Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 569smooth mapswhich defines D M by the equalityH M = h 0 + span R {h 1 , h 1 , ..., h n },D M = B # (dH M ),where we used the notation dH M to denote the set ∪ h∈HM dh. We also usethe notation H M = h 0 + H ∆ for an affine space of smooth maps where h 0is a smooth map and H ∆ a linear space of smooth maps.Having defined Hamiltonian control systems we turn to their trajectoriesor solutions: A smooth curve γ : I → M, I ⊆ R + 0 is called a trajectory ofcontrol system Σ = (U, M, H), iff there exists a curve γ U : I → U satisfying[Tabuada and Pappas (2001)]ẏ(t) = F (γ(t), γ U (t)), for every t ∈ I.Now, given a Hamiltonian control system and a desired property, an abstractedHamiltonian system is a reduced system that preserves the propertyof interest while ignoring modelling detail (see [Tabuada and Pappas(2001)]). Property preserving abstractions of control systems are importantfor reducing the complexity of their analysis or design. From an analysisperspective, given a large scale control system and a property to be verified,one extracts a smaller abstracted system with equivalent properties.Checking the property on the abstraction is then equivalent to checking theproperty on the original system. From a design perspective, rather thandesigning a controller for the original large scale system, one designs a controllerfor the smaller abstracted system, and then refines the design to theoriginal system while incorporating modelling detail.This approach critically depends on whether we are able to constructhierarchies of abstractions as well as characterize conditions under whichvarious properties of interest propagate from the original to the abstractedsystem and vice versa. In [Pappas et al. (2000)], hierarchical abstractionsof linear control systems were extracted using computationally efficient constructions,and conditions under which controllability of the abstracted systemimplied controllability of the original system were obtained. This ledto extremely efficient hierarchical controllability algorithms. In the samespirit, abstractions of nonlinear control affine systems were considered in[Pappas and Simic (2002)], and the canonical construction for linear systemswas generalized to nonlinear control affine systems.