# Data-Driven Stability Assessment of Multilayer Long Short-Term Memory Networks
## 4 Results and Discussion
### Additional Design Considerations  
In this section, practical considerations regarding the generation and the scaling of the dataset, as well as the choice of the input signals are reported.  
1. Ensuring that the training dataset is composed by different signals in terms of magnitude and type is a key factor to obtain an NN capable of generalising the output. Additionally, other experimental results proved to be key to improve the ability of the network to learn different dynamic behaviour with corresponding improvements in the associated poles. During the design of the datasets, two main factors emerged, which are capable of jeopardising the accuracy of the identified poles: * the presence of ramps with low slopes; * the presence of sinusoidal inputs with long periods. More generally, the presence of inputs exciting only slow dynamics is believed to cause issues while using _tfest_. To avoid the issues associated with the sinusoidal inputs, it is possible to set a lower boundary to the period of those signals. By defining the period of the chosen sinusoidal inputs \(T_{sin}\) and the period of the simulation run \(T_{run}\), it is possible to impose: \(T_{sin}\geqslant T_{run}\). Regarding the slope of the ramp (\(m_{r}\)), no analytical boundary was identified, and a trial-and-error approach led to finding different values for each model simulated (for reference: \(m_{r}\geqslant 0.1\) (N) and \(m_{r}\geqslant 0.5\) (Nm) for the linear and nonlinear models, respectively).
2. It is also important to recall that the dataset size affects the training capabilities of the DNN. In this application, the dataset, formed by 500 runs (corresponding to 500 different inputs), has \(10^{6}\) points. Previous training tests carried out with a smaller dataset did not show satisfactory identification performances. Ensuring that the training set is large enough is therefore an important aspect of the procedure.
3. The last factor affecting the performance of the framework is identified as the scaling of the dataset. The input gate of the LSTM cells is sensible relative to the magnitude of the input signal: if the latter is excessive, the gate can saturate and show degrading identification performances. The input time series can be bounded between \([0,1]\) or \(\pm 3\sigma\) (with \(\sigma\) denoting the standard deviation) by means of a normalisation or  
Figure 11: Nonlinear model (overdamped case)—system with noise.  
a standardisation, respectively. For this work, the dataset was standardised since the method proved to be more robust with respect to the presence of outstanding outliers. For instance, the presence of a single step signal with a final value one order of magnitude higher than the average value of the rest of the dataset in a normalised dataset could lead the NN to assign a higher weight to some neurons in order to minimise the error for that case. That would in turn cause a degradation of the performance for the signals with a smaller magnitude. On the contrary, the standardisation of the dataset allows detecting when the high value signal is outside the \(\pm 3\sigma\) interval of the dataset and consequently assigning to it a relative lower weight during the training step.