robust digital control approach for high performance tunneling current measurement system

Control Engineering Practice 20 (2012) 643–653

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Control Engineering Practice

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Robust digital control approach for high performance tunneling currentmeasurement system

Irfan Ahmad a, Alina Voda a,n, Gildas Besanc-on a,b, Gabriel Buche a

a GIPSA-lab, Control Systems Department, Grenoble INP/UJF, BP 46, 38402 Saint-Martin d’H�eres, Franceb Institut Universitaire de France, France

a r t i c l e i n f o

Article history:

Received 17 November 2010

Accepted 29 February 2012Available online 1 April 2012

Keywords:

Tunneling current

Robust digital control

Nano-positioning

Piezoelectric actuator

Scanning tunneling microscope

Sensitivity function shaping

61/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.conengprac.2012.02.010

esponding author.

ail addresses: [email protected]

[email protected] (A. Voda),

[email protected] (G. Besanc-

[email protected] (G. Buche).

a b s t r a c t

This paper is devoted to the digital control system design for high performance measurement of

tunneling current. A common approach for such applications is to use a conventional Proportional

Integral (PI) control. In this paper, a robust digital design method is instead considered, based on

combined pole placement with sensitivity function shaping, and allowing for better performance

tuning in terms of precision, robustness and disturbance rejection. The resulting control scheme looks

like some enhanced PID controller, and is validated over an experimental setup, developed in GIPSA-lab

(Grenoble Image Parole Signal Automatique) research center. The corresponding simulation and experi-

mental results show improved performances with respect to those obtained with the more conven-

tional PI control technique.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Tunneling current is a quantum mechanical effect: an electronhas a non-zero probability of tunneling through a potential barrier(Landau & Lifshitz, 1977). This phenomenon of tunneling currentwas first observed in early 1980s by Gerd Binnig and HeinrichRohrer, and appears when an extremely sharp metallic electricallycharged tip is approached at the vicinity of the conductive samplesurface (distance between tip apex and sample surface in the rangeof 0:1�1� 10�9 m) (Chen, 2008). An important application of thistunneling current with the ability to scan the tip against the samplesurface was the invention of the scanning tunneling microscope(STM) (Binning & Rohrer, 1986). It was the first member of thefamily of Scanning Probe Microscopes (SPMs) that can characterizethe surface morphology with atomic resolution exploiting differentphysical phenomena, and in which the so-called Atomic ForceMicroscope (AFM) has received a particular attention (even fromthe very beginning Rugar & Hansma, 1990).

Tunneling current is also used to measure accelerations downto submicro-g (Liu et al., 1998; Liu & Kenny, 2001; Rockstad et al.,1996) and to sense sub-micrometer displacements (Blanvillain,Voda, Besancon, & Buche, 2009; Bocko, 1990; Ekinci, 2005). Since

ll rights reserved.

-inp.fr (I. Ahmad),

on),

the distance between tip apex and sample surface must be lessthan 1� 10�9 m to get the tunneling effect, ultrahigh positioningaccuracy together with high bandwidth are here major chal-lenges. This means that control has an important role to play insuch tunneling applications. The control scheme in most of themis mainly composed of a sensor for the tunneling current mea-surement and also a regulation feedback loop, having a piezo-electric actuator attached to the tip in order to move it preciselyin appropriate direction. Most of such controllers in commercialapplications (and more generally in commercial SPMs for similarcontrol problems) reduce to simple ones (mainly Proportional-Inte-gral, PI), where the parameters are fixed manually by the operator. Onthe other hand, a lot of advances have been provided in SPM controlproblems by the scientific community (more particularly in AFM)over the last two decades (see references below).

In the same spirit of control improvement, the present work isdevoted to the inspection of a robust design approach for the controlof tunneling phenomenon in the presence of a piezoelectric actuatorand a sensor for the tunneling current measurement. An experimentalsetup has been developed in that respect by the control group ofGIPSA-lab (Grenoble Image Parole Signal Automatique Lab), in order toanalyze the influence of different control techniques on tunnelingcurrent measurement. This setup works at ambient atmosphere andis based on STM principles, although the purpose is not necessarily totake images of the surface. Tunneling current is the only measure-ment in vertical z-direction of STM. During scanning, when theseparation between the sample surface and the tip decreases orincreases due to variations in the sample topography, the control

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I. Ahmad et al. / Control Engineering Practice 20 (2012) 643–653644

signal regulates the tunneling current by actuating the tip away fromor towards the sample surface.

Notice that in STM applications for instance, the use of simplecontrollers with manual tuning can result in imaging perfor-mances which are not satisfactory (Anguiano, Oliva, & Aguilar,1998) and this has already motivated dedicated studies: a feed-back loop of STM in vertical z-direction with some stabilityconditions has been presented in Oliva, Anguiano, Denisenko,Aguilar, and Pena (1995) but this analysis was limited to theclassical PI control technique, and based on a simplified version ofthe system model. A step variation in sample surface has beenstudied in Bonnail (2001) and Bonnail, Tonneau, Jandard,Capolino, and Dallaporta (2004) and a VSC (variable structurecontrol) design methodology in the presence of PI control hasbeen proposed in order to avoid the tip collision with the samplesurface. However, tunneling current being of order of nano-amperes, the presence of different sources of noise (Bordoni &Karim, 1994) (thermal noise, shot noise, 1/f noise, quantizationnoise, etc.) as well as sample surface variations highly influencethe precision of its measurement. In addition, nonlinearities andphysical limitations in the control loop are also limiting factors forthe performances, which still motivate further studies.

Notice also that similar problems and control studies aboutpiezoelectric actuation for nanopositioning can be found invarious other SPM applications, with purposes of performanceimprovement. Even though not explicitly handling tunnelingeffect, some of them can be recalled here for the sake ofcomparison: Salapaka, Sebastian, Cleveland, and Salapaka (2002)for instance have addressed a robust H1 control design approachfor the lateral motion of AFM tip in order to achieve both highbandwidth and high precision; Sebastian and Salapaka (2003,2005) have considered a two-dimensional large-range nanoposi-tioning system and analyzed the performance with GloverMcFarlane loop-shaping and robust H1 control design; Bhikkaji,Ratnam, Fleming, and Moheimani (2007), Bhikkaji, Ratnam, andMoheimani (2007) have highlighted the problem of lightlydamped low frequency resonant mode of piezoelectric actuator andproposed a solution using a positive velocity and position feedbackcontrol design methodology; Some inversion-based feedforwardcontrol approach in order to enhance the tracking bandwidth hasbeen analyzed in Aphale, Devasia, and Moheimani (2008); Thevertical direction control of AFM-scanner with classical PI has beenpresented in Schitter et al. (2007); High speed imaging of fragilesamples with AFM using some dynamic PID controller has beendiscussed in Kodera, Sakashita, and Ando (2006); In order to enhancethe imaging speed of AFM with precision, a model-based open-loopcontrol has first been analyzed (Schitter & Stemmer, 2004); and veryrecently, a model-based feedback controller with dual (instead ofsingle) actuation has been studied in Kuiper and Schitter (in press).A good survey on control issues for nanopositioning in terms ofhardware design and control methodologies is proposed in Devasia,

Fig. 1. Complete simulation model w

Eleftheriou, and Moheimani (2007), while an overview dedicated toAFM can be found in Abramovitch, Andersson, Pao, and Schitter(2007), and about issues in video-rate scanning speed for various SPMapplications in Schitter and Rost (2008) and Rost et al. (2009).

But the phenomenon of tunneling current (in vertical z-direction as in STM) has not really been analyzed with somemodern robust control design methodology in any of thosereferences.

The goal of the present work is thus to investigate a robustdesign approach in this context, choosing a method developedsince several years in GIPSA-lab (Landau & Karimi, 1998) andbased on combined pole placement with sensitivity functionshaping, to provide a control design for a better measurementof tunneling current. The desired performance and stabilityrequirements are expressed by means of constraints on the shapeof closed-loop sensitivity functions. To the authors’ knowledge,such a control design methodology has never been experimentedfor tunneling current measurement systems so far. The readerscan also refer to Voda (2010) and references therein for recentrelated works.

The working principle with complete description of the con-sidered experimental setup is given in Section 2. The systemmodeling for controller design is provided in Section 3. Section 4then presents the control problem formulation, desired perfor-mances, robust digital controller design and its performanceanalysis. In particular a comparison in simulation with resultsobtained via a classical PI controller is included, as well as adiscussion at the light of a full PID approach. Experimental resultsto validate the plant model and to analyze the performance ofcontrollers are presented in Section 5. Finally, Section 6 drawssome conclusions.

2. System description

2.1. Working principle

The complete closed-loop control scheme which will be hereconsidered is presented in Fig. 1. The working principle of an STMin vertical z-direction is based on the measurement and control ofthe tunneling current ðitÞ produced between the sharp metallic tipand a biased ðvbÞ sample surface when the distance (d) betweenthem is less than 1� 10�9 m. This tunneling current exponen-tially depends on the distance between tip and sample surfacewith the following nonlinear relation:

itðtÞ ¼ g � vb � e�k�dðtÞ ð1Þ

where g and k (depending on work functions F of the tipand sample surface) are constants. Controlling this tunnelingcurrent ðitÞ by keeping the distance (d) constant in the presenceof external disturbances (noise (n), surface variations ðzSÞ, etc.)

ith block diagram representation.

I. Ahmad et al. / Control Engineering Practice 20 (2012) 643–653 645

can be considered as the main objective of the feedback controlsystem in vertical z-direction.

A feedback loop constantly monitors the tunneling current ðitÞ.The tunneling current sensor is a current to voltage convertor(CVC) with high gain which converts the small tunneling currentinto a voltage ðv3Þ. This voltage is subject to measurement noise(n) resulting into the available voltage for feedback ðvyÞ. On theother hand, a piezoelectric actuator is attached to the tip to moveit in appropriate direction according to the applied voltage ðv2Þ inorder to keep the distance (d) – or in other words the tunnelingcurrent ðitÞ – constant at its desired value. The nonlinear hyster-esis phenomenon is expected to be negligible here for the piezo-electric actuator since the amplitude of input voltage ðv2Þ is verysmall for the vertical motion (z-direction) (Bonnail, Tonneau,Capolino, & Dallaporta, 2000). A voltage amplifier is used beforepiezoelectric actuator at the output of controller and the posi-tion effect of the piezoelectric actuator (z) directly defines thedistance ðdðtÞ ¼ d0þzSðtÞ�zðtÞÞ between tip and sample surface(where d0 is the initial distance between tip and sample surface,and zS is an external unknown disturbance, typically due tosurface variations).

2.2. Experimental setup

An experimental setup (Fig. 2) has been developed in GIPSA-lab,Grenoble (Blanvillain, 2010) to observe the tunneling current andto validate the proposed control scheme. This setup is based onthe STM working principle. Very sharp tips (platinum/iridium,work function F¼ 5:6 eV) are prepared by electrochemical reac-tions. The complete procedure for preparing tips is not in thescope of this paper but related relevant works can be found inLibioulle, Houbion, and Gilles (1995), Weinstein, Slutzky,Arenshtam, and Ben-Jacob (1995), Sorensen, Hvid, Mortensen,and Morch (1999), and Rogers, Shapter, Skinner, and Gascoigne(2000). The tip is fixed in a holder which is attached to thepiezoelectric actuator to move the tip in vertical z-direction. Thepiezoelectric actuator (Piezomechanics/PSt150) has a resonancefrequency of 120 kHz, a gain of 1.2 nm/V and an input voltage

Fig. 2. Experimental platform developed in GIPSA-lab.

range from �20 V to 130 V. The output of the controller is bet-ween 710 V and a voltage amplifier (Piezojena/ENV300) having again of 15 V/V and a bandwidth of 4 kHz provides appropriatesignals to the piezoelectric actuator. The change in distance (d)between tip and sample surface modifies the tunneling currentðitÞ and a tunneling current sensor (CVC) (home-made) of band-width 13 kHz and gain of 109 V=A gives the signal to the con-troller. The CVC is fixed close enough to the tip to minimize themeasurement noise (n). The sample surface (gold, work functionF¼ 5:4 eV) is placed over a small bench which can be movedlaterally very precisely in x- and y-directions with the help of twomicrometer screws having a travel range of 13 mm and sensitivityof 0:5 mm. Three other micrometer screws having the same travelrange of 13 mm and sensitivity of 0:6 mm are attached to thepiezoelectric actuator platform to move the tip manually invertical z-direction. These sensitivities of the screws are basedon a 11 rotation of the adjustment micrometer screws. Thesemicrometer screws and camera with telecentric zoom help theoperator to bring the tip manually close to the sample surface sothat the distance between them is in the range of few micro-meters. The subsequent tip approach mechanism is done with thehelp of piezoelectric actuator until the tunneling current ðitÞ isobtained. The whole experimental setup is placed over an anti-vibration table (Microworld). The control scheme is implementedin a computer (Development PC, processor 2.5 GHz) connectedwith another computer (Target PC, processor 3.2 GHz) throughEthernet. The Target PC has acquisition card (PCI DAS1602/16,8 differential inputs, 2 outputs, and 16 bits resolution) connectedto the experimental setup via Analog-Digital converters andincluding an anti-aliasing filter (with a bandwith of 10 kHz).

3. System modeling

The overall system consists of the voltage amplifier, the verticalz-piezoelectric actuator, the physical law which gives the relation-ship between tunneling current and the distance between tip andsample surface, and finally the current sensor (CVC).

The voltage amplifier has been modeled by linear first orderdynamics as

_x1ðtÞ ¼�ov � x1ðtÞþv1ðtÞ

v2ðtÞ ¼ Gvov � x1ðtÞ ð2Þ

where x1 is the state variable of the system, v1 and v2 are inputand output of the voltage amplifier respectively, ov is thebandwidth and Gv is the gain of the voltage amplifier.

One of the advantages of using piezoelectric actuators is thatunder some experimental conditions their dynamics can be wellapproximated by linear models (Bhikkaji et al., 2007), like thefollowing second order linear model that has been used in thispaper:

_x2ðtÞ ¼�2zoa � x2ðtÞ�o2a � x3ðtÞþv2ðtÞ

_x3ðtÞ ¼ x2ðtÞ

zðtÞ ¼ Gao2a � x3ðtÞ ð3Þ

where x2 and x3 are the state variables of the system, v2 and z areinput and output of the piezoelectric actuator respectively, z isthe damping, oa the bandwidth and Ga the gain of the piezo-electric actuator.

The physical law between tunneling current ðitÞ and distance (d)has been modeled by an exponential static nonlinearity as given byexpression (1).

The current sensor (CVC) has been modeled by linear first orderdynamics as follows:

_x4ðtÞ ¼�oc � x4ðtÞþ itðtÞ

Table 1System parameters with values used for simulation.

Symbols Description

vb Bias voltage (1.025 V)

g Constant (0.0011)

k Constant (1.65 A�1)

Gv Gain of the voltage amplifier (15 V/V)

ov Bandwidth of the voltage amplifier (4 kHz)

Ga Gain of the piezoelectric actuator (1.2 nm/V)

oa Bandwidth of the piezoelectric actuator (120 kHz)

z Damping of the piezoelectric actuator (0.9)

Gc Gain of the current sensor (109 V/A)

oc Bandwidth of the current sensor (13 kHz)

d0 Initial distance (1 nm)

ieq Equilibrium tunneling current (1 nA)


v3ðtÞ ¼ Gcoc � x4ðtÞ ð4Þ

where x4 is the state variable of the system, it and v3 are input andoutput of the current sensor respectively, oc is the bandwidth andGc the gain of the current sensor.

Now, the output of the overall system (see Fig. 1) can bewritten as

vyðtÞ ¼ v3ðtÞþnðtÞ ð5Þ

where vy is the output of the overall system and n is themeasurement noise. The detailed discussion about sources of thismeasurement noise can be found in Lagoute (2003).

To summarize, the overall global nonlinear system (order 4) isgiven by


_x2ðtÞ ¼ Gvov � x1ðtÞ�2zoa � x2ðtÞ�o2a � x3ðtÞ

_x3ðtÞ ¼ x2ðtÞ

_x4ðtÞ ¼�oc � x4ðtÞþgvb � e�kðd0þ zSðtÞ�Gao2

a �x3ðtÞÞ

vyðtÞ ¼ Gcoc � x4ðtÞþnðtÞ

8>>>>>><>>>>>>:

ð6Þ

Now, with the purpose of a linear control design, this globalnonlinear model (6) needs to be transformed into an appropriatelinear system.

A common approach for linearization by physicists in thisspecific situation is to use a logarithmic amplifier after the currentsensor in order to deal with the static exponential nonlinearity(Oliva et al., 1995). This linearization approach is usually based onthe following two assumptions:

�
Neglecting dynamics of the current sensor, considered asconstant; �
Fig. 3. Control design model G, controller K and feedback loop.

Neglecting the presence of noise (n) between exponential andlogarithmic nonlinearity;

and may bring problems related to this additional (nonlinear)device.

In the present paper instead, a first order linear approximationapproach is used to get rid of the exponential nonlinearity aroundan equilibrium point, which is a way to avoid the above men-tioned limitations (in the sense that the noise effect is kept as anadditive output one, and the current sensor dynamics can easilybe taken into account). The linearized equation corresponding tothe nonlinear Eq. (1) is

itðtÞ ¼ ieq�kieqðdðtÞ�deqÞ ð7Þ

where ieq is the equilibrium tunneling current and deq thecorresponding equilibrium distance (for a zero equilibrium dis-turbance zs).

With notations of model (6), the distance variation d�deq in (7)can easily be expressed in terms of variations in x3 as well as zs.

Hence the overall global system (order four) becomes


_x2ðtÞ ¼ Gvov � x1ðtÞ�2zoa � x2ðtÞ�o2a � x3ðtÞ

_x3ðtÞ ¼ x2ðtÞ

_x4ðtÞ ¼�c1Gao2a � x3ðtÞ�oc � x4ðtÞþc1 � zSðtÞ

vyðtÞ ¼ Gcoc � x4ðtÞþnðtÞ

8>>>>>><>>>>>>:

ð8Þ

where c1 ¼�k � ieq is a constant and variables now stand forvariations w.r.t. the equilibrium corresponding to ieq. This inducesan approximation error in the model only due to the abovelinearization, that is roughly of order ðd�deqÞ

2 which is prettynegligible, and can be handled by the robust control which will beproposed next anyway.

Complete details on system identification of the model consid-ered in this paper can be found in Blanvillain (2010). All parametervalues of (8) are given in Table 1.

4. Digital controller design

Before the synthesis of the digital controller, it is necessary todefine the control problem in this particular application and alsothe desired performance. For this purpose, the complete simula-tion model (Fig. 1) is transformed into an appropriate feedbackcontrol design model (Fig. 3) with linear time invariant discretetransfer functions where K represents the controller, Gf representsthe feedforward dynamics with the voltage amplifier, the piezo-electric actuator and the physical law between tunneling currentand distance, and Gb represents the feedback dynamics of tunnel-ing current sensor (CVC). The controlled output in this case istunneling current ðitÞ and it is required to find out the influence ofall external inputs (desired tunneling current, surface variationsand noise) over this controlled output.

With usual notations, the structure of a linear time invariantdiscrete time model (G) used for digital controller design is:

Gðz�1Þ ¼z�dl Bðz�1Þ

Aðz�1Þð9Þ

where dl¼ delay (in number of sampling periods), Bðz�1Þ ¼

b1z�1þ � � � þbnBz�nB , and Aðz�1Þ ¼ 1þa1z�1þ � � � þanA

z�nA .


The coefficients of the above polynomials (b1, . . . ,bnBand

a1, . . . ,anA) are found from the state-space model (8) and trans-

formation into digital transfer function. A unit sampling delayðdl ¼ 1Þ has been considered for the above model of the plant.An RS-type polynomial controller is proposed (according tocontrol scheme of Fig. 3) with controller polynomials Rðz�1Þ andSðz�1Þ as follows:

Rðz�1Þ ¼ r0þr1z�1þ � � � þrnRz�nR ð10Þ

Sðz�1Þ ¼ 1þs1z�1þ � � � þsnSz�nS ð11Þ

4.1. Control problem formulation

As already mentioned, the control ‘high performance’ objec-tives for the considered system of tunneling current are

�
High measurement (or equivalently current control) accuracy. � Good robustness and stability margins. � Large closed-loop bandwidth. � Noise attenuation at system input (STM application).
From Fig. 3, this can be formulated as a disturbance rejectionproblem: variations in the sample surface ðzSÞ as well as measure-ment noise (n) are indeed considered as external disturbances,where the first one can be considered as a slowly varyingdisturbance and the latter one can be considered as a fastlyvarying disturbance. For the control purpose, these disturbancesmust be rejected by moving the tip in appropriate direction withthe help of the piezoelectric actuator so that the tunneling currentðitÞ should always remain constant at its desired value. In addi-tion, as usual in control design, the control variable itself must notbe too much affected by measurement noises, and this is evenmore important for STM applications (since the control is alsoused for imaging purposes). All those control objectives will betackled altogether by the proposed design approach which isdescribed hereafter.

4.2. Desired performance

The stability and performance requirements can be expressedby means of constraints on the shape of closed-loop sensitivityfunctions (Landau & Zito, 2006). The sensitivity functions arerepresented on Fig. 3:

The transfer function between the disturbance zS and thecontrolled output it through constant c1 is given by the output

sensitivity function

Soðz�1Þ ¼

Aðz�1ÞSðz�1Þ

Aðz�1ÞSðz�1Þþz�dl Bðz�1ÞRðz�1Þð12Þ

The transfer function between the noise n and the systeminput v1 is given by the input sensitivity function

Kðz�1ÞSoðz�1Þ ¼

Aðz�1ÞRðz�1Þ


The transfer function between the filtered disturbance n

through noise transfer Gn and the controlled output it is givenby the complementary sensitivity function

Tðz�1Þ ¼Bðz�1ÞRðz�1Þ


For sufficient stability margins, the maximum of outputsensitivity function Soðz�1Þ should be less than 6 dB which willensure a good robustness margin as well. It can be called

Constraint 1 for the synthesis of the controller

JSoðz�1ÞJ1r6 dB, 8o ð15Þ

In the same way, the maximum of complementary sensitivityfunction Tðz�1Þ should be less than 3.5 dB, referred to as Con-

straint 2

JTðz�1ÞJ1r3:5 dB, 8o ð16Þ

To avoid instability due to saturation effects in the electronic part,the maximum of input sensitivity function Kðz�1ÞSoðz�1Þ should beless than 20 dB which will be called Constraint 3 for the synthesisof controller

JKðz�1ÞSoðz�1ÞJ1r20 dB, 8o ð17Þ

For measurement accuracy, a maximum of 710% variations isallowed in the desired tunneling current. This performancecriterion will define the maximum allowed error voltage ðveÞ, orthe lower limit (�20 dB) for sensitivity function Soðz�1Þ in themeasurement bandwidth ðoMÞ, where oM defines the maximumallowed variations 400 rad/s of sample surface ðzSÞ with a max-imum amplitude of 0.5 A. To limit the influence of surfacevariations ðzSÞ on the controlled output ðitÞ in order to achievethe desired measurement accuracy, the transfer function betweensample surface ðzSÞ and controlled output ðitÞ is considered whichis given by c1 � Soðz�1Þ as shown in Fig. 3, where c1 is a knownconstant. According to the desired accuracy in measurementbandwidth, this constraint (Constraint 4) can be formulated as

9c1 � Soðz�1Þ9r�20 dB, 0roroM ð18Þ

) 9Soðz�1Þ9r�27:2 dB, 0roroM ð19Þ

In addition, the transfer function between noise (n) and controlledoutput ðitÞ has to be considered and it is given by �Gnðz�1ÞTðz�1Þ

(see Fig. 3), where Gnðz�1Þ is a known transfer function. Accordingto the desired performance in terms of measurement accuracy,this constraint (Constraint 5) can be written as

9Gnðz�1ÞTðz�1Þ9o0 dB, o4oM ð20Þ

) 9Tðz�1Þ9o1

Gnðz�1Þ

��, o4oM ð21Þ

In order to limit the influence of noise ðnÞ on the system inputðv1Þ, the transfer function between them Kðz�1ÞSoðz�1Þ can beitself limited, for instance below 1% of the noise level, correspond-ing to a Constraint 6

9Kðz�1ÞSoðz�1Þ9o�40 dB, o4oM ð22Þ

The sampling frequency ðf SÞ chosen for all signals is 30 kHz andthe desired closed-loop bandwidth is of the same order as the oneof the voltage amplifier (4 kHz).

All these desired performances and constraints, summarized inTable 2, are used in coming sections for controller design and forperformance analysis of the closed-loop system.

4.3. Digital control design using pole placement with sensitivity

function shaping technique

A robust digital RS controller is designed using pole placementcombined with the shaping of sensitivity functions. Details aboutthis control design methodology can be found in Landau andKarimi (1998) and Landau and Zito (2006). It is adopted here sinceit takes into account simultaneously robustness and performancespecifications for the closed loop.

The shaping of sensitivity functions is done by appropriateselection of the desired closed loop poles and the introduction ofpre-specified filters in the controller. Those filters are introduced

Table 2Desired performances and corresponding constraints for the controller synthesis.

Desired performances Controller design constraints

Stability and robustness

DMZ0:5, DtZTS (sampling period) JSoJ1r6 dB, 8oDGZ2, Df4291 JTJ1r3:5 dB, 8oAvoid actuator saturations JKSoJ1r20 dB, 8o

Measurement accuracy710% variations of tunneling current 9So9r�27:2 dB, 0roroM

Noise attenuation at controlled output9T9o

1

Gn

��, o4oM

Closed-loop bandwidth Closed-loop dominant poles at 4 kHz

Noise attenuation at system input 9KSo9o�40 dB, o4oM

DM¼modulus margin, Dt¼ delay margin, DG¼ gain margin, Df¼ phase margin.


in Rðz�1Þ and Sðz�1Þ as follows:

Rðz�1Þ ¼HRðz�1ÞR0ðz�1Þ ð23Þ

Sðz�1Þ ¼HSðz�1ÞS0ðz�1Þ ð24Þ

where HRðz�1Þ and HSðz

�1Þ correspond to pre-specified fixedfilters. The desired closed loop poles are defined by a polynomialPðz�1Þ of the form

Pðz�1Þ ¼ PDðz�1Þ � PF ðz

�1Þ ð25Þ

where PDðz�1Þ and PF ðz

�1Þ correspond to the dominant andauxiliary closed loop poles respectively. The unknown polyno-mials R0ðz�1Þ and S0ðz�1Þ are obtained as solutions of the poly-nomial equation

Pðz�1Þ ¼ Aðz�1ÞSðz�1Þþz�dl Bðz�1ÞRðz�1Þ ð26Þ

with Pðz�1Þ as in (25), Rðz�1Þ and Sðz�1Þ as in (23) and (24).The controller polynomials Rðz�1Þ and Sðz�1Þ (more precisely

R0ðz�1Þ and S0ðz�1Þ) are the minimal degree solutions of (26).As the desired closed-loop bandwidth is of the same order as

the one of the voltage amplifier (4 kHz), a simple linear timeinvariant discrete time model is considered for synthesis of thecontroller, where the dynamics of the piezoelectric actuator(bandwidth 120 kHz) and the tunneling current sensor (CVC)(bandwidth 13 kHz) are taken as constant gains (coefficients ofthe system model Gðz�1Þ (8) used for the synthesis of thecontroller: b1¼233.8, a1 ¼�0:433 and dl¼1). However, theobtained digital controller is validated in simulation (Section 4.5)with the complete continuous non-linear time invariant model asgiven in (6). It is further validated with experimental results aspresented in Section 5.

The designed controller characteristics are:Closed-loop dominant poles PD: These dominant poles are

placed at 4 kHz with damping coefficient of 0.9 so that theclosed-loop natural frequency remains almost the same as theopen-loop one.

Closed-loop auxiliary poles PF: A single high frequency real poleis added at �0.6 in order to improve the robustness of thefeedback loop.

Controller fixed part HS: An integrator is used for HS fixed partof the controller.

Controller fixed part HR: A real zero has been introduced at 0:5f S

in order to shape the input sensitivity function and to open theloop at Shannon frequency.

The obtained controller polynomials after solving (26) are

Rðz�1Þ ¼ 2� 10�3þð1:17� 10�3

Þz�1�ð0:83� 10�3Þz�2

Sðz�1Þ ¼ 1þð153:75� 10�3Þz�1�ð706:8� 10�3

Þz�2�ð446:94� 10�3Þz�3

(

ð27Þ

The closed loop sensitivity functions are plotted in the nextsection with real-time experimental data for complete analysis.

4.4. Digital PI control design

For the sake of comparison, a digital PI controller is designedby classical pole placement technique, keeping in mind the samerobustness and stability margins as in the case of RS controldesigned by pole placement with sensitivity function shapingmethod. The resulting PI controller polynomials are

Rðz�1Þ ¼ 0:21� 10�3þð0:21� 10�3

Þz�1

Sðz�1Þ ¼ 1�z�1

(ð28Þ

The performance comparison between the two designed con-trollers (27) and (28) is done firstly in simulation and then on theexperimental setup.

4.5. Simulation results

After the control design, the controller performance is vali-dated with a simulation model, having actual non-linearities(exponential, saturations), measurement noise (n) and physicallimitations in closed-loop, aiming at representing a real system asclose as possible. The validation of controller with such a simula-tion model is an important step before experimental validation.

Fig. 4 shows the simulation result with the classical PI controllerand with the proposed RS controller in the presence of surfacevariations ðzSÞ (first graph) with a frequency of 100 rad/s and anamplitude of 0.5 A. The two horizontal dotted lines in the secondand third graphs represent the acceptable bounds of 710% varia-tions in tunneling current ðitÞ. The desired tunneling current value is0.25 nA. It can be observed that the tunneling current variationremains within the desired limits with both designed controllers.

Now, if a simulation is performed with a slightly higherfrequency of 400 rad/s of surface variations ðzSÞwith an amplitudeof 0.5 A, as presented in Fig. 5 (first graph), it can be observed thatthe variation in tunneling current ðitÞ still remains within accep-table bounds with the proposed RS controller (third graph) but itbecomes unacceptable with classical PI controller (second graph).These results show the possible higher speed of STM operationwith the proposed RS controller than with the conventional PIcontroller. All these simulations are performed in the presence ofsensor noise (n) of 10 mV=

ffiffiffiffiffiffiffiHzp

.Finally, the performances of controllers are analyzed with

random surface variations ðzSÞ (first graph in Fig. 6). Again,variations in tunneling current ðitÞ can be observed to be of lowermagnitude with the proposed RS controller (third graph) thanwith the classical PI controller (second graph). The reason can beinvestigated by comparing the closed loop sensitivity functions asthey carry much information about the disturbance rejection.These closed loop sensitivity functions are identified with real-time experimental data in next section and those simulationresults are confirmed with experimental ones.

Remark 1. Notice that the PI control case is chosen as a referencebecause it corresponds to the most commonly available controllerin commercial SPMs. In some cases, the classical PID – furtherincluding a derivative action – is also provided, which should beappropriately tuned, and it can be noticed that the RS controllerresulting from the tuning procedure presented here above finallymeets some PID-like structure, further including one additionalfilter. The presented robust methodology then appears as someenhanced-PID-like tuning procedure, but with the additional filterobviously providing more control degrees than a classical PID. Forinstance, when simply focusing on the measurement accuracyrequirement for the considered closed-loop bandwidth, it appears

Fig. 5. Simulation results: comparison between PI and RS controller in the

presence of sinusoidal surface variations ðzSÞ of frequency of 400 rad/s, an

amplitude of 0.5 A and measurement noise (n) of 10 mV=ffiffiffiffiffiffiHzp

.

Fig. 6. Simulation results: comparison between PI and RS controller in the

presence of random surface variations ðzSÞ.Fig. 4. Simulation results: comparison between PI and RS controller in the

presence of sinusoidal surface variations ðzSÞ of frequency of 100 rad/s, an

amplitude of 0.5 A and measurement noise (n) of 10 mV=ffiffiffiffiffiffiHzp

.


that a standard PID can only meet either accuracy or bandwith,but not both of them simultaneously.

5. Experimental validation

After simulations, the control scheme previously highlighted isvalidated on the experimental platform (Fig. 2) in order to

observe the real-time variations of tunneling current ðitÞ. Themajor difficulty is the repeatability of good operating and envir-onmental conditions to obtain the tunneling current. Once thetunneling current is obtained then the feedback control system istested. Since the platform works at ambient atmosphere, addi-tional external environmental disturbances can be presentbeyond the disturbances considered for simulations (measure-ment noise (n) and sample surface variations ðzSÞ). The controllermust indeed be robust and have a capability to reject the externaldisturbances in order to achieve the desired performances. More-over, experimental identification of the system model and of theobtained closed-loop sensitivity function is performed to validatethe designed feedback control loop.

5.1. Identification of plant model

The open loop system (plant) between the output of thecontroller ðv1Þ and the measured voltage ðvyÞ is identified by thetechniques of identification in closed loop. The objective of theidentification in closed loop is to find the best plant model whichminimizes the prediction error between the measured output ofthe true closed loop system and the predicted closed loop output.All details with procedures and algorithms about identification inclosed loop can be found in Landau and Zito (2006) (the inter-ested readers can also look at Langer & Landau, 1996 for anexample of application to a flexible structure).

An external excitation signal of PRBS type (Pseudo-RandomBinary Sequence) is superposed to the reference. The character-istics of this PRBS signal are chosen as: amplitude¼70.1 V,number of registers¼10, sampling frequency¼30 kHz and fre-quency divider¼2. The identification method CLOE (Closed LoopOutput Error) is used to identify the model of the system (plant).The theoretical gain of the open loop system (plant) which includesvoltage amplifier, piezoelectric actuator, the physical tunnelingcurrent phenomenon and the current sensor (CVC), is 309 V/V andthe identified gain of the open loop system is 316 V/V. Theidentified bandwidth of the open loop system is also very similarto the theoretical one ð � 4 kHzÞ. Fig. 7 shows the Bode plot of the


identified model, the complete linearized simulation model(Eq. (8)) and the design model (used for the synthesis of thecontroller) of the open loop system (plant). The small differencebetween the identified model and the design one will be handledby the robustness of the designed controller.

5.2. Identification of closed loop sensitivity functions

The capability of the control system to reject all externaldisturbances can be analyzed by closed loop sensitivity functionsas mentioned in Section 4.2. Obviously the RS controller havingbeen designed with appropriate sensitivity shaping, the expectedshapes are indeed obtained with the design model, and for thisreason are not included here. In order to get a closer view of someactual robustness w.r.t. unmodeled disturbances instead, theclosed loop sensitivity functions are experimentally identified asthis was done with the model itself. To that end, the externalexcitation PRBS signal is again superposed to the reference withthe same characteristics as mentioned above. Based on the experi-mental data (the reference voltage ðvref Þ, error voltage ðveÞ, theoutput of controller ðv1Þ and the measured voltage ðvyÞ) in thepresence of the proposed RS controller designed by pole placementwith sensitivity function shaping and the conventional PI control-ler, the identification method ELS (Extended Least Squares) is usedto identify the closed loop sensitivity functions.

Fig. 8 shows the closed loop output sensitivity functionðSoðz�1ÞÞ with both controllers. This sensitivity function showsthe relationship between the disturbance of surface variations ðzSÞ

and the tunneling current ðitÞ as mentioned in Section 4.2 (seeFig. 3). A much stronger attenuation of this disturbance at lowfrequencies can be observed with the proposed RS controllerdesigned by pole placement with sensitivity function shapingthan with the conventional PI controller, which was evident withthe simulation results as well. It can be noticed that the proposedRS controller can attenuate more any external environmentaldisturbance than the conventional PI controller in order toachieve less fluctuations of tunneling current ðitÞ.

Fig. 9 shows the closed loop complementary sensitivity functionðTðz�1ÞÞ with both controllers, corresponding to the relationshipbetween the measurement noise (n) and the tunneling current ðitÞas mentioned in Section 4.2 (see Fig. 3). Measurement noise (n) isconsidered as a high frequency disturbance and it can be observed

Fig. 7. Bode diagram of plant ðGÞ model.

that both controllers attenuate well this disturbance, while the RScontroller even provides a larger closed loop bandwidth.

Fig. 10 finally shows the closed loop input sensitivity functionðKSoðz�1ÞÞ with both controllers. From the discussion of Section4.2 (see Fig. 3), it appears that the constraints to avoid actuatorsaturations and to limit the influence of noise (n) on the systeminput ðv1Þ are fully met with both the conventional PI controllerand the proposed RS controller.

5.3. Steady state current control

After arriving in the tunneling region (distance (d) between tipapex and sample surface lower than 1� 10�9 m), differentdesired values of tunneling current ðitÞ are given and the resultingvariations in the measured tunneling current are examined.

Fig. 11 shows the measured tunneling current ðitÞ with theproposed RS controller designed by pole placement with sensi-tivity function shaping and with the conventional PI controller.The desired tunneling current value is here set to 0.25 nA. It canbe observed that tunneling current ðitÞ variations are indeed lowerwith the proposed RS controller than with the conventional PI

Fig. 8. Experimentally identified closed loop output sensitivity function ðSoÞ.

Fig. 9. Experimentally identified closed loop complementary sensitivity function ðTÞ.


controller. The standard deviation of the measured tunnelingcurrent is 10.8 pA with the proposed RS controller and 24.3 pAwith the PI controller. The tunneling current variations with theproposed RS controller remain within more or less 710% varia-tions of desired value which is in good accordance with simula-tion results.

Next, the tunneling current ðitÞ variations are examined with aslightly larger desired value which requires to move the tipprecisely closer to the sample surface. Fig. 12 shows the behaviorof tunneling current ðitÞ with the proposed RS controller and withthe conventional PI controller when the desired value of tunnel-ing current is set to 0.75 nA. It can be observed again that thetunneling current ðitÞ variations are lower with the proposed RScontroller than with the conventional PI controller. The standarddeviation of the measured tunneling current is 22.7 pA with theproposed RS controller and 46.2 pA with the PI controller. It can

Fig. 10. Experimentally identified closed loop input sensitivity function ðKSoÞ.

Fig. 11. Comparison of experimental results with reference tunneling current of

0.25 nA.

Fig. 12. Comparison of experimental results with reference tunneling current of

0.75 nA.

also be observed that with both controllers, the tunneling currentðitÞ variations are a little larger than in the previous case. This canbe explained by stronger mechanical noise.

5.4. Current reference tracking

Tunneling current ðitÞ behavior can also be observed in thepresence of step variations of the desired tunneling current.Figs. 13 and 14 show the tunneling current ðitÞ variations withthe proposed RS controller and with the conventional PI con-troller respectively. It can be noticed very clearly that at each stepchange of the desired tunneling current, the tunneling current ðitÞvariations with the proposed RS controller is lower than the oneswith the conventional PI controller.

Finally, the power spectral densities of the measured tunnelingcurrent ðitÞ are analyzed (Fig. 15) with the proposed RS controllerand with the conventional PI controller. The much strongerattenuation of disturbances with the proposed RS controller thanwith the conventional PI controller is indeed confirmed, particu-larly at low frequencies. It can be noticed that some peaks appearat some frequencies with both controllers, which will requirefurther analysis in future developments.

6. Conclusion and perspectives

In this paper, a system of tunneling current measurement hasbeen briefly presented, a corresponding dynamic modeling hasbeen proposed and a related control problem with desiredmeasurement performance has been formulated. Then, measure-ment requirements have been translated into control require-ments and a modern robust digital RS controller has beendesigned by combined pole placement with sensitivity functionshaping method. A comparison has been performed with themore conventional PI control design methodology. Results withboth control techniques have been experimentally validated.Simulations as well as experimental results have finally con-firmed better performances in terms of precision and disturbance

Fig. 14. Experimental result with PI control having step variations of desired

tunneling current.

Fig. 15. Comparison of power spectral densities of tunneling current with

proposed RS control designed by pole placement with sensitivity function shaping

and with conventional PI control.

Fig. 13. Experimental result with RS control designed by pole placement with

sensitivity function shaping having step variations of desired tunneling current.


rejection with the proposed controller for the system of tunnelingcurrent measurement.

It can be noticed that on this basis, the performance of theproposed controller can be enhanced, by introducing some notchfilters in the controller in order to take care of the peaks observedin the spectral analysis of the measured tunneling current forinstance. For future work as well, it will be interesting to improveelectronics on the one hand (so that all bandwidths becomecomparable to that of the currently used piezo-actuator), andanalyze the performance of tunneling current measurementsystem when the sample surface is oscillating with high frequen-cies on the other hand.

Acknowledgement

The authors would like to sincerely thank Dr. Sylvain Blanvillainof GIPSA-lab, Grenoble for his helpful discussions and an activeparticipation for the development of experimental setup used forthe validation of proposed control scheme.

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