# Survey on Modelling and Techniques for Friction Estimation in Automotive Brakes
## 2 Techniques for the Estimation of BLCF in Automotive Disc Brakes
### Neural Networks for BLCF Estimation  
As seen previously, owing to the manifold phenomena affecting the brake operation and the awkwardness of rendering each of them, the existing analytical formulations provide unsatisfactory results. Moreover, considering that the models' parameters must be identified over again in case of a change in the plant characteristics, it is clear that the models are unreliable for online applications. These issues led to another approach, namely artificial neural networks, which has been introduced as a tool for systematic studies of the influencing variables, such as temperature, speed and load, for the prediction of BLCF. It is a valid tool for predicting tribological properties thanks to its ability to cluster and associate experimental data. Friction is highly dependent on the brake linings' chemical composition, environmental conditions and operating conditions; thus the application of neural networks could be useful to unveil the tangled correlations among the several parameters. Figure 9 shows an example of neural network that embeds the abovementioned dependences for BLCF calculation. This technique differs from traditional modelling approaches since they are trained to learn the right solutions rather than being designed to model specific phenomena. As shown in Figure 9, a neural network is composed of nodes (neurons), responsible for the processing of the information through the layer (e.g., the kind of analytical dependence among the variables) and the branches (synapses) that instead are responsible for the transmission of the signal between the layers (e.g., the relative weight of the identified analytical relationship). The neurons between the input and the output layers constitute hidden layers that serve to add non-linearity to the system and ramify the interactions among the variables of the previous layer. It is obvious that the network performance depends on its structure in terms of both the number of neurons per each layer and the number of hidden layers. In [60], the author focuses on the analysis of the influence of the neural network architectures on its generalization capability. It is known that an insufficient number of neurons in the hidden layers leads to an inability to solve the problem; on the other hand, too many neurons lead to overfitting and consequently to the lessening of the generalization capability. The transmission of the signals is performed in analogy with the way the biological neural system operates. The signals are generated in the neurons when the information coming from the previous layer exceeds a certain threshold (bias). Once the signal is generated, it is transmitted through the synapses to the next layer; the synapses modulate the relative importance of the signals flowing between two layers. Drawing on the analytical representation, the threshold crossing is modelled through an activation function, generally a sigmoid function; the transmission across the layer is realized through a weighted summation of the signals coming from the previous layer. Generally, a neural network configuration of three layers is used because, as assessed, it ensures good mapping ability and, at the same time, low computational burden. The output of each neuron \(j\), Equation (5), stems from a weighted summation that passes through the activation function in Equation (6):  
\[O_{j}=\varphi\left(\sum_{i=1}^{n}w_{ij}O_{i}\right) \tag{5}\]  
\[(z)=\frac{1}{1+e^{-z}}, \tag{6}\]  
Figure 9: Artificial neural network for BLCF estimation. where \(O_{i}\) is the output of the neurons of the previous layer; \(n\) is the number of inputs to the neuron; and \(w_{ij}\) is the weight between neurons \(i\) and \(j\). As mentioned above, the activation function \(\varphi\) is generally represented by a sigmoid function but any differentiable function is suitable for the backpropagation application.