A PDE approach to the derivation of the Brayton-Moser form for power-shaping control

Power-shaping control is an extension of energy-balancing passivity-based control based on a particular form of the dynamics, the Brayton-Moser form. One of the main difficulties in this approach is to write the dynamics in the suitable form since it requires the solution of a partial differential equation (PDE) system with an additional sign constraint. Here a general methodology is described for solving this partial differential equation system. The set of all solutions to the PDE system is given as the solution of a linear equation system. A necessary condition is given so that a solution of the linear system which meets the sign condition exists. This methodology is illustrated on a chemical reactor example, where the physical knowledge of the system is used to find a suitable solution.