Ivancevic_Applied-Diff-Geom

576 Applied Differential Geometry: A Modern Introductionloop functor N L : CAT ⇒ CAT ∗ between two mutually dual nonlinearcategories CAT and CAT ∗ . We apply the natural process Ξ, separately(1) To the feedforward decomposable systemS ≡ (X, A, U, B, Y, C) in Vect, and(2) To the feedback morphism K = e −1 ◦ m −1 : Y → U in Vect ∗ .Under the action of the natural process Ξ, the linear feedforward systemdynamics (X, A) in Vect transforms into a nonlinear feedforward Ξ−dynamics(Ξ[X], Ξ[A]) in CAT , represented by a nonlinear feedforward decomposablesystem, Ξ[S] ≡ (Ξ[X], Ξ[A], Ξ[U], Ξ[B], Ξ[Y ], Ξ[C]).The reachability map transforms into the input process Ξ[e] = Ξ[A] ◦Ξ[B] : Ξ[U] −→ Ξ[X], while its dual, observability map transforms into theoutput process Ξ[m] = Ξ[C] ◦ Ξ[A] : Ξ[X] −→ Ξ[Y ]. In this way the totalresponse of the linear system r = m ◦ e : U → Y in Mor(Vect) transformsinto the nonlinear system behavior, Ξ[r] = Ξ[m] ◦ Ξ[e] : Ξ[U] −→ Ξ[Y ] inMor(CAT ). Obviously, Ξ[r], if exists, is given by a nonlinear Ξ−-transformof the linear state equations (4.66–4.67).Analogously, the linear feedback morphism K = e −1 ◦ m −1 : Y → Uin Mor(Vect ∗ ) transforms into the nonlinear feedback morphism Ξ[K] =Ξ[e −1 ] ◦ Ξ[m −1 ] : Ξ[Y ] → Ξ[U] in Mor(CAT ∗ ).In this way, the natural system process Ξ : L ⇛ N L is established.That means that the nonlinear loop functor L = Ξ[L] : CAT ⇒ CAT ∗ isdefined out of the linear, closed–loop, continual–sequential MIMO–system(4.66).In this section we formulate the nonlinear loop functor L = Ξ[L] :CAT ⇒ CAT ∗ for various hierarchical levels of muscular–like FC.4.9.6.2 Spinal Control LevelOur first task is to establish the nonlinear loop functor L = Ξ[L] : EX ⇒EX ∗ on the category EX of spinal FC–level.Recall that our dissipative, driven δ−Hamiltonian biodynamical systemon the configuration manifold M is, in local canonical–symplectic coordinatesq i , p i ∈ U p on the momentum phase–space manifold T ∗ M, given by