## 1 Introduction  
Machine learning techniques, when they are applied to dynamic systems, are preferred to have in their turn a dynamic structure, namely the variable time being included in the algebraic structure of the model. More specifically, in the case of Artificial Neural Networks, the dynamics of the model are obtained by including delay blocks in the structure [1]. The dynamics can be introduced in the neural network by maintaining its feedforward structure (Time Delay Neural Networks--TDNN [2, 3, 4, 5]), or by introducing feedback [6]. In the latter case, the delays are mandatory, otherwise, the calculation of the neuron output cannot be resolved. Both feedforward and feedback neural networks are suitable for modeling dynamic systems; thus, the choice of which paradigm to use depends on the requirements of the problem to be addressed and the available resources. The feedforward paradigm has the advantage of leveraging the same algorithms for training static NNs. In particular, in the case the delay blocks are foreseen only in a delay line at the input (Focused Time Delay Neural Networks--FTDNN [7]), the downstream part of the NN is structured as a static one, and so any static paradigm can be implemented. A different training strategy is adopted depending on the stationary or non-stationary behavior of the system to be modeled. In the first case, the whole evolution of the system, assumed as a training set, can be used to train the NN iteratively in batch mode, in the same way the static NNs are trained. In case the physical system is not stationary, the batch mode is no longer suitable, and the NN model must be adapted dynamically during the evolution of the system [8]. To this purpose, the training set at each iteration is constituted only by the last few samples of the physical signal, while the sensitivity of the model with respect to the past samples tends to vanish with time. From this point of view, the training strategy is the same as adapting filters [9, 10], with the added value of exploiting nonlinearity. Froma topological point of view, FTDNNs can be seen as the cascade of a Finite Impulsive Response (FIR) linear filter with a Multi-Layer Perceptron [11]. As for the FIRs, these kinds of NNs have a short-term memory, represented by the samples stored in the delay line, and a long-term memory, represented by the weights of the connections. The FIR filters take their name from the fact that the impulsive response has a duration equal to the number of delays in the delay line (Memory Depth), after which it is null. This implies that a proper number of delays must be defined a priori to guarantee that the dynamics of the system under study will be properly modeled. As such, since the dynamics are unknown a priori, a trial-and-error procedure is adopted to design the delay line. As an alternative, feedback can be introduced in the structure of the NNs [12; 13; 14]. Different strategies are used depending on whether the feedback connects each pair of neurons in the network, or neurons belonging to different layers, or if the output of the NN is fed back to the input, or, finally, if the feedback is localized within the neurons. This last category is called Locally Recurrent Neural Networks (LRNNs [15; 16]), and for many applications, it represents the best compromise between performance and computational burden. The advantage of feedback is that the memory depth can be arranged by modifying a parameter, rather than changing the topology of the NN. The drawbacks are a larger computational cost with respect to feedforward NNs, and the fact that they are subject to instabilities [4; 17]. LRNNs allow one to limit the computational cost, but the stability issue remains. The global structure of these networks is the same as that of a Multi-Layer Perceptron (MLP), but internally they are structured as an Infinite Impulsive Response (IIR) linear filter, optionally combined with an FIR filter, while the nonlinearity is placed downstream of the linear filter.