Computing dynamic output feedback laws

The pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed-loop system has a desired set of eigenvalues. Converting this problem into a question of enumerative geometry, efficient numerical homotopy algorithms to solve this problem for general multiple-input-multiple-output systems have been proposed recently. Despite the wider application range of dynamic feedback laws, the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before. In this note, we present symbolic-numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine. We report on numerical experiments with our publicly available software PHCpack and illustrate its application on various control problems from the literature.