The method of controlled Lagrangians given by Bloch, Leonard, and Marsden provides a means for stabilizing relative equilibria -- motions in a mechanical system which are stationary in some of the variables -- by directly controlling the non-stationary variables. For example, one might wish to stabilize the angle of an inverted pendulum relative to a cart (to which it is attached), by controlling the position of the cart.
In the method of Bloch, Leonard, and Marsden, this is done by modifying the kinetic energy of the Lagrangian (kinetic - potential energy) in such a way that the dynamics associated with the modified Lagrangian without control are exactly the dynamics of the original Lagrangian with control. This modification leads to certain requirements, called matching conditions, which must be satisfied. This puts restrictions on the types of systems for which this method can be used.
My current research involves a modification to this method which relaxes these matching conditions. This allows the method to be applied to a broader class of problems, and also allows more freedom in the design of stabilizing controllers.
Chang, D.E., Long, D., Zenkov, D. [2009] On Embedding of Control Systems into Euclidean Space Proc. ICROS-SICE, 2009, 1542-1546.
Long, D.A., Bloch, A.M., Marsden, J.E., Zenkov, D.V. [2008] Relaxed Matching for Stabilization of Mechanical Systems Proc. MTNS 18, 2008.Long, D., Zenkov, D.V. [2007] Relaxed Matching for Stabilization of Relative Equilibria of Mechnical Systems Proc. CDC 46, 6238--6243. (pdf)
White, G., Madison, E., Lentz, M., Long, D. [2001] Simulation as a Production Employee Training Tool: A Real-World Case Study National Meeting of the American Society of Business and Behavioral Science, 2001.
Classes taught include: College Algebra, Finite Mathematics, Precalculus, and Calculus I & II