This document summarizes the identification of a tenth-order HDD servosystem model using two methods: sine sweep and average ETFE. Sine sweep accurately identified the model by increasing the input magnitude to reduce noise effects, though it was time consuming. Average ETFE also produced accurate results by averaging multiple experiments, reducing noise through averaging. Both methods yielded pole estimates close to the true model, demonstrating their effectiveness in system identification of the high-bandwidth, high-order HDD servosystem model.

1. System Identi

2. cation (Final Project)
May 1, 2013
Teng-Hu Cheng
1 Motivation
The recent demands for high-capacity memory storage and limited physical size urge the HDD
(Hard disk drives) to increase areal density, which requires the HDD servosystem to have high
control performance. To achieve this, mode-switching control (MSC) structure is widely used
and it consists of two control modes, track-seeking and track-following. The track-seeking con-
troller drives the magnetic head to the neighbor of the desired track, while the track-following
controller regulates the position and velocity of the head on the desired track for consequent
READ/WRITE operation. However, the high bandwidth requirement makes the system hard
to be identi

3. ed in dierent aspects. In this project, we use two dierent methods, Sine sweep
and Average ETFE, to identify the system model and to assess the results of using the two
methods.
1.1 Practical Issues
Due to the high bandwidth requirement of the servosystem, the limited natural sampling fre-
quency puts limitation on how high the bandwidth of the controller can be as well as the
continuous model. However, in this project, we don't have to worry about the practical issue
since we only run simulation to identify the model in this project.
1.2 Simulation Issues
The model to be identi

4. ed is a tenth-order system with relative degree of 3, and the system
has a high bandwidth according to the dynamics. So we can expect that at high frequency the
magnitude of output from the system will be pretty small, hence the measurement noise can
possibly dominate the measurement which makes it hard to identify the system.
Figure 1: HDD components
1

5. 2 Model
2.1 Block Diagram
The HDD system consists of MSC structure can be represented via block diagram as shown
in Fig. 2. In the block diagram, the error signal is generated by subtracting the target track
command by the measured position with noise, and the controller is taking the position error
for computation and produces a desired voltage output to drive the VCM to compensate the
error. In practical case, the system to be identi

6. ed includes not only the mechanical part, but
also the actuactor and the power ampli

7. er. As indicated in the

8. gure, G(s) involves the three
dynamics[2].

10. Figure 2: Block diagram
2.2 Sampling Frequency
Considering avoiding aliasing, the sampling frequency should be at least half the bandwidth of
the plant. In this project, the system bandwidth is around 1.5kHz. According to the rule of
thumb, sampling frequency is taking 5-10 times the bandwidth of the plant, so the sampling
frequency is set to be 10kHz in this project.
2.3 Transfer Function
The transfer function of model indicated in Fig. 2 is [1]:
G(s) =
s7 + 681240s6 + 2:463e9s5 + 1:427e15s4 + 6:79e17s3 + 2:518e23s2 + 1:018e25s + 4:878e28
s10 + 3274s9 + 4:02e9s8 + 9:7e12s7 + 2:75e18s6 + 2:039e21s5 + 3:87e26s4 + 2:2e28s3 + 7:47e31s2 + 8:95e32s
(1)
by using the sampling time as 104sec. or sampling frequency 10kHz, the discrete-time transfer
function can be represented as
G(z) =
1:66e12z9 + 2:63e12z8 1:5e11z7 + 2:07e11z6 1:17e11z5 4:77e12z4 + 1:6e11z3 1:35e11z2 + 2:9e12z + 1:2e12
z10 4:514z9 + 8:587z8 9:031z7 + 5:728z6 2:487z5 + 2:273z4 4:362z3 + 5:123z2 3:038z + 0:7208
(2)
The bode plot of G(s) is shown in Fig. 3.
2

11. 100
0
-100
-200
-300
-400
Magnitude (dB)
0
10
2
10
4
10
6
10
8
10
-45
-90
-135
-180
-225
-270
Phase (deg)
Bode Diagram
Frequency (rad/s)
Figure 3: Bode plot of G(s)
3 Methodology
From the homework3 we knew that the sine sweep has the best capability of identifying the
system by reducing the eect of measurement noise, so we

12. rst use this method to identify this
high-order system.
3.1 Sine Sweep Method (non-parametric)
Sine sweep method has both advantages and disadvantages. This method is conducted by
repeating n times of experiments with sine input signal of unique frequency for each experiment,
and the n number of frequencies of the overall experiments should cover the frequency range of
the model. The resolution of the bode plot can be increased at which the variation is high by
increasing the number of experiments n, but the price to pay is time consuming.
On the other hand, the relative magnitude of output and noise should be considered. Con-
sidering the model in (1), the relative degree is 3 and the frequency range of the model is around
0-1 kHz. At low input frequency the magnitude of the output can be large but small at high
frequency, where the noise can dominate the output and causes inaccurate modeling.
For n experiments, the input sine signal in ith experiment can be expressed as
uk = sin(
ik); 1 i n:
In this project, the sampling time is T = 0:0001 sec, or sampling frequency f=10kHz. The
discrete and continuous frequency conversion can be expressed as
(rad=sample) $
T
= 104(rad=sec:)
The experiment parameters can be summarized in Table 1.
3

13. Table 1. Experiment parameters (Sine Sweep)
Number of experiments (n) 2000
Sampling time (T) 104 [sec.]
Input frequency range (
i)
i 2
0; 104
rad/s
frequency resolution 5 rad/sec.
Input Data length 20000
Number of transient data being cut 5000
Deviation of noise 0.1
Since all parameters are set except the magnitude of the input signal, we now consider it as a
variable, and see how the magnitude aects the modeling accuracy. Begin from input magnitude
of 100, the comparison of the discrete-time bode plots between true and estimate models are
shown in Fig. 4.
1010
10-10
10-4 10-3 10-2 10-1 100 101
100
Magntue
true
estimated
200
100
0
-100
-200
Phase
true
estimated
10-4 10-3 10-2 10-1 100 101
Figure 4: Bode plot of true and estimate model with control input magnitude 105.
It is obvious that the mismatch between the two bode plot is larger and not acceptable, and
at this stage we can only speculate that the noise might corrupt the model estimate.
To verify this prediction, the magnitude of the control input is further increased to 105, and
the comparison of the discrete-time bode plots is shown in Fig. 5.
4

14. 105
10-5
10-10
10-4 10-3 10-2 10-1 100 101
100
Magntue
true
estimated
200
100
0
-100
-200
Phase
true
estimated
10-4 10-3 10-2 10-1 100 101
Figure 5: Bode plot of true and estimate model with control input magnitude 105.
In Fig. 5, the improvement of the estimate in both low and high frequencies can be seen, so
the speculation is reasonable. To further increase the modeling accuracy, the input magnitude
is increased to 1010, and the result is shown in Fig. 6.
10-5
10-10
10-4 10-3 10-2 10-1 100 101
100
Magntue
true
estimated
200
100
0
-100
-200
Phase
true
estimated
10-4 10-3 10-2 10-1 100 101
Figure 6: Bode plot of true and estimate model with control input magnitude 1010.
The true and estimate models match quite well in Fig. 6. However, the error is relatively
large in low frequency range ( 102). The reason that I can think of is that the frequency
resolution of sine sweep is too low, which is around 5rad/sec. By further increase the number
5

15. of experiments to n = 20k, the improvement in low frequency can be observed in Fig. 7.
10-5
10-10
10-4 10-3 10-2 10-1 100 101
100
Magntue
true
estimated
200
100
0
-100
-200
Phase
true
estimated
10-4 10-3 10-2 10-1 100 101
Figure 7: Bode plot of true and estimate model with control input magnitude 1010 and re

16. ne
freq. resolution.
The identi

17. ed model in Fig. 7 is ^ G(z) =
^b
^a , where ^a and ^b
are
^b
=
7:6e6; 0:053; 0:048; 0:0647; 0:1952; 0:2342; 0:1534; 0:0161; 0:0450; 0:0148
106
^a = [1; 2:9055; 3:5316; 2:3861; 0:9088; 0:4847; 1:3427; 2:0657; 1:3385; 0:2793; 0:0002] ;
and the corresponding poles are:
[0:7120 + 0:5204i; 0:7120 0:5204i] ; [0:1857 + 0:9756i; 0:1857 0:9756i] ;
[0:9996 + 0:0007i; 0:9996 0:0007i] ; [0:7842 + 0:5657i; 0:7842 0:5657i] ; 0:3912; 0:0008:
The poles of the true model G(z) are:
[0:7118 + 0:5204i; 0:7118 0:5204i] ; [0:1861 + 0:9755i; 0:1861 0:9755i] ;
[0:9968 + 0:044i; 0:9968 0:044i] ; [0:7864 + 0:5714i; 0:78464 0:5714i] ; 1; 0:9988:
Comparing the two sets of poles, all poles are quite close except the

18. nal two, although the
bode plots are quite close.
3.2 Average ETFE
In the previous section, sine sweep method shows a good capacility to estimate the model, but
the disadvantage is high cost (time and energy). Furthermore, due to the relative degree and
high bandwidth of the system, the noise is rejected by increasing the magnitude of the sine input
in high frequency range, and this is sometimes infeasible in practice. To reduce the magnitude
of input signal, average ETFE can be applied.
6

19. In average ETFE, the noise is rejected by means of averaging the transfer function by dierent
sets of experiments so the noise is reduced to 1
n by averaging n times of experiments. In
each experiment, the control input signal is consisting of a series of sine signals with dierent
frequncies which uniformly cover the system bandwidth. The matlab command of used to
generate the input signal is:
u = idinput(length(t);0 sine0; [0 1]; [1; 1]; [1000; 10; 1]);
Another experiment parameters are listed in Table 2.
Table 2. Experiment parameters (Ave ETFE)
Number of experiments (n) 50, 500, 2000
Sampling time (T) 104 [sec.]
Input magnitude 1
Input Data length 3000
Number of transient data being cut 0
Deviation of noise 0.1
Dierent from the sine sweep method, the magnitude of the input signal in Table 2. is
reduced to 1, and there are 4 sets of Number of experiments (n), which are used to compare
the accuracy trend on the model estimate. Start with n = 50, the comparison of the bode plots
are shown in Fig. 8.
10-5
10-6
10-7
10-8
Magnitude
true
estimated
10-2 10-1 100 101
200
100
0
-100
-200
Phase
true
estimated
10-2 10-1 100 101
Figure 8: Bode plot of true and estimate model using ave. ETFE with n = 50.
Obviously, the estimating result is not good enough especially at the low frequency range.
To improve the estimation, the number of experiments is increase to n = 500, and the result is
shown in Fig. 9.
7

20. 10-5
10-6
10-7
10-8
Magnitude
true
estimated
10-2 10-1 100 101
200
100
0
-100
-200
Phase
true
estimated
10-2 10-1 100 101
Figure 9: Bode plot of true and estimate model using ave. ETFE with n = 500.
From Fig.9, the model estimate has been improved both in low and high frequency range,
but there is still room for improvement. To further improvement the result, n is now set to
2000, and the result is shown in Fig. 10.
10-5
10-6
10-7
10-8
Magnitude
true
estimated
10-2 10-1 100 101
200
100
0
-100
-200
Phase
true
estimated
10-2 10-1 100 101
Figure 10: Bode plot of true and estimate model using ave. ETFE with n = 2000.
The poles of the estimated model ^ G(z) are:
[0:5061 + 0:6317i; 0:5061 0:6317i] ; [0:7018 + 0:5030i; 0:7018 0:5030i] ;
8

21. [0:0845 + 0:7275i; 0:0845 0:7275i]; [0:1657 + 0:4512i; 0:1657 0:4512i] ; 1:0038; 0:9078:
The identi

22. ed poles have one unstable pole 1.0038, which is very close to the stability margin,
and others are very close to the true model.
4 Conclusion
Comparing the identi

23. cation results, each method demonstrates a good estimate capability when
noise with the same deviation exists in both systems. But both methods require multiple sets
of data from separate experiments in order to attenuate the noise. Moreover, each of them have
dierent advantages and disadvantages according to the experiments in this project.
4.1 Sine Sweep method
For sine sweep method, the estimated model is more accurate no matter in high and low fre-
quencies, and the magnitude and phase plot are matched very well. But the price to pay is that
it requires a lot of sets of experiments to collect the date for estimate and that is sometimes
not allowable in practical cases. If the sets of experiments are not enough, the resolution in
frequency space would not be high, and that might cause inaccurate modeling in low frequency
range. Moreover, the magnitude of the control input signal need to be large at high frequency
for high relative order or high bandwidth system, so that can also be restrictive in practical.
4.2 Average ETFE method
However, in this case, the estimated result is still excellent but not as well as sine sweep method.
The bode plot shows that the estimated model is very close to the true model in either low or high
frequency regions, but the magnitude of the control input signal is relatively small compare with
the sine sweep method, so this is more feasible in practical even though multiple experiments
are still required. In addition, there is no resolution problem as in sine sweep case when the
number of experiments decrease. But decreasing number of experiments aects the level of
noise attenuation, which can be proved by inspecting Fig 8. to Fig. 10. For n = 50 2000, the
estimated models are always in the acceptable ragne, and the dierence between n = 500 and
n = 2000 is not too much. So the average ETFE method seems to provide a more reliable and
feasible way to estimate system model than the sine sweep method as long as the experiments
in this project are concerned.
References
[1] Hamid D. Taghirad and Ehsan Jamei. Robust performance veri?cation of adaptive robust
controller for hard disk drives. IEEE Transactions on Industrial Electronics, 55(1):448{456,
2008.
[2] Takashi Yamaguchi, Hidehiko Numasato, and Hiromu Hirai. A mode-switching control for
motion control and its application to disk drives: Design of optimal mode-switching condi-
tions. IEEE/ASME Trans. Mechatron., 3(3):202{209, 1998.
9