Performance enhancement of hard disk drives through data-driven control design and population clustering

Precision Engineering 56 (2019) 267–279

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Performance enhancement of hard disk drives through data-driven control design and population clustering☆

T

Saeid Bashash∗, Shahriar Shariat Department of Mechanical Engineering, San Jose State University, San Jose, CA 95192, USA

ARTICLE INFO

ABSTRACT

Keywords: Hard disk drives Precision positioning Data-driven control Population clustering

This paper presents a frequency-domain data-driven control design method for improving the robustness and performance of a large population of hard disk drives. First, a multi-rate notch filter design method is presented to prevent the aliasing of post-Nyquist frequency resonant modes. The tradeoff between aliasing and system phase loss due to the multi-rate notch filters is investigated and discussed using a set of real HDD plant data. A data-driven servo control design framework for a dual-stage HDD system is then proposed, where the stability, robustness, and performance properties of the closed-loop system are simultaneously computed and optimized. The validity of the proposed notch filter and controller design methods is evaluated through a set of time-domain simulations. Finally, a clustering-based control optimization method is discussed to further improve the population-level system performance. Simulation results indicate that the proposed data-driven controller with frequency domain plant clustering is an effective approach for designing high-performance servo controllers for HDDs.

1. Introduction

population into smaller groups of similar plants. The dual stage HDD servo system falls in the category of the dualinput single-output (DISO) system. Several design techniques in the multi-input multi-output (MIMO) domain have been used to design servo controllers for HDDs, including LQR/LTR [1], H∞ [2], and μsynthesis [3,4]. Moreover, the single-input single-output (SISO) methods have been applied through certain feedback loop architectures and design procedures such as the master-slave architecture [5], the PQ design method [6], the parallel architecture [7], and the decoupling technique [8]. In the latter case, the sensitivity transfer function of the individual actuators in the dual stage system are approximately decoupled, thereby enabling independent SISO design for each actuator. In this paper, the sensitivity decoupling method is used to develop the controller for the HDD servo system. This method has been used in both industry and many research studies [9–11]. Recently, with the advancement of computing technologies and numerical optimization algorithms, data-driven methods have become a popular choice in designing feedback control systems. In such methods, experimentally-collected data can be used either for identifying parametric models, or in lieu of the physical/analytical models for direct data-driven control design. Comparison of the data-driven and model-based control design suggests that the former has certain advantages in scenarios where control performance is sensitive to

This paper presents a novel data-driven approach for servo control optimization of a population of hard disk drives (HDDs), and improving the overall population performance through plant clustering techniques. Data-driven control design in this paper refers to the design of parametric controllers using non-parametric (numerical) frequencydomain plant data. This research is motivated by the ever increasing demand for high capacity HDDs for personal and enterprise-class data storage needs. With the advent of IoT, social media, and cloud analytics, the digital universe has undergone an exponential growth, while the production rate of storage devices has been evolving at a much slower pace due to the development and manufacturing limitations. Therefore, maximizing the storage capacity of the individual devices is an important enabler for keeping up with the pace of data generation. Improving the positioning accuracy of the read-write head on the spinning media has always been one of the key areas of improvement for the HDD data track density. To develop highly optimized HDD servo controllers, this paper proposes using multi-objective numerical optimization for the multi-rate notch filters as well as the servo control loops for the voice coil motor (VCM) and the PZT micro actuator (MA) systems. Additional population-level performance improvement can be attained by developing multiple controllers based on clustering of the This paper was recommended by Associate Editor Chinedum Okwudire. Corresponding author. E-mail address: [email protected] (S. Bashash).

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https://doi.org/10.1016/j.precisioneng.2018.12.007 Received 12 September 2018; Received in revised form 14 November 2018; Accepted 14 December 2018 Available online 17 December 2018 0141-6359/ © 2018 Elsevier Inc. All rights reserved.

Precision Engineering 56 (2019) 267–279

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modeling accuracy [12]. Since modeling error is always present in the characterization and identification of physical systems, data-driven methods can potentially outperform model-based designs. In the HDD design domain, experimental frequency response data (FRD) provides a valuable non-parametric model for frequency domain loop shaping. Particularly, variations in the plants due to manufacturing tolerances can be effectively captured in a large pool of spectral data. Non-parametric FRD models can be mixed with parametric controllers to render frequency-domain closed-loop system characteristics such as the sensitivity transfer function shape and stability margins. To design controllers for non-parametric (data-driven) models, two design frameworks have been pursued. The most common approach is to formulate a convex optimization problem to minimize a certain cost function such as the L2/L∞ norm of the sensitivity transfer function error from a desired one under a PID control law [13], or robust controller design in the Nyquist diagram using linear/quadratic programming and H∞ controller design [14–16]. The second approach is to formulate a non-convex multi-objective optimization problem, including all the required robustness and performance objectives. Computer simulations are then used to evaluate the candidate controllers and optimize them to achieve the desired design goals [17]. This approach enables comprehensive feedback loop optimization for different types of objectives and lower/upper bound specs, for a population of plants, thereby being well-suited for production-level design. The main drawbacks of this approach include computational time and lack of guaranteed convergence to the global optimum. The computational burden can be mitigated by deploying efficient coding and optimization algorithms in fast computers. The second problem can be mitigated by initializing the controller parameters in the neighborhood of the optimal design, which would require some expertise in controller tuning. In this paper, a simulation-based control optimization problem is formulated and evaluated to allow the controller design and analysis for a large pool of plant data based on objectives defined by industry. One of the key challenges in the HDD control design is the presence of high-frequency resonant modes beyond the Nyquist frequency of the closed-loop system. These dynamics fold back to the low-frequency region due to aliasing effect, potentially resulting in performance loss or instability [18]. Designing multi-rate notch filters to suppress post-Nyquist frequency modes is an important step in HDD control development. Simulation-based data-driven control is well-suited for designing multi-rate notch filters for a large population of plants. The main optimization tradeoff is to balance the folding error with the phase loss introduced by the multi-rate notch filters to the main control loop [19]. In this paper, a full data-driven control optimization for the dual stage HDD servo systems is developed and investigated. First, a multirate notch filter optimization problem is formulated and evaluated for a real set of HDD plant data. Then, the control optimization problem is formulated to optimize the VCM and MA controllers to achieve required robustness and performance targets set by the manufacturer. Finally, a novel clustering technique is applied to further maximize the performance of the population by designing multiple controllers for different clusters within the population instead of only one controller for the entire population. The remainder of the paper is organized as follows: In Section II, the data-driven multi-rate notch filter design is presented. Section III develops the dual stage servo loop optimization formulations and evaluations. Section IV presents time domain simulations to further support the frequency domain designs. Section V proposes the plant clustering technique for the population performance enhancement. Finally, Section VI summarizes the paper's conclusions.

Fig. 1. Control architecture of the digital dual stage HDD servo system.

architecture [8]. In this figure, Cv(z) and Cm(z) are the digital controllers for the VCM and the MA loops, respectively, with Pv(s) and Pm(s) representing their corresponding continuous-time plant dynamics. The output of the MA controller is fed into a plant estimator, Em(z), to provide the VCM loop with an approximate position of the MA actuator. Each controller's output is passed through an up-sampler filter, Hnx, which holds and resamples the input signal with n times faster rate (nx). The resampled control signal is then fed into a high-rate notch filter set, Nv/m,nx(z), to mitigate the aliasing of the post-Nyquist plant modes. The nx notch filter output is ultimately sent to the plant through a zero-order-hold DAC system. It is important to note that the represented block diagram in Fig. 1 is a simplified version of the actual system in practice, where multiple layers of low-frequency calibration functions are included to compensate for the drive-to-drive variations, temperature fluctuations, and nonlinearities. Moreover, there are several feedforward trajectories injected into various points of the control loop during track seeking process. Nonetheless, the main control design task for the track following process relies on the optimization of the dual-stage control system shown in Fig. 1. To simplify the analysis and design of the VCM and MA controllers, it is common practice to convert the multi-rate control system into an equivalent single-rate system. The formula for this conversion is provided in Section III. The digital plants Pv(z) and Pm(z) represent the converted singlerate plant dynamics. The accuracy of conversion is dependent on the effectiveness of the nx notch filters in suppressing the aliased modes. The sensitivity transfer function of the equivalent single-rate dual stage actuator (DSA) system is given by:

Sd =

E 1 = R 1 + Cv Pv + Cm Pm + Cm Em Cv Pv

(1)

where all the transfer functions are in the z domain. If we choose MA estimator to be the exact model of the MA plant, i.e., Em = Pm, then we can recast Eq. (1) as:

Sd =

1 = Sv Sm (1 + Cv Pv )(1 + Cm Pm)

(2)

where Sv and Sm represent the individual sensitivity transfer functions of the VCM and the MA loop. This way, the sensitivity transfer functions are decoupled, and the controllers can be designed separately. In practice, since Em is an approximation of the MA plant, the decoupling is not perfectly accomplished. Therefore, Eq. (1) is used to analyze the dual stage system as is. However, one can still optimize the controllers individually before fine-tuning for the full dual-stage design. A sequential design process is used in this paper to design the VCM controller first, and then the MA controller, as discussed in Section IV. To develop and evaluate the data-driven control framework, a set of spectral plant data provided by Western Digital Corporation, a leading HDD manufacturer, is used in this study. Fig. 2 shows the Bode diagram of 500 VCM and MA plants collected at 4x sampling rate. As per requirement by the project sponsor, the frequency axis has been normalized with respect to the Nyquist frequency of the system ( Nq) and the magnitude has been shifted to represent the data near zero (dB) magnitude. Besides, the frequencies of the resonant modes have been perturbed to the right or the left by small amounts to further cover the

2. HDD control architecture and spectral plant data This section briefly reviews the control architecture for the HDD servo loop studied in this paper, followed by a set of real plant data used for control design and optimization. Fig. 1 shows the block diagram of a dual stage servo system based on the sensitivity decoupling 268

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Fig. 3. Example of folding a resonant system's frequency response from 4x to 1x sampling region with and without a multi-rate notch filter.

plot with a resonant mode at 2.5 Nq . Folding the frequency response from the original 4x into the 1x region would result in an aliased resonant mode at 0.5 Nq . The aliased resonant mode is mitigated substantially if the original resonant mode is cascaded with notch filter in the 4x region before being folded. As can be seen, folding from the 4x into the 1x region introduces a significant phase drop due to the zeroorder hold (ZOH) conversion process. Introducing the 4x notch filter at 2.5 Nq would further drop the folded system's phase at lower frequencies. It is worthwhile to note that, a notch filter can be created by a second order transfer function as follows:

Fig. 2. Spectral frequency response data of 500 HDD plants collected at 4x sampling rate: (a) Normalized VCM plants, and (b) normalized MA plants.

original data. This modification may introduce marginal impacts on the results of this paper, but does not change the design process and the expected outcomes. As seen from the FRD plots, significant variation is present in the plant population due to collection of the data from different drives, heads, disk positions, and temperatures. The controller must retain robustness with respect to the entire population. In practice, a larger pool of data is considered for the intermediate stage control development process. The final controller optimization is carried out on a large set of boundary samples belonging to drives intentionally designed at the boundaries of the manufacturing tolerances. This way, the controller robustness is further enhanced before being implemented in the final product. An obvious observation from the VCM and MA plant data in Fig. 2 is the presence of several high-frequency resonant modes beyond the Nyquist frequency of the system. Therefore, the design of high-rate notch filters (4x in this case) plays an important role in the control design process. In the next section, a data-driven optimization method is presented to design the multi-rate notch filters, followed by the design of the dual stage servo controllers in the subsequent sections.

N (s ) =

s2 + 2 s2 + 2

N

Ns

D

Ds +

+

2 N 2 D

(3)

If the numerator and denominator frequencies are identical (i.e., = D ), the resulting notch filter will be symmetric with the DC gain of 1. If D > N , the resulting filter is a notch filter. Otherwise it will be a peak filter. In this paper, we use symmetric notch filters only, for the simplicity of design, and discretize the continuous-time notch transfer function using the pole-zero matching technique for digital implementation. Moreover, the damping ratios N and D are calculated through a map which relates them to the depth and width of the resulting notch filter. Therefore, each filter is described by three parameters, depth, width, and center frequency. The width and center frequency must be constrained to be positive during the optimization process in order for the filter to remain stable. Since the main focus of this effort is to develop a numerical notch filter optimization framework for experimentally-measured plant data, a numerical formulation for computing the folding error is presented. The plant model is assumed to be a frequency-dependent complex function, Pnx ( ) = A ( ) e j ( ) , where A ( ) and ( ) are frequency-dependent numerical magnitude and phase functions, respectively. To quantify the folding error magnitude, the first step is to remove the nx ZOH effect from the notched plant and introduce the 1x ZOH response to it as follows: N

3. Data-driven multi-rate notch filter design Increasing the sampling frequency of the HDD control loop comes at the cost of consuming the data storage space to accommodate for more servo ID (SID) fields. Multi-rate servo design has therefore been introduced to compensate the resonant modes beyond the Nyquist frequency of the system without the need for increasing the main sampling frequency of the system [20]. Different methods have been used to design multi-rate notch filter for HDDs including nonlinear optimization [21], interlacing method [22], and state observer-based design [23,24], among others. This paper presents a data-driven method for robust multi-rate filter design for a population of HDDs. This method builds on a previous work where multiple tradeoffs where investigated in the optimization of multi-rate notch filters for HDDs [19].

Pˆnx ( ) = Pnx ( ) Nnx e

j T n

1x Gho (j ) nx Gho (j )

(4)

nx (j ) is where Nnx represents the digital multi-rate notch filter, and Gho the ZOH frequency response function defined as: nx Gho (j ) =

3.1. Aliasing effect in multi-rate digital systems

1

e j T/n j T /n

(5)

where T is the 1x sampling time, and n is sampling rate ratio. The resulting frequency response function, Pˆnx , can then be folded

Fig. 3 shows an example of a digital system's frequency response 269

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3.2. Frequency-domain notch filter optimization The main tradeoff in the multi-rate notch filter optimization process is to balance the folding error magnitude with the notch filters phase loss to minimize the impact of the notch filters on the main control loop's phase margin. Assuming a single set of notch filters is to serve a population of Np plants, the objective of the optimization problem objective in this study is to:

minimize (Phase (Nnx )|@ q

subject to: max ( Efold, i ) < qmin q qmax

into the 1x region. To do so, the frequency response values at frequencies beyond Nq are augmented to those within the 1x region: (6)

Nq

where Efold is the folding error computed from:

Efold ( ) =

n 2

1

k=1

Pˆnx ((2k + 1) ) +

n 2 k=1

Pˆnx (2k

Nq

), 0 <

Nq

2

{1,2, …, Np} (8)

where q is the notch filter parameter set, including the depths, widths, and frequencies of all the notch filters considered in the design, and δ is the allowed upperbound of the folding error for all the plants in the population. The objective of the optimization problem outlined in Eq. (8) is to minimize the phase drop of the cascaded notch filters at a particular frequency ( 0 ) while maintaining the folding error below a certain threshold. The frequency 0 should be selected in the low frequency range where the phase of the notch filters drops uniformly over frequency. Any frequency within 10–90% of the lowest notch filter frequency would be an acceptable choice. Fig. 5 shows the magnitudes of the folding error for the VCM plants in the absence of multi-rate notch filters, which have been converted to percentage through division by the average gain of the plants near the Nyquist frequency. For some of the plants in the population, the folding error magnitude can raise to as high as 600% with respect to the baseline magnitude, as can be seen from the figure. This value must be reduced significantly through the design of multi-rate notch filters for the successful implementation of the controller. To optimize the multi-rate notch filters for folding error mitigation, 8 notch filters are used in this study. Each filter is described by three parameters: center frequency, depth, and width. Hence, a total of 24 parameters are iterated over during the optimization process. The notch filters are initialized almost uniformly across the frequency range within the 2x-4x Nyquist intervals (see Fig. 6(a)). The optimization constraints are added to the main cost function using quadratic penalty functions with large weight values. MATLAB's “fminsearch” function, which is based on the Nelder-Mead simplex algorithm [25], is used to carry out the optimization. The optimization objectives and constraints are set as follows: The frequency at which the notch phase loss was evaluated (i.e., 0 ) is set to be 0.1 Nq , which is near the gain crossover frequency of the VCM control loop at which the phase margin is calculated. The folding error upperbound (i.e., δ) is set to 20% of the baseline plant gain at Nyquist frequency. Notch frequencies, depths, and widths were constrained to 1–4 Nq , 2–40 dB, and 0.05–0.5 Nq , respectively. Fig. 6 shows the optimization results after 10,000 counts of function evaluations. As can be seen, the notch filters have departed from the initial set and resided mainly on the resonant modes immediately after the Nyquist frequency. The notched plants (i.e., Pnx Nnx ) are shown in Fig. 6(b), which indicate the resonant modes have been considerably suppressed beyond the Nyquist frequency. The folding error magnitude shown in Fig. 6(c) also indicates that the folding error magnitude has been brought down below or near the desired target through the optimized notch filters. The main optimization objective, that is the notch phase loss, is minimized to around 4.6 degrees at 0.1 Nq . Suppressing the folding error further will result in additional phase loss which could negatively impact the controller's phase margin. Fig. 7 shows the tradeoff between the folding error magnitude and the notch filter phase loss at 0.1 Nq , as well as the resulting notch filters optimized for different folding error upper bounds. Several other tradeoffs exist in the multi-rate notch filter design for servo systems, as discussed in detail (for the PZT micro actuator system) in Ref. [19].

Fig. 4. Conversion of the plant from nx to 1x sampling rate and the corresponding folding error.

P1x ( ) = Pˆnx ( ) + Efold ( ), 0 <

i

= 0)

(7)

where Pˆnx represents the complex conjugate of Pˆnx . The first term on the right side of Eq. (7) represents the superposition of the post-Nyquist frequency response function within the odd Nyquist intervals, e.g., 3x (2 3 Nq) , 5x, etc. The second term represents the superposition of the frequency response function within the even Nyquist intervals after being mirrored with respect to the corresponding Nyquist frequency and converted to complex conjugate form as presented in (7). Fig. 4 shows the magnitude plots of the 4x plant (Pnx ) shown in Fig. 3, alongside the notched plant (Pnx Nnx ), the notched plant after ZOH rate conversion (Pˆnx ), the folding error (Efold ), and the obtained 1x plant (P1x ). As can be seen, the ZOH rate conversion brings down the post-Nyquist magnitude response since the magnitude of the 1x ZOH transfer function diminishes faster than that of a 4x ZOH transfer function over frequency. The folding error magnitude plot shows that the leftovers of the notched resonant mode are superimposed from the 3rd Nyquist region onto the 1x plant as expected. The folding error formulation presented in this section allows us to develop a numerical optimization algorithm for designing multi-rate notch filters in a systematic way, as explained next.

Fig. 5. Percentage of folding error with respect to the DC gains of the plants, in the absence of multi-rate notch filters. 270

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4. Data-driven dual stage servo loop optimization In this section, a robust control optimization process is formulated and presented for the dual-stage HDD servo system. The structure of the controller transfer functions are shown in Fig. 8. The VCM controller comprises a gain, a set of pole-zero filters, and a set of (single-rate) notch filters. Moreover, the VCM controller is equipped with an integral controller to attenuate low frequency disturbances such as air pressure. The MA controller comprises a gain, a set of pole-zero filters, a set of peak filters, and a set of notch filters. The pole-zero filters in both VCM and MA controllers are used to shape the low-to-mid-frequency region, and the notch filters are tuned to suppress the high-frequency resonant modes within the 1x Nyquist region. The peak filters in the MA controller provide additional suppression effect over the concentrated midfrequency disturbances such as disk vibrations modes. To develop a systematic control optimization framework for the dual stage HDD system, a set of design objectives must be specified. Table 1 lists some of the closed-loop system properties to be achieved by the controller for the HDD plants investigated in this paper. These objectives are usually set by the manufacturer based on the market conditions as well as prior development experiences. The main optimization objectives include the stability margins, the loop gain crossover frequency, the non-repeatable position error magnitude, and the upper bound limits on the sensitivity transfer function. The loop gain crossover frequency is a measure of the system bandwidth. The sensitivity transfer function must satisfy certain upper bounds at different frequency regions. Two low frequency and high frequency upper bounds are provided in Table 1. Typically, a more aggressive upper bound is chosen for the high frequency region to mitigate uncertainties present in the resonant modes. A more detailed upper bound function for the sensitivity transfer function will be illustrated in the results section. Using the controller structures shown in Fig. 8, and adopting the sensitivity decoupling architecture (Fig. 1), a data-driven control design framework is presented next, to meet the desired objectives listed in Table 1.

Fig. 6. Multi-rate notch optimization results: (a) The initial and optimal notch filter set, (b) notched plants, and (c) folding error percentage with respect to the plants' DC gains.

4.1. Data-driven control formulation To develop a control design process for the dual stage HDD system, a multi-objective optimization problem is formulated based on the objectives listed in Table 1, as follows:

minimize q

s. t .

qmin

N i=1

M J (q) j = 1 ij

q

qmax

(9)

where q is the vector of the controller parameters including the DC and integral gains of the controllers as well as the parameters of the pole-

Fig. 7. Tradeoff between folding error mitigation and notch phase loss minimization: (a) The tradeoff curve, and (b) the resulting notch filters for the different folding error upper bounds.

Fig. 8. Controller structures for the VCM and the MA actuators. 271

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Table 1 Main VCM and DSA control design objectives. Objective

VCM

DSA

Gain Margin (Min) Phase Margin (Min) Crossover Freq. (Min)

4 dB 30 Deg 0.1 Nq

4 dB 30 Deg 0.2 Nq

Position Error Signal (Max) Low-freq. Sens. TF limit (Max) High-freq. Sens. TF limit (Max)

– 10 dB 3 dB

1/4 VCM 12 dB 6 dB

zero, notch, and peak filters; i and j represent the indices of the plants and the objective functions, respectively, with N and M being their respective total numbers; Jij is the cost associated with the dissatisfaction of objective j for plant i. This cost is calculated through a quadratic penalty function as follows:

Jij (q) =

wj

(

Fij (q) Fj, d

0,

),

Fj, d 2

Fij (q)

Fj, d

Fij (q) < Fj, d

(10)

where wj is the weight associated with objective j, Fij is the objective function value for plant i and objective j (e.g., gain margin), and Fj,d is the desired value of objective function j. To solve the above non-convex optimization problem, the same numerical optimization process applied to the multi-rate notch filter design problem is used as well. The closed-loop system properties such as the stability margins and sensitivity transfer function limits are computed for all the FRD plants in the design batch under the same controller choice. These properties are then compared to the desired targets to compute the cost function based on Eq. (10). The individual cost functions are then aggregated into a single cost function according to Eq. (9) and optimized numerically by iterating over the controller parameters. It is important to note that the penalty function formulated in Eq. (10) treats the desired objective as an upper bound. Therefore, for some of the objectives such as stability margins and loop gain crossover frequency, where a lower bound is desired, a negative sign must be applied to both the desired and the computed values. It is also important to note that, the stability of the closed-loop system during the optimization process is evaluated by simulating the system using a simple parametric plant model, e.g., a rigid body model for the VCM actuator with the same low-frequency gain and phase characteristics. The simplified parametric model provides a basic check on the closed-loop system poles to make sure they are within the unit circle. If any of the poles falls out of the unit circle, the optimization imposes a large penalty to the cost function to reject the corresponding design candidate. Additional stability improvement for the closed-loop system is obtained by imposing sufficient phase and gain margin lower bounds during the optimization.

Fig. 9. VCM control optimization results: (a) Frequency response of the initial and optimized controllers, (b) the initial sensitivity TF, (c) the optimal sensitivity TF, and (d) comparison of the initial and optimized stability margins and loop gain crossover frequencies.

convex, the optimization may converge to a local minimum. Therefore, the controller parameters must be initialized in way that the optimization starts from a stable closed-loop system with relatively desirable properties. The initial controllers are tuned manually by placing the notch filters on the target resonant modes and adjusting other parameters to provide reasonable properties. Fig. 9 shows the VCM loop optimization results after 10,000 counts of function evaluations through the Nelder-Mead simplex search algorithm. The weights are chosen to be the same for all objectives (wj = 1 for all j). The frequency responses of the initial and optimal controllers are shown in Fig. 9(a). The initial and optimal sensitivity TFs are shown

4.2. Numerical optimization results For the VCM loop, in addition to the DC gain and the integrator gain, 2 pole-zero and 10 notch filters are used to shape the low-frequency region and attenuate the high-frequency resonant modes. Therefore, the optimization space includes a total of 36 parameters. For the MA controller, 2 pole-zero, 3 peak, and 10 notch filters are used in addition to the DC gain, resulting in 44 optimization parameters. Due to the large number of optimization parameters, the VCM control loop is optimized first followed by the MA controller. Since the proposed optimization problem is nonlinear and non-

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Fig. 10. Normalized PES spectrum and cumulative sum for the optimized VCM control loop.

in Fig. 9(b) and (c), respectively. The applied sensitivity TF upper bound function is also plotted in the figure. It is clear that optimized sensitivity TF satisfies the desired upper bound function better than the initial hand-tuned controller. The comparison of the stability margins and the loop gain crossover frequency are shown in Fig. 9(d) in the form of cumulative distribution function (CDF). We can see that the optimized controller yields more favorable results compared to the initial controller. It is important to note that some of the plants in the population do not satisfy all the objectives simultaneously, mainly because of the fact that the desired objective values are rather ambitious to push the system performance higher. Besides, the tradeoffs between the objectives limit the degree to which the optimization can satisfy all the desired properties. A clustering-based design is proposed in Section VI to further enhance the population-level performance of the system. The optimization of the MA controller is performed similarly. One difference is the addition of the position error signal (PES) upper bound to the list of desired objectives. As given in Table 1, the design goal is to bring the dual stage PES level down to 0.25% of that of the VCM loop. This leap in the performance is obtained by the MA actuator's high bandwidth. To incorporate the PES objective in the optimization problem, an average open-loop disturbance spectrum is used based on real drivelevel measurements. Fig. 10 shows the normalized PES power spectral density (PSD) and its cumulative sum for the VCM system obtained by injecting the open-loop disturbance spectrum to the optimized VCM control loop. The low frequency components of the PES are related to air flutter created by disk rotation, and the high frequency components are mainly resulted from disk vibrations and arm bending/torsion modes. The dashed line in Fig. 10 indicates the target upper bound for the DSA loop. To achieve this performance level, significant reduction in the magnitude of sensitivity transfer function below the DSA crossover frequency is required. Using peak filters to locally suppress the disk vibration disturbances helps get closer to the desired performance level. The results of the MA controller optimization are shown in Fig. 11. Comparison of the initial and optimized MA controllers in Fig. 11(a) demonstrates significant differences across the frequency spectrum. The peak filters used in the design stand out in the low to mid-frequency range. Similar to the VCM design, the optimized DSA sensitivity transfer function fits better under the specified upper bound as can be seen from Fig. 11 (b) and (c). The stability margin and crossover frequency distributions in Fig. 11 (d) become narrower after optimization with improved average values for gain margin and cross over frequency. The average phase margin distribution has shifted in the undesirable direction, but its lower tail has improved.

Fig. 11. MA control optimization results: (a) Frequency response of the initial and optimized MA controllers, (b) the initial DSA sensitivity TF, (c) the optimal DSA sensitivity TF, and (d) comparison of the initial and optimized DSA stability margins and loop gain crossover frequencies.

Significant improvement is also observed by the comparing the PES spectrum under the optimized DSA controller (shown in Fig. 12) with that of the VCM controller (shown in Fig. 10). Although the PES cumulative sum is slightly above the desired level, there is nearly 70% reduction in the PES level compared to the VCM loop due to the added bandwidth through secondary actuator and the local disturbance suppressions through the MA peak filters. The optimized MA sensitivity transfer function defined as 1/(1+CmPm) is shown in Fig. 13, where the

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Fig. 14. System identification of a sample (a) VCM and (b) MA plant data.

Fig. 12. Comparison of the DSA PES spectra (a) before and (b) after the optimization of the MA controller.

optimization. Increasing the number of poles would generally improve the accuracy of the fit, but beyond a certain value, it results in numerical instability. Therefore, a reasonable plant order must be selected for the best fit. Fig. 14 shows the frequency response of the identified models for a selected VCM and MA plant against the real data. The number of poles was set to 30 for the VCM model and 40 for the MA model based on a few trials and errors. As can be seen, the identified models accurately capture the low frequency gain and phase data as well as the main resonant modes. These models enable the assessment of the developed filters and controllers in the time domain for further assurance and/or tuning. Fig. 13. Frequency response of the MA sensitivity TF using optimized controller.

5.2. Multi-rate simulations In order to evaluate the performance of the multi-rate notch filters, a multi-rate state space simulation is conducted by converting the identified models, multi-rate notch filters, and controllers to state space representations, and simulating them together. Only the VCM loop results are provided in this paper in the interest of space. To excite the high frequency resonant modes in the absence and presence of multi-rate notch filters with different suppression levels, the unit impulse function is used as the reference input, i.e., r(k) = δ(k). Fig. 15 shows the resulting response of the VCM output after the low frequency modes are settled. As can be seen from Fig. 15(a), the high frequency oscillations are significant in the absence of multi-rate notch filters. When the notch filters are present, different vibration magnitudes are obtained depending on the imposed level of the folding error suppression. The lower the folding error upper bound, the smaller is the magnitude of vibrations. However, as discussed in Section III, a more aggressive suppression of the high-frequency modes would result in more phase loss and performance drop of the main controller. The upper bound of 20% seems to provide a reasonable tradeoff between the folding error and the phase loss, as shown earlier in Fig. 7. To provide a better picture of the vibrations beyond the Nyquist frequency, the frequency contents of the time-domain signals for different cases are obtained through the Fast Fourier Transform method, and compared in Fig. 15(b).

error attenuation is mainly achieved at frequencies below 0.2 Nq . To further investigate the performance of the developed multi-rate notch filters and controllers, a set of time-domain simulations are presented in the next section. 5. Time-domain simulations To evaluate the multi-rate notch filters and dual stage system controllers, a time-domain study is carried out in this section. First, a frequency-domain system identification procedure is applied to obtain approximate models for the VCM and MA actuators. Then, a multi-rate simulation is carried out to evaluate the multi-rate notch filters. Finally, the tracking performances of the VCM and DSA controllers are evaluated. 5.1. System identification of HDD plants Identification of the VCM and MA plant models is carried out through MATLAB's “tfest” function from the System Identification Toolbox. This function receives the plant's frequency response data in the form of a complex vector, and the number of model poles, and returns a transfer function fit obtained through least squares 274

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and DSA sensitivity transfer functions are shown in Fig. 16. A multi-frequency reference trajectory is generated and applied to evaluate the tracking performance of the controllers. The reference trajectory contains sinusoidal sub-trajectories at 0.01 Nq , 0.02 Nq , and 0.1 Nq frequencies with different magnitude and phase values. From the sensitivity transfer functions, it is expected that the controllers will be able to track the low-frequency components more accurately. Fig. 17 shows the trajectory tracking results, where the VCM controller is able to reduce the tracking error down to a certain level, with MA controller providing further attenuation. The DSA loop with active VCM and MA controllers provides a high low-frequency tracking accuracy. However, its performance is limited in tracking the high-frequency component of the reference trajectory. 5.4. Comparison of the data-driven controller with a model-based controller In this section, a representative model-based controller is designed and compared to the proposed data-driven controller. For brevity, only the VCM loop is considered for this comparative study. For the identified VCM plant model (shown in Fig. 14), an H∞ optimal loop shaping controller is designed using MATLAB's “loopsyn” function, which synthesizes a controller to approximate a desired loop shape transfer function (i.e., Plant × Controller) for the system. The desired loop shape transfer function is obtained through an optimization process to achieve similar closed-loop system properties to those set for the datadriven controller (listed in Table 1). Since the plant model has multiple sharp resonant modes, the loop shaping synthesis becomes a numerically ill-conditioned problem, which cannot be solved due to round-off error. To avoid this problem, an inverse plant model is developed as a part of the controller to cancel the stable poles and the minimum-phase zeros of the plant. For the nonminimum phase zeros, which cannot be cancelled directly, a set of stable damped poles are included in the inverse model to ensure its causality. The plant and the inverse model are cascaded and reduced to the minimal realization form, and then used for the controller synthesis. The controller synthesis process runs successfully on the reduced-order model. The resultant controller is then cascaded with the developed inverse plant model to obtain the final controller transfer function. To demonstrate the developed model-based controller, the controller is applied to the parametric model (resulted from system identification) as well as the non-parametric model (original frequency response data). Fig. 18(a) shows the frequency response plots of the resulting sensitivity transfer functions. Moreover, this figure shows the sensitivity transfer function response for the data-driven controller applied to the non-parametric model for comparison. As can be seen from the figure, the parametric model results in a much smoother sensitivity transfer function compared to the non-parametric model for the same controller. This implies to one of the weaknesses of the modelbased design process, which ignores some of the plant details lost during the system identification process. The comparison between the model-based and the data-driven controllers shows that these controllers exhibit very similar low-frequency characteristics, since they are both designed based on similar criteria. However, there are certain differences across the frequency range, which could differentiate their population-level performance. Fig. 18(b) shows the time-domain simulations of the model-based controller on the parametric model. Compared to the simulations in the previous section, a similar tracking performance is obtained for the model-based controller. The tracking error comparisons show that the model-based controller performs slightly better than the data-driven controller for this specific reference trajectory. However, the modelbased controller does not achieve a sufficient robustness level for the entire population as shown in Fig. 18(c), and is therefore disqualified for product-level implementation. Other design techniques such as μsynthesis may mitigate the population-level robustness deficiency problem for model-based controllers. However, the characterization of the

Fig. 15. Unit impulse response of the VCM control loop: (a) Time domain trajectories, and (b) frequency contents of the time-domain trajectories, for different suppression levels of folding error.

Fig. 16. VCM and DSA Sensitivity TFs for the identified plant models.

5.3. Single-rate controller simulations To evaluate the tracking performance of the designed VCM and MA controllers in the time domain, the identified plant models are cascaded with the multi-rate notch filters with 20% suppression level and converted to single-rate transfer functions. The closed-loop VCM and DSA control loops are then constructed and simulated. The resulting VCM

Fig. 17. Multi-frequency reference tracking simulation through the VCM and dual-stage systems. 275

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Fig. 19. Clustering of HDD plants based on (a) VCM and (b) DSA gain and phase margin distributions.

and the population is partitioned into the selected number of clusters using MATLAB's “kmeans” function. This function is based on the widely-used local search algorithm proposed in Ref. [26] with additional improvements at the initialization stage through the k-means++ scheme [27]. Fig. 19 shows the clustering of the HDD population into 5 clusters in the gain and phase margin space. The VCM clusters are partitioned along the gain margin axis due a wider gain margin distribution range, whereas the DSA clusters are mainly partitioned along the phase margin axis. The next step is to optimize the controllers for each cluster separately, and compare them with the case of using a single controller for the entire population. To provide a fair comparison, the previously optimized controller is used as the initial design for all the clusters, and the same number of optimization iterations is applied to all cases, ranging from 1 cluster (i.e., the entire population) to 5 clusters. The clustering-based control optimization results are shown in Fig. 20, where the gain margin, phase margin, and crossover frequency distributions are plotted and compared using one to five controllers for the population. As can be seen, the segments of the distributions that are fallen below the desired targets under the single controller design have moved toward the desired values under the clustering-based multi-controller design. To provide a better picture of the improvements, the average improvements below the desired targets are calculated and compared, as shown in Fig. 21. For the VCM loop, the average phase margin, gain margin, and crossover frequency values below the desired targets improve up to 0.1%, 1.7%, and 0.8% with 5 clusters. For the DSA loop, these values are 5.5%, 0.9%, and 0.7%, respectively. Evidently, further improvements are expected with higher number of clusters.

Fig. 18. Model-based loop shaping controller design for the VCM loop: (a) Frequency response plots of the sensitivity transfer functions using parametric and non-parametric (FRD) models, (b) time-domain reference tracking simulation, and (c) population-level behavior.

uncertainty in an analytical form would add further complexity to the design process. 6. Clustering-based performance enhancement Designing a single controller for the entire HDD population is limited due the presence of a tradeoff between the robustness and performance objectives. This tradeoff is stronger when there is a significant diversity in the plant population. In this section, a novel method is proposed to further improve the performance and robustness of the HDD population. The proposed method relies on the effective clustering of the population into smaller groups of similar plants, and designing different controllers for the different clusters. This way, the tradeoff between the controller robustness and performance is mitigated due to the fact that the controllers are designed for a less diverse set of plants. Therefore, both properties can be improved. Two clustering criteria are investigated in this paper, one relying on the gain and phase margin distributions, and the other based on the sensitivity TF shapes. 6.1. Gain/phase margin-based clustering

6.2. Sensitivity TF-based clustering

The first clustering scheme studied in this paper is a 2-dimensional partitioning based on the gain and phase margin distributions using the k-means clustering method. Each sample in the population is characterized based on two observation points, i.e., gain and phase margin,

The second approach proposed for the clustering of the HDD plants is to use the sensitivity transfer function shapes as the clustering metric. Since the phase and gain margins are calculated only at two frequency points, they may not reflect the differences between the plants at other 276

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Fig. 22. Sample sensitivity-based clustering of (a) VCM and (b) MA plants into 4 clusters.

Fig. 20. Gain and phase margin-based clustering and optimization of (a) VCM and (b) DSA loops with multiple controllers.

Fig. 23. Sensitivity-based clustering and optimization of (a) VCM and (b) DSA loops with multiple controllers.

Fig. 21. Average improvements below desired targets: (a) VCM and (b) DSA.

frequencies. Therefore, calculating the cluster distances based on the entire frequency spectrum would provide a better metric for the determining the similarity level within the clusters. The magnitude of the sensitivity transfer function at different frequencies is used here to calculate the Euclidean distance for the k-means clustering process. The VCM and MA sensitivity transfer functions are used separately to cluster the VCM and MA plants. A sample sensitivity-based clustering result is shown in Fig. 22, where the sensitivity transfer functions have been partitioned into 4

groups of similar clusters. A careful look into each group indicates that there are certain features that are present and shared within the same cluster only. To quantify the improvements provided by the proposed sensitivity-based clustering scheme, the controllers are separately optimized for each cluster. The resulting distributions shown in Fig. 23 indicate that the gain margin, phase margin, and crossover frequency distributions improve significantly below the desired target levels in both VCM and DSA loops. It is important to note that dividing a large population of plants into 277

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7. Conclusions The proposed data-driven control design method presented in this paper is an effective approach for developing realistic servo controllers for HDDs manufactured in large volumes. Frequency-domain multiobjective numerical optimization provides an effective tool for the simultaneous optimization of different closed-loop system objectives such as stability margins, sensitivity transfer function upper bounds, and PES spectrum. In addition, partitioning the HDD plant population into smaller groups of similar plants thorough spectral clustering enables additional improvement in the servo system properties. Future work will focus on exploring other clustering schemes and investigating the robustness of the developed controllers for off-the-cluster plants. Acknowledgment Authors would like to acknowledge the funding and data provided by Western Digital Corporation. Authors would also like to thank Drs. Ehsan Keikha, Min Chen, Wei Xi, and Fred Hong for their support and helpful discussions. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.precisioneng.2018.12.007.

Fig. 24. Comparison of the improvements in the gain margin, phase margin, and crossover frequency and their associated optimization cost for (a) VCM loop, and (b) DSA loop, for different clustering schemes.

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multiple groups of smaller size would automatically reduce the diversity within each group. Therefore, the benefits resulted from clustering could in part be due to the reduction of the number of plants each controller has to deal with. To differentiate this effect, a random clustering process is also carried out and compared with the proposed clustering schemes. In the random clustering process, the population is divided into multiple clusters without any similarity measures within each cluster. To compare the different clustering schemes, two comparisons are made: First, the overall improvement in the average values of the gain margin, phase margin, and crossover frequency below the desired target values are compared similar to the previous comparisons, except that the improvements are lumped into a single average value. Moreover, the quadratic cost associated with violating the desired targets are compared as well, similar to the cost function used for the control optimization process (i.e., Eq. (10)). This metric provides a better measure for the effectiveness of each clustering scheme, because it takes into account the number of samples under the desired targets as well, unlike the average values. The comparison of the random, margin-based, and sensitivity-based clustering schemes are provided in Fig. 24. It can be seen from the random (or blind) clustering results that a part of the improvement received from the proposed clustering practice is in fact coming from the reduction of the plant population for each controller. Additionally, from the VCM results, we observe a consistent improvement in the average and quadratic cost values moving from the random clustering scenario to the margin-based and to the sensitivity-based clustering schemes. However, the DSA results indicate that margin-based clustering outperforms sensitivity-based clustering in the average value improvements, but not in the quadratic cost improvements. Given the fact that the quadratic cost function provides a better measure of the overall robustness and performance of the closed-loop system, we can conclude that sensitivity-based clustering is superior to margin-based clustering for the HDD plant data investigated in this study. 278

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