Phase portrait approximation using dynamic neural networks

In this paper it is shown that a nonlinear system with an isolated equilibrium can be identified using two kinds of Dynamic Neural Networks, the Hopfield type (DNN-H) and the Multilayer type (DNN-M). The identified models are in continuous time. The equilibrium point is assumed to be the origin, without loss of generality, and it is assumed to be stable. Two different approaches are used to identify the nonlinear system, the first one is a phase portrait identification and the second one is the identification of the input/output response. The phase portraits for the two models are plotted and the eigenvalues of the Jacobian of the plant and the Jacobian of the models are calculated to make comparisons. Both networks can be used to identify nonlinear systems. The DNN Multilayer type is used to keep the number of states at a minimum, i.e., the number of states of the network is the same as the plant, the approximation capabilities of the DNN-M are increased by increasing the number of neurons of the internal multilayer network keeping the number of states constant. The DNN Hopfield type is used to obtain a state representation without hidden units but the number of states is not minimum, that is, the DNN-H equations are simpler but the number of states may be higher than the number of states of the plant.