Input-output linearisation of nonlinear systems with ill-defined relative degree: the ball and beam revisited

We establish that for SISO systems lack of well-defined relative degree is not an obstacle to exact inversion. Sufficient conditions for the existence of exact linearising inputs are derived. Exact tracking solutions of a number of example systems, including the ball and beam system studied by Hauser et al. (1992), are analysed. It is shown that linearising inputs may be: 1) highly non-unique, 2) discontinuous and include impulses, and 3) fragile. A framework is established for approximate linearisation. whereby exact linearising inputs are formulated as the limits of sequences of realisable, arbitrarily accurate linearising inputs. This framework is constructive in nature and of key importance when the exact linearising inputs are, for example, unrealisable