FINITE ZERO STRUCTURE OF LINEAR PERIODIC DISCRETE-TIME-SYSTEMS

Periodic ordered sets of structural indices and a new and simpler characterization of the notions of invariant zero, transmission zero, pole and eigenvalue are introduced for a linear periodic discrete-time system. Also the notions of input and output decoupling zeros (together with their ordered sets of structural indices) are extended to this type of system and are shown to have a meaning and relations with the structural properties of the system, wholly similar to the time-invariant ones. The ordered sets of structural indices of non-zero poles, eigenvalues and zeros of any type are independent of time, while those (and even the existence) of the null invariant zero, transmission zero, input and output decoupling zero and pole can depend on time, as well as those of the null eigenvalue (although its algebraic multiplicity is independent of time). The ordered sets of structural indices of invariant zeros are not altered by a linear periodic state feedback (as well as those of input decoupling zeros) and coincide with those of the geometric notion of zero.