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This work discusses the construction of an observer for language-driven control systems. Such systems accept symbolic inputs corresponding to feedback control laws together with logical conditions which determine when each control law is to be applied. Here, we explore the problem of how to identify the symbolic string driving the system through observation of its output. We discuss the identification of individual “instructions” from which inputs are composed, as well as the recovery of more global features of the input. These features can be quite complex depending on the richness of the grammar. Our approach is motivated by settings where language-driven systems (e.g., mobile robots) must cooperate “silently”, so that a newcomer who wants to be useful must first determine what others in the team are trying to accomplish. Our ideas are illustrated in a series of numerical experiments involving symbolic control of linear systems using the motion description language MDLe.
In this paper, we provide a new formulation for the generalized periodic Toda lattice. Since the work of Kostant, Adler and Symes, it has been known that the Toda lattice is related to the structure of simple Lie algebras. Indeed, the non-periodic and the periodic Toda lattices can be expressed as Hamiltonian systems on coadjoint orbits: the former of a simple Lie group and the latter of the associated loop group. Alternatively, the non-periodic Toda lattice was expressed as a gradient flow on an adjoint orbit of a simple Lie group by Bloch, Brockett and Ratiu. Based on the description of certain gradient flows on adjoint orbits in affine Lie algebras as double bracket equations, we show that the periodic Toda lattice also admits a canonical gradient formulation and relate it to the structure of affine Kač-Moody algebras.
We investigate a generalization of Brockett’s celebrated double bracket flow that is closely related to matrix Riccati differential equations. Using known results on the classification of transitive Lie group actions on homogeneous spaces, necessary and sufficient conditions for accessibility of the generalized double bracket flow on Grassmann manifolds are derived. This leads to sufficient Lie–algebraic conditions for controllability of the generalized double bracket flow. Accessibility on the Lagrangian Grassmann manifold is studied as well, with applications to matrix Riccati differential equations from optimal control.
Robot navigation over large areas inevitably has to rely on maps of the environment. The standard manner in which such maps are defined is through geometry, e.g. through traversability grid maps or through a division of the environment into free-space and obstacle-space. In this paper, we combine certain aspects of the geometric maps, through the notion of distinctive places, with a topological description of how these places are related. What is novel is the idea that the adjacency relation is defined by the existence of a control law that drives the robot between topologically connected places. Moreover, these maps can be automatically constructed based on the premise that the nodes correspond to places associated with a hightened control activity.