Realizability of precompensators in linear multivariable systems: A structural approach

In this work, given a linear multivariable system, the problem of static state feedback realization of dynamic compensators is considered. Necessary and sufficient conditions for the existence of a static state feedback that realizes the dynamic compensator (square or full column rank compensator) are stated in structural terms, i. e., in terms of the zero-pole structure of the compensator, and the eigenvalues and the row image of the controllability matrix of the compensated system. Based on these conditions a formula is presented to find the state feedback matrices realizing a given compensator. It is also shown that the static state feedback realizing the compensator is unique if and only if the closed-loop system is controllable.