ROBUST ESTIMATION AND ADAPTIVE CONTROLLER TUNING FOR VARIANCE MINIMIZATION IN SERVO SYSTEMS

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1 009 JSME-IIP/ASME-ISPS Joint Conferene on Miromehatronis for Information and Preision Equipment (MIPE 009) June 7-0, 009, Tsukuba International Congress Center, Ibaraki, Japan ROBUST ESTIMATIO AD ADAPTIVE COTROLLER TUIG FOR VARIACE MIIMIZATIO I SERVO SYSTEMS Raymond A. de Callafon Jaob Shool of Engineering, Department of MAE and Center for Magneti Reording Researh University of California San Diego 9500 Gilman Dr, La Jolla, CA USA allafon@usd.edu ABSTRACT Improving storage density by means of an enhaned servo system typially requires the redution of variane of the Position Error Signal (PES). The variane of the PES is determined by both period and non-repeatable disturbanes for whih the harateristis are often not known a-priori during the servo algorithm design. Moreover, the servo ontrol algorithm is often limited to a standard Proportional, Integral and Derivative (PID) ontroller as more ompliated algorithms are viewed to be less robust. In this doument it is shown how a standard (PID) servo ontrol algorithm an be augmented an additional feedbak loop that an be tuned automatially by estimating the atual disturbane spetra seen in the PES. Adaptation to the disturbane spetra is done in lieu of possible model unertainties in the servo atuator, guaranteeing stability robustness. As suh, the ontrol algorithm provides a Robust Estimation and Adaptive Controller Tuning (REACT) to disturbane spetra to minimize PES variane in high performane servo systems in data storage appliations.. ITRODUCTIO Optimizing servo ontrollers for high performane data storage appliations require an intriate tuning performane in terms of Position Error Signal (PES) variane minimization, while maintaining robustness. PES variane minimization is required to improve storage density, while performane robustness is the mathematial onept of guaranteeing that a single and well designed algorithm is guaranteed to yield the same performane on a large lass of servo atuator similar dynamial properties. Obviously, performane and robustness are often onfliting requirements [], unleashing the need to ompromise performane to ensure, at least, stability robustness. In addition, for optimal tuning of the servo algorithm in terms of PES variane minimization, a detailed model is needed of all possible disturbanes that will be present in the servo loop and ontribute to PES variane. Suh a model an be formulated reasonably well for repeatable disturbanes, but the knowledge on random or non-repeatable disturbanes is often laking and only beomes available after the servo algorithm has been implemented. To failitate automati tuning of the servo ontrol algorithm to the disturbanes seen in the servo loop this doument proposes a methodology for Robust Estimation and Adaptive Controller Tuning (REACT) [] to minimize PES variane in high performane servo systems for data storage appliations. The algorithm formulates the tuning as a perturbation on an existing standard Proportional, Integral and Derivative (PID) ontroller and formulates onditions for stability robustness by onsidering a speial so-alled Youla [3] and dual-youla [4] parametrization and possible unertainty in the atuator dynamis. This doument gives an overview of the main idea behind the REACT algorithm to tune servo ontrollers for data storage systems. Appliations of REACT in the field of Ative oise Control an be found in [5], while an appliation to a Linear Tape Open (LTO) drive system is presented at the MIPE 09 onferene [6]. The outline of this doument is as follows. Setion first gives an overview of the Youla parametrization used to formulate the perturbation of the servo ontroller. This is followed in Setion 3 by presenting the results on how model unertainty in atuator dynamis an be inorporated to ensure stability robustness during servo ontroller tuning. Setion 4 and 5 show how the parametrization an be used to tune the servo ontrol algorithm diretly on the basis of data measured diretly in an operational servo loop and an example is given in Setion 6.. COTROLLER PARAMETRIZATIO A well-know result in ontroller design and optimization is the Youla parametrization [3] that allows the parametrization of the lass of all stabilizing feedbak ontrollers C for a given dynamial system G by a single stable dynamial system. The parametrization is given

2 009 JSME-IIP/ASME-ISPS Joint Conferene on Miromehatronis for Information and Preision Equipment (MIPE 009) June 7-0, 009, Tsukuba International Congress Center, Ibaraki, Japan in terms of oprime fatorization of the dynamial system G and the feedbak ontroller C that are defined as follows for Single Input, Single Output systems. Definition (right oprime fators): Consider the pair of stable transfer funtions (, D). The pair (, D) is a right oprime fatorization (rf) of G if the following three items hold: a. there exists a pair of stable transfer funtion ( X, Y ) suh that X + YD, b. det{ D } 0 and. G D Aording to Definition, oprime fators (, D) are always stable transfer funtions and item a. guarantees they do not have ommon unstable zeros, whereas item b. and. guarantee that G an be written as the produt of and the inverse of D. A similar definition an also be given for the oprime fators (, D ) of the feedbak ontroller C D. With the definition of oprime fators, the Youla parametrization [3] applied to a servo system reads as follows. Definition (Youla paramerization): Let (, D) be a rf of an atuator transfer funtion G and let (, D ) be a rf of a an initial feedbak ontroller C that stabilizes G. Then all ontrollers C of the form C D + D and D D where is any stable transfer funtion, stabilize the same atuator transfer funtion G. The Youla parametrization is a powerful result, as it allows us to parametrize all stable ontrollers for a speifi servo atuator dynamis G via (righ oprime fators and a stable perturbation. Hene, as long as we keep the transfer funtion stable, the newly tuned or perturbed ontroller C D in Definition will still stabilize the atuator dynamis G. The use of oprime fators is required for unstable atuator or ontroller dynamis to avoid the anellation of unstable poles and zeros in our feedbak loop during ontroller perturbations. However, the Youla parametrization an be simplified in ase both the atuator dynamis G and the initial feedbak ontroller C are known to be stable. With a stable atuator G, we obtain the trivial hoie (, D) ( G,) for the rf of G, while for a stable initial ontroller C we may hoose (, D ) ( C,) for the rf of C. In that ase, the newly tuned or perturbed ontroller C D in Definition will simplify to C () ( C + ) /( G) () and writing the newly tuned or perturbed ontroller C as a feedbak onnetion of the stable transfer funtion and the atuator model G. Again, as long as is stable the ontroller C stabilizes G. As a final note, the assumption of a stable atuator G also allows the initial ontroller C to be hosen as C 0 as a stable atuator G does not require a ontroller for stabilization. In that ase C an be simplified further to C /( G) (3) However, any information on an initially designed ontroller C used in the stabilization or ontrol of the atuator dynamis G an be used in the parametrization of the newly tuned or perturbed ontroller C. 3. ACTUATOR UCERTAITY The Youla parametrization as given in Definition assumes no modeling error or unertainty on the (stable) atuator dynamis G to ensure stability of the newly tuned or perturbed ontroller C in () or (). As a result, the only requirement for stability robustness is the stability of the perturbation in (). In ase of atuator unertainty, an additional onstraint on the atual size of will have to be imposed to guarantee stability robustness. The bound on the size of for stability robustness depends on how the unertainty on the atuator dynamis G is desribed, e.g. additive or multipliative unertainty. However, desribing the unertainty on the atuator dynamis G also in a oprime framework allows a lear omputation of the upper bound on the size of the perturbation. Following the ideas of a double-youla parametrization [7], the following main result is given here. Corollary (stability robustness for dual-youla): Let (, D) be a rf of an nominal atuator transfer funtion G and let (, D ) be a rf of a an initial feedbak ontroller C that stabilizes G. ow onsider a stable unertainty perturbing G D to G given by G D + D and (4) D D where is unknown but bounded by an H -norm < γ. Then the unertain atuator dynamis G the ontroller perturbation C given in () yields a stable feedbak system for all < γ if and only if is stable and γ. The result in Corollary indiates that next to the stability requirement of mentioned in Definition we now also have a size onstraint on measured by an H -norm. If we have no unertainty, γ and only the requirement on the stability of remains. The unertainty on the nominal atuator dynamis G is strutured aording to a dual-youla parametrization, as indiated in Figure. Although unfamiliar at first, it has been shown to have many favorable properties [7] ompared to standard additive or multipliative unertainties. Similar to the argumentation used for the Youla parametrization, it an be seen that a stable nominal model G a rf (, D) ( G,) an

3 009 JSME-IIP/ASME-ISPS Joint Conferene on Miromehatronis for Information and Preision Equipment (MIPE 009) June 7-0, 009, Tsukuba International Congress Center, Ibaraki, Japan initial ontroller C 0 redues the unertainty desription in (3) bak to a simple additive unertainty G G +. However, any information on an initially designed ontroller C used in the stabilization or ontrol of the atuator dynamis G an be used in the unertainty desription of the unertain atuator dynamis G. 4. AFFIE OPTIMIZATIO ext to providing a parametrization of all stabilizing ontrollers, the Youla parametrization in (), and its dual form for unertainty representation, provides an another advantage for ontroller adaptation: all losed-loop transfer funtion are linear in the ontroller perturbation of (). This an be seen as follows. Consider the output sensitivity funtion or transfer funtion S from the input disturbane d to the output signal or PES e in Figure given by S /( + G C) (5) Assuming the nominal atuator model G is stable, allows substitution of C C given in () to modify the sensitivity funtion S in (5) to S ( G ) /( + G C) G ) (6) S Fig. : Blok diagram of ombined ontroller perturbation aording to () and atuator unertainty given in (4). and indiating that the losed-loop transfer funtion that models the disturbane rejetion (sensitivity funtion) is linear in the ontroller perturbation parameter. A similar result an also be obtained when using the full freedom in right oprime fatorization, allowing the nominal atuator model also to be unstable. The Youla parametrization allows affine optimization tehniques to, for example, minimize the -norm or -norm of the (weighted) sensitivity funtion S as a funtion of. Moreover, diret minimization of the variane of the PES e an be used for a -norm, allowing diret data-based losed-loop tuning of a minimum variane ontroller [8] omputed via the ontroller perturbation parameter. For that purpose, the transfer funtion ( must also parametrized in a linear affine form and here ( is hosen to be parametrized via a nth tap disrete-time Finite Impulse Response (FIR) filter n ( θ ) θ (7) k k q where q denotes the time shift operator and θ the k parameters to be estimated in the FIR filter. The parametrization of the ontroller C ( in () and ( θ ) in (7) provides a rih lass of ontrollers that does not neessarily require a speifi parametrization of noise models to tune the feedbak ontroller [9]. 5. CLOSED-LOOP AD DATA-BASED TUIG For omputational purposes, the variane of the PES e given in Figure is given by var{ e} e ( (8) t and omputed over a final time interval of data points. To minimize the variane of the PES e as depited in Figure and to use the affine optimization result given in (6) and (7) we need aess to speifi losed-loop signals for diret data-based tuning of the ontroller perturbation parameter ( θ ). Assuming disturbanes that influene the variane of the PES e our as an additive output disturbane d as indiated in Figure, we see that e( S ( (9) and substitution of S in (9) yields e( t, θ ) d ( ( θ ) G ( d ( (0) so that minimization of the -norm of the time domain signal e ( t, θ ) using the affine parametrization in (7) requires aess to the two losed-loop signals y ( and v( G ( d ( G ( y( () The signal y( given in () is readily available, as this is the PES when the initial ontroller C is implemented in the feedbak loop or ( θ ) 0. For the omputation of the signal v( we need the (perturbed) atuator model G ( that an be approximated by the nominal model G( that is used also for the omputing the perturbation of the ontroller C to C given in (). The availability of the signals y( and v( defined in () now allows the minimization of the variane of the PES as funtion of the parameter θ of the FIR filter ( θ ) via ˆ θ arg min θ t [ y( ( θ ) v( ] and beomes a standard Least Squares (LS) optimization for the affine parametrization of ( θ ) given in (7). The LS optimization an be implemented reursively [8] to failitate adaptive tuning of the ontroller, but speial

4 009 JSME-IIP/ASME-ISPS Joint Conferene on Miromehatronis for Information and Preision Equipment (MIPE 009) June 7-0, 009, Tsukuba International Congress Center, Ibaraki, Japan onditions on the rate of hange in the parameters θ has to be imposed to guarantee stability of the resulting time varying system []. It should be noted that the LS optimization aims at minimizing the variane of the PES resulting in large ontrol signals for the minimum variane ontroller [8]. To avoid large ontrol signal, an additional penalty on the ontrol signal u( C ( () an be imposed by inluding a (filtered version) of u( in () in the minimization of (). Alternatively, ontrol signals an be limited by adding an additional fixed filter stable and stably invertible F( into the ontroller perturbation F ( θ ) ( θ ) F ( (3) and possibly inluding the inverse of F( in the filtering of y( as used in the LS optimization. 6. EXAMPLE Pratial implementations of REACT have been reported in [5] and at the urrent MIPE 09 onferene [6]. To illustrate the main onepts in this paper, a simulation example is used to illustrate the power of the REACT algorithm, as it allows diret tuning of the feedbak ontroller respet to disturbanes. For the example in this paper we onsider the Zero Order Hold (ZOH) equivalent of a ontinuous-time 4 th order servo atuator model sampled at 0 khz Kω ω G ( s) ( s + βωs + ω )( s + β ωs + ω ) 3 K 00, ω 0 rad/s, ω 0 rad/s, β and β 0. (4) that models a low frequeny frition or flex able mode and a high frequent poorly damped resonane mode. The atuator model is ontrolled by a st order disrete-time Proportional Derivative (PD) ontroller sampling at 0kHz and given by q C ( 0 (5) q 0.95 reating a feedbak loop a gain margin of 9dB at approx. 83 rad/s and a phase margin of 55.8 deg at approx. 00 rad/s. For simulating the effet of REACT, a low-pass filtered unit variane white noise ε ( together a sinusoidal signal of 0Hz are used to reate an additive disturbane d ( L( ε ( + 0.sin(π 0 (6) where the low-pass filter L ( is a 4 th order disrete-time Butterworth filter a utoff frequeny of 5 rad/s. Simulating the performane of the initial ontroller C ( in (5) in feedbak the ZOH equivalent of G (s) in (4) yields the disturbane and PES e( depited in Figure. Although there is Fig. : Simulation results for rejetion of disturbane given in (6) for the initial ontroller C( given in (5). Top figure: atual disturbane and PES e(. Bottom figure: ontrol signal u(. disturbane attenuation, the existing PD ontroller C( in (5) has obviously not been optimized for the non-repeatable and low frequent periodi disturbanes seen in the feedbak loop. To optimize the feedbak ontroller using the REACT algorithm, the ontroller C ( in (5) is perturbed to C ( in () only a 0 th order FIR filter ( θ ) in (7). To limit the ontrol signal u( during the variane minimization of REACT, the filter F( in (3) is hosen as a low-pass 4 th order disrete-time Butterworth filter a utoff frequeny of 000π rad/s. Simulating the performane of the perturbed ontroller C ( in () in feedbak the ZOH equivalent of G (s) in (4) yields the end result depited in Figure 3. Fig. 3: Simulation results for rejetion of disturbane given in (6) for the optimized REACT ontroller C ( using a n0 order FIR filter in (7). It an be seen from the simulation that REACT improves the variane of the PES e(, at only a small inrease of

5 009 JSME-IIP/ASME-ISPS Joint Conferene on Miromehatronis for Information and Preision Equipment (MIPE 009) June 7-0, 009, Tsukuba International Congress Center, Ibaraki, Japan the ontrol signal u(. A omparison of the Bode plots of the initial ontroller C( in (5) and the ontroller C ( in () is given in Figure 4. Fig. 4: Amplitude (top) and phase (bottom) Bode plot of the initial ontroller C( and the optimized ontroller C ( found via REACT. From Figure 4 it an be observed that the ontroller C ( found after the optimization has inreased the overall gain to provide non-repeatable disturbane rejetion and reated additional gain at low frequenies to target the low frequeny disturbanes. The improvement in disturbane rejetion an also be seen in Figure 5 in whih a omparison is made between the sensitivity funtions in (5) and S ( in (6). The additional gain at low frequenies in C ( emulates an integrator that was missing in the PD ontroller C( and now reates a feedbak loop a gain margin of 8.5dB at approx rad/s and a phase margin of 5.3 Fig. 5: Amplitude Bode plot of the initial sensitivity funtion in (5) and the optimized sensitivity funtion S ( in (6) found via REACT. deg at approx. 865 rad/s. The advantage is that the resulting ontroller C ( was found by automati tuning based on data obtained from the losed-loop system using the initial ontroller C(. 7. SUMMARY AD COCLUSIOS An initial (PID) servo ontrol algorithm an be augmented a Youla parametrization based perturbation to formulate a self-tuning algorithm for a servo ontroller. For atuator dynamis that an be modeled by a stable transfer funtion, the parametrization is formulated as a feedbak loop that uses the atuator model and a free, but stable, perturbation transfer funtion given by an Finite Impulse Response (FIR) filter. The parameters of the FIR filter an be found by an affine optimization based on losed-loop data to optimally tune the perturbed feedbak ontroller. Unertainty on the atuator model an be inorporated by bounding the allowable ontroller perturbation to provide a Robust Estimation and Adaptive Controller Tuning (REACT) to disturbane spetra to minimize PES variane in high performane servo systems in data storage appliations. REFERECES [] Zhou, K., Essentials of Robust Control, Prentie-Hall, Upper Saddle River, J, USA, 998. [] Kinney, C.E. and de Callafon, R.A., Robust Estimation for Automati Controller Tuning Appliation to Ative oise Control. To appear in Speial Edition to Celebrate the Retirement of Prof. Okko Bosgra, Springer, 009. [3] Youla, D.C., Bongiorno, J.J., Jabr, H.A., Modern Wiener-Hopf design of optimal ontrollers - Part I: The single-input-output ase, IEEE Trans. on Automati Control, Vol., pp. 3-3, 976. [4] Van den Hof, P.M.J., Shrama, R.J.P., Bongers, P.M.M., On nominal models, model unertainty and iterative methods in identifiation and ontrol design, Leture otes in Control and Information Sienes, Vol. 9, pp , 994. [5] Kinney, C., Villalta, A., and de Callafon, R., Ative oise Control of a Cooling Fan in a Short Dut, Pro. oisecon008-asme CAD, Dearborn, Mihigan, July 8 th -30 th, 008, CAD [6] Kinney, C.E. and de Callafon, R.A., A Signifiant Improvement to Tape Drive PES by Caneling LTM a Robust, High Performane Controller, Pro. of MIPE 09 onferene, Ibaraki, Japan, 009. [7] Douma, S.G. Van den Hof, P.M.J. and Bosgra, O.H. Controller tuning freedom under plant identifiation unertainty: double Youla beats gap in robust stability, Automatia, Vol. 39, pp , 003. [8] Astrom, K.J. and Wittenmark, B. On Self Tuning Regulators, Automatia, Vol. 9, pp 85-99, 973. [9] Patete, A., Furuta, K. And Tomizuka, M. Self-tuning ontrol based on generalized minimum variane riterion for auto-regressive models, Automatia, Vol. 44, pp , 008.