Fractional-Order Proportional Derivative Controller Synthesis and ...

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 22, NO. 1, JANUARY 2014

281

Fractional-Order Proportional Derivative Controller Synthesis and Implementation for Hard-Disk-Drive Servo System Ying Luo, Tao Zhang, BongJin Lee, Changik Kang, and YangQuan Chen Abstract— In this brief, a fractional-order (FO) proportional derivative (PD) controller is designed in a systematic way based on the frequency responses data model for the hard-disk-drive (HDD) servo system. The open-loop system using the designed FO controller is with a flat-phase feature around the gain crossover frequency, which obtains the robustness to loop-gain variations. Thus, the control performance can be made uniform with more consistency than the optimized traditional integer-order controller when the loop gain changes. The theoretical foundation of the FO controller design is presented first. Thereafter, the implementation details of the designed controller in the realtime HDD servo system are introduced, which are important for the real applications of the FO controllers in practice. From the experimental validation, the presented FO-PD controller and the design synthesis are found to be efficient in improving the track-following control performance for the HDD servo system. Index Terms— Flat phase, fractional-order (FO) controller implementation, hard-disk drive (HDD), iso-damping, proportional derivative (PD) controller, robustness, servo system, systematic design.

I. I NTRODUCTION ARD-DISK DRIVES (HDD) have been widely used as a data-storage medium not only for computers but also for many other date-processing devices [1], [2]. The HDD servo mechanism is a high-accuracy control system which plays a critical part in the increasing high-density and highperformance requirements of HDDs [3], [4]. In an HDD servo system, a voice coil motor (VCM) as an actuator is mechanically connected to the heads for reading and writing data. The VCM horizontally activates the heads to the target track on the surface of the disks as the recording medium, where the data reading and writing can be implemented [1]. In all the hardware components in an HDD servo system, there exist offsets and gain variations, e.g., the VCM motor and driver, combo circuits, different heads and disks, and also different track widths on the disks. For example, the length of magnetic fringing field and magnetic flux density are not

H

Manuscript received December 24, 2011; accepted April 14, 2012. Manuscript received in final form January 6, 2013. Date of publication February 12, 2013; date of current version December 17, 2013. Recommended by Associate Editor T. Parisini. Y. Luo is with the Department of Electrical and Computer Engineering, Utah State University, Logan, UT 84321 USA, and also with Samsung Information Systems America, San Jose, CA 95134 USA (e-mail: [email protected]). T. Zhang, B. Lee, and C. Kang are with Samsung Information Systems America, San Jose, CA 95134 USA (e-mail: [email protected]; [email protected]; [email protected]). Y. Q. Chen is with the Mechatronics, Embedded Systems, and Automation (MESA) Laboratory, and with the ME and EECS Graduate Programs, School of Engineering, University of California, Merced, CA 95343 USA. He was with the Department of Electrical and Computer Engineering, Utah State University, Logan, UT 84321 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2013.2239111

uniform in the VCM, and therefore the force from magnetic interaction between the moving-coil magnetic field and the permanent magnets varies [5]. Meanwhile, the flying height of the head, external disturbances, and temperature changes also affect the system gain significantly [6], [7]. Consequently, the loop gain of the HDD servo system is far from constant, the tracking control performance varies easily, and even the stability margins are not reliable with significant loop-gain variations. Therefore, the effect of the loop-gain changes has to be suppressed in HDD servo systems to achieve consistent tracking performance over all track locations. In order to minimize the effect of loop-gain variation, many algorithms have been proposed, which can be classified into two categories. One is concerned with calibration methods [5], [8], [9], in which automatic gain control methods have been widely applied for the loop-gain calibration. However, after the servo gain calibration is fixed, the gain variations due to environmental changes cannot be handled. In [7], a model reference adaptation scheme is proposed for the servo loop-gain calibration in HDD servo systems. Another way is to design robust controllers for rejecting the loop-gain variations [10]–[12]. In [12], a method is investigated to adjust the controller gain automatically for maintaining the open-loop gain in the presence of plant gain variation and to enhance the robustness of feedback loop. In [13], a novel robust structure of the model reference adaptive control is presented for fieldoriented control drives with significant loop-gain variations when the plant inertia is large. Meanwhile, fractional-order (FO) calculus applications have been attracting more and more attention in the system identification and control domain [14]–[17]. Fractional calculus theory has a long history of more than 300 years. Many real objects normally can be more accurately characterized with the noninteger-order dynamic model based on fractional calculus than the traditional integer-order model [14], [18]–[20]. From the system control point of view, it is clear that properly designed FO controllers can outperform the optimized integer-order controllers, as shown in many works [15], [16], and [21]–[25]. Especially, the FO proportionalintegral-derivative controller design and implementation is becoming a mature technique to be applied in real industrial environments [23], [26]–[29]. In this brief, based on the frequency responses data (FRD) model, the synthesis and implementation of an FO proportional-derivative (PD) controller are presented for HDD servo systems. The open-loop system using designed FO controller has the unique feature of flat-phase, e.g., “isodamping” [30], to get the robustness w.r.t. the system loopgain variations. Thus, the control performance can be more consistent than with the optimized traditional integer-order

1063-6536 © 2013 IEEE

controller when the loop-gain changes. This FO controller is designed for the track-following control in the seek tracking control course of the HDD servo. First, the basic methodology of the FO controller design synthesis is introduced. Thereafter, the implementation details of the real-time HDD servo system are presented, which are important for the real applications of FO controllers in practice. From the experiments, the proposed FO-PD controller and the design scheme have been validated to be efficient in improving the track-following control performance of an HDD servo system. The contributions of this brief are as follows: 1) the FO-PD controller design scheme is based on the general FRD model, which is not constrained to model structures; 2) the FO-PD controller is designed and implemented in a real industrial system with details on readjustment and recalculation for achieving the desired specifications; and 3) the effectiveness and advantage of the designed FO-PD controller have been validated in a real industrial HDD servo system for the first time. The rest of this brief is organized as follows. In Section III, the theoretical foundation of the FO PI/D controller design for achieving the robustness w.r.t. the loop-gain variations is introduced. Then, the FO operator implementation in real industrial systems is presented in Section IV. Since differences exist between the ideal and implemented FO operator, the FO controller design scheme is adjusted in Section V. Experimental validation is demonstrated in Section VI. Conclusions are given in Section VII.

Magnitude (dB)


Phase (deg)

282

100 80 60 40 20 0 -20 -40 -60 -80 -100 0 -180 -360 -540 -720 -900 0 10

10

1

10

2

10

3

10

4

Frequency (Hz)

Fig. 1.

Measured FRD model of an HDD platform.

of f (τ ), and (x) is the Gamma function with the definition ∞ (x) = t x−1 e−t dt. 0

From [14, (2.253)] p

L{0 Dt f (t); s} = s p F(s) −

n−1

s p−k−1 f (k) (0)

(2)

k=0

where n − 1 < p ≤ n. For more details on fractional calculus fundamentals, the reader may refer to [14].

II. F UNDAMENTAL OF F RACTIONAL C ALCULUS The idea of fractional calculus has been known since the development of the regular (integer-order) calculus, with the first reference probably being associated with Leibniz and L’Hôpital in 1695 where the half-order derivative was mentioned. Fractional calculus is a generalization of integration and differentiation to the noninteger-order fundamental operator r a Dt , where a and t are the limits of the operation. The continuous integro-differential operator is defined as ⎧ r r d /dt , (r ) > 0 ⎪ ⎪ ⎨ 1, (r ) = 0 r a Dt = t ⎪ −r ⎪ ⎩ (dτ) , (r ) < 0 a

where r is the order of the operation; generally r ∈ R but r could also be a complex number. In this brief, the following Caputo definition is adopted for fractional derivative, which allows utilization of initial values of classical integer-order derivatives with known physical interpretations [14]: t f (n) (τ ) dα f (t) 1 α = dτ (1) 0 Dt f (t) = dt α (n − α) 0 (t − τ )α+1−n where n is an integer satisfying n − 1 < α ≤ n, α is the order of the fractional derivative, f (n) (τ ) is the nth derivative

III. FO C ONTROLLER D ESIGN W ITH “F LAT P HASE ” In this section, the methodology of the FO PI/D controller systematic design is presented. The open loop with the designed FO controller is with the flat-phase feature for isodamping property. Therefore, the servo system can be more robust to loop-gain variations and obtain more consistent servo performance from track to track and from head to head than with the traditional integer-order controller in an HDD servo mechanism. The formulation of the proposed FO controller is C(s) = K p (1 + K d s r )

(3)

where, if r ∈ (0, 2), the designed controller is FO-PD and, if r ∈ (−2, 0), the designed controller is FO-PI. The plant model of an HDD servo system is presented as P(s), which can be a FRD model as shown in Fig. 1 directly measured from the real HDD, without any fitting or approximation. The FO controller (3) is designed following three tuning specifications as follows [20]. 1) Gain crossover frequency specification |C( j ω)P( j ω)||ω=ωg = 1 where ωg is the gain crossover frequency. 2) Phase margin specification

(C( j ω)P( j ω))|ω=ωg = φm − π

LUO et al.: FRACTIONAL-ORDER PD CONTROLLER FOR HARD-DISK-DRIVE SERVO SYSTEM

283

where φm is the desired phase margin. 3) Flat-phase specification (C( j ω)P( j ω)) |ω=ωg = 0 dω which means that the open-loop system using the designed FO controller is with iso-damping property, e.g., the system is robust to loop-gain variations, and the servo performance can be made significantly uniform. The initial design procedures of the FO controller can be summarized as follows, without loss of generality. 1) Given the parametric model or the nonparametric model P(s) of the plant. 2) Given the desired gain crossover frequency ωg . 3) Given the desired phase margin φm . 4) According to the plant model phase (P( j ωg )), and the FO controller phase (C( j ωg )) at the gain crossover frequency ωg , get a first relationship in terms of K d and r from Specification 2. 5) According to the flat-phase Specification 3, the phase of the open-loop system is flat around the gain crossover frequency ωg , and obtain the second relationship between K d and r . 6) From the two relationships of K d and r in (iv) and (v), the two parameters K d and r of the FO controller can be solved in theory. Although an analytical solution is difficult to arrive at, a numerical method can be used to get the parameters [20]. 7) With the plant gain |P( j ωg )| and the FO controller gain |C( j ωg )| at the gain crossover frequency ωg , get the equation in terms of the K d , r , and K p

Kp =

1 2 |P( j ωg )| 1 + K d ωg2r + 2K d ωrg cos(r π/2)

using the solved values of K d and r in (vi), the third parameter K p can be calculated. Therefore, the three parameters of the FO controller can be obtained, with the desired flat-phase feature for the open-loop system. IV. I MPLEMENTATION OF THE FO C ONTROLLER With the given plant model, the FO controller can be designed following the three specifications, satisfying the desired gain crossover frequency, phase margin, and flat-phase feature. The implementation of the designed FO controller in the real-time servo system is critical to achieving the expected control performance benefits. The key point of the FO controller application is the approximation implementation of the FO operator s r , where r ∈ (−2, 2). In this section, the FO operator s r is implemented by the impulse response invariant method [31], which can avoid the constraint of the frequency range. A. Phase Loss from Sampling Delay According to the Bode plot of the implemented FO operator s r, the desired gain and phase can be satisfied with the

Fig. 2. Open-loop Bode plot with the designed FO controller in the continuous-time domain.

continuous approximation by the high-order transfer function. However, in the course of discretization, which is necessary for the controller implementation in a real system, the phase of the designed controller is lost, especially in the highfrequency range close to the Nyquist frequency. This phase loss is unavoidable from the theory of discretization because of the sampling time delay. In this brief, the following FO controller design example on the HDD servo system is illustrated to make the statement clear. In order to compare the FO controller with the original integer-order controller, the gain crossover frequency is set as ωg = 1400 Hz, and the phase margin is set as φm = 35°. According to the procedures in Section III, the FO controller can be designed as C(s) = 0.90814(1 + 9.5398 × 10−5 s 1.0611)

(4)

with the fraction order r = 1.0611 ∈ (0, 2), and an FO-PD controller is obtained. The open-loop Bode plot with the designed FO-PD controller in continuous form is shown in Fig. 2. It can be seen that the gain crossover frequency, phase margin, and the “flat phase” specifications are all satisfied. In the approximate implementation of the FO operator s 1.0611, the eighth-order transfer function is used following the impulse-response-invariant implementation method [31], and the detailed transfer function is G s 1.0611 =

Asr Bsr

where Asr = z 8 − 5.26z 7 + 11.63z 6 − 13.99z 5 + 9.872z 4 − 4.097z 3 +0.9427z 2 − 0.1027z + 0.003426 −5 8

−5 7

(5) −5 6

Bsr = 1.961 × 10 z − 7.981 × 10 z + 12.97 × 10 z −10.6 × 10−5 z 5 + 4.442 × 10−5 z 4 − 8.129 × 10−6 z 3 +1.123 × 10−7 z 2 + 9.75 × 10−8 z − 3.156 × 10−9 . (6)

284


After the discretization of the designed FO controller, the sampling delay is induced into the system, and the phase of the implemented FO controller is delayed. The sampling frequency is 23.914 kHz, i.e., the sampling period is Ts = 0.0418 ms. The signal period with the gain crossover frequency is T p = 0.7143 ms, and the lost phase of the FO operator (s r ) from discretization can be calculated as 0.0418 × 10−3 2π 360° Ts = = 10.5336°. × × −3 Tp 2 0.7143 × 10 2

(7)

According to the designed FO controller (4), the Bode plots of the FO operator s 1.0611 are presented and compared in Fig. 3. It can be seen that the phase loss from the continuous (blue line) FO operator to the discretized (red line) FO operator at 1.4 kHz is around 8.1°. This phase loss is smaller than the calculated 10.5336° above. The reason is that the discretized gain of s 1.0611 is slightly higher than that in continuous form around the gain crossover frequency, which can also be seen in Fig. 3. The phase delay is reduced by about 2.5◦ . Thus, the exact phase loss of the implemented FO operator is around 8° at the gain crossover frequency, which is validated in Fig. 3. In the frequency range ω ∈ (1k, 2k) Hz of Fig. 3, the gain difference between the continuous and discretized FO operators is almost constant around 0.2 dB. So, 2.5° phase difference can be obtained in this frequency range, which is the gain crossover frequency chosen. Therefore, the exact phase loss of the implemented FO operator at the frequency ω can be calculated as follows: 360° − 2.5° (8) φloss = ωTs × 2 where Ts is the sampling period. Then, the phase delay θd of the implemented FO controller at the designed gain crossover frequency ωg can be calculated according to the phase loss of the implemented FO operator. The frequency response of the continuous FO-PD controller C1 (s) is (9) C1 ( j ω) = K p (1 + K d Ae j α ) where A = ωr , α = πr/2. The phase of C1 is θ1 = arctan(

180° K d A sin α )× . 1 + K d A cos α π

(10)

The frequency response of the implemented FO-PD controller C2 (s) with phase loss of the FO operator is C2 ( j ω) = K p (1 + K d Ae j (α−δ))

(11)

where A = ωr , α = πr/2, δ = φlost . The phase of C2 is

K d A sin(α − δ) 180° θ2 = arctan . (12) × 1 + K d Acos(α − δ) π Then, the phase delay θd can be calculated as θd = θ1 − θ2 .

Approximated True

(13)

When ω = 1400 × 2π (rad/s), the phase delay of the implemented FO controller is θd = 59.4149°−53.9271° = 5.4877°. The simulated Bode plots in Fig. 4(a) and (b) validate this calculation result.

(a)

Approximated True

(b) Fig. 3. Bode plot comparison of the continuous true and discretizationapproximated FO operator s 1.0611 .

Because of the phase loss from the discretization, the gain crossover frequency, phase margin, and flat-phase specifications cannot be satisfied in open-loop Bode plot as shown in Fig. 4(a) and (b). B. Gain Boosting from Discretization In the course of the discretization of the FO operator s r , not only is a phase delay induced, but also the magnitude is slightly boosted. The increase is around 0.2 dB in the frequency range (1k, 2k) Hz, where the gain crossover frequency is chosen from. So, the gain change kd of the implemented FO-PD controller from the discretization can be calculated. The frequency response of the continuous FO-PD controller C1 (s) is C1 ( j ω) = K p (1 + K d Ae j α )

(14)

where α = πr/2, A = ωr . The gain of C1 (s) is

k1 = 20log10 K p (1+ K d A cos(α))2 +(K d A sin(α))2 . (15)


285

Approximated True

(a)

(b)

Fig. 4. Bode plot comparison of the FO-PD controller and open-loop system. (a) Bode plot comparison of the continuous and discretized FO-PD controllers. (b) Open-loop Bode plot with discretized FO-PD.

The frequency response of the implemented FO-PD controller C2 (s) with phase loss of the FO operator is C2 ( j ω) = K p (1 + K d Ae jβ )

(16)

where α = πr/2, β = α − φlost , A = ωr . The gain of C2 (s) is

k2 = 20log10 K p (1+ K d A2 cos(β))2 + (K d A2 sin(β))2 (17) where α = πr/2, β = α − φlost , ksrd = 0.20, A2 = ωr × 10ksrd /20 . Then, the gain change kd can be obtained as kd = k1 − k2 .

(18)

When, ω = 1400 × 2π (rad/s), the gain change of the implemented FO-PD controller is kd = 3.7211−4.4363 = −0.7152. This calculation result can be validated in the simulation Bode plot of Fig. 4. V. R EADJUSTMENT FOR THE D ESIGNED FO-PD C ONTROLLER According to the presentation in Section IV for the implementation of the designed FO-PD controller, the intially designed and implemented FO-PD controller after discretization cannot satisfy the proposed three specifications in Section III. Therefore, the designed FO-PD controller needs to be adjusted with predicting and considering the phase loss and gain boosting in advance, so as to satisfy the desired specifications after the discretization.

Fig. 5.

Slope of the open-loop magnitude in frequency range (1k, 2k).

margin setting for the FO-PD controller design. The desired phase margin can be adjusted as φm = φm + θ d .

(19)

If the desired final phase margin is φm = 35°, the phase loss can be calculated in advance as θd = 5.4877°. Therefore, the phase margin needs to be set as φm = 35° + 5.4877° = 40.4877°.

(20)

A. Phase Margin Readjustment With Phase Loss Prediction

B. Gain Crossover Frequency Readjustment With Gain Boosting Prediction

As presented in Section IV-A, the phase delay of the implemented FO-PD controller or the open-loop system can be calculated according to the phase loss of the implemented FO operator s r . So, this phase loss can be built into the phase

The gain change of the implemented FO-PD controller can be calculated as kd = −0.7152 which is mentioned in Section IV-B. Meanwhile, according to Fig. 5, the slope of the open-loop magnitude in the frequency range (1k, 2k) Hz

286


−5

x 10 18

From Specification (II) From Specification (III) − Line 1 From Specification (III) − Line 2

16 14

Kd

12 10

The solution is obtained from this intersection

8 6 4 2 0

Fig. 6.

0.8

Open-loop Bode plot with adjusted FO-PD controller.

is around 0.0063 dB/Hz. Then, the frequency offset ωos can be calculated as ωos = −0.7152/0.0063 = −113.5238 Hz.

(21)

So, the gain crossover frequency setting for the FO-PD controller design needs to be adjusted as ωg = ωg + ωos = 1400 − 113.5 = 1286.5 Hz.

(22)

C. Phase Slope Readjustment With Phase Loss Slope Prediction Since the phase delay can be calculated as in Section IV-A, the phase slope change can also be calculated according to the derivative of the phase delay w.r.t. the frequency ω d(ωTs × 360° d(φlost ) 2 − 2.5°) = dω d(ω) Ts 360° (°/Hz) = (s). = Ts × 2 2

(23)

(24)

For the approximation implementation of the FO operator s 1.1411 , the eighth-order transfer function is also used following the impulse-response-invariant implementation method [31], and the detailed transfer function is G s 1.1411 =

Asrr Bsrr

1

1.1

1.2 r

1.3

1.4

1.5

1.6

Fig. 7. Numerical solution from the intersection of the curves in terms of r and K d .

where Asrr = z 8 − 5.299z 7 + 11.82z 6 − 14.35z 5 + 10.23z 4 − 4.3z 3 Bsrr

+1.005z 2 − 0.1117z + 0.003846 (25) = 8.653 × 10−6 z 8 − 3.509 × 10−5 z 7 + 5.707 × 10−5 z 6 −4.726 × 10−5 z 5 + 2.085 × 10−5 z 4 − 4.69 × 10−6 z 3 +4.966×10−7z 2 −3.556×10−8z +2.752×10−9. (26)

Using this discretized FO-PD controller, the open-loop Bode plot can be drawn as shown in Fig. 6. It can be seen that the desired phase margin, gain crossover frequency, and flat-phase feature are all satisfied with the discretization implementation of the designed FO-PD controller. D. FO-PD Controller Design and Implementation Procedure Summary

So, the phase derivative w.r.t. the frequency at the gain crossover frequency point needs to be set as −Ts × (360°/2)(°/) Hz rather than zero, to compensate for the phase loss in the discretization. This compensation can guarantee the flat-phase specification in the implemented control system with discretized FO-PD controller. According to the readjustments for phase margin, gain crossover frequency, and phase slope, the FO-PD controller can be redesigned as Cre (s) = 0.86878(1 + 4.7552 × 10−5 s 1.1411).

0.9

After the discussion for the implementation details of the designed FO controller, the final design and implementation procedure of the FO controller can be summarized as follows. 1) Given the parametric model or the non-parametric model of the plant. 2) Given the gain crossover frequency. 3) Given the phase margin. 4) Gain crossover frequency readjustment with gain boosting prediction. 5) Phase margin adjustment with phase loss prediction. 6) Phase slope readjustment with phase slope change prediction. 7) From Specification 2, get one relationship between K d and r , and a curve in terms of these two parameters can be drawn with a numerical method, as shown by the red line in Fig. 7. 8) From Specification 3, obtain the other relationship between K d and r which is a second-order equation; therefore the other two curves can be drawn as the blue and green lines in Fig. 7.


(a)

287

(b)

Fig. 8. Open loop and error sensitivity Bode plot comparisons between the designed FO-PD and original controllers. (a) Open-loop Bode plot comparison. (b) Error sensitivity Bode plot comparison with loop gain variations.

9) According to the intersection of two curves in Fig. 7 from steps 7 and 8, the parameters K d and r can be fixed. 10) From Specification 1, and the solution of K d and r in step 9, the third parameter K p can be determined. Therefore, the FO-PD controller is implemented satisfying the preselected gain crossover frequency, phase margin, and the flat-phase requirement in a real HDD servo system. VI. E XPERIMENT With the designed FO-PD controller following the details presented in this brief, the open-loop Bode plot comparison of the designed FO controller and original integer-order controller can be seen in Fig. 8(a). The red line is for the open-loop Bode plot with the designed FO-PD controller, and the blue line is for the original integer-order controller. This original controller used for the comparison with the designed FO-PD is optimized by the loop-shaping according to the performance requirements of the customers. The flat-phase feature of the open-loop system with the designed FO-PD controller can be seen clearly in Fig. 8(a). Meanwhile, from Fig. 8(b), it can be seen that the sensitivity function plots around the gain crossover frequency are maintained well even under loop-gain variations with the designed FO-PD controller, which illustrates its better robustness over the traditional controller. The advantage of the designed FO-PD controller can be validated by the experimental demonstration presented in Figs. 9 and 10, and Tables I and II for the track-following and the throughput performances, respectively. A. Original Integer-Order Controller Design In order to compare it with the designed FO-PD controller fairly, the original-integer order controller is optimized according to the random neighborhood search method [6], [32],

where the gain crossover frequency and phase margin are set as the same values as those for the FO controller design, i.e., ωg = 1400 Hz and φm = 35°. B. Track-Following Performance In track-following control of the HDD servo system, the track misregistrations (TMR) are measured using the designed FO-PD and original controllers with loop-gain variations in Fig. 9. In order to verify the robustness to the loop-gain variations with the designed FO-PD controller and compare it with the original controller fairly, the loop gain is artificially tuned by changing the proportional coefficient of the controller, e.g., K p in (3). In Fig. 9, the blue line stands for the TMR without-loop gain variation. The green and red lines represent the TMR with −20% and +20% loop-gain variations, respectively. It is clear that the three TMR lines in Fig. 9(b) are much more concentrative than in Fig. 9(a), i.e., the track-following performance using the designed FO-PD controller is much more robust to the loop-gain variations than with the original controller. This performance advantage with the designed FO-PD controller is due to the open-loop flatphase “iso-damping” property using the proposed controller design scheme in this brief. From disturbance sensitivity point of view, in Fig. 8(b) we can see that the sensitivity hump around the gain crossover frequency using the designed FOPD controller is maintained better than the original controller under loop-gain variations. So, the sensitivity change with disturbance and noise from loop-gain variations using the designed FO-PD is smaller than while using the original controller, which also proves the robust performance improvement by using the designed FO-PD controller. The detailed numerical comparison for the track-following performance is shown in Table I. It can been seen that the overall tracking performance using the designed FO-PD with loop-gain

288


TABLE I TMR P ERFORMANCE I MPROVEMENT 100% Gain

120% Gain

80% Gain

TMR (original)

13.53

12.59

18.60

TMR (FO-PD)

11.49

11.40

12.66

Improvement

15.05%

9.45%

31.94%

TABLE II S EEK S ETTLE P ERFORMANCE I MPROVEMENT

I/O transfer average value

25

7 20

Magnitude

5

15 sigma = 17.06 (OrgC 100%Gain, C:OD/H:All) sigma = 17.86 (OrgC 120%Gain, C:OD/H:All) sigma = 22.34 (OrgC 80%Gain, C:OD/H:All) 10

3

Cumulative Sum

6

2 5 1

0

0

2000

4000

6000 Frequency (Hz)

FO-PD

Improvement

124.5IOPS

130.1IOPS

5.6 IOPS (4.5%)

C. Throughput Performance

(a) 8

4

Original

8000

10000

0 12000

(b)

I/O transfer speed (MB/s)

Fig. 9. TMR comparison with original controller and the designed FO-PD controller. (a) Original controller. (b) FO-PD controller.

As shown in Fig. 9, the TMRs with loop-gain variations are compared using two controllers. However, the loop-gain variations are produced artificially for simplicity and directly verifying the robustness of the system, which are not the real loop-gain variations in the operations of an HDD. Actually, the loop-gain variations are severe in HDDs with the operations from head to head and from track to track. In order to validate the real benefit of the designed FO-PD controller, the I/O throughput performance is tested with the real loopgain variations in the read/write operations in an HDD. In Fig. 10, the blue dot-line stands for the I/O transfer efficiency with original controller, and the red dot-line represents that with the designed FO-PD controller. From Table II, the I/O transfer performance achieves about a 4.5% improvement. The throughput performance with the designed FO-PD controller is obviously better than with the original optimized one. VII. C ONCLUSION This brief presented a design synthesis for an FO-PD controller to achieve two preselections (phase margin and gain crossover frequency) and flat-phase requirement for the HDD servo system. The details of the design and implementation of the proposed FO controller in industrial HDD servo system were presented clearly. The designed FO controller is more robust w.r.t. the loop-gain variations and makes the servo performance more consistent in an HDD tracking system than the original optimized-integer order controller. Drive-level test results showed that both TMR and throughput performance could be improved with the designed FO-PD controller in the HDD servo system. R EFERENCES

Transfer length (KB)

Fig. 10. I/O transfer performance comparison between the original controller and designed FO-PD controller.

variations is more consistent than that using the original controller.

[1] A. A. Mamun, G. Guo, and C. Bi, Hard Disk Drive: Mechatronics and Control, Boca Raton, FL, USA: CRC Press, 2007. [2] K. Peng, B. M. Chen, T. H. Lee, and V. Venkataramanan, Hard Disk Drive Servo Systems (Advances in Industrial Control), 2nd ed. New York, USA: Springer-Verlag, 2006. [3] R. Conway, J. Choi, R. Nagamune, and R. Horowitz, “Robust trackfollowing controller design in hard disk drives based on parameter dependent Lyapunov functions,” IEEE Trans. Magn., vol. 46, no. 4, pp. 1060–1068, Apr. 2010. [4] C. K. Thum, C. Du, B. M. Chen, E. H. Ong, and K. P. Tan, “A unified control scheme for track seeking and following of a hard disk drive servo system,” IEEE Trans. Control Syst. Technol., vol. 18, no. 4, pp. 294–306, Mar. 2010.


[5] C. I. Kang and M. Abed, “Servo loop gain identification and compensation in hard disk head-positioning servo,” IEEE Trans. Magn., vol. 34, no. 4, pp. 1889–1891, Jul. 1998. [6] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 3rd ed. Reading, MA, USA: Addison-Wesley, 1998. [7] F. C. Wang, Q. W. Jia, C. F. Wang, and J. Y. Wang, “Servo loop gain calibration using model reference adaptation in HDD servo systems,” in Proc. Chin. Control Decision Conf., Guilin, China, Jun. 2009, pp. 3377–3381. [8] E. Banta, “Analysis of an automatic gain control (AGC),” IEEE Trans. Autom. Control, vol. 9, no. 2, pp. 181–182, Apr. 1964. [9] Q. W. Jia and G. Mathew, “A novel AGC scheme for DFE read channels,” IEEE Trans. Magn., vol. 36, no. 5, pp. 2210–2212, Sep. 2000. [10] J. R. Corrado and W. M. Haddad, “Static output feedback controllers for systems with parametric uncertainty and controller gain variation,” in Proc. Amer. Control Conf., vol. 2. Jun. 1999, pp. 915–919. [11] P. K. Dash, S. Morris, and S. Mishra, “Design of a nonlinear variablegain fuzzy controller for FACTS devices,” IEEE Trans. Control Syst. Technol., vol. 12, no. 3, pp. 428–438, May 2004. [12] K. S. Kim and K. H. Rew, “Enhancing robustness by feedback loop gain adjustment,” in Proc. Int. Conf. Control Autom. Syst., Oct. 2007, pp. 2876–2879. [13] S. R. Bowes and J. Li, “New robust adaptive control algorithm for highperformance AC drives,” IEEE Trans. Ind. Electron., vol. 47, no. 2, pp. 325–336, Apr. 2000. [14] I. Podlubny, Fractional Differential Equations. New York, USA: Academic Press, 1999. [15] B. M. Vinagre and Y. Q. Chen. (2002). The 41st IEEE CDC2002 Tutorial Workshop 2 [Online]. Available: http://mechatronics.ece.usu.edu/ foc/cdc02 [16] A. Oustaloup, Ed., Proceedings of the Second IFAC Symposium on Fractional Dierentiation and its Applications (FDA06). Oxford, U.K.: Elsevier Science, Jul. 2006. [17] Y. Q. Chen, I. Petras, and D. Y. Xue, “Fractional order control—A tutorial,” in Proc. Amer. Control Conf., St. Louis, MO, USA, 2009, pp. 1397–1411. [18] J. C. Wang, “Realizations of generalized Warburg impedance with RC ladder networks and transmission lines,” J. Electrochem. Soc., vol. 134, no. 8, pp. 1915–1920, 1987. [19] S. Westerlund and L. Ekstam, “Capacitor theory,” IEEE Trans. Dielectr. Electr. Insul., vol. 1, no. 5, pp. 826–839, Oct. 1994.

289

[20] Y. Luo and Y. Q. Chen, “Fractional-order [proportional derivative] controller for a class of fractional order systems,” Automatica, vol. 45, no. 10, pp. 2446–2450, 2009. [21] I. Petras, “The fractional-order controllers: Methods for their synthesis and application,” J. Electr. Eng., vol. 50, nos. 9–10, pp. 284–288, 1999. [22] I. Podlubny, “Fractional-order systems and P I λ D μ controller,” IEEE Trans. Autom. Control, vol. 44, no. 1, pp. 208–214, Jan. 1999. [23] C. A. Monje, B. M. Vinagre, V. Feliu, and Y. Q. Chen, “Tuning and autotuning of fractional order controllers for industry applications,” Control Eng. Pract., vol. 16, no. 7, pp. 798–812, 2008. [24] Y. Luo, Y. Q. Chen, H. S. Ahn, and Y. G. Pi, “Fractional order robust control for cogging effect compensation in PMSM position servo systems: Stability analysis and experiments,” Control Eng. Pract., vol. 18, no. 9, pp. 1022–1036, 2010. [25] Y. Luo, Y. Q. Chen, and Y. G. Pi, “Fractional order ultra low-speed position servo: Improved performance via describing function analysis,” Int. Soc. Autom. Trans., vol. 50, no. 1, pp. 53–60, 2011. [26] B. J. Lurie, “Three-parameter tunable tilt-integral-derivative (TID) controller,” U.S. Patent 5 371 670, Dec. 6, 1994. [27] A. Oustaloup, X. Moreau, and M. Nouillant, “The CRONE suspension,” Control Eng. Pract., vol. 4, no. 8, pp. 1101–1108, 1996. [28] H. S. Li, Y. Luo, and Y. Q. Chen, “A fractional order proportional and derivative (FOPD) motion controller: Tuning rule and experiments,” IEEE Trans. Control Syst. Technol., vol. 18, no. 2, pp. 516–520, Mar. 2010. [29] Y. Luo, C. Y. Wang, Y. Q. Chen, and Y. G. Pi, “Tuning fractional order proportional integral controllers for fractional order systems,” J. Process Control, vol. 20, no. 7, pp. 823–831, 2010. [30] Y. Q. Chen and K. L. Moore, “Relay feedback tuning of robust PID controllers with ISO-damping property,” IEEE Trans. Syst., Man Cybern. B, Cybern., vol. 35, no. 1, pp. 23–31, Feb. 2005. [31] Y. Q. Chen. (2008). Impulse Response Invariant Discretization of Fractional Order Integrators/Differentiators Compute a Discrete-Time Finite Dimensional (z) Transfer Function to Approximate s r with r a Real Number [Online]. Available: http://www.mathworks.com/matlabcentral/ fileexchange/21342 [32] C. I. Marrison and R. F. Stengel, “Robust control system design using random search and genetic algorithm,” IEEE Trans. Autom. Control, vol. 42, no. 6, pp. 835–839, Jun. 1997.