Level 3: Black Box Models

At the next level are black box models, which are basically used as data prediction mechanisms. They have a parametric form (polynomial, transfer function, etc.) but there is little physical information that can be gleaned from their outputs.

The black box model means that we know only the input x(t) and the output y(t) of the system, but the system itself is not known. The behaviour of the system is derived using data prediction mechanics such as polynomials or transfer functions.

Examples for black box models are [1]:

• Look-up table models,

• Curve fitting,

• Transfer function,

• Neuronal networks.

x(t) y(t)

b₀, b₁, … b_m

Figure 7: Black box model: example of curve fitting by polynomial interpolation.

Curve Fitting:

One commonly used data prediction method is fitting of various types of functions, e.g., polynomials, exponential functions or sine and cosine functions, to the mea-sured data. In many cases, a polynomial of a certain degree will be appropriate as a rudimentary model, as illustrated in Fig.(7). A simple example of a curve-fitting model is that of a straight line. Here the polynomial equation is given by

y = b₀+ b₁x (9)

where b1 is the slope of the straight line. For more complex data, such as a simple curve, the degree of the polynomial equation must be expanded to a higher order m as follows.

y = b₀+ b₁x + ... + b_mxm (10) Park et al. [31] measured the deformation of a steel beam using a strain gauge fibre-optic sensor and extracted the information about the deformation from the measured data using a 2^nd order polynomial equation.

Neural Networks

One damage feature extraction method which can be associated with a black box model is the method of artificial neural networks (ANNs) or only neuronal net-works. This method has been frequently used in the last few decades. ANNs are biologically inspired artificial intelligence representations that mimic the function-ality of the nervous system. Similar to biological neural networks, ANNs consist of a set of transfer functions which can be discontinuous, (e.g., symmetrized Heavi-side step function), continuous or linear. The goal of the ANN is to solve problems in the same way as the human brain, and this means that the ANN must learn from defined and known data in order to be able to interpret measured data. Even though the ANN is biologically inspired, the underlying physical process, in the best case, may be approximated by polynomials. One of the first know applica-tions of Neuronal Networks for damage feature extraction was invented in 2001 by Sohn et al [32]. They present a Neuronal Network algorithm in their paper with the general purpose of feature extraction and data reduction that contain the maximum amount of information from the original data set. For the fundamentals of ANNs, see, for example, Haykin [33].

Due to its popularity, the ANN method is increasingly finding its way into SHM applications. For example, Su and Ye [34] extracted spectrographic features from Lamb wave signals in the time-frequency domain to construct a damage pa-rameters database (DPD). The DPD was then used off-line to train a multilayer feed forward ANN under the supervision of an error back-propagation algorithm.

In his book, Giurgiutiu [5] gave a short introduction to ANNs and their possible uses in SHM applications. Based on the example of measurement data from elec-tromechanical impedance (EMI) experiments on thin circular aluminium plates with different damage locations, the functionality of ANNs in applications with EMI measurements was demonstrated. Lam et al. [35] used a combination of the pattern recognition method and the Bayesian ANN design method to form a practical SHM methodology. In their damage detection method, they proposed Ritz vectors as pattern features to detect the damage location and severity; these changes correspond to the differences between the measured Ritz vectors for the

undamaged and the possibly damaged structures. The functionality of the method was demonstrated using analytical models. Bao et al. [36] described in their work an approach of reducing the noise within measured data by transforming signals in images and training a deep neural network for anomaly classification. For ma-chine health monitoring Zhao et al. [37] used different methods of deep-learning.

The compared and discussed the results in order to give a overview of possible application. Cha et al. [38] invented in their work a visual inspection system in combination with a deep-learning algorithm to avoid the human factor during visual inspection.

Transfer Function

Another possibility for predicting the behaviour of the system without knowing the physical principles of the system, is to derive a so-called transfer function from the input and output data, (e.g., [39]). In principle, the dynamic behaviour of a linear system can be described by differential equations in the time domain,

a_ndⁿy where x is the input and y is the output of the linear system. When the initial state of the linear system is assumed to be without energy and furthermore the Laplace transform [40] (y(s) = L{y(t)}, x(s) = L{x(t)}) is applied to the differential equation the above differential equation changes to,

a_n· sⁿ· y(s) + a_n−1· sⁿ⁻¹· y(s) + ... + a₁· s · y(s) + a₀· y(s) =

b_m· s^m· x(s) + b_m−1 · s^m−1· x(s) + ... + b₁· s · x(s) + b₀ · x(s). (12) The quotient ^y(s)_x(s) forms now the transfer function G(s),

G(s) = y(s)

x(s) = b_m· s^m+ b_m−1· s^m−1+ ... + b₁· s + b₀

a_n· sⁿ+ a_n−1· sⁿ⁻¹+ ... + a₁· s + a₀ = Z(s)

N (s). (13) The transfer function of a linear system is thus a rational function,

G(s) = b(s)

a(s) (14)

where the polynomials a(s) and b(s) are given by equation (12) and the transfer function is not any more depending on the input- and out signals.

Much literature can be found on the topic of transfer functions for damage feature extraction. Here, only the papers that are the most relevant from the point of view of the author are briefly mentioned.

Zang et al. [41] presented a possible method for correlating measured frequency responses from multiple sensors and proposed to use these correlated responses as an indicator for structural damage detection. They developed one transfer func-tion for the pristine state of the structure and a second transfer funcfunc-tion for all subsequent responses of the structure, and correlated both transfer functions. Ex-periments on real structures showed that the algorithm required further investi-gation, leading to the combination of the transfer function method with neural networks. Lanza di Scalea et al. [42] estimated in their work the dynamic trans-fer function between two outputs of a linear system subjected to an uncontrolled and generally unknown excitation and possible uncorrelated noise present at both outputs. They demonstrated that by taking in account the normalizing factor for the two outputs the linear system works stable also for unknown excitation.

Summary: Black Box Models

The examples illustrated above represent only a few data prediction methods, and the list of suitable methods can be expanded to include others, such as least mean square estimation or more sophisticated methods.

Black box models are very often used for applications on civil structures like bridges and one of the reason is the problem of the identification of the physi-cal properties of the structures because their development phase was many years (decades) ago. Examples for black box models used for the monitoring of bridges can be found in e.g. [43] or [44]. Park et al. [45] used displacement measurement taken by different linear sensor for performing a inverse structural analysis of a steel beam.

Nevertheless, there is still little physical information that can be gleaned from the output of black box models.