Parameter identification using the skewed Kalman Filter

Transcription

1 Parameter identification using the sewed Kalman Filter Katrin Runtemund, Gerhard Müller, U München

2 Motivation and the investigated identification problem Problem: damped SDOF system excited by a sewed process wanted: estimation of the stiffness, damping paramter a 0 =/m and a 1 =c/m given: mathematical model of the system the statistical characteristics of the input force wind data was downloaded from the internet database: Database of Wind Characteristics located at DU, Denmar, from the Oa Cree wind farm site the true system response w(t) was simulated using the wind data

3 Wind speed measurement mast 1; heigth 79 m; frequency 8 Hz; duration 600s Statistics mean 0 m/s; variance 0,69 m/s; sewness -1,99 website: of DU

4 Overview Sewed Kalman Filter Extension of the sewed Kalman Filter for parameter identification problems Identification results of the application Conclusion

5 Sewed Kalman Filter

6 Sewed Kalman Filter (skf) rend & Noise Kalman Filter Sewed Kalman Filter introduced by P. Naveau, M. Genton and X. Shen Paper: A sewed Kalman Filter, Journal of Multivariate Analysis, 2004 based on the closed sew-normal distribution (CSN) introduced by G. González-Farías, J.A. Domínguez-Molina, A. K. Gupta (2004) Figure: M. G. Genton: Sew-Elliptical Distributions and their Applications A Journal Beyond Normality 1/19

7 Closed sew-normal distribution (CSN) family of distributions including the normal one, but with 3 additional parameters allows a continuous variation from a normal to a half-normal distribution closed under linear transformation marginalization conditioning summation* 2/19

8 Closed sew-normal distribution (CSN) CSN probability density function of a n-dimensional random vector X X ~ CSN n,m μ, Σ, D, ν, Δ p n,m x = φ n x; μ, Σ Φ m D x μ ; ν, Δ Φ m 0; ν, Δ + DΣD p n,m x = φ n x; μ, Σ Φ m D x μ ; ν, Δ mean values: Φ m 0; ν, μ ϵ Δ R+ n and DΣD ν ϵ R m φ n ; η, Ω = N η, Ω covariance matrices: Σ ϵ R n n and Δ ϵ R m m ΦΦ n n ; ; η, η, Ω Ω : : cumulative distribution distribution function function arbitrary matrix D ϵ R m n C = Φ m 0; ν, Δ + DΣD = const. D regulates sewness continuous variation from the normal (D = 0) to a half normal PDF μ, Σ location and scale parameters ν closure under conditioning Δ closure under marginalization C closure under summation 3/19

9 Standard Kalman Filter he standard KF is based on a linear state space model linear system equation: x x Bf Sw w ~ N 0, observation equation: z Ax the state follows a first order hidden Marov process represented as Bayesian Dynamic Networ 1 v v ~ N 0, v w 4/19

10 Implementation of sewness into the state space model Alternative 1: Introducing the sewness in the sytem equation by modeling the input force as additive CSN noise: f ~ CSN,,D,, x sewed x 1 noise Bf Sw sewed problem: the sum of two CSN of dimension (n, m) is of dimension (n, 2m) rapid increase of the dimension of the matrices step D, Δ at each time 5/19

11 Alternative 2: splitting up the state in a linear and a sewed part generation of the sewed process where D,,D,, ~ CSN D Y X D D D D, N ~ Y X 1 y y s Implementation of sewness into the state space model 1 Ly y x = u, s 6/19 noise sewed linear v s Au Gx w u S z S u 1 v Ax w f x z S B x 1

12 Alternative 2: splitting up the state in a linear and a sewed part generation of the sewed process step 1: time-update step 2: determine the joint distribution step 3: conditioning Implementation of sewness into the state space model noise sewed linear v s Au Gx w u S z S u 1 v Ax w f x z S B x 1 1 L y y 1 1 1, 1 1, 1 1 L L L L, L ~ N y y,,d,, ~ CSN D 1 y y s 7/19 x = u, s

13 Bayesian approach of the sewed Kalman filter 1. Initialising the filter using the prior PDF p u 1, y 1 z 1: 1 he filter is initialized by the multivariate norm ibuted 2. posterior Prediction: PDF of time-update u 1, y 1 z 1: 1 conditional nown previous from on time the observations step. In the up prediction to time step th estimate -1 of the using multivariate the system normal equations distributed p u variable, y, y 1 z 1: 1 conditional on the observ by 1: 1 the up multivariate to time normal 1 is distributed determined posterior using eq. PDF observing (4). of uafter 1 z,, y 1 the z 1: 1 measurement nown from lielihood the p, ythe, y3. prediction 1, zmeasurement 1: 1 step is the obtained prior update estimate from after the of observing error the Amultivariate eμ u, 1 = z : S normal Ε s z 1: 1 distributed between variable the predicte he onditional incoming from on measurement the the observations prediction using z 1: 1 error the conditional up e to = time z Ε expectation As z μ1 1: 1 u, 1 is determined of Bthe Ε priori s z using 1: 1 PDF. eq. Applying (4). After the Ba asurement e the posterior lielihood PDF of p the z, updated u, y, y 1 parameters, z 1: 1 uof is the zobtained 1: linear and the from using sewed the part error observation y ez 1: = can z be deriv his 1 between leads equation to the eq. predicted (6) and the incoming measurement using the conditional expectation ri PDF. Applying the Bayes rule the posterior PDF of the updated parameters of the linear part y4. z 1: Applying can be derived the Bayes [1]. his Rule leads leads to eq. (6) to the posterior PDF p u, y, y 1 z 1: = p z u, y, y 1 p u, y, y 1 z 1: 1 p z z 1: 1 8/19

14 Bayesian approach to the sewed Kalman filter p u, y, y 1 z 1: = p z u, y, y 1 p u, y, y 1 z 1: 1 p z z 1: 1 5. Marginalization leads to the posterior PDF of the linear part p u z 1: 6. Marginalization and Conditioning with respect to leads to the p u posterior PDF of the sewed, y, y part 1 z 1: = p z u, y, y 1 p u, y, y 1 z 1: p s z 1: p z z 1: 1 Advantages of the method: computationally efficient: the closure properties of the CSN allow to reduce the filter algorithm to the recursive calculation of the parameters At each time step a different sewness can be implemented by changing the matrix of the process y L y L 1 9/19

15 PREDICION (IME UPDAE) Project the state ahead (prior estimate) Linear part μ u, = μ u, 1 Σ uu, = Σ uu, 1 + S Σ ww, 1 S Sewed part μ y, = L μ y, 1 Σ yy, = L Σ yy, 1 L + Σ ηη, D = Σ yy, 1 L 1 Σ yy, ν = μ y, 1 D μ y, Δ = Σ yy, 1 D Σ yy, D CORRECION (MEASUREMEN UPDAE) Compute the Kalman gain matrix of the linear and sewed part K = Σ uu, A Σ vv, + A Σ uu, A + S Var[s z 1: 1 ]S 1 K s, = C S Σ vv, + A Σ uu, A + S Var[s z 1: 1 ]S 1 Update estimate with measurement (posterior estimate) e = z A μ u, S E s z 1: 1 Linear part μ y, = μ u, + K e Σ uu, = Σ uu, K A Σ uu, Sewed part μ y, = μ y, + K s, e Σ yy, = Σ yy, K s, S C D = Σ yy, L 1 Σ yy, ν = μ y, 1 +C 1 C 1 K s, e D μ y, Δ = Σ yy, D Σ yy, D 10/19

16 PREDICION (IME UPDAE) Project the state ahead (prior estimate) Linear part μ u, = μ u, 1 Σ uu, = Σ uu, 1 + S Σ ww, 1 S Sewed part μ y, = L μ y, 1 Σ yy, = L Σ yy, 1 L + Σ ηη, D = Σ yy, 1 L 1 Σ yy, ν = μ y, 1 D μ y, Δ = Σ yy, 1 D Σ yy, D CORRECION (MEASUREMEN UPDAE) Compute the Kalman gain matrix of the linear and sewed part K = Σ uu, A Σ vv, + A Σ uu, A + S Var[s z 1: 1 ]S 1 K s, = C S Σ vv, + A Σ uu, A + S Var[s z 1: 1 ]S 1 Update estimate with measurement (posterior estimate) e = z A μ u, S E s z 1: 1 Linear part μ y, = μ u, + K e Σ uu, = Σ uu, K A Σ uu, Sewed part μ y, = μ y, + K s, e Σ yy, = Σ yy, K s, S C D = Σ yy, L 1 Σ yy, ν = μ y, 1 +C 1 C 1 K s, e D μ y, Δ = Σ yy, D Σ yy, D Sewed Kalman Filter Kalman Filter 10/19

17 Extended sewed Kalman Filter (EsKF)

18 Parameter identification using the Extended Kalman Filter (EKF) Example: damped single degree of freedom system wanted: estimation of the stiffness, damping paramter a 0 =/m and a 1 =c/m u u Sw 1 Problem: he transfer matrices,-1, B,-1, S,-1 depend on the unnown parameters: = e δ t δ ω d sin ω d t + cos ω d t ω 0 2 ω d sin ω d t 1 ω d sin ω d t cos ω d t δ ω d sin ω d t a a m c 2 m d /19

19 Parameter identification using the Extended Kalman Filter (EKF) step 1: Extending the state by the parameter vector p to include the model parameters to be identified x ext, = x p = w v a 0, a 1, extended state space model x ext, = f x 1, p 1, u 1, w 1, w p, 1 z ext, = A ext, x ext, + v step 2: Approximation of the nonlinear function f by a first order aylor series near the current state estimate xˆ 1 x ext, f x 0, 1 + f x 0, 1 x 0i, 1 x 0i, 1 x 0i, 1 x 0i, 1 x 0, 1 = x 1 p 1 u 1 w 1 w p, 1 x 1 p 1 u = x 0, 1 12/19

20 Parameter identification using the Extended Kalman Filter (EKF) step 2: Approximation of the nonlinear function f by a first order aylor series near the current state estimate xˆ 1 x ext, f x 0, 1 + ext, x 1 x 1 p 1 p p, 1 ε ext,x, 1 + B ext, u 1 u 1 ε ext,u, 1 + S ext, w 1 w p, 1 ε ext,w, 1 ext, = p 1 J f, p 1 with J f, p I 1 = f x 0, B ext, = B p p 1 x0i, 1 S ext, = S p 1 J f, w p, 1 with J f, w p, 1 = f x 0, I 2 2 w p, 1 x 0i, 1 Jacobian Matrix w a J f p = 0 w a 1 v a 0 v a 1 13/19

21 Parameter identification using the Extended sewed Kalman Filter (EsKF) in case of the sewed state space model, the linearization is done around the current state of the linear part linear part : sewed part : u y ext, L ext, -1 y 1 u ext, 1 S ext, -1 s w ext, y y 1 D z Gx A ext u A ext S ext, -1 s v A ext the linearized extended state space model can now be investigated using the introduced sewed Kalman filter algorithm where the time-variante transfer matrices, S extended matrices ext,, S ext, are replaced by the 14/19

22 Identification results

23 Approximation of the PDF of the measurement wind data 15/19

24 Identification results identified identification true initial initial error parameters error values values EKF EsKF EKF EsKF a 0 [s 2 ] % 20,40 101,84 79,6 % 2 % a 1 [s 1 ] 0,2 0,3 50 % 1,18 0, % 5% 16/19

25 Predicted displacement and generated load 17/19

26 Conclusion he method is computational efficient he method wored well for the introduced identification problem he CSN allows to describe the characteristics of the regarded sewed wind data he form of the temperal evaluation of the wind data was approximated well, but the amplitude was wrong by a factor 100 Further wor Find the error.any comments? Investigate the sensitivity of the parameter identification results depending on the initial conditions Extending the method to a multi-degree of freedom system 19/19

27 hans! Any questions or comments? 19/19

28 PREDICION (IME UPDAE) Project the state ahead (prior estimate) Linear part μ u, = μ u, 1 Σ uu, = Σ uu, 1 + S Σ ww, 1 S Sewed part μ y, = L μ y, 1 Σ yy, = L Σ yy, 1 L + Σ ηη, D = Σ yy, 1 L 1 Σ yy, ν = μ y, 1 D μ y, Δ = Σ yy, 1 D Σ yy, D CORRECION (MEASUREMEN UPDAE) Compute the Kalman gain matrix of the linear and sewed part K = Σ uu, A Σ vv, + A Σ uu, A + S Var[s z 1: 1 ]S 1 K s, = C S Σ vv, + A Σ uu, A + S Var[s z 1: 1 ]S 1 Update estimate with measurement (posterior estimate) e = z A μ u, S E s z 1: 1 Linear part μ y, = μ u, + K e Σ uu, = Σ uu, K A Σ uu, Sewed part μ y, = μ y, + K s, e Σ yy, = Σ yy, K s, S C D = Σ yy, L 1 Σ yy, ν = μ y, 1 +C 1 C 1 K s, e D μ y, Δ = Σ yy, D Σ yy, D 18/19