Abstract

Most algebraic calculations which one sees in linear systems theory, for example in IEEE TAC, involve block matrices and so are highly non-commutative. Thus conventional commutative computer algebra packages, as in Mathematica and Maple, do not address them. Here we investigate the usefulness of non-commutative computer algebra in a particular area of control theory − singularly perturbed dynamic systems − where working with the non-commutative polynomials involved is especially tedious. Our conclusion is that they have considerable potential for helping practitioners with such computations. Commutative Gröbner basis algorithms are powerful and make up the engines in symbolic algebra packages' Solve commands. Non-commutative Gröbner basis algorithms are more recent, but we shall see that they, together with an algorithm for removing redundant equations, are useful in manipulating the messy sets of non-commutative polynomial equations which arise in singular perturbation calculations. We use the non-commutative algebra package NCAlgebra and the non-commutative Gröbner basis package NCGB which runs under it on two different problems. We illustrate the method on the classical state feedback optimal control problem, see [1], where we obtain one more (very long) term than was done previously. Then we use it to derive singular perturbation expansions for the relatively new (linear) information state equation.

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Mathematics

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The definitive version is available at http://www3.interscience.wiley.com/journal/72517344/abstract.

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URL: https://digitalcommons.calpoly.edu/math_fac/13