Matching conditions and geometric invariants for second-order control systems

Working independently, a number of researchers studying Lagrangian control systems have proposed control designs for which the motion of the controlled system is itself characterized by a system of Euler-Lagrange equations. Of particular interest are the structured feedback designs proposed by Bloch, Leonard, and Marsden and the oscillatory open-loop designs proposed by Baillieul. In this paper, we discuss a certain condition, which we have called the input symmetry condition and which plays a role in both these designs. In the case of structured feedback, the condition ensures that the closed-loop system will remain a Lagrangian system, and that the corresponding Euler-Lagrange equations will be consistent with the dynamics of the original uncontrolled Lagrangian system. For the open-loop designs, the condition plays a role in proving stability of critical points of an energy-like function called the averaged potential. The condition also appears in a third context-namely in establishing conditions for reducing so-called acceleration controlled Lagrangian systems to velocity controlled Lagrangian systems. A major contribution of this note is a simple low-dimensional physical example for which the symmetry condition is not satisfied. A detailed analysis is provided