A Discrete Time Optimal Control Solution for Hard Disk Drives Servo ...

22nd IEEE International Symposium on Intelligent Control Part of IEEE Multi-conference on Systems and Control Singapore, 1-3 October 2007

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A Discrete Time Optimal Control Solution for Hard Disk Drives Servo Design Shaohua Hu1, Zhiqiang Gao2 precisely within the target track once it gets there. For simplicity, they may be viewed, in general, as transient and steady state phases (modes), respectively. In control design for these two modes, there are different priorities in accuracy and speed, and is therefore handled by two different controllers in the MSC framework. As the operation transitions from one mode to the other, so do the controllers. The transition is usually hard coded as "if… then…" statement in controllers but it could also be implemented by making the control law as one seamless function of the tracking error, see, for example, [7]. However different they are in appearance, the fundamentals stay the same: the control design is carried out separately for the two modes and later combined in implementation. With the HDD evolving towards smaller size, larger volume and faster access speed [6], the existing control techniques may not be adequate, particularly concerning the two key issues: performance and simplicity.

Abstract: This paper concerns with the challenge in high performance motion control systems, typified by the computer hard disk drive positioning control problem. Not only such systems seek very high accuracy, there is also an insatiable appetite for speed. To make the problem more challenging, the design has to address the actuator saturation issue. This is, in essence, a time optimal control problem and a novel solution, a discrete time optimal control law, is proposed in this paper. It is shown that such solution is 1) a single controller that unifies the separate track-seeking and track-following control functions in existing methods; 2) robust against unknown resonant modes; and 3) easy to implement and tune. The new control law is compared with existing methods and the results are quite promising in simulation studies of two different hard disk drives. Key Words: Time Optimal Control, Hard Disk Drives, Servo Systems, Discrete Time Optimal Control, Unified Servo Control.

Performance: On paper, the HDD servo design problem and Time Optimal Control (TOC) formulation seem to be a perfect match. The plant is approximately a double integrator, the actuator is amplitude limited, and the minimal transient time is sought. In fact, many researchers have proposed TOC based methods for HDD, see, for example, [8-12]. While the TOC solution could be good for the track-seeking mode, the harsh control switching needs to be softened for the tracking-following mode. As results, the proximate TOC was often used at the expense of performance degradation. For example, the method proposed in [7] was shown to have over 30% shorter settling time than that of the approximate TOC in [1]. It makes one wonder how much optimality is left after all the approximations were made. Perhaps it is time to reexamine the approximation of TOC and to propose a better solution.

I. Introduction What makes the servo control problem associated with computer hard disk drive (HDD) intriguing is that the performance is never good enough. We always try to cramp more data into a smaller disk at an ever faster rate. In HDD, data is stored on a magnetic medium divided into tracks. To store more data, tracks are made increasingly narrower, which in turn requires the read and write (R/W) heads to be positioned more accurately. But to achieve faster data access, the HDD assembly is required to move more quickly, which tends to excite the actuator and Hard Disk Assembly (HDA) resonance and thus causing the R/W heads to oscillate. Other factors such as the bias torque, frictions, windage, positional repeatable runout and various noises can also degrade the control performances. Furthermore, the power amplifier saturation and the low sampling rate proved to be significant design constraints for the HDD servomechanisms. Understandably, position servo has become one of the bottleneck problems in HDD technology and much effort has been devoted to this subject, see, for example, [1-12] and references therein. A conventional strategy used in existing methods is known as Mode Switching Control (MSC) [2], where the problem is subdivided into trackseeking and track-following modes. The former refers to the phase where the R/W head moves from its initial position to a target track, while the latter refers to keeping it 1

2

Simplicity: Using two different control schemes in MSC makes the servo design complex and hard to tune. Furthermore, the switching from the track-seek to trackfollowing mode tends to cause undesired transient at the settling phase. Such transients make the effective seek time longer. In order to obtain a smooth transition at the hand-off, some special measures or complex schemes such as Initial Value Compensation (IVC) [2] have to be employed. The complexity in the control scheme also leads to difficulty in tuning. For example, in implementing an H∞ controller, much effort went into picking the weighting coefficients by

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Note that the sign function in (2.2) makes u(t) takes on only the extreme values, r or –r, therefore the name bang-bang control. While this control law is the fastest among all controllers in transient convergence time, its bang-bang nature is harsh for the actuators. Such problem may not exist in a discrete time optimal control Consider a discrete time double integral plant ⎡1 h ⎤ ⎡0⎤ x(k + 1) = ⎢ (2.3) ⎥ x(k ) + ⎢ h ⎥ u (k ) , |u(k)| ≤ r 0 1 ⎣ ⎦ ⎣ ⎦ where h is the sampling interval. The time optimal control problem can be stated as: given the plant (2.3) and its initial state x(0), determining the control signal sequence, u(0), u(1), …, u(k), such that the state x(k) is driven back to the origin in a minimum number of steps, subject to the constraint of |u(k)| ≤ r, i.e., find u(k*), |u(k)| ≤ r, such that k*=min{k|x(k)=0}. Note that, by nature, the solution is obviously not bang-bang at least close to the origin.

trial and error. Again one wonders, instead of MSC, perhaps we can find a single controller that can deal with both modes in HDD servo. Furthermore, perhaps this controller can be made easily to tune. These challenges lead us back to the basics of time optimal control problem. The performance degradation in the proximate TOC solution comes from the trade-off between the performance and smoothness of the control signal. It is well known that the TOC solution is bang-bang control in continuous time, but not necessarily so in discrete time case. On the other hand, finding a closed-form solution for the discrete TOC (DTOC) problem proved to be challenging. It was not until recently a closed-form DTOC law was found [14]. The detailed proof was shown in [15], and the properties and applications were shown in [16]. In this paper we attempt to show that the DTOC method is a promising solution for the HDD servo problem. The new controller, denoted as the Time Optimal Unified Servo Controller (TOUSC) [13], distinguishes itself from existing methods in many aspects. As it turns out, this single controller handles both track seeking and following modes quite well. It obtains a very short seek time and a good position and torque disturbance rejection in the trackfollowing mode. Most notably, the TOUSC does not have overshoot in it head position output, as expected from the time optimal control solution. The paper is organized as follows. The introduction of the discrete time optimal control strategy [15] and its extension to a non-unity gain double integral system are presented in Section II. In Section III, the control structure of TOUSC is given and corresponding simulations are conducted on a 13kTPI HDD model. The TOUSC is also compared with existing control algorithms such as the Proximate Time-Optimal Servomechanism (PTOS) and the Composite Nonlinear Feedback (CNF) [7] in a 25kTPI HDD model in simulation. Finally, the concluding remarks are included in Section IV.

2.1 The Closed-form DTOC Solution The solution of this discrete time optimal problem recently introduced in [14-16] is u = − rsat ( a ( x1 , x2 , r , h), hr ) (2.4)

where 2 2 ⎧ h r + 8r | y | − hr 2 ⎪x + sign( y ),| y |> h r a ( x1 , x2 , r , h) = ⎨ 2 2 2 ⎪ x + y / h, | y |≤ h r ⎩ 2

⎧ sign( s ) s > δ ⎪ ⎨ s s ≤δ ⎪⎩ δ This control law can be easily implemented in a digital computer as

y=x1+hx2,

sat ( s , δ ) =

u = fhan (( x1 , x 2 , r , h ) d = rh ; d 0 = hd

II. Discrete Time Optimal Control

y = x1 + hx 2

For HDD, the rigid body plant is close to a double integral open loop system with motor torque as the input and position as the output. The design goal is to make the output go to a desired location as soon as possible (in minimal time) subject to the input saturation (motor torque is bounded). Therefore, the recent advances in the area of time optimal control [14-16], as shown below, may very well lend a hand in managing the HDD control problem. Consider the double integral plant with the actuator saturation: ⎡0 1 ⎤ ⎡0⎤ x (t ) = ⎢ x(t ) + ⎢ ⎥ u (t ),| u (t ) |≤ r (2.1) ⎥ ⎣0 0⎦ ⎣1 ⎦ It is well known that the control law that drives the state from any initial condition back to the origin in minimum time is [17-20] x (t ) x 2 (t ) u (t ) = − rsign ( x1 (t ) + 2 ) (2.2) 2r

a0 =

d 2 + 8r | y |

a0 − d ⎧ sign ( y ), | y |> d 0 ⎪x + a=⎨ 2 2 ⎪⎩ x 2 + y / h , | y |≤ d 0 ⎧ r sign ( a ), | a |> d ⎪ fhan = − ⎨ a | a |≤ d ⎪⎩ r d ,

(2.5)

Note that this time optimal control law is not bang-bang Furthermore, for double integral plants with a non-unity gain, such as ⎡0 1 ⎤ ⎡0⎤ x (t ) = ⎢ x(t ) + ⎢ ⎥ u (t ),| u (t ) |≤ r (2.1a) ⎥ ⎣0 0⎦ ⎣b ⎦

let v=bu, and |v(k)| ≤ br, (2.1a) is a unit gain double integral plant with respect to v. And the corresponding DTOC law is (2.4a) v= -brsat( a ( x1 , x2 , br , h) , hbr) or,

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The linear current estimator (LCE) [1] was proposed to reduce the phase lag in the conventional digital state estimators. Assume that the plant is

(2.5a)

where the function fhan( ) is defined in (2.5). •

⎧ X ( k + 1) = Φ X ( k ) + Γ u ( k ) ⎨ ⎩ y ( k ) = HX ( k )

2.2 Tuning of DTOC The parameters of DTOC law are r and h in (2.4) and they have strict physical meanings, i.e. r is the physical saturation limit of the actuator and h is the step size in the discrete control system. The control law (2.4) was derived constructively in [14,15] and its properties and applications are discussed in [16]. Most notably, for the plant in (2.3), this controller has the fastest transient response of all controllers, it never overshoots, and its control signal is smooth. In linear control system, the settling time is inversely proportional to the closed-loop bandwidth, which is often used as the yard stick in comparing different controllers. In this sense, the DTOC has the highest bandwidth among all controllers, subject to the actuator saturation constraint. The intuition developed in linear control design also predicts that high bandwidth controller is prone to sensor noises. This proves to be the case for DTOC as well in simulation studies. Considering that DTOC requires the velocity measurement that is often noisy, it becomes apparent that a balance needs to be established between the aggressiveness of the controller and its noise sensibility. One may observe from (2.4) that the DTOC law is linear (therefore smooth) in the area in the x1-x2 phase plane defined by | a ( x1 , x2 , r , h) |≤ hr and |x1+hx2| ≤ h2r (2.6)

(2.9)

where u and y are the input and output, respectively, X is the state variable, and Ф, Г and H are constant matrices of appropriate dimensions. The current estimate Xˆ ( k ) is given as (2.10) Xˆ (k ) = X (k ) + Lc [ y(k ) − HX (k )] where Lc is the observer gain and X ( k ) is the predicted estimate based on a model prediction from the previous time estimate, that is X ( k ) = ΦXˆ ( k − 1) + Γu ( k − 1) (2.11) For the 13kTPI HDD plant described in Appendix, the rigid-body model in discrete state equation with a sample period of 66.67 μs is ⎡1 1.6207 ⎤ ⎡1 0.8104⎤ , ⎡1.5953 ⎤ ,Γ = (2.12) Ψ=⎢ Φ=⎢ ⎥ ⎥ ⎢1.9686 ⎥ 1 ⎦ 1 ⎦ ⎣0 ⎣ ⎦ ⎣0 The observer gain is chosen as Lc = [0.79 0.25] (2.13) This current estimator, defined in (2.10) and (2.11), shows marked improvement over the standard Luenberger observer in terms of phase lag in the estimated state. With the sampling period a limiting factor in this problem, this saving in phase lag proves to be significant, as shown in simulation study later.

It was suggested and verified in simulation [16] that enlarging this area will help to reduce noise sensitivity while retaining the performance benefits of DTOC. This can be done by replacing h in (2.4) with h = k h h, k h ≥ 1 ,

Remarks: 1. The TD can be viewed as a special nonlinear filter that not only reduces the noise in the input signal but also provides its approximate differentiation. It does so without the knowledge of the mathematical model of the dynamic system where the signal was produced. Its tuning parameter, M, is analogous to the bandwidth of a linear filter. Furthermore, M has a clear physical meaning as the limit on the second derivative of the filtered output. 2. The LCE is an improved Luenberger observer in discrete time. It requires the mathematical model but delivers an estimate with less phase lag. 3. Having good velocity estimation is critical for time optimal control. There were other methods considered in addition to the two above but not selected for a variety of reasons concerning performance and implementation.

and kh becomes the only tuning parameter for DTOC. 2.3 Velocity Estimation More often than not the velocity is not measured and the control law based on it, such as DTOC, requires a good estimation of it. The estimator should have fast convergence time and the good noise sensitivity. Two candidates are considered here: the tracking differentiator (TD) [14] and the current estimator [1]. The control law in (2.4) was originally intended as a means for nonlinear differentiation and the result was TD [14]. TD in discrete time form is given as ⎡1 h ⎤ ⎡0⎤ (2.7) v( k + 1) = ⎢ v( k ) + ⎢ ⎥ u ( k ) , |u(k)| ≤ r ⎥ ⎣0 1 ⎦ ⎣h⎦ with u = − Msat ( a (v1 − s, v2 , M , h), hM ) (2.8) T where s is the input signal, v=[v1, v2] the state, h the sampling period, and sat ( a (v1 , v2 , r , h), hr ) is defined in (2.4). The output of TD, v1 and v2, are approximations of s and s , respectively. The filter parameter M has the physical meaning of  s≤M .

III. Time Optimal Unified Servo Control

In this section, we'll try to devise a solution for HDD that has better performance and is easy to implement. We'll start with the analysis of the plant dynamics, based on which a control strategy is formed. The new control method will be evaluated in simulation.

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and disturbances will have to be determined in a realistic simulation study.

3.1 Plant Dynamics A typical HDD servo system includes a power amplifier (PA) and the actuator, which consists of a voice coil motor (VCM) and the R/W head assembly. The design objective is to make the R/W head position, y(t), follow a given reference, or setpoint, ysp. The voltage control signal applied to the power amplifier produces an armature current in the VCM, which in turn produces mechanical torque that moves the R/W head assembly. The parameters of one particular HDD is given in Appendix and will be used as a test bed. Assume that a current loop is used in PA and its current output is approximately equal to the input voltage, u. Then the HDD plant can be described as  y = f (t , y , y ,...w) + bu , |u|≤ r (3.1) where b=KtRhead/Jtotal with Kt, Rhead, and Jtotal given in Appendix, r is the maximum torque in VCM, and f (t , y , y ,...w) represents the combined effects of back electromotive force (e.m.f.), frictions, PA voltage saturation, and hundreds of high-frequency actuator/assembly resonance during an R/W head positioning operation, some of them are shown in Appendix. w in (3.1) denotes disturbances, including the position disturbance and torque disturbance.

3.3 Implementation of a TD based TOUSC The TOUSC is quite easy to implement and tune. The main issue in its implementation is to obtain good velocity estimation when it is not directly measured. There are two mechanisms suggested in section II, which lead to two different ways of implementing TOUSC. The first, as shown in Figure 1, is based on the TD. In this case, the filter parameter M needs to be adjusted appropriately to insure that the estimate v1 and v2 track e and e closely.

Figure 1 Structure of the initial TOUSC system The TOUSC was setup and verified on a 13KTPI industrial hard disk drive model in Simulink. The simulation model incorporated all hard disk drive plant parameters listed in the Appendix. An industry standard sampling rate of 15kHz was used. The D/A output has a saturation of ±1.9A, which corresponds to the current saturation limit, r, in TOUSC, and the power amplifier’s maximum output voltage is ±12.0V. For a 10,000-track seek distance, the step response curves are plotted in Figure 2 and 3. The settling time is 11.3 ms (0.04 tracks) and the steady state error is 0.03 tracks. A series of similar simulations showed that the control method does not generate overshoot and has good control performances for any reasonable track distance for short to long. In simulation the initial value v1(0) in TD should be the same as the desired track number to be moved.

3.2 Problem Solving Strategy There are two school of thoughts in dealing with the plant in (3.1). The majority of control research is devoted to the method of first determining f (t , y , y ,...w) thought system identification techniques, including adaptive techniques, and then synthesize a controller based on it. Another school of thoughts [15] is to start with what is known, i.e.  y = bu , |u|≤ r (3.2) and design an aggressive control law, which, perhaps, can overcome the effects of f (t , y , y ,...w) and achieve satisfactory performance without the detailed knowledge of the plant. Consider the design objective of driving error to zero in shortest time possible under the actuator constraint, the control problem of (3.2) bears amazing resemblance to that of TOC. We therefore decide to investigate the DTOC method as a potential solution to the HDD servo problem. For the sake of simplicity, let the setpoint ysp be a constant and let e=ysp-y, then, e = −bu , |u|≤ r (3.3) Define (3.4) x(t ) = ( x1 (t ), x 2 (t )) T = ( −e(t ),−e(t )) T which puts (3.3) in the form of (2.1a). The corresponding DTOC law is easily determined from (2.5a) as u = (1/ b) fst (−e, −e, br , h) (3.5) This control law is denoted as time optimal unified servo control (TOUSC). Note that the TOUSC is derived based on the time optimal principal for a double integral plant. This generally holds for all servo positioning systems. Whether or not this control law can adequately deal with the unknown dynamics

Position (tracks) 12000 10000 8000 6000 4000 2000 0

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0.005

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10000.1 Zoom In

10000.05 10000 9999.95 9999.9 9999.85 0.01

0.011

0.012

0.013

0.014

0.015 Time (sec.)

Figure 2 Position response for 10,000-track distance It was discovered in simulation that TOUSC performs better if h is replaced by kh*h with k h > 1 . Intuitively, this expands the linear control region [13] and makes the control

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signal smoother. The suitable coefficient kh was chosen as 6.5.

This allows the TOUSC to be more aggressive (kh is lowered from 6.5 to 3.5). Overall, the simulation results support the initial conjecture that the DTOC method is a good fit for the HDD servo problem.

C o ntro l (A ) 2 1

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9999.95

Figure 3 Control command and power amplifier voltage

9999.9 9999.85 0.008

Remarks: 1. The time optimal control strategy, TOUSC, handles the complexity of the HDD dynamics rather well in the above simulation study. The control signal is quite smooth and the accuracy is excellent. 2. The settling time and steady state oscillation are areas for improvement. Note that the phase lag in the feedback loop often is responsible for the oscillation, an estimator with less phase lag will likely to help.

0.009

0.01

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0.012

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0.014 0.015 Time(sec.)

Figure 5 Position response of the modified TOUSC Control (A) 2 1 0 -1 -2

3.4 Implementation of a LCE based TOUSC The TOUSC that is based on LCE [13] is shown in Figure 4. The idea is that the LCE perhaps will help to reduce the phase lag and therefore reduce the amount of oscillations in help to reduce the phase lag and therefore reduce the amount of oscillations in y. This idea will soon be put in a test.

0

0.005

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PA Voltage (Volts) 10 5 0 -5 -10 -15

0

Figure 6 Control signal and voltage of the modified TOUSC 3.5 Comparison of TOUSC with PTOS and CNF To compare the control performances of TOUSC with those of the PTOS and the CNF[7], the LCE-based TOUSC is applied to a fourth-order 25kTPI HDD model used in [7]: Y (s) 6 . 4013 × 2 . 467 × 10 15 (3.6) =

Figure 4 TOUSC system structure using a current estimator The simulation of the LCE-based TOUSC was carried out on the same platform as in the previous test described above. The 10,000-track step response and control signal are shown in Figure 5 and 6, respectively. The settling time is 8.4 ms, a 25% improvement over that in the TD based TOUSC. The magnitude of the steady state oscillation is also decreased by a factor of three. Notice that kh was set to 3.5 and all eight high frequency resonant modes in Table I in Appendix were included in the simulation. The effectiveness of the LCE was clearly shown in Figure 7 and Figure 8. Compared to the true e(t ) , the

U (s)

s 2 ( s 2 + 2 . 513 × 10

3

s + 2 . 467 × 10

8

)

where u is the actuator input (in Volts) bounded as |u|≤3.0 V and y is the R/W head position (in μm). The position responses and control voltages for 100 μm and 300 μm seek distances are shown in Figure 9. Here we also used the sampling rate of 10kHz. Lc=[0.99 0.25] and Kh=2.5. The simulation results including those of the PTOS and CNF in [7] are seen in Table I. The settling time is defined the same as that in [7], or the total time to move the R/W head from the initial position to within 0.2 μm steady error around the final target. The TOUSC improves the settling time

current estimator reduces the phase lag in eˆ(t ) dramatically. 293

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performance by 10 to 20% over the CNF and by 30 to 45% over the PTOS. e-dot (rad/sec) 20

controller has only one tuning parameter which can be easily adjusted in implementation. Two velocity estimation techniques were tested, with the linear current estimator method showing clear advantage in reducing the phase lag. Overall, the proposed TOUSC technique appears to be a strong candidate in industrial servo applications, particularly the HDD servo systems.

e-dot and its estimates

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S olid line: Y s p = 100 um Das hed line: Y s p = 300 um

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Figure 9 Normalized responses with TOUSC in simulation

e-dot E stim ate o f e -dot by TD

-9

Acknowledgement The authors would like to thank Dr. Lin Yang of Western Digital Corporation and Professor Jingqing Han of Chinese Academy of Sciences for their great helps in the course of this research.

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Figure 8 e(t ) and its estimates (zoomed in)

References

Table I Settling Time From Simulation Results Seek (µm)

Settling Time (ms)

Length

PTOS

CNF

TOUSC

100

5.6

3.7

2.9

300

5.9

4.3

3.9

[1] Gene F. Franklin, J. David Powell and Michael L. Workman, “Digital Control of Dynamic Systems”, 3rd edition, Reading, MA, Addison-Wesley, 1998. [2] T. Yamaguchi, K. Shishida, "A Mode-Switching Controller With Initial Value Compensation For Hard Disk Drive Servo Control", Control Eng. Practice, Vol. 5, No.11, pp.1525-1532, 1997. [3] Young-Hoon Kim. Seung-Hi Lee, “An approach to dual-stage servo design in computer disk drives”, IEEE Trans. On Control Systems Technology, Vol.12, No.1,Jan. 2004. [4] Roberto Horowitz, Bo Li, “Adaptive Track-Following Servos for Disk File Actuators”, IEEE Trans. On Magnetics, Vol.32, No.3, May 1996. [5] Li Yang, M. Tomizuka, “Short seeking by multirate digital controllers for computation saving with initial value adjustment”, IEEE/ASME Trans. On Mechatronics, Vol.11, No.1, pp.9-16, Feb. 2006. [6] R. W. Wood, and H. Takano, “Prospects for magnetic recording over the next 10 years, IEEE International

IV. CONCLUSION

A discrete time optimal control approach is proposed for the HDD servo design. It yields a single controller solution that covers the entire track seeking, settling and following operation. No controller switching or separate design for different operating modes is necessary. The new controller is designed with little knowledge of the plant but tested with a full simulation model of HDD that has, among other components, eight resonant modes. Its performance is quite promising in terms of short settling time, high accuracy and robustness against unmodeled dynamics. The new 294

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Magnetics Conf., San Diego, California, USA, pp.98, May, 2006. [7] V. Venkataramanan, K. Peng, B. M. Chen, and T.H. Lee, “Discrete-time composite nonlinear feedback control with an application in design of a hard disk drive servo system”, IEEE Trans. On Control Systems Technology, Vol.11, No.1, 2003. [8] L.Y. Pao and G. F. Franklin, "Proximate time-optimal control of third-order servomechanisms," IEEE Trans. Automat. Contr., vol. 38, pp. 560-580, Apr. 1993. [9] W.N. Pattern, H.C. Wu, and L. White, "A minumum time seek controller for a disk drive," IEEE Trans. On Magnetics, Vol.31, pp.2380-2387, May 1995. [10] H.T. Ho, "Fast bang-bang servo control", IEEE Trans. On Magnetics, Vol.33, pp.4522-4527, Nov. 1997 [11] M. Iwashiro, M. Yatsu, and H. Suzuki, "Time optimal track-to-track seek control by model following deadbeat control," IEEE Trans. On Magnetics, Vol.35, pp.904-909, Mar 1999. [12] B.K. Kim et al, "Robust time optimal control design for hard disk drives," IEEE Trans. On Magnetics, Vol.36, pp.35908-3607, Sept. 1999. [13] Shaohua Hu, “On High Performance Servo Control Algorithms for Hard Disk Drive”, Doctoral Dissertation, Cleveland State University, 2001. [14] Jinqing Han, and Lulin Yuan,. “The Discrete Form of Tracking-Differentiator”, J. Sys. Sci. & Math. Scis., (Chinese), 19(3), pp.268-273, 1999. [15] Zhiqiang Gao, “On Discrete Time Optimal Control: A Closed-form Solution”, Proceedings of the 2004 American Control Conference, pp.52-59, June 30-July 2, 2004, Boston. [16] Z. Gao and S. Hu, "On Properties and Applications of A New Form of Discrete Time Optimal Control Law," Proceedings of IEEE IAS Annual Meeting, Seattle, Oct. 2004. [17] M. Athans and P. Falb, Optimal Control, McGraw-Hill Book Company, New York, 1966. [18] D.E. Kirk, Optimal Control Theory, Prentice Hall, 1970. [19] H.S. Tsien and Jian Song, Engineering Cybernetics, Science Press, 1980 (Chinese). [20] Sun Jian and Hang King-Ching, “Analysis and Synthesis of Time Optimal Control Systems,” Proc. of the Second IFAC Congress, Basel, 1963.

Parameter

Power Amplifier

Actuator

Resonance

Rcoil Lcoil Kpa emax Slewrate Jtotal Kt Rhead TPI Curclip f1 f2 f3 f4 f5 f6 f7 f8 b1 b2 b3 b4 b5 b6 b7 b8

ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8

APPENDIX The model of the VCM actuator of an industrial 13.0kTPI hard disk drive is described by 8 bi Y ( s ) K t Rhead = + ∑ 2 2 2 I c ( s ) J total s i =1 s + 2ξ i ω i s + ω i where ic is the current command (in ampere) and y is the position output (in meter). The model parameters are listed in the following Table:

G ( s) =

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Description

Nominal value

Coil resistance Coil inductance Control gain Saturated voltage DAC Rate limit Moving inertia Torque constant Head radius Tracks density DAC saturation 1st frequency 2nd frequency 3rd frequency 4th frequency 5th frequency 6th frequency 7th frequency 8th frequency 1st coupling coefficient 2nd coupling coefficient 3rd coupling coefficient 4th coupling coefficient 5th coupling coefficient 6th coupling coefficient 7th coupling coefficient 8th coupling coefficient 1st damping ratio 2nd damping ratio 3rd damping ratio 4th damping ratio 5th damping ratio 6th damping ratio 7th damping ratio 8th damping ratio

5.9Ω 0.368mH 1.0×106 12.0 Volts 10,000A/s 2.54×10-6 Kg.m2 0.075 N.m/A 1.9 Inches 13,000 TPI ±1.9A 4,500 Hz 5,400 Hz 5,550 Hz 5,670 Hz 7,300 Hz 7,450 Hz 8,000 Hz 9,650 Hz 1700 260 300 45 100 105 105 455 0.018 0.025 0.025 0.001 0.010 0.010 0.010 0.013