Bilinear control systems for a problem of optimal crane control

Abstract

The paper deals with two-mass dynamic "trolley – cargo" movement model that represents
operation of an overhead crane with a variable length rope. A problem of finding the optimal time
control of the overhead crane, carrying loads from one point to another, are set. Two new
approaches to solve this problem are suggested and considered. For the first approach, when it is
assumed that the rope length is varying according to some predetermined law, the model remains in
a linear form, but the system, describing its motion, turns into the system with variable coefficients.
For the second approach, assuming that the rope length can be controlled, the model reduces to the
form of the so-called biaffine system. It is obtained a solution of the second order equation with
variable coefficients for the case when the rope length is a piecewise linear function that is known
in advance. Under consideration of the corresponding biaffine system with the factors used in the
work we have not been able to show the fact of system controllability over the entire space. Further
research of the biaffine and bilinear control systems is required to obtain positive results.