Fig. 2. Time-Optimal control profiles with both acceleration. Fig. 4. Sloped fast control profiles with both acceleration

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1 VIBRATION SUPPRESSION CONTROL PROFILE GENERATION FOR HARD DISK DRIVE FLEXIBLE ARM LONG SEEK POSITION CONTROL Li Zhou, Eduardo A. Misawa School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK , USA Astract: A control profile is generated which suppresses all the resonant dynamics in a hard disk drive flexile arm. This control profile has oth the drive voltage and velocity constraints which are required in hard disk drive long seek control. The control profile is generated from the sloped fast acceleration command and the viration suppression shape filter technology. The simulation results for hard disk drive long seek control illustrate the effectiveness of the proposed method. Copyright c 5 IFAC Keywords: Flexile dynamic system, residual viration control, hard disk drive actuator Acceleration profile High frequency Kv structure R(s) s Velocity Kp s Fig.. A typical mechanical flexile system.. INTRODUCTION Position Control of flexile structures has een extensively studied in recent years. Flexile structures such as high-speed disk drive actuators require extremely precise positioning under very tight time constraints. Whenever a fast motion is commanded, residual viration in the flexile structure is induced, which increases the settling time. One solution is to design a closed-loop controller to damp out virations caused y the command inputs and disturances to the plant. However, the resulting closed-loop response may still e too slow to provide an acceptale settling time, and the closed-loop control is not ale to compensate for high frequency residual viration which occurs eyond the closed-loop andwidth. An alternative approach is to develop an appropriate reference trajectory that is ale to minimize the excitation energy imparted to the system at its natural frequencies. Supported y the National Science Foundation, grant numer , and Seagate Technology LLC of Oklahoma City, Oklahoma. Fig. shows a typical mechanical flexile system, where s is an integrator, K v is a velocity constant gain, and K p is a position constant gain. The high frequency modes can e descried as a transfer function R(s) = lim ns n + n s n + + s+ n a ns n +a n s n + +a s+ in which an infinite numer of lightly damped resonant structures is possile. The goal of viration suppression trajectory generation is to find a fast input trajectory, under some physical constraint, with minimum possile residual viration. In the previous study (Zhou and Misawa, 5), a control profile is generated which suppresses all the resonant dynamics in a flexile dynamic system. The proposed methods (Zhou and Misawa, 5) develop a viration suppression control profile in the hard disk drive short seek control. In (Zhou and Misawa, 5c), a viration suppression control profile generation with oth acceleration and velocity constraints is studied. The proposed method (Zhou and Misawa, 5c) develops a viration suppression control profile for hard disk drive long seek control. The control profile has oth the drive current (or acceleration) and velocity constraints. In real application, the drive current does no saturate. It is the applied drive voltage that saturates. This paper presents a viration suppression

2 Acceleration Velocity Position Acceleration Velocity Position Fig.. -Optimal control profiles with oth acceleration and velocity constraints. Va Vc V current acceleration Kc L.s+R Ke s / velocity Fig. 3. The voice coil servo motor dynamics. s / position control profile generation method with oth the drive applied voltage and velocity constraints. Fig. 4. Sloped fast control profiles with oth acceleration and velocity constraints. 3. VIBRATION SUPPRESSION CONTROL PROFILE GENERATION WITH BOTH VOLTAGE AND VELOCITY CONSTRAINTS FOR A FLEXIBLE SYSTEM In this section, a viration suppression control profile generation with oth applied voltage and velocity constraints for a flexile system is induced y using the viration suppression shape filter technique (Zhou and Misawa, 5a). 3. Calculating the Numer of the Sloped Positive Acceleration Command Samples to Reach the Velocity Constraint. LONG SEEK CONTROL PROFILE WITH BOTH APPLIED VOLTAGE AND VELOCITY CONSTRAINTS For a purely rigid ody, it can e inferred that the timeoptimal acceleration profile with velocity constraint is composed y three parts. First, acceleration is commanded which always reaches the maximum limit. Secondly, when the maximum velocity is reached, the acceleration command ecomes zero. In this situation, the rigid ody is cruising with a constant velocity. The third part is a deceleration command which always reaches the minimum limit. Fig. shows typical timeoptimal control profiles with oth acceleration and velocity constraints. Fig. 3 shows a simplified hard disk drive voice coil servo motor dynamics. The applied voltage V a is the sum of the control voltage V c and the ack-emf voltage V. The control voltage in terms of motor current command i is V c = Ri + L di dt, where L is the armature inductance and R is the armature resistance. The ack-emf voltage in terms of the arm velocity vel is V = K e vel, where K e is the ack-emf constant. Since the ack-emf voltage is proportional to the velocity, a sloped acceleration command can e designed to overcome the effect of the ack-emf voltage as shown in Fig. 4. The slope needs to e chosen such that the maximum allowale applied voltage is met for as long as possile ut not saturated. In this section, the numer of the sloped positive acceleration command samples is calculated. The constraint of the acceleration u[k] is assumed to e u[k] A max. The maximum velocity is assumed to e V max and the sampling period is assumed to e T s. The relationship etween the acceleration command u[k] and the velocity v[k] is given as V (z) K a z U(z) = z, where K a is a constant gain. The difference equation etween acceleration u[k] at the discretetime instant kt s and velocity v[k] at the discrete-time instant kt s is given as v[k] = K a u[k ] + v[k ]. If the initial velocity v[] is assumed to e zero, the velocity at the discrete-time instant kt s can e k computed as v[k] = K a i= u[i]. The sloped positive acceleration command u[k] is descried as u[k] = A max k S, k =,, m, where S is the acceleration decrease per sample. As a result the following equation holds, m V max = K a i= m u[i] = K a m i= (A max i S), = K a (A max + S/)m K a Sm /. Hence, m is the least positive solution of a secondorder polynomial equation K a Sm / K a (A max + S/)m+V max =. The numer of the sloped positive acceleration command samples can e calculated as m = floor (m) ()

3 and the maximum velocity V rmax from () is V rmax = K a (A max +S/)m K a Sm / V max. () 3. Calculating the Numer of the Zero Acceleration Command Samples When the rigid ody reaches the maximum velocity constraint descried in (), the rigid ody is cruising at the constant velocity V rmax as shown in Fig. 4. If the position movement is assumed to e P max, the numer of the zero acceleration command samples is calculated. The[ state-space ] model [ of ] the rigid ody p[k + ] p[k] is descried as = G + K v[k + ] v[k] Hu[k], [ ] [ ] T where G =, H = s /, p[k] is the position T s at the discrete-time instant kt s, v[k] is the velocity at the discrete-time instant kt s, and K is a constant gain. The acceleration command u has the following format u =[A max, A max S,, A max (m )S, m,,,, n A max, (A max S), (A max (m )S) ]. m If the initial position p[] and velocity v[] are assumed to e zero, the position and velocity at the discrete-time instant kt s can e computed as (Ogata, 995) [ p[k] v[k] ] [ ] = G k p[] v[] G i K Hu[k i ], k + i= k = G i K Hu[k i ]. i= So at the discrete-time instant (m + n)t s, [ ] m p[m + n] +n = G i K v[m + n] Hu[m + n i ], i= = K m T s (A max + S Sm )(m + n). If the position at the discrete-time instant (m +n)t s is imposed to e P max, i.e. K m T s (A max + S Sm )(m + n) = P max, then n = P max K m T s (A max + S Sm ) m. (3) Generally the aove n is not an integer. Let n = floor(n) + α, where α = n floor(n) and α <. The numer of zero acceleration command samples can e chosen to e n = floor(n) +. (4) In the aove implementation, since the resultant numer of zero acceleration command n is generally greater than the required fractional numer of samples n, the resultant position at the end of the acceleration command is greater than the required position constraint which is P max. Fig. 5 shows the calculated fractional numer of the maximum velocity profile. The time interval etween the final maximum velocity impulse V rmax and the next velocity impulse is αt s which is less than one sampling period T s. Fig. 6 shows the modification of the integer numer of the maximum velocity profile from (4). Compared with Fig. 5, the summation of velocity impulses in Fig. 6 is increased y ( α)v rmax per sample. The additional velocity impulse summation can e compensated for y slightly modifying the velocity impulses. The acceleration command corresponding the velocity profile in Fig. 6 is u =[A max, A max S,, A max (m )S, m,,,, n A max, (A max S), (A max (m )S) ]. m The velocity profile from (5) can e descried as v[] =, (5) k v[k] = K a u[i], k =,, m + n, v[m + n ] =. i= The aove velocity profile can e modified to v [] =, v [k] = v[k] ( α)v rmax m + n, k =,, m + n, v [m + n ] =. The integral of the modified velocity impulses is exactly the same as the required integral of the velocity impulses in Fig. 5. The resultant modified acceleration command corresponding to (5) is u [] = A max ( α)v rmax K a (m + n ), u [k] = u[k], k =,, m + n, u [m + n ] = [A max ( α)v rmax K a (m + n ) ]. (6) In (4), if the resultant integer numer n of the zero acceleration command is less than, then the acceleration and the velocity limits are not required to achieve the position constraint. In this situation, to guarantee the position constraint, either a reduced acceleration limit or a reduced velocity limit may e implemented. It is easy to understand that the resultant maximum

4 V rmax α Vrmax Fig. 5. The calculated fractional numer of the maximum velocity profile. V rmax Vrmax Magnitude (db) Frequency (rad/sec) Fig. 8. Bode magnitude of the resonance structure. α ( α).5 Fig. 6. The modification of the integer numer of the maximum velocity profile. Fast acceleration command Viration suppression shape filter Roust viration suppresion command Fig. 7. Generation of a viration suppression command. velocity from the modified acceleration command (6) is slightly less than V rmax in (). 3.3 Viration Suppression Profile Generation with Both Acceleration and Velocity Constraints Since the sloped fast acceleration command is generated in the previous section, a viration suppression command can e generated as shown in Fig. 7. The viration suppression command is the convolution of the sloped fast command and the viration suppression shape filter. The viration suppression shape filter in Fig. 7 is simply descried in (Zhou and Misawa, 5a). In (Zhou and Misawa, 5a), it shows that the Input Shaping R (Singer and Seering, 99) is a special case of a non-continuous impulse function ased viration suppression shape filter. Different from the Input Shaping R, the viration suppression shape filter in (Zhou and Misawa, 5a) is generated from a continuous function, so it is ale to suppress the high frequency resonance modes esides canceling the low frequence resonance modes. However, the Input Shaping R are not ale to suppress the unmodeled high frequency virations if they are designed ased on a low frequency resonance mode (Zhou and Misawa, 5a). 4. SIMULATION RESULTS FOR HARD DISK DRIVE LONG SEEK CONTROL Consider the following flexile system which is emedded in a hard disk assemly, H(s) = K c K v Input Shaping R is a registered trademark of Convolve, Inc. in the United States. Current (Amps) x 3 Fig. 9. Sloped fast current command with the velocity constraint. K p R(s) s, where the input is the current signal in amps and the output is the position signal in tracks. The variale K c =.3 tracks/sample amp is a constant gain from current to acceleration, K v = 5 4 samples sec is the velocity gain, K p = 5 4 samples sec is the position gain, and R(s) is a resonance structure. The Bode magnitude plot of a reduced order (8 th ) R(s) is shown in Fig. 8. This resonance transfer function R(s) was derived from the flexile arm of an open disk drive at the Oklahoma State University Advanced Controls Laoratory. The resonance modes change drastically due to variation of the mode parameters. On the Bode plot, the peaks of the frequency response may shift oth in frequency and in amplitude. The maximum velocity constraint is V max = 3 tracks/sample, the applied voltage constraint is V a = volts, the long seek position movement is P max = 3 4 tracks, the sampling period is T s = 5 seconds, the maximum current is chosen to e A amx =.3 amp, and the slope value is chosen to e S =.5 tracks/sample sample. Fig. 9 shows the sloped fast current command with the velocity constraint. Fig. shows the resultant velocity signal. Fig. shows the resultant position signal. Fig. shows the position signal near the target track. The interval of Y axis in Fig. is scaled to exactly tracks and it shows that the residual viration exists for a long period of time after the end of the current command (6.3 msec). To suppress the residual viration, a rectangle ased shaper filter (Zhou and Misawa, 5a) is designed ased on the first resonance mode in the flexile

5 4.9 Velocity (Tracks/sample) Viration suppression shape filter x 3 Fig.. The velocity signal with the sloped fast current. 3.5 x 4 Fig. 3. Rectangle ased shape filter ased on resonance parameter ω = 6. 3 rad/sec and ζ = Current (Amps) Fig.. The position signal with the sloped fast current command x Fig. 4. Viration suppression current command. 4 3 Velocity (Tracks/sample) Fig.. The position signal near the target track. system. The first resonance mode has the parameter ω = 6. 3 rad/sec and ζ =.7. Fig. 3 shows the resultant viration suppression shape filter. Fig. 4 shows the viration suppression current command. Fig. 5 shows the resultant velocity signal. Fig. 6 shows the resultant position signal near the target track. The interval of Y axis in Fig. 6 is scaled to exactly tracks. Although the residual viration due to the first resonance mode has een canceled, a large viration still exists after the end of the current command. This residual viration is caused y the second resonance mode in the flexile system. To suppress the residual viration of the second resonance mode, a rectangle ased shaper filter (Zhou and Misawa, 5a) is designed ased on the second resonance mode in the flexile system. This mode has the parameter ω =. 4 rad/sec and ζ =.8. Fig. 7 shows the resultant viration suppression shape filter ased on the second resonance mode. Comining the shape filter in Fig. 3 and the shape filter in Fig. 7 results in a new shape filter as shown in Fig. 8. The resultant new viration suppression shape Fig. 5. Velocity signal with the viration suppression current command Fig. 6. Position signal near the target track. filter in Fig. 8 cancels the residual viration due to oth the first and the second resonance modes. Fig. 9 shows the viration suppression current command. Fig. shows the resultant velocity signal. Fig. shows the resultant position signal near the target track. The interval of Y axis in Fig. is scaled to exactly track. It is ovious that the residual viration due to oth the first and the second resonance modes is canceled and the residual viration due to all the high

6 .45 4 Viration suppression shape filter Velocity (Tracks/sample) x 4 Fig. 7. Rectangle ased shape filter ased on resonance parameter ω =. 4 rad/sec and ζ = Fig.. Velocity signal with the viration suppression current. 3.5 Viration suppression shape filter x 3 Fig. 8. Viration suppression shape filter to cancel oth the first resonance mode and the second resonance mode Fig.. Position signal near the target track..5 Current (Amps) x 3 Fig. 9. Viration suppression current command. frequency modes is also suppressed. Fig. shows the drive applied voltage signal due to the drive current command and it shows that the maximum allowale applied voltage is met ut not saturated. The future work will include how to automatically select the slope parameter S and drive current limit A max given the velocity, position and applied voltage constraints. 5. CONCLUSIONS In this examination, a viration suppression control profile is generated with oth the drive voltage and velocity constraints. The simulation results of the hard disk drive long seek control show the effectiveness of this method. The proposed methods apply to other flexile dynamic system long seek control prolem. The methods in this paper are patented (pending). Commercial use of these methods requires written permission from the Oklahoma State University. Applied volatage (Volts) Fig.. Applied drive voltage signal due to the drive current command. REFERENCES Ogata, K. (995). Discrete- Control Systems. Prentice Hall, Inc., Englewood Cliffs, NJ. Singer, N. C. and W. P. Seering (99). Preshaping command inputs to reduce system viration. ASME, Journal of Dynamic Systems, Measurement, and Control, Zhou, L. and E. A. Misawa (5a). From Input Shaping R and OATF to viration suppression shape filter. To appear in 5 American Control Conference. Zhou, L. and E. A. Misawa (5). Generation of a viration suppression control profile from optimal energy concentration functions. To appear in 5 American Control Conference. Zhou, L. and E. A. Misawa (5c). Viration suppression control profile generation with oth acceleration and velocity constraints. To appear in 5 American Control Conference.