Note that since feedback control is included in wha.t is considered the plant, ~ is not likely to be very large at frequencies within the bandwidth of the closed loop, so the nominal closed loop gains are close to the actual gains. When the inverse (gn) is used, the closed loop is less robust to uncertainty than the (zpe) and (sec) inverses.
3.2 Estimated Inverses
3.3 Inverses with Reduced Order Repetitive Control
To start, three different choices for G inv are compared when used in conjunction with the basic repetitive control internal model (80 memory delay chain and q = (z + 6 + z-1)/8). The bode plots of the three inverses, and magnitude plots of (z-d - GpGinv)q in each case are shown in Fig. 5. Notice that the fifth order ZPE based inverse (zpe) is nearly indistingushable from its second order approximation (sec) in the frequency domain. If the approximate inverse is set to a fixed gain (gn) , the difference in phase is significant at higher frequencies. The DC gaiml of all of these inverses are set such that the DC gain of GpG inv = O.S. This results in a slightly less aggressive repetitive controller (known as a result of practical experience). The lower plot in Fig. 5 illustrates the nomina.l robust stability of the system. Notice that the gain inverse (gn) brings the system closer to instability than the ZPE and its approximate. Choosing d so that the phases of z-d and GpGinv are close helps the closed loop system meet the stability requirements . Table 1 further compares the three proposed inverses. The nominal attenuation (in dB) of the closed loop system, and the size of the smallest uncertainty,
Similar comparisons to those above are made when the delay chain is decimated by a factor of 4. Frequency response data is therefore valid only up to one fourth of the Nyquist frequency associated with the measurement rate. Every 4th output of G inv is used to drive the internal model (decimation). Unless otherwise noted, the output of the repetitive controller is linearly interpolated between successive elements in the table. In Fig. 6, the upper plot is of the same robust stability indicator we have used before, now applied to a reduced order repetitive control system with decimation and interpolation. It is qualitatively very similar to the lower plot of Fig. 5. This is expected. The inverses are being implemented at the measurement rate of the plant: they are not affected by the memory reduction of the internal model. The upper plot in Fig. 7 shows a comparison of the averaged experimental time traces for repetitive control reduced by a factor of 4 with Ginv being set to both the (sec) and (gn) inverses described above. Both control structures achieve the tracking goal. Errors at the 9th harmonic persist when (gn) is used. This persistence is related to the fact that this harmonic is the Nyquist frequency for the reduced sample rate.
895
Copyright 1999 IF AC
ISBN: 008 0432484
COST EFFECTIVE REPETITIVE CONTROLLERS FOR DATA STO ...
14th World Congress ofTFAC
Table 2. Comparison between q's harmonic
q4
q128
qlS
une 0.91
att
unc
74
0.90
0.84
att 61 37
0.73
50
0.71
att 101 72
2
0.81
23
0.56
35
0.51
54
3
0.95
13
0.44
25
0.33
41
0
une 0.93
1
3.4 The Effect of q The development in section 1.2.2 suggested that the filter q can play a large role in both the stability robustness and the performance of repetitive controllers. The center plot in Fig. 6 shows how the stability of a reduced order repetitive controller (r = 4) is affected by different choices of the filter q. The inverse dynamics used for these comparisons is the cheap one, (gn). The filter q is implemented at 1/4 the measurement rate. The three q's compared are all easily implemented on a digital CQInputer because the divisions are all by powers of 2. They are, with mildest roHoff listed last: q4 = (z + 2 + z~1)/4
+ 14 + z-1)/16 q128 = (z + 126 + z-l )/128 q16
=
(z
The effect of q on the stability robustness is seen in the figure. Notice that the magnitude moves closer to 1 as q's with less roUoff are selected. The classic tradeoff of feedback control, sensitivity vs. complementary sensitivity, appears. This tradeoff is apparent in the calculated performance and uncertainty tolerances analagous to Table 1 shown in Table 2. The lower plot in Fig. 7 displays experimental corroboration of the data in the table. The traces shown are the average periodic error over 12 revolutions. Notice that although the fundamental component is completely gone from the trace for q = Q128, again the 9th harmonic is very visible.
3.5 Interpolation and A!iasing The most likely reason for the appearance of higher harmonic components in the errors is aliasing. By picking every 4th output of the inverse plant model, higher frequency components are aliased. By simulating the response to a 7th harmonic repetitive disturbance with amplitude 10 tracks, this aliasing effect is readily visible in the upper plot of Fig. 8: both 7th and 11th harmonics are present in the error signal. Holding the repetitive signal instead of interpolating increases the magnitude of the errors. The lower plot in Fig. 6 shows the effect of interpolation on the stability robustness of the reduced by 4 repetitive controller with inverse (gn) and q = (z+30+z- 1 )/32. Using the same repetitive control parameters, the effect of not interpolating the repetitive control
~
0
.
Fourier coeff ot simulated PES.
10 5 :;;; 0 ?A .f:I 1 ~ 0.5
•
~
hClld
inle,,;
I
--.J. x 0
Fourier coeff of experimental
•o!IJllljI_ : w5¥ •• 5R!I __~.~.
'ES..
1
. ." J hold interp
O--"-'~.--"--'LJL..!
25 30 20 15 coefficient Fig. 8. Experimental and simulated evidence of aliasing
0
5
10
output is examined experimentally. The Fourier coefficients are plotted in the lower half of Fig. 8. More fundamental frequency error is present when the repetitive control output is not interpolated. Furthermore, without interpolation, errors at the aliased equivalents of the fundamental frequency, the 18th and 20th harmonics, are also large. This effect is much smaller when interpolation is used. A direction for future work is to incorporate antialiasing into the decimation.
4. CONCLUSION The work presented in this paper has been motivated by the goal of making repetitive control more accessible to applications by reducing the computational and/or memory costs ofimplementation. The simulation and experimental results demonstrate that this goal can be achieved without sacrificing the objectives of repetitive controL In particular, by using a simple gain instead of a fifth order inverse and one quarter the memory of a standard repetitive controller, almost no performance at the fundamental repetitive frequency was lost. The frequency domain analysis of the reduced order repetitive controllers was restricted by the Nyquist frequency associated with the new sampling rate. Moving away from frequency domain ideas, and into operator based performance characterizations will be the next step toward making fair comparisons of the proposed designs. It may also allow for less ad-hoc choices of interpolation and decimation filters. 5. REFERENCES Chiu, Tsu-Chih et al. (1993). Compensation for repeatable and non-repeatable tracking errors in disk file systems. Proceedings of the JSME International Conference on Advanced Mechatronics pp. 710-717. Tomizuka., M. (1987). Zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems, Measurement, and Control109, 65~68. Tomizuka, M., T-C. Tsao and K-K. Chew (1989). Analysis and synthesis of discrete-time repetitive controllers. ASME Journal of Dynamic System, Measurement, and Control III , 353358.
896
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ISBN: 008 0432484