The Method of Controlled Lagrangians with Symmetry and its Applications
(対称性を持った制御ラグランジアンの手法とその応用）

The method of controlled Lagrangians has been developed to stabilize Lagrangian mechanical systems 
with symmetry by shaping the kinetic energy by feedback control. The new Lagrangian, which is called a 
controlled Lagrangian, is constructed by changing the kinetic energy of a given Lagrangian in such
a manner that the symmetry is preserved. Since the new system transformed by the feedback is also a 
Lagrangian mechanical system without control input, the stabilization can be understood by using 
energy-based Lyapunov functions. The procedure for finding such a controlled Lagrangian is called 
matching. In various documents about this method, systems have been studied with abelian symmetry 
only. The main aim of this paper is to study controlled Lagrangians with non-abelian symmetry.
Sufficient conditions for a matched controlled Lagrangian are obtained and applied to two examples;
one is the inverted pendulum on a cart which has abelian symmetry and the other is the twisted Kane-Scher
model which has non-abelian symmetry. The example of the inverted pendulum on a cart shows that the method 
of controlled Lagrangians is a generalization of the pole placement method for linear control systems to 
the energy-based Lyapunov method for non-linear mechanical control systems.