On Lagrangian Dynamics and Its Control Formulation
S. H. Lam
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, \epsilon, 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 $\epsilon$. An example is worked out using both the
open-loop and the closed-loop control formulations.
Revised February, 1998.
Published in Applied Mathematics and Computations, 91,
pp. 259-284, Elsevier Science, 1998.
to obtain a copy in Acrobat
.pdf format (700K).