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Stabilization of nonlinear RLC circuits: power shaping and passivation

Project members: D. Jeltsema, J.M.A. Scherpen, Romeo Ortega (Supelec, France)

Sponsored by: Marie Curie Control Training Site (CTS)

Passivity is a fundamental property of dynamical systems that constitutes a cornerstone for many major developments in systems and control theory, including optimal ($\mathcal{H}_2$ and $\mathcal{H}_\infty$) control, realization theory and adaptive control. Passivity has also been instrumental to reformulate, in an elegant and unifying manner, the central problem of feedback stabilization -- either in its form of feedback passivation for general nonlinear systems or as energy-shaping control for systems with physical structures.

It is well-known that arbitrary interconnections of passive (possibly nonlinear) resistors, inductors and capacitors define passive systems with supply rate the product of the external sources voltages and currents, and storage function the total stored energy. Interestingly, 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 circuits 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 as recently proposed in the literature. 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. 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.

Additional research includes the extension of our results beyond the realm of RLC circuits, e.g., to mechanical or electromechanical systems. A related question is whether we can find Brayton-Moser like models for this class of systems.


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