Representation and Control of Brayton—Moser Systems using a Geometric Decomposition

This paper considers the problem of representing a sufficiently smooth control affine system as a structured Brayton–Moser system and to use the obtained structure to stabilize a desired equilibrium of the system. The present note proposes a geometric decomposition technique to express a given vector field as a Brayton–Moser system with desired structure. The proposed method is based on a decomposition of a differential one-form that encodes the divergence of a given vector field into its exact and anti-exact components, by using a homotopy operator, and into its co-exact and anti-coexact components, by introducing a dual homotopy operator. This enables one to compute, via integration, the potential and the structure generating the drift vector field. By identification of the obtained structure and a desired structure, it is therefore possible to study the feedback realization problem of the Brayton–Moser structure and feedback stabilization of control affine systems. Application of the proposed constructive approach to the control of the three-dimensional rigid-body problem is presented to illustrate the propose approach.