On Lagrangian Dynamics and Its Control Formulations

Abstract This paper exploits the equivalence between constrained dynamics problems and a class of tracking control problems. An alternative derivation of the modern formulation of Parczewski and Blajer (1989) is presented, including some new analysis on the “almost singular” case, followed by a critique of the classical Lagrangian formulation. In addition, a new formulation which allows the constraints to be satisfied approximately is presented. A general closed-loop control law which depends on a user-specified threshold of constraint errors, 2, is proposed, transforming the default DAE problem into a stiff ODE initial-value problem which can be routinely solved on computers. It is shown that certain explosively unstable open-loop systems may be stabilized by the use of closed-loop control using a small but finite 2. An example is worked out using both the open-loop and the closedloop control formulations.