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.