In contrast to Fourier transforms, wavelet transforms can pro-vide detailed local information of signals. Based on the wavelet transform, wavelets can approximate any finite energy function. This paper proposes wavelet networks to approximate periodic functions. Accordingly, a wavelet network controller is developed to eliminate repetitive error caused by repetitive disturbance. Experiments for flying height control of a disk drive system are implemented to demonstrate the present method in comparison with another method using Fourier series based network control.
Index Terms—Bender, disk drive, flying height control, PZT,
repetitive error, wavelet network (WN), wavelet theory.
I. INTRODUCTION
T
RACKING control commonly appears in servomecha-nisms where high accuracy is demanded. Tracking errors can be classified into two types: repetitive and nonrepeti-tive errors. In general, repetinonrepeti-tive tracking errors appear due to tracking periodic inputs or disturbances. Many methods have been proposed to deal with repetitive errors, such as in-ternal-model-based repetitive control, observer-based learning control [1], and neural-network-based control techniques [2]. On the other hand, the wavelet transform emerges as a new powerful tool for signal processing and function approximation. Comparing with the short-time Fourier transform that gives signal information with a fixed time window, the wavelet trans-form gives detailed local intrans-formation of a signal by translating and narrowing a time window for detecting high-frequency behavior while widening one for investigating low-frequency behavior. Hence, on a time-frequency plane, wavelet transforms achieve a better compromise between time and frequency do-mains [3]. The repetitive error due to spindle motor rotation of disk drives lies in the low-frequency region. Wavelets with a large time window can detect low-frequency information of the repetitive error, while maintaining good time localization property. Inspired by the wavelet transform theory and neural networks, to deal with disk drives, this paper develops a wavelet network controller using wavelets to generate periodic control signals that can cancel repetitive errors. Experimental results and simulation results are presented to demonstrate the effec-tiveness of the proposed method.Manuscript received February 11, 2004. Manuscript received in final form September 7, 2005. Recommended by Associate Editor D. W. Repperger. This work was supported by the National Science Council in Taiwan, R.O.C., by Grant NSC94-2752-E009-009-PAE.
The authors are with the Department of Mechanical Engineering, Na-tional Chiao Tung University, Hsinchu 30010, Taiwan, R.O.C. (e-mail: [email protected]).
Digital Object Identifier 10.1109/TCST.2005.860523
Fig. 1. Experimental setup for PZT-disk system.
II. PZT-DISKSYSTEM
In order to control the flying height of a pickup head to follow disk deformation of a disk drive in this paper, an apparatus in-cluding a PZT bender and a disk drive is shown in Fig. 1. The control algorithm is executed on a PC and its output via a PZT driver controls the bending displacement at the tip of the PZT bender. When the spindle motor rotates at a constant speed, the disk surface deformation is composed of a repetitive com-ponent and a nonrepetitive component . The flying height is measured by using a laser Doppler vibrometer as the feedback signal.
The block diagram of the present control system is depicted in Fig. 2 where denotes the desired flying height. The wavelet network (WN) control signal is used to cancel repetitive disk surface deformation that causes repetitive errors. During disk rotation, the disk surface deformation is treated as an output disturbance in controlling bending displacement of the PZT bender tip to follow the disk surface. The PZT driver that amplifies the control voltage can be treated as a con-stant gain . A linear model of the plant, i.e., the PZT bender, is identified and Bode plots of open-loop gain are depicted in Fig. 3. To stabilize the closed-loop system, a compensator is designed so that becomes a type 1 system with the gain margin and phase margin of 20.5 dB and 67.7 deg, respectively. According to Fig. 2, the flying height of the PZT bender tip is expressed by
(1) Dealing with the third term on the right-hand side (RHS), in order to cancel the repetitive disturbance , the WN con-troller proposed in the next section is trained until its control
signal achieves .
Accord-ingly, the repetitive error caused by the repetitive disturbance 1063-6536/$20.00 © 2006 IEEE
Fig. 2. Block diagram of PZT-disk system.
Fig. 3. Compensator design in frequency domain.
can be eliminated and the flying height depends on the de-sired flying height subject to remaining nonrepetitive distur-bance .
III. WAVELETNETWORKCONTROL
In contrast to Fourier series expansions that express a peri-odic function in terms of trigonometric series, a wavelet series expansion expresses a finite energy function with a series set ob-tained by dilating and translating a zero-mean mother wavelet. Concerning the wavelet theory, mathematical foundations of wavelets were developed by Grossmann and Morlet [4]. Mallat [5] presented wavelet representations to approximate signals at a given resolution. The wavelet transform and the short-time Fourier transform (STFT) were compared by Daubechies [3]. The wavelet transforms better compromise in localization be-tween time and frequency domains than the STFT. The wavelet network was proposed by Zhang and Benveniste [6] to approxi-mate nonlinear functions. Delyon et al. [7] carried out accuracy analysis to approximate a continuous function using a wavelet network. In contrast to wavelets, a biased wavelet has a nonzero mean and can better reproduce signal components that are in the low-frequency region on the time-frequency plane since the nonzero mean enlarges low-frequency gain [8]. In this paper for disk drives, the spindle motor speed is 90 Hz which lies in the low-frequency region among 5-kHz measurement bandwidth in experiments. Hence, this paper employs biased rather than un-biased wavelets and a wavelet network controller is developed.
A. Biased Wavelet
A set of biased wavelets is defined by [8]
(2)
where , i.e., , denotes a mother
wavelet and and are dilation and translation parameters of the biased wavelet, respectively. In contrast to wavelets, biased wavelets have an adjustable nonzero mean bias function
that improves representation capability of biased wavelet expansions, which better reproduce signal components in the low-frequency region on the time-frequency plane than wavelet expansions. The set reduces to a wavelet set when the bias parameter is set to zero, since . Any finite en-ergy function can be expanded using the biased wavelet set or the wavelet set . In general, are not lin-early independent, which means that both and are frames for rather than bases.
B. Wavelet Network Approximation
Based on the wavelet transform, the wavelet network pro-posed in [6] is of the form
(3) where is a wavelet, and denote dilation and translation parameters, respectively, are weightings, and is an offset to help deal with nonzero mean functions. In order to achieve better approximation, particularly for signals that lie in the low frequency region on the time-frequency plane, this paper pro-poses a biased wavelet network on the basis of biased wavelet
sets , defined as
(4) This paper will use (4) in constructing biased wavelet networks.
A Mexican hat wavelet is
The wavelet networks cannot be directly applied to approx-imate a periodic function , since is not a finite energy function, i.e., . However, if the periodic function
with a period satisfies an integration condition
(6) one can find a finite energy function that satisfies
when and has a wavelet network approximation . Hence, the periodic function can be rewritten as
(7) and a wavelet network approximation is expressed as
(8) In other words, the wavelet network is trained to learn the finite energy function instead of the original periodic function . In practice, it is not necessary to exactly define the finite energy function . The wavelet network is just trained to approximate some function that is equal to in the domain regardless of . In other words, parameters of are updated until approximates in the domain . Hence, the finite energy function can be any finite energy function that satisfies
(9) Besides, the definition of in the domain is arbi-trary and of no significance during training.
D. Approximation by Periodic Continuous Function
If a periodic function is continuous, i.e., . As a consequence, its corresponding approximation has to satisfy . On the other hand, a biased wavelet in (5) decreases quickly and is negligible when , where is prescribed as 4 in this paper. The biased wavelets when play a major role in approximation. Hence, this paper proposes a wavelet network approximation for
(10) and has . Parameters in are trained to approximate the finite energy function defined in (9). The biased wavelet decreases quickly and the value of when can be treated as zero. Hence, the approximation effect of when can be negligible and the wavelet sets
(11) in (10) reduce to , which is a periodic function defined as shown in (12) at the bottom of the page. The periodic function reproduces the major portion of the wavelet with a period . Truncation of the wavelet leads to
with . When the dilation parameter ,
the major portion of the wavelet is when and equals when tends to become zero. As a result, the wavelet approximation (8) becomes
(13) where are defined in (12).
E. Wavelet Network Controller
According to (13), a WN controller with nodes in the single hidden layer can be expressed as
(14) The WN control signal is a continuous function with a period and has an adjustable parameter vector defined as
. The total number of parameters in
Fig. 4. Comparison of flying height errors during learning using biased and unbiased wavelets.
is . In this paper, a sampling frequency is pre-scribed as 10 kHz and can be updated online using a gradient training rule to minimize the error function
(15)
where denotes the system error, is a sampling step, and is the initial step in two periods. The error function is the squared sum of . The parameter is in turn updated to become a new value
(16) where is the updated value of in the th iteration, is the variation of and a signum function deter-mines the training direction of .
In order to examine the stability of the proposed scheme in Fig. 2, a nonnegative function is defined as , where denotes in the th iteration a pa-rameter error vector relative to an optimal papa-rameter vector . If the nonrepetitive disturbance is a random distribu-tion during learning and will not affect training direcdistribu-tion in (16), will be limited to the neighborhood of zero; i.e.,
where is the
perturbation of with respect to the nonrepetitive disturbance . Hence, parameters are limited to the neighborhood of the optimal parameters and the WN control signal is bounded. For a bounded , the system stability depends on the design of the compensator in Fig. 2. According to (15) and (16), the gradient training rule will converge slowly if the learning step is small. However, a large leads to large oscillation in . Hence, the learning step is prescribed as a large con-stant in the beginning to reduce the system error quickly. Afterwards, a smaller is used to improve precision.
F. Simulation Results
Fig. 4 compares learning histories of WN controllers between biased and unbiased wavelets. The latter does without bias pa-rameters, i.e., in (12). The root-mean-square (rms) value of the flying height error is donated as .
Fig. 5. Learning histories in experiments. (a) rms(e). (b) a ; b ; c ; q , and.
Both WN controllers have three neurons in the single hidden layer and have to follow a measured disk surface deformation with peak-to-peak variation of 9 m at 5400 rpm. Before learning, parameters in the unbiased WN controller are pre-scribed the same as those in the biased WN controller except . Fig. 4 shows that the biased WN controller not only results in smaller error, but also faster convergence. Hence, the biased wavelet sets are used for WN controllers in experiments.
IV. EXPERIMENTALRESULTS
In order to demonstrate the effectiveness of the proposed WN controller, experiments are carried out, where a PZT bender is commanded to follow surface deformation of a hard disk ro-tating at 5400 rpm. A WN is online trained by prescribing a desired flying height . The WN has three neurons in the single hidden layer. Fig. 5(a) shows that the rms error converges after 20 s. Fig. 5(b) depicts learning histories of the first neuron in the hidden layer, and . In the presence of the compensator , the WN controller is not enabled until s. As shown in Fig. 6, flying height variation rapidly reduces to m in 11 ms. Fig. 7 compares power spectrums of the flying height error when the WN controller is disabled and when enabled. As a consequence, the resultant flying height error power re-duces 87, 25, and 31 dB at 90, 180, and 270 Hz, respectively, due to the WN controller. Hence, the present WN controller suc-cessfully cancels the repetitive disturbance and let the PZT bender follow the disk surface.
Fig. 6. Measured control results when the PZT bender follows a hard disk with desired flying heightr .
Fig. 7. Comparison of power spectrums for flying height error among WN control, FSBN control, and feedback control without WN and FSBN.
Fig. 8. Comparison of the WN control signalr , the FSBN control signal ^r , and disk deformationd, where l is half period.
In addition to the WN controller, Fig. 7 also compares with a Fourier series-based network (FSBN) controller [9] defined as
(17)
where denotes the number of harmonic frequencies; , and are Fourier coefficients. For comparison, the FSBN controller yields that replaces the WN control signal in Fig. 2. The FSBN controller results in a flying height that varies between m in contrast to m by using WN. The FSBN controller reduces flying height error power 57, 52, and 5 dB at 90, 180, and 270 Hz, respectively.
with a 90 phase delay of the open-loop system in the low frequency region as shown in Fig. 3. The FSBN con-trol signal behaves similarly to the WN control signal in , but oscillates in . Therefore, the WN control performs better than FSBN control on the time axis responding to the disk deformation .
V. CONCLUSION
Based on a proposed wavelet network, this paper has pre-sented a WN controller to reduce repetitive error in disk drives. The WN controller, which uses the system time in (12) as the only input signal in the input layer, can generate the required control signal that has the same period as the disturbance pe-riod. The repetitive error caused by the repetitive disturbance is hence reduced. Experimental results have validated the pro-posed WN controller, which effectively reduces repetitive flying height error.
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C. M. Chang received the B.S. degree in mechanical
engineering from National Taiwan University of Sci-ence and Technology, Taipei, in 1997 and the M.S. and Ph.D. degrees in mechanical engineering from the National Chiao Tung University, Taiwan, in 2000 and 2005, respectively.
Since 2005, he has been with the AU Optronics Corporation (AUO), Taichung, Taiwan. His interests include control, optomechatronics, and display tech-nology.
T. S. Liu received the B.S. degree from National
Taiwan University in 1979, the M.S. and Ph.D. degrees from the University of Iowa, Ames, in 1982 and 1986, respectively, all in mechanical engineering.
He has been with Department of Mechanical Engineering, National Chiao Tung University, Taiwan, since 1987. He was a visiting researcher with the Institute of Precision Engineering, Tokyo Institute of Technology, in 1991. He was also a visiting researcher with the Institute of Precision Engineering, Swiss Federal Institute of Technology, Zurich, in 1998. He is currently a Professor with the Department of Mechanical Engineering, National Chiao Tung University. His research interests are in optical disk drives, data storage, and dynamics and control of mechatronic systems.
