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Industrial noise can be successfully mitigated with the combined use of passive and Active Noise Control (ANC) strategies. In a noisy area, a practical solution for noise attenuation may include both the use of baffles and ANC. When the operator is required to stay in movement in a delimited spatial area, conventional ANC is usually not able to adequately cancel the noise over the whole area. New control strategies need to be devised to achieve acceptable spatial coverage. A three-dimensional actuator model is proposed in this paper. Active Noise Control (ANC) usually requires a feedback noise measurement for the proper response of the loop controller. In some situations, especially where the real-time tridimensional positioning of a feedback transducer is unfeasible, the availability of a 3D precise noise level estimator is indispensable. In our previous works [1,2], using a vibrating signal of the primary source of noise as an input reference for spatial noise level prediction proved to be a very good choice. Another interesting aspect observed in those previous works was the need for a variable-structure linear model, which is equivalent to a sort of a nonlinear model, with unknown analytical equivalence until now. To overcome this in this paper we propose a model structure based on an Artificial Neural Network (ANN) as a nonlinear black-box model to capture the dynamic nonlinear behaveior of the investigated process. This can be used in a future closed loop noise cancelling strategy. We devise an ANN architecture and a corresponding training methodology to cope with the problem, and a MISO (Multi-Input Single-Output) model structure is used in the identification of the system dynamics. A metric is established to compare the obtained results with other works elsewhere. The results show that the obtained model is consistent and it adequately describes the main dynamics of the studied phenomenon, showing that the MISO approach using an ANN is appropriate for the simulation of the investigated process. A clear conclusion is reached highlighting the promising results obtained using this kind of modeling for ANC.

Considering performance requirements, requested in many current applications that use mathematical models, the behavior of the most physical phenomena can be represented by linear systems. The procedures for parametric identification for linear systems are well established and show many theoretical and practical results [3,4]. Some systems fail to have their behavior well described by linear models if their frontiers or ranges of values where they are excited are extended. In these cases, it is necessary to use a nonlinear model, and the identification of nonlinear systems using neural networks has been attracting interest and it has been applied successfully elsewhere [5-7].

Modeling techniques which use Artificial Neural Networks (ANNs) have been widely investigated and successfully applied to identification and control problems over the last twenty years. Specifically in vibro-acoustic systems, ANNs have been used in speech recognition [

ANN is also being used for applications in Active Noise Control (ANC). Bambang [

We present in this paper a methodology for building an ANN model to estimate the noise level in a certain spatial region subjected to noise emissions from a single vibrating source. The proposed model is designed to run in real-time, providing noise level estimation to be used by an ANC control system.

The neural network is trained to estimate the noise level at any point in the contained space in our acoustic system and uses as variables of input of the spatial coordinates of that point and the vibration signal, which is measured at the primary source.

The objective function used in network training is the sum of square errors (difference between the measured value for the noise level and the value predicted by the model in each spatial point). A set of experimental data is chosen for training, using the least squares metric and the obtained neural network is validated through simulations, comparing predictions with another set of data obtained from our experimental platform.

The vibro-acoustic system under study is composed of a centrifugal pump installed in a room (

details of both the accelerometer and the microphone installation. Considering the data collected by the accelerometer installed in the pump, and varying the position of a microphone per 350 predetermined points in the room (they are identified by its coordinates

;

and), 350 pattern pairs were collected which represent the dynamic of the vibro-acoustic transmission between the input signal that comes from the accelerometer (u) and the output signal, which comes from the microphone (y). The collected set of pairs defines the group of standard that will be used to train a recurrent ANN network that can best describe the dynamic of the vibroacoustic transmission in the proposed experimental platform.

Since Rosenblatt [

• one hidden layer with S_{I} sigmoid neurons with biases b^{1} which are associated with each neuron;

• one layer with S_{L} output neurons activated by a linear function with biases b^{2}, which are associated to each neuron.

By using a sufficient number of neurons (S_{I}) in the hidden layer of a two-layer network it is possible to approximate any function with a finite number of discontinuities within an accuracy that is arbitrarily specified [24,25]. This structure is shown in

The chosen orders for u and y in

(a) (b) (c)

Magalhaes [

(a) (b)

allows future implementation of this model structure in control systems in real-time.

For the purpose of comparing the present structure with that obtained in the work of Magalhaes [

where R is the number of network inputs and S_{I} is the number of neurons of the hidden layer. In this work, was adopted in order to keep the resulting number of parameters less than 135, as stated earlier. With the input layer defined according to the

Once the network topology is characterized, it is necessary to establish the training procedure, which was formulated through the optimization procedure shown in

Thus, the training of the network was formulated as a general problem of nonlinear optimization with constraints [27,28]:

where x is the vector of parameters of length n, is the objective function, which gives a scalar value, and the vector function gives a vector of length m

containing the values of equalities and inequalities constraints which are evaluated at x. Constraints are usually used to achieve certain desired properties for the network or to restrict the search region to avoid convergence problems. For both the topology and the experimental data used in our ANN model, no constraints were necessary because the optimization algorithm behaved smoothly in most of the runs.

The solutions of the Khun-Tucker (KT) [

The training was performed using MATLAB^{®}. The availability of a good initial estimate is an important requisite for success in solving these problems. In this case, an initial guess was used to train the model according to the scheme shown in

As stated earlier, it is necessary to establish a metric (criterion) to be used in the objective function. This metric can have a decisive influence in the determination of the optimal point as the optimization procedure will seek the parameters that lead the model to the best possible performance in relation to these established criteria. In this article Euclidean distance is used [

is commonly used in the treatment of causal events in order to obtain an objective function that gives the distance of functional responses (responses of the models) in relation to its target profile (experimental output). Euclidean distance is also commonly called the error vector norm (EVN) and is given by:

for the time domain, and by:

for the frequency domain, where SP and TP are the numbers of points in space and in time, respectively, y and Y are the outputs in the time and in the frequency domains respectively, the subscripts Mod and Exp refer to the model and to the experimental output respectively, the subscript s defines a specific spatial position, k and ω are the discrete time and frequency, respectively, the subscripts TD and FD refer to the domains of time and frequency, respectively, and h symbolizes the functional relationship between ω and k.

In the optimization procedure used here (Figures 6 and 7), both metrics established by the objective functions described in Equations (4) and (5) were tested, as well as their combinations, obtaining equivalent models. Here, for the purpose of numerical comparison between the different modeling approaches, Equation (4) was assumed.

The resulting models were used to predict the dynamic and spatial behavior of the system´s output signal (acoustic power measured by microphone). From Figures 8-10 the best and the worst results of the models in the estimation of the output signal for plans Z=1, 3 and 5 can be seen, as well as a qualitative comparison with estimated results with the models that were obtained in the work of Magalhaes [

• Smaller errors are obtained when considering the delays in each XZ plane to identify the models in the various grid points collected;

• According to the results of Magalhaes [

respective Euclidean norm values of the experimental signal (the objective function, which is used in the networks training, is the average of the square root of the sum of square errors);

• The ANN model performed equivalent to the ARX model (27 MRTFs, 135 parameters), with the advantage that it used fewer parameters (80 parameters).

This paper presented the development of an Artificial Neural Network (ANN) to describe the vibrate-acoustic transmission between a primary source of noise and a receiver in a room. The obtained dynamic ANN model captured the main dynamics of the system and performed equivalent to the performance presented by an ARXinterpolated model [

The authors acknowledge CAPES and CNPq (Brazilian federal research agencies) for their financial support.