Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 5394.9.2 Feedback LinearizationRecall that the core of control theory is the idea of the feedback. In case ofnonlinear control, this implies feedback linearization.Exact Feedback LinearizationThe idea of feedback linearization is to algebraically transform the nonlinearsystem dynamics into a fully or partly linear one so that the linearcontrol techniques can be applied. Note that this is not the same as a conventionallinearization using Jacobians. In this subsection we will presentthe modern, geometrical, Lie–derivative based techniques for exact feedbacklinearization of nonlinear control systems.The Lie Derivative and Lie Bracket in Control Theory. Recall(see (3.7) above) that given a scalar function h(x) and a vector–field f(x),we define a new scalar function, L f h = ∇hf, which is the Lie derivativeof h w.r.t. f, i.e., the directional derivative of h along the direction of thevector f. Repeated Lie derivatives can be defined recursively:L 0 f h = h, L i f h = L f(L i−1f) (h = ∇L i−1f)h f, (for i = 1, 2, ...)Or given another vector–field, g, then L g L f h(x) is defined asL g L f h = ∇ (L f h) g.For example, if we have a control systemẋ = f(x),y = h(x),with the state x = x(t) and the output y, then the derivatives of the outputare:ẏ = ∂h∂xẋ = L f h, and ÿ = ∂L f h∂x ẋ = L2 f h.Also, recall that the curvature of two vector–fields, g 1 , g 2 , gives a non–zero Lie bracket, [g 1 , g 2 ] ( (3.7.2) see Figure 4.2). Lie bracket motions cangenerate new directions in which the system can move.In general, the Lie bracket of two vector–fields, f(x) and g(x), is definedby[f, g] = Ad f g = ∇gf − ∇fg = ∂g∂x f − ∂f∂x g,