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Classical Tuning of Force Feedback Control forNanopositioning Systems with Load VariationsDOI:10.1016/j.ifacol.2016.10.674

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Citation for published version (APA):Kara Mohamed, M., & Heath, W. (2017). Classical Tuning of Force Feedback Control for Nanopositioning Systemswith Load Variations. In IFAC-PapersOnLine: 7th IFAC Symposium on Mechatronic Systems MECHATRONICS2016 — Loughborough University, Leicestershire, UK, 5—8 September 2016 Elsevier BV.https://doi.org/10.1016/j.ifacol.2016.10.674Published in:IFAC-PapersOnLine

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Download date:17. Aug. 2021

Classical Tuning of Force Feedback Controlfor Nanopositioning Systems with Load

Variations

Mohamed Kara-Mohamed and William P. Heath ∗

∗ School of Electrical and Electronic EngineeringSackville Street Building

The University of ManchesterManchester, M13 9PL, UK

Mohamed.karamohamed@manchester.ac.ukWilliam.Heath@manchester.ac.uk

Abstract: In nanopositioning systems, force feedback control has been proposed as an advancedcontrol technique to enhance the bandwidth of the system and improve the tracking performance.In direct tracking with force feedback control, the architecture employs an inner force feedbackloop to damp the first resonance peak and enhance the bandwidth. The position feedback loopis then used to enhance the tracking performance of the overall system. This paper discussesthe practical issues associated with the control design of force feedback. The paper presentshardware results to support the analysis and proposes a systematic tuning method to retain theadvantages of the force feedback control in the face of load variations.

Keywords: Nanopositioning, Control, Force Feedback, Dual Sensor Technology

1. INTRODUCTION

Nanopositioning stages are used in many applications suchas scanning probe microscopy, atomic force microscopy,lithography and imaging to generate mechanical displace-ment in microscopic scale (Devasia et al., 2007). Thesedevices, along with their applications, have introducedfundamental change in several scientific areas includingbiology, chemistry, physics and materials science (see forinstance Jandt et al., 2000; Bushan, 2010; Fleming andLeang, 2014, and references therein). In general, thesedevices employ piezoelectric actuators due to their highresolution and compact size (Xu, 2015). However, piezo-electric nanopositioning stages have challenging character-istics such as creep and hysteresis that limit the capabil-ities and may degrade the performance (Eielsen et al.,2014). Moreover, the existence of lightly damped reso-nances in all types of nanopositioning systems, whichresults from the interaction between the platform and thestiff support flexures, represents a main disadvantage thatlimits the bandwidth of the closed-loop system (Salapakaet al., 2002). Practically speaking, the bandwidth is lim-ited by the first resonance peak. It has been reported inthe literature (e.g., Fleming (2010)), that with feedbackonly control the bandwidth is usually set to be no morethan 2% of the first resonance peak. This is to maintaingood performance of the closed-loop system and preserverobustness against load variations. This is a major unde-sirable limitation for many applications (Clayton et al.,2009). For instance, one of the recent applications that

? This work is financially sponsored by both Elektron Technologyand EPSRC (EP/K503782/1) Project: IAA-087-2015, Concept andFeasibility Study.

require high bandwidth nanopositioning systems are hard-disk drives (HDD) (Bushan, 2010). In response to thisproblem several control methods have been proposed inthe literature (and are implemented in practice) to dampthe first resonance peak and enhance the bandwidth of theclosed-loop system, (see for instance Eielsen et al., 2014;Aphale et al., 2007; Das et al., 2014; Fleming and Leang,2014). The proposed fixed-structure control systems can bedivided into four categories; feedback control, feedforwardcontrol, iterative control and sensorless control (Devasiaet al., 2007). A combination of feedback and feedforwardcontrol is also reported by Kara-Mohamed et al. (2015)with promising results. Force feedback technique has beenintroduced by Preumont (2006); Fleming (2010) and Flem-ing and Leang (2010) as a new control system to enhancetracking and vibration control in nanopositioning stageswith significant improvement in bandwidth. This paperfocuses on the practical implementation aspects of theforce feedback control and produces a case study withexperimental results for a short range stage with forcefeedback control.

In Section 2, the force feedback control technique is re-viewed briefly. Section 3 produces general practical tuningguideline on how to consider frequency response along withtime domain analysis to tune the force feedback controland maintain good tracking performance of the stage. InSection 4 a case study is produced to showcase an exampleof designing force feedback control in the presence of loadvariations. The conclusion of the paper is presented inSection 5.

2. FORCE FEEDBACK CONTROL

Force feedback control is based on adding a sensor inaddition to the displacement sensor to measure the forceapplied by the piezoelectric actuator. Accordingly, one ofthe control architectures that has been suggested by Flem-ing (2010) is to utilise two feedback loops; see Fig 1.The inner control loop involves the measurement from theforce sensor and is designed to achieve damping controland enhance the bandwidth. The outer loop involves thedisplacement feedback and is used for position tracking.This stands in contrast to “dual sensor control” (as definedby Fleming, 2010) where knowledge of the relative gainof the position and force response is required to achievecorrect steady state tracking.

r Stage +

Force sensor 𝐶𝐶𝑓𝑓

p

vf - -

u x 𝐶𝐶𝑝𝑝

Fig. 1. Control architecture of the force feedback controlwith direct tracking.

Assuming linearity, the transfer function matrix of thestage from the actuator input u to the two outputs of thesystem is:

G =

[GpGf

], (1)

where Gp is the transfer function from the actuator inputu to the displacement position p and Gf is the transferfunction from the actuator input u to the force sensoroutput Vf .

The closed-loop response from the command signal r tothe displacement position of the stage p is given by:

Trd =GpCfCd

1 +GpCfCd +GfCf. (2)

The transfer function from the output of the PI compen-sator x to the position of the stage p is given by:

Txp =GpCf

1 +GfCf. (3)

Preumont (2006) and Fleming (2010) argue that the keyproperty of the system Gf is that its frequency responsehas phase lying in the range [0◦, 180◦]. This means thatthe system can be damped using simple integral control.If the inner controller is chosen to be an integrator of theform Cf (s) = 1

τfsthen the resulting forward loop transfer

function GfCf has an infinite gain margin and a phasemargin of 90◦ (i.e., it is passive). An ‘optimal’ value for τfcan be chosen based on a root locus argument assumingsecond order dynamics (Preumont, 2006; Fleming, 2010).In practice, digital implementation and/or any slight mis-alignment of the sensor invalidates the assumption of pas-sivity. In addition, the presence of higher order dynamics

and the requirement for good performance against varyingloads invalidates any ‘optimal’ tuning based on root locusarguments. Nevertheless the control structure works welland can be easily tuned via classical techniques and/orsimple step response tests. For most applications it issufficient to choose the outer loop controller as a PI ofthe form Cp = kp + 1

τps.

3. TUNING PROCEDURE

Fleming (2010) does not discuss the combined tuningof the force feedback loop and the outer loop. From apractical point of view, tuning of both loops is vital. Wepropose in this section general guidelines of how the innerand outer loop of the system should be tuned in orderto exploit the advantages of force feedback control. Inparticular we discuss how both time domain and frequencydomain requirements should be addressed in the presenceof load variations.

3.1 Integral time constant τf

The parameter τf serves to increase the effective dampingof the position response. The best practice for tuningthis parameter is simply to apply a step command to theinternal loop on x with the outer loop open (i.e., bypassingthe PI compensator) and to measure the output from thedisplacement sensor. For example, one can start with asmall value for the integral time constant (where clearlythe system will exhibit resonant behaviour) and thenslowly increase the time constant until the step response issatisfactory. Increasing τf above a certain range will bringresonance again in the response. Therefore, τf should betuned in the range between these two resonant values toachieve the best possible damping. Frequency response ofthe internal loop from x to p should be used to confirmthe quality of the time domain tuning of τf . As a measureof robustness, the chosen value τf should be tested withreasonable load on the stage. Trials confirm that doing thisfor one particular load gives good response across all loadslower than the tested load.

3.2 The outer loop control parameters

The parameters of the external PI loop can be tuned inthe normal way of tuning a PI compensator for stages withno force feedback loop. However, a few points should betaken into consideration when tuning the PI controller fora system fitted with force feedback.

• The force feedback control can achieve higher band-width and therefore the PI integral time constant τpshould be tuned to maintain the high bandwidth ofthe system.• The presence of the force sensor and the internal

force feedback loop makes the stage stiffer. Thisshould be taken into consideration when tuning thePI controller which requires higher proportional gainkp than the case of a similar stage without a forcefeedback sensor. However, increasing the proportionalgain will magnify the impact of the sensor noiseon the response of the system. Generally speaking,increasing the proportional gain kp will reduce the rise

time at the expense of increased noise amplification,oscillations and overshoot.• Usually the requirement is to achieve fast and accu-

rate response over a range of loads. In this aspect,the design in the frequency domain is advantageousas several models (corresponding to different loads)can be overlain. Specifically we recommend drawingthe Nichols charts of TxdCp for various loads as thecorresponding M-circles give the peak gain of theclosed-loop response from set-point to position.

• As noted by Eielsen et al. (2014), the plant is SIMOand therefore the results of Heath and Gayadeen(2011) give a useful graphical robustness measure ofthe closed-loop response. A similar measure (withsuitable weighting functions) is used by Eielsen et al.(2014) to evaluate the force feedback control struc-ture. Therefore, we need to look at the response of

| GfCf

1+GfCfCp+GfCf| + | GpCfCp

1+GpCfCp+GfCf| across all fre-

quencies.

4. CASE STUDY

In this section we demonstrate the tuning procedure dis-cussed in the previous section on the control of a stage fromQueensgate: the NPS-X-15A. The stage NPS-X-15A is asingle axis system and has a range of ±7.5µm. The stageis driven by the controller NPC5200 from Elektron Tech-nology which enables the implementation of PI control orPI + Notch in addition to the force feedback control. Thestage is pre-fitted with force feedback control which canbe disabled or enabled. The NPC5200 controller can beconnected to the PC and it uses the interface softwareQueensgate Nanobench to adjust its settings. The suppliedinput to the stage is the voltage for the piezoelectricactuator and the output is the measured displacement.The force sensor output is used internally for the forcefeedback loop.

The control structure of the stage is represented in Fig 2.The structure includes a low-frequency bypass as recom-mended by Fleming (2010) in the context of the alternativedual sensor structure. Complementary low and high passfilters of identical crossover frequency are needed becauseat low frequencies the piezoelectric force sensor is notsensitive and the generated signal of the sensor is propor-tional to the displacement measurement. Therefore, thefeedback signal is generated from the force sensor abovethe crossover frequency and from the input signal belowthe crossover frequency.

r Stage

+ Force sensor

Cp C

f

kf

p

vf - -

ku

L pass filter -------------- H pass filter

u

+

x

Fig. 2. Force feedback control architecture of the stageNPS-X-15A.

The scaling gains kf and ku on the force feedback loop areused to match the scale of the force sensor output Vf withthe scale of the stage actuator input u (Fleming, 2010). Forsimplicity of tuning, we recommend setting kf to unity andselecting ku so that the response matches the force outputat low frequencies. Given this, and assuming satisfactorydesign of the complementary filters, the control structureof the system can be simplified to be as in Fig 1.

To obtain the model of the stage over the frequency rangeof interest, a pseudo-random binary sequence (PRBS)signal of 0.5 Vpp (peak-to-peak magnitude) is applied tothe stage input and the output is measured, see Kara-Mohamed et al. (2015) for full details of the systemidentification procedure followed. The signal is applied tothe system and the output is measured via the NationalInstruments DAQ card NI PCI-6154 with a sampling rateof 10 kHz. The signals are processed and the model isobtained using the subspace method via MATLAB SystemIdentification Toolbox.

One of the main benefits of force feedback technologyis robustness to load variations. For robustness consid-eration, we tune the stage for the nominal case with noload and then we load it with various loads and observethe performance of the designed controller against loadvariations. Here we illustrate the results for two cases; noload and 800 g load. The 800 g load represents 80% of themaximum load the stage can tolerate and this is a highrobustness measure.

The step response of Gf shows that the gain of theforce sensor is not unity and hence in order to tune thecomplementary filter a calibration gain of ku is needed.

We identify the open-loop position response and obtain itsfrequency domain model for the nominal case and for theloaded case as expressed in Fig 3. The data acquisitionfrom the force sensor is not readily available and hence weare unable to produce a direct frequency response of Gf .Nevertheless Gf can easily be inferred from the frequencyresponse of the open-loop position response and the innerloop response measured from x to p since

Gf = (CfGp − Txp)/(kfCfTxp).

Using the step response method described in the tuningprocedure, we can get a satisfactory response of theinternal loop for τf = 9 × 10−5 as given in Fig 4. Thisis followed by measuring the frequency response of theloop in order to confirm the quality of our tuning. Fig 5demonstrates the measured closed-loop frequency responseof the internal loop from x to p for the nominal case withno load and with the case of 800 g load.

From the frequency response of the internal loop system,we can now reconstruct the frequency response of transferfunction for the force sensor Gf . Fig 6 depicts this fre-quency response for the nominal case with no load and forthe loaded case. Based on Gf , the reconstructed frequencyresponse of the internal open-loop system CfGf is shownin Fig 7. This figure shows that even though the systemCfGf is not completely passive across all frequencies inpresence of load, it is still almost passive in the frequencyregion of interest which matches the proposed theory offorce feedback control. However, extra care needs to betaken when tuning the PI controller as the system is not

Mag

nitu

de (

dB)

-40

-30

-20

-10

0

10

10-1 100 101 102 103 104 105

Pha

se (

deg)

-360

-270

-180

-90

0

90

Frequency (rad/s)

Fig. 3. Open-loop frequency response for Gp: unloadedstage (blue solid line) and loaded stage with 800g (reddashed line).

Time(ms)0 0.5 1 1.5

Pos

ition

(V)

-1

0

1

2

3

4

5

Fig. 4. Step response of the internal loop from x to p withτf = 9× 10−5. The tuning is for unloaded stage.

completely passive and hence an infinite gain margin anda phase margin of 90◦ for the closed-loop system are notguaranteed.

We continue the control design by choosing a satisfactoryPI controller of Cp = 1 + 1

6×10−4s . Fig 8 demonstrates themeasured closed-loop frequency response with this tuningand Fig 9 shows the step response of the closed-loop systemfor a step of 1Vpp(peak-to-peak) for the nominal case.As suggested in the tuning procedure, a Nichols chart ofCpTxd to confirm the quality of the control design andthe robustness to load variation is given in Fig 11. Theconstant magnitude circles (M-circles) determine the peakgain of the closed-loop response. This plot offers a robust-ness measure to structured uncertainty. The impact of theload is clear by shifting the closed-loop response tword theinstability point of (0dB,−180◦). However, the system isstill stable with good performance. Fig 10 shows the stepresponse of the closed-loop system for as step of 1Vpp(peak-to-peak) for the case of 800 g load. Moreover, Fig 12 shows

Mag

nitu

de (

dB)

-40

-30

-20

-10

0

10

10-1 100 101 102 103 104 105

Pha

se (

deg)

-225

-180

-135

-90

-45

0

Frequency (rad/s)

Fig. 5. Identified frequency response of the internal closed-loop system from x to p, with no load (blue solid line)and with 800 g load (red dashed line).

102 103 104 105

Mag

itude

(dB

)

-40

-20

0

20

40

Frequency (rad/s)102 103 104 105

Pha

se(d

eg)

-100

0

100

200

Fig. 6. Reconstructed frequency response of the forcesensor open-loop system Gf : with no load (blue solidline) and for the case of 800 g load (red dashed line).

the sensitivities | GfCf

1+GfCfCp+GfCf|, | GpCfCp

1+GpCfCp+GfCf| and

their absolute sum as described in the tuning procedure.The sensitivities in Fig 12 are useful as a robustnessmeasure to unstructured uncertainty (Eielsen et al., 2014;Heath and Gayadeen, 2011).

The closed-loop response to noise is an important designconsideration. Fig 13 evaluates the noise response of theclosed loop system for the designed controller using an in-terferometer which provides a measure that is independentof the control system sensors. The Fast Fourier Transform(absolute value) of the noise is provided in Fig 14. Bothfigures confirm good noise performance of the tuned forcefeedback control with an average noise of 70.5 pm.

102 103 104 105

Mag

itude

(dB

)

-20

0

20

40

Frequency (rad/s)102 103 104 105

Pha

se(d

eg)

-200

-100

0

100

200

Fig. 7. Reconstructed Frequency response of the internalopen-loop system CfGf : with no load (blue solid line)and for the case of 800 g load (red dashed line).

Mag

nitu

de (

dB)

-60

-40

-20

0

20

10-1 100 101 102 103 104 105

Pha

se (

deg)

-360

-270

-180

-90

0

Frequency (rad/s)

Fig. 8. Closed-loop frequency response of the X15 stagewith force feedback control: with no load(blue solidline) and with 800 g load (red dashed line)

5. CONCLUSION

Force feedback technology has been introduced as an ef-fective control solution to achieve accurate, robust andfast nanopositioning stages. However, to exploit the advan-tages of this technology, the configuration requires carefultuning and implementation in order not to excite thehigh frequency resonant peaks. This paper produces apractical tuning method for the force feedback control.The demonstrated example proves that with the suggestedtuning procedure, a closed loop bandwidth close to the firstresonant peak can be achieved. The tuning provides alsogood robustness to load variations which is an importantmeasure for any control design for nanopositioning stages.

time(s)0.26 0.27 0.28 0.29 0.3 0.31 0.32

posi

tion(

V)

-0.6

-0.4

-0.2

0

0.2

0.4

Fig. 9. Step response of the closed-loop system withthe designed force feedback control- no load. Thereference signal (red dashed line) and the closed-loopresponse with with no load on the stage (blue solidline).

Time(ms)0.26 0.27 0.28 0.29 0.3 0.31 0.32

Pos

ition

(V)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Fig. 10. Step response of the closed-loop system withthe designed force feedback control- with load. Thereference signal (red dashed line) and the closed-loopresponse with 800 g load on the stage (blue solid line).

-360 -315 -270 -225 -180 -135 -90 -45 0-60

-50

-40

-30

-20

-10

0

10

20

30

40

-60 dB

-40 dB

-20 dB

-12 dB

-6 dB

-3 dB

-1 dB

0 dB

0.25 dB

0.5 dB

1 dB

3 dB

6 dB

Open-Loop Phase (deg)

Ope

n-Lo

op G

ain

(dB

)

Fig. 11. Nichols Chart for the open loop system CpTxp:the nominal case with no load (blue with cross (×)marks) and the case of 800 g load (red with circle (o)marks).

Frequency (radians/sample)102 103 104 105

Gai

n (d

B)

-35

-30

-25

-20

-15

-10

-5

0

5

10

Fig. 12. The two sensitivities of the closed loop sys-

tem: S1 = | GfCf

1+GfCfCp+GfCf| (blue solid line),

S2 = | GpCfCp

1+GpCfCp+GfCf| (black dashed-dotted line)

and their absolute sum (dashed red line).

-300

-200

-100

0

100

200

300

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

posit

ion

(pm

)

Time (s)

Fig. 13. Noise performance of the closed loop systemwith force feedback. Data is obtained using a laserinterferometer.

-10

0

10

20

30

40

50

60

70

80

90

0 10000 20000 30000 40000 50000

|FFT

|

Frequency (Hz)

Fig. 14. Fast Fourier Transform (absolute value) of thenoise signal with force feedback control.

ACKNOWLEDGEMENT

The authors would like to thank Elektron TechnologyPLC and its Queensgate Engineering Department for theirvaluable support of this work.

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