Stabilization of nonlinear systems via potential-based realization

This paper considers the problem of representing a sufficiently smooth control affine system as a structured potential-driven system and to exploit the obtained representation for stability analysis and state feedback controller design. To recover the advantages of those representations for the stabilization of general nonlinear systems, the present note proposes a geometric decomposition technique, based on the Hodge decomposition theorem, to re-express a given vector field into a potential-driven form. Using the proposed decomposition technique, stability conditions are developed based on the convexity of the computed potentials. Finally, damping feedback stabilization is studied in the context of the proposed decomposition by using damping feedback to reshape the Hessian matrix of the obtained potential. Examples are given throughout the paper to illustrate the proposed approach.