2015 IEEE International Symposium on Intelligent Control (ISIC) Part of 2015 IEEE Multi-Conference on Systems and Control September 21-23, 2015 Sydney, Australia Amplitude Control of the Cantilever of the Transverse Dynamic Force Microscope Thang Nguyen4, Christopher Edwards3, Guido Herrmann1 , Toshiaka Hatano1 , Stuart C Burgess1, and Mervyn Miles2 the tip of the cantilever is close to the sample surface, the amplitude of the oscillation of the cantilever tip becomes smaller due to the interaction force, which is a shear force created by the viscous and elastic characteristics of the ordered water layer The small decreasing amplitude can become difficult to detect due to sensor noise and decreasing signal-to-noise ratio If a feedback loop is utilized to manipulate the amplitude of the oscillation of the cantilever at the top, to maintain the amplitude of oscillation at the tip constant, then in principle the signal-to-noise ratio is kept constant, even for increased shear forces close to the specimen In this case, the change in the required control signal represents an indirect measure of the interaction force, and hence, for the proximity of the tip to the surface Moreover, for a large cantilever-specimen distance, the amplitude of the cantilever will not become excessively large so that any interaction of the cantilever with the ordered water layer is kept minimal Thus, any possible, accidental damage due to the impact of the cantilever tip onto the specimen can be avoided Abstract— In this paper, the problem of amplitude control of the cantilever in a Transverse Dynamic Force Microscope (TDFM) is considered The dynamics of the cantilever are initially presented as a partial differential equation which is subsequently approximated by a finite dimensional model The unknown shear force parameters which affect the amplitude of the cantilever are estimated by an adaptive scheme A controller which utilizes the parameter estimates of the cantilever shear force model is proposed to regulate the amplitude of the cantilever This is achieved by an adaptive internal model based feedforward approach using a novel estimation scheme, to create an overall feedforward model with unity gain To counter the effect of model uncertainty of the cantilever, a feedback scheme senses the difference between the expected unity gain model and the actual cantilever tip position, and feeds back this error dynamically Numerical simulations are presented to illustrate the effectiveness of the method I I NTRODUCTION The Atomic Force Microscope (AFM) is an important tool for investigating specimens at the nano scale [1], [2], [3] The image of the specimen is obtained from data generated from a cantilever oscillated at a high frequency Usually, the cantilever is positioned horizontally and oscillated perpendicular to the plane of the specimen These AFMs operate in a so-called tapping or contact mode (for non-oscillating cantilevers), but the physical interaction between the tip of the cantilever and the specimen can cause physical damage to soft (biological) specimens A novel AFM developed at the University of Bristol, called a Transverse Dynamic Force Microscope (TDFM) was proposed to circumvent this disadvantage In this device, the cantilever is oriented vertically and perpendicular to the sample surface [4], [5] This setup facilitates non-contact operation, avoiding any possible collisions between the tip and the sample The position of the top of the cantilever is excited sinusoidally at constant amplitude (using a light high-bandwidth piezo-actuator) and the tip of the cantilever (at the bottom) oscillates with the same frequency, but its amplitude depends on the interaction force between the tip and a thin, ordered water layer which even exists for specimens under ambient conditions (i.e subject to typical levels of humidity and air pressure) When Fig Various models of the cantilever dynamics of an AFM have been investigated for different situations, however, these are all variations of a 4th order partial differential equation (PDE), based on a Euler-Bernoulli beam model In this paper, the model for the cantilever dynamics of the TDFM has specific damping effects and an unique interaction force due to the unconventional operation mode of the TDFM compared to other work Here an approximate finite dimensional LTI model is obtained using the approach in [6] Based on this model, a scheme for controlling the amplitude of the cantilever is proposed using information from the model, together with estimates of the parameters of the unknown shear force which are obtained from the schemes in [7] The main contribution of this paper is the construction of the control law for the amplitude control problem of the This research is supported under the EPSRC grants EP/I034882/1 & EP/I034831/1 Department of Mechanical Engineering, University of Bristol, University Walk, Bristol, BS8 1TR, UK; Corresponding Author at Bristol: G.Herrmann@bris.ac.uk Centre for Nanoscience and Quantum Information, University of Bristol, Tyndall Avenue, Bristol, BS8 1FD, UK College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF, UK; Corresponding Author at Exeter: C.Edwards@exeter.ac.uk School of Electrical Engineering, International University, Vietnam National University- HCM city, Thu Duc, Ho Chi Minh city, Vietnam: thangnt@hcmiu.edu.vn 978-1-4799-7788-8/15/$31.00 ©2015 IEEE Simplified schematic of the VOPM together with SEW system 701 cantilever Unlike [8], [9] which combine adaptive laws with a robust H2 /H∞ internal model controller scheme, here an adaptive estimation scheme with an internal model based controller is employed to address the problem r ue + II P ROBLEM F ORMULATION Gc(s) In this paper, the proposed model of the cantilever is given by the following partial differential equation [4], [5]: ∂ EI0 (Y + α Y ) + SYă + wY = ∂ζ4 Adaptive Algorithm ur G(s) + u H(s) (1) together with boundary conditions Fig (2) (3) ζ =L − κ Y (L) Schematic of the control system A Approximate model As developed in [6], using the ideas of [14] the Laplace transform of Y (L,t) subject to (1)-(5) satisfies (4) Yˇ (L, s) (5) Gu (L, s)u(s) ˇ + G f (L, s) fˇ(s) = (7) where the transfer functions are In Equations (2)-(5), E represents Young’s modulus, α is the loss factor, I0 is the secondary moment of area, S is the cross-sectional area, ρ is the density of the cantilever material, the constant γ is the external damping coefficient, w is the width, and L is the length of the cantilever The variable ζ denotes position along the cantilever axis, whilst Y represents transversal displacement at a point along the cantilever The signal u(t) is the excitation applied at the top of the cantilever, and f (t) is the interaction shear force between the cantilever tip and the specimen As in [4], the interaction force is assumed to be a combination of viscous and elastic forces: ∂Y ∂t + e Y (ζ = 0) = u(t), ∂Y (ζ = 0) = 0, ∂ζ ∂ 2Y (ζ = L) = 0, ∂ζ2 ∂ 3Y EI0 (ζ = L) = − f (t), ∂ζ f (t) = −ν Y(L,t) - (cosh(η L) + cos(η L)) cosh(η L) cos(η L) + (8) cosh(η L) sin(η L) − sinh(η L) cos(η L) EI0 η (cosh(η L) cos(η L) + 1) (9) Gu (L, s) = G f (L, s) = with η4 = − ρ Ss2 + γ ws EI0 (1 + α s) (10) In (7), u(s), ˇ fˇ(s), and Yˇ (L, s) represent the Laplace transforms of the inputs u(t), f (t), and Y (L,t) Recently in [6], a framework was presented to obtain a finite dimensional LTI approximate model for Gu (L, s) and G f (L, s) Specifically, Gu (L, s) and G f (L, s) are first presented in terms of sums of infinite series Then, in principle, a linear finite dimensional approximate LTI model can be obtained by retaining only a finite number of the terms in the infinite series In [6], it was shown that Gu (L, s) and G f (L, s) can be approximated by the following transfer functions (6) where ν and κ represent dissipative and elastic constants respectively The shear force in (6) represents the interaction between the cantilever tip and the sample, which is assumed to be a function of the distance between the tip and the sample The coefficients κ and ν are time-varying but can be continuously monitored using adaptive schemes such as those in [7] Whilst the model has similarities to the ones in [10], [11], [12], [13], the boundary conditions used here are different and reflect the unique nature of Bristol’s TDFM In this paper, an approximate finite dimensional model for the cantilever dynamics will be employed, as derived in [7] Based on this model, a control law u(t) will be designed to control the amplitude of the cantilever and maintain it at a specified level, despite changes in the shear force EI0 (xk )3 α1 (Lxk )φk (L) + Cn ρ S(βk )1/2 pk (s) k=1 n Gu,n (L, s) = ∑ and n G f ,n (L, s) = φ (L) k , ∑ ρ Sp k (s) (11) (12) k=1 where xk is a positive solution of III M AIN R ESULTS cos(x) cosh(x) + = 0, In this section, the approximate model obtained from [7] will be described Based on this approximate model, a control algorithm is derived, which employs the estimates of the shear force parameters, obtained by (two) adaptive schemes the α1 (Lxk ), φk (L), βk , pk (s) are given in [6], and the coefficients n α1 (Lxk )φn (L) (14) Cn = − ∑ 1/2 k=1 xk (βk ) 702 (13) B Control design In the Laplace transform model of the cantilever in (7), Gu (L, s) and G f (L, s) are replaced by G˜ u (s) and G˜ f (s) where G˜ u (s) G˜ f (s) := Gu,n (L, s) := G f ,n (L, s) In this section, a control law for amplitude regulation of the cantilever will be developed The control structure is depicted in Fig It consists of two parts The first part is given by a feedforward component C(θ , s) This feedforward component is designed to enforce C(θ , s)G(s) has unit gain, ensuring that the output signal Yˆ (L,t) has amplitude (for a correct model of the cantilever dynamics: see Fig 2) The feedforward component of C(θ , s) is enhanced by an online identifier of the viscous and elastic component coefficients of the shear force In a second step, robustness to model uncertainty of the cantilever model G(s) is obtained The output of an internal model of the nominal dynamics C(θ , s)G(s) is compared with the actual cantilever tip position Y (L,t) to activate a closed-loop scheme using a compensator Gc (s) This ensures that the constant amplitude is retained even for uncertain cantilever parameters Let C(θ , s) be the transfer function of the feed forward compensator chosen as (15) (16) Suppose N1 (s) G˜ u (s) = Gu,n (L, s) = D(s) (17) N2 (s) G˜ f (s) = G f ,n (L, s) = D(s) (18) N1 (s) = k1 (sn + bnsn−1 + + b1), (19) N2 (s) = k2 (sn−2 + cn−2sn−3 + + c1) (20) and where n C(θ , s) = and D(s) = sn + an sn−1 + + a1 (21) (22) Note that G˜ u (s) is relative degree whilst the relative degree of G˜ f (s) is By assumption, from (6) fˇ(s) = −(ν s + κ )Yˇ (L, s), N1 (s) D(s) + (ν s + κ )N2(s) nz (23) H(s) = ν κ (24) (25) pz i=1 i=1 n p p ∏ (s + zi ) (28) where H1 (s) = p ∏ (s + zni ) ∏ (s − zi ) G(s) = i=1 nz which is the product of C(θ , s) and G(s) from (26) and (27) This of course only holds if the model of G(s) in (26) perfectly captures the dynamics of the cantilever The transfer function (28) guarantees that the gain of the controlled system is unity when the input reference is sinusoidal However, technically θ is unknown and hence, (27) is not realizable Here, (27) will incorporate an estimate of θ (written θˆ ) obtained from the method proposed in [7] From (24) write H1 (s) (29) G(s) = H3 (s) + (ν s + κ )H2(s) where κ and ν are the coefficients in the interaction force model in (6) From early work in [15], [6], [7], it was shown that a typical cantilever model exhibits nonminimum phase behaviour Hence, G(s) can be rewritten as nz (27) i=1 Define a new parameter vector as θ= pz p ∏ (s + zni ) ∏ (s + zi ) i=1 i=1 ∏ (s − zi ) where fˇ(s) is the Laplace transform of the shear force f (t) Hence, the overall transfer function uˆ to Yˆ (L, s) is G(s) = i=1 nz (under the assumption θ is known) Note the transfer function C(θ , s) has stable poles The transfer function Gc (s) in the feedback loop in Fig is chosen so that the closed loop characteristic equation + Gc(s)G(s) = is Hurwitz From (27), the transfer function relating the sinusoidal reference r(t) to the tip Yˆ (L, s) is the all-pass transfer function Hence, similar to (7), we get Yˆ (L, s) ≈ G˜ u uˆ + G˜ f fˆ ∏ (s + pi (θ )) N1 (s) N2 (s) D(s) , H2 (s) = , H3 (s) = , ψ (s) ψ (s) ψ (s) (30) and (26) ψ (s) = sn + ψnsn−1 + + ψ1 ∏ (s + pi (θ )) (31) i=1 is an arbitrary, fixed Hurwitz polynomial Introduce two filters for the input and output as where zni are zeros with negative real parts, zip are zeros with positive real parts, and pi (θ ) are the poles of G(s) which depend on θ It is assumed that for appropriate values of θ , the poles of (26) lie in the LHS of the complex plane w˙ u (t) = w˙ y (t) = 703 Aψ wu (t) + bψ u(t) Aψ wy (t) + bψ Y (L,t) (32) (33) where wu and wy Aψ = − ψ1 are the state vectors and 0 . . , bψ = . 0 − ψn C Compensator realisation With the estimate θˆ from (45), the controller in (27) becomes C(θˆ , s), which is time-varying A realisation of the controller in state space form is given by (34) x˙c (t) ur (t) y1 (t) = k2 c1 bn − ψn wu (t), cn−2 (36) wy , (37) cn−2 y f (t) = y(t) + a1 − ψ1 an − ψn wy (t) u(t) = ur (t) + ue (t) (35) wy , c1 y2 (t) = k2 (47) The control input is Define u f (t) = k1 u(t) + k1 b1 − ψ1 = Ac xc (t) + Bc(θˆ )r(t) = Cc (θˆ )xc (t) + Dc r(t) (48) where ue (t) is the output of the transfer function Gc (s) whose input is e(t) As proved in [7], if the spectral density of u(t) contains at least two points, then φ (t) is persistently exciting This guarantees M(t) > Hence, the estimate θˆ from (45) is well defined, and from (47), and (38) ur (t) = Cc (θˆ )e−Ac (t−t0 ) xc (t0 ) where y(t) = Y (L,t) Then, t ψ (t) = φ (t)θ (39) ψ (t) = y f (t) − u f (t) (40) φ (t) = y2 (t) y1 (t) (41) e−Ac (t−τ ) Bc (θˆ )r(τ )d τ + Dc r(t).(49) +Cc (θˆ ) t0 where For the finite time estimation, (47) is a switching linear system which has two linear time-invariant state space forms The transfer function from r(t) to the control input u(t) is and the parameter θ is defined in (25) Following the approach in [16], introduce a filtered regressor matrix N(t) ˙ = −kFF N(t) + kFF φ T (t)ψ (t), N(0) = N(t) u(s) C(θˆ , s) + Gc (s)H(s) = r(s) + Gc (s)G(s) Since r(t) and its integral are sinusoidal, their spectral density contain two points [17] Also, (49) is a stable system, hence, its output will have sinusoidal properties Furthermore, (50) implies u(t) is sinusoidal at steady state Hence, the spectral density of u(t) has at least two points This guarantees that θˆ = θ in finite time (42) where kFF ∈ R+ is a forgetting factor Also consider a filtered regressor matrix M(t) which is the solution to the following differential equation ˙ = −kFF M(t) + kFF φ T (t)φ (t), M(0) = M(t) (43) IV N UMERICAL E XAMPLE The cantilever in Bristol’s TDFM is made of Silicon Nitride (Si3N4) and the physical parameters are given as follows: Young’s modulus E = 210Gpa, density ρ = 3100kg/m3, length L = 28µ m, width w = 2µ m, thickness tc = 200nm The cross sectional area is As = Wtc and the second moment of area is I = 1/12Wtc3 The loss factor is α = 7.39 × 10−9 The frequency of the excitation signal is ω = 2.131 × 106rad/s which is near the first resonance frequency The chosen amplitude for the reference signal is d0 = 1.8nm For the parameters of the estimator kFF = 500, K(0) = I2 (the identity matrix of order 2), ψ1 = a1 , ψ2 = a2 , ψ3 = a3 , ψ4 = a4 , δ = 10−6 The transfer function Gc (s) is chosen as ∗ 104 (51) Gc (s) = s + 2.2 ∗ 103 In this paper, ten sets of κ and ν corresponding to ten different tip-to-surface distances (nm) are considered in the following table These parameters have been collected from experiments using Bristol’s TDFM They are employed here to mimic the dynamics of the cantilever when parameters change In the simulations, two scenarios are considered: in the first there is no noise; and in the second, both the input and output are influenced by band-limited white noise with zero mean Then as shown in [16], N(t) = M(t)θ (44) Hence, a finite time estimation is given as θˆ (t) = M −1 (t)N(t) det(M(t)) < δ , otherwise (45) where δ is an arbitrarily small positive parameter which decides the finite-time convergence of the parameter estimate With the estimate θˆ , the transfer function relating the ideal sinusoidal reference r(t) to the tip is C(θˆ , s)G(s) = H(s)Gc (s)G(s) + C(θˆ , s)G(s) + Gc(s)G(s) (50) (46) If G(s) perfectly models the dynamics of the cantilever and θˆ = θ , then the transfer function (46) will be identical to (28) since in this situation C(θˆ , s)G(s) = H(s) In the case when G(s) does not perfectly capture the dynamics of the cantilever, the regulatory performance of the controller depends on Gc (s), and can be tuned by the choice of Gc (s) The feedback compensator Gc (s) needs to be robust to changes in the cantilever dynamics and in changes in the shear force The convergence of θˆ to θ will be discussed in the next subsection 704 TABLE I S HEAR FORCE PARAMETERS W R T TIP - TO - SURFACE DISTANCES tip-to-surface distances (nm) 1.5 2.5 3.5 4.5 5.5 κ (kg/s2 ) ∗ 10−4 2.5 ∗ 10−4 2.5 ∗ 10−4 1.5 ∗ 10−4 10−4 10−4 10−4 0.5 ∗ 10−4 0.5 ∗ 10−4 0.2 ∗ 10−4 finite dimensional model which has been derived from the irrational transfer function associated with the 4th order PDE representation The control structure is based on an internal model scheme, where the control law is parameterised in terms of the unknown shear force parameters, which are estimated using an adaptive scheme Numerical examples have illustrated the performance of the proposed controller It has been shown that the control signal can be used to obtain a measure of the distance of the cantilever tip to the specimen ν (s−1 ) 1.2 ∗ 10−9 10−9 0.9 ∗ 10−9 0.85 ∗ 10−9 0.82 ∗ 10−9 0.8 ∗ 10−9 0.75 ∗ 10−9 0.72 ∗ 10−9 0.68 ∗ 10−9 0.5 ∗ 10−9 R EFERENCES [1] G Binnig, C Quate, and C Gerber, “Atomic force microscope,” Physical Review Letters, 1986 [2] K A Ramirez-Aguilar and K L Rowlen, “Tip characterization from afm images of nanometric spherical particles,” Langmuir, vol 14, no 9, pp 2562–2566, 1998 [3] R Howland and L Benatar, Practical Guide to Scanning Probe Microscopy Park Scientific Instruments: Sunnyvale, CA, USA, 1993 [4] M Antognozzi, “Investigation of the shear force contrast mechanism in transverse dynamic force microscopy,” Ph.D dissertation, Univeristy of Bristol, 2000 [5] M Antognozzi, A Humphris, and M Miles, “Observation of molecular layering in a confined water film and study of the layers viscoelastic properties,” Applied Physics Letters, vol 78, no 3, pp 300–302, 2001 [6] T Nguyen, C Edwards, G Herrmann, T Hatano, S Burgess, and M Miles, “Cantilever dynamics modelling for the transverse dynamic force microscope,” in Proceedings of the 2014 Conference on Decision and Control (CDC), Dec 2014 [7] ——, “Adaptive estimation of the shear force in the cantilever dynamics of the transverse dynamic force microscope,” in 2014 IEEE International Symposium on Intelligent Control (ISIC), Oct 2014, pp 542–547 [8] A Datta and J Ochoa, “Adaptive internal model control: {H2} optimization for stable plants,” Automatica, vol 34, no 1, pp 75 – 82, 1998 [9] A Datta and L Xing, “Adaptive internal model control: H infin; optimization for stable plants,” IEEE Transactions on Automatic Control, vol 44, no 11, pp 2130–2134, Nov 1999 [10] M Fardad, M Jovanovic, and M Salapaka, “Damping mechanisms in dynamic mode atomic force microscopy applications,” American Control Conference, 2009 ACC ’09., pp 2272–2277, 2009 [11] B Jacob, C Trunk, and M Winklmeier, “Analyticity and Riesz basis property of semigroups associated to damped vibrations,” Journal of Evolution Equations, vol 8, no 2, pp 263–281, 2008 [12] B Guo, “Riesz basis approach to the stabilization of a flexible beam with a tip mass,” SIAM Journal on Control and Optimization, vol 39, no 6, pp 1736–1747, 2001 [13] M Tucsnak and G Weiss, Observation and control for operator semigroups Springer, 2009 [14] R F Curtain and H Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory New York: Springer-Verlag, 1995 [15] T Nguyen, S G Khan, C Edwards, G Herrmann, R Harniman, S C Burgess, M Antognozzi, and M Miles, “Shear force reconstruction in a vertically oriented probe microscope using a super-twisting observer,” in Conference on Decision and Control, Dec 2013, pp 4266 – 4271 [16] J Na, M Mahyuddin, G Herrmann, and X Ren, “Robust adaptive finite-time parameter estimation for linearly parameterized nonlinear systems,” in 32nd Chinese Control Conference, 2013, pp 1735–1741 [17] S Sastry and M Bodson, Adaptive Control: Stability, Convergence, and Robustness Prentice-Hall, 1989 and variance 1.25 Such a level of noise is usually observable in the sensor dynamics of the TDFM system Figures and show the time evolution of u(t) and Y (L,t) for the noise-free case for the first set of κ and ν values i.e at a tip-to-specimen distance of 1.5nm The amplitude of the cantilever is 1.8 nm, which is exactly the amplitude of the reference signal The settling time is about 0.005 ms Figure depicts the relationship between the amplitude of the control effort and the tip-to-surface distance for the noise-free case, and demonstrates that the control effort compensates for the changes in the shear force parameters Figure show the time evolution of Y (L,t) for the noisy case for the first set of κ and ν values (a tip-to-specimen distance of 1.5nm) The effect of noise is seen by comparing Figure with Figure The settling time is about 0.005 ms Figure describes the relationship between the amplitude of the control effort and the distance to the surface, which is similar to the noise free case The closer to the specimen, the larger the control effort which is necessary to keep the control output at the required value Figure illustrates the cantilever amplitude with respect to changes in the tip to surface distance (nm) Due to the effect of the noise, the amplitude of the cantilever is slightly larger than 1.8 nm (which is the amplitude of the reference signal) This implies that the proposed control method is still effective in the presence of noise Now consider the case where the cantilever parameters are slightly perturbed: E = 209Gpa, ρ = 3090kg/m3, L = 27.5µ m, w = 1.95µ m, tc = 190nm, α = 7.4 × 10−9 The time evolutions of u(t) and Y (L,t) for the first set of κ and ν values (a tip-to-specimen distance of 1.5nm) with perturbed cantilever parameters are shown in Figures 11 and 9, and are similar to those for the unperturbed case The control efforts depicted in Figures 10 and 12 are slightly larger than those for the unperturbed cantilever parameters Likewise, the amplitudes of the output are slightly larger than those for the unperturbed case However the simulations show that the proposed scheme possesses some robustness u(nm) V C ONCLUSIONS −1 The problem of controlling the amplitude of the tip of the cantilever in a TDFM has been studied in the paper This has been done to retain constant signal-to-noise ratio when measuring the cantilever tip position and to avoid too large tip position amplitudes, minimizing possible specimen damage A control law has been constructed based on an approximate −2 0.05 0.1 0.15 t(ms) Fig The evolution of u(t) for the noise free case for the first shear force parameters 705 2 y(nm) y(nm) −4 −2 −2 0.05 0.1 −4 0.15 0.05 Control amplitude (nm) 0.15 0.1 0.05 2.5 3.5 4.5 tip to surface distance (nm) 5.5 Control amplitude (nm) Control amplitude (nm) 0.2 0.1 0.05 1.5 2.5 y(nm) −2 −2 −4 0.05 0.1 0.15 0.05 Control amplitude (nm) 0.15 0.1 0.05 2.5 3.5 4.5 tip to surface distance (nm) 5.5 Cantilever amplitude (nm) 1.5 0.5 2.5 3.5 4.5 tip to surface distance (nm) 5.5 0.1 0.05 1.5 2.5 3.5 4.5 tip to surface distance (nm) 5.5 Fig 12 The relation of the amplitude of the sinusoidal control signal u(t) and the tip-to-surface distance for the noisy case with perturbed cantilever parameters 2 0.2 0.15 Fig The relation of the amplitude of the sinusoidal control signal u(t) and the tip-to-surface distance for the noisy case 1.5 Control amplitude (nm) 0.2 0.15 Fig 11 The evolution of the output y(t) for the noisy case for the first shear force parameters with perturbed cantilever parameters Fig The evolution of the output y(t) for the noisy case for the first shear force parameters 1.5 0.1 t(ms) t(ms) Control amplitude (nm) 5.5 Control amplitude (nm) Cantilever amplitude (nm) Cantilever amplitude (nm) 3.5 4.5 tip to surface distance (nm) Fig 10 The relation between the amplitude of the sinusoidal control signal u(t) and the tip-to-surface distance for the noise free case with perturbed cantilever parameters y(nm) 0.2 0.15 Fig The relation between the amplitude of the sinusoidal control signal u(t) and the tip-to-surface distance for the noise free case −4 0.15 Fig The evolution of y(t) for the noise free case for the first shear force parameters with perturbed cantilever parameters Fig The evolution of y(t) for the noise free case for the first shear force parameters 1.5 0.1 t(ms) t(ms) Fig The relation of the output amplitude y(t) and the tip-to-surface distance for the noisy case 1.5 0.5 1.5 2.5 3.5 4.5 tip to surface distance (nm) 5.5 Fig 13 The relation of the output amplitude y(t) and the tip-to-surface distance for the noisy case with perturbed cantilever parameters 706 ... along the cantilever The signal u(t) is the excitation applied at the top of the cantilever, and f (t) is the interaction shear force between the cantilever tip and the specimen As in [4], the. .. for the transverse dynamic force microscope, ” in Proceedings of the 2014 Conference on Decision and Control (CDC), Dec 2014 [7] ——, “Adaptive estimation of the shear force in the cantilever dynamics... 1.5nm The amplitude of the cantilever is 1.8 nm, which is exactly the amplitude of the reference signal The settling time is about 0.005 ms Figure depicts the relationship between the amplitude of