Researchers: Rolf Johansson, Anders Robertsson in cooperation with Prof. A. Shiriaev, Umeå University, Swedish Research Council 2006-2008, Ref. 2005-4182, 2008-4369
This aim of this project is to develop feedback control laws for nonlinear dynamical systems represented by the classical Euler-Lagrange equations. We consider the systems with the number of actuators being less than the number of its degrees of freedom (DOF) by one. Examples of such dynamical systems are ubiquitous, for instance, a cart-pendulum system (2 DOF correspond to position of the cart and angle of the pendulum, 1 actuator produces the force applied to the cart) and a model of a ship on a plane (3 DOF; 2 actuators). The two problems, approached in the project, are: how to derive a simple and efficient algorithm of motion planning for such a under-actuated systems and how to make a pre-planned motion orbitally stable in the closed loop. It is well known that feedback control design for under-actuated systems is inherently difficult task since not every desired motion is feasible for a system with not actuated DOF. Our controller design approach is based on the idea of virtual holonomic constraint: geometrical relations imposed between generalized coordinates, which are made invariant for the closed loop system. Exploiting this idea, we have obtained series of preliminary results, in particular, on reducibility of dynamics, integrability of zero dynamics, extension of the famous Lyapunov lemma on presence of center in a nonlinear system, constructive procedure for exponential orbital stabilization of pre-planned motions, extensions to hybrid dynamical systems.
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