Robust H Control of a Scanning Tunneling Microscope under Parametric Uncertainties

Transcription

1 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 FrC08.6 Robust H Control of a Scanning Tunneling Microscope under Parametric Uncertainties Irfan Ahmad, Alina Voda and Gildas Besançon Abstract This paper is devoted to the control system design for high performance scanning tunneling microscope (STM). A common approach by scanning probe community is to use conventional proportional integral (PI) control design to control the vertical movement of STM tip (z-direction). In this article, a modern H control design is analyzed in order to obtain the dual purpose of ultrahigh positioning accuracy with high bandwidth. Uncertainty model, based on experimental analysis of tunneling characteristics and parametric description of the STM, and norm-bounded real perturbations are considered, and an H controller is designed by following the desired control objectives. A performance and robustness analysis is finally performed to test robust stability and performance of STM. Keywords : Scanning tunneling microscope, Nano-technology, Precision positioning, H control, Uncertainty model, Parametric variations, Robustness. I. INTRODUCTION Scanning Tunneling Microscope (STM) is an important tool in obtaining images of the material samples with atomic resolution which was first invented by Gerd Binnig and Heinrich Rohrer in early 1980s [1]. STM works on tunneling phenomenon which occurs when an extremely sharp metallic electrically charged tip approaches at the vicinity of the conductive sample surface (the distance between tip apex and sample surface in the range of m) [2]. Because of the atomic resolution, the STM has vast applications in different domains and the ultrahigh positioning accuracy with high bandwidth are the great challenges. As the distance between STM tip apex and surface is less than m to get the tunneling effect, the electronic control in vertical z-direction is very critical in order to get a good image quality of the surface, in the presence of external disturbances. 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 necessarily to the reality [3]. The feedback loop of STM in vertical z-direction with some stability conditions has been presented in [4]. All such analysis The authors are with GIPSA-lab, Control System Department, ENSE3, BP 46, Saint Martin d Hères, France. ( irfan.ahmad, alina.voda, gildas.besancon@gipsa-lab.inpg.fr) are 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 [5] and a VSC (variable structure control) design methodology in the presence of PI control is proposed in order to avoid STM tip collision with sample surface. A control design methodology based on pole placement with sensitivity function shaping using second order digital notch filter is proposed in [6] for the feedback control system of STM (the general description of this control design methodology is given in [7], [8]) but the proper tuning of the control parameters in order to follow the design constraints can be a difficult task for STM operators. According to author s knowledge, no one else before analyzed the performance of STM in vertical z-direction with modern H control design framework, although it has been discussed in case of Atomic Force Microscope (AFM) control [9]. The control of AFM for vertical z-direction is discussed in [10], [11] with classical PI (PID) control technique as well. So, we first tried to investigate the performance of STM, keeping in mind the dual purpose of ultrahigh positioning accuracy with high bandwidth, with modern H approach and we concluded better performance than classical PI control design methodology [12]. But still there was a need of further analysis with uncertainty model designed by considering variations in all parameters of STM nominal model. Then, robust stability and performance must be analyzed in addition to nominal stability and performance of the system which can be critical depending on the range of parametric variations. 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 analyze the scanning tunneling microscope (STM) performance with nominal and uncertainty model in the H framework. A complete system overview with corresponding simulation model and the tunneling characteristics of STM experimental setup is given in Section II. The nominal and uncertainty model are discussed in Section III. Section IV then presents the control problem formulation with desired performances. H control design applied to STM is discussed in section V. Simulation results to validate the controller and to analyze the robust stability and performance of STM are presented in Section VI. Finally, Section VII draws some conclusions /10/$ AACC 6555

2 Fig. 1. Complete simulation model for STM closed-loop system A. Working Principle II. STM SYSTEM DESCRIPTION Scanning tunneling microscope (STM) works by scanning a very sharp metal wire tip over a surface. By bringing the tip very close to the conducting sample surface (less than m), and by applying an electrical voltage v b (v b = 0.1 V ) to the tip or sample, a small tunnel current (i t ) is produced between STM tip and sample surface. This tunnel current depends exponentially on the distance (d) between STM tip and sample surface with following nonlinear relation : i t = i o e α Φ d (1) where Φ is the work function, d is the variations in the distance (d), α is a constant term (α = ev 1 Å 1 ) and i o is the initial tunnel current when d = 0. Controlling this tunnel current (i t ) by keeping the distance (d) constant ( d = 0) in the presence of external disturbances (sensor noise (n), surface variations (z S ) 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. A feedback loop constantly monitors the tunneling current (i t ). The current amplifier (pre-amplifier or I- V converter) converts the small tunneling current into a voltage (v 3 ) with a gain (R) of Ω (bandwidth 15 kh z). This pre-amplifier is usually the most important source of noise (n) in the loop. A logarithmic amplifier (bandwidth 60 kh z) is used in the feedback loop to deal with exponential nonlinearity and to make the entire electronic response linear (approximately) with respect to the distance (d). The output (v y ) is given by the following nonlinear relation : v y = K L log 10 ( v3 E L where K L is the conversion factor (2.5 V ) and E L is the sensitivity of log amplifier (0.001 V ). The piezoelectric ) (2) actuator attached with STM tip to move it in appropriate direction according to the applied voltage (v 2 ) in order to keep the distance (d) constant. An amplifier (Gain = 4) of bandwidth 100 kh z is used before piezoelectric actuator as the output of controller (v 1 ) can be between ±10 V. The piezoelectric actuators are now widely used for high positioning accuracy at nanometer and sub-nanometer resolution with high bandwidths [13], [14]. One of the advantages of using piezoelectric actuators is that under certain experimental conditions their dynamics can be well approximated by linear models [15], that s why a second order linear model can be used for piezoelectric actuator : G a (s) = ( 1 ω 2 0 ) s 2 + γ 0 ( 1 Qω 0 ) s + 1 where γ 0 is the sensitivity ( mv 1 ), ω 0 is the resonance frequency (28 kh z) and Q is the quality factor (4.5) of piezoelectric actuator model. The nonlinear phenomenon like hysteresis is not expected for the piezoelectric actuator as the amplitude of input voltage (v 2 ) is very small for the vertical movement (z-direction) of STM tip. The output of the piezoelectric actuator (z) is used to find out the distance (d) between STM tip and sample surface (z s ) from d = z 0 z z s (Fig. 1) where z 0 is the initial position of STM tip when no voltage is applied to the piezo. B. Tunneling Characteristics of STM with Experimental Setup Some experimental tests of STM are performed in Institut Néel at room temperature in order to observe the tunneling characteristics with the change in the distance between STM tip and sample surface. These results helped us to observe the tunneling phenomenon and to identify its characteristics. Different tests are performed by moving the STM tip vertically upward ( m away from the surface) and vertically downward ( m towards the (3) 6556

3 surface) in order to observe the variations in the tunneling current with the distance. Test results are plotted in Fig. 2. The variations in the different tunnel current profiles are because of the different initial conditions and also the environmental condition which includes temperature variations, vibrations, noise etc. We tried to simulate the tunneling characteristics and identified the value of the work function Φ. We observed that with work function Φ = 0.5 ev, our simulation approximately follows the tunnel current profile obtained from the experiment (Fig. 2). For simulations and synthesis of control, we have taken the experimentally identified value of work function and large variation (20% from the nominal value) of Φ is considered for uncertainty model of the system. The parameter values of piezoelectric actuator and amplifiers in the closed loop, used in this article, are taken from the experimental setup. C. Control Design Model The complete simulation model (Fig. 1) needs to be transformed into an appropriate linear design model (Fig. 3) which is required for the linear controller design for feedback control system of STM. A first order linear approximation approach is used to linearize both non-linearities Eq. (1) and Eq. (2) independently around their equilibrium points (d 0 and v 30 ). The corresponding linearized equations are respectively : Fig. 2. Experimental and simulated response of tunnel current i t = c 1 + c 3 c 2 d (4) v y = c 4 c 6 + c 5 v 3 (5) Fig. 3. Design model for closed-loop system of STM where c 1,c 2,...,c 6 are constants which depend on the parameters of Eq. 1 and Eq. 2 and defined by the following relation : c 1 = i o e α Φ d 0 c 2 = α Φ c 1 c 3 = d 0 c 2 ( ) v30 = K L log 10 c 4 K L c 5 = v 30 ln (10) K L c 6 = ln (10) Now, the feedback dynamics (Fig. 1) where tunnel current (i t ) is converted into a voltage (v y ) can be represented by a 2 nd order linear model H(s). H(s) = E L (6) c ω 1 ω 2 s 2 + (ω 1 + ω 2 ) s + ω 1 ω 2 (7) where c is a constant term depending on parameters of Eq. 4 and Eq. 5 and ω 1, ω 2 are two bandwidths of the preamplifier and logarithmic amplifier. After linearization, the equivalent linear control design model is given in Fig. 3 where G(s) represents the 3 r d order linear model including piezo pre-amplifier and piezoelectric actuator model and G n is a constant term which represents the noise (n) transfer. III. NOMINAL AND UNCERTAINTY MODEL The nominal model of the plant G 0 (s) is obtained considering the values of physical parameters of STM taken from the experimental setup. Then, its worst case behavior is analyzed by taking into account the variations of the model parameters inside a given range of values. The nominal values of STM model, together with their percentage variation are shown in Table I. In order to check the robust stability (RS) and robust performance (RP) conditions, a set of plants Π may be represented by multiplicative uncertainty with a scalar weight W I (s) using : G (s) = G 0 (s)(1 +W I (s) I (s)) (8) TABLE I VALUES OF THE STM MODEL PARAMETERS TOGETHER WITH THEIR % VARIATION Parameter Value % variation ω 0 [kh z] 28 ± % γ 0 [mv 1 ] ± % Q 4.5± % Φ [ev ] 0.5 ± % R [Ω] ± % ω 1 [kh z] 15 ± % 6557

4 where (j ω) 1, ω represents the normalized real perturbations and G Π. In case of multiplicative uncertainty model, the relative error function can be computed as : l I (ω) = max G (j ω) G 0 (j ω) G Π G 0 (j ω) (9) and with rational weight W I (j ω) is chosen as follows : W I (j ω) l I (ω), ω (10) Relative errors l I (j ω) together with the rational weight W I (j ω) for 729 possible combinations of the STM physical parameters are plotted in Fig. 4. The weighting function W I (j ω) is chosen equal to a third order transfer function in a way that it includes the set of all possible plant models. W I (s) = ((1/ω B )s + A)(s 2 + 2ζ 1 ω n s + ω 2 n ) ((1/(ω B M))s + 1)(s 2 + 2ζ 2 ω n s + ω 2 n) (11) The values considered for all parameters of W I (s) are given in Table II. Using the small gain theorem, the condition for Robust Stability (RS) is given by [16] : RS T < 1, W I ω (12) Considering the performance specifications in terms of the sensitivity function, the condition for Robust Performance (RP) is obtained as [16] : RP W 1 S + W I T < 1, ω (13) where S and T are sensitivity function and complementary sensitivity function respectively. IV. CONTROL PROBLEM 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 (z S ) by keeping the distance (d) constant between STM tip and sample TABLE II PARAMETERS FOR THE RATIONAL WEIGHT W I (s) ( THE VALUES OF THE FREQUENCIES ARE EXPRESSED IN RAD/SEC) A M ζ 1 ζ 2 ω n ω B surface. It can also be formulated as a disturbance rejection problem. The variations in the sample surface (z S ) 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 ( m). The main objective of the control system is to achieve better performance of STM in terms of high positioning accuracy ± m in the presence of good robustness margin ( S 6 db and T 3.5 db and stability margins (gain margin > 6 db and phase margin > 30 o ). Such positioning accuracy is required with high closedloop bandwidth in order to get fast scan speed as slower scan speed can introduce certain considerable points, like drift. The desired performance is required with the maximum continuous surface variations of frequency r ad/sec having amplitude m in the presence of sensor (pre-amplifier) noise (n) of 45 mv / H z. V. H CONTROL DESIGN APPLIED TO STM The desired performances are imposed on the closedloop sensitivity functions using appropriate weighting functions and then the mixed-sensitivity H control design methodology is adopted to fulfill the requirements. The functions W 1, W 2 and W 3 weight the controlled outputs y 1, y 2 and y 3 respectively (Fig. 3) and should be chosen according to the performance specifications. The generalized plant P (Fig. 5) (i.e. the interconnection of the plant and the weighting functions) is given by : y 1 W 1 W 1 H W 1 HG n W 1 HG y 2 y 3 = W 2 0 W 3 0 W 3 G V E } I H HG n {{ HG } P V REF Z S n V 1 Thus, the H control problem is to find a stabilizing controller K (s) which minimizes γ [16] such that : W 1 S W 1 HS W 1 HG n S W 2 K S W 2 HK S W 2 HG n K S < γ (14) W 3 GK S W 3 S W 3 G n T We have chosen the weighting functions as follows : Fig. 4. Relative plant errors l I (j ω) and rational weight W I (j ω) for 729 possible combinations of the STM physical parameters W 1 (s) = (1/M s) s + ω s s + ω s ɛ s (15) 6558

5 Fig. 5. Generalized design model for closed-loop system of STM Fig. 7. Closed-loop sensitivity function (KS) with H control W 2 (s) = s + (ω u/m u ) ɛ u s + ω u (16) W 3 (s) = s + (ω t /M t ) ɛ t s + ω t (17) The values considered for all parameters of above three weighting functions in order to achieve desired performance are given in Table III. After computation, the TABLE III PARAMETERS FOR PERFORMANCE WEIGHTING FUNCTIONS W 1 (s), W 2 (s) AND W 3 (s) ( THE VALUES OF THE FREQUENCIES ARE EXPRESSED IN RAD/SEC) Fig. 8. Closed-loop sensitivity function (T) with H control M s ɛ s ω s M u ɛ u ω u M t ɛ t ω t minimal cost achieved for STM feedback control system was γ = 1.6 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 A. Control Loop Performance Analysis The weighting functions (W 1, W 2 and W 3 ) were designed considering the requirement of high positioning accuracy ± m with high bandwidth and good robustness. The proposed control technique achieves all Fig. 6. Closed-loop sensitivity function (S) with H control the requirements as the obtained sensitivity functions fairly match the desired loop shaping (Fig. 6-8). From robustness point of view, we obtained good modulus margin as S = 1.55 db and T = 0.08 db and good stability margins (gain margin = 10.1 db and phase margin = 66.1 o ). The obtained closed-loop bandwidth is r ad/sec which ensures the required good performance with fast variations ( r ad/sec) in the sample surface (z S ). Similarly, all other constraints in terms of better noise (n) rejection and to avoid actuator saturations are fully met with proposed control technique. VI. SIMULATION RESULTS After the control design and closed-loop analysis, we can now validate its performance with a simulation model, having actual non-linearities, sensor noise (n) and physical limitations in closed-loop, aiming at representing a real system as close as possible. Fig. 9 shows the simulation result with the proposed H control technique in presence of surface variations z S with a frequency of r ad/sec and an amplitude of m, and in the presence of sensor noise (n) (in the pre-amplifier) of 45 mv / H z. The dotted lines represent the positioning accuracy (acceptable bounds) of ± m. It can be observed that the movement of the STM tip remains within the desired limits. Finally, we have verified the robust stability (RS) and robust performance (RP) conditions as mentioned in Eq. 6559

6 Fig. 9. Simulation result with H control having surface variations z S of frequency of r ad/sec, an amplitude of m and in the presence of sensor noise (n) of 45 mv / H z Fig. 10. System robust stability tested with H control technique 12 and Eq. 13. Simulation result shows that robust stability (RS) condition is satisfied (Fig. 10) i.e. the closed-loop system remains stable for all perturbed plants around the nominal model up to the chosen worst-case model uncertainty. Fig. 11 shows that robust performance (RP) achieved according to the condition given in Eq. 13. VII. CONCLUSIONS This article has dealt with H control design for STM, under high performance specifications and real parametric uncertainties. Tunneling characteristics of STM are analyzed experimentally and an important parameter Φ is identified. Performance in terms of high positioning accuracy (± m) with high bandwidth and good robustness is achieved. Simulation results show that RS and RP conditions are fulfilled under parametric variations of the system model. Real-time experimental validation of the H controller is currently in progress. Acknowledgement : The authors would like to sincerely thank Prof. Hervé Courtois for providing the access to an experimental STM setup of the Institut Néel in Grenoble. Also we would like to thank M. Thomas Quaglio from Institut Néel for helping in all experimental tests and his useful discussions. Fig. 11. System robust performance tested with H control technique REFERENCES [1] G. Binning and H. Rohrer, Scanning Tunneling Microscopy, IBM J. Res. Develop., vol. 30, pages , 1986 [2] C.J. Chen, Introduction to scanning tunneling microscopy, 2nd edition, Oxford science publications, 2008 [3] E. Anguiano, A.I. Oliva and M. Aguilar, Optimal conditions for imaging in scanning tunneling microscopy : Theory, Rev. Sci. Instrum., Vol. 69 (2), pages , November 1998 [4] A.I. Oliva, E. Anguiano, N. Denisenko, M. Aguilar and J.L. Pena, Analysis of scanning tunneling microscopy feedback system, Rev. Sci. Instrum., Vol. 66 (5), pages , May 1995 [5] N. Bonnail, D. Tonneau, F. Jandard, G.A. Capolino and H. Dallaporta, Variable structure control of a piezoelectric actuator for a scanning tunneling microscope, IEEE transaction on Industrial Electronics, Vol. 51 (2), pages , April 2004 [6] I. Ahmad, A. Voda and G. Besançon, Controller design for a closedloop scanning tunneling microscope, 4th IEEE Conference on Automation Science and Engineering, Washington DC, USA, pages , August 2008 [7] I.D. Landau and A.Karimi, Robust digital control using pole placement with sensitivity function shaping method, Int. J. Robust Nonlinear Control, vol. 8, pages , 1998 [8] H. Prochazka and I.D. Landau, Pole placement with sensitivity function shaping using 2nd order digital notch filters, Automatica 39 (2003), pages , February 2003 [9] S. Salapaka, A. Sebastian, J.P. Cleveland and M.V. Salapaka, High bandwidth nano-positioner : A robust control approach, Rev. Sci. Instrum., Vol. 73 (9), pages , September 2002 [10] G. Schitter, K.J. Astrom, B.E. DeMartini, P.J. Thurner, K.L. Turner and P.K. Hansma, Design and Modeling of a High-Speed AFM-Scanner, IEEE transaction on Control Systems Technology, Vol. 15 (5), pages , September 2007 [11] D.Y. Abramovitch, S.B. Andersson, L.Y. Pao and G. Schitter,A Tutorial on the Mechanisms, Dynamics, and Control of Atomic Force Microscopes, American Control Conference, NY, USA, pages , July 2007 [12] I. Ahmad, A. Voda and G. Besançon, H controller design for high performance scanning tunneling microscope, 48th IEEE Conference on Decision and Control, Shanghai, China, December 2009 [13] M.E. Taylo, Dynamics of piezoelectric tube scanners for scanning probe microscopy, Rev. Sci. Instrum., Vol. 64 (1), pages , 1993 [14] G. Schitter, and A. Stemmer, Identification and open-loop tracking control of a piezoelectric tube scanner for high-speed scanning probe microscopy, IEEE transaction on Control System Technology, vol 12 (3), pages , 2004 [15] B. Bhikkaji, M. Ratnam, A.J. Fleming and S.O.R. Moheimani, Highperformance control of piezoelectric tube scanners, IEEE transaction on Control Systems Technology, Vol. 15 (5), pages , September 2007 [16] Skogestad, S and Postlethwaite, Multivariable feedback control : analysis and design, John Wiley and Sons,