Algorithms for Transformation into the Extended Jordan Controllable and observable forms

In this paper, we first propose an algorithm that permits to derive a transformation allowing to transform a controllable pair into the Extended Jordan Controllable Canonical (EJCC) form. The proposed EJCC form is, in fact, an extension of the Brunovsky and the Jordan form. Next, by duality we derive an algorithm that permits to transform an observable pair into the Extended Jordan Observable Canonical (EJOC) form. A detailed justification of the proposed algorithms is provided and some examples are given to show the validity of the algorithms.