Estimator-Based Sliding Mode Control of an Drive under Shock and Vibration

Yu Zhou,1 Maarten Steinbuch,2 Senior Member, IEEE, and Dragan Kostić,3 Student Member, IEEE 1Electrical Development Department, Philips Optical Storage, 620A Lorong 1 Toa Payoh, Singapore 319762 2,3Dynamics and Control Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology (TU/e), P.O.Box 513, 5600 MB Eindhoven, The Netherlands Email: [email protected], [email protected], and [email protected]

Abstract acceleration feed forward to both focus and radial servo loop to counteract the external disturbances. In combination with a A more robust servo control system using Sliding Mode number of modifications on the mechanism like pre-loading the Control to handle shock and vibration disturbances for upper porous bearing of the spindle motor these ideas were implemented and the mute level could be increased by a factor of systems is presented in the paper. An two. However, pre-loading the spindle motor will shorten the estimator-based SMC controller is used in the radial servo- motor lifetime, especially with the increase of the disc speed. loop instead of the traditional PID controller. Simulation Another way to increase the drive’s insensitivity to external and experimental results show a significant improvement of shocks and vibration disturbances is to develop a more robust the drive’s anti-shock performance in the radial direction. stable servo control system so that the laser spot stays on track at The same algorithm can be applied to the focus servo-loop. all times.

Keywords: First appeared in the early sixties [4], the sliding mode control has been widely studied recently and has been successfully applied to robot manipulators, high-performance electric motors, automotive Servo, Anti-Shock, Sliding Mode Control, DVD engines and power systems [5,6], etc., due to its notable advantage of insensitivity to the disturbance and system uncertainties. This paper applies the SMC technique to the two-stage servo tracking 1. Introduction system of optical disc drive to improve the product shock and vibration performance. An observer based discrete-time sliding With the introduction and development of high density and high mode controller has been developed to control the fine actuator capacity optical discs (like digital versatile disc DVD), and with with improved tracking shock and vibration performance. increasing demands on higher data transfer rate, it becomes more challenging to provide the system margins which are necessary for reliable data playback. From a servo point of view, one of the major obstacles for reliability of read-out data is given by the 2. Problem Formulation internal and external disturbances. The most important disturbances present in optical disc mechanism are rotation of the Figure 1 illustrates the simplified block diagram of the spot disc, eccentricity and track irregularities, mechanical vibration and position control system during tracking for both focus and radial. shocks, and positioning sensing noise. The relative laser spot position error signal e(s) is detected by the optical pick up G1(s). The commonly used controller K(s) and the Control systems subject to periodic disturbances may well benefit actuator driver G2(s) feed the system with the currents. H(s) from the use of repetitive [1] and learning feedforward control. presents the transfer function from the control current to the radial But, maintaining the laser spot within acceptable limits of track or focus spot position. d(s) represents dynamic disturbances center under shock condition, especially when the shock or generated within the drive or from the external environment. This vibration lasts for a few milliseconds, is still a significant technical mainly includes radial and vertical track positions deviations challenge. As one of the importance quality ratings for the coming from disc unbalance, eccentricity, unroundness, etc., and system, a lot of effort has been put to improve the external disturbances from mechanical shocks and vibrations. The system shock immunity, especially, for optical data drives, reference signal r(s) predefines the reference situation at the disc. portable, Car CD/DVD players, etc. Some earlier research work It is given by the disc reflective laser in case of the focus control showed that, in order to obtain sufficient shock immunity, the loop, and by the center of the read-out track in case of the radial damping of the suspension should be higher and the servo gain at loop. Due to the spiral shaped track of the optical disc itself, the low frequencies should be sufficiently higher. Other research [2,3] laser spot along radial direction is controlled by the sledge- on the anti-shock system design for the Car CD system employed actuator radial loop with this PID-based controller to control the actuator fine displacements and the sledge positioning system to move the actuator outward at a slower space during tracking. According to the theory of Variable Structure Systems [4,5,9], the These two control systems form the two-stage servo tracking variations in the plant parameters and modeling uncertainties are system in the optical disc drive. matched only on the control channels. This means that by proper 0 selection of the control law, a total invariance to disturbances and Switch Control Electronics on PCB Tracking + parameter variations can be achieved on the sliding surface. For Xa - Sledge Sledge the discrete-time SISO system described by Eq. (1), the objective Seeking Track Controller Driver is to get the state x(k) of the actuator to track a desired time- counter varying state r(k) in the present of model uncertainty and disturbances. Consider a smooth sliding surface defined by: e(s) u(s) PID Controller Actuator Driver r(s) + K(s) G2(s) S(k) = ge(k) - = d(s) e(k) r(k) - x(k) (2) Xs + Position Detector Xa Radial Actuator = G1(s) H(s) where g is the constant row vector selected such that S(k) 0 + defines a stable sliding surface or sliding mode in the state space, the actuator desired tracking position invariant to disturbances or Sledge OPU Motor dynamic uncertainties.

The convenient reaching condition for the discrete system to Figure 1. Simplified block diagram of the radial control guarantee the existence of the ideal sliding mode is given by [10]:

Disturbances caused by the deviation from the nominal position, S(k +1) = (1- η)S(k) - εsgn(S(k)); ε > 0, η > 0 (3) rotation of the disk, eccentricity and track irregularities, etc., can be well regulated or controlled by the present PID controller and where ε is the control gain of SMC, and η is a positive constant some learning control algorithm [1,7,8] during tracking. However, affecting the response in the reaching phase. The system states the conventional PID controllers and the learning algorithms are will move monotonically toward the sliding surface from any no more effective to overcome the non-linear movement of the initial state when the reaching condition is met. The sliding track on the disc relative to the laser spot in the present of external manifold is attractive under the given reaching condition, which shock and mechanical vibration disturbances. Robustness to the can be proven using the Lyapunov Stability Theory [5]. modeling errors is also not guaranteed in the entire operating range. All these uncertainties of the system lead to the application Substituting equations (2) and (1) into (3), the equivalent control of SMC techniques to the servo system design. An observer based law to steer the errors from any finite value to the sliding surface SMC controller is developed here to replace the traditional linear and keep the state on the surface in the face of unknown PID controller for the optical disc drive servo system. Here, the disturbances can be then given by: radial direction, which has been proved to be the most critical, is investigated. The same algorithm is applicable to the focus loop. − u(k) = (gb ) 1{g[((1−η)I − A )x(k) d d (4) +ηr(k) + ∆r(k) − d(k)]+ ε sgn(S(k))} 3. Design of SMC for Radial Servo System

where ∆r(k) = r(k +1) − r(k) . As can be seen in the equation The discrete-time state equations of the radial actuator for a general optical drive can be described as: (4), the control switching across the surface S(k) = 0 is necessarily imperfect in implementation. This would lead to + = + + x(k 1) A d x(k) b d u(k) d(k) control chattering, which is highly undesirable in practice since it (1) y(k) = c x(k) involves high control activity and may excite unmodeled high- d frequency dynamics. Smoothing out the control discontinuity in a thin boundary layer neighboring the switching surface is generally Where x(k) = [x(k) V (k)]T is the state vector consisting of the used to eliminate the control chattering. The equivalent control law can be then expressed as: radial actuator position x [m] and the radial actuator velocity V

[m/s] at discrete time kT . Notice that this second order model = −1 −η − +η describes the rigid body dynamics of the actuator only. u(k) (gb d ) {g[((1 )I A d )x(k) r(k) Measurements show that in the frequency range of interest this is a S(k) (5) + ∆r(k) − d(k)]+ εsat( )} valid assumption (see also Figure 2). u is the scalar input. d Φ represents disturbances coming from mechanical shock and vibration, and is bounded according to the commercial where Φ is the boundary layer thickness, and sat(.) is the specification. y is a scalar output. A d , b d , and c d are saturation function. constant matrices/vectors of appropriate dimensions and are However, the control law in (4) is not implementable because of assumed to be known exactly. The pair ( A d , b d ) is controllable, the present of the unknown disturbance d(k) . Assume that the and ( A d , c d ) is observable. disturbance is bounded and considerably slower than the sampling frequency 1/ T , which holds for the optical drive design by the 4. SMC Controller for the DVD Driver specification. Assume the difference between d(k −1) and d(k) As mentioned above, the observer-based SMC controller to is of O(T) [11], then, the value d(k) at time k can be considered compensate for the external shock disturbance is inserted into the to be close to the value at time k −1. Estimating d(k) from radial actuator control loop of the two-stage servo system in the drive. Figure 2 shows the Bode plot of the radial actuator for the d(k −1) , we have Philips commercial DVD product Mercury II. Matrices corresponding to the system (1) are: d(k) ≈ d(k −1) (6) d(k −1) = x(k) − A x(k −1) − b u(k −1)  1 T   2  d d = = KT / 2 A d  2  b d   −ω T − 2ξω T  KT −ξω KT 2  Substituting the above expression into the control law (5) gives: 0 0 0 = [] c d 1 0 −1 S(k) u(k) = (gb ) {εsat( ) − g[(ηI + A )x(k) 2 d Φ d (7) where K is the dynamic gain of 22.786 m/s /V between the + − +η + ∆ + − output of the controller and the output of the actuator in radial A d x(k 1) r(k) r(k) b d u(k 1)]} direction. ω0 is the fundamental resonance frequency (rigid body mode) of the actuator of 45 Hz in radial direction, ξ is the For the optical drive system, as stated in the previous section, only associated relative damping of 0.076. T is the sampling period in the position error signal is available. The velocity signal of the [sec]. d(k) is the perturbation such as shock and vibration, and is actuator, however, is not sensed due to technical or economical reasons. Furthermore, sensor noise adding to the position error bounded according to the commercial specification. signal will also corrupt the measurement results. A state observer Figure 2. Bode plot of the radial actuator for the Mercury II drive [11] is used here to obtain the best estimate/prediction of the states from a record of noisy measurements with fast convergence.

Assume that the sensor noise is white Gaussian zero-mean stationary signal with known covariance given below:

E{n(k)} = 0, (8) + T = δ − E{n(k)n(k 1) } R 0 (k 1)

where R 0 is positive definite. The state observer is given by:

+ = + + − xˆ(k 1) A d xˆ(k) b d u(k) L(y(k) c d xˆ(k)) = − + + (A d Lc d )xˆ(k) b d u(k) Ly(k) = Σ T L c d R 0 (9) where L is the observer gain, designed by for instance the LQE The two-stage radial control loop for the Philips Mercury II drive method. The SMC control law for the actuator loop can be then is designed such that the sledge does the stepping whenever the modified as: radial actuator position with respect to the sledge edge is larger than a certain threshold, otherwise, the sledge will remain in hold ˆ mode. The movement of the actuator is thus controlled to always −1 S(k) u(k) = (gb ) {εsat( ) − g[(ηI + A )xˆ(k) follow the spiral trajectory track on the disc. The reference can be d Φ d described as: + − +η + ∆ + − A d xˆ(k 1) r(k) r(k) b d u(k 1)]}

(10) xd (k)  pf rotTk r(k) =   =   Vd (k)  pf rot  where Sˆ(k) = gxˆ(k) , xˆ(k) is the estimate of state x(k) at time (11) k . where p is the radial track distance, frot is the disc rotating For the conventional two-stage radial loop of the optical disc frequency. The usually defined radial error signal (RES), which is drive, the actuator position with respect to the sledge can be detected by the optical pick-up unit and feedback to the digital obtained and fed back to the sledge controller by reading the controller through signal- preprocessor equals = − integrator value from the PID-based actuator controller. In the xd (k) - x(k) pf rotTk x(k) . observer-based SMC for the radial tracking loop developed here, the actuator position can be easily obtained from the observer. The initial value of the observer gains are determined by the LQE vibrator table in the direction that the vibration is incited, here, the method and the final gain values for the Mercury II drive radial radial direction of the loader is sensed. The drive’s playing actuator are decided via pole placement by trial and error as: performance (normally called playability) under shock is measured by doing BLER (Block Error Rate)+ measurement. It Lres=1.3e4; Lv=1.7241e6 means playing the whole disc without any C2 uncorrectable error under shock. The tested Mercury II driver without SMC controller The equivalent dynamics on the sliding surface for d(k) = 0 is can only stand up to 4gm/300ms shock with no C2 error. With the SMC controller, the driver can play without any C2 error under − e(k +1) = [A + b (gb ) 1 g((1−η)I − A )]e(k) + 7gm/300ms shock. Figure 6 shows the radial error signal with and d d d d without SMC controller under 7gm/300ms. The measured radial − Sˆ(k) actuator sensitivity is around 0.65µm/V during playing at 1.2X b (gb ) 1[εsat( ) + g((1−η)I − A )~x(k)] (12) d d Φ d DVD. The typical track pitch of DVD disc is 0.74 µm. As can be = seen from the plots, the peak off-track value without and with the ge(k) 0 SMC controller is reduced from 28.1% to 8.7%. Figure 7 showed the BLER plot used to evaluate the playability performance. The Since the state estimation error ~x(k) = x(k) − xˆ(k) tends meaning of the 9 plots from the top to the bottom is as follows: ˆ BLER count; PI Cor. (C1 correction); PI Peak (maximum C1 asymptotically to zero independently on the function S and error error); UCPI (uncorrectable C1 error); PO Cor.(C2 correction); PO variable e, and the Sˆ(k) also asymptotically tends to zero (see peak (maximum C2 error); UCPO (Uncorrectable C2 error – not reliable in the present application); FLAG (C2 uncorrectable error Appendix), the above system equation can be rewritten as: flag); and 100*Overspeed (scanning speed of disc compared to the

− 1X scanning speed). Inspection of the front part of the plot reveals e(k +1) = [A + b (gb ) 1g((1−η)I − A )]e(k) d d d d (13) a significant C2 uncorrectable error with the original PID cont- ge(k) = 0 roller (see the error FLAG plot in the figure) when 7gm/300ms is shock applied. While the second part of the BLER shows no C2 The g should be chosen such that the above system is stable, uncorrectable error (no error FLAG) with SMC controller. These e(k) = x(k) − r(k) → 0 as k → ∞ , that is, the tracking error results are in agreement with the simulation results. converges asymptotically to zero. The coefficients of the switching function for the radial actuator during tracking for Mercury II drive are:

gres=1.e2; gv=1.6e4; ε =600 where, the control gain ε of SMC is chosen such that the whole system has a crossover frequency (determined using frequency response measurements, hence linearizing the nonlinear system as in the describing function method) close to that of the original PID controller, that is 2.2kHz, when the radial error is within the boundary area. Here, a boundary of 1000 is used. When the system is operating outside the boundary layer, a higher SMC control gain ε is used, as discussed in the Appendix. For the system under the bounded disturbance d(k) , the tracking error will converge to the bounded area as stated in the previous section.

5. Simulations and Experimental Results

Figure 3 shows the simulation block diagram. A formalized acce- Figure 3. Matlab Simulation diagram for PID radial tracking leration profile of a half-sine is chosen to represent the typical control and SMC control shock disturbances in Audio/Video applications. Figure 4 shows that peak off-track value of the original PID controller is 34.6% and it is reduced to 17.7% when the SMC controller is used.

Figure 5 shows the experimental setup for the testing. The shock disturbance is generated by the vibration test system - vibrator V100 from Gearing & Watson Electronics Ltd. A formalized acceleration profile of half-sinoid is chosen to represent the typical +BLER measurement is a general measurement to indicate the shock disturbances in Audio/Video applications. It is added to the overall playing quality of the drive. It counts erroneous data DVD Mercury II driver through the vibrator table. The spectral blocks during playing or tracking. The C2 error FLAG indicate content and accuracy of the vibration/shock excitation of the uncorrectable errors count after error correction action in the vibrator table is ensured by feeding back the accelerometer signal engine level. If uncorrectable error occurred, drive will delivers to the vibration controller. The accelerometer is connected to the unreliable data to the interface [13]. 40 6. Conclusion

30 PID Controller

From the above simulation and experimental results done on the 20 SMC Controller Philips commercial DVD driver Mercury II, the Observer-based

10 SMC with different control gain to compensate high vibration and

shock do show a high level of immunity to unexpected external 0 disturbance. Playability testing result in radial direction shows that

Offtrack (%) Offtrack the shock performance specification can be improved from -10 4gm/300ms to 7gm/300ms. This method will improve the compact

-20 disc systems, especially those with high requirements on the anti-

shock performance, like portable CD/DVD player, Car CD/DVD -30 players, etc., without any increase of the material or process cost.

-40 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 However, the present work on the SMC controller with different Time (sec.) controller gains, that switches the threshold when the position Figure 4. Simulation results of radial error signal off-track (in %) error is within and outside of the boundary layer, is based on the with original PID controller and SMC controller under shock experience observation. As a result, it might lead to wrong adjudge of disc defect as shock or vibration. For the disc defect, the correct action should come from the error correction and data processing part. How to precisely detect or predict the mechanical shock is natural topic for further investigation.

Appendix

− Theorem: Assume that the eigenvalues of (A d Lc d ) are

chosen inside the unit circle. Then the discrete-time system (1)

subject to the control law (10) with the use of observer (9) is

asymptotically stable, if there is no uncertainty and sensor noise,

that is, Sˆ(k) → 0 as k → ∞ if d(k) = 0 and n(k) = 0 .

Proof: From Eq.(3) (11), it can be easily deduced that

Sˆ(k) Figure 5. The experiment setup for the shock test Sˆ(k +1) = (1−η)Sˆ(k) −εsat( ) + gLc ~x(k) Φ d (14) − Since the eigenvalues of (A d Lc d ) are inside the unit circle in 0.6 0.4 the complex plane, the state estimation error ~x(k) tends 0.2

force (V) ~ 0 asymptotically to zero. That is, after a short transient, x(k) ≈ 0 , -0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 therefore, the Eq. (14) can be written as:

0.2 time (s) ˆ S(k) Sˆ(k +1) = (1−η)Sˆ(k) −εsat( ) (15) 0.1 Φ 0 2 RES (V) = ˆ Define the Lyapunov Function as: V (k) S (k) , then -0.1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 V (k +1) −V (k) = Sˆ 2 (k +1) − Sˆ 2 (k) 0.2 time (s) Sˆ (k) 0.1 =[(1− η)Sˆ (k) − εsat( )]2 − Sˆ 2 (k) Φ 0 RES (V) ˆ (16) -0.1 ˆ S(k) 2 ˆ 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 ≤ [S(k) − εsat( )] − S (k) time (s) Φ

Sˆ (k) Sˆ (k) = ε 2sat( ) 2 − 2εsat( )Sˆ (k) Figure 6. The experimental testing results for the radial error Φ Φ signal with and without SMC controller 2 ˆ For S(k) > Φ , because ε > 0 and Φ is positive value, it can be readily verify that

Sˆ(k) V (k +1) −V (k) < ε[ε − 2Sˆ(k) sgn( )] < 0 provided that: Φ 0 < ε < 2Φ . It follows that the trajectory always converges to the boundary References with thickness Φ. [1] M. Steinbuch and G. Schootstra, “Robust Repetitive Control”, , When inside the boundary layer Sˆ(k) < Φ in Proc. 1st IFAC conference on Mechatronic Systems; Editors: R. Isermann, Darmstadt, Germany, (2000) ε [2] M.L. Norg, M. Steinbuch, H. Rumpf, “Feed Forward control to Sˆ(k +1) = (1−η − )Sˆ(k) < Sˆ(k) (17) Φ enhance CD skip performance”, in Proc. 1998 Soc. of Automotive Engineers (SAE) Intenational Congress and Exposition; Detroit, ˆ provided that 0< ε < Φ . Therefore S(k) tends asymptotically to United States, pp.6, (1998) zero. [3] M.F. Heertjes, F. Sperling, M.J.G. van de Molengraft, “Computing periodic solutions for a CD-player with impact using Substituting (7) into (9) yields the equivalent dynamics as: piecewise linear shooting”, in Proc. 40th IEEE Conference on Decision and Control (CDC); Editors: J. Jim Zhou, Orlando, − Sˆ(k) xˆ(k +1) = Γ xˆ(k) − b (gb ) 1[εsat( ) + Florida, United States, 2195-2200, (2001). eq d d Φ (18) [4] V.I. Utkin, “Sliding modes in Control and Optimization”, − Sˆ(k) Springer, Berlin, 1991 g(I − A )~x(k)] ≈ Γ xˆ(k) − b (gb ) 1εsat( ) n d eq d d Φ [5] J.J.E. Slotine and W. Li, “Applied Nonlinear Control”, Englewood Cliffs, NJ: Prentice-Hall, 1991. = − where Γ eq A d Lc d . Combined with the above theorem, [6] V.A. Taran, “Improving the dynamic properties of automatic control systems by means of nonlinear corrections and variable xˆ(k) will converge to zero asymptotically. structure”, Automatic, Remote Control, 25, 1996, p.140-149 [7] H.G.M. Dotch, H.T.Smakman, P.M.J. Van den Hof, and Corollary: The necessary and sufficient condition for the sliding M.Steinbuch, “Adaptive repetitive control of a compact disc surface to be attractive under the reaching condition (15) is: mechanism”, Proc. 1995, IEEE Conference on Decision and Control, New Orleans, Dec. 1995, pp.1720-1725 0 < ε < 2Φ When the states are outside the boundary [8] Y. Zhou, G. Leenknegt, M. Steinbuch, “Tracking learning 0 < ε < Φ When the states are inside the boundary feedforward control for high speed CD-ROM”, in Proc. 1st IFAC conference on Mechatronic Systems; Editors: R.Isermann, Darmstadt, Germany, 961-966, (2000) [9] G. Weibing, Y. Wang, and A. Homaifa, “Discrete Variable Structure Control System”, IEEE Trans. On Industrial Electronics, Vol.42, No.2, 1995 [10] K.D. Yong,., Utkin, V.I., and Ozguner, U., “A Control Engi- neers Guide to Sliding Mode Control”, Proc. of the 1996 IEEE workshop on variable structure systems, 1996. (pp.1-14) [11] D.G. Luenberger, “Introduction to Dynamic Systems: Theory, Models & Applications”, Wiley, 1986. [12] K. Furuta, “Sliding Mode Control of a Discrete System”, System & Control Letters, Vol. 14, 1990. (pp 145-152). [13] H. Hoeve., Timmermans, J., and Vries, L.B. “Error Correction and Concealment in the Compact Disc System”, Philips Technical Review, Vol.40, No.6, pp.166-172, 1982.

Figure 7. The BLER plot on DVD-SL disc when playing at 1.2X The front part present the BLER plot with the original PID controller, while the back part show the BLER with SMC controller