Causal polynomial pole assignment and stabilisation of SISO strictly causal linear discrete processes

摘要

The paper considers the pole assignment and the stabilisation problem under the practically essential constraint, that the designed controllers have to be causal, because only in that case they will be realisable in real time. It is shown that in case of a strictly proper process, when the set of solutions of the pole assignment or stabilisation problem is not empty, then it always contains a non-empty subset of non-causal controllers. Moreover, according to the set of causal controllers, we have one of the three situations: (1) the set is empty, (2) the set has actually one element and (3) there exists a subset of causal solutions. The paper provides complete solutions for both problems in the form of instructions for building the corresponding sets of causal controllers. Drawing on examples of a double integrator with delay, the procedures are illustrated.