Optimal Plant Shaping for High Bandwidth Disturbance Rejection in Discrete Disturbance Observers

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1 Optimal Plant Shaping for High Bandwidth Disturbance Rejection in Discrete Disturbance Observers Xu Chen and Masayoshi Tomiuka Abstract The Qfilter cutoff frequency in a Disturbance Observer DOB) is restricted by the model mismatch between the plant and its nominal model. DOB Robustness is usually poor when the cutoff frequency of Qfilter increases. In this paper, robustness of the DOB is improved by shaping the true plant with a cascaded frequency shaping filter, while the nominal plant model remains fixed. Once the shaping filter is optimally designed based on DOB robust stability, the cutoff frequency of Qfilter can be greatly increased without loss of stability. The proposed algorithm is implemented on a typical tracking problem, i.e., the spiral writing in hard disk drives. I. INTRODUCTION Fast and accurate tracking control under the presence of disturbances is a central problem in mechacal positiong systems. High bandwidth and high robustness are two main objectives in the control of these devices. One effective way for disturbance attenuation is to use a disturbance observer DOB), which estimates the disturbance and feeds the negative of the estimated signal back to the plant input to accomplish the disturbance rejection. Examples of successful DOB applications include hard disk drives [] [3], optical disk drives [4], linear motors [5], [6], and positiong tables [7]. A standard DOB dynamically links together the actual plant, the inverse of a nominal plant model and the socalled Qfilter, which is normally a lowpass filter. The cutoff frequency, ω c, of the Qfilter plays an essential role in DOB performance and robustness. From the command following point of view, the DOB loop approximates the nominal plant model below the Qfilter s cutoff frequency, and approximates the true plant when the magtude of Qfilter drops to a small value. From the disturbance rejection point of view, the higher the cutoff frequency ω c, the larger the disturbance rejection capacity. However, robustness also decreases as ω c increases. Therefore, the cutoff frequency of Qfilter is usually restricted to maintain the system robustness. A typical example of this tradeoff can be found in [3]. For a robust motion control system, DOB, feedback and feedforward control strategies are usually combined. In such a case, DOB is applied as an inner loop Xu Chen is with the Mechacal Systems Control Laboratory, Department of Mechacal Engineering, Uversity of Califora, Berkeley, CA, 947, USA maxchen@me.berkeley.edu Masayoshi Tomiuka is with the Faculty in the Department of Mechacal Engineering, Uversity of Califora, Berkeley, CA, 947, USA tomiuka@me.berkeley.edu robust compensator to attenuate the low frequency disturbance and to provide the required robust performance of the feedforward controller. An effective feedforward controller requires an accurate model within an as large frequency region as possible, which in turn translates to the requirement of a higher Qfilter cutoff frequency. Increasing the disturbance rejection capacity while maintaing the robust stability is thus a key point to achieve high performance tracking. There have been many attempts to address the DOB robustness problem. The existing investigations have mainly been rooted in the designs of the Qfilter [8], [9] and the outer loop controllers []. In this paper, the robustness improvement of DOB is considered from another aspect. It is proposed to modify the dynamics on the side of the true plant. The robustness of DOB is improved by shaping the true plant with a cascaded filter, while the nominal plant model remains fixed. Once an optimal shaping filter is designed that mimies the model uncertainties, the cutoff frequency of Qfilter can be greatly increased without loss of stability. The remainder of this paper is orgaed as follows. Section II provides a review of the standard digital disturbance observer. The proposed DOB based on optimal plant shaping algorithm is presented in Section III. The efficacy of the proposed method is shown in Section IV, along with a typical tracking problem in hard disk drive systems. Section V concludes the paper. II. REVIEW OF THE STANDARD DISCRETE DOB Discrete DOB will be reviewed here briefly. Further discussions can be found in [7], [], []. A. Standard Discrete DOB Structure Consider a plant G p s), with a ero order holder equivalence G p ), and a loworder nominal model G n ) = m G ). A standard digital DOB can be constructed as shown in Fig. a), where Q ) is the lowpass Qfilter, G ) is the inverse of the nominal plant without delays, uk), wk), yk), and nk) are the control input, the plant input disturbance, the plant output, and the measurement noise. A stable inverse model G ) is needed in the above DOB design. If G ) has mimal phase, its inverse can directly be assigned, if not, stable inversion techques For the DOB design, we intentionally factorie out the delay terms m. G ) is the factoried nominal plant for DOB inverse.

2 such as the ZPET method [3] should be applied. Figure a) can be transformed to an equivalent form as shown in Fig. b). It is straightforward to derive the following transfer functions from u k), nk), and wk) to the output yk): G ) G ) p G ) u y = G ) G p ) G ) m) Q )) G ) G ) p Q ) ny = G ) G p ) G ) m) Q )) G ) G ) p G ) Q ) m) wy = G ) G p ) G ) m) Q )3) The output yk) can thus be expressed as: yk) = G u y ) u k)g wy ) wk)g ny ) nk). 4) u k) u k) Fig.. uk) Q ) m wk) G a) Standard form m Q ) ) G ) Q ) P G ) G ) P y k) nk) wk) uk) yk) b) Equivalent form nk) Block diagram of the standard digital DOB. In the low frequency region where Q ), if the delay is small so that m )G, then G wy ) in 3), G ny ) in ), and G u y ) G n in ). The disturbance wk) is thus attenuated and the entire DOB loop behaves like the nominal plant G n ). This observation makes it possible to design other feedback or feedforward controllers simply based on the low order nominal model. In the high frequency region where Q ), DOB is essentially inactive, therefore G u y ) G p ), G wy ) G p ), and G ny ), which indicates that the high frequency sensor noise is attenuated. B. Stability Robustness When there are unmodeled dynamics in the plant, the cutoff frequency of the Qfilter should be carefully selected. Consider G p ) as a perturbed version of the nominal model m G ), and the unmodeled dynamics as a multiplicative perturbation ), G p ) = m G ) )). 5) From ), the closed loop characteristic polynomial is given by A c ) = G ) G p ) G ) m) Q ) 6) Substituting 5) to 6), we have A c ) = G n ) m ) Q )). 7) From 7), a sufficient condition for DOB to be stable is that the eros of G n ) are all in the ut circle and that e jω) Q e jω) < ω. 8) Selection of the nominal plant model has been discussed at the beginng of this subsection. The difficulty for maintaing stability is mainly caused by the unmodeled dynamics ), as we observe in the following section. III. DOB BASED ON OPTIMAL PLANT SHAPING In practice, many mechacal systems have good models in the midlow frequency range and fixed high frequency resonance modes. Examples of such systems include but not limited to servo track writers [4], optical disk drives [4], hard disk drives [5], and directdrive positiong tables [7]. The major resonance modes usually can be well modeled as second order damping systems with transfer functions k i / s ζ i ω ) is ω i. For example, in Hard Disk Drives HDDs), the Voice Coil Motor VCM) actuator system [6] typically has a frequency response as shown in the solid line in Fig.. Magtude db) Phase deg) Frequency H) Plant Nominal Plant Model Fig.. Frequency responses of a VCM actuator system and the nominal plant model. When DOB is applied to systems having resonances as discussed above, the nominal model is usually selected ignoring the resonance modes, to achieve a stable inverse nominal model G ), and to simplify the design of feedback or feedforward controllers outside

3 the DOB loop. It is desired that sensors for output measurement are accurate, such that sensor noise contributes little to the output error. Under such conditions, it may be preferred that the cutoff frequency of Qfilter be chosen as high as possible. However, according to the DOB stability condition in 8), the ignored resonance modes will severely limit the bandwidth of Qfilter and thus the disturbance rejection capacity of DOB. Figure 3 shows the Qfilter designed in a standard DOB where the plant is the VCM actuator system in Fig., sampled at.64 kh. It is seen that in order to maintain the DOB stability, the Qfilter s cutoff frequency ω c is limited to be below 3 H. To achieve a larger ω c, we hereby propose a new modified disturbance observer structure by shaping the plant to mimie the modeling error. Magtude db) / Q 3 4 Frequency H) Fig. 3. Magtude responses of Q ) and / ) in the standard DOB: cutoff frequency of Qfilter restricted to be below 3 H. A. Controller Structure and Properties Figure 4 shows the structure of the proposed DOB. Different from the standard form, a shaping filter G s ) is cascaded preceding the plant G p ). Defin ing the shaped plant Ḡ ) p G ) p G ) s, we now have G ) Ḡ ) p G ) u y = G ) [ Ḡ p ) G n )] Q ), Ḡ ) p Q ) G ny ) = G ) [ Ḡ p ) G n )] Q ), G wy ) = G ) p G )[ Q ) m] G ) [ Ḡ p ) G n )] Q ). u k) Fig. 4. uk) Q ) G ) s m wk) G ) P G ) y k) 9) nk) Block diagram: DOB based on optimal plant shaping. TABLE I Properties of the Proposed DOB When m. Qfilter Q ) Q ) G ) u y G ) wy G ) u y G ) n G ) wy G u y Ḡ ) p G ) wy G ) p G ny ) G ny ) G ny ) Parallel to the analysis in Section II, Table I can be constructed. It is seen that the modified DOB possesses all the advantages of a standard DOB, i.e., low frequency disturbance rejection, nominal model following, and high frequency sensor noise attenuation. Moreover, superior to the standard disturbance observer, the modified DOB has the following additional properties: ) Resonance attenuation: from Table I, in the high frequency region, the DOB loop behaves like G u y ) Ḡ p ) = G p )G s ). If a shaping filter is properly constructed, the resonance modes can then be well attenuated. ) Enlarged disturbance rejection bandwidth: by careful design, G s ) can actively reduce the mismatch between Ḡ p ) and G n ). After optimiation of G s ), the cutoff frequency of Q ), and thus the DOB disturbance rejection bandwidth, can be greatly enlarged. B. Stability Robustness Similar to Section II, we have the following results for the proposed DOB structure: The closed loop characteristic polynomial: Ā c ) = G ) [ Ḡ p ) G n )] Q ). ) Let Ḡ ) p be subjected to multiplicative model uncertainties: where Ḡ p ) = G n ) )), ) ) = G ) p G ) s G ) n G n ). ) The stability of the modified DOB requires G ) n to have no eros outside the ut circle and that e jω) Q e jω) < ω. 3) Equation 3) appears to have a similar structure with 8), except that here ) is the multiplicative model uncertainty between Ḡ p ) and G n ), while in the standard DOB ) = [ G ) p G )] n /G ) n. Assuming that the nominal plant model remains the same in the two cases, in the proposed DOB, by cascading the shaping filter to the plant, we make the magtude of e jω ), and thus that of e jω) Q e jω) smaller. Therefore, the Qfilter can have a higher cutoff frequency in the proposed method.

4 C. Shaping Filter Design Define x as the coefficient vector to be optimied in the shaping filter. Based on the previous discussion, x should be chosen such that: Jx) = Gs x,e jω )G p e jω ) G n e jω ) 4) is mimied over the frequency range where there is strong model mismatch between G p ) and G n ). For the plant in Fig., direct inspection gives that the range [ kh, 3 kh] needs to be taken into the optimiation. Notice that 4) contains two optimiation problems, the magtude matching to mimie G s e jω )G p e jω ) G n e jω ), and the phase matching to mimie phase G s e jω )G p e jω ) ) phase G n e jω ) ). For the magtude matching, when resonances are the main sources of model mismatch, the structure of the shaping filter can be a combination of several notch filters, i.e., m s ζ i ω i s ω i G s s) = s x i ω i s ω, 5) i i= where ζ i and ω i are the damping ratio and center frequency of the i th resonance, and are obtained through system identification. x i x i > ζ i for notch filters) is the notch coefficient that needs tung. It is easier to perform optimiation in the continuous time domain, by noting that the magtude response at frequency ω k is given by ) m ω G s jωk = i= i ω k ω i ω k ) 4 ω i ω k ζ i ) 4 ω i ω k x i. 6) Taking the log to normalie the ut to db, we have f k y ) = log Gs jωk ) = Mk m log { } A k B k y i, 7) where y i = x i, A k =, ω i k) ω Bk = 4 ω i ω k, and M k = m i= log{ } A k B k a i. Noting that log Gs jω)g p jω) = log G s jω) log G p jω), we transform the magtude matching in 4) to the following simplified version: i= min f y ) log G n jω) log G p jω) ) y R m 8) subject to ζ i < y i <, i =,,...,m 9) where f y ), log G n jω), and log G p jω) are column vectors with each of their elements being the magtude response at selected frequency points. This is a nonlinear least squares problem. When the number of parameters to be identified is small, performing direct line search to y i over the region [ζ,], i.e., evaluating i the objective function at a variety of parameter values in the input space, can lead us to the mimal. Otherwise, nonlinear programming based on combined gradient and Newton methods can be applied. The latter can be performed in MATLAB using the nonlinear least squares data fitting function lsqcurvefit. The G s s) derived above is then digitied to G s ), using MATLAB s cd function with the pole ero match method matched). The phase match problem can now be addressed by adding delay j or advance elements j into G s ), and examing directly the phase difference between G s e jω )G p e jω ) and G n e jω ) in the frequency response. For causality, when j is needed, instead of modifying G s ), j can be added to G n ), since mimiing the phase difference between j G s )G p ) and G ) n is equivalent to mimiing the phase difference between G s )G p ) and j G ) n. However, to achieve a DOB with high performance, the delay elements should not be too many []. IV. CASE STUDY The proposed algorithm is implemented in a typical tracking problem, the spiral writing in hard disk drives. The spiral writing task is performed on a servo track writer, which uses the VCM actuator system for positiong. The servo writer head is expected to follow a reference trajectory as shown in Fig. 5 when moving from the disk outer diameter to the inner diameter. The goal is to achieve constant velocity as soon as possible with smooth settling and mimum tracking error. The well formulated opensource HDD benchmark tool in [6] was used in simulations. The plant has a frequency response as shown in Fig.. The disturbances include the torque disturbance, disk fluttering, the repeatable runout, and the measurement noise. The system has a sampling time of sec. [track] [tracks/sec] x 5 Desired Position x 5 Desired Velocity time[sec] Fig. 5. Reference trajectory for spiral writing: the actuator starts acceleration at.5sec, quickly reaches a constant speed for spiral writing, and finally decelerates to ero speed. The Qfilter used in this paper is given by Qs) = 3τs τs ) 3, )

5 where τ is a time parameter that determines the cutoff frequency. It has been suggested in [] that such a Qfilter results in balanced low frequency and high frequency performances. The Qfilter was digitalied using bilinear transformation for implementation. The solid line and the dotted line in Fig. 6 show the frequency responses of the plant and the nominal plant. There are three dominant resonant modes in the plant. The shaping filter is the product of three notch filters. In the magtude matching, the notch coefficients were all itialied at., and converged to the optima [.43,.36,.364] T. Magtude db) Phase deg) Bode Diagram 8 nominal plant optimally shaped plant 36 unshaped plant shaping filter Frequency H) Fig. 6. Frequency responses of the plant, optimally shaped plant, the shaping filter, and the nominal plant. Figure 7 shows graphically the robust stability conditions in the proposed DOB structure. Compared with Fig. 3, the high frequency model uncertainties have been greatly reduced, and the Qfilter cutoff frequency can now increase from 3 H to as large as H. Magtude db) 5 5 / Q 3 4 Frequency H) Fig. 7. Magtude responses of Q ) and / ) in the proposed DOB: Qfilter cutoff frequency can expand to as large as H. The shaping filter brought some phase drop starting at around kh. Phase matching was thus performed after the magtude matching. Note that digital disturbance observers accept only integer amounts of delays. Two delay steps were added to achieve the good phase match up to about 3 kh, which is high enough for normal servo operation. The nominal plant finally developed a form as a double integrator with two delay steps. The overall controller structure is shown in Fig. 8. The proposed DOB was applied as an inner loop controller. The DOB loop was then stabilied by a PD type feedback controller G PD ), to meet the stability and the bandwidth requirements. The disturbance observer makes the DOB loop behave like the nominal plant G n ), and remain robust in the presence of low frequency disturbances. Therefore, below the cutoff frequency of Q ), the feedback closed loop transfer function will be close to G closed ) = G n )G PD ) G n )G PD ). ) Finally, a feedforward ero phase error tracking [3] controller G ZPET ) was constructed based on G closed ), to achieve high performance tracking. y d k) G G ) s G ZPET ) G PD ) Q ) m wk) P ) G ) dk) yk) nk) Fig. 8. Overall controller structure applying the DOB based on optimal plant shaping. wk), dk), and nk) are respectively the input disturbance, the output disturbance, and the sensor noise. For comparison, another simulation was performed with the proposed DOB replaced by the standard DOB. Figure 9 shows tracking errors of the system in the two cases. The maximum error dropped from. track pitch TP) in the standard DOB to.3 TP in the proposed DOB, implying a 38% improvement. The standard deviation of the error decreased from.58 to.39. A spectrum analysis of the error signals further reveals the superiority of the proposed algorithm. We can see from Fig., that compared to the standard DOB, the proposed algorithm had a much higher disturbance rejection band. Errors below H were greatly reduced due to the high cutoff frequency of Qfilter. Finally, the magtude responses from the input disturbance to the plant output are shown in Fig.. The response in the proposed DOB exhibits a much deeper magtude drop in the middle frequency region, resulting in a larger disturbance rejection capacity. A small high frequency region gets slightly amplified, due to the waterbed effect that it is theoretically impossible for the closed loop transfer function to simultaneously have small magtudes at all frequencies. However, since the low frequency disturbance dominates, overall the error is better attenuated. V. CONCLUSION This paper has presented a new method to expand the DOB disturbance rejection capacity. When low fre The feedback controller in this case is the same PD controller cascaded with several notch filters.

6 Fig.. position error TP) position error TP) Signal Power Signal Power Magtude db)... position error Time sec)... a) applying the standard DOB: σ =.58 position error Time sec) b) applying the proposed DOB: σ =.39 Fig. 9. Time traces of the tracking errors. PES Amplitude Spectrum 3 4 Frequency [H] a) applying the standard DOB PES Amplitude Spectrum 3 4 Fig w/o DOB Frequency [H] b) applying the proposed DOB Frequency spectra of the tracking error. Frequency response: input disturbance to DOB output DOB w/ optimal plant shaping Frequency H) standard DOB Magtude response from input disturbance to plant output. quency disturbance dominates in a control system that has high frequency resonances, the proposed algorithm provides more freedom for designg the cutoff frequency of the Qfilter. Compared with the standard DOB, simulation showed sigficant reduction in the tracking error. ACKNOWLEDGMENTS This work was supported by the Computer Mechacs Laboratory CML) in the Department of Mechacal Engineering, Uversity of Califora at Berkeley. References [] M. White, M. Tomiuka, and C. Smith, Rejection of disk drive vibration and shock disturbances with a disturbance observer, in Proc. 999 American Control Conference, vol. 6, 999, pp [] J. Ishikawa and M. Tomiuka, Pivot friction compensation using an accelerometer and a disturbance observer for hard disk drives, IEEE/ASME Transactions on Mechatrocs, vol. 3, no. 3, pp. 94, 998. [3] M. White, M. Tomiuka, and C. Smith, Improved track following in magnetic disk drives using a disturbance observer, IEEE/ASME Transactions on Mechatrocs, vol. 5, no., pp. 3,. [4] K. Yang, Y. Choi, and W. K. Chung, On the tracking performance improvement of optical disk drive servo systems using errorbased disturbance observer, IEEE Transactions on Industrial Electrocs, vol. 5, no., pp. 7 79, Feb. 5. [5] K. Tan, T. Lee, H. Dou, S. Chin, and S. Zhao, Precision motion control with disturbance observer for pulsewidthmodulateddriven permanentmagnet linear motors, IEEE Transactions on Magnetics, vol. 39, no. 3 Part, pp , 3. [6] S. Komada, M. Ishida, K. Ohshi, and T. Hori, Disturbance observerbased motion control of direct drive motors, IEEE Transaction on Energy Conversion, vol. 6, no. 3, pp , 99. [7] C. Kempf and S. Kobayashi, Disturbance observer and feedforward design for a highspeeddirectdrive positiong table, IEEE Transactions on Control Systems Technology, vol. 7, no. 5, pp , 999. [8] C. Wang and M. Tomiuka, Design of robustly stable disturbance observers based on closed loop consideration using hinfity optimiation and its applications to motion control systems, in Proc. 4 American Control Conference, 4, pp [9] Y. Choi, K. Yang, W. Chung, H. Kim, and I. Suh, On the robustness and performance of disturbance observers for secondorder systems, IEEE Transactions on Automatic Control, vol. 48, no., pp. 35 3, 3. [] B. Kim and W. Chung, Advanced disturbance observer design for mechacal positiong systems, IEEE Transactions on Industrial Electrocs, vol. 5, no. 6, pp. 7 6, 3. [] C. Kempf and S. Kobayashi, Discretetime disturbance observer design for systems with time delay, in Proc th International Workshop on Advanced Motion Control, vol., Mar 996, pp vol.. [] T. Murakami and K. Ohshi, Advanced motion control in mechatrocsa tutorial, in Proc. IEEE Workshop on Intelligent Motion Control, vol., 99, pp [3] M. Tomiuka, Zero phase error tracking algorithm for digital control, Journal of Dynamic Systems, Measurement, and Control, vol. 9, no., pp , 987. [4] B. Baker and A. Moraru, Servotrackwriter with improved positiong system, IEEE Transactions on Magnetics, vol. 33, no. 5, pp , Sep 997. [5] M. Kobayashi, S. Nakagawa, T. Atsumi, and T. Yamaguchi, Highbandwidth servo control designs for magnetic disk drives, in Proc. IEEE/ASME International Conference on Advanced Intelligent Mechatrocs, vol.,, p. 4âĂŞ9. [6] M. Hirata, Nss benchmark problem of hard disk drive system,