## Chapter 41

Preface

To what extent can we make our A, B, C, D linear systems and operator calculations easier with
computer assistance? The standard symbol manipulators Mathematica, Maple, and Macsyma do not do
noncommutative algebra intelligently, so it is hard to find out. Our group wrote a package NCAlgebra
which runs under Mathematica and which is a reasonable beginning.

We see this package as a competitor to the yellow pad. Once you get used to it this might be
considerably more effective for hand calculations of modest size. Like Mathematica the emphasis is on
interaction with the program and flexibility.

I now prefer it to a yellow pad for many types of calculation and can recommend it to anyone who is
good with Mathematica or alternatively is familiar with computers to the extent of doing a substantial
amount of word processing. Putting a graduate student with computer talent on NCAlgebra also might
work well. Our experience at UCSD is that students find it easy to learn. Indeed anyone can play with
our package effortlessly, but one would have to be cautious about committing to a long term
research project using NCAlgebra without being able to make additional commands on his
own.

SYStems is based on NCAlgebra, a collection of ”functions” for Mathematica designed to facilitate
manipulation and reduction of noncommutative algebraic expressions. Specifically, it allows computer
calculation in an algebra with involution. Such computations are common in many areas but our
background is operator theory and engineering systems so we are aiming at research in these areas rather
than at the complete treatments of the basics in these subjects.

SYSTEMS is a package which runs under NCAlgebra. It is in a primitive form with its main contents
being a file for doing H^{∞} control for linear and certain nonlinear systems. This originated with a paper
(Aug91; by Ball, Helton and Walker [BHW]) which solves a collection of nonlinear H^{∞} control problems.
The formulas in that paper are executable inside NCAlgebra which greatly facilitates further research
along these lines.