Parameterisation of stabilising controllers with precompensators

Within the framework of the factorisation approach, the author presents parameterisation methods for stabilising controllers. The parameterisations of this study are characterised by an ideal finitely generated by some strictly causal stable transfer functions. Several typical examples of the parameterisations are also presented, namely (i) all stabilising controllers such that their relative degrees are more than or equal to some fixed number in the continuous-time linear time-invariant (LTI) system model; (ii) all stabilising controllers such that they must have at least some fixed number of delay operators in the discrete-time LTI system model; (iii) all stabilising controllers such that they must have some delay operators in the multidimensional system model. The parameterisation method of this study is also applied to (iv) a system that is stabilisable but that does not admit a doubly coprime factorisation.