LDV control over compact riemannian manifolds

Linear Dynamically Varying (LDV) systems are a subset of Linear Parameter Varying (LPV) systems characterized by parameters that are dynamically modeled. An LDV system is, in most cases of practical interest, a family of linearized approximations of a nonlinear dynamical system indexed by the point around which the system is linearized. LDV systems emerge quite naturally in a generic trajectory tracking problem in which the tracking error is modeled as an LDV system and the tracking controller is tuned so as to minimize an LQ performance. This paper focuses on tracking a natural trajectory of a nonlinear dynamical system running over a Riemannian manifold. The associated LDV tracking error system runs over the tangent bundle and LQ minimization secures asymptotic tracking of the nonlinear trajectory. The LDV tracking controller is provided by the solution to a partial differential Riccati equation (PDRE), itself related to a linear partial differential Hamiltonian operator. The index of the latter operator reveals some ergodic properties of the reference flow.