The link between IDA-PBC and the direct Lyapunov approach for underactuated mechanical systems

Abstract

Interconnection and Damping Assignment - Passivity Based Control (IDA-PBC) provides a means for designing stabilizing control laws for underactuated, nonlinear mechanical systems. It has been shown to encompass several separately published methodologies that also treat the same class of systems. A key step in the design process is the use of the annihilator operator for finding the governing differential and partial differential equations which determine the equivalent mass matrix and potential energy of the stabilized system. In the Direct Lyapunov Approach (DLA), the annihilator operator is not used to develop the new mass matrix which has the beneficial result of producing a control law that is linear in the velocities. It is shown through a projection of the IDA-PBC fundamental relation that the DLA method can be developed. This process also demonstrates that the DLA produced system is Lagrangian and that relations exist that relate the matrices of the original and stabilized systems. An example supports the results derived in the text.