Fast Seek Control for Flexible Disk Drive Systems

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1 Fast Seek Control for Flexible Disk Drive Systems with Back EMF and Inductance Chanat La-orpacharapan and Lucy Y. Pao Department of Electrical and Computer Engineering niversity of Colorado, Boulder, CO and Abstract This paper derives shaped time-optimal servomechanism (STOS) controllers for flexible disk drive systems considering the effect of back electromotive force (back EMF) and coil inductance. To eliminate the vibration due to an unwanted flexible mode, input shaping is used to alter the time-optimal velocity phase-plane profile that moves the third-order nonflexible dynamics from rest to rest. Because the closed-form solution for the time-optimal control of the third-order nonflexible dynamics is not known, the shaped time-optimal velocity phase-plane profiles are derived numerically. By using a curve-fit polynomial to approximate these shaped timeoptimal velocity phase-plane profiles, we obtain an analytical expression for the STOS control. Simulations show that the curve-fit polynomials give high accuracy and the STOS control yields near time-optimal performance without unwanted residual vibration.. Introduction It is important that servomechanisms effect minimal time responses to set point changes. In disk drive systems, the voice coil motor is used to move the read/write head to the desired track as fast as possible. Decreasing time response has generally been accomplished by decreasing the system mass through the use of lighter materials, but this comes at a cost of servomechanism structures becoming flexible. While rapid and precise control of flexible structures has been an active research area, there remain many challenging problems due to the increasing demands on the control systems in a number of applications [, 5]. Time-optimal feedback control laws for single-mode undamped flexible systems have been derived to simultaneously account for both the rigid dynamics and the flexible dynamics [, ]. nfortunately, these truly time-optimal feedback control laws are not easily generalizable for damping, different acceleration and deceleration rates, slew rate limits, or additional flexible modes. Near time-optimal feedback controllers such as the proximate time-optimal servomechanism (PTOS) and extended This work was supported in part by a Thai Government Scholarship, the S National Science Foundation (Grant CMS-459), the Colorado Center for Information Storage, and a niversity of Colorado Faculty Fellowship. x ref xe Σ y y STOS Control Flexible Structure Pos Vel Fig. : Closed-loop system with a shaped time-optimal servomechanism (STOS) controller. PTOS (XPTOS) have been developed in [3,,, 3, 6]. PTOS does not explicitly account for flexible dynamics while XPTOS accounts for flexible dynamics only in the final settling phase of the controller. When the flexible dynamics become more significant, accounting for the flexible dynamics only in the final settling phase is not adequate, and it becomes necessary to account for the flexible dynamics during the entire slewing motion. Ho has used input shaping to account for the flexible dynamics during the entire slewing motion [4]. The feedforward command is derived by shaping the bang-bang input that is based on the open-loop deadbeat design for the pure rigid body system. The control command does not have an analytical solution and it is necessary to numerically re-compute the shaped feedforward command for each set point change. A shaped time-optimal servomechanism (STOS) technique for flexible structures based on shaping time-optimal phase-plane trajectories has recently been developed [6-9]. The approach is to consider the rigid and flexible dynamics separately. The unwanted flexible mode is accounted for by shaping the time-optimal control that moves the rigid-body portion of the system from rest to rest with an input shaper. The resulting STOS control leads to an altered phase-plane velocity profile such that vibration is eliminated. Fig. illustrates a closed-loop system with a STOS controller. While yielding near time-optimal performance, the STOS approach is more mathematically tractable and is generalizable for considering damping, different acceleration and deceleration rates, slew rate limits, or additional flexible modes. The STOS control laws use a phase-plane similar to that

2 used in PTOS or XPTOS. Hence, implementing the STOS control laws in applications using controllers similar to the PTOS or XPTOS controllers should not need significant reconfigurations. Moreover, the STOS control can address dominant flexible modes that can not be addressed by the PTOS or XPTOS control. Further, compared to feedforward input shaping, the shaped phase-plane control laws automatically handle some set point changes without having to re-compute the shaped feedforward command. The STOS control laws are derived for flexible structures consisting of a pure rigid-body in [6] and a second-order rigid body with damping such as in disk drive systems that have back electromotive force (EMF) dynamics in [7-8]. In this paper, we extend the STOS control approach to further account for coil inductance dynamics which leads to systems having third-order rigid body dynamics. Since the time-optimal control for the third-order rigid body system does not have an analytical solution, the velocity phase-plane profiles resulting from the shaped time-optimal control have to be derived numerically. By using a polynomial curve fit to approximate the shaped time-optimal velocity phase-plane profiles, we show that an analytical expression can be obtained for the STOS control law. This paper is organized as follows. In Section, we derive the STOS control law for flexible structures consisting of the third-order rigid body and one damped flexible mode. Section 3 discusses some implementation issues and presents simulation results using the STOS control laws derived in Section. Finally, concluding remarks are summarized in Section 4.. Seek Feedback Control The read/write arm motion in disk drive systems is generally controlled in two stages: a seek stage that slews the read/write head to the desired track followed by a settling stage that regulates the read/write head on the desired track. In this paper, we focus on the seek motion. The rigid body dynamics of the read/write arm of a disk drive system driven by a voice coil motor during the seek motion (block diagram illustrated in Fig. ) can be described as [4] L di dt Ri V b V i J ẍ r K t i where x r, i, V b, V i, L, R, J, and K t are the angle of the read/write arm (rad), the current (A), the back EMF force (V), the input voltage (V), the coil inductance (H), the coil resistance (Ω), the moment of inertia (N m s ), and the torque constant (N m A), respectively. A servomechanism where the rigid dynamics is the system () consisting of one integrator and two real poles, and where the flexible dynamics consist of one dominant flexible mode with damping can be modeled using a state space dif- () + volt Ls R K b K t J s K f s ζω n s ω n s + pos Fig. : Block diagram of a single-mode flexible structure driven by a voice coil motor. R, L, K t, J, and K b are coil resistance, coil inductance, torque constant, moment of inertia, and back EMF gain, respectively. ferential equation as follows: where A and a 3 ẋ Ax Bu () x x r i x f ẋ f a 3 a 3 a 33 B ω n ζω n b 3 K t J a K b 3 L a 33 L b 5 K f b 3 b 5 The state x represents angle, x r and x f, current i, and angular velocities, and ẋ f, where rigid body and flexible body are denoted by the subscripts r and f, respectively. ω n is the natural frequency and ζ is the damping ratio of the flexible mode. The trajectory of the states x r,, and i due to a constant input at time t can be described as x r i λ eλ t λ λ λ λ eλ t λ a 3 b 3 a 33 λ λ i λ e λ t λ e λ t λ λ a 3 i b 3 a 3 t a 33 b a 3 λ λ λ e λ t λ e λ t a 3 a 3 λ λ e λ t e λ t λ λ a 3 b 3 b 3 R L x r + e λ t e λ t λ λ a 3 i b 3 a 3 λ e λ t λ e λ t λ λ i (3) where x r,, and i are the states x r,, and i at the initial time, respectively. λ and λ are the real finite poles of the rigid body portion and are λ a 33 a 33 4a 3 a 3

3 * a a a a t t t 3 T t T (b) Shaper = a a a a t T t t 3 t 3 T t T (a) Time-optimal input (c) Shaped input Fig. 3: nshaped and shaped input profiles, where T π. The time-optimal input (Fig. 3a) that drives the thirdorder rigid-body portion of the system from rest to rest is u t t t t t t t (4) t t 3 The switching times t, t, and t 3 can be obtained from the following constraint equations [4]: λ e t e λ t λ e t 3 λ e t e λ t λ e t 3 (5) t t t 3 a 3 X b 3 where and X are the maximum control value of the voice coil actuator and the maneuver size, respectively. The shaped input is the result of the convolution of the time-optimal control with the two impulses of the input shaper (see Fig. 3b) that have amplitudes of a and a where [5] and a M a M M ζπ M exp (6) ζ and occur at times and T π, where ω n ζ is the damped natural frequency. Assuming that the damped natural frequency is large enough such that the shaper length π is smaller than the lengths t, t t, and t 3 t of the first, second, and third pulses of the unshaped time-optimal input, the shaped timeoptimal input (Fig. 3c) is u t π a t π t t a a t π t a a t π t a t 3 π t t ω t d t π t t ω t d t 3 t t 3 π (7) For different shaped profiles such as when t t 3 t or when t π t t, the shaped time-optimal t π input is different than in (7) but the corresponding STOS control can be derived similarly as detailed below for the shaped time-optimal input in (7). The time-optimal control is such that as the maneuver size X increases, all the pulse lengths t, t t, and t 3 t increase. For disk drive system parameters, the pulse length t 3 t is usually the smallest of the three pulse lengths. The dominant flexible mode in disk drives is typically at a relatively high frequency (-6 khz [4]), so the shaper length π is very small, and thus the minimum maneuver size X min that leads to π t 3 t is also very small. For seek motions, the move size is generally larger than X min and hence (7) would be applicable. Applying the shaped time-optimal control (7) to the system () which has the parameter values of the disk drive system in [4], as given in Table, we have the x r phaseplane trajectories as shown in Fig. 4. It is clear that the phaseplane can be divided into seven regions corresponding to different values of the shaped input. The first switching curve is when the shaped time-optimal input changes its value from a to at the switching time π. The state at this switching time is independent of the move distance X and the switching curve is the

4 a a S 3 Velocity of Rigid Body ẋr (rad/s) a a Error in Position of Rigid Body ( 3 rad) x Fig. 4: The solid lines are phase-plane trajectories resulting from the shaped time-optimal input. The dashed lines divide the plane into regions representing different values of the control input to be applied. following horizontal line (see Figs. 4 and 5): S λ e πλ λ e πλ a b 3 a 3 λ λ a a a b 3 a 3 (8) Similarly, the last switching curve is when the shaped time-optimal input changes its value from to a at the switching time t 3. The state at this switching time is independent of the move distance X implying that this switching curve is also a horizontal line (see Figs. 4 and 5) and is S 6 λ e πλ λ e πλ a b 3 a 3 λ λ a b 3 a 3 (9) The second through fifth switching curves can be solved numerically. First, after solving for the time-optimal control from (5), the x r and states at the switching times t, t π, t, and t π due to the shaped time-optimal control can be determined. The polynomial functions of these switching curves can then be obtained by a curve-fit approximation. Further discussion on determining the curve-fit polynomials is given later in Section 3 for the particular disk drive system parameters of Table. The derivation of the switching curve equations for maneuvers in the negative direction can be computed similarly. S S f x S 4 S S 5 6 f Fig. 5: Entire phase-plane: the control input is assigned to be, f,, and f in the first through fourth quadrants, respectively. In fact, the switching curves for the fourth quadrant are antisymmetric to the switching curves for the second quadrant (see Fig. 5). For rest-to-rest motion, the trajectory ideally never enters the first or third quadrant. However, as discussed in [6-9], in practice, overshoot may occur due to discrete sampling times or other implementation issues as well as disturbances or nonideal behavior. Thus we assign control values to these regions (see Fig. 5) so that they will cause the system to move into the second and fourth quadrant as quickly as possible. The goal is that the control action in the second and fourth quadrants brings the system to the origin in the respective quadrant. For all four quadrants, the STOS control law can be expressed as a logic expression as follows: u f f ()

5 u xr ẋr i ẋ f x f Time (s) Fig. 6: Typical time response due to the STOS control law. u x r i x f and ẋ f are V, rad, rad/s, A, 5 rad, and rad/s, respectively. where f 4 F F a A A B B A a a C C B F F a a E E D G G E a G D D C A S B S C S 3 C S 3 D S 4 E S 5 G S 6 F x () and x r x ref is the error in position of the rigid body. S and S 6 are given in (8) and (9), respectively. S, S 3, S 3, S 4, and S 5 are the curve-fit polynomials of the second through fifth switching curves. S 3 represents the third switching curve S 3 as a function of, and S 3 represents the third switching curve S 3 as a function of. x at the third switching time t π for a maneuver of distance X min. As described earlier, X min is the minimum maneuver size that leads to π t 3 t. f is the switching function for the second quadrant, and f is the switching function for the fourth quadrant and can be described as f f () Table : Voice coil motor and flexible structure parameter values for a disk drive system [4]. The modal frequency is ω n π f n rad/s. For these parameter values, λ 4896 s, λ 3 9 s, and b 3 b 5 5 (V s in the state-space model of (). parameter K t K b disk drive value N m/a V s K f 5 /(V s ) R L J Vmax f n Ω 3 H 5 kg m m V 5 khz ζ 5 3. Discussion and Simulation Results Fig. 6 shows a typical time response of the derived STOS control law applied to a disk drive read/write arm driven by a voice coil motor which has a model as shown in Fig. with the system parameter values given in Table. The shaped voltage command u saturates at Vmax. We used the Newton- Raphson technique to solve for the time-optimal switching times from (5) and then computed the x r and states due to the shaped time-optimal control input at the switching times t, t π, t, and t π. The curve-fit polynomials for the switching curves S, S 3, S 3, S 4, and S 5 are derived by a least-squares technique using third-, third-, third-, first-, and first-order polynomials, respectively. The simulations show that the read/write head is moved from rest to rest to the desired track and at the end of the move there is no residual vibration. The STOS control yields a near time-optimal settling time. Solving for the time-optimal control for the third-order rigid body is much simpler than solving for the time-optimal control for the full fifth-order system consisting of both the third-order rigid body and the second-order flexible body. The number of pulses in the time-optimal control for the fifthorder flexible system can be different from one maneuver to another, sometimes making the numerical solution of the time-optimal control very difficult. Thus, the STOS control approach developed in this paper is more feasible, especially when the unwanted flexible mode is a very high frequency mode since the resulting maneuver time will be very close to time-optimal. We have used the x r phase-plane in deriving the STOS controller in this paper because previous techniques such as the PTOS and XPTOS [3, 5] also use this rigid body x r phase-plane, even though additional dynamics (such as flexible modes) exist. However, our approach takes into

6 account the dominant flexible mode in developing the STOS controller for the entire slewing motion of the system. As a result, the developed STOS controller has the capability to address the dominant flexible mode where the PTOS or XPTOS control might not be able to. Although they are rather complex, ()-() provide analytical expressions for the STOS control law. In practice, the STOS control law can be easily implemented via a table: given a position error and velocity, the table provides the control signal to be applied. In applications where extreme accuracy is required, the move distance can be divided into several ranges so that the curve-fit polynomials can better approximate the shaped phase-plane profiles in each range. This will lead to multiple control laws, one for each maneuver range. Each control law can be computed using the procedure described in Section. These multiple control laws can then be implemented using multiple tables: given the move distance X, the table is chosen; then given the position error and the velocity, that table provides the control signal to be applied. For systems having multiple flexible modes, the STOS control law still cancels the vibration due to the modeled flexible mode, and it will not amplify the vibration due to the other unmodeled modes (any more than compared to not shaping the control for the modeled flexible mode) [5]. Addressing more flexible modes in the STOS control law will lead to more switching curves and regions in the phase-plane. However, the ideas described here can still be used to develop control laws for multiple mode flexible systems, and in practice the STOS control law can similarly be easily implemented via tables. 4. Conclusions Shaped time-optimal servomechanism (STOS) control laws have been derived for flexible disk drive systems considering the effect of back EMF and coil inductance, which leads to systems having third-order rigid-body dynamics. The STOS approach accounts for the flexible body dynamics by using an input shaping technique to shape the numerically determined time-optimal control for the third-order rigid body dynamics. The resulting shaped velocity phase-plane profiles are the desired trajectories that the STOS controller causes the system to follow. Hence, while the developed STOS controller is based on the rigid body x r phase-plane trajectories, the STOS control and trajectories in this phase-plane are shaped using knowledge of the additional dynamics in the system consisting of a damped flexible mode. The shaped velocity phase-plane profiles were derived numerically since the time-optimal control for the third-order rigid body system has to be solved numerically. By using polynomial curve fits, we have derived an analytical expression for the STOS control law. Simulations show that first- through third-order polynomials are sufficient for providing accurate curve fits, and the derived STOS control law is shown to yield near time-optimal performance without unwanted residual vibration. References [] E. Barbieri and. Ozguner. A New Minimum-Time Control Law for a One-Mode Model of a Flexible Slewing Structure, IEEE Trans. Aut. Ctrl., 38(), Jan [] W. J. Book. Controlled Motion in an Elastic World, ASME J. Dyn. Sys., Meas., and Ctrl., 5(), June 993. [3] G. F. Franklin, J. D. Powell, and M. L. Workman. Digital Control of Dynamic Systems, Reading, MA: Addison- Wesley, 998. [4] H. T. Ho. Fast Servo Bang-Bang Seek Control, IEEE Trans. Mag., 33(6), Nov [5] J. L. Junkins and Y. Kim. Introduction to Dynamics and Control of Flexible Structures, AIAA, Washington, D.C., 993. [6] C. La-orpacharapan and L. Y. Pao. Control of Flexible Structures with a Projected Phase-Plane Approach, Proc. Amer. Ctrl. Conf., Arlington, VA, June. [7] C. La-orpacharapan and L. Y. Pao. Shaped Phase- Plane Control for Flexible Structures with Friction, Proc. Amer. Ctrl. Conf., Anchorage, AK, May. [8] C. La-orpacharapan and L. Y. Pao. Shaped Control for Damped Flexible Structures with Friction and Slew Rate Limits, Proc. IEEE Conf. Dec. and Ctrl., Las Vegas, NV, Dec.. [9] C. La-orpacharapan and L. Y. Pao. Shaped Time- Optimal Control for Disk Drive Systems with Back EMF, Slew Rate Limits, and Different Acceleration and Deceleration Rates, Proc. Amer. Ctrl. Conf., Denver, CO, June 3. [] F. Li and P. M. Bainum. Analytical Time-Optimal Control Synthesis of Fourth-Order System and Maneuvers of Flexible Structures, J. Guid., Ctrl., and Dyn., 7(6), Nov.- Dec [] W. S. Newman. Robust Near Time-Optimal Control, IEEE Trans. Aut. Ctrl., 35(7), July 99. [] L. Y. Pao and G. F. Franklin. Proximate Time-Optimal Control of Third-Order Servomechanisms, IEEE Trans. Aut. Ctrl., 38(4), April 993. [3] L. Y. Pao and G. F. Franklin. The Robustness of a Proximate Time-Optimal Controller, IEEE Trans. Aut. Ctrl., 39(9), Sept [4] W. N. Patten, H. C. Wu, and L. White. A Minimum Time Seek Controller for a Disk Drive, IEEE Trans. Mag., 3(3), May 995. [5] N. C. Singer and W. P. Seering. Preshaping Command Inputs to Reduce System Vibration, ASME J. Dyn. Sys., Meas., and Ctrl., (), Mar. 99. [6] M. L. Workman and G. F. Franklin. Implementation of Adaptive Proximate Time-Optimal Controllers, Proc. Amer. Ctrl. Conf., Atlanta, GA, June 988.

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