Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
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H∞ Controller Design for High-Performance Scanning Tunneling Microscope Irfan Ahmad, Alina Voda and Gildas Besançon Abstract— Ultrahigh positioning accuracy with high bandwidths are the great challenges in the field of nanopositioning and scanning systems. This article talks about the controller design for a closed-loop scanning tunneling microscope to deal with such challenges, considering fast continuous variations in sample surface with the presence of noise in the loop. The desired performances in vertical Z-direction of the scanner are imposed on the closed-loop sensitivity functions using appropriate weighting functions and then a mixed-sensitivity H∞ controller is designed. The results are compared with the conventional proportionalintegral control design commonly used by the scanning probe community, underlining the improvements obtained in terms of high precision with high bandwidth. Keywords : Scanning tunneling microscope, Tunneling, Nano-technology, Precision positioning, H∞ control, Closedloop sensitivity functions, Simulation.
I. INTRODUCTION In the beginning of the 1980s, Gerd Binnig and Heinrich Rohrer experienced the phenomenon of tunnel current between a metallic electrically charged tip and a conductive sample surface when the tip is approached at the vicinity of the surface (distance between tip apex and sample surface in the range of 0.1 − 1 × 10−9 m). This phenomenon combined with the ability to scan the tip against the sample surface gave birth to Scanning Tunneling Microscope (STM) [1]. It was the first member of the family of scanning probe microscopes (SPM) that can characterize surface morphology with atomic resolution. These days, the STM has vast applications in different domains and the high positioning accuracy with high bandwidth are some main requirements in certain domains. The electronic control of STM is mainly composed of a sensor for the tunnel current measurement and also a regulation feedback loop, having piezoelectric actuator attached with STM tip to move it in appropriate direction. The piezoelectric actuators are now widely used for linear motion with high positioning accuracy at nanometer and subnanometer resolution with high bandwidths [2]. Presently, in most commercial equipments of STM, only simple types of controllers (proportional-integral (PI) or proportionalintegral with derivative (PID) control) are implemented to control the movement of STM tip in vertical Z-direction where parameters of such controllers are fixed manually by the operator. In such operation modes, the imaging process can not be optimum and the image does not correspond The authors are with GIPSA-lab, Control System Department, ENSE3, BP 46, 38402 Saint Martin d’Hères, France. (email: irfan.ahmad, alina.voda,
[email protected])
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
necessarily to the reality [3]. Since the distance between STM tip apex and surface is less than 1 × 10−9 m to get the tunneling effect, the electronic control is very critical in order to get a good image quality of the surface, in the presence of external disturbances. The feedback loop of STM, to control the movement of STM tip in vertical Z-direction, with some stability conditions has been presented in [4], [5]. All such analysis has done with simple classical PI (PID) control technique with a simplified version of the system model. There is no discussion about noise in that work as well. A step variation in sample surface is studied in [6], [7] and a variable structure control (VSC) design methodology in the presence of PI control is proposed in order to avoid STM tip collision with sample surface, but there was still a need of STM performance analysis and improvements in terms of positioning accuracy with high bandwidth and its tradeoff with loop stability and robustness, in view of using it with fast variations (continuous) in sample surface. A control design methodology based on pole placement with sensitivity function shaping using second order digital notch filter is proposed in [8] for the feedback control system of STM and the general description of this design methodology is given in [9], [10]. The desired performances are expressed in terms of templates on the shape of closed-loop sensitivity functions but the proper tuning of the control parameters in order to follow these templates can be a difficult task for STM operators. The control of Atomic Force Microscope (AFM) for vertical Z-direction is discussed in [11], [12] with classical PI (PID) control technique. A robust H∞ design methodology has been studied in [13] for the case of AFM control which focused on reference tracking in lateral direction without detailed analysis on the weighting function design. The presence of noise, non-linearities and physical limitations in the control loop are always the limiting factors to be considered in order to get desired performances. The goal of the present work is thus to propose an H∞ control design for the vertical Z-movement of STM tip in order to improve performances in terms of positioning accuracy and closed-loop bandwidth, inspite of limitations mentioned above. These desired performances and robustness requirements are imposed on the closed-loop sensitivity functions using appropriate weighting functions which become the part of the generalized plant and then the mixed-sensitivity H∞ control design methodology is adopted to fulfill the requirements. A complete system overview with corresponding simulation model is given in Section II. The control design model is established in Section III. Section IV then presents
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Fig. 1.
Complete simulation model for STM
TABLE I S YSTEM PARAMETERS WITH VALUES USED FOR SIMULATION
the control problem formulation with desired performances, mixed-sensitivity H∞ controller synthesis with the designing of weighting functions and the obtained performances. Simulation results to validate the controller and also its comparison with conventional PI control technique are presented in Section V. Finally, Section VI draws some conclusions.
Symbols σ0 vb Φ KL EL R Q γ0 ω0 z0 d0 v30 ω1 ω2 ω3
II. SYSTEM OVERVIEW Scanning tunneling microscope (STM) operates by bringing a small metallic electrically charged tip (probe) close enough to the conducting surface of a sample so that a small tunnel current (it ) is produced between STM tip and sample surface. This tunnel current depends on the distance (d) between STM tip and sample surface with non-linear relation (1) : √ it = σ0 · vb · e−1.025 Φ·d (1) where σ0 , vb and Φ are constants which are defined in Table I (values taken from [5]). Controlling this tunnel current (it ) by keeping the distance (d) constant in the presence of external disturbances (sensor noise (n), surface variations (zS ) etc.) is the main objective of the feedback control system of STM. A complete overview of the closed-loop control scheme which will be here considered is presented in Fig. 1. The tunnel current occurring in STM is very small, typically from 0.01 × 10−9 to 10 × 10−9 A. The current amplifier (preamplifier or I-V converter) is thus an essential element of STM, which converts the small tunneling current (it ) into a voltage (v3 ). This pre-amplifier has a finite bandwidth of 600 kHz and it is usually the most important source of noise (n). We have seen in (1) the nonlinear relation between the tunneling characteristics and the distance. To make the entire electronic response linear (approximately) with respect to distance (d), a logarithmic amplifier is attached to the output of pre-amplifier. The output of logarithmic amplifier (vy ) is given by the nonlinear relation (2) : |v3 | vy = KL · log10 (2) EL
Description Proportionality constant (0.5 Ω−1 ) Bias voltage (0.1 V ) Work function (4 eV ) Conversion factor of log amplifier (2.5 V ) Sensitivity of log amplifier (0.001 V ) Resistance for I-V conversion (1 × 109 Ω) Quality factor of piezo (4.5) Sensitivity of piezo(40 A0V −1 ) Resonance frequency for piezo (40 kHz) Initial position of STM tip (12 A0 ) Equilibrium point for exponential non-linearity (8 A0 ) Equilibrium point for logarithmic non-linearity (3.77 V ) Bandwidth of pre-amplifier (600 kHz) Bandwidth of logarithmic amplifier (60 kHz) Bandwidth of amplifier before piezo (100 kHz)
where KL and EL are defined in Table I. This logarithmic amplifier has a finite bandwidth of 60 kHz and its output can be between 0 − 10 V . These all tasks are performed by measurement electronics in the feedback loop (Fig. 1). The output voltage (vy ) of logarithmic amplifier is now given to control electronics which sends a required voltage (v1 ) to an amplifier (gain = 19) and then to Z-piezoelectric actuator. This voltage (v1 ) can be between ±10 V . The tip of STM which is connected with piezoelectric actuator will start moving towards the required position according to the applied voltage (v2 ) in order to keep the distance (d) constant (0.8 × 10−9 m) in the presence of external disturbances (variation in sample surface (zS ), noise (n)). One of the advantages of using piezoelectric actuators is that under certain experimental conditions their dynamics can be well approximated by linear models [14], that’s why a second order linear model is used for piezoelectric actuator as given in relation (3) : γ0 (3) Ga (s) = 1 1 2+ s s + 1 2 Qω ω 0
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0
ThCIn2.16 where γ0 , ω0 and Q are defined in Table I. The nonlinear phenomenon like hysteresis is not expected for the piezoelectric actuator as the amplitude of input voltage (v2 ) is very small for the vertical movement (Z-direction) of STM tip. The output of piezoelectric actuator (z) is used to find out the distance (d) between STM tip and sample surface (zS ) from d = z0 − z − zS (Fig. 1) where z0 is the equilibrium position of STM tip when no voltage is applied to piezo. III. CONTROL DESIGN MODEL In this section, the complete simulation model (Fig. 1). is transformed into an appropriate linear control design model which is required before designing the controller for feedback control system of STM. The final simulation with actual non-linearities in the presence of sensor noise (n) in the closed-loop will validate the controller and will help us to observe the results close to the real system of STM. In all previous analysis on controller design for STM, the effect of two non-linearities (exponential with relation (1) and logarithmic with relation (2)) are linearized for control design model by coupling them directly through I-V converter (1×109 Ω), neglecting the dynamics of the two amplifiers (pre-amplifier and logarithmic amplifier) and the presence of noise (n) between these two non-linearities. Instead, here we have chosen to rely on a computed first order linear approximation of the overall equations to linearize both nonlinearities (1) and (2) independently around their equilibrium points (d0 and v30 ) without neglecting the dynamics of the two amplifiers and the presence of noise (n). The linearized equations corresponding to (1) and (2) are respectively : it = c1 + c3 − c2 · d
(4)
vy = c4 − c6 + c5 · v3
(5)
where all c1 , c2 , . . . , c6 are constants which depends on the parameters of (1) and (2) and defined by the following relation (6) : √
c1 c2 c3 c4 c5 c6
= σ0 · vb√ · e−1.025· Φ·d0 = 1.025 Φ · c1 = d0 · c2 = KL · log10 = =
V30 EL
(6)
KL V30 ·ln(10) KL ln(10)
here σ0 , vb , Φ, d0 , KL , EL and v30 are STM design parameters and defined in Table I. After linearization, the equivalent control design model is given in Fig. 2 where G(s) represents the 3rd order model including STM and piezoelectric actuator, H(s) represents the 2nd order model (7) for the feedback dynamics where tunnel current (it ) is converted in to an approximately linear voltage (vy ), and Gn represents the noise (n) transfer. c · ω1 · ω2 H(s) = 2 (7) s + (ω1 + ω2 ) s + ω1 · ω2 where c is a constant term depending on parameters of (4), (5) and ω1 , ω2 are two bandwidths defined in Table I. Three
Fig. 2.
Design model for closed-loop system of STM
weighting functions W1 (s), W2 (s) and W3 (s) are designed for H∞ controller analysis which is explained in detail in next section. IV. MIXED-SENSITIVITY H∞ CONTROLLER DESIGN In this section, we will design a mixed-sensitivity H∞ control in order to achieve the desired performances of the STM. Before going to the synthesis of the controller, first we need to explain well the control problem in case of STM feedback loop and also the performance which we want to achieve. A. Control Problem Formulation and Desired Performance For a feedback control of the STM, the control problem can be formulated as a tracking problem where the STM tip tracks the unknown sample surface (zS ) by keeping the distance (d) constant between STM tip and sample surface. It can also be formulated as a disturbance rejection problem. The variations in the sample surface (zS ) and also noise (n) are considered as external disturbances where the first one can be considered as a slow varying disturbance and the latter one can be considered as a fast varying disturbance. These disturbances are rejected by moving the STM tip in appropriate direction so that the distance (d) should always remain constant at its desired value (0.8 × 10−9 m). Our main objective is to achieve better performance of STM in terms of high positioning accuracy ±8 × 10−12 m with high closed-loop bandwidth, in the presence of good robustness margin (kSk∞ ≤ 6 dB and kT k∞ ≤ 3.5 dB where S and T are sensitivity function and complementary sensitivity function respectively) and stability margins (gain margin > 6dB and phase margin > 30˚). Such positioning accuracy is required with the maximum continuous variations of sample surface 1 × 104 rad/sec having amplitude 8 × 10−10 m in √ the presence of sensor (pre-amplifier) noise (n) of 45 mV / Hz. The reference input voltage (vre f ) corresponds to the desired distance (d) between STM tip and sample surface, hence the feedback voltage (vy ) always try to follow the reference input voltage (vre f ) in the presence of above mentioned disturbances to keep the distance (d) constant at desired value (0.8 × 10−9 m).
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ThCIn2.16 B. Controller Synthesis The desired performances are imposed on the closed-loop sensitivity functions using appropriate weighting functions and then the mixed-sensitivity H∞ control design methodology is adopted to fulfill the requirements. The closedloop sensitivity functions are classically given by following relation (8) : S(s) = 1/(1 + K(s) · G(s) · H(s)) KS(s) = K(s) · S(s) T (s) = K(s) · G(s) · H(s)/(1 + K(s) · G(s) · H(s))
Fig. 3.
Generalized plant with controller
(8)
(3) : W3 is designed to impose limitations on complementary sensitivity function (T ) and it is given as [15] :
The functions W1 , W2 and W3 weight the controlled outputs y1 , y2 and y3 respectively (Fig. 2) and should be chosen according to the performance specifications. The generalized plant P (Fig. 3) (i.e. the interconnection of the plant and the weighting functions) is given by :
s + (ωt /Mt ) (12) εt · s + ωt where Mt = 1.5 to have a good robustness margin (i.e. kT k∞ ≤ 3.5 dB) for all frequency range, ωt = 1 × 107 rad/sec to attenuate the noise (n) at high frequencies with εt = 1. After computation, the minimal cost achieved for STM feedback control system was γ = 1.42 which means that the obtained sensitivity functions match nearly the desired loop shaping. The obtained sensitivity functions with the desired loop shaping in terms of weighting filters are shown in Fig. 4-6.
y1 W1 y2 0 = y3 0 VE I |
W1 H 0 −W3 H
−W1 HGn 0 0 −HGn {z P
VREF −W1 HG W2 ZS W3 G n V1 −HG }
Thus, the H∞ control problem is to find a stabilizing controller K(s) which minimizes γ [15] such that :
W1 S W1 HS −W1 HGn S
W2 KS W2 HKS W2 HGn KS < γ (9)
W3 GKS −W3 S
−W3 Gn T ∞
The obtained controller K(s) has the same number of state variables as P. So, the choice of the weighting functions is an important issue in the H∞ control problem. We have chosen the weighting functions as follows : (1) : W1 is used to impose the desired performance specifications on closed-loop sensitivity function S [15], that is : (1/Ms ) s + ωs W1 (s) = (10) s + ωs · εs
W3 (s) =
C. Control Loop Performance Analysis The weighting functions were designed considering the requirement of high positioning accuracy ±8 × 10−12 m with high bandwidth in the presence of good robustness so we can say that the proposed control technique will achieve all requirements as the obtained sensitivity functions fairly match the desired loop shaping (Fig. 4-6). For robustness point of view, we obtained good modulus margin as kSk∞ = 2.4 dB and kT k∞ = 0.08 dB and good stability margins (gain margin = 14.7 dB and phase margin = 66.1˚). The obtained closed-loop bandwidth is 6.1 × 105 rad/sec which will ensure the required good performance with fast variations (1×104 rad/sec) in sample surface (zS ). Similarly, all other constraints in terms of better noise (n) rejection and to avoid actuator saturations are fully met with proposed control technique. For the comparison purpose, a standard PI
where Ms = 2.0 to have a good robustness margin (i.e. kSk∞ ≤ 6 dB) for all frequency range, ωs = 4 × 105 rad/sec to have a good attenuation of disturbances from low frequency up to ωs and εs = 0.012 to reduce the steady-state error in the presence of maximum allowed variations in the sample surface (8 × 10−10 m). (2) : W2 is designed to respect the actuator limitations for STM. It is chosen as follows [15] : W2 (s) =
s + (ωu /Mu ) εu · s + ωu
(11)
where Mu = 3.2 to impose limitation on the maximum value of controller output up to the frequency ωu which is chosen ωu = 1 × 107 rad/sec and εu = 1 to limit the effect of measurement noise (n) at high frequencies. Fig. 4.
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Closed-loop sensitivity function (S) with H∞ control
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Fig. 5.
Fig. 7.
Closed-loop sensitivity function (KS) with H∞ control
controller is designed which can be represented by following relation (13) : 1 (13) K(s) = KP 1 + sTi where KP and Ti represents the proportional and integral action respectively. These two parameters are tuned according to standard pole placement technique. With this classical PI control technique, the obtained closed-loop bandwidth is (4.1 × 104 rad/sec) which is much less than the bandwidth obtained with H∞ control technique. The gain margin obtained with PI control technique is 5.5 dB which will not allow us to increase more the controller gains or to increase more the closed-loop bandwidth. This means that fast variations in sample surface (zS ) with the positioning accuracy of ±8 × 10−12 m is not possible with classical PI control technique (Fig. 7) which is evident in our simulation results as well. V. SIMULATION RESULTS After the control design and closed-loop analysis, we can now validate its performance with a simulation model aiming at representing a real system as much as possible. For this purpose, in this section, we will validate our controller with
Fig. 6.
Closed-loop sensitivity function (T) with H∞ control
Closed-loop sensitivity function (S) comparison
the simulation model (Fig. 1) having actual nonlinearities, presence of noise (n) and physical limitations and then will compare it with other control techniques. Fig. 8 shows the first simulation result with proposed H∞ control technique having surface variations ZS = 1 × −10 m with sensor (pre104 rad/sec, amplitude 8 × 10√ amplifier) noise (n) of 45 mV / Hz. The dotted lines represents the positioning accuracy (acceptable bounds) of ±8 × 10−12 m. We can observe that the movement of STM tip remains within the desired limits. Now, Fig. 9 compares the results of proposed H∞ and classical PI control technique where surface variations (zS ) has amplitude 4 × 10−10 m with the same frequency and sensor noise (n) as in case of Fig. 8. The result shows that with PI control technique, the variations in distance (d) go outside the acceptable bounds which means that desired positioning accuracy is not possible with PI control technique. To see the rapidity of these controllers, a step variation of sample surface (4 × 10−10 m) is simulated (Fig. 10) where we can see that with H∞ control, STM tip comes back to desired position to keep the distance (d) constant much faster than classical PI control technique. Finally, we compare the result of H∞ control with RS control,
Fig. 8. Simulation result with H∞ control having surface variations √ ZS = 1 × 104 rad/sec, amplitude 8 × 10−10 m with sensor noise 45 mV / Hz
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Fig. 9. Comparison result between H∞ and PI control having surface 4 −10 m with sensor noise variations √ ZS = 1 × 10 rad/sec, amplitude 4 × 10 45 mV / Hz
which was proposed in [8]. We can see that H∞ control rejects noise (n) better than RS control as shown in Fig. 11. VI. CONCLUSIONS In this article, we have designed a mixed-sensitivity H∞ control technique for feedback loop of STM and analyzed its performance in terms of high positioning accuracy with high bandwidth with the requirements of good robustness as well and finally compare its results with commercially used classical PI control technique for STM. This shows how STM performances can indeed be significantly improved by appropriate control design, in terms of high positioning accuracy (±8 × 10−12 m) with high bandwidth (obtained closed-loop bandwidth is almost 15 times better than the bandwidth obtained with PI control technique), which means that fast and large sample surface variations can be treated much better in that case. Further analysis with plant uncertainties as well as an experimental validation of the proposed control scheme should be the part of future studies.
Fig. 10. Comparison result between PI and H∞ control having step surface variations of 4 × 10−10 m
Fig. 11. STM tip movement with RS √ and H∞ control having no surface variations with sensor noise 45 mV / Hz
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