An exact characterization of linear systems with zeros everywhere in the complex plane

In this note, an exact characterization of autonomous linear multivariable systems with zeros everywhere in the complex plane is given in terms of the system matrices. Such systems are called degenerate. An important consequence of this characterization is that nondegeneracy is equivalent to the concept of functional reproducibility, introduced by Brockett and Mesarović. Based on this equivalence, a new test for nondegeneracy in terms of the functional reproducibility matrix is derived. An example is included to show that controllability, observability, and output-controllability are not sufficient to guarantee the nondegeneracy of a linear system.