Ergodic and Topological Aspects of Linear Dynamically Varying (LDV) Control

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9802594JonckheereWhen confronted with the problem of controlling a nonlinear dynamical system with unknown parameters, it is tempting to compute a family of linearized approximations, derive the LQ, H-infinity or Mu controller around each operating point, and then "stitch together" all of the locally stabilizing compensators in a globally working scheme. One such approach has recently been referred to as Linear Parametrically Varying (LPV) control where the parameters are of unknown dynamics, but constrained to lie in a bounded set, in which their variation should be slow enough to ensure stability of the adaptive scheme. Motivated by problems as tracking of transitive orbits, this proposal develops the so-called Linear Dynamically Varying (LDV approach, in which the parameters are dynamically modeled, the dynamics of the variation of the parameters is incorporated in the linear LQ or H-infinity design, resulting in a design that need not be locally stable, but that is guaranteed to be globally stable. Mathematically, in discrete-time (rep. continuous time), approach is characterized by functional (rep. partial differential) Riccati equations and linear matrix inequalities in sharp contrast with the "state dependent" Riccati equation of the traditional LPV theory. The major focus of attention is on the case of parameters running in a compact set and this quite naturally endows the LDV svstem with ergodic properties. Ergodic theory is "put to work" to develop a computational scheme for solving functional Riccati equations that relies crucially on the Poincare recurrence scheme. Other fixed point, continuous and differentiable selections, and Leray-Schauder degree methods are proposed as well. An example of the manifestation of the ergodic properties of the design is a "self-similar" solution to the functional Riccati equation. Next, while the LPV approach has focused on parameters running in a subset of the Euclidean space where the reference axes are taken for granted, the LDV approach on the other hand focuses on parameters running on a nontrivial (incontractible) manifold, and existence of the reference axes cannot be taken for granted. The relevant global topological property is parallelizability, that is, existence of a smooth orthonormal reference frame in the tangent space to the manifold, relative to which the linearlized state space equation are written. In case the state manifold is not parallelizable, the guiding idea to find parallelizable covering manifold on which the LDV system runs. Finally, an attempt to classifly these kind of control problems using the Godbillion-Vey characteristic classes of higher-codimenional foliation is proposed. Finally, the Gelfand-Feigin-Fuks theory of variation of characteristic classes is proposed to measure robustness and control authority. ***