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In terms of the original system matrices, a new criterion is given for the I-controllablisability of descriptor linear systems. The range of the dynamical orders that the resultant closed-loop system can achieve is also characterised. A general parameterisation of all the I-controllablising controllers that make the resultant closed-loop system have the desired dynamical order is presented. The design is based on orthogonal matrix transformations that can be implemented in a numerically stable way.