The Pieri homotopy algorithm for solving the pole placement problem in the control of linear systems

Abstract:

To control the behavior of a linear system of first-order differential equations, the pole placement problem asks to find a feedback law so that the derived closed-loop system has a prescribed set of eigenvalues. In the geometric formulation of this problem, we are asked to find m-planes in (m+p)-space that meet mp given p-planes nontrivially. In enumerative geometry, the Schubert calculus provides us with ways to count the number of solutions. In order to find the solutions, we must solve a polynomial system. As existing solvers fail to exploit the structure of the polynomial system, Huber, Sottile and Sturmfels proposed the so-called Pieri homotopies which lead to an optimal method, i.e.: every solution path in the deformation generically leads to one feedback law. In a joint work with Birk Huber (SIAM J. Control Optim. 38(4): 1265-1287, 2000), the Pieri homotopies were extended to solve the dynamic pole placement problem.

Workshop on Algorithms, Computational Complexity, and Models of Computation for Nonlinear and Multivariate Problems, July 16-20, Mt Holyoke College.