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 Multi-Input-Multi-Output (MIMO) 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 paper 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 and illustrate its application on various control problems from the literature.
2000 Mathematics Subject Classification. Primary 93B55. Secondary 14Q99, 65H10, 68W30, 93B27.
Key words and phrases. Control of linear systems, dynamic output feedback, Multi-Input-Multi-Output (MIMO) systems, numerical homotopy algorithms, numerical Schubert calculus, pole placement, symbolic-numeric computations.
IEEE Transactions on Automatic Control 49(8):1393-1397, 2004.