On the Method of Interconnection and Damping Assignment Passivity-Based Control for the Stabilization of Mechanical Systems

TL;DR

This paper enhances the IDA-PBC method for stabilizing mechanical systems by removing redundancies, simplifying equations, and providing verifiable stabilizability criteria, making it as powerful as the controlled Lagrangian approach.

Contribution

It introduces a quadratic gyroscopic force approach to eliminate redundancy and simplifies the matching PDEs in IDA-PBC, expanding its applicability and robustness.

Findings

Redundancy in the skew-symmetric interconnection matrix is removed for systems with more than two degrees of freedom.

Simplification of the matching PDEs by eliminating gyroscopic forces.

Verifiable criteria for Lyapunov and exponential stabilizability of Hamiltonian systems.

Abstract

Interconnection and damping assignment passivity-based control (IDA-PBC) is an excellent method to stabilize mechanical systems in the Hamiltonian formalism. In this paper, several improvements are made on the IDA-PBC method. The skew-symmetric interconnection submatrix in the conventional form of IDA-PBC is shown to have some redundancy for systems with the number of degrees of freedom greater than two, containing unnecessary components that do not contribute to the dynamics. To completely remove this redundancy, the use of quadratic gyroscopic forces is proposed in place of the skew-symmetric interconnection submatrix. Reduction of the number of matching partial differential equations in IDA-PBC and simplification of the structure of the matching partial differential equations are achieved by eliminating the gyroscopic force from the matching partial differential equations. In addition, easily verifiable criteria are provided for Lyapunov/exponential stabilizability by IDA-PBC for all linear controlled Hamiltonian systems with arbitrary degrees of underactuation and for all nonlinear controlled Hamiltonian systems with one degree of underactuation. A general design procedure for IDA-PBC is given and illustrated with examples. The duality of the new IDA-PBC method to the method of controlled Lagrangians is discussed. This paper renders the IDA-PBC method as powerful as the controlled Lagrangian method.