On the equivalence of two nonlinear control approaches: Immersion and invariance and IDA-PBC

In this paper we compare the two well-known nonlinear control design techniques Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) and Immersion and Invariance (I&I) at the example of the so-called Acrobot underactuated mechanical system. The immersion and matching equations in both approaches have a similar structure which is exploited to derive equivalent control laws, each of them providing a different perspective on the stabilization problem. 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 that appear in the I&I framework. 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 particular choice of the I&I design parameters in the present example, including the unconventional definition of coordinates on the invariant manifold, provides an interesting re-interpretation of the IDA-PBC control law from the I&I perspective. Finally, a discussion on the equivalence of the two approaches is presented by examining the cases of linear mechanical systems with one unactuated pivot as well as of general linear mechanical systems.