Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 551Controllability ConditionNonlinear controllability is an extension of linear controllability. Thenonlinear MIMO systemẋ = f(x) + g(x) uis controllableif the set of vector–fields {g, [f, g], ..., [f n−1 , g]} is independent.For example, for the kinematic car system of the form (4.53), the nonlinearcontrollability criterion reads: If the Lie bracket tree:g 1 , g 2 , [g 1 , g 2 ], [[g 1 , g 2 ], g 1 ], [[g 1 , g 2 ], g 2 ], [[[g 1 , g 2 ], g 1 ], g 1 ],[[[g 1 , g 2 ], g 1 ], g 2 ], [[[g 1 , g 2 ], g 2 ], g 1 ], [[[g 1 , g 2 ], g 2 ], g 2 ], ...– has full rank then the system is controllable [Isidori (1989); Nijmeijer andvan der Schaft (1990); Goodwine (1998)]. In this case the combined input⎧(1, 0), t ∈ [0, ε]⎪⎨(0, 1), t ∈ [ε, 2ε](u 1 , u 2 ) =(−1, 0), t ∈ [2ε, 3ε]⎪⎩(0, −1), t ∈ [3ε, 4ε]gives the motion x(4ε) = x(0) + ε 2 [g 1 , g 2 ] + O(ε 3 ), with the flow given by(see (3.49) below)() nF [g1,g2]t = lim F√ −g2 F√ −g1 F√ g2F√ g1.n→∞ t/n t/n t/n t/nDistributionsIn control theory, the set of all possible directions in which the systemcan move, or the set of all points the system can reach, is of obviousfundamental importance. Geometrically, this is related to distributions.Recall from subsection 4.7 above that a distribution ∆ ⊂ X k (M) on asmooth nD manifold M is a subbundle of its tangent bundle T M, whichassigns a subspace of the tangent space T x M to each point x ∈ M in asmooth way. The dimension of ∆(x) over R at a point x ∈ M is called therank of ∆ at x.A distribution ∆ is involutive if, for any two vector–fields X, Y ∈ ∆,their Lie bracket [X, Y ] ∈ ∆.A function f ∈ C ∞ (M) is called an integral of ∆ if df(x) ∈ ∆ 0 (x) foreach x ∈ M. An integral manifold of ∆ is a submanifold N of M suchthat T x N ⊂ ∆(x) for each x ∈ N. A distribution ∆ is integrable if, forany x ∈ M, there is a submanifold N ⊂ M, whose dimension is the same