Singularly perturbed control systems using non-commutative computer algebra

Most algebraic calculations which one sees in linear systems theory, for ex ample in IEEE TAG, involve block matrices and so are highly non-commutative . Thus conventional commutative computer algebra packages, as in Mathematic a and Maple, do not address them. Here we investigate the usefulness of non -commutative computer algebra in a particular area of control theory-singul arly perturbed dynamic systems-where working with the non-commutative polyn omials involved is especially tedious. Our conclusion is that they have con siderable potential for helping practitioners with such computations. Commu tative Grobner basis algorithms are powerful and make up the engines in sym bolic algebra packages' Solve commands. Non-commutative Grobner basis algor ithms are more recent, but we shall see that they, together with an algorit hm for removing "redundant equations", are useful in manipulating the messy sets of non-commutative polynomial equations which arise in singular pertu rbation calculations. We use the non-commutative algebra package NCAlgebra and the non-commutative Grobner basis package NCGB which runs under it on t wo different problems. We illustrate the method on the classical state feed back optimal control problem, see [1], where we obtain one more (very long) term than was done previously. Then we use it to derive singular perturbat ion expansions for the relatively new (linear) information state equation. Copyright (C) 2000 John Wiley & Sons, Ltd.