Polynomial matrix approach to tracking control of non-linear systems

To characterise complete controllability of a time-varying linearised error system of a given non-linear system, this study focuses the attention on hyper-regularity of a polynomial matrix derived from the non-linear system and introduces the concept called controllable trajectory. It is shown that if a polynomial matrix derived from a given non-linear system is hyper-regular, then every linearised system along any controllable trajectory is completely controllable. Through a non-holonomic mobile robot example, it is demonstrated that if a polynomial matrix derived from a given non-linear system is hyper-regular, and if a reference trajectory is the state part of a periodic controllable trajectory, a trajectory tracking control is achieved by a two-degree-of-freedom controller design method.