Homogeneity with incremental degrees and global stabilisation of a class of high-order upper-triangular systems

The problem of global asymptotic stabilisation is investigated for a class of upper-triangular systems with uncontrollable linearisation. A notion of generalised homogeneity is introduced in this article, providing a new insight and a deeper perspective on how to categorise nonlinear terms into the higher-order and equal-order terms in a non-traditional way. The delicate analysis and new categorisation of the vector field make it possible to encompass more general nonlinearities. A globally stabilising controller is recursively constructed by a bottom-up, step-by-step procedure, containing the equal-order terms as its indispensable components. It is illustrated that the results obtained in this article incorporate and extend existing results for the global stabilisation of upper-triangular systems.