A. Poulimenos, M. Spiridonakos, and S. Fassois

Transcription

1 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS: AN OVERVIEW AND COMPARISON A. Poulimenos, M. Spiridonakos, and S. Fassois DEPARTMENT OF MECHANICAL & AERONAUTICAL ENGINEERING, GR 265 PATRAS, GREECE sms ISMA26 International Conference on Noise and Vibration Engineering Leuven, Belgium, September 26

2 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 1 TALK OUTLINE 1. INTRODUCTION 2. THE PRESENT PARADIGM (CASE STUDY) 3. THE GENERAL TARMA MODEL REPRESENTATION 4. MODEL PARAMETER ESTIMATION 5. MODEL STRUCTURE ESTIMATION 6. NON-STATIONARY VIBRATION MODELLING RESULTS 7. CONCLUDING REMARKS

3 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 2 1. INTRODUCTION The General Problem Modelling and analysis of non-stationary vibration. Problem Characteristics Non-stationary stochastic signals are characterized by time-varying statistics require time-frequency methods for their analysis. Mechanical Vibration Acceleration (cm/sec 2 ) Output signal (location 3) Time (sec)

4 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 3 Problem Significance Non-stationary random vibration is commonly encountered in practice: Seismic & structural systems Sea vessels Rotating machinery Automotive & aircraft systems Robotic devices and so on Methods for Non-stationary Random Vibration Modelling and Analysis Non-parametric (Hammond & White 1996; Cohen 1994; Spanos & Failla 25) Parametric (Poulimenos & Fassois 26; Niedzwiecki 2; Kitagawa & Gersch 1996; Owen et al. 21; Cooper & Worden 2) Spectrogram (STFT) Cohen class of distributions Wavelet-based methods TARMA & State-space Unstructured parameter evolution Stochastic parameter evolution Deterministic parameter evolution

5 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 4 Why Parametric Methods? They offer a number of advantages over non-parametric counterparts: Representation parsimony Improved accuracy and resolution Improved tracking of the time-varying dynamics Flexibility in analysis Aims of the Work 1. Critical overview of parametric time-domain methods for non-stationary random vibration modelling and analysis. 2. Comparative assessment of the methods via Monte-Carlo experiments (simulated non-stationary vibration signal with precisely known characteristics).

6 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 5 2. THE PRESENT PARADIGM (CASE STUDY) THE PROBLEM: Modeling, analysis and prediction of non-stationary vibration. The Underlying System m 3 k 3 (t) c 3 x 3 Time-varying stiffnesses k 2 (t) and k 3 (t): k i (t) = k i, + k i,1 sin(2πt/p i,1 ) + k i,2 sin(2π/p i,2 ) m 2 k 2 (t) m 1 k 1 c 2 x 2 x 1 r(t) Symbol Value m 1.5 kg m kg m 3 1. kg c 2.5 N/(m/s) c 3.3 N/(m/s) k 1 3 N/m k 2 (t) k 2, = 1 (N/m), k 2,1 = 6, k 2,2 = 2 (N/m) P 2,1 = 17, P 2,2 = 85 (s) k 3 (t) k 3, = 12 (N/m), k 3,1 = 72, k 3,2 = 24 (N/m) P 3,1 = , P 3,2 = (s)

7 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 6 System Properties 6 5 ω n3 Frequency (Hz) ω n2 1 ω n1 The Non-stationary Vibration Signal Time (sec) STFT Displacement Time (sec)

8 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 7 3. THE GENERAL TARMA MODEL REPRESENTATION A TARMA(n a, n c ) model is of the form: x[t] + n a a i [t] x[t i] i=1 }{{} AR part = e[t] + n c c i [t] e[t i], i=1 }{{} MA part t t, e[t] NID(, σ 2 e[t]) t : normalized discrete time n a, n c : orders of the AR/MA polynomials x[t] : the non-stationary response signal a i [t], c i [t] : AR/MA time-varying parameters e[t] : innovations (residual) sequence Classes of Parametric Methods Parameter evolution Representation parsimony Dynamics evolution UNSTRUCTURED low slow STOCHASTIC low slow/medium DETERMINISTIC high slow/medium/fast

9 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 8 4. MODEL PARAMETER ESTIMATION Estimation of the AR, MA parameter vector and the residual variance at all time instants for selected model form and structure [ ] T, θ[t] = a 1 [t]... a na [t]. c }{{} 1 [t]... c nc [t] σ 2 }{{} e [t], (t = 1,..., N) AR parameters MA parameters }{{} residual variance Unstructured Parameter Evolution (UPE-TARMA) Models The AR/MA parameters do not have a structured time-dependency, and are free to change with time. Their estimation may be achieved either by locally stationary or recursive methods. Locally stationary methods [Short Time ARMA (ST-ARMA) estimation] Conventional, stationary ARMA modelling to successive, short (approximately stationary), time segments. Acceleration a) b) Time (sec)

10 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 9 Recursive methods The AR/MA parameter vector estimate θ[t] at each time instant t is recursively updated at the time instant t + 1 that the next signal sample becomes available. 1 Acceleration Time (sec) t 1 t ˆθ[t] = ˆθ[t 1] + k[t] ê[t t 1] }{{} prediction error θ[t] = [ a 1 [t]...a na [t]. c 1 [t]... c nc [t] ] T Exponentially Weighted Prediction Error criterion: ˆ θ[t] = arg min θ[t] t λ t τ e 2 [τ, θ τ 1 ] τ=1 The RML-TARMA method is presently used (Ljung 1999).

11 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 1 RML-TARMA Estimation Estimator update: ˆθ[t] = ˆθ[t 1] + k[t]ê[t t 1] Prediction error: ê[t t 1] = x[t] ˆx[t t 1] = x[t] φ T [t]ˆθ[t 1] Gain: P [t 1]ψ[t] k[t] = λ + ψ T [t]p [t 1]ψ[t] Covariance update: P [t] = 1 ( P [t 1] P [t ) 1]ψ[t]ψT [t]p [t 1] λ λ + ψ T [t]p [t 1]ψ[t] Filtering: ψ[t] + ĉ 1 [t 1]ψ[t 1] ĉ nc [t 1]ψ[t n c ] = φ[t] A-posteriori error: ê[t t] = x[t] φ T [t]ˆθ[t] φ[t] = [ ] T x[t 1]... x[t n α ]. ê[t 1 t 1]... ê[t n c t n c ] Initialization: ˆθ[] =, P [] = αi with α designating a large positive number. The signal and a-posteriori error initial values are set to zero.

12 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 11 Stochastic Parameter Evolution (SP-TARMA) Models (Kitagawa & Gersch 1996) The AR/MA parameters are considered as being random variables allowed to change with time, with their evolution subject to stochastic smoothness constraints: κ α i [t] = w αi [t] κ c i [t] = w ci [t], ( = 1 B).2.3 α i Time (sec) SP-TARMA models may be set into linear state space form: t 2, t 1, t α i [t] = 2 α i [t 1] α i [t 2] + w αi [t] (κ = 2) z[t] = F z[t 1] + G w[t], x[t] = h T [t,z t 1 ] z[t] + e[t] z[t] = [ a 1 [t]... a na [t] c 1 [t]...c nc [t].....a 1 [t κ+1]...a na [t κ+1] c 1 [t κ+1]... c nc [t κ+1] ] T Estimation of the state vector z[t] at each time instant may be achieved by the Kalman Filter. A smoothed estimate ẑ[t N] of the state vector may be obtained via a backward smoothing algorithm.

13 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 12 Kalman Filter and Backward Smoothing for the Estimation of SP-TARMA Models (normalized form) Time update (prediction): State prediction: ẑ[t t 1] = Fẑ[t 1 t 1] Prediction error: ê[t t 1] = x[t] h T [t]ẑ[t t 1] Covariance update: P [t t 1] = F P [t 1 t 1]F T + G Q[t]G T Observation update (filtering): Gain: k[t] = P [t t 1]h[t] State update: ẑ[t t] = ẑ[t t 1] + k[t]ê[t t 1] Covariance update: P [t t] = ( I k[t]h T [t] ) P [t t 1] ( ) 1 h T [t] P [t t 1]h[t] + 1 Smoothing: A[t] = P [t t]f T P 1 [t + 1 t] ẑ[t N] = ẑ[t t] + A[t] (ẑ[t + 1 N] ẑ[t + 1 t]) ( ) P [t N] = P [t t] + A[t] P [t + 1 N] P [t + 1 t] A T [t] P [t t] = P [t t] σe[t], P [t t 1] 2 = P [t t 1], Q[t] σe[t] 2 = Q[t] σe[t] = σ2 w[t] I 2 σ }{{} e[t] 2 na ν[t] Initialization: ˆθ[] =, P [] = αi with α designating a large positive number.

14 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 13 Deterministic Parameter Evolution (FS-TARMA) Models (Grenier 1989, Poulimenos & Fassois 26) The AR/MA parameters and innovations variance are deterministic functions of time, belonging to specific functional subspaces: F AR = {Gba (1)[t],..., G ba (p a )[t]}, F MA = {Gbc (1)[t],..., G bc (p c )[t]}, F σ 2 e = {Gbs (1)[t],..., G bs (p s )[t]} α i Time (sec) G ba (1)[t] G ba (2)[t] G ba (2)[t] Time (sec) 2 2 G ba (4)[t] G ba (5)[t] G ba (6)[t] Time (sec) 1 Time (sec) a i [t] = 6 j=1 a i,j G ba (j)[t], a i,j = [ ] T Estimation of the model parameters a i,j, c i,j, s j (coefficients of projection), may be based upon a Prediction Error criterion (non-quadratic problem). The original PE problem may be tackled by: linear multistage methods: 2SLS method, P-A method recursive methods: Recursive Extended Least Squares (RELS) method

15 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 14 Multistage Identification of FS-TARMA Models The Two-Stage Least Squares (2SLS) method The Polynomial-Algebraic (P-A) method NO x[t] (t = 1,...,N) Inverse Function Estimation AR/MA Projection Coefficient Estimation î ˆϑ RSSn RSS n 1 ε RSS n 1 YES Innovations Variance Estimation ˆϑ, ŝ A truncated FS-TAR model is estimated through OLS. The FS-TARMA model is approximated by replacing the past, but not the current, values of the prediction error e[t, ϑ] by the obtained residual series e[t, î]. The estimation of the AR/MA coefficient of projection vector ϑ is then reduced to a quadratic optimization problem. NO x[t] (t = 1,...,N) Inverse Function Estimation Initial AR/MA Projection Coefficient Estimation Signal Filtering Final AR/MA Projection Coefficient Estimation RSSn RSS n 1 ε RSS n 1 YES Innovations Variance Estimation A truncated FS-TAR representation is estimated through OLS. Initial estimates of the AR and MA coefficients of projection are obtained based on the definition of the inverse function. A filtered signal x[t] is obtained by consecutive filtering operations. The final AR coefficients of projection are estimated based on the filtered signal. The final MA coefficient of projection estimates are obtained through deconvolution. ˆϑ, ŝ

16 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS MODEL STRUCTURE ESTIMATION Estimation of the model orders and additional structural parameters for a selected model class M = {n a, n c } M SP = {n a, n c, κ} M FS = {na, n c, F AR, F MA, F σ 2 e } n a, n c : AR/MA orders κ : smoothness constraints order F : functional subspace General approach: minimization of the Bayessian Information Criterion (BIC) or the log-likelihood function. BIC = [ N 2 ln 2π N ( ln(σe[t]) 2 + e2 [t] σe[t] 2 ) ] t=1 }{{} ln L(x N ) + lnn 2 d: the number of independently estimated model parameters d

17 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 16 Integer Optimization Scheme (Poulimenos and Fassois 23) Minimization of the BIC is achieved via a direct, hybrid, two-phase optimization scheme: PHASE I. Coarse ( global ) optimization. is applied to the complete search space aiming at locating promising regions. Approach: Genetic Algorithms PHASE II. Fine ( local ) optimization. operates in the neighborhood of each Phase I result aiming at locating the exact optimum point. Approach: Backward Regression

18 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS NON-STATIONARY VIBRATION MODELLING RESULTS Model structure selection: (a) TARMA(n,n) models (n = 2, 4,...) are estimated (b) MA order reduction is considered Criteria: BIC and log-likelihood function BIC BIC AR order MA order

19 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 18 Identified Models Model Class Identification Method Method Characteristics Identified Model Unstructured ST-ARMA M = 31; m = 1; PE estimation ST-ARMA(6,3) Parameter (Levenberg-Marquardt: Evolution termination rule ˆϑ i ˆϑ i 1 < 1 6 ) RML-TARMA λ =.978; initl. covariance α = 1 4 RML-TARMA(6,3) Stochastic SP-TARMA ELS-like estimation; α = 1 4 SP-TARMA(6,3) κ = 2 Parameter ˆν = Evolution SP-TARMA (smoothed) Deterministic FS-TARMA (2SLS) n i = 2; QR implementation of OLS FS-TARMA(6,3) [41,2,19] Parameter FS-TARMA (2SLS-PE) Evolution (Levenberg-Marquardt) FS-TARMA (RELS) initl. covariance α = 1 4 SP-TARMA(6, 3) models: α i [t] = 2 α i [t 1] α i [t 2] + w αi [t], c i [t] = 2 c i [t 1] c i [t 2] + w ci [t] FS-TARMA(6, 3) models: trigonometric functional subspaces

20 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 19 Model-Based Natural Frequency Estimates 6 : Theoretical : FS TARMA (2SLS PE) : FS TARMA (RELS) : SP TARMA (smoothed) ω n3 5 Frequency (Hz) ω n2 1 ω n Time (sec)

21 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 2 Model-Based Natural Frequency Estimates 6 : Theoretical : RML TARMA : SP TARMA ω n3 5 Frequency (Hz) ω n2 1 ω n Time (sec)

22 PARAMETRIC T IME -D OMAIN M ETHODS FOR N ON -S TATIONARY R ANDOM V IBRATION I DENTIFICATION AND A NALYSIS 21 Model-Based Frozen Vibration Analysis ST-ARMA(6,3) S TOCHASTIC M ECHANICAL S YSTEMS & AUTOMATION (SMSA) L ABORATORY RML-TARMA(6,3) U NIVERSITY OF PATRAS

23 PARAMETRIC T IME -D OMAIN M ETHODS FOR N ON -S TATIONARY R ANDOM V IBRATION I DENTIFICATION AND A NALYSIS 22 Model-Based Frozen Vibration Analysis SP-TARMA(6,3) S TOCHASTIC M ECHANICAL S YSTEMS & AUTOMATION (SMSA) L ABORATORY SP-TARMA(6,3) (smoothed) U NIVERSITY OF PATRAS

24 PARAMETRIC T IME -D OMAIN M ETHODS FOR N ON -S TATIONARY R ANDOM V IBRATION I DENTIFICATION AND A NALYSIS 23 Model-Based Frozen Vibration Analysis FS-TARMA(6,3) (2SLS) S TOCHASTIC M ECHANICAL S YSTEMS & AUTOMATION (SMSA) L ABORATORY FS-TARMA(6,3) (RELS) U NIVERSITY OF PATRAS

25 PARAMETRIC T IME -D OMAIN M ETHODS FOR N ON -S TATIONARY R ANDOM V IBRATION I DENTIFICATION AND A NALYSIS 24 Model-Based Frozen Vibration Analysis FS-TARMA(6,3) (2SLS-PE) S TOCHASTIC M ECHANICAL S YSTEMS & AUTOMATION (SMSA) L ABORATORY Theoretical U NIVERSITY OF PATRAS

26 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS 25 Comparative Results RSS/SSS (%) CPU time (normalized %) 1 2 1% %.15%.5%.86% 42.46% 17.57% ST ARMA RML RARMA SP TARMA SP TARMA (smoothed) FS TARMA (2SLS) FS TARMA (2SLS PE) FS TARMA (RELS) Parametric Methods Main Characteristics TARMA Representation Modelling/Prediction Vibration Computational Ease Method Parsimony Accuracy Analysis Simplicity of Use ST-ARMA n/a RML-TARMA SP-TARMA SP-TARMA (smoothed) FS-TARMA (2SLS) FS-TARMA (2SLS-PE) FS-TARMA (RELS)

27 PARAMETRIC TIME-DOMAIN METHODS FOR NON-STATIONARY RANDOM VIBRATION IDENTIFICATION AND ANALYSIS CONCLUDING REMARKS 1. Parametric methods for non-stationary random vibration identification and analysis were presented and their effectiveness was demonstrated. 2. The methods belong to one of the following major classes: (a) Unstructured Parameter Evolution (UPE) (b) Stochastic Parameter Evolution (SPE) (c) Deterministic Parameter Evolution (DPE) 3. The best overall performance was achieved by methods from the DPE class, followed by methods from the SPE class, and finally, the UPE class. This may be attributed to the increased structure of DPE models which leads to statistical parsimony and is in accord with the time-evolution of the actual structural dynamics. 4. Despite the progress achieved, a lot of work is necessary for addressing the numerous open issues.