Energy Shaping of Mechanical Systems via PID Control and Extension to Constant Speed Tracking

In a recent contribution, it was shown that a class of mechanical systems, which contains many practical examples, can be stabilized via energy shaping without solving partial differential equations. The proposed controller consists of two terms, a partial linearizing state-feedback and a linear PID loop around two new passive outputs. In this brief note we prove that the first, admittedly non-robust, step can be obviated leaving only the linear PID. A second contribution of the note is to propose a slight modification to the controller to go beyond regulation tasks-being able to follow ramp references in the actuated coordinates.