Computer simplification of engineering systems formulas

The three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA (the 3 M's), but they are weak in the area of noncommutative operations. This article reports on applications of a powerful tool: a noncommutative version of the Grobner basis algorithm. The commutative version of this algorithm is implemented on each of the 3 M's. It has many applications ranging from solving systems of equations to computations involving polynomial ideals. Our application to the simplification of expressions which occur in systems theory is unique. We describe the Grobner basis for several elementary situations which arise in systems theory. These give a "complete" set of simplifying rules for formulas which arise in these situations. We have found that this process elucidates the nature of simplifying rules and provides a practical means of simplifying some types of complex expressions. The research required the use of software suited for computing with noncommuting symbolic expressions. This system uses a new approach to the development of mathematical software. It provides the flexibility needed for experimentation with algorithms, data representation, and data analysis. In another direction, NCAlgebra packages for Mathematica extend many of Mathematica's commands to symbolic expressions in noncommutative algebras. We have incorporated in these packages some of the results on simplification described in this paper.<
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