Extension of pid regulators to dynamical systems on smooth manifolds (M-PID)

This paper outlines possible extensions of a classical proportional-integral-derivative (PID) scheme to abstract systems whose state-spaces are smooth manifolds (i.e., state-manifolds). In particular, we first present a ``direct"" extension of a PID controller for first-order systems where the control actions are direct translations to manifolds, via canonical identifications. We show that one such direct extension is, however, unfeasible. Next, we suggest a specific extension, which arises as a nonstraightforward and careful definition of error terms, which leads to a consistent definition of M-PID controller. On the basis of the successful, novel definition of PID controller, we proceed in extending its design from first-order to second-order systems in two different versions, namely, position and velocity control. To complete this endeavor, we prove analytically that both position and velocity control may be achieved through the proposed approach.