1,024 research outputs found
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
The computational complexity of the Chow form
We present a bounded probability algorithm for the computation of the Chow
forms of the equidimensional components of an algebraic variety. Its complexity
is polynomial in the length and in the geometric degree of the input equation
system defining the variety. In particular, it provides an alternative
algorithm for the equidimensional decomposition of a variety.
As an application we obtain an algorithm for the computation of a subclass of
sparse resultants, whose complexity is polynomial in the dimension and the
volume of the input set of exponents. As a further application, we derive an
algorithm for the computation of the (unique) solution of a generic
over-determined equation system.Comment: 60 pages, Latex2
Tracking p-adic precision
We present a new method to propagate -adic precision in computations,
which also applies to other ultrametric fields. We illustrate it with many
examples and give a toy application to the stable computation of the SOMOS 4
sequence
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