Exponential Convergence Bounds in Least Squares Estimation: Identification of Viscoelastic Properties in Atomic Force Microscopy

Transcription

1 Exponential Convergence Bounds in Least Squares Estimation: Identification of Viscoelastic Properties in Atomic Force Microscopy Michael R. P. Ragazzon, J. Tommy Gravdahl, Kristin Y. Pettersen Department of Engineering Cybernetics, NTNU Norwegian University of Science and Technology IEEE Conference on Control Technology and Applications (CCTA), Kohala Coast, Hawai i, August 27-30, 2017

2 1 Introduction Using atomic force microscopy (AFM) for studying soft, biological material has become increasingly popular in recent years.

3 1 Introduction Using atomic force microscopy (AFM) for studying soft, biological material has become increasingly popular in recent years. A recent modeling and identification approach allows recording of viscoelastic properties of the sample.

4 1 Introduction Using atomic force microscopy (AFM) for studying soft, biological material has become increasingly popular in recent years. A recent modeling and identification approach allows recording of viscoelastic properties of the sample. Exponential convergence of parameters are guaranteed but even exponential convergence can be slow.

5 1 Introduction Using atomic force microscopy (AFM) for studying soft, biological material has become increasingly popular in recent years. A recent modeling and identification approach allows recording of viscoelastic properties of the sample. Exponential convergence of parameters are guaranteed but even exponential convergence can be slow. Would like a pre-determined upper bound on how fast parameters converge.

6 2 Introduction Atomic Force Microscopy (AFM) Laser z-scanner Cantilever Photo detector Sample xy-scanner

7 2 Introduction Atomic Force Microscopy (AFM) Laser z-scanner Cantilever Photo detector Sample xy-scanner Can use AFM as a force sensor.

8 3 Introduction Sample Modeling z y x Lumped spring-damper elements along the lateral xy-axes. Spatially resolved by moving AFM tip over each element and indenting.

9 4 Introduction Time interval between indentations specified by operator.

10 4 Introduction Time interval between indentations specified by operator. Thus, parameters are time-varying, piecewise constant, at a user-defined constant interval. 0.5 θ (t) θ(t) 0 θ i θ i 0.5 t i t i+1 t i+2 t i+3 Time

11 5 Introduction 0.5 θ (t) θ(t) 0 θ i θ i 0.5 t i t i+1 t i+2 t i+3 Time Goal: Parameter estimation Determine the time interval necessary at each identation tap in order to guarantee convergence to some specified accuracy.

12 6 Outline Introduction RLS Estimation Parameter Convergence Example: Viscoelastic Identification Conclusions

13 6 Recursive Least Squares (RLS) Estimation Plant model and estimation model given by z = θ T φ, (1) ẑ = θ T φ, (2) Update law of the recursive least-squares algorithm with forgetting factor given by θ = Pεφ (3) where Ṗ = βp PφφT P m 2 (4) ε = z ẑ m 2 (5) m 2 = 1 + αφ T φ (6)

14 7 Persistency of Excitation The regressor vector φ is said to be persistently exciting (PE) if there exists constants, α 0, α 1, T 0 > 0 such that α 0 I 1 t+t0 φφ T dτ α 1 I, t 0. (7) T 0 t where I is the identity matrix and α 0 is known as the level of excitation.

15 8 Outline Introduction RLS Estimation Parameter Convergence Example: Viscoelastic Identification Conclusions

16 8 Constant Parameters Lemma 1: Bounds on P(t) If m, φ L, φ is PE, and θ is constant, then the least squares algorithm given by (5)-(4) guarantees the following bounds on P(t): with γ 1 I P(t) γ 2 I, t 0 (8) γ 1 = (λ min (P 0 ) 1 + (αβ) 1) 1 (9) { } m 2 γ 2 = max, λ max(p 0 ) e βt 0 (10) α 0 T 0 where m 2 = sup t m 2 (t) and λ min ( ), λ max( ) denotes the minimum and maximum eigenvalue, respectively.

17 9 Constant Parameters Exponential convergence can be proved using the Lyapunov-like function where θ θ θ and Γ = P 1 (t). V = θ T Γ θ 2. (11)

18 10 Constant Parameters Lemma 2: Exponential Convergence If m, φ L, φ is PE, and θ is constant, then the least squares algorithm guarantees that V from (11) decreases according to with 0 < γ < 1, where V (t + T 0 ) γv (t), t 0 (12) γ = 1 µ 1 + βt 0 (13) α 0 T 0 γ 1 µ = 2 m 2 + φ 4 T0 2γ2 2 (14) and φ = sup t φ.

19 11 Constant Parameters Theorem 1: Parameter Convergence Let m, φ L, φ be PE, and θ constant. Then, the least squares algorithm guarantees θ(t) ae λ(t t 0) θ(t 0 ), t t 0 (15) for any t 0 0, where the constants a > 1, λ > 0 are given by a = γ2 γγ 1, λ = log γ 2T 0. (16) Furthermore, if t 0 = 0, a less conservative bound is given by γ2 a = γλ min (P 0 ). (17)

20 12 Time-Varying Parameters 0.5 θ (t) θ(t) 0 θ i θ i 0.5 t i t i+1 t i+2 t i+3 Time θ (t) = θ i, {t, i} : t [t i, t i+1 ), i N, t i = it (18) θ i θ i 1 θ, i. (19)

21 13 Time-Varying Parameters Theorem 2: Time Interval for Convergence Let m, φ L, φ be PE, and the parameter vector θ (t) be described by (18) and satisfy (19). Then, the least squares algorithm guarantees ( ) θ i ae λt i+1 ( ) ae λt i 1 θ(0) + ae λt 1 ae λt θ, i N. (20) Furthermore, if ae λt < 1, then, for large i, θ i R θ (21) where R = ae λt / ( 1 ae ) λt. Conversely, for some specified R > 0, T is given by T = λ 1 a (R + 1) log. (22) R

22 14 Outline Introduction RLS Estimation Parameter Convergence Example: Viscoelastic Identification Conclusions

23 14 Cantilever Dynamics Detector Mirror z U Cantilever base Cantilever dynamics, second-order oscillator: Z Tip D Cantilever deflection Ms 2 Z CsD KD = (c s + k ) δ, (23) z 1 h 3 δ 3 z 2 R z 3 Sample x 1 x 2 X x x 3

24 15 Parametric System The parametric system (1) can now be set up as follows, θ = [ c k ] T (24) φ = [ sδ δ ] T /Λ(s) ( ) (25) z = Ms 2 Z CsD KD /Λ(s) (26) ( 2 1/Λ(s) = 1/ ωc 1 s + 1) (27)

25 16 Tuning and Convergence Rate θ 10 4 t = 1T 0 t = 5T 0 t = 10T 0 t = 15T 0 t = 20T Upper limit on convergence of parameter error after t = nt 0, as a function of β, with t 0 = 0, p 0 = Circles mark minimum points. β

26 17 Tuning and Convergence Rate 10 1 T [s] R = 1 R = 0.1 R = 0.01 R = Time-varying parameters: Minimum estimation interval T necessary to guarantee parameter estimate to within R θ of real parameters, as a function of β. β

27 18 Simulation Setup U F ts Sample k, c Cantilever dynamics Z D D U Signal filtering z φ Least squares estimator k, c

28 19 Simulation Results c [Ns/m] Real k [N/m] Real Estimated 1 Estimated Time [s] Time [s]

29 20 Simulation Results 10 6 Real error Upper bound θ R θ 10 0 θ(t) Time [s] Parameter upper bound versus simulated error, with R =

30 21 Simulation Results 10 6 Real error Upper bound θ 10 0 θ(t) Time [s] Upper bound with covariance reset between intervals.

31 22 Ongoing Experimental Results

32 c (Ns/m) k (N/m) Y [µm] Introduction RLS Estimation Parameter Convergence Example: Viscoelastic Identification Conclusions 22 Ongoing Experimental Results Elastic Modulus Pa Time (s) X [µm]

33 23 Outline Introduction RLS Estimation Parameter Convergence Example: Viscoelastic Identification Conclusions

34 23 Conclusions Upper bound on estimation error in RLS found.

35 23 Conclusions Upper bound on estimation error in RLS found. Time interval found for convergence in the case of piecewise constant parameters.

36 23 Conclusions Upper bound on estimation error in RLS found. Time interval found for convergence in the case of piecewise constant parameters. Results shown to be useful for tuning.

37 23 Conclusions Upper bound on estimation error in RLS found. Time interval found for convergence in the case of piecewise constant parameters. Results shown to be useful for tuning. Applied to viscoelastic identification in AFM already proven useful in experiments.

38 23 Conclusions Upper bound on estimation error in RLS found. Time interval found for convergence in the case of piecewise constant parameters. Results shown to be useful for tuning. Applied to viscoelastic identification in AFM already proven useful in experiments. Questions?