Noncommutative Computer Algebra in the Control of Singularly Perturbed Dynamical Systems
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Most algebraic calculations which one sees in linear systems theory, for example in IEEE TAC, involve block matrices and so are highly noncommutative. Thus conventional commutative computer algebra packages, as in Mathematica and Maple, do not address them.
Here we investigate the usefulness of noncommutative computer algebra in a particular area of control theory-singularly perturbed dynamic systems-where working with the noncommutative polynomials involved is especially tedious. Our conclusion is that they have considerable potential for helping practitioners with such computations. For example, the methods introduced here take the most standard textbook singular perturbation calculation, [KK086], one step further than had been done previously.
Commutative Groebner basis algorithms are powerful and make up the engines in symbolic algebra packages’ Solve commands. Noncommutative Groebner basis algorithms are more recent, but we shall see that they are useful in manipulating the messy sets of noncommutative polynomial equations which arise in singular perturbation calculations. We use the noncommutative algebra package NCAlgebra and the noncommutative Groebner basis package NCGB which runs under it.
J. W. Helton, F. Dell Kronewitter, and Mark Stankus. "Noncommutative Computer Algebra in the Control of Singularly Perturbed Dynamical Systems" Proceedings of the 38th IEEE Conference on Decision & Control, Phoenix, Arizona.. Dec. 1999.
Available at: http://works.bepress.com/mstankus/5