On the Stabilization of the Flexible Manipulator - Liapunov based Design. Robustness.

Abstract

This work deals with dynamics and control of the flexible manipulator viewed as a system with distributed parameters. It is in fact described by a mixed problem (with initial and boundary conditions) for a hyperbolic partial differential equation, the flexible manipulator being assimilated to a rod. As a consequence of the deduction of the model via the variational principle of Hamilton from Rational Mechanics, the boundary conditions result as “derivative” in the sense that they contain time derivatives of higher order (in comparison with the standard Neumann or Robin type ones). To the controlled model there is associated a control Liapunov functional by using the energy identity which is well known in the theory of partial differential equations. Using this functional the boundary stabilizing controller is synthesized; this controller ensures high precision positioning and additional boundary damping. All this synthesis may remain at the formal level, mathematically speaking. The rigorous results are obtained by using a one to one correspondence between the solutions of the boundary value problem and of an associated system of functional differential equations of neutral type. This association allows to prove in a rigorous way existence, uniqueness and well posedness. Moreover, in several cases there is obtained global asymptotic stability which is robust with respect to the class of nonlinear controllers - being in fact absolute stability. The paper ends with conclusions and by pointing out possible extensions of the results.