Differential Neural Networks Observers: Development, Stability Analysis and Implementation

The control and possible optimization of a dynamic process usually requires the complete on-line availability of its state-vector and parameters. However, in the most of practical situations only the input and the output of a controlled system are accessible: all other variables cannot be obtained on-line due to technical difficulties, the absence of specific required sensors or cost (Radke & Gao, 2006). This situation restricts possibilities to design an effective automatic control strategy. To this matter many approaches have been proposed to obtain some numerical approximation of the entire set of variables, taking into account the current available information. Some of these algorithms assume a complete or partial knowledge of the system structure (mathematical model). It is worth mentioning that the influence of possible disturbances, uncertainties and nonlinearities are not always considered. The aforementioned researching topic is called state estimation, state observation or, more recently, software sensors design. There are some classical approaches dealing with same problem. Among others there are a few based on the Lie-algebraic method (Knobloch et. al., 1993), Lyapunov-like observers (Zak & Walcott, 1990), the high-gain observation (Tornambe 1989), optimization-based observer (Krener & Isidori 1983), the reduced-order nonlinear observers (Nicosia et. al.,1988), recent structures based on sliding mode technique (Wang & Gao, 2003), numerical approaches as the set-membership observers (Alamo et. al., 2005) and etc. If the description of a process is incomplete or partially known, one can take the advantage of the function approximation capacity of the Artificial Neural Networks (ANN) (Haykin, 1994) involving it in the observer structure designing (Abdollahi et. al., 2006), (Haddad, et. al. 2007), (Pilutla & Keyhani, 1999). There are known two types of ANN: static one, (Haykin, 1994) and dynamic neural networks (DNN). The first one deals with the class of global optimization problems trying to adjust the weights of such ANN to minimize an identification error. The second approach, exploiting the feedback properties of the applied Dynamic ANN, permits to avoid many problems related to global extremum searching. Last method transforms the learning process to an adequate feedback design (Poznyak et. al., 2001). Dynamic ANN’s provide an

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