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 [J. Appl. Math. 56 (1989) 676–684] 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, ϵ, 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 ϵ. An example is worked out using both the open-loop and the closed-loop control formulations.