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1 2002

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On a neural control close to time-optimal of a non-linear and discontinuous object

Hejmo W., Majewska K.

Systems Science

2002

Vol. 28, nr 2

51-58

The paper deals with the time-optimal problem of the controlled object the dynamics of which is given by the mapping: x = y, y = f (x) + u, |u| [left angle bracket]or= 1, with selected discontinuous function of motion resistance f. For any target state the time optimal trajectory may be formed by the solution induced by the "bang-bang" control function with one, two or three switching operations. The switching curve, generally discontinuous, cannot be defined in open, algebraic form. The unique one way of its shape formation is the numerical approach. Moreover, in the time-optimal process the phenomenon of nonunique solution may appear, the trajectories of which reach the target state along a totally different way in the same minimum time. We are able to determine the value of the time optimal control function belonging to each state in the state plane, thus, feedback system may be theoretically constructed. Unfortunately, because of non-algebraic formulas determining the states in which the switching operations should be executed and, moreover, because a singular phenomenon of non-unique time optimal solutions may appear, the standard concept of regular closed-loop system synthesis becomes inappropriate. In order to synthesize the engineering closed-loop control structure, there is suggested an idea of the controller of the three-layer feed-forward back propagation network. The construction and the training of the neural network have been used with Levenberg-Marquardt method. A comparison between theoretical minimum time of target reaching and time of target reaching in the system created in accordance with the neural system proposed is made.