## 3 Air Data System Description and Functions
### Synthetic Sensor Description  
The proposed ADS is a single line replaceable unit (LRU) embedding a synthetic sensor, dedicated to AoA and AoS estimation based on a patented technology at TRL6 [34]. A prototype overview is represented in Figure 2a. The synthetic sensors are essentially state observers for which the A/C flight dynamic model is replaced by a model based on neural networks [35]. Exploiting the advantages of the fly-by-wire (FBW) technology, the air data system will receive, as input, consolidated data from the avionic bus (introduced in Section 3.2) to be fused with measured ones (introduced in Section 3) in order to estimate AoA and AoS with high reliability, as schematically presented in Figure 3b. The synthetic sensors rely on a pretrained (i.e., deterministic) neural network and the use of A/C data from the attitude and heading reference system, primary surface commands/deflections, and a Global Navigation Satellite System (GNSS). The neural network consists of a biased linear combination of nonlinear activation functions. Each activation function is driven by a biased linear combination of the output of the preceding nodes. The multilayer perceptron (MLP) can be described as a nonlinear map between the input and the target. The training stage has the aim to find weights of the network that better fit the desired map. The MLP approach is mathematically proven using the universal approximation theorem. In fact, it is proven that any continuous function of \(n\) real variables, with support in the unit hypercube, can be uniformly approximated by finite superposition of a fixed, univariate function that is discriminatory [36]. Figure 2: Overview of the proposed air data system (ADS): architecture and parts. The synthetic sensors deal with a very straightforward model, suitable for real-time and cost-effective innovative avionic systems. Consider the following assumption on AoA and AoS valid:  
\[\alpha_{\text{SS}} =\hat{\alpha}+\Delta\alpha \tag{1}\] \[\beta_{\text{SS}} =\hat{\beta}+\Delta\beta \tag{2}\]  
where \(\hat{\alpha}\) and \(\hat{\beta}\) are initial estimations obtained with flight mechanics equations, whereas \(\Delta\alpha\) and \(\Delta\beta\) are the differences between the linear estimations and the true values of Figure (a)a. According to a patented procedure [37], \(\hat{\alpha}\) and \(\hat{\beta}\) are augmented with the evaluation of \(\Delta\alpha\) and \(\Delta\beta\) based on two MLPs, which process measurements obtained with non-protruding sensors (except for the Pitot tube and TAT). \(\hat{\alpha}\) and \(\hat{\beta}\) can be evaluated as follows:  
\[\hat{\alpha}=\theta\ -\gamma \tag{3}\]  
\[\hat{\beta}=K\frac{n_{y}}{q_{c}} \tag{4}\]  
where \(\theta\) stands for the pitch angle, \(\gamma\) stands for the flight path angle, \(n_{y}\) is the proper acceleration as measured by the accelerometer along the \(Y_{B}\) axis, and \(q_{c}\) is the impact pressure. \(K\) is an A/C constant derived from flight mechanic considerations. Mathematical demonstrations exist [36; 38; 39; 40; 41; 42] about the MLP performing as a universal approximator. During the training procedure, the weights of the linear combinations are estimated when solving the nonconvex problem of the error function optimization, for which different heuristic rules exist. The Levenberg-Marquard (LM) algorithm is used in this work. The complete input vector needed by the synthetic sensors includes data from the GPS (providing \(V_{down}\)), the ADS, and the attitude and heading reference system (AHRS). The synthetic sensors considered in this work have the following characteristics:  
* feed-forward neural network,
* one hidden layer with 24 neurons,
* neurons with sigmoidal activation functions,
* one output layer with a single (or double for the VS-A&S) linear neuron, and
* limited output during the operative life. Figure 3: Aircraft model and ADS scheme.