Power-shaping control: Writing the system dynamics into the Brayton–Moser form

Power-shaping control is an extension of energy-balancing passivity-based control that is based on a particular form of the dynamics, the Brayton–Moser form. One of the main difficulties in this control approach is to write the dynamics in the suitable form since this 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. Furthermore a necessary condition is given so that a solution of the linear system which meets the sign condition exists. This methodology is illustrated on the case of a chemical reactor, where the physical knowledge of the system is used to find a suitable solution.