On dynamics and control for a system of conservation laws - the case of the overhead crane

It is considered a non-standard model for the overhead (gantry) crane where two basic simplifying assumptions are overruled: negligible distributed mass of the rope and negligible acceleration of the load mass with respect to the gravity acceleration. Especially the last assumption (valid at each time moment) is difficult to check theoretically. The model thus obtained completes a previous one and is described by a partial differential equation of hyperbolic type in one space variable, with space varying parameters. Its boundary conditions are controlled ordinary differential equations. We have thus a boundary controlled system with distributed parameters. There is discussed the simplifying role of the zeroing of the small parameters: a well known model with lumped parameters is re-discovered. The second part of the paper is concerned with control laws analysis and synthesis. The basic tool is a control Liapunov function (c.l.f.) suggested by the identity of the energy integral. The feedback control law is synthesized at the formal level. Afterwards the well-posedness and the stability of equilibria are discussed for the closed loop system.