Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 543(1) L g L k f h(x) = 0 for all x in a neighborhood of x o and all k < r − 1; and(2) L g L r−1fh(x o ) ≠ 0.For example, controlled Van der Pol oscillator has the state–space form[] [ ]xẋ = f(x) + g(x) u =202ωζ (1 − µx 2 1) x 2 − ω 2 + u.x 1 1Suppose the output function is chosen as y = h(x) = x 1 . In this case wehaveL g h(x) = ∂h∂x g(x) = [ 1 0 ] [ ]0= 0, and1L f h(x) = ∂h∂x f(x) = [ 1 0 ] [ ]x 22ωζ (1 − µx 2 1) x 2 − ω 2 = x 2 .x 1MoreoverL g L f h(x) = ∂(L f h)∂x g(x) = [ 0 1 ] [ ]0= 1,1and thus we see that the Vand der Pol oscillator system has relative degree2 at any point x o .However, if the output function is, for instance y = h(x) = sin x 2 , thenL g h(x) = cos x 2 . The system has relative degree 1 at any point x o , providedthat (x o ) 2 ≠ (2k + 1) π/2. If the point x o is such that this condition isviolated, no relative degree can be defined.As another example, consider a linear system in the state–space formẋ = A x + B u, y = C x.In this case, since f(x) = A x, g(x) = B, h(x) = C x, it can be seen thatL k f h(x) = C A k x,L g L k f h(x) = C A k B.and therefore,Thus, the integer r is characterized by the conditionsC A k B = 0, for all k < r − 1C A r−1 B ≠ 0,otherwise.It is well–known that the integer satisfying these conditions is exactly equalto the difference between the degree of the denominator polynomial and the