Multi Channel Output Only Identification of an Extendable Arm Structure Under Random Excitation: A comparison of parametric methods

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1 Multi Channel Output Only Identification of an Extendable Arm Structure Under Random Excitation: A comparison of parametric methods Minas Spiridonakos and Spilios Fassois Stochastic Mechanical Systems & Automation (SMSA) Laboratory Department of Mechanical & Aeronautical Engineering University of Patras, GR 265 Patras, Greece Tel/Fax: {mspirid, fassois}@mech.upatras.gr Internet: Abstract The problem of multi channel output only identification of the time varying dynamics of an extendable prismatic arm structure under random excitation is considered. The identification is based on three simultaneously measured non stationary vibration response signals obtained during a single experiment in which the arm extends from its fully retracted to its fully extended position. Non stationary Functional Series Vector Time-dependent AutoRegressive (FS VTAR) and AutoRegressive Moving Average (FS VTARMA) methods are used, while critical comparisons with corresponding Pseudo- Linear Regression (PLR VTAR and PLR VTARMA) methods employing recursive models are also made. The superiority of the Functional Series methods, and particularly of the FS VTARMA method, in capturing the time varying dynamics is demonstrated. I. INTRODUCTION This paper concerns the output only identification and dynamic analysis of the non stationary dynamics of time varying structures based on multi channel random vibration response signals. Time varying structures are often encountered in space applications. Prime examples include deployable robotic arms [1], other deployable structures like antennas, solar arrays, truss structures with scissors like elements [2], as well as rockets (where the fuel being consumed is a significant portion of the total mass) [3]. In all of these cases the structural dynamics change with time during the structure s deployment and/or operation. The identification of a model for the time varying structural dynamics is necessary for a number of purposes, including dynamic analysis, the refinement/updating of previously developed finite element type models, diagnostics and control. As a consequence, the identification problem is receiving increased attention in recent years (see the survey of Poulimenos and Fassois [4]). The identification problem may be posed either in an output only or in an input output framework, depending upon whether or not the excitation signals are available. An output only framework, making no use of the excitation, is employed in the present study. Moreover, time varying identification methods may be classified as either non parametric or parametric, with the latter offering advantages in terms of compactness of representation (parsimony), as well as in terms of achievable accuracy and resolution. Parametric models are also generally better suited to dynamic analysis and characterization, the refinement of analytical models, design modifications, simulation, excitation characterization, as well as prediction and control [4], [5]. Parametric identification methods are often based on Time dependent AutoRegressive Moving Average (TARMA) models or their extensions [4], [6]. These constitute conceptual extensions of their conventional (time-invariant) counterparts, in that their parameters and innovations variance are allowed to evolve with time. A notable class of parametric methods postulates deterministic parameter evolution on the model parameters these are referred to as Functional Series (FS) type methods and are known to provide high accuracy and model compactness [4], [6, ch. 6]. FS type methods have been successfully applied in a number of studies over the past few years, such as the analysis of mechanical vibration in rotating machinery [7], modelling and analysis of a simulated planar manipulator [5], modelling and vibration analysis of a time varying bridge like structure [8], [9], and others. Vector FS VTARMA identification is postulated in [9], []. This present study focuses on the output only identification of the time varying dynamics of an extendable prismatic arm structure based on three vibration response signals. The structure has been used in recent studies by the authors: In [11] the output only identification based on a single vibration response channel and scalar FS TAR/TARMA models is considered. The same scalar problem is treated in [12], where a detailed comparison of scalar FS TAR/TARMA estimation methods is performed. Damage detection in the structure is considered in a companion paper [13], within which the output only multi channel identification problem is briefly treated, but only for pure AR (FS-VTAR) models. The present paper focuses on the output only multi channel RAST 29, 4th International Conference on Recent Advances in Space Technologies Istanbul, Turkey, June 29

2 Fig mm Output 3 AC motor (back) Output 2 Output 1 Shaker The extendable arm structure and the experimental setup. TABLE I MAIN DIMENSIONS FOR THE EXTENDABLE ARM STRUCTURE. Length (mm) Width (mm) Height (mm) Frame Beam Acceleration (m/s 2 ).2 (a).2.4 (b).4.6 (c) Frequency (Hz) (d) (e) (f) Fig. 2. (a)-(c) The non stationary vibration signals, and (d)-(f) 2D plots of the non parametric estimates of their time dependent power spectral density functions (STFT method). windowing) yields the time dependent Power Spectral Density (PSD) functions of Fig. 2(d)-(f), from which non stationarity in the frequency content is also evident. identification of the extendable arm structure time varying dynamics via Functional Series VTAR and VTARMA (FS VTAR and FS VTARMA) methods. A detailed assessment of their achievable accuracy, performance characteristics, and overall pros and cons is made. Moreover, critical comparisons with corresponding recursive Pseudo Linear Regression VTAR and VTARMA (PLR VTAR and PLR VTARMA) methods [14, pp ] are also performed and conclusions are drawn. II. THE LABORATORY TIME VARYING STRUCTURE A. The Laboratory Setup The extendable arm structure is shown in Fig. 1. It consists of a fixed-free outer frame and an inner beam sliding in it, driven by an AC motor. The arm main dimensions are summarized in Table I. The structure is subject to zero mean Gaussian random force excitation which is vertically exerted via an electromechanical shaker equipped with a stinger (Fig. 1). The structural vibration in the vertical direction is measured at three selected locations via lightweight piezoelectric accelerometers, and the obtained signals are conditioned and subsequently driven into a SigLab 2-42 data acquisition module. B. The Non Stationary Vibration Signals The motion scenario considered is the extension of the arm from its fully retracted to its fully extended position at a constant speed of u 8 mm/s. The vertical vibration (acceleration) signals are sampled at f s = 128 Hz, each one being N = 11, 52 samples (9 s) long. The study focuses on the 2 Hz frequency range, hence the signals are properly high pass filtered. The measured signals are depicted in Fig. 2(a)-(c) and are evidently variance non stationary. Non parametric time frequency analysis based on the Short Time Fourier Transform (STFT; moving window of 512 samples, Hamming data III. FUNCTIONAL SERIES VTARMA MODELLING Functional Series Vector Time-dependent AutoRegressive Moving Average (FS-VTARMA) models constitute conceptual extensions of their conventional (stationary) Vector ARMA counterparts [15], in that their parameters are explicit functions of time, belonging to functional subspaces spanned by selected deterministic functions (basis functions). A k variate FS-VTARMA(n a, n c ) [pa,p c,p s] model, with n a, n c designating its AutoRegressive (AR) and Moving Average (MA) orders, respectively, and p a, p c, p s the AR, MA, and innovations covariance matrix functional basis dimensionalities, respectively, is of the form: n a n c x[t] + A i [t] x[t i] = e[t] + C i [t] e[t i] } {{ } AR part } {{ } MA part A[B, t] x[t] = C[B, t] e[t] (1) with t designating normalized discrete time, x[t] (k 1) the nonstationary response signal vector (k designates the number of response channels), e[t] (k 1) the (unobservable) innovations (residual) sequence which is uncorrelated and characterized by zero-mean and time-dependent non-singular (and generally non-diagonal) covariance matrix Σ[t] (k k). A i [t] (k k) and C i [t] (k k) stand for the model s AR and MA time-dependent parameter matrices and B for the backshift operator (B i x[t] = x[t i]). A[B, t] and C[B, t] designate the AR and MA time-dependent matrix polynomial operators, respectively: A[B, t] = n a I k + A i [t] B i, C[B, t] = n c I k + C i [t] B i (2) The model parameter matrices, along with the innovations time-dependent covariance matrix, belong to functional sub-

3 spaces with respective bases: F AR = { Gba(1)[t],..., G ba(p a)[t] } F MA = { Gbc(1)[t],..., G bc(p c)[t] } F Σ = { Gbs(1)[t],..., G bs(p s)[t] } (3a) (3b) (3c) where the indices b a (i)(i = 1,...,p a ), b c (i)(i = 1,...,p c ), and b s (i)(i = 1,..., p s ) designate the functions (from a properly ordered functional set, such as Chebyshev, Legendre, trigonometric, and so on) that are included in each basis. The elements of the time-dependent model parameter matrices, along with the innovations time-dependent covariance matrix, may be thus expressed as: p a A i [t]{a i l,m [t]} : ai l,m [t] = a i,j l,m G b a(j)[t] (4a) j=1 p c C i [t]{c i l,m[t]} : c i l,m[t] = j=1 p s c i,j l,m G b c(j)[t] Σ[t]{s l,m [t]} : s l,m [t] = s j l,m G b s(j)[t] j=1 (4b) (4c) with l, m = 1,...,k, and a i,j l,m, ci,j l,m, sj l,m designating the AR, MA and innovations covariance matrix coefficients of projection, respectively. The FS-VTARMA model is thus parameterized in terms of the time-invariant projection coefficients a i,j l,m, ci,j l,m, sj l,m, while a specific model structure, say M, is defined by the model orders n a, n c, and the functional subspaces F AR, F MA and F Σ. Note that the FS VTAR model structure may be obtained as a special case by omitting the MA part of the complete FS VTARMA model in (1). A. Model Parameter Estimation Model parameter estimation refers to the determination, for a given model structure, of the AR/MA parameter and innovations covariance matrix projection coefficient vectors: ϑ = [ a 1,1 1,1... ana,pa k,k. c 1,1 1,1... ] T, [ ] cnc,pc k,k s = s 1 1,1...s ps T k,k based on available signal samples x N 1 = {x[1],...,x[n]}. The estimation of ϑ may be based on the Prediction Error (PE) principle, according to which a quadratic functional of the model s one-step-ahead prediction error e[t, ϑ] (residual) sequence is minimized: ˆϑ = arg min ϑ N e T [t, ϑ] e[t, ϑ] (5) t=1 with arg min designating argument minimizing and e[t, ϑ] being obtained as: n a n c e[t, ϑ] = x[t]+ A i [t, ϑ] x[t i] C i [t, ϑ] e[t i, ϑ] (6) Although, for FS-VTAR models (ϑ a) the dependence of e[t, a] on a is linear, this in not the case for full FS-VTARMA models, in which the non-linear dependence of the residual series upon the MA coefficients of projection vector c leads to a non quadratic optimization problem [4]. This problem may be tackled via iterative non linear optimization techniques. These are, nevertheless, amenable to wrong convergence problems which are due to the potential existence of several local minima in the PE criterion. For this reason rather accurate initial guess parameter values are typically required. In this study, initial parameter values are obtained via the Two Stage Least Squares (2SLS) method [9] which resorts on a sequence of linear operations. The covariance matrix of the estimated FS-VTARMA residuals [e[t, ˆϑ] in (6)] is subsequently obtained via a moving average filter (sliding time-window) as follows: Σ[t] = 1 2M + 1 t+m τ=t M e[τ, ˆϑ] e T [τ, ˆϑ] (7) with 2M + 1 designating the window length. The corresponding projection coefficient vector s may be subsequently obtained by solving a set of equations in the least squares sense. B. Model Structure Selection Given a basis function family (such as Chebyshev, Legendre, trigonometric, and so on), model structure estimation refers to the estimation of the set of integers: M = {n a, n c, p a, p c, p s, b a (j), b c (j), b s (j)} (8) Model structure estimation may be viewed as a discrete variable selection problem that may be tackled via an integer optimization method [4] utilizing a Genetic Algorithm (GA) that minimizes the Bayesian Information Criterion ( fitness function) [9]: BIC = 1 N ( ln Σ[t, s] + e T [t, ϑ] Σ 1 [t, s] e[t, ϑ] ) 2 t=1 + lnn d + Nk ln 2π (9) 2 2 with d designating the number of independently adjusted (estimated) model parameters. IV. STRUCTURAL IDENTIFICATION AND ANALYSIS A. Identification Results RESULTS FS VTARMA identification is considered using functional subspaces spanned by sine and cosine functions (selected to reflect smooth variability in the dynamics as indicated by the non-parametrically estimated PSD functions of Fig. 2): [ κ π t N 1 ] [ ] κ π t G [t] = 1, G 2κ 1 [t] = sin, G 2κ [t] = cos N 1 () with κ = 1, 2,.... For the model structure selection problem FS VTARMA(n, n) (n = 2,...,) models are considered, with the integer optimization scheme of subsection III-B

4 BIC AR/MA order Fig. 3. FS-VTARMA order selection: BIC ( ) and RSS/SSS ( ) values versus AR/MA order. RSS/SSS (%) RSS/SSS (%) Number of parameters (a) 6.5 (b) PLR VTAR(11) FS VTAR(9) [,] PLR VTARMA(7,4) FS VTARMA(7,4) [7,2,] Fig. 4. Comparison of the identified non stationary models: (a) Prediction error (RSS/SSS) values, and (b) number of model parameters utilized only for the determination of the AR/MA and innovations covariance functional subspaces. Thus, for each FS VTARMA(n, n) model a GA is used for the estimation of the set {p a, p c, p s, b a (j), b c (j), b s (j)}. The BIC criterion and the sum of squares of the prediction errors (Residual Sum of Squares, RSS) normalized by the sum of squares of the signal samples (Series Sum of Squares, SSS) for the obtained FS VTARMA(n, n) (n = 2,...,) models are shown in Fig. 3. Note that in the present (multi channel or vector) case the RSS is defined as the aggregate (over all response channels) residual sum of squares. Similarly for the SSS. Thereby, a FS VTARMA(7, 7) model is initially selected while further investigation for the reduction of the MA order leads to the final selection of a FS VTARMA(7, 4) [7,2,] model for representing the time varying structure. This model features trigonometric (sine and cosine) functional subspaces with p a = 7 (b a = [, 1, 2, 3, 4, 7, 9]), p c = 2 (b c = [, 2]), and p s = (b s = [, 1,...,9]). For purposes of comparison, a FS VTAR model and recursive VTAR and VTARMA models (also referred to as VRAR and VRARMA models) are also estimated. FS VTAR model parameter estimation is based on minimization of the PE criterion of (5). Its functional subspaces are also spanned by the trigonometric (sine and cosine) functions of () and its structure is selected via the scheme of subsection III-B. On the other hand, the recursive models are estimated via the Pseudo-Linear Regression (PLR) method [14, pp ]. Notice that for improved accuracy a forward, a backward, and a final forward pass are made for the signals, while the model orders and forgetting factor λ are selected based on minimization of the RSS (note that the BIC cannot be formally used with recursive models). The characteristics of the identification methods and the finally obtained models are summarized in Table II (α refers to the initial diagonal element of the covariance matrix associated with the PLR method). The RSS/SSS values for each of the four estimated models are presented in Fig. 4(a). Evidently, the identified FS models attain better predictions (lower RSS/SSS values) than their PLR counterparts, with the best predictive performance being achieved by the FS VTARMA model. The parametrization parsimony attained by the FS models is compared (in terms of estimated model parameters) to that of their PLR counterparts in Fig. 4(b). While the FS VTAR model is fully described by 9 parameters (coefficients of projection) and the FS VTARMA by 63, the PLR VTAR and PLR VTARMA models require 1,14,48 parameters. This excessive difference is due to the need of the recursive PLR models to store their parameter and innovations covariance matrix values for each time instant. B. Model-Based Analysis Results Non stationary vibration analysis based on the estimated VTAR and VTARMA models is now considered. The vector vibration response frozen PSD matrix is obtained as: S F (ω, t) = 1 2π Ψ[e jωts, t] Σ[t] Ψ T [e jωts, t] (11) where Ψ[e jωts, t] = A 1 [e jωts, t] C[e jωts, t]. In this expression the model parameter matrices and innovations covariance matrix are replaced by their respective estimates, ω designates frequency in rad/s, T s the sampling period in s, and j the imaginary unit. The frozen time PSD matrices corresponding to the estimated PLR VTARMA(7, 4) and FS VTARMA(7, 4) [7,2,] models are, along with the non parametric PSD matrix estimate (STFT method), presented in Fig. 5. Note that in TABLE II IDENTIFICATION METHODS, THEIR SELECTED CHARACTERISTICS, AND THE IDENTIFIED MODELS Identification Method Method Characteristics Identified Model PLR VTAR λ =.992, α = 4, M = 256 PLR VTAR(11) PLR VTARMA λ =.993, α = 4, M = 256 PLR VTARMA(7,4) FS VTAR OLS, QR implementation of OLS, M = 256 FS VTAR(9) [,] FS VTARMA 2SLS, 2 iterations, QR implementation of OLS, M = 256 FS VTARMA(7,4) [7,2,] Levenberg Marquardt non linear optimization

5 ωni [t] = ln λi [t] (rad/s), ζi [t] = cos arg ln λi [t] (12) Ts with λi designating the i-th discrete-time frozen pole. The FS VTARMA(7, 4)[7,2,] based natural frequency estimates (with ζi [t] < 15%) are, along with their PLR VTAR(11), FS VTAR(9)[,], and PLR-VTARMA(7, 4) model counterparts, depicted in Fig. 6. The VTAR models provide various additional (false or computational) modes, while the VTARMA model estimates track their non-parametric counterparts better. Nevertheless, the PLR-VTARMA(7, 4) based estimates exhibit increased scatter. It is also noted that, the estimates of the vector FS VTARMA(7, 4)[7,2,] model, although they introduce some additional modes in the frequency area above 35 Hz, are in good overall agreement with those obtained by the univariate FS TARMA(9, 9)[4,3,5] model in [12]. V. C ONCLUDING REMARKS The identification of a time varying extendable arm was considered. The identification was based on a recently introduced FS VTARMA method. The identified FS VTARMA(7, 4)[7,2,] model was contrasted to a simpler FS VTAR(9)[,] model, as well as recursive PLR VTAR(11) and PLR VTARMA(7, 4) models. The estimated FS VTARMA model was shown to provide PSD and modal parameter estimates that are in good agreement with the non parametric PSD estimates. It was also shown to surpass its counterparts in terms of predictive ability and tracking accuracy for the frozen time structural dynamics. Overall, the results demonstrate the effectiveness of the parametric FS VTARMA method and its potential for highly parsimonious (compact) and accurate identification, as well as for model based dynamic analysis of time varying structures based on multi channel (vector) measurements under unobservable random excitation. ACKNOWLEDGMENT The support of this work by the Greek General Secretariat for Research & Development (Italy Greece joint research and FS VTAR(9)[,] PLR VTAR(11) Frequency (Hz) FS VTARMA(7,4)[7,2,] PLR VTARMA(7,4) 4 Frequency (Hz) this figure Sii denotes a diagonal element (the auto spectral density of the i th response signal), while Sij (i 6= j) denotes the magnitude of an off diagonal element (magnitude of the cross spectral density relating the i th and j th response signals). It may be observed that the PLR VTARMA(7, 4) model based estimates seem to exhibit significant scatter in both the frequency and magnitude content of the spectral densities, which seems to contradict the expectedly smooth evolution of the dynamics. On the other hand, the FS VTARMA(7, 4)[7,2,] model based PSD matrix estimate seems to exhibit good tracking for both the resonances and antiresonances. Overall, the identified FS VTARMA model provides much more clear, smooth and informative PSD estimates than its PLR VTARMA and VTAR counterparts. The structure s frozen-time natural frequencies and damping ratios may be also obtained as: Fig. 6. Frozen time PLR VTAR(11), FS-VTAR(9)[,], PLR VTARMA(7, 4), and FS VTARMA(7, 4)[7,2,] based natural frequency estimates along with the S11 (ω, t) STFT based PSD estimate (background). (Only natural frequencies with damping ratio < 15% are depicted.) technology program, project 85-e) is gratefully acknowledged. R EFERENCES [1] S. Dwivedy and P. Eberhard, Dynamic analysis of flexible manipulators, a literature review, Mechanism and Machine Theory, vol. 41, no. 7, pp , 26. [2] G. Tibert, Deployable Tensegrity Structures for Space Applications, Ph.D. dissertation, Royal Institute of Technology, Department of Mechanics, 22. [3] M. Basseville, A. Benveniste, M. Goursat, L. Hermans, L. Mevel, and H. 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6 }{{} }{{} FS VTARMA(7, 4)[7,2,] PLR VTARMA(7, 4) STFT }{{} Fig. 5. Non-parametric STFT based and frozen time PLR VTARMA(7, 4) and FS VTARMA(7, 4) [7,2,] based PSD estimates. [13], Vibration based fault detection in an extendable prismatic link structure via non-stationary FS-VTAR models, in Proceedings of the International Operational Modal Analysis Conference, Ancona, Italy, 29. [14] T. Söderström and P. Stoica, System Identification. Prentice Hall, [15] V. Papakos and S. Fassois, Multichannel identification of aircraft skeleton structures under unobservable excitation: a vector AR/ARMA framework, Mechanical Systems and Signal Processing, vol. 17, pp , 23.