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Benjamin Potsaide-mail: potsab@rpi.eduJohn Ting-Yung Wen1Directore-mail: wenj@rpi.eduCenter for Automation Technologies and Systems(CATS),Rensselaer Polytechnic Institute,Troy, NY 12180Mark Unrathe-mail: unrathm@esi.comDavid Watt2e-mail: david.watt@sri.comMehmet Alpaye-mail: alpaym@esi.comElectro Scientific Industries, Inc.,Portland, OR 972291High Performance MotionTracking Control for ElectronicManufacturingMotion control requirements in electronic manufacturing demand both higher speeds andgreater precision to accommodate continuously shrinking part/feature sizes and higherdensities. However, improving both performance criteria simultaneously is difficult because of resonances that are inherent to the underlying positioning systems. This paperpresents an experimental study of a feedforward controller that was designed for apoint-to-point motion control system on a modern and state of the art laser processingsystem for electronics manufacturing. We systematically apply model identification, inverse dynamics control, iterative refinement (to address modeling inaccuracies), andadaptive least mean square to achieve high speed trajectory tracking. The key innovations lie in using the identified model to generate the gradient descent used in the iterative learning control, encoding the result from the learning control in a finite impulseresponse filter and adapting the finite impulse response coefficients during operationusing the least-mean-square update based on position, velocity, and acceleration feedforward signals. Experimental results are provided to show the efficacy of the proposedapproach, a variation of which has been implemented on the production machine.关DOI: 10.1115/1.2789467兴Keywords: precision motion, iterative learning, inverse dynamics, motion control, adaptive FIR filterIntroductionA key challenge facing electronic manufacturers today is that asfeature sizes continuously decrease and component counts increase, maintaining or boosting production rates requires improvement in both the speed and precision of the manufacturing equipment. However, improving both of these performance criteriasimultaneously is difficult because higher speeds and accelerations tend to excite structural modes in the systems, which aretypically lightly damped. For electronic memory repair 关1兴 andmicrovia drilling applications 关2兴, these unwanted vibrations reduce accuracy and detrimentally affect settling times. Consequently, feedforward compensation schemes to augment the feedback controller are becoming an increasingly critical componentof the overall control architecture to achieve today’s demandingperformance requirements.Various feedforward control schemes have been proposed tosuppress undesired vibration. The simplest is to generate a smoothtrajectory with bounded velocity, acceleration, and possibly evenhigher motion derivatives 共e.g., trapezoidal and S-curve velocityprofiles are available in most industrial motion controllers兲 关3兴.This approach simply limits the bandwidth of the input trajectoryand does not take any model information into account. When ananalytic model 共linear or nonlinear兲 is available, various modelinversion techniques have been proposed 关4–8兴. When the modelis nonminimum-phase 共i.e., the zero dynamics are unstable兲, thecausal inverse is unstable. A noncausal stable inverse may beused, but the output would be phase shifted with respect to thedesired output, resulting in added motion time or tracking error.1Also at Department of Electrical, Computer, Systems Engineering.Also at SRI International.Contributed by the Dynamic Systems, Measurement, and Control Division ofASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 6, 2006; final manuscript received March 2, 2007.Review conducted by Santosh Devasia. Paper presented at the IEEE Conference onControl Applications, Taipei, Taiwan, September 2004.2The inverse dynamics approach requires a high fidelity nonlinearmodel, which is usually unavailable in practice. Furthermore, ifnonlinearity is present, the computation could also be quite demanding. A similar model inversion approach for nonminimumphase discrete time linear time invariant 共LTI兲 systems has alsobeen proposed to achieve the overall zero phase shift 关9–11兴.However, it is not possible to simultaneously achieve zero phaseas well as unity gain 共since that would require causal inversion ofa nonminimum-phase transfer function兲. To avoid explicit modelinversion, an H model matching approach 关12,13兴 may be usedto find the feedforward controller. As the bandwidth of the weighting functions in the objective function increases, the solution converges to the causal inversion. If the system has a few dominantvibrational modes at known frequencies, then a finite impulse response 共FIR兲 filter may be designed to cancel vibration after afinite period of time 关14–18兴. However, the approach does notgeneralize to more complex systems or nonlinear systems.In this paper, we develop and demonstrate an adaptive feedforward control approach to significantly improve the tracking performance of a modern and state of the art electronics processingmachine used for microvia laser drilling operations. The plant is asingle-input/single-output 共SISO兲 positioning system and we assume that a suitable feedback controller has already been implemented, as shown in Fig. 1. This controller has been carefullydesigned to ensure stability even in the presence of slight plantvariations with the remaining design freedom used to achievegood tracking and settling performance. However, the performance of the production feedback controller alone was not acceptable, which motivated this study of adaptive feedforward compensation.The laser scanning configuration used in the production machine and the experimental testbed in our laboratory are shown inFig. 2. In the machine, a beam steering mirror redirects a collimated laser beam toward a scan lens, which collects the laser lightand focuses the beam onto the working area. This scan lens obeysan f- relationship between the angle of the incoming light beamand the position of the focused spot on the surface to be processedJournal of Dynamic Systems, Measurement, and ControlCopyright 2007 by ASMENOVEMBER 2007, Vol. 129 / 767Downloaded 28 Dec 2007 to 128.113.122.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

Fig. 1 Feedback and feedforward control architecture共i.e., the position of the spot in x or y coordinates is directlyproportional to the angle of the associated beam steering mirror, ,not the tangent of as is usually the case兲 关19兴. For reference, thetransformation to convert between rotary joint angle and linearspot position is given byx 0.1667 m x rady 0.1667 m y rad共1兲The angular position of the beam steering mirror is measuredthrough a high resolution rotary joint encoder and, in actuality, thecontroller acts to direct the angle of the steering mirror. However,we have chosen to perform all calculations with respect to theprojected laser spot coordinate system as this is the true criticaldimension of interest during the laser processing operation and tobe consistent with the existing production closed loop controllerspecifications.The existing closed loop control system and the data acquisitionsystem operate at a sampling frequency of 200 kHz, which waschosen to effectively address the high frequency oscillatory modesof the system. To avoid internal saturation within the feedbackcontroller and electronics of the servo amplifier, the trajectorymust obey certain velocity and acceleration constraints. A maximum velocity vmax 5 m / s and maximum acceleration amax 1000g 9800 m / s2 were chosen based on the thermal/electroniclimitations of the hardware and the requirements of the electronicmanufacturing process. A high fidelity model that includes theseeffects has been considered, but the approach was rejected due tothe model complexity. Note that these values are associated withthe trajectory of the projected laser spot on the working surfaceand the transformation back to the rotary coordinate system isgiven by Eq. 共1兲. The motion range varies from 200 m to5000 m and the motion objective is to minimize the settling timein point-to-point motion subject to the velocity and accelerationconstraints. We transform the problem into a trajectory trackingproblem by generating a desired output trajectory that satisfies thevelocity and acceleration constraints, and then designing the feedforward controller to track this desired trajectory as closely aspossible. If only velocity and acceleration bounds are imposed,the minimum time trajectory would be of a trapezoidal velocity共rectangular acceleration兲 profile. If additional jerk bounds arealso required, the acceleration profile is trapezoidal 共this is theso-called S-curve velocity profile兲. In our experimental study, wehave found that a half-sine acceleration profile gives the best performance. All the experimental data presented in the paper arebased on this trajectory profile.Our design approach starts by identifying a LTI model bychoosing the input amplitude in the linear regime. The LTI modelis then used in two ways: first generate a nominal feedforwardbased on its approximate inverse, and then use it as the gradient inthe iterative learning control 共ILC兲 algorithm to obtain the smalladditional corrections for a set of motion trajectories covering therange of move lengths expected during operation. The correctiveinput generation is encoded in a FIR filter 共for computational efficiency兲 with the nominal values of the coefficient obtained based768 / Vol. 129, NOVEMBER 2007Fig. 2 Laser scanner configuration and experimental testbed.The f- lens provides a flat image field and a linear relationshipbetween x, y coordinates and x, y, respectively.on the learning results. The FIR coefficients are adaptively updated in run time using the least-mean-square 共LMS兲 algorithmbased on the position, velocity, and acceleration tracking errors.The use of inverse dynamics for the nominal feedforward allowsILC and adaptive LMS to provide only a small correction. This isimportant for the convergence of ILC, since only the approximategradient 共the LTI model instead of the true linear time varying共LTV兲 , gradient兲 is used, and adaptive for LMS, since the corrective signal is assumed to be linear in the FIR coefficients.Iterative learning was first proposed for robot tracking controlin Ref. 关20兴, and has been extensively applied to motion controlproblems 共see Ref. 关21兴 for a survey and references therein兲. Wehave chosen the gradient algorithm 关22–25兴, which is similar tothe Newton algorithm based ILC in Refs. 关8,26,27兴. We use theidentified LTI model in the gradient update rather than the trueLTV gradient. The LTI model is a reasonable approximation fortwo reasons: The desired trajectory is chosen 共with velocity andacceleration bounds兲 to avoid thermal/electronic saturation of thesystem, and the required correction by ILC is of very small amplitude. Using the LTI model to approximate the true LTV gradient has also been used in Ref. 关27兴. Adaptive feedforward controlhas also been extensively applied to motion control and vibrationsuppression 关28–30兴. We apply adaptation to the FIR coefficientsfor the desired position, velocity, and acceleration separately, andshow that the position adaptation loop reduces steady state errors,velocity adaptation reduces tracking error in the high velocity zeroacceleration regions, and acceleration adaptation reduces trackingerror in high acceleration/deceleration regions.Transactions of the ASMEDownloaded 28 Dec 2007 to 128.113.122.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

The contribution of this work is to systematically integrate existing techniques in system identification, inverse dynamics control, gradient based ILC, and adaptive LMS to achieve rapid andprecise motion in an industry control system. The novelty in ourapproach includes the use of the identified LTI model for bothinverse dynamics and gradient based learning control, the use ofILC result to initialize the adaptive LMS algorithm, and the use ofposition, velocity, and acceleration feedforward filter adaptation tospecifically target steady state, high velocity, and highacceleration/deceleration regions. Experimental results based onan industrial strength controller are also included to show theresults and effectiveness of this approach.The rest of this paper is organized as follows. Section 2 presents our methods for system identification and our identifiedmodel. Section 3 shows the result of applying an inverse filterbased on the identified system as the first component of the feedforward controller. Section 4 presents an iterative refinement 共i.e.,ILC兲 based approach for producing a corrective input to the feedforward controller using a model-based gradient descent algorithm. Section 5 generates a FIR approximation to the iterativerefinement corrective input that facilitates real-time correctionsfor arbitrary move lengths. Section 6 shows how we adapt the FIRcoefficients during run time using position, velocity, and acceleration tracking errors and a gradient update algorithm. The finalexperimental results for tracking a random move length trajectoryare shown and summarized in Sec. 7. Conclusions and futurework are presented in Sec. 8.2IdentificationLTI model identification based on input/output responses maybe performed in either the time or frequency domains. An approximately 50 KHz noise signal was present in the position measurement that adversely affected the effectiveness of the time domain approach. Therefore, we decided to use the frequencydomain subspace identification method 关31,32兴. By weighting thefrequencies appropriately, we were able to emphasize the frequency band of interest while rejecting the noise in our identification process.We first obtained the experimental frequency response of thesystem by stimulating the hardware with a known input signalthen collecting and processing the experimental response. Inputexcitation is an important consideration in system identification.Common choices include impulse 共could be approximately generated through an impact hammer兲, pulse train, sine sweep, pseudorandom binary sequence 共PRBS兲, and Schroeder-phase signal关33兴. Important attributes of a good excitation signal include excitation of the frequencies of interest, small enough amplitude toavoid saturation and other nonlinear effects, and large enoughamplitude for good signal-to-noise ratio 共SNR兲.Working with production hardware that had limited communications ability, we were forced to implement within the memoryand storage constraints of the digital signal processor 共DSP兲 basedclosed loop controller board. The operating frequency of 200 kHzwith only 20,000 data points available for the arbitrary input waveform and response wave form limited the complete excitation andresponse signals to 0.1 s in duration. Note, however, that the index into the wave form could be reset to the start of the signalupon reaching the end, which allowed for the signal to be repeatedindefinitely. This capability suggested that signals that are continuous across the wraparound would be particularly desirable inorder to allow for the initial transients to decay and the system toreach a steady state. A sinusoidal signal with a period equal to0.1 s 共frequency of 10 Hz兲 satisfies these criteria, as well as allhigher order harmonics. These limitations discouraged the use of asine sweep as the rate of frequency change was too rapid for thesystem to reach steady state and the signal is not continuousacross the wraparound.Journal of Dynamic Systems, Measurement, and ControlBoth applying a series of single frequency sinusoidal signals ata time and applying a Schroeder-phase excitation gave particularly useful frequency response results when using multiples ofthe fundamental 10 Hz frequency and allowing the system toreach a steady state through several repetitions of the signal. Thesingle frequency sinusoidal signal approach provided a much better SNR but was time consuming as a separate experiment wasperformed for every frequency of interest. The Schroeder-phasesignal was attractive since it could excite multiple frequencieswhile satisfying a specified time domain amplitude bound. Wefound that the single frequency sinusoidal approach provided anexcellent experimental frequency response, while the Schroederphase approach proved most useful for obtaining a very quick andsometimes noisy snapshot of the frequency response. For this reason, we chose the single frequency sinusoidal approach to generate the experimental frequency response used for system identification and we gathered data at 20 Hz intervals from 0 Hz to20 kHz.In system identification, it is important to consider possibleinput/output delays. If the experimental system contains a puredelay, direct application of finite dimensional LTI identificationwill often approximate the delay with nonminimum-phase zerosand additional poles 共as in the Padé approximation of the delay兲.The approximation can be avoided by including a pure time delayin the identification procedure by time shifting the output signalrelative to the input signal before performing the identification.Including the delay in this manner can result in identified modelsof lower order and better agreement between simulated and experimental responses 共no artificial undershoot兲. Ultimately, whenwe implement the inverse dynamics controller, the nonminimumphase zeros can compromise the tracking performance, while thepure delay simply leads to a time shift in the response.The production controller did include a pure digital delay forinternal signal synchronization and explicitly including the delayin our identification procedure was critical for obtaining the desired performance results. Notice how the identified plant modelexhibits significant nonminimum phase-zero behavior when identified with no delay, while including a delay of 16 samples 共at5 s sampling period兲 shows minimal nonminimum-phase zerocontribution 共see Fig. 3 for comparison兲. For the final identifiedmodel used in the controller design, we included a 16-sampledelay pure delay and identified a 16th order model, Ĝ. The identified model is compared to the experimental response with respect to the frequency response in Fig. 4. We verified the modelby returning to the time domain and comparing simulated andexperimental step responses, as shown in Fig. 5.3Inverse Dynamics ControlThe next step is to use the identified model Ĝ to construct anapproximate inverse dynamics filter Ĝ†. Two aspects of this procedure need to be treated with care. First, if Ĝ is strictly proper, itsinverse is improper. Therefore, higher order derivatives of y deswould be needed to implement Ĝ†y des in practice. In our case, thisis not a problem because the identified model has the same number of poles and zeros. However, Ĝ has a few nonminimum-phasezeros with small real parts and therefore a stable causal inversedoes not exist. There are several alternative approaches. Onecould apply the noncausal 共but stable兲 inverse of Ĝ to y des 关5兴.This would effectively delay the output response by shifting thetime origin. Since the real part of the zero is small, this delaycould be significant. The zero phase error tracking control methodwas proposed in Ref. 关9兴, which multiplies the nonminimumphase zeros by its mirror image to achieve the zero overall phaseshift. The gain, however, would deviate from 1, which would thenrequire further compensation. We have chosen to replace the unstable zeros by their stable mirror images, and then invert theNOVEMBER 2007, Vol. 129 / 769Downloaded 28 Dec 2007 to 128.113.122.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

Fig. 3 Pole/zero comparison between zero-delay and 16sample delay identified modelsresulting minimum-phase transfer function 共a similar method hasalso been suggested in Ref. 关34兴兲. The approximate inverse, Ĝ†, isthen applied to the desired output y des to generate the commandinput u, as shown in Fig. 6. The transfer function from y des to y isan all pass function, which consists of a pair of high frequency共about 7 KHz兲 lightly damped poles and their mirror image zeros.As a result, the phase delay of the output with respect to thedesired trajectory 共based on the half-sine acceleration profile兲 isalmost negligible. In simulation, the output tracks the desired trajectory as expected. However, when this controller was implemented on the physical experiment, large tracking errors near theentry into and inside the settling zone 共low velocity and highacceleration兲 are observed. Since this occurs in the low velocityregion, the cause is likely the nonlinear friction in the positioner.The experimental result of the 500 m move is shown in Fig. 7.A 1 m band is shown in this figure and represents the maximum acceptable settling specification. We concluded that thoughthe model matches the linear 共small amplitude兲 behavior well, it isnot good enough to meet the desired performance specification inpractice.4Fig. 4 Gain/phase comparison between experimental data andidentified model with 16-sample delayThe overall control architecture is shown in Fig. 8. Note that y desis delayed to match with the pure delay in the actual plant. Thebasic algorithm that we have implemented is summarized below:Iterative RefinementThe discrepancy between the experimental and simulated responses is due to the mismatch between the model and the physical system. To correct for this mismatch, we modify the input tobe u Ĝ†y des u, where u is obtained iteratively based on theoutput tracking error. We use the gradient ILC algorithm 关22–25兴to generate u with the identified LTI system Ĝ to approximatethe gradient operator 共as in Ref. 关27兴兲. The derivation of the algorithm and its convergence property are given in Appendix A.1.770 / Vol. 129, NOVEMBER 2007Fig. 5 Step response comparison between experimental dataand identified model with 16-sample delayFig. 6 Inverse dynamics feedforward control architecture using the identified modelTransactions of the ASMEDownloaded 28 Dec 2007 to 128.113.122.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

Fig. 7 Experimental results using the inverse dynamics filterfor a 500 m move lengthGiven y* ª 兵y*共ti兲 : i 苸 0 , 1 , . . . , N其 and u0 ª 兵u0共ti兲 : i 苸 0 , 1 , . . . , N其共generated from inverse dynamics兲. Set u u01. Apply u to the physical system and obtain the outputsequence y ª 兵y共ti兲 : i 苸 0 , 1 , . . . , N其.2. Update u by adding a corrective term uគ Ĝ*共yគ yគ *兲Fig. 9 Procedure of calculating G*e共2兲where Ĝ* is the adjoint of Ĝ, and may be set as a sufficientlysmall constant or found by using a line search 共which wouldrequire additional runs兲.3. Iterate until 储y y*储 or 储 u储 becomes sufficiently small.signal does not correspond to mirror movement but to a repeatableelectrical coupling effect between the sensor and actuator electronics and other noises.The key step in the above algorithm is the updated equation 共2兲.Let the state space parameters of Ĝ be 共A , B , C , D兲. The adjointĜ* is given by 共 AT , CT , BT , DT兲, but it must propagate backward in time from the zero state 共see Appendix A.2兲. To implement Ĝ* y, we use the time reversal filtering approach as in Ref.关35兴: First reverse y to be backward in time, filter it forward intime through the filter 共AT , CT , BT , DT兲 共for SISO, it is the same asĜ兲, and then reverse the result in time again. The procedure isillustrated in Fig. 9.As suggested by our industrial collaborators and also independently published by other researchers 关36兴, the forward filteringpart may be performed using the actual system 共by feeding thetime reversed error signal into the plant as the input兲, thus avoiding using the analytic gradient altogether.The result of iterative refinement for the move lengths of500 m and 1000 m is shown in Fig. 10. In both cases, theoutput tracking error essentially converged after eight iterations.The very high frequency oscillation 共around 50 kHz兲 in the output5Finite Impulse Response ApproximationExcellent tracking performance is obtained by using iterativerefinement, but the update process is nonreal time, i.e., the outputerror of a complete run is needed to update the input at any giventime. To allow for the real-time trajectory tracking, we use theresults from iterative refinement to train a filter that maps y des tothe corrective input, u. Many filter parametrizations are possible;we decided on the FIR filter structure due to its efficient real-timeimplementation in the DSP real-time controller, its guaranteed stability 共adaptation of IIR filters can be dangerous because adaptation of the coefficients can result in an unstable filter兲, and theease for finding the filter coefficients. The overall control architecture is shown in Fig. 11. In principle, it is possible to approximate the inverse dynamics portion by a FIR filter as well. However, since this approximation needs to hold for a much largerinput range, a high order FIR filter and higher real-time computational load would be required. In the adaptive case, as in the nextsection, there may be additional convergence issues associatedwith the update of a large number of coefficients and the possiblenonlinear effect due to the large input range.We parametrize the corrective input by the following FIR filter:*** uk w1y k n w2y k n . . . wny k n11 12 1Fig. 8 Inverse dynamics combined with iterative refinementfeedforward control architectureJournal of Dynamic Systems, Measurement, and Control共3兲where n n1 n2 is the order of the filter and n1 is the look-aheadhorizon. The filter coefficients, 兵wi , i 1 , . . . , n其, may be obtainedthrough a least-squares fit to the 共y des , u兲 data obtained fromiterative refinement. The idea is to choose a sufficiently rich set ofy des so that the FIR filter is applicable to the set of desire trajectories and move lengths of interest. To find the filter coefficients,NOVEMBER 2007, Vol. 129 / 771Downloaded 28 Dec 2007 to 128.113.122.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

Fig. 10 Experimental results of applying iterative refinement to 500 m and 1000 mmovesment inputs very well. Also note that the high frequency noisegenerated by the iterative refinement has been eliminated by theleast-squares fit operation. The output response for the 500 mmove is shown in Fig. 13. The use of an FIR approximation doesan overall good job at reproducing the iterative refinement results,but does result in slightly larger errors, especially in terms of thesteady state value.first write the FIR equation aswhere 共k1 , k2兲 共range of data used for the fit兲, n 共order of FIR兲, andn1 共look-ahead horizon兲 are parameters to be chosen. The leastsquares solution of the filter coefficients, w, are given byw Y †U共4兲†where Y is the Moore–Penrose pseudoinverse of Y. Note that Uand Y contain the iteratively refined results of a wide range ofmove lengths that are concatenated together. Thus, a single FIRfilter is generated to work well in a least-squares sense over thecomplete range of expected move lengths.Through experimentation, we decided on a 100-tap 共n 100兲FIR filter with 1 look-ahead step 共n1 1兲. For real-time implementation, the FIR runs at 20 kHz and is upsampled to 200 kHzthrough sample and hold 共the inverse dynamics filter runs at thefull 200 kHz兲. The iterative refinement data for move lengths from200 m to 700 m at 100 m increments were used to fit theFIR filter coefficients. The comparison between u for the iterative refinement case and FIR filter case for a 500 m move isshown in Fig. 12, where the FIR reproduces the iterative refine-Fig. 11 Feedforward control architecture using a combinationof inverse dynamics and the FIR filter772 / Vol. 129, NOVEMBER 20076Adaptive Finite Impulse Response FilterThe FIR approach as described requires fitting the filter coefficients to the ideal input/output responses pregenerated by the iterative refinement process applied to a range of move lengths.This presents two drawbacks: 共1兲 As seen in the previous section,the FIR fit is not perfect, which results in a larger output trackingerror; 共2兲 when the system changes over time, the performance ofthe FIR filter will likely degrade. To address these issues, weinclude velocity and acceleration feedforward terms and apply theadaptive LMS controller 关37兴 to adjust the FIR filter coefficientsbased on the output tracking error in real time. This adaptiveFig. 12 Command input comparison between inverse dynamics, iterative refinement, and FIR filter: 500 m moveTransactions of the ASMEDownloaded 28 Dec 2007 to 128.113.122.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

Fig. 13 Output comparison between iterative refinement andFIR filter: 500 m moveapproach is equivalent to the standard gradient parameter estimation algorithm, as shown in Appendix B. The resulting overallcontroller architecture is shown in Fig. 14.We investigated an adaptive LMS architecture involving threeFIR filters, with the corresponding inputs being the desired position, velocity, and acceleration: uk wTx yគ desk nx wvTyគ̇ desk nv wTa yគ̈ desk na111共5兲whereyគ desj ª 兵y des共ti j兲:i 0,1, . . . ,Nx其共6兲yគ̇ desj ª 兵ẏ des共ti j兲:i 0,1, . . . ,Nv其共7兲yគ̈ desj ª 兵ÿ des共ti j兲:i 0,1, . . . ,Na其共8兲The coefficient update using adaptive LMS is given by wx x共y y *兲Ĝdyគ * wv v共y y *兲Ĝdyគ̇ * wa a共y y *兲Ĝdyគ̈ *共9兲where Ĝd is the identified LTI syst

analytic model linear or nonlinear is available, various model inversion techniques have been proposed 4-8 . When the model is nonminimum-phase i.e., the zero dynamics are unstable , the causal inverse is unstable. A noncausal stable inverse may be used, but the output would be phase shifted with respect to the

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