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 Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While dynamic feedback laws offer a wider range of use, 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.Comment: 20 pages, 3 figures; the software described in this paper is publicly available via http://www.math.uic.edu/~jan/download.htm

Galois groups of Schubert problems via homotopy computation

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_6006.Comment: 17 pages, 4 figures. 3 references adde

Numerical Schubert calculus

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gr\"obner basis for the Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.Comment: 24 pages, LaTeX 2e with 2 figures, used epsf.st

Modernizing PHCpack through phcpy

PHCpack is a large software package for solving systems of polynomial equations. The executable phc is menu driven and file oriented. This paper describes the development of phcpy, a Python interface to PHCpack. Instead of navigating through menus, users of phcpy solve systems in the Python shell or via scripts. Persistent objects replace intermediate files.Comment: Part of the Proceedings of the 6th European Conference on Python in Science (EuroSciPy 2013), Pierre de Buyl and Nelle Varoquaux editors, (2014

Numerical Schubert Calculus by the Pieri Homotopy Algorithm

[[abstract]]Based on Pieri's formula on Schubert varieties, the Pieri homotopy algorithm was first proposed by Huber, Sottile, and Sturmfels [J. Symbolic Comput., 26 (1998), pp. 767-788] for numerical Schubert calculus to enumerate all p-planes in Cm+p that meet n given planes in general position. The algorithm has been improved by Huber and Verschelde [SIAM J. Control Optim., 38 (2000), pp. 1265-1287] to be more intuitive and more suitable for computer implementations. A different approach of employing the Pieri homotopy algorithm for numerical Schubert calculus is presented in this paper. A major advantage of our method is that the polynomial equations in the process are all square systems admitting the same number of equations and unknowns. Moreover, the degree of each polynomial equation is always 2, which warrants much better numerical stability when the solutions are being solved. Numerical results for a big variety of examples illustrate that a considerable advance in speed as well as much smaller storage requirements have been achieved by the resulting algorithm.[[notice]]補正完畢[[incitationindex]]SCI[[booktype]]紙本[[booktype]]電子

Some real and unreal enumerative geometry for flag manifolds

We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative geometric problems on flag manifolds may have all their solutions be real and modify this method to show that another class may have no real solutions, which is a new phenomenon. This method originated in a numerical homotopy continuation algorithm adapted to the special Schubert calculus on Grassmannians and in principle gives optimal numerical homotopy algorithms for finding explicit solutions to these other enumerative problems.Comment: 19 pages, LaTeX-2e; Updated and final version. To appear in the issue of Michigan Mathematical Journal dedicated to Bill Fulto