Fast servo bang-bang seek control

4522 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 6, NOVEMBER 1997 Fast Servo Bang–Bang Seek Control Hai T. Ho, Member, IEEE Abstract— In this paper, we present a new seek algorithm formulated by combining the time-optimal control and input shaping methods. The algorithm achieves near optimal bang–bang performance with minimal excitation of the resonance mode. The computation requirements are minimal, and therefore, the algorithm is attractive for practical design and applications. We shall focus on the application of hard disk drives. Index Terms— Disk drive, seek, servo. I. INTRODUCTION ESONANCES exist in every mechanical motion system, and in many applications, they hinder the ability of the servo system to seek in the least amount of time. In a disk drive, the actuator arm with the read/write head mounted at its tip is a prevalent source of resonance. The ability for the actuator to seek from one track to another quickly and quietly is very important because the data retrieval performance of the drive is directly affected by how fast the head seeks from one track to another. Furthermore, as the track per inch (TPI) density continues to increase, the behavior of the resonance becomes more dominant and, therefore, needs to be dealt with effectively. The dominant flexible modes are typically in the range of 2–6 Khz. The closed-loop servo system provides control of the actuator to seek from one track to another in the shortest time and then regulate the head position to follow the track with minimum variance. During seeking, the actuator gets driven by a bang–bang current profile to achieve timeoptimal control, but due to the presence of resonance, the ideal bang–bang profile needs to be smoothed out, particularly at the end (arrival Stage 4 in Fig. 4), so that there is less excitation of the resonance. Consequently, longer seek time results. The seek algorithm that has been widely used in the industry for years and was mathematically formalized by Workman [1], [2] is called the proximate-time-optimal-servomechanism (PTOS). This has an exponential decay at the arrival stage resembling a first order low-pass filter. Another seek algorithm that has been used is a two-step deadbeat arrival. This is essentially staircasing the arrival so that the two intermediate smaller steps would lessen the resonance excitation, but only to a limited extent. There are other known advanced seek control methods such as fuzzy logic based [3], [4], refinement of the PTOS [5], third-order bang–bang [6], and input frequency shaping [7]; none of these, however, focus directly on simultaneous minimization of rise time and resonance excitation. Thus, that was the motivation of this study. R Manuscript received May 17, 1996; revised June 4, 1997. The author is with Maxtor Corporation, Longmont, CA 80501 USA (e-mail: hai [email protected]). Publisher Item Identifier S 0018-9464(97)06895-7. Fig. 1. Dynamic system response to step command. In the past few years, there has been a proliferation of applications using command input shaping in motion control systems to suppress resonances. The input shaping method was introduced by Singer and Seering in the late 1980’s [8] and is a method to shape the input to a mechanical system to avoid exciting the resonances. This method is different from known conventional methods [7], [9] because it is formulated in the time rather than the frequency domain, and hence, it has unique advantages such as simplicity and faster rise time than the conventional exponential decay. The implementation of the shaping filter is in the form of a simple finite impulse response (FIR) filter with as few as one delay tap. This simplicity makes it extremely attractive for use in the disk drive servos because of the limited processor bandwidth. Note that the objective of this paper is to formulate the proposed seek algorithm and not so much to conduct a comparison study between it and the other existing methods. However, we will mention some basic characteristics and features of some existing algorithms. Section II presents the basic idea of shaping filter, servo dynamics, and briefly covers the seek algorithms that are found in commercial disk drives. In Section III, we formulate the new seek algorithm and present simulation results. The paper ends with the conclusion in Section IV. II. BACKGROUND A. Input Shaping Filter We give a step command driving a dynamic system with a structural mode, as shown in Fig. 1, the response to a step input will have ringing, as shown in Fig. 2(a). Most of the methods to reduce the ringing are based on using an intermediate filter to smooth out the step. Some of the common schemes are: 1) low-pass filtering the step, as shown in Fig. 2(b); 2) staircasing the step, as shown in Fig. 2(c); 3) notch filtering the step, as shown in Fig. 2(d). Next, the 1-tap FIR input shaping filter [8] response is shown in Fig. 2(e). Here, the elimination of the ringing is from the cancelation of the responses to the two sub-step levels. The staircase command is a result of feeding the step through a 1-tap FIR filter 0018–9464/97$10.00  1997 IEEE (2.4) HO: FAST SERVO BANG–BANG SEEK CONTROL 4523 where the coefficients are designed using a priori knowledge of the resonance mode as [8] (2.5a) where (2.5b) (a) (b) (c) (d) and the sampling period of the FIR is determined as (2.5c) In comparing the effectiveness of these smoothing filters, we observe the following. i) Resonance suppression: The notch and the shaping FIR have superior resonance suppression over the low-pass and staircase filters. ii) Rise time: The shaping FIR has the fastest rise time, which is four times less than that of the notch. The short rise time is very advantageous in the arrival stage of the seek profile. iii) Implementation: The 1-tap FIR is the simplest to implement. iv) Robustness: Fig. 3 shows that the FIR maintains over 70% suppression, whereas the resonance frequency varies over the range of ±20%. However, the notch filter maintains over 80% suppression over a wider range of frequency. It is clear that the notch and lowpass filters are more robust in exchange for slower rise time. More vigorous comparison of the shaping filter with various low-pass and notch filters can be found in [10], where similar conclusions were made. In Section III, we shall formulate the bang–bang seek algorithm that incorporates the 1-tap shaping filter. (e) Fig. 2. (a) u(t) and y (t) without filter F = 1: (b) u(t) and y (t) with fourth-order LPF: F (s) = 1=(s + 1)4 ;  = 7=!r 1  2 : (c) u(t) and y (t) with arbitrary staircase: F (z ) = :3z02 + :3z 01 + :4; Ts = 223 6sec. (d) u(t) and y (t) with Notch filter: F (s) = (s2 + !r2 )=s2 + 4!r (1 :0826)s + !r2 : (e) u(t) and y (t) with 1-tap shaping FIR: F (z ) = :523 + :476z 01 ; Ts = 161 sec. 0 0 B. Plant Dynamics The complete model of the disk drive servo dynamics can have an order of up to 40 to describe dynamics of the VCM, amplifier, PES channel, HDA dynamics, and more. However, in designing the controller, many of the dynamics are ignored or approximated. In particular, the seek algorithm design typically uses the following approximated model of the plant dynamics: Fig. 3. Sensitivity of shaping filter to drift in the mode frequency. described in state-space form as (2.7) where (2.6) where (2.8) the the the the the position error; voltage control input; power amplifier dynamics; VCM dynamics; actuator containing resonance dynamics. The most simplified form of (2.6) is the rigid body double integrator. In the discrete domain, the double integrator can be with vector containing the servo position error and velocity, and is the sampling period. The model (2.8) implies that the amplifier dynamics are constant, the motor , and back emf are negligible, and the actuator resonances are ignored. However, in order to achieve good seek performance, the effects of these components need to be accounted for in the design. 4524 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 6, NOVEMBER 1997 that Fig. 2(b) and (e) showed that exponential signal is not as efficient as input shaping in terms of rise time. The next type of seek (Type III) is based on Stage 4 open-loop arrival. Normally, the nominal values of the rigid body parameters of the plant are known pretty accurately. Therefore, to take advantage of this, open-loop deadbeat control of the arrival in two steps (samples) can be done. The arrival control law for a Type III seek profile is Fig. 4. Three different bang–bang seek profiles. C. Common Seek Algorithms Three different seek (acceleration) profiles are shown in Fig. 4. The Type I is the ideal time-optimal bang–bang profile. Types II and III are commonly used in commercial disk drive servo systems. The profile is broken into four stages: Stage 1) the usually saturated acceleration; Stage 2) the transition from acceleration to deceleration; Stage 3) the tracking deceleration; Stage 4) known as the arrival stage responsible for bringing the head to zero position error and zero velocity before switching to track following (or settling) mode. The track-following mode typically uses a PID or lead-lag compensator for regulating the head at the final position [1], [11]. A desirable seek control is one that achieves the most accurate arrival, least resonance excitation, and shortest rise time. The Type I seek profile corresponds to the time-optimal seek control law [12] (2.9a) (2.9b) where is the deceleration rate during Stage 3. The sgn function in (2.9a) would surely cause chatter and excite the resonance modes. In particular, during the arrival Stage 4, if there is sufficient resonance excitation, then the servo position would exhibit ringing behavior and would require extra time for the track following controller to dampen out the ringing. A simple modification of (2.9) leads to a more practical Type II seek profile known as the PTOS [2]. Shown in Fig. 4, the bang–bang profile is smoothed to lessen the excitation of the resonance modes. The seek control law for Type II is [2] (2.10a) (2.10b) Here, the control voltage is produced from the saturation function and, therefore, eliminates most of the chatter effect. In addition, when the head is near the target position the control function now allows for exponential decay of the control voltage. This is why the control law in (2.10) is more common in practice. The rate of the decay is controlled by the gain but since the velocity loop of the servo system is a Type I system and is data sampled, needs to be kept small to avoid large velocity tracking error. In addition, recall track follow (or settle) mode (2.11) to yield (2.12) with the constraint that the two subdeceleration levels not exceed a maximum allowable level, i.e., (2.13) In (2.12), the arrival Stage 4 begins at where is chosen to satisfy the constraint (2.13). is the sampling period and is also the time between two consecutive servo sectors. It can be readily shown that the two zero arrival targets in (2.12) can be achieved with the two design variables and via the normalized double integrator plant model (2.8). In contrast to the time-optimal Type I control, here, the two intermediate deceleration levels are for the purpose of reducing the resonance excitation. However, the excitation suppression is only limited because the resonances were not specifically accounted for in the formulation. III. NEW SEEK ALGORITHM In this section, we formulate a seek algorithm that incorporates the shaping filter described in Section II. The aim of this seek algorithm is to simultaneously minimize the resonance excitation and the rise time. We call this Type IV, and the seek acceleration profile is shown in Fig. 5. Stages 1 and 2 both have intermediate steps introduced by the shaping filter. Stage 4 begins at , and the control law consists of five deceleration steps – but only two of these five, and , are the design variables to achieve deadbeat arrival. The and are intermediate step responses other three steps of the shaping filter with input and It is clear at this point that this seek algorithm is similar to Type III, where the arrival is based on open-loop deadbeat design. The objective here is to choose such that the following constraints are satisfied. i) ii) iii) iv) zero arrival velocity; zero arrival position error; Step changes are specified by FIR to account for resonance; saturation limit (3.1a) (3.1b) (3.1c) (3.1d) The reason for selecting five-steps is as follows. Steps and force arrival conditions for velocity and position to be zero. HO: FAST SERVO BANG–BANG SEEK CONTROL 4525 and are intermediate steps produced by The steps the FIR filter. First, recall the discrete state space model of the plant in (2.8) (3.2) with the sampling period determined by (2.5c). This system has the state solution in the form (3.3) To satisfy (3.1c), we make and That is steps of and the intermediate FIR Fig. 5. The new Type IV seek profile with shaping filter. (3.4) and are determined by where the FIR filter coefficients (2.5a). To satisfy (3.1a) and (3.1b), we simply set Last, to satisfy (3.1d), we need to choose the time when the descent occurs so that the magnitude for the five levels stay within the limit. If we introduce two new variables as (3.10) (3.5) then (3.8) can be written as Combining (3.3)–(3.5), we get (3.11) (3.6) and we impose limits on and as (3.12) where the available initial conditions are (3.7) Hence, the remaining independent two deceleration levels can be found by manipulating (3.6) to yield (3.8) where (3.9a) Note that there are only two real-time multiplications in , and the rest can be computed off-line and stored in memory. It can be shown that the elements of and are (3.9b) If and are within the limits, then and will automatically be within limits because they are intermediate steps of , and . After substituting (3.11) into (3.12), it follows that Stage 4 begins at time [see (3.13) at the bottom of the page]. The inequalities in (3.13) portray a phase plane region that corresponds to the allowable deceleration levels with amplitude less than An example of this region is shown in Fig. 6. The time-optimal deceleration profile that the servo velocity tracks during Stage 3 is also shown here. As soon as the sampled velocity and the position error enters the region, the arrival descent can immediately begin, in which is marked. If, for some reason, the descent begins outside of the region, then there will be suboptimal response, i.e., longer seek time. This is especially concerning for short seek, where the derate value is smaller; however, short seeks usually deal with their own version of the seek algorithm. It is important to mention here that even though this new seek algorithm involves a five-step descent as opposed to the aforementioned two-step descent, this does not mean that the five-step algorithm prolongs the seek time. The first reason is that the descent phase plane region for the five-step is much larger, and therefore the descent begins sooner. The second reason is that the sampling time period of the five-step (3.13) 4526 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 6, NOVEMBER 1997 Fig. 9. Commanded velocity and the VCM velocity. Fig. 6. Descent phase plane region and the velocity profile. Fig. 7. A 700-track seek control voltage of the new seek algorithm. Fig. 8. Acceleration response of VCM. , which is typically less algorithm during the descent is The third reason is that the first two deceleration than and are typically equal or larger than which levels In means that the descent does not effectively occur until Fig. 6 derate refers to the ratio of the deceleration rate over the maximum allowable deceleration rate, i.e., The ratio is also shown. Remarks: During the deceleration tracking Stage 3, the system state tracks the profile (2.9b) shown by the line with arrow in Fig. 6. At some point the sampled velocity and position error enter the descent region; hence, Stage 4 begins. If the entry point is before the tangent point, then all the five deceleration levels will have the same polarity. This is desirable because there will be no large abrupt step change that will cause noticeable current rise time introduced by the amplifier, which would cause less accuracy in arriving at the TABLE I SIMULATION PARAMETERS zero target. In addition, during Stage 3, if the system state does not track the profile accurately, then the entry point may be after the tangent point. Equation (3.13) reveals that lowering the derate value will allow the region envelope to be larger, thereby increasing the robustness of getting the entry point to occur before the tangent point. A simulation using the complete model of the disk drive servo plant was conducted. The model includes the effects of the resonance, VCM inductance, resistance, back emf, and current amplifier. The values for these parameters are given in Table I. The sampling rate was based on 60 sectors per revolution with the disk spinning at the rate of 4480 r/min. The track density was set at 4232 tracks/in. The dominant resonance mode was at 3100 Hz with 0.03 damping. Using the new seek algorithm, the servo was commanded to seek 700 tracks. Fig. 7 shows the control voltage of the seek profile with the five-step arrival. Note that every step change in this profile is characterized by the input shaping filter designed for the resonance mode. Therefore, we can see that the acceleration response in Fig. 8 has minimal ringing. In actual implementation, there will be some amount of ringing because of the nonideal environment and the uncertainty of the resonance characteristics. However, as long as the mode stays within ±20% of the nominal, then at least 70% of the ringing should be suppressed, as earlier shown in Fig. 3. The continuous velocity and the sampled commanded velocity are shown in Fig. 9. Here, we see that the servo velocity tracks the commanded velocity until the arrival Stage 4, and it also exhibits minimal ringing. Hence, Fig. 7 shows that very HO: FAST SERVO BANG–BANG SEEK CONTROL little time is compromised for smoothing the seek profile to avoid exciting the resonance mode. The arrival position and velocity in this simulation were 0.03 track and 0.1 track/sect, respectively. IV. CONCLUSIONS The FIR shaping filter was incorporated in the design of a bang–bang seek algorithm. A seek can be completed faster because minimum time was spent to smooth out the profile to avoid exciting the resonance. The key tradeoff is the robustness of the seek arrival. This algorithm is effective for suppressing a specific limited band of frequency where the resonances exist. Precise knowledge of the dominant resonance mode is utilized, and as long as the mode stays within ±20% of the nominal; then, simulation shows that at least 70% of the ringing is suppressed. REFERENCES [1] G. Franklin, D. Powell, and M. Workman, “Case design: Disk drive servo,” in Digital Control of Dynamic Systems. Reading, MA: Addison-Wesley, 1990, sec. 12.7. [2] M. Workman, “Adaptive proximate time-optimal servomechanisms,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 1987. [3] S. Yoshida and N. Wakabayashi, “A fuzzy logic controller for a rigid disk drive,” IEEE Cont. Syst. Mag., June 1992. 4527 [4] S. Yoshida, M. Tokura, M. Imura, and N. Wakabayashi, “Bang–bang seek controller for HDD with fuzzy algorithm,” J. Magn. Jpn., vol. 6, Mar. 1991. [5] T. Lee, T. Low, and A. Al-Mamun, “DSP-based seek controller for disk drive servomechanism,” IEEE Trans. Magn., vol. 29, Nov. 1993. [6] K. Ananthanaratanan, “Third-order theory and bang–bang control of voice coil actuator,” IEEE Trans. Magn., vol. 18, May 1982. [7] H. Meckl and K. Kudo, “Input shaping technique to reduce vibration for disk drive heads,” in Proc. 1st Conf. Motion Vibration Contr., Yokohama, Japan, Sept. 1992. [8] N. Singer and W. Seering, “Preshaping command inputs to reduce system vibration,” ASME J. Dynamic Syst., Meas. Contr., vol. 112, pp. 76–81, Mar. 1990. [9] P. Meckl and W. Seering, “Reducing residual vibration in systems with uncertain resonances,” IEEE Contr. Syst. Mag., Apr. 1988. [10] W. Singhose, N. Singer, and W. Seering, “Comparison of comand shaping method for reducing residual vibration,” in Proc. 1995 Euro. Contr. Conf., 1995. [11] R. Oswald, “Design of a disk drive file head-positioning servo,” IBM J. Res. Dev., Nov. 1974. [12] A. Bryson and Y. Ho, Applied Optimal Control. Washington, DC: Halsted, 1975. Hai Ho (M’95) received the B.S.E.E., M.S.E.E., and Ph.D. degrees from the University of Colorado, Boulder, in 1988, 1989, and 1994, respectively. He is currently with Maxtor Corporation, Longmont, CO, working as a Servo Technologist in the advanced technology group. He spent the last ten years in the data storage industry and has six issued/pending patents in the areas of cartridge library systems and disk drive servo architecture. He has written more than a dozen technical papers in the areas of HDD servo and neural networks.