Polynomial homotopies on multicore workstations
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
| Hi-index | 0.00 |
The output feedback 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. In 1981, R. W. Brocket and C. I. Byrnes published the theoretical relations between the static output feedback pole placement problem and enumerative geometry problem. In 1996, M. S. Ravi, J. Rosenthal, and X. Wang generalized the theory to solve the dynamic output feedback pole placement problem for general Multi-Input-Multi-Output (MIMO) systems. The output feedback pole placement problem was still stated as an open problem at this time, since there was no algorithm to solve it. In 1998, B. Huber, F. Sottile, and B. Sturmfels proposed “numerical Schubert calculus”, which led to an efficient homotopy algorithm to solve the enumerative geometry problem. In this thesis, we present symbolic-numeric algorithms to turn the solutions of the homotopies to the output feedback laws of the pole placement problem. Despite the wider application range of the dynamic output 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. Since the numeric calculation of the Smith normal form is essential for minimal realization of the output feedback laws and the inverse of a polynomial matrix, an original algorithm through computing the extended GCD of two polynomials with the root matching method is implemented. To efficiently solve high dimensional problems with a very large number of solutions, a parallel Pieri homotopy algorithm is designed and implemented. We successfully applied our numerical approach for solving the output feedback pole placement problem to several applications from the literature. Both the realization algorithm and parallel Pieri homotopies are implemented in PHCpack (ACM TOMS 795), a publicly available software package for solving polynomial system with homotopy methods.