Reconstruction of multiple impact forces by wavelet relevance vector machine approach

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Reconstruction of multiple impact forces by wavelet relevance vector machine approach Souleymane Samagassi a, Abdellatif Khamlichi a,n, Abdellah Driouach a, Eric Jacquelin b,c,d a

Communication Systems and Detection Laboratory, Faculty of Sciences at Tetouan, BP. 2121, Tetouan 93002, Morocco University of Lyon, F-69622 Lyon, France IFSTTAR, LBMC, UMR_T9406, Bron, France d University Lyon 1, Villeurbanne, France b c

a r t i c l e i n f o

abstract

Article history: Received 9 August 2014 Received in revised form 31 May 2015 Accepted 13 August 2015 Handling Editor: I. Trendafilova

In this work, reconstruction of forces generated by multiple impacts occurring on linear elastic structures has been achieved through wavelet relevance vector machine approach to inverse problem solution. The posterior density of probabilities function integrating both the likelihood and prior random information was considered for the particular case of a system being affected by an additive random measurement noise. Compressed sensing technique was used to provide an adequate sparse representation of the inverse problem through using projection on a wavelet basis. Bayesian hierarchical modeling according to the relevance vector machine approximation was then applied in order to estimate the forces generated by impact events. The obtained results were remarkably good as the reconstructed forces were found to be very close to the original forces at the system input. The method was found also to provide a way for localization of impact forces. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction In structural health monitoring, on-line evaluation of the forces produced during impact events is useful. As direct measurement of these forces is not feasible, indirect methods constitute a practical way to get this information by solution of an inverse problem [1–3]. Various methods dealing with reconstruction of the forces characteristics ensuing impact on an elastic structure, by using response measurement of the structure at some of its points, have been introduced in the last decades [4–8]. Sanchez and Benaroya [9] have given recently a detailed review of these methods. For noise free linear impact situations, the time domain approach to force reconstruction has been considered by using an explicit discrete model of the structure. A Toeplitz like time response matrix representing the system dynamics is obtained by operating time discretization of the continuous convolution problem. This can be performed either by time transfer function approach or by means of the state space formulation. As the obtained matrix is generally ill-posed, regularization of the deconvolution operation is needed in order to achieve adequate estimation of a usable solution of the inverse problem.

n

Corresponding author. Tel.: þ212 600769960. E-mail address: [email protected] (A. Khamlichi).

http://dx.doi.org/10.1016/j.jsv.2015.08.014 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

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S. Samagassi et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Among the deterministic techniques that have been proposed for regularization, one finds singular value decomposition and generalized singular value decomposition of the Toeplitz like matrix, which are followed by Tikhonov procedure in order to stabilize the inversion operation [4,6]. In most of the common approaches to force reconstruction, a deterministic structural model is used in order to describe the system behavior. This model can be obtained analytically, by means of the finite element method or identified experimentally. Deterministic methods are however too approximate because they assume some perfect knowledge of material properties, geometric parameters and boundary conditions. In real problems these data cannot be exactly determined and the structural model may be affected by various sources of uncertainties. Furthermore, the structure response which is measured by means of imperfect sensors may be perturbed by noise. To emphasize these essential features related to reconstruction of forces that are generated by impacts occurring on real structures, the problem has been formulated within the convenient framework of stochastically oriented approaches, such as Bayesian based methods [10–13]. The Bayesian approach allows for the propagation and quantification of the errors on the reconstructed forces by means of the posterior density of probabilities function that takes into account these uncertainties in a quite natural manner. The prior available information and the posterior information provided by response measurement are then more adequately used and estimation of the inverse problem solution is possible via the posterior density of probabilities function [14]. When the priori distribution parameters are not available, hierarchical relationships between these parameters and some parameters called hyperparameters can be introduced. The hyperparameters appearing in the last hierarchical level are assumed to be known, so they form part of the prior available information. These hyperparameters are problem dependent and can be identified from adequate experiment, or by means of extensive simulations conducted on the problem to be solved. Sometimes they can be approximated just in an uninformative way. Another key issue in the inverse problem of multiple force reconstruction is related to the aim of performing this assignment by using only a small number of sensors. In this case the information is not exhaustive and the obtained Toeplitz like system is largely underdetermined. Compressed Sensing (CS) can then be used for finding the sparsest solution to this particular system of linear equations [15–17]. The principle of sparsity can be viewed as the feature that a natural signal has to be compressible; in the sense that it has sparse or approximately sparse representation when expressed in an appropriate basis. Sparsity expresses the idea that the information rate of a continuous-time signal may be much smaller than that suggested by its bandwidth. Sparsity regularization can be possible by implementing a change of variables so that the coefficients of the model with respect to the new basis will be sparse [16]. The wavelet derived basis is usually used in this context [18]. Solution of the obtained sparse inverse problem by using the well known Markov Chain Monte Carlo based approaches might need acquisition of large amount of data. Moreover, in these approaches the design of appropriate random sampling techniques is required in order to get useful information about the mean and variance of the unknown. To decrease the number of these samples, the single vector machine (SVM) was introduced [19]. This method focuses on the quality rather than on the quantity of samples as it works with only a small number of extreme cases that are sampled so as to be close enough to the decision boundaries [20]. For this reason, the SVM has been widely used as an effective solution approach to inverse problems with sparse sensing data [21] and was found to yield more accurate results than the conventional approaches [22]. The SVM based approach has shown however some limitations. These are associated to its need for Mercer kernels and the definition of the error/margin trade-off parameter [23]. New alternative methods were recently introduced and were found to be more attractive with regard to these concerns. A Bayesian extension of the SVM, called the relevance vector machine (RVM) was developed in the context of image classification [23]. This method requires fewer training cases than the SVM and enables to circumvent most of the previous shortcomings. Significant reduction in the computational complexity of the decision function, more sparseness of the solution and the ease of free parameters tuning can then be achieved through the RVM. These features make this method suitable for real-time applications [24]. The sparse multinomial logistic regression (SMLR) is another method that was introduced recently for image classification [25]. However, this last method has shown some deficiencies in terms of performance as it tends mostly to use extreme cases that lay away from the decision domain [26]. The authors of the previous reference have made a comparison between the SVM, the SMLR and the RVM. They found that the RVM produces the most accurate results while this method involves the smallest available training cases. For almost the same accuracy, the SVM had required 4 times more useful training cases than the RVM. In this work, reconstruction of multiple forces is considered by assuming the structure response to be reduced to only one measurement provided by a single sensor. The problem to be solved is largely undetermined in these conditions. The Bayesian hierarchical approach according to wavelet relevance vector machine (WRVM) method is applied. The term wavelet in the WRVM is related to the wavelet based basis used to transform the system matrix to a sparse representation. The problem is assumed to be formulated in the time domain under a sparse form obtained through using the Daubechies wavelet transform. Solution of the sparse inverse problem is performed by means of the sparse Bayesian learning conducted according to compressed sensing (CS) and RVM algorithm [27]. The time signal representing the multiple forces generated by impact is then extracted from the actual strain measurement performed at a given point of the structure. This procedure is finally customized to conduct impact localization when the impact points are assumed to be unknown.

Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

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2. Multiple force reconstruction in elastic impact Force reconstruction is a problem that has usually non-unique solutions. To overcome this ill-posedness, additional constraints are typically applied in order to transform it into a well-defined problem which exhibits a unique useful solution. Current reconstruction methods of forces are classified as deterministic and probabilistic. A detailed review giving the state of the art in this field was given by Sanchez and Benaroya [9]. The authors have emphasized the fact that probabilistic reconstruction techniques account for noise effect and led often to more concise results, in a sense that they are self-regularizing. Zhang et al. [10] have used the Bayesian approach to reconstruct forces for two situations: force reconstruction with uncertain frequency response functions, and with uncertain modal parameters. Applying Bayesian techniques, the force was reconstructed using only the available measured data without the need for additional constraints to stabilize the results. The authors have illustrated the advantages of the Bayesian approach as compared to the Tikhonov regularization method. They showed that formally these two methods are equivalent. They noted however that the Bayesian method has the ability of achieving inherently regularization of the problem, whereas Tikhonov regularization requires additional effort to determine a suitable regularization parameter. Considering the situation where multiple forces are to be simultaneously reconstructed, the deterministic approaches have revealed some limits as the regularization parameter is not obvious to identify, due to the presence of multiple edges in the L-curve for example. Bayesian based approaches have also revealed their limits when dealing with the problem of multiple forces as the required amount of information to perform accurate reconstruction is excessively high. A large number of sensors are needed. The lack access to observation data is the common case in practice because of the restriction imposed on the number of measurement sensors that could be implemented. This puts a drastic limitation on solution to the force reconstruction problem that involves more than one single force; this occurs even when using the most effective Bayesian approaches. The measurement signal is generally a sparse vector and ill-conditioning of the inverse problem impedes then accurate computation of the solution. The transient force signals that are generated by impacts have generally short time durations, so these signals are also sparse. All these facts yield that the actual inverse problem is sparse in nature, so use of appropriate approaches that have the capacity to deal with regularization in this specific context is required. The main ideas introduced in the following consist of applying the sparse representation concept to the multiple forces reconstruction problem in order to transform it into a convenient form which is more suitable for the inversion operation and prone to provide increased accuracy. In the following, the Zhang approach [10] is adapted to the time domain formulation of the force reconstruction problem. Then, the sparse representation of the problem is derived. This is achieved by means of projection of the unknown global force vector on a wavelet basis. The final step of the reconstruction procedure consists of performing the sparse signal reconstruction by using the Bayesian hierarchical WRVM method. 2.1. Formulation of the inverse problem Let us consider an elastic linear structure undergoing an impact event and assume under this impact that N f localized concentrated forces are generated. The structure response is assumed to be measured by means of one sensor located at a given known position. The aim is then to determine the points of application of these forces as well as their time dependency by solution of an inverse problem where the information provided by this sensor is used. Force reconstruction problems can be formulated by means of adequate physical equations, called the forward model. This relates the global vector of forces acting on the system, F, to observations collected in a set of measurement data, Y. Measured data can be in general strains, accelerations or displacements. For clarity of the presentation, let us assume that forward modeling of the system is free from model uncertainties. Taking the common case of linear elastic structures, the transient dynamical equations that determine the dynamic response of the structure under the action of any given general time-dependent transient loads write M z€ þ C z_ þKz ¼ f

(1)

where M is the mass matrix, C the damping matrix, K the stiffness matrix, z the displacement vector and f the vector of applied loads. Eq. (1) can be obtained for instance by using the finite element method or any other method of discretization. Denoting
t the state vector Z ¼ z_ z and considering a measurementY, Eq. (1) can be put under the following form of a state-space model ( Z_ ¼ AZ þ Bf (2) Y ¼ CZ þDf Under zero initial conditions, the application of the Laplace transform to the continuous-time state-space model (2) yields ( sZðsÞ ¼ AZðsÞ þBf ðsÞ (3) YðsÞ ¼ CZðsÞ þ Df ðsÞ where the bar stands for the Laplace transform. Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

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By eliminating the states ZðsÞ, the following input–output relation is readily obtained YðsÞ ¼ gðsÞf ðsÞ

(4)

1

with gðsÞ ¼ C ðsI  AÞ B þD. The frequency response function (FRF) is defined as the transfer function in which the complex Laplace variable is pffiffiffiffiffiffiffiffi restricted to purely imaginary values s ¼ jω with j ¼  1. The practical relevance of the FRF lies in the fact that it is easily identified from the measured time data by using non-parametric methods. The FRF is obtained as gðjωÞ ¼ C ðj ω I  AÞ  1 B þD

(5)

Now, returning back to the time domain and assuming the force vector to be causal, the measurement is related to the force by the following convolution product Z

t

YðtÞ ¼ gðtÞnf ðtÞ ¼ 0

gðt  τÞf ðτÞdτ

(6)

where g is the time transfer function obtained as the inverse Fourier transform of g. Upon time discretization and summation by using the rectangles formula in Eq. (6), the following Toeplitz like equation is obtained GF ¼ Y

(7)

where F A ℝm is the global vector of the discrete applied forces, Y A ℝk is the column vector that collects measurements provided by the sensor, G A ℝk  ℝm is the matrix collecting the time transfer matrices between the points of application of forces and sensor location. In these notations m ¼ Nf  Nt and k ¼ N t where N t designates the number of time steps used for sampling the continuous response measured by the sensor. The elastic system is thus shown to be modeled by the matrix, G, which relates the global vector of forces F to the acquired measurement data Y according to the linear algebraic Eq. (7). The terminology common to inverse problems [28] refers to F as the model and to the equation GF ¼ Y as the mathematical model where G is the measurement matrix. In the general case where the structure is not discretized according to the finite element method, deriving the matrix G might involve discretization of an ordinary differential equation, discretization of a partial differential equation, evaluation of a convolution integral, or application of a non-parametric algorithm such as in blind convolution or experimental identification procedures. The forward problem as expressed by Eq. (7) is to find Y given F. The focus in the following is on solution of the inverse problem which consists of finding F given Y. Additionally, the objective is to deal with the reconstruction of a set of impact forces F by using only one sensor, that is to say in conditions where k o m holds. In this case, it is well known from linear algebra that Eq. (7) has infinitely many solutions and is largely indeterminate. This is a key issue since there will be commonly an infinite number of mathematical models aside from F true which fit the data, Y, assumed to be perfectly measured. Another important issue in practice is that actual observations always contain some amount of noise. Noise may arise in particular because of numerical round-off or nonlinearities that have not been taken into account for instrument readings. The data can thus be thought of to be consisting of noiseless observations from a perfect experiment providing, Y true ¼ GF true , plus a noise component η such that GF true þ η ¼ Y

(8)

where η A ℝk is a random vector representing measurement noise. Because there may be many models F that adequately fit the data Y. It is essential to know the feature of what solution has been obtained, or how good it is with regards to physical evidence and its ability to adjust accurately the data. If the mathematical model of the system is incorrect or the data contain large amplitude noise, there may be no model that exactly fits the data. We have then an ill-posed problem for which existence of a solution is just unfeasible. Multiple exact solutions to the inverse problem can also exist. This may happen even when an infinite number of exact data points are provided for Y. Such a situation reflects the fact that there exist other freeloading solutions besides F true that exactly satisfy GF ¼ Y true . This will occur in particular if the number of forces to be reconstructed exceeds one. In this case, the system GF ¼ Y true is rectangular and the matrix G is rank deficient, so it has a nontrivial null space. Any model F 0 that lies in the null space of G is solution to GF 0 ¼ 0 and by superposition, any linear combination of these null space models can be added to a particular model that satisfies GF ¼ Y true . Regularization is considered then in order to get the usable solution among all the mathematical possible ones. As for real systems the matrix G might also be polluted, one has two ways in order to take into account system matrix uncertainties. The first method uses model uncertainty propagation based approaches that enable to estimate variability of G following those of the basic design parameters. The second method consists in applying posterior noise on the matrix G in order to take into account globally and non-parametrically the various physical perturbations, modeling errors as well as material and geometrical variability. Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

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The propagation based method gives raise to major difficulties and is out of the scope of the present work. Thus, in the following, only the second approach is considered and the system matrix is assumed to be perturbed by a general white Gaussian noise. 2.2. Regularization A key aspect of many inverse problems is that they are ill-posed. The process of computing an inverse solution can be in many situations extremely unstable. A small change in measurement produced by noise, a small η in Eq. (8), can lead to a huge change in the estimated model. However, it is commonly possible to stabilize the inversion process by imposing additional constraints that bias moderately the solution. This process is generally referred to as regularization and is often essential to producing a usable solution. A reasonable approach to finding the best approximate solution to an inconsistent system of linear equations, such that described by Eq. (8), is to find a vector F that minimizes some error measures. These last called residuals are calculated from the differences between the observations and the theoretical predictions provided by the forward problem. A traditional method that has been widely applied consists in finding the model F that minimizes the Euclidian norm of the residual vector, i.e. the least squares solution. The problem takes then the form of the following mathematical program minimize with ‖Y  GF‖22 ¼

nt  P i¼1

Y i  ðGFÞi

2

‖Y  GF‖2

(9)

where the theoretical predictions are denoted ðGFÞi for each time step.

An alternative procedure that is less sensitive to data points that are exaggeratedly discordant with the mathematical model due for example to accidental errors [28], is the following robust estimation procedure which takes also the form of a minimization problem: minimize with ‖Y  GF‖1 ¼

‖Y  GF‖1

(10)

nt   P Y i ðGFÞ . i

i¼1

For a general linear least squares problem having the form of Eq. (9), there may be infinitely many least squares solutions. If we consider that the data contain noise, then there is no possibility to find a single point in fitting such noise exactly. On the contrary, there can be many solutions that can adequately fit the data in the sense that ‖Y GF‖2 is small enough. This defines a pertinent approach as weakening the formulation enables to avoid the contradiction which is deferred by noise effect. In zeroth-order Tikhonov regularization, the solutions are considered with ‖Y GF‖2 r δ, and the selection of the one that minimizes the 2-norm of F is performed. This correspond to minimize

‖F‖2

subject to

‖Y  GF‖2 r δ

(11)

where δ is a regularization parameter. The motivation for minimizing the 2-norm of F is to arrive at a solution that contains just sufficient features to adequately fit the noisy data. Allowing a poorer fit to the data enables then fitting these data with a smaller norm model. The selection of the best value of the regularization parameter δ can be performed by an adequate criterion [28]. Applying the method of Lagrange multipliers to the constrained optimization problem (11), one arrives at the following unconstrained problem [29]:  minimize ‖Y  GF‖22 þ α2 ‖F‖22 (12) where α is a regularization parameter. In practice, it happens that many of the unknown model components will be zero. Rather than using Tikhonov regularization to minimize ‖F‖2 , as stated in Eq. (11), one may choose to minimize the number of nonzero entries in F to obtain a sparse model. The notation ‖F‖0 is commonly used to denote the number of nonzero entries in F, note however that ‖‖0 does not satisfy the requirements for a vector norm. We can formulate then a corresponding regularized inverse problem as minimize

‖F‖0

subject to

‖Y  GF‖2 r δ

(13)

Unfortunately, these kinds of optimization problems can be extremely difficult to solve. A surprisingly effective alternative to (13) is to instead find the least squares solution that minimizes ‖F‖1 minimize

‖F‖1

subject to

‖Y  GF‖2 r δ

(14)

These 1-norm regularized models have the tendency of being sparse, and this becomes even more prominent in higher dimensions. The heuristic approach of minimizing ‖F‖1 instead of ‖F‖0 works very well in practice and recent work [30] has demonstrated reasonable conditions under which the solution to the 1-norm regularized problem is identical to or at least close to the solution of the regularized problem (13). Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

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Using the standard approach of moving the constraint into the objective function by means of Lagrange multipliers, we can select a positive regularization parameter, α, such that Eq. (14) is equivalent to  (15) minimize ‖Y  GF‖22 þ α‖F‖1 This is a convex optimization problem that can be solved efficiently by various algorithms [17] as long as the dimension of the problem remains small. When the number of unknowns becomes high, appropriate methods should be introduced to get a usable solution of Eq. (15). Compressed sensing is an adequate technique that is largely used to get a better conditioned problem. 2.3. Compressed sensing Compressed sensing (CS) is a simple method for finding the sparsest solution to some underdetermined system of linear equations having the form of Eqs. (7) or (8). CS appears to be an effective solution approach particularly in situations where the number of sensors is limited due to implementation constraints or cost; and when the sensing process provides a small number of measurements due to a poor sampling rate [18]. The principle of sparsity depicts the propensity that a natural signal has to be compressible; in the sense that it has sparse or approximately sparse representation when expressed in an appropriate basis. Sparsity expresses the idea that the information rate of a continuous-time signal may be much smaller than that suggested by its bandwidth; and that a discrete-time signal will depend on a number of degrees of freedom which are relatively much smaller than suggested by its actual length. In many situations, models are not directly sparse. Sparsity regularization will of course produce poor model recovery in this case. However, it is possible in some cases to implement a change of variables so that the coefficients of the model with respect to the new basis will be sparse [31]. Let us consider the discrete signal F in Eqs. (7) or (8) and an associated discrete signal X which is r-sparse in an appropriate basis Θ. This means that X contains no more than r non-zero elements or significant components with typically r⪡m. In vector–matrix notation, this property can be expressed by a projection taking the following form: F ¼ΘX

(16)

where X A ℝ and Θ A ℝ  ℝ . The columns of Θ represent m vectors defining an orthonormal basis. For most natural force signals F that are piecewise smooth, the wavelet transform [18,32] has been shown to yield sparse representation. Specifically, if X represents the wavelet coefficients for the signal F which is assumed to be piecewise smooth, then the error ‖X X r ‖2 has been shown to decay quickly with increasing r, in which X r represents X with the smallest ðm  rÞ coefficients set to zero. This implies that only a small number r of dominant wavelet coefficients are required to approximate F well. t t Since Θ is an orthogonal basis matrix which satisfies Θ Θ ¼ ΘΘ ¼ Idm , Eq. (16) can be easily inverted and m

m

m

X ¼ tΘ F

(17)

Y ¼ G Θ X þη

(18)

Y ¼ AX þ η

(19)

t

where ð Þ stands for the transpose operation. Using Eq. (17), Eq. (8) becomes

or equivalently with A ¼ GΘ A ℝ  ℝ . It should be noted that only the largest r coefficients X r have a strong dependency on the forces signal F, so that the approximation to F based on X r yields a good representation with r⪡m. Because of this, one may be tempted to perform the CS measurements adaptively by centering the course on the dominant coefficients X r . In practice however, one does not measure explicitly the dominant coefficientsX r ; rather a weighted combination of all transform coefficients X as shown by Eq. (19) is provided and X r is obtained by solution of an inverse problem. Several techniques to approximate X r accurately based on the measurements Y are available [33–37]. A crucial advantage of CS is that the weights coefficients of matrix A should not be adapted to the signal F under test. The CS theory indicates that a fixed set of weights may be used for all signals F that are within a given class. The fundamental question that naturally arises from signal model (19) is how many measurements should one collect so that these measurements will be sufficient to recover signal X r and then the forces F from Eq. (16). Candès and Wakin [17] addressed this question and showed that there exists a four-to-one practical rule which can be employed for exact reconstruction. They stated that one needs in general about four incoherent measurements per unknown nonzero term of F such as k

m

k Z 4r

(20)

regardless of the signal’s dimension m. Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

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An important feature of compressed sensing is that it is robust in the sense that small perturbations in the data cause small deviations in the reconstruction. Small perturbations in the data refer generally to signals that are not exactly sparse but nearly sparse or to the presence of noise in measurement. Signal X in Eq. (19) may be estimated from noisy measurement Y by solving the convex minimization problem, called second-order cone program, as follows: minimize ‖X‖1

subject to

‖AX  Y‖2 r δ

where δ is a bound of the amount of noise in the data. The regularized problem associated to the problem (21) can be written as  min ‖Y  AX‖22 þ α‖X‖1

(21)

(22)

The recovery of a sparse solution using Eq. (22) depends on the details and appropriateness of the particular G and Θ defining the matrix A. The analysis and implementation of compressive schemes become relatively simple if the measurement matrix is random [38]. To solve the minimization problem as defined by Eq. (22), the Bayesian hierarchical approach in its version called wavelet relevance vector machine is considered in the following. The robustness of the CS relies heavily on a notion called restricted isometry property (RIP) [39]. If the system matrix A verifies this property then any small perturbation of measurement Y resulting from noise will yield barely a small perturbation on the reconstructed sparse signal X: Considering the particular problem of force reconstruction, it has not yet been proven mathematically that the matrix A verifies the RIP. So, robustness can only be guessed numerically. This kind of robustness analysis can also be performed when the matrix A is polluted by modeling errors such that the actual system matrix is A þ δA. 2.4. Bayesian hierarchical approach In the Bayesian approach the model is not deterministic, but is rather a random variable, and the solution takes the form of a probability distribution for the model parameters called the posterior distribution. This probability distribution can then be used to answer probabilistic questions about the model [40]. With reference to Eq. (8), let us denote the prior distribution by π ðf Þ, and assume that we are interested in computing the conditional probability distribution, π ðyf Þ, that, given a particular model f , corresponding observation data, y, will be observed. Given a prior distribution, the conditional posterior distribution of the model parameters given the data is sought  after. This posterior probability distribution for the model parameters will be denoted by π ðf yÞ. Bayes' theorem relates the prior and posterior distributions in a way that makes the computation of π ðf yÞ possible. It states that   π ðyf Þπ ðf Þ (23) π ðf yÞ ¼ c  where the constant c is introduced to normalize the conditional distribution π ðf yÞ so that its integral in model space is one. This can be computed as Z  c¼ π ðyf Þπ ðf Þdf (24) Because c is not needed to determine the posterior distribution, Eq. (23) is often written as a statement of proportionality   π ðf yÞ p π ðyf Þπ ðf Þ (25)  The probability distribution π ðf yÞ does not provide a single model that can be considered to be the answer. However, in cases where a single  out representative model is wanted, it may be appropriate to identify the one corresponding to the largest value of π ðf yÞ. This model is referred to as the maximum a posteriori (MAP) model. Another possibility is to select the mean of the posterior distribution. In situations where the posterior distribution is normal, the MAP and posterior mean models will be identical. In general, the computation of a posterior distribution using Eq. (23) can be difficult. The chief difficulty lies in evaluating the integral in Eq. (24). This integral often has very high dimensionality, and numerical integration techniques may thus be computationally costly. There are a number of useful special cases in which the computation of the posterior distribution is greatly simplified. But, in general Bayesian based approaches result to be more pertinent in this context. In the following, the impact forces vector F will be reconstructed by a hierarchical Bayesian approach via an efficient algorithm based on RVM method.  m  m Let us consider a set of input vectors F j j ¼ 1 along with the corresponding measurements called targets Y j j ¼ 1 and an explicit dependency function between them of the form yðξÞ ¼

m X

Kðξ; ξj ÞX j

(26)

j¼1

Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

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ς

Strain gauge F1 (t)

F2 (t) …

Y(t)

FNf (t)

O

ξf 2

ξs

ξf 1

… ξfNf

ξ

L Fig. 1. A beam having uniform cross section and simply supported on its both ends is subjected to multiple impact forces; ξf d designates an impact point location and ξs the strain gauge position.

 m where X j j ¼ 1 are the model weights, ξ is an arbitrary point of the elastic structure, ξj the point of application of force component F j and m the problem size. Using Eqs. (7) and (16), the kernel function Kð U ; U Þ is given by Kðξ; ξj Þ ¼ Gðξ; ξj ÞΘ

(27) 

In supervised learning with respect to Eq. (26), the training dataset of input-target pairs F j ; Y j

m j¼1

is used to build the

dependency of the targets on the inputs with the ability of predicting accurate values of y for unseen values of ξ. Various methods were proposed in the literature to deal with this problem [41,42]. The Relevance Vector Machine (RVM) [43] is an enhanced probabilistic sparse kernel model where a Bayesian approach to learning is adopted. A prior over the weights governed by a set of hyperparametrs is introduced, one associated with each weight, and their most probable values are iteratively estimated from the data. Sparsity is often achieved as in practice the posterior distributions of weights are sharply peaked around zero. The procedure for obtaining the hyperparametrs values and the weights is recalled in the following.    m Giving a dataset F j ; Y j and assuming the standard formulation π ðyξÞ  NðyyðξÞ; τ2 Þ, such that the mean for a given j¼1

ξ is modeled by yðξÞ as defined in Eq. (26), the likelihood of the dataset can then be written as pðyjx; τ2 Þ ¼ 

 1 2 m=2 exp  2 ‖y Ax‖ 2τ 2 π τ2 1

(28)

where y ¼ ðy1 ; :::; ym Þ, x ¼ ðx1 ; :::; xm Þ and A A ℝm  ℝm is the design matrix with Aij ¼ Kðξi ; ξj Þ. As direct maximum-likelihood estimation of X and τ2 from Eq. (28) may lead to over fitting problems, smoother Gaussian prior over the weights was instead introduced in the form m

π ðxjσ Þ ¼ ∏ NðX i j0; σ i 1 Þ

(29)

j¼1

with σ is a vector of m hyperparameters σ i . This introduction of an individual hyperparameter for every weight is the key feature of the model (RVM) as it yields useful sparsity properties. The posterior over weights is then obtained according to Bayes rule as

  1 1 1 t ðx  π ðxy; σ ; τ2 Þ ¼ exp  γ Þ ϒ ðx  γ Þ (30) 2 ð2π Þm=2 with 

ϒ ¼ τ  2t AA þ Σ

1

γ ¼ τ  2 ϒ t AY in which Σ ¼ diagðσ 1 ; :::; σ m Þ. By integrating the weights, the marginal likelihood or evidence for the hyperparameters is obtained as

 1=2

1   1 2 1t 2  1t  1t π ðyσ ; τ2 Þ ¼ τ I þ A Σ A exp  y τ I þ A Σ A y m m 2 ð2π Þm=2

(31) (32)

(33)

where I m is the identity of order m. Ideal Bayesian inference needs defining hyperpriors over σ and τ2 , and then integrating out the hyperparametrs. However, this is bulky and a simplified approach is proposed [44] where the marginal likelihood as given in Eq. (33) is optimized  with respect to σ and τ2 . This is equivalent to finding the maximum of π ðσ ; τ2 yÞ assuming a uniform hyperprior or uninformative situation. Predictions are made then according to Eq. (30) where the obtained optimal parameters are used. Values of σ and τ2 which maximize Eq. (33) cannot be obtained directly and iterative re-estimation of σ is needed. This is performed by means of direct differentiation of Eq. (33) which gives

σ new ¼ j

1  σ j ϒ jj

γ 2j

(34)

Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

S. Samagassi et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

9

The quantity in the numerator of the second half of Eq. (34) can be interpreted as a measure of how accurate is determined each parameter X j by the data [44]. For the noise variance, the re-estimate is provided by ðτ2 Þnew ¼

‖y  Ax‖2 m P J ð1  σ j ϒ jj Þ

(35)

j¼1

 In practice, during this process of re-estimation, many of the σ j approach infinity, so that π ðxj y; σ ; τ2 Þ presents an infinite peak at zero. The associated kernel can then be eliminated from the problem. In the following the WRVM is used under its version called the fast marginal likelihood maximization for sparse Bayesian models [27]. 3. Results and discussion Although the method can be applied in general to any elastic structure for which the matrix G in Eq. (8) is available, the WRVM algorithm will be tested on a planar beam having a symmetric section and loaded orthogonally to its mean fiber, in its plane of symmetry. Fig. 1 depicts the considered beam which is assumed to be made from a homogeneous elastic linear material and having a uniform cross section. The beam section is assumed to be rectangular. This enables to derive simple analytical time transfer functions between any excitation point where a transverse force f ðξf ; tÞ is applied and any response point where the longitudinal normal strain yðξs ; tÞ is assumed to be measured. The measurement point is assumed to be located on the upper beam fiber where a strain gauge sensor is placed.

It is straightforward to derive for the simply supported beam considered here that the Toeplitz like matrix G ξf ; ξs , giving the discrete time response in terms of longitudinal axial strain of the upper fiber Y xs as function of the discrete force vector F xf according to Eq. (7), writes     modes

π2h N X nπξf nπξs 2 sin gðωn ; δn ; ði ℓÞΔTÞ G ξf ; ξs ¼ n sin L L ρSL3 n ¼ 1 with gðωn ; δn ; ði ℓÞΔTÞ ¼ 0 if i oℓ, and



gðωn ; δn ; ði ℓÞΔTÞ ¼

ΔT exp  δn ωn ði ℓÞΔT qffiffiffiffiffiffiffiffiffiffiffiffiffi ωn 1  δ2n



 qffiffiffiffiffiffiffiffiffiffiffiffiffi  2 if i Zℓ sin ωn 1  δn ði ℓÞΔT

(36)

(37)

and the rotational frequencies are sffiffiffiffiffiffi n2 π 2 h E ωn ¼ 3ρ 2 L2

(38)

15 Original Reconstructed

Force (N)

10

5

0 0

200

400 600 Time step

800

1000

Fig. 2. Reconstructed transient impact forces; in discontinuous red the original signal and in black the reconstructed signal.

Table 1 Error as function of the noise power put on the measurement. Noise power σ 2η (%)

0.1

0.5

1

Error (%)

4.1

4.7

6.9

5 28

Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

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Table 2 Error as function of the noise power put on the system matrix; measurement noise power was fixed at σ 2η ¼ 0:5%. Noise power σ 2G (%) Error (%)

0.1

0.5

1

4.6

5.6

7.7

5 53

where i; ℓ A f1; 2; :::; Nt g, E is the Young’s modulus, ρ the density, hthe beam height, L the beam length, δn the modal damping associated to mode number n, Δt the time step and Nmodes the total number of modes selected owing to modal truncation. Nmodes is determined from a convergence study or can be estimated by considering the highest frequency contained in the impact signal. The time step is chosen in order to satisfy Shannon sampling condition with regards to the maximum frequency contained in the excitation, even if this condition is not necessary. Two cases are studied: A beam subjected to two impact concentrated forces and a beam with eight impact concentrated forces. All the forces are assumed to be orthogonal to the beam main fiber. Effect of noise and system matrix variability will be studied in this configuration. The second case of study is intended to solve approximately the localization problem which appears in the general case where the points of application of impact forces are not known a priori. The applied forces are assumed to be of the general form FðtÞ ¼ φt 2 e  ψ t

(39)

where the parameters φ and ψ are related to the maximum point of the force signal ðt max ; f max Þ by φ ¼ f max e and ψ ¼ 2=t max , where e is the Neper number. Let us consider the following geometric and material properties for the beam: L ¼ 1 m, h ¼ 5  10  3 m, E ¼ 7:06 10 10 Pa, ρ ¼ 2660 kg m  3 and uniform damping coefficient δn ¼ 5  10  3 . The five first frequencies of the pinned–pinned beam are: f 1 ¼ 11:68 Hz, f 2 ¼ 46:72 Hz, f 3 ¼ 105:1 Hz, f 4 ¼ 186:9Hz and f 5 ¼ 292 Hz. The total duration of calculation was fixed at T c ¼ 0:1 s and the time step was set to the value Δt ¼ 1:9512  10  4 s. The time step was fixed so as to satisfy Shannon’s sampling condition which is here equivalent to Δt o Δt max ¼ 1:7123  10  3 s, where Δt max ¼ 1=ð2f 5 Þ. The applied force spectrum is taken with a maximum frequency which is smaller than f 5 . A Daubechies wavelet based transformation is applied for sparsity requirement in order to write Eq. (8) under the sparse form as in Eq. (19). Ten levels were used and the total number of points was fixed at 4096. The mother wavelet applied for this transformation was Daubechies of order 4. Each force was disretized by using 512 time steps uniformally distributed in the time interval of computation ½0; T c . 2

=t 2max

3.1. Case of study #1: reconstruction of two forces Here, time histories of the applied forces are assumed to be generated by the constants φ1 ¼ 106 and ψ 1 ¼ 200, φ2 ¼ 106 and ψ 2 ¼ 400 according to Eq. (39). Their points of applications are assumed to be given respectively by ξf 1 ¼ L=10 and ξf 2 ¼ 4L=10. A single sensor is used for measurement and its location is fixed at ξs ¼ L=2. Due to the superposition principle, the inverse problem to solve in this case writes " #
 F1 G1 G2 ¼Y (40) F2 h and the matrix A is given by G1 Θ

i G2 Θ .

Applying the WRVM algorithm of Section 2.4, Fig. 2 shows the reconstructed global force signal as obtained by using the Bayesian hierarchical approach and the CS-WRVM based algorithm. The global force time signal contains two parts: the first half interval gives the force F 1 on the time domain ½1; 512 and the second half interval the force F 2 on the time domain ½513; 1024 where the unit of time correspond to the time step Δt ¼ 1:9512  10  4 s. Fig. 2 shows that reconstruction is quite good for the two forces applied simultaneously on the elastic beam even when using just a single sensor. It should be noticed that trying to reconstruct directly the forces without performing the wavelet transformation yields a very bad result and reconstruction results to be impossible. 3.2. Case of study #2: effect of noise and model uncertainties on reconstruction of forces To test robustness to noise of the proposed multiple forces reconstruction procedure based on the Bayesian hierarchical approach and the CS-WRVM algorithm, further simulations were performed. The same data than that given in Section 3.1 were used. In the first test the system model corresponding to Eq. (40) was perturbed by an additive white Gaussian noise as follows: " #
 F1 G1 G2 (41) ¼ Y þη F2 Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

S. Samagassi et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

11

15 Original Reconstructed

Force (N)

10

5

0 0

1000

2000 Time step

3000

4000

Fig. 3. Reconstructed transient impact forces; in discontinuous red the original signal and in black the reconstructed signal.

where η is a white Gaussian noise of power σ 2η . The maximum error was calculated here according to the uniform norm. Table 1 gives the obtained results as function of σ 2η . In the second test the system model of Eq. (40) was perturbed according to the following equation: " # h i F1 G1 þ δG1 G2 þ δ G2 (42) ¼ Y þη F2 where δG1 and δG2 are also chosen to be distributed according to a white Gaussian noise of the same power σ 2G . Fixing the noise power at σ 2η ¼ 0:5%, Table 2 gives the obtained results as function of σ 2G . 3.3. Case of study #3: approximate localization of impact Here, it is assumed that the two points of the applied forces corresponding to the previous case of study number 1 are unknown. So, impact identification involves both force localization and time history reconstruction. For this purpose the beam is meshed by using a set of ten points. In each point a virtual force is applied, even if its magnitude can strictly vanish when the point is free from any applied external force. The points are assumed to be regularly distributed on the beam and separated by the distance L=10. Since only the eight interior points are active, the inverse problem to be solved writes under the following form: 2 3 F1
6 7 G1 ::: G8 4 ⋮ 5 ¼ Y (43) F8 h The matrix A corresponds here to A ¼ G1 Θ

:::

i G8 Θ .

Applying the hierarchical Bayesian algorithm of Section 2.4, Fig. 3 gives the reconstructed global force signal as obtained by the CS-WRVM algorithm. For comparison the original impact forces as given by their analytical expressions were also plotted. The first obtained peak corresponds to the force F 1 which is applied at ξf 1 ¼ L=10 and the second one to the force F 4

applied at ξf 4 ¼ 4L=10. Fig. 3 shows that reconstruction of the global force vector was quite good by using just a single sensor. One should observe that this reconstruction strategy solves simultaneously the localization problem as the null forces are automatically reconstructed. Here also reconstruction is not possible without compressed sensing and without the use of the sparse transformation by means of matrix Θ. The construction of the wavelet transformation for high values of integer J, representing a particular power of number 2: the exponent in 2J , by using standard algorithm like the one used in this work can create saturation of the machine memory. This occurred for J Z13 on a PC of 1.94 Go of RAM. This issue has still to be addressed in order to release this complication. This will put a strict limitation on the procedure introduced in this work when considering for instance harmonic forces, as the observation period should be lengthened. In these conditions, the sparcity feature will also be lost. 4. Conclusion A Bayesian hierarchical approach based on compressed sensing and wavelet relevance vector machine was used in order to solve the multiple forces reconstruction problem. Both reconstruction of forces with well known impact points problems Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i

12

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and the general situation where locations of impact points are part of the problem were solved by the proposed method. The obtained results were very good in terms of accuracy and computational cost. The numerical study conducted showed robustness of the proposed Bayesian procedure. Bounded errors on the system matrix and the measurement were found to yield small perturbation of the calculated solution. Even if a simple elastic beam was investigated, generalization of this method to more general structures can be achieved without major difficulties; the only problem to decide is to be able to perform the wavelet transformation without memory saturation as more points are then required. References [1] E. Hensel, Inverse Theory and Applications for Engineers, Prentice-Hall, 1991. [2] R. Ghosh, N. Dilip, Methods of Inverse Problems in Physics, CRC Press, 1991, 504. [3] P.E. Hollandsworth, H.R. 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Please cite this article as: S. Samagassi, et al., Reconstruction of multiple impact forces by wavelet relevance vector machine approach, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.08.014i