Stabilization of nonlinear RLC circuits: power shaping and passivation

In this paper we prove that for a class of RLC circuits with convex energy function and weak electromagnetic coupling it is possible to "add a differentiation" to the port terminals preserving passivity - with a new storage function that is directly related to the circuit power. The result is of interest in circuit theory, but also has applications in control problems as it suggests the paradigm of power shaping stabilization as an alternative to the well-known method of energy shaping. We show in the paper that, in contrast with energy shaping designs, power shaping is not restricted to systems without pervasive dissipation and naturally allows to add "derivative" actions in the control. These important features, that stymie the applicability of energy shaping control, make power shaping very practically appealing, as illustrated with examples in the paper. To establish our results we exploit the geometric property that voltages and currents in RLC circuits live in orthogonal spaces, i.e., Tellegen's theorem, and heavily rely on the seminal paper of Brayton and Moser in 1964.