Equivalence of Immersion and Invariance and IDA-PBC for the Acrobot

In this note the two well known nonlinear control design techniques Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) and Immersion and Invariance (I&I) are compared through the example of the so-called Acrobot underactuated mechanical system. Equivalences of both procedures become obvious from the corresponding immersion and matching equations. In particular, the coordinate change which renders the potential energy matching PDE in IDA-PBC an ordinary differential equation is used to define the immersion map in I&I. It is shown that the energy shaping part of the IDA-PBC controller makes the closed-loop system an interconnection of two lower-dimensional port-Hamiltonian (pH) systems in the on- and off-manifold coordinates. The effect of damping injection output feedback can be identified with dissipation in the off-manifold part of the interconnected system. Dissipation is propagated to the on-manifold part which results in asymptotic stability of the system's equilibrium. The analysis in the present work provides an interesting interpretation of the effect of the IDA-PBC control law using the I&I framework.