We study the feedback group action on multi-input nonlinear control systems with uncontrollable mode. We follow slightly an approach proposed in Kang and Krener [W. Kang and A. J. Krener, SIAM J. Control. Optim., 30 (1992), pp. 1319–1337] which consists of analyzing the system and the feedback group step by step. We construct a normal form which generalizes, on one hand, the results obtained in the single-input case and, on the other hand, those recently obtained by the same author in the controllable case. We illustrate our results by studying the Caltech Multi-Vehicle Wireless Testbed (MVWT) and the prototype of Planar Vertical TakeOff and Landing aircraft (PVTOL). We also study the notion of bifurcation of controllability for systems with one nonzero uncontrollable mode. We first show that the equilibria for those systems is a p-dimensional submanifold (p equals number of inputs). Provided that one term in their normal form is nonzero, we show that these systems are linearly controllable, hence stabilizable, at any nearby equilibrium point of the origin.
Tall, Issa A. "Feedback Classification of Multi-Input Nonlinear Control Systems." (Jan 2005).