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An Algorithm for Local Transverse Feedback Linearization

Given a multi-input, nonlinear, time-invariant, control-affine system and a controlled invariant, closed, embedded submanifold $\mathsf{N}$, the local transverse feedback linearization (TFL) problem seeks a coordinate and feedback transformation such that, in transformed coordinates, the dynamics governing the system's transverse evolution with respect to $\mathsf{N}$ are linear, time-invariant and controllable. The transformed system is said to be in the TFL normal form. Checkable necessary and sufficient conditions for this problem to be solvable are known, but, unfortunately, the literature does not present a prescription that constructs the required transformation for multi-input systems. In this article we present an algorithm that produces a virtual output of suitable vector relative degree that, using input-output feedback linearization, puts the system into TFL normal form. The procedure is based on dual conditions for TFL and is fundamentally different from existing methods, such as the GS and Blended algorithms, because of the "desired" zero dynamics manifold $\mathsf{N}$. The proposed algorithm is the first to take into consideration the desired zero dynamics manifold.