Topography and Force Estimation in Atomic Force Microscopy by State and Parameter Estimation

Transcription

1 Topography and Force Estimation in Atomic Force Microscopy by State and Parameter Estimation Michael R. P. Ragazzon, J. Tommy Gravdahl, Kristin Y. Pettersen, Arnfinn A. Eielsen Department of Engineering Cybernetics, Norwegian University of Science and Technology ACC Chicago, July 1 3, 2015

2 1 Introduction AFM AM Loop Function generator Piezo Laser Lock-in amplifier Phase Amplitude Cantilever Photo detector Sample xyz-scanner z PID controller Setpoint

3 2 Introduction AFM AM Loop Setpoint Amplitude estimation Sensor Controller Output image Cantilever response Piezo actuator F Tip-sample interaction force D θ Sample height

4 3 Introduction AFM AM Loop Setpoint Amplitude estimation Sensor Controller Output image Cantilever response Piezo actuator F Tip-sample interaction force D θ Sample height

5 4 Introduction AFM AM Loop Setpoint Amplitude estimation Sensor Controller Output image Cantilever response Piezo actuator F Tip-sample interaction force D θ Sample height

6 5 Introduction AFM AM Loop Setpoint Amplitude estimation Sensor Controller Output image Cantilever response Piezo actuator F Tip-sample interaction force D θ Sample height

7 6 Introduction AFM AM Loop Setpoint Amplitude estimation Sensor Controller State and parameter estimator ˆθ Output image Cantilever response Piezo actuator F Tip-sample interaction force D θ Sample height

8 7 Introduction We will employ observers to directly estimate the topography By considering the cantilever dynamics and interaction force Two observers were designed for the same purpose: A nonlinear observer with well-defined exponential stability results. An extended Kalman Filter for comparison.

9 8 Outline Introduction System Modeling Observer Design Simulation Results Conclusion and Further Work

10 8 System Modeling F exc x 1 G F ts( ) D θ G: Cantilever model from applied force, to tip deflection. Second order harmonic oscillator. F ts: Tip-sample interaction force: Modeled by Lennard-Jones potential.

11 9 System Modeling Lennard-Jones potential [ σ 2 F ts(d) = k 1 D σ 8 D 8 ] (1) 2 F ts Force (nn) Tip-sample distance, D (nm)

12 10 System Modeling State-space form System can be expressed as an extended state-space model suitable for the state- and parameter estimator. [ ] [ ] [ ] [ ] [ ] ẋ A E x B 0 = + u + d (2) φ 0 0 φ 0 1 [ ] [ ] A = ω0 2, B = E = 1, C = [ 1 0 ] (3) 2ζω 0 m

13 10 System Modeling State-space form System can be expressed as an extended state-space model suitable for the state- and parameter estimator. [ ] [ ] [ ] [ ] [ ] ẋ A E x B 0 = + u + d (2) φ 0 0 φ 0 1 [ ] [ ] A = ω0 2, B = E = 1, C = [ 1 0 ] (3) 2ζω 0 m where the state x (x 1, x 2 ) T represent the cantilever deflection and the deflection velocity the Lennard-Jones force F ts has been introduced as a state φ d is the time-derivative of φ and the input u is the driving force of the cantilever.

14 11 Outline Introduction System Modeling Observer Design Simulation Results Conclusion and Further Work

15 11 Nonlinear Observer Based on Håvard F. Grip s article: Estimation of States and Parameters for linear systems with nonlinearly parameterized perturbations (2011). u System Modified high-gain observer (y, ˆx, ˆφ) Parameter estimator ˆθ Structure of the state- and parameter estimator. From (Grip et al., 2011).

16 12 Nonlinear Observer Modified High-Gain Observer Observes the state (ˆx, ˆφ) as if the parameter estimate ˆθ is perfectly known. ˆx = Aˆx + Bu + E ˆφ + K x(ε)(y Cˆx) ż = g ˆθ g θ x Kx(ε)(y Cˆx) + K φ(ε)(y Cˆx) (4) ˆφ = g(ˆx 1, ˆθ) + z Need to determine K x(ε), K φ (ε) such that the error dynamics of the observer are input-to-state stable.

17 12 Nonlinear Observer Modified High-Gain Observer Observes the state (ˆx, ˆφ) as if the parameter estimate ˆθ is perfectly known. ˆx = Aˆx + Bu + E ˆφ + K x(ε)(y Cˆx) ż = g ˆθ g θ x Kx(ε)(y Cˆx) + K φ(ε)(y Cˆx) (4) ˆφ = g(ˆx 1, ˆθ) + z Need to determine K x(ε), K φ (ε) such that the error dynamics of the observer are input-to-state stable. Result [ ] 4.0ɛ 1 K x(ε) = 5.04ɛ 2 (5) K φ (ε) = 4.0ω0ɛ ζω 0 ɛ ɛ 3 (6)

18 13 Nonlinear Observer Parameter Estimator Need to find an update law ˆθ = u θ (ν, ˆx, ˆφ, ˆθ) (7) such that the origin of the error dynamics θ = u θ (ν, ˆx, ˆφ, θ θ) (8) is uniformly exponentially stable whenever ˆx = x and ˆφ = φ.

19 13 Nonlinear Observer Parameter Estimator Need to find an update law ˆθ = u θ (ν, ˆx, ˆφ, ˆθ) (7) such that the origin of the error dynamics θ = u θ (ν, ˆx, ˆφ, θ θ) (8) is uniformly exponentially stable whenever ˆx = x and ˆφ = φ. Assumption 6 There exist a differentiable function V u : R 0 (Θ Θ) R 0 and positive constants a 1,..., a 4 such that for all (t, θ) R 0 (Θ Θ), a 1 θ 2 V u(t, θ) a 2 θ 2 V u (t, t θ) Vu θ (t, θ)u θ (ν, x, φ, θ θ) a 3 θ V u θ (t, θ) a 4 θ Furthermore, the update law (7) ensures that if ˆθ(0) Θ, then for all t 0, ˆθ Θ.

20 13 Nonlinear Observer Parameter Estimator Need to find an update law ˆθ = u θ (ν, ˆx, ˆφ, ˆθ) (7) such that the origin of the error dynamics θ = u θ (ν, ˆx, ˆφ, θ θ) (8) is uniformly exponentially stable whenever ˆx = x and ˆφ = φ. Assumption 6 satisifed by the following update law: ( u θ (ν, ˆx, ˆφ, ˆθ) = Proj ΓM(ν, ˆx, ˆθ)( ˆφ g(ν, ˆx, ˆθ) ) ] M = [tanh(m 1 2 Mmax rate(ˆd D M )) + 1 (9) (10)

21 14 Nonlinear Observer Assumptions Bounded topography, well defined input signal and time-derivative

22 14 Nonlinear Observer Assumptions Bounded topography, well defined input signal and time-derivative Operation in noncontact mode allows the force profile to be represented by a monotonic function

23 14 Nonlinear Observer Assumptions Bounded topography, well defined input signal and time-derivative Operation in noncontact mode allows the force profile to be represented by a monotonic function Known Lennard-Jones parameters Parameters appear linearly in Lennard-Jones function Method can easily be extended to simultaneously determine these.

24 15 Nonlinear Observer Stability Theorem Theorem 1 (Abbreviated) If all assumptions are satisfied and ˆθ(0) Θ, there exists 0 < ε 1 such that for all 0 < ε ε, the origin of the error dynamics of the observer and parameter estimator is exponentially stable.

25 16 Outline Introduction System Modeling Observer Design Simulation Results Conclusion and Further Work

26 16 Simulation Results Height (nm) Position (nm) Topography estimate with output noise. θ ˆθ ˆθ ekf

27 17 Simulation Results 0 Force (pn) Position (nm) φ ˆφ ˆφ ekf Estimated interaction force φ with noise.

28 18 Outline Introduction System Modeling Observer Design Simulation Results Conclusion and Further Work

29 18 Conclusion New dynamic mode imaging method. Cantilever dynamics and interaction force "inverted" to estimate topography. Avoids bandwidth-limiting amplitude estimation. Exponentially stable observer.

30 19 Further Work Experimental results. Include estimation of Lennard-Jones parameters. Use observer in feedback loop for control. Proper analysis of closed-loop bandwidth.

31 19 Further Work Experimental results. Include estimation of Lennard-Jones parameters. Use observer in feedback loop for control. Proper analysis of closed-loop bandwidth. Questions?

32 20 References Håvard Fjær Grip, Ali Saberi, and Tor A Johansen. Estimation of states and parameters for linear systems with nonlinearly parameterized perturbations. Systems & Control Letters, 60(9): , Michael R P Ragazzon, J Tommy Gravdahl, Kristin Y Pettersen, and Arnfinn A Eielsen. Topography and Force Imaging in Atomic Force Microscopy by State and Parameter Estimation. In American Control Conference, 2015, 2015.