Implicit IDA-PBC for Underactuated Mechanical Systems: An LMI-based Approach

Recently, the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) methodology has been extended to underactuated mechanical systems in implicit port-Hamiltonian representation. The method is not restricted to holonomic systems, does not require a positive-definite target inertia matrix and, under general conditions, avoids the need for solving partial differential equations. In this paper we simplify the conditions for (local) stability and present equivalent matching equations. In addition, we exploit the inherent polynomial structure of implicit systems modeled in Euclidean space, such that the implicit IDA-PBC problem can be cast as a linear matrix inequality problem. The method is applicable to desired Hamiltonians with arbitrary polynomial order. The proposed methodology is validated on the portal crane and the cart-pole system.