Ivancevic_Applied-Diff-Geom

554 Applied Differential Geometry: A Modern Introduction4.9.4 Geometrical Control of Mechanical SystemsMuch of the existing work on control of mechanical systems has reliedon the presence of specific structure. The most common examples of thetypes of structure assumed are symmetry (conservation laws) and constraints.While it may seem counter–intuitive that constraints may helpin control theory, this is sometimes in fact the case. The reason is thatthe constraints give extra forces (forces of constraint) which can be usedto advantage. probably, the most interesting work is done from the Lagrangian(respectively Hamiltonian) perspective where we study systemswhose Lagrangians are ‘kinetic energy minus potential energy’ (resp. ‘kineticenergy plus potential energy’). For these simple mechanical controlsystems, the controllability questions are different than those typically askedin nonlinear control theory. In particular, one is often more interested inwhat happens to configurations rather than states, which are configurationsand velocities (resp. momenta) for these systems (see [Lewis (1995);Lewis and Murray (1997)]).4.9.4.1 Abstract Control SystemIn general, a nonlinear control system Σ can be represented as a triple(Σ, M, f), where M is the system’s state–space manifold with the tangentbundle T M and the general fibre bundle E, and f is a smooth map, suchthat the following bundle diagram commutes [Manikonda (1998)]ψE✲ T M❅❅ ππ M❅❅❘ ✠Mwhere ψ : (x, u) ↦→ (x, f(x, u)), π M is the natural projection of T M onM, the projection π : E → M is a smooth fibre bundle, and the fibers ofE represent the input spaces. If one chooses fibre–respecting coordinates(x, u) for E, then locally this definition reduces to ψ : (x, u) ↦→ (x, ψ(x, u)),i.e.,ẋ = ψ(x, u).The specific form of the map ψ, usually used in nonlinear control, is