A Dynamical Systems view of Feedback Synthesis

Abstract

Control theory and dynamical systems are closely intertwined fields. Pontryagin's maximum principal offers a strong connection by providing a constructive way to synthesize feedback control laws via lifting the control system to a (Hamiltonian) dynamical system. Feedback laws can then be encoded as invariant manifolds of this induced dynamical system - under the condition that these manifolds project diffeomorphically back to the base control system. While feedback stabilization is impossible for many control systems, the above procedure can still be carried out. In this setting, the invariant manifold no longer projects diffeomorphically which results in the emergence of caustics. This paper offers an overview of the above connection by translating target sets from controls to isotropic submanifolds in symplectic geometry and associates feedback controllability to singularities of the induced invariant manifolds. Multiple low-dimensional examples are included to elucidate the theory.