3006 lud wick
TRANSCRIPT
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 14
A USERrsquoS GUIDE TO REPETITIVE CONTROL SYSTEMS AND THEINTERNAL MODEL PRINCIPLE
Stephen J Ludwick and Joseph A ProfetaAerotech Inc
Pittsburgh PA USA
ABSTRACTRepetitive control refers to a family of algorithmsthat are particularly useful when either the servocommand or disturbance is largely periodicExamples are very common in machine toolsdata storage systems and sensor testing Anyoscillatory or rotational motion generates someperiodic error in both the active and ancillarymotion axes The Internal Model Principle ofcontrol theory states that algorithms designed toperfectly reject input signals must contain a
model of that input and thus a repetitivecontroller contains a periodic signal generatorThe most general cases of repetitive control canbe difficult to stabilize If however the algorithmsare limited to operate within the existing servosystem bandwidth the tuning becomes muchmore straightforward and the algorithms becomeuseful tools for a servo system designer
THE INTERNAL MODEL PRINCIPLE The Internal Model Principle of control theory isa deceptively simple but powerful concept Firstformalized in the mid-1970rsquos [1] it can be
loosely stated as requiring that an algorithmcontain a generator (or model) of any inputsignal if that input is to be tracked with identicallyzero steady state error Figure 1 illustrates thisconcept with a block diagram For there to bezero error between the commanded referenceand measured signals then the controlalgorithm must be able to self-generate thissignal in the absence of any further input
A familiar example of the Internal ModelPrinciple applied in practice is through the use ofan integrator (I) term in the common PIDcontroller Consider the case of a linear-motor-driven positioning stage modeled as a free masswith a control force applied to it Proportionaland Derivative control alone are sufficient tostabilize the system but any constantdisturbance force (due to the process gravitycables and etc) requires some error betweenthe reference and measured positions in orderfor the spring-like proportional control term togenerate an output A constant disturbance is
modeled as a step input with a Laplacetransform of 1s Adding this term an integratorto the control algorithm allows the output to growto a constant value as required to cancel thedisturbance and achieve zero steady-state error
The Internal Model Principle is very general butspecific realizations of it appear frequently inprecision motion control applications Any input(whether a command trajectory or a disturbance)that repeats with some known period can be
addressed with a controller that contains aperiodic signal generator These are therepetitive controllers that will be discussed in thenext section If these inputs are frequency-limited they can be represented as a summationof sinusoids In that case we approach themwith harmonic cancellation algorithms that applythe Internal Model Principle with a series ofoscillators in the control algorithm
FIGURE 1 The Internal Model Principle requiresthat the controller contain a model of the inputsignals in order to be able to generate theappropriate output in the absence of any steady-state forcing error
REPETITIVE CONTROLLERS A periodic signal generator in the feedbackcontrol algorithm satisfies the Internal ModelPrinciple and allows for perfect tracking ofperiodic commands and perfect rejection of
periodic disturbances The class of controlalgorithms that collectively addresses thisproblem is called repetitive control Thesealgorithms first appeared in the literature in theearly 1980rsquos with a paper by Inoue [2] who usedthe Internal Model Principle as the basis for aldquocontroller for repetitive operationrdquo The authorsused a controller with a delay element in thefeedback loop to form a periodic signalgenerator In the continuous-time domain
P(s)C(s)R(s)
W(s)
E(s) = 0ΣΣ Z(s)
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 24
however the time delay element correspondedto a controller with an infinite number ofmarginally-stable poles This is illustrated inFigure 2 A signal with arbitrarily-sharptransitions in the time domain requires a high-bandwidth signal generator capable of creatingthis high-frequency content Stabilizing thesesystems is challenging since the high-frequencycontroller poles tend to interact with unmodeledor variable dynamics in the mechanical structureof a servomechanism which can lead toinstability A similar analysis for repetitivecontrol algorithms in the discrete-time domainshows the same problem [3]
FIGURE 2 A delay element in the feedback loopof a continuous-time control algorithm satisfiesthe Internal Model Principle for periodic inputsbut effectively contains a high (theoreticallyinfinite) number of oscillators to replicate anarbitrary periodic input
The relationship between the repeatingsequence in the time domain and the pole
locations in the frequency domain can beunderstood by recalling Fourier series analyses Any periodic signal can be equally-wellrepresented as a summation of simpleoscillating functions ndash namely sinusoids Thusthe repetitive controllers when applied to linearsystems can be viewed as a series of single-frequency oscillators added to the controlalgorithm to cancel an input that is itself asummation of single-frequency sinusoids Thisinterpretation is valuable since it allows us touse a familiar tool the Bode diagram fordetermining stability margins and the steady-
state response of these systems
HARMONIC CANCELLATIONWe refer to the special case of repetitive controlapplied to just a limited number of discretefrequencies as harmonic cancellation Thesecases are very common in precision motioncontrol applications and include
bull Force and torque ripple
bull Unbalanced payloads on rotary axes
bull Cyclic command profiles
bull Screw lead and gear pitch
bull Link-style cable carrier systems
Notice that some disturbances can be periodic intime while others are periodic on displacementand thus the specific frequency can vary In thefollowing analysis we will assume constant-speed operation with a known frequency
FIGURE 3 The harmonic cancellation algorithmC(s) is implemented in a plug-in architectureThis keeps Gc(s) the standard PID controllerunchanged and allows the harmonic cancellationalgorithms to readily enabled and disabled
We have found it useful to implement harmoniccancellation algorithms in a ldquoplug-inrdquo style thatallows it to be easily enabled and disabled asrequired The block diagram in Figure 3represents the harmonic cancellation algorithmas the standard PID controller as
and the plant (nominally a free mass) as
In keeping with the Internal Model Principle the
harmonic cancellation algorithm contains paralleloscillators one for each frequency containedwithin the disturbance signal
We can see the effect of the harmoniccancellation algorithm in the frequency domainby looking at the case of a single-frequencydisturbance Each individual oscillator in thealgorithm has a continuous-time Laplace-transform representation of
983139983151983155 983155983145983150
983086 (1)
Figure 4 shows a frequency response plot of theharmonic cancellation algorithm as the gain termsweeps from zero (disabling the oscillator)through higher values The key point to note isthat the magnitude is infinite at the oscillatorfrequency Recalling our early example of thefamiliar integrator providing zero steady stateerror to constant disturbances we can interpret
+
+-L 0
t
InitialFunction
0
e-Ls
PeriodicSignal
Ims
Res
jω1
-jω1
2jω1
-2jω1
ω1 = 2π L
ΣU(s)
E(s)
Delay Element
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 34
the harmonic cancellation block as simply anintegrator at a non-zero frequency
The tracking error due to disturbances isidentically zero at the oscillator frequency Wecan see this by referring back to the blockdiagram of Figure 4 and calculating that thetracking error due to a disturbance reduces to
983089
983089 983089 (2)
Evaluating this expression at the oscillatorfrequency
983089
983089 983089 983089infin 983088
(3)
we arrive at identically zero steady-state error to
disturbances at the frequency of interest Asimilar analysis shows unity response with zerophase shift between the commanded and actualposition profiles when the goal is to track aperiodic profile
FIGURE 4 Bode plots of a single-frequencyharmonic cancellation term with feedthroughshow the very high (infinite) magnitude at theoscillator frequency
In this section we have demonstrated that anoscillator in the control algorithm acts as anldquointegratorrdquo term to signals at the particularoscillator frequency Applying multipleoscillators in parallel allows the cancellation ofmore-complex waveforms and approaches thegeneral case of the full repetitive controllersThese controllers are implemented in a ldquoplug-inrdquo
manner that allows the standard PID controlgains to remain unchanged We use familiarfrequency-domain tuning tools to determinestability margins (crossover frequency phasemargin and gain margin) when applyingharmonic cancellation algorithms Generallyhowever due to the very limited frequencyrange that the harmonic cancellation algorithm ismost active over these systems arestraightforward to tune as long as the correctionfrequency is well below the system crossoverfrequency Figure 5 shows an experimentalopen-loop frequency response of an examplesystem with the harmonic cancellation algorithmactive The dominant peaks in the loop gainbelow the system crossover frequency areclearly visible and we can also see that theireffect is sufficiently localized such that gain andphase at the crossover frequency are relativelyunaffected
FIGURE 5 This open-loop Bode plot of a linearstage shows the strong increase in loop gain atthe cancellation frequencies (10 Hz and 20 Hz)and that they have minimal influence on theresponse near the 55 Hz crossover frequency
APPLICATION EXAMPLESThe overall concepts of the Internal ModelPrinciple repetitive control and harmoniccancellation once understood are broadlyapplicable One example is in the control of theread-write arm in hard disk drives [4] The disksdo not spin on a perfectly true axis but repetitivecontrol applied to the synchronous portion of theerror motion improves the ability of the head totrack the motion
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 44
Fast tool servo mechanisms used in asymmetricturning operations benefit from the application ofrepetitive control as well [5][6] When turningthe surface of a toric shape (such as the moldfor a contact lens that corrects for astigmatism)the cutting tool essentially returns the samepoint with each revolution of the spindle Thisperiodic toolpath can be decomposed into itsFourier series coefficients with harmoniccancellation oscillators applied to each of these
Aerotech Inc designs and builds precisionmotion control systems including themechanics drives and control algorithms Wehave found repetitive controllers to be usefulenough to include them as a standard featureThe challenge for us was not in the algorithmsthemselves (there is a 30 year deep library oftechnical publications to build on) but inpackaging the most-useful features into a user-
friendly interface accessible to someone withoutan extraordinary level of training Onestraightforward application was to a system witha horizontally-mounted rotary stage Thecustomer needed to improve their velocitystablity and we found that unbalance (once perrevolution) and motor pole pitch (nine times perrevolution) were the dominant terms This isapparent in Figure 6 which shows the positionerror measured while the stage rotated at 60RPM We applied harmonic cancellationalgorithms at these frequencies and reduced theroot mean square tracking error from 33 arc-sec
to 17 arc-sec a 12x reduction
FIGURE 6 Harmonic cancellation algorithmsapplied to a horizontally-mounted rotary stagereduced the root mean square tracking error byapproximately 12x at a 60 RPM speed Thedominant errors were the payload unbalanceand torque variations at the motor pole period
CONCLUSIONSIn this paper we presented the global concept ofthe Internal Model Principle of feedback controlsystems and showed how this led to thedevelopment of repetitive controllers and aneven simpler case of harmonic cancellationalgorithms Periodic disturbances arecommonplace in precision motion controlapplications and understanding when andwhere these algorithms can be applied gives thecontrol systems engineer an additional tool thatis both effective and can be analyzed with well-understood frequency-domain techniques
REFERENCES[1] Francis BA and Wonham WM The Internal
Model Principle of control theory Automatica 197612457-465
[2] Inoue T Nakano M and Iwai S Highaccuracy control of servomechanism for
repeated contouring In Proceedings of the10
th Annual Symposium Incremental
Motion Control Systems and Devices1981285-291
[3] Tomizuka M Tsao T-C and Chew K-K Analysis and synthesis of discrete-timerepetitive controllers ASME Journal ofDynamic Systems Measurement andControl 1989111353-358
[4] Chen YQ Moore KL Yu J and Zhang TIterative learning control and repetitivecontrol in hard disk drive industry ndash atutorial International Journal of Adaptive
Control and Signal Processing200722(4)325-343
[5] Ludwick SJ Chargin DA Calzaretta JATrumper DL Design of a rotary fast toolservo for ophthalmic lens fabricationPrecision Engineering 199923(4)253-259
[6] Lu X and Trumper DL Ultra fast tool servosfor diamond turning CIRP Annals200554(1)383-388
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 24
however the time delay element correspondedto a controller with an infinite number ofmarginally-stable poles This is illustrated inFigure 2 A signal with arbitrarily-sharptransitions in the time domain requires a high-bandwidth signal generator capable of creatingthis high-frequency content Stabilizing thesesystems is challenging since the high-frequencycontroller poles tend to interact with unmodeledor variable dynamics in the mechanical structureof a servomechanism which can lead toinstability A similar analysis for repetitivecontrol algorithms in the discrete-time domainshows the same problem [3]
FIGURE 2 A delay element in the feedback loopof a continuous-time control algorithm satisfiesthe Internal Model Principle for periodic inputsbut effectively contains a high (theoreticallyinfinite) number of oscillators to replicate anarbitrary periodic input
The relationship between the repeatingsequence in the time domain and the pole
locations in the frequency domain can beunderstood by recalling Fourier series analyses Any periodic signal can be equally-wellrepresented as a summation of simpleoscillating functions ndash namely sinusoids Thusthe repetitive controllers when applied to linearsystems can be viewed as a series of single-frequency oscillators added to the controlalgorithm to cancel an input that is itself asummation of single-frequency sinusoids Thisinterpretation is valuable since it allows us touse a familiar tool the Bode diagram fordetermining stability margins and the steady-
state response of these systems
HARMONIC CANCELLATIONWe refer to the special case of repetitive controlapplied to just a limited number of discretefrequencies as harmonic cancellation Thesecases are very common in precision motioncontrol applications and include
bull Force and torque ripple
bull Unbalanced payloads on rotary axes
bull Cyclic command profiles
bull Screw lead and gear pitch
bull Link-style cable carrier systems
Notice that some disturbances can be periodic intime while others are periodic on displacementand thus the specific frequency can vary In thefollowing analysis we will assume constant-speed operation with a known frequency
FIGURE 3 The harmonic cancellation algorithmC(s) is implemented in a plug-in architectureThis keeps Gc(s) the standard PID controllerunchanged and allows the harmonic cancellationalgorithms to readily enabled and disabled
We have found it useful to implement harmoniccancellation algorithms in a ldquoplug-inrdquo style thatallows it to be easily enabled and disabled asrequired The block diagram in Figure 3represents the harmonic cancellation algorithmas the standard PID controller as
and the plant (nominally a free mass) as
In keeping with the Internal Model Principle the
harmonic cancellation algorithm contains paralleloscillators one for each frequency containedwithin the disturbance signal
We can see the effect of the harmoniccancellation algorithm in the frequency domainby looking at the case of a single-frequencydisturbance Each individual oscillator in thealgorithm has a continuous-time Laplace-transform representation of
983139983151983155 983155983145983150
983086 (1)
Figure 4 shows a frequency response plot of theharmonic cancellation algorithm as the gain termsweeps from zero (disabling the oscillator)through higher values The key point to note isthat the magnitude is infinite at the oscillatorfrequency Recalling our early example of thefamiliar integrator providing zero steady stateerror to constant disturbances we can interpret
+
+-L 0
t
InitialFunction
0
e-Ls
PeriodicSignal
Ims
Res
jω1
-jω1
2jω1
-2jω1
ω1 = 2π L
ΣU(s)
E(s)
Delay Element
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 34
the harmonic cancellation block as simply anintegrator at a non-zero frequency
The tracking error due to disturbances isidentically zero at the oscillator frequency Wecan see this by referring back to the blockdiagram of Figure 4 and calculating that thetracking error due to a disturbance reduces to
983089
983089 983089 (2)
Evaluating this expression at the oscillatorfrequency
983089
983089 983089 983089infin 983088
(3)
we arrive at identically zero steady-state error to
disturbances at the frequency of interest Asimilar analysis shows unity response with zerophase shift between the commanded and actualposition profiles when the goal is to track aperiodic profile
FIGURE 4 Bode plots of a single-frequencyharmonic cancellation term with feedthroughshow the very high (infinite) magnitude at theoscillator frequency
In this section we have demonstrated that anoscillator in the control algorithm acts as anldquointegratorrdquo term to signals at the particularoscillator frequency Applying multipleoscillators in parallel allows the cancellation ofmore-complex waveforms and approaches thegeneral case of the full repetitive controllersThese controllers are implemented in a ldquoplug-inrdquo
manner that allows the standard PID controlgains to remain unchanged We use familiarfrequency-domain tuning tools to determinestability margins (crossover frequency phasemargin and gain margin) when applyingharmonic cancellation algorithms Generallyhowever due to the very limited frequencyrange that the harmonic cancellation algorithm ismost active over these systems arestraightforward to tune as long as the correctionfrequency is well below the system crossoverfrequency Figure 5 shows an experimentalopen-loop frequency response of an examplesystem with the harmonic cancellation algorithmactive The dominant peaks in the loop gainbelow the system crossover frequency areclearly visible and we can also see that theireffect is sufficiently localized such that gain andphase at the crossover frequency are relativelyunaffected
FIGURE 5 This open-loop Bode plot of a linearstage shows the strong increase in loop gain atthe cancellation frequencies (10 Hz and 20 Hz)and that they have minimal influence on theresponse near the 55 Hz crossover frequency
APPLICATION EXAMPLESThe overall concepts of the Internal ModelPrinciple repetitive control and harmoniccancellation once understood are broadlyapplicable One example is in the control of theread-write arm in hard disk drives [4] The disksdo not spin on a perfectly true axis but repetitivecontrol applied to the synchronous portion of theerror motion improves the ability of the head totrack the motion
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 44
Fast tool servo mechanisms used in asymmetricturning operations benefit from the application ofrepetitive control as well [5][6] When turningthe surface of a toric shape (such as the moldfor a contact lens that corrects for astigmatism)the cutting tool essentially returns the samepoint with each revolution of the spindle Thisperiodic toolpath can be decomposed into itsFourier series coefficients with harmoniccancellation oscillators applied to each of these
Aerotech Inc designs and builds precisionmotion control systems including themechanics drives and control algorithms Wehave found repetitive controllers to be usefulenough to include them as a standard featureThe challenge for us was not in the algorithmsthemselves (there is a 30 year deep library oftechnical publications to build on) but inpackaging the most-useful features into a user-
friendly interface accessible to someone withoutan extraordinary level of training Onestraightforward application was to a system witha horizontally-mounted rotary stage Thecustomer needed to improve their velocitystablity and we found that unbalance (once perrevolution) and motor pole pitch (nine times perrevolution) were the dominant terms This isapparent in Figure 6 which shows the positionerror measured while the stage rotated at 60RPM We applied harmonic cancellationalgorithms at these frequencies and reduced theroot mean square tracking error from 33 arc-sec
to 17 arc-sec a 12x reduction
FIGURE 6 Harmonic cancellation algorithmsapplied to a horizontally-mounted rotary stagereduced the root mean square tracking error byapproximately 12x at a 60 RPM speed Thedominant errors were the payload unbalanceand torque variations at the motor pole period
CONCLUSIONSIn this paper we presented the global concept ofthe Internal Model Principle of feedback controlsystems and showed how this led to thedevelopment of repetitive controllers and aneven simpler case of harmonic cancellationalgorithms Periodic disturbances arecommonplace in precision motion controlapplications and understanding when andwhere these algorithms can be applied gives thecontrol systems engineer an additional tool thatis both effective and can be analyzed with well-understood frequency-domain techniques
REFERENCES[1] Francis BA and Wonham WM The Internal
Model Principle of control theory Automatica 197612457-465
[2] Inoue T Nakano M and Iwai S Highaccuracy control of servomechanism for
repeated contouring In Proceedings of the10
th Annual Symposium Incremental
Motion Control Systems and Devices1981285-291
[3] Tomizuka M Tsao T-C and Chew K-K Analysis and synthesis of discrete-timerepetitive controllers ASME Journal ofDynamic Systems Measurement andControl 1989111353-358
[4] Chen YQ Moore KL Yu J and Zhang TIterative learning control and repetitivecontrol in hard disk drive industry ndash atutorial International Journal of Adaptive
Control and Signal Processing200722(4)325-343
[5] Ludwick SJ Chargin DA Calzaretta JATrumper DL Design of a rotary fast toolservo for ophthalmic lens fabricationPrecision Engineering 199923(4)253-259
[6] Lu X and Trumper DL Ultra fast tool servosfor diamond turning CIRP Annals200554(1)383-388
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 34
the harmonic cancellation block as simply anintegrator at a non-zero frequency
The tracking error due to disturbances isidentically zero at the oscillator frequency Wecan see this by referring back to the blockdiagram of Figure 4 and calculating that thetracking error due to a disturbance reduces to
983089
983089 983089 (2)
Evaluating this expression at the oscillatorfrequency
983089
983089 983089 983089infin 983088
(3)
we arrive at identically zero steady-state error to
disturbances at the frequency of interest Asimilar analysis shows unity response with zerophase shift between the commanded and actualposition profiles when the goal is to track aperiodic profile
FIGURE 4 Bode plots of a single-frequencyharmonic cancellation term with feedthroughshow the very high (infinite) magnitude at theoscillator frequency
In this section we have demonstrated that anoscillator in the control algorithm acts as anldquointegratorrdquo term to signals at the particularoscillator frequency Applying multipleoscillators in parallel allows the cancellation ofmore-complex waveforms and approaches thegeneral case of the full repetitive controllersThese controllers are implemented in a ldquoplug-inrdquo
manner that allows the standard PID controlgains to remain unchanged We use familiarfrequency-domain tuning tools to determinestability margins (crossover frequency phasemargin and gain margin) when applyingharmonic cancellation algorithms Generallyhowever due to the very limited frequencyrange that the harmonic cancellation algorithm ismost active over these systems arestraightforward to tune as long as the correctionfrequency is well below the system crossoverfrequency Figure 5 shows an experimentalopen-loop frequency response of an examplesystem with the harmonic cancellation algorithmactive The dominant peaks in the loop gainbelow the system crossover frequency areclearly visible and we can also see that theireffect is sufficiently localized such that gain andphase at the crossover frequency are relativelyunaffected
FIGURE 5 This open-loop Bode plot of a linearstage shows the strong increase in loop gain atthe cancellation frequencies (10 Hz and 20 Hz)and that they have minimal influence on theresponse near the 55 Hz crossover frequency
APPLICATION EXAMPLESThe overall concepts of the Internal ModelPrinciple repetitive control and harmoniccancellation once understood are broadlyapplicable One example is in the control of theread-write arm in hard disk drives [4] The disksdo not spin on a perfectly true axis but repetitivecontrol applied to the synchronous portion of theerror motion improves the ability of the head totrack the motion
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 44
Fast tool servo mechanisms used in asymmetricturning operations benefit from the application ofrepetitive control as well [5][6] When turningthe surface of a toric shape (such as the moldfor a contact lens that corrects for astigmatism)the cutting tool essentially returns the samepoint with each revolution of the spindle Thisperiodic toolpath can be decomposed into itsFourier series coefficients with harmoniccancellation oscillators applied to each of these
Aerotech Inc designs and builds precisionmotion control systems including themechanics drives and control algorithms Wehave found repetitive controllers to be usefulenough to include them as a standard featureThe challenge for us was not in the algorithmsthemselves (there is a 30 year deep library oftechnical publications to build on) but inpackaging the most-useful features into a user-
friendly interface accessible to someone withoutan extraordinary level of training Onestraightforward application was to a system witha horizontally-mounted rotary stage Thecustomer needed to improve their velocitystablity and we found that unbalance (once perrevolution) and motor pole pitch (nine times perrevolution) were the dominant terms This isapparent in Figure 6 which shows the positionerror measured while the stage rotated at 60RPM We applied harmonic cancellationalgorithms at these frequencies and reduced theroot mean square tracking error from 33 arc-sec
to 17 arc-sec a 12x reduction
FIGURE 6 Harmonic cancellation algorithmsapplied to a horizontally-mounted rotary stagereduced the root mean square tracking error byapproximately 12x at a 60 RPM speed Thedominant errors were the payload unbalanceand torque variations at the motor pole period
CONCLUSIONSIn this paper we presented the global concept ofthe Internal Model Principle of feedback controlsystems and showed how this led to thedevelopment of repetitive controllers and aneven simpler case of harmonic cancellationalgorithms Periodic disturbances arecommonplace in precision motion controlapplications and understanding when andwhere these algorithms can be applied gives thecontrol systems engineer an additional tool thatis both effective and can be analyzed with well-understood frequency-domain techniques
REFERENCES[1] Francis BA and Wonham WM The Internal
Model Principle of control theory Automatica 197612457-465
[2] Inoue T Nakano M and Iwai S Highaccuracy control of servomechanism for
repeated contouring In Proceedings of the10
th Annual Symposium Incremental
Motion Control Systems and Devices1981285-291
[3] Tomizuka M Tsao T-C and Chew K-K Analysis and synthesis of discrete-timerepetitive controllers ASME Journal ofDynamic Systems Measurement andControl 1989111353-358
[4] Chen YQ Moore KL Yu J and Zhang TIterative learning control and repetitivecontrol in hard disk drive industry ndash atutorial International Journal of Adaptive
Control and Signal Processing200722(4)325-343
[5] Ludwick SJ Chargin DA Calzaretta JATrumper DL Design of a rotary fast toolservo for ophthalmic lens fabricationPrecision Engineering 199923(4)253-259
[6] Lu X and Trumper DL Ultra fast tool servosfor diamond turning CIRP Annals200554(1)383-388
8182019 3006 Lud Wick
httpslidepdfcomreaderfull3006-lud-wick 44
Fast tool servo mechanisms used in asymmetricturning operations benefit from the application ofrepetitive control as well [5][6] When turningthe surface of a toric shape (such as the moldfor a contact lens that corrects for astigmatism)the cutting tool essentially returns the samepoint with each revolution of the spindle Thisperiodic toolpath can be decomposed into itsFourier series coefficients with harmoniccancellation oscillators applied to each of these
Aerotech Inc designs and builds precisionmotion control systems including themechanics drives and control algorithms Wehave found repetitive controllers to be usefulenough to include them as a standard featureThe challenge for us was not in the algorithmsthemselves (there is a 30 year deep library oftechnical publications to build on) but inpackaging the most-useful features into a user-
friendly interface accessible to someone withoutan extraordinary level of training Onestraightforward application was to a system witha horizontally-mounted rotary stage Thecustomer needed to improve their velocitystablity and we found that unbalance (once perrevolution) and motor pole pitch (nine times perrevolution) were the dominant terms This isapparent in Figure 6 which shows the positionerror measured while the stage rotated at 60RPM We applied harmonic cancellationalgorithms at these frequencies and reduced theroot mean square tracking error from 33 arc-sec
to 17 arc-sec a 12x reduction
FIGURE 6 Harmonic cancellation algorithmsapplied to a horizontally-mounted rotary stagereduced the root mean square tracking error byapproximately 12x at a 60 RPM speed Thedominant errors were the payload unbalanceand torque variations at the motor pole period
CONCLUSIONSIn this paper we presented the global concept ofthe Internal Model Principle of feedback controlsystems and showed how this led to thedevelopment of repetitive controllers and aneven simpler case of harmonic cancellationalgorithms Periodic disturbances arecommonplace in precision motion controlapplications and understanding when andwhere these algorithms can be applied gives thecontrol systems engineer an additional tool thatis both effective and can be analyzed with well-understood frequency-domain techniques
REFERENCES[1] Francis BA and Wonham WM The Internal
Model Principle of control theory Automatica 197612457-465
[2] Inoue T Nakano M and Iwai S Highaccuracy control of servomechanism for
repeated contouring In Proceedings of the10
th Annual Symposium Incremental
Motion Control Systems and Devices1981285-291
[3] Tomizuka M Tsao T-C and Chew K-K Analysis and synthesis of discrete-timerepetitive controllers ASME Journal ofDynamic Systems Measurement andControl 1989111353-358
[4] Chen YQ Moore KL Yu J and Zhang TIterative learning control and repetitivecontrol in hard disk drive industry ndash atutorial International Journal of Adaptive
Control and Signal Processing200722(4)325-343
[5] Ludwick SJ Chargin DA Calzaretta JATrumper DL Design of a rotary fast toolservo for ophthalmic lens fabricationPrecision Engineering 199923(4)253-259
[6] Lu X and Trumper DL Ultra fast tool servosfor diamond turning CIRP Annals200554(1)383-388