Ivancevic_Applied-Diff-Geom

546 Applied Differential Geometry: A Modern Introductionlinearizable part plus nonlinear terms of highest possible order around anequilibrium point or an equilibrium manifold. However, in many applicationsone requires a large region of operation for the nonlinearizable system.In such a case, demanding the nonlinear terms to be neglected to be of highestpossible order may, in fact, be quite undesirable. One might prefer thatthe nonlinear terms to be neglected be small in a uniform sense over theregion of operation. In tis section we propose an approach to approximatefeedback linearization that uses a change of coordinates and a preliminaryfeedback to put a system (4.47) in a perturbed Brunovsky form,ż = Az + B u new + P (z) + Q(z) u new ), (4.49)where P (z) and Q(z) vanish at z = 0 and are ‘small’ on M. We get upperbounds on uniform norms of P and Q (depending on some measures ofnoninvolutivity of D) on any compact, contractible M.A different, indirect approach was presented in [Banaszuk and Hauser(1996)]. In this section, the authors present an approach for finding feedbacklinearizable systems that approximate a given SISO nonlinear systemon a given compact region of the state–space. First, they it is shown thatif the system is close to being involutive, then it is also close to being linearizable.Rather than working directly with the characteristic distributionof the system, the authors work with characteristic 1−forms, i.e., with the1−forms annihilating the characteristic distribution. It is shown that homotopyoperators can be used to decompose a given characteristic 1−forminto an exact and an antiexact part. The exact part is used to define achange of coordinates to a normal form that looks like a linearizable partplus nonlinear perturbation terms. The nonlinear terms in this normalform depend continuously on the antiexact part, and they vanish wheneverthe antiexact part does. Thus, the antiexact part of a given characteristic1−form is a measure of nonlinearizability of the system. If the nonlinearterms are small, by neglecting them we get a linearizable system approximatingthe original system. One can design control for the original systemby designing it for the approximating linearizable system and applying itto the original one. We apply this approach for design of locally stabilizingfeedback laws for nonlinear systems that are close to being linearizable.Let us start with approximating characteristic 1−forms by exact formsusing homotopy operators (compare with equation (3.44) above). Namely,on any contractible region M one can define a linear operator H that sat-