Closed-loop structure of decouplable linear multivariable systems

Considering a controllable, square, linear multivariable system, which is decouplable by static state feedback, we completely characterize in this paper the structure of the decoupled closed-loop system. The family of all attainable transfer function matrices for the decoupled closed-loop system is characterized, which also completely establishes all possible combinations of attainable finite pole and zero structures. The set of assignable poles as well as the set of fixed decoupling poles are determined, and decoupling is achieved avoiding unnecessary cancellations of invariant zeros. For a particular attainable decoupled closed-loop structure, it is shown how to find the corresponding state feedback, and it is proved that this feedback is unique if and only if the system is controllable.