Parametrization of IDA-PBC by assignment of local linear dynamics

Abstract

In this paper the open problem of finding a suitable parametrization for Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) is addressed. IDA-PBC is motivated by the physical notion of storage, exchange and dissipation of energy. However, interconnection and damping matrices are fixed before the closed loop energy is determined. The interdependence of design parameters and energy impedes an a priori assessment of closed loop dynamics. We propose a systematic reduction of the set of constant design parameters in terms of solvability of the matching PDEs and closed loop stability. The constant parametrization simplifies the solvability conditions for the PDEs. A comparison of desired linearized dynamics with the linearized closed loop system yields a parametrization of IDA-PBC that ensures stability of the controlled system with a nonlinear Lyapunov function, while desired local dynamic behavior is realized. As a result of a linear coordinate transformation, the matching of the linearized dynamics is expressed by a linear system of equations. Its solution for suitable parameters obviates the tedious definiteness inspection of the resulting energy.