An artificial neural network is basically a computational system which receives an input, processes the data, and provides an output. Designing the neural network requires choice of the topology, performance (transfer) function, learning algorithm and criteria for stopping the learning process. The neural network constructed for this problem, shown in figure 8, has three input signals time and the angle of rotation of the satellite and a coefficient k j , which is necessary to map output signal with the corresponding timestamp t j . Two hidden layers contain 7 and 10 neurons respectively. Only one output is necessary in the considered case. It is a rotation angle of the satellite in some discrete time t j which is also equivalent to a decision variable X j of the dynamic optimisation task. After the inputs have been entered into the neural network, the response is calculated using a transfer function.
For numerical simulations Encogag library and its .NET implementation has been applied. As the transfer functions the hyperbolic tangent and linear functions were used, for the hidden layer and the output layer respectively. The multilayer neural network was trained using three different supervised algorithms: backpropagation learning algorithm, the Levenberg-Marquardt, the Resilient Propagation and the Quick Propagation. Although the results obtained were different in each case, they always fulfilled the requirements given in the optimisation problem. The best accuracy of the neural network was obtained using the Resilient Propagation
Figure 8. The topology of the neural network.
After the training process an acceptable accuracy of network response has been obtained. Fig. 9 and Fig. 10 show courses of the drive function calculated from the optimisation task by means of the Nelder-Mead method (a) and using the neural network (b) for data taken from learning set. Fig. 11 and Fig. 12 show courses of the tip displacement of outer panels in the local coordinate system of the central body before optimisation (a), obtained from the dynamic optimisation (b) and the neural network (c) for data taken from the learning set.
Figure 9. Courses of the drive function calculated by the Nelder-Mead method (a) and the neural network (b);
Figure 10. Courses of the drive function calculated by the Nelder-Mead method (a) and the neural network (b).
Figure 11. Courses of outer panels tip displacement in the local coordinate system of the central body before (a), after (b) optimisation and obtained from neural network (c).
The trained and validated neural networks can be used in real-time prediction of driving functions. After applying the driving function obtained from the trained neural network, vibrations of the panels are about 5 times smaller than before optimisation in the considered case.
Although the analysis presented in the paper was concerned only with one type of a satellite and manoeuvre, the method presented can be used in the analysis of any flexible multibody system. In order to control the stability in all conditions, the training set of the optimal solution has to include results of the direct optimisation for other possible types of manoeuvres.