Polpac: MATLAB Toolbox for Computing with Polynomial Matrices

Introduction

Polynomial matrices play an important role in control theory. They are used in the design of multi-variable linear control systems in the frequency domain, but also in the design of robust controllers of multi-variable non-linear control systems.

As versions of MATLAB older than Version 5 were not equipped to dealing directly with such data structures, I decided in the late eighties, to develop a version of a polynomial matrix toolbox for MATLAB based on regular matrices. To this end, polynomial matricas had to be mapped onto regular matrices.

I decided to offer my class on Numerical Methods for Control during one year as a project-oriented class. In that year, the class was attended by 15 students. I subdivided the students into five groups. The Polpac Toolbox was developed jointly by my students and myself as part of that class.

A major goal of Polpac was to be able to observe and understand the error propagation during the execution of algorithms. To this end, all of the Polpac routines were coded in three variants that differed in the way, in which the polynomials were represented. In mode 1, polynomials are represented through their coefficients. In mode 2, they are being captured by storing their zeros, whereas in mode 3, each polynomial is represented by a set of supporting values.

MATLAB offers today data structures that would drastically simplify such a development. Algorithms, coded by making use of these data structures, would also be much easier to read and maintain. However, I never took the time to rewrite the Polpac Toolbox from scratch making use of these new features.

For this reason, the Polpac Toolbox is still based on Version 4 of MATLAB. The functions can be called without any problems also from newer versions of MATLAB. However, the proper maintenance of the library is problematic.

Historical Development

The error propagation of numerical algorithms based on polynomial matrices had already been analyzed in the early eighties by Magnus Rimvall and myself in a number of semester and diploma projects at ETH Zurich. In those days, our interest fell on representation by means of supporting values. This representation enabled us to introduce redundancy that could be exploited in minimizing the error amplification during continued polynomial matrix operations. One student showed that it is numerically best to choose the supporting values equidistantly spaced along the unit circle of the complex plane.
In 1988, I developed a first version of a Polpac toolbox together with my students of the class on Numerical Methods for Control. The software was developed in CTRL-C, a predecessor of today's widely distributed MATLAB software.
In 1991, Sung-Do Chi developed a new representation by supporting values in the frequency domain that exhibited promising numerical properties [1].
In 1997, the Polpac toolbox was ported from CTRL-C to Version 4 of MATLAB. This is the version of the software that is being made available through this web page. ()

Most Important Publications

Cellier, F.E., and S.D. Chi (1991), Numerical Properties of Trajectory Representations of Polynomial Matrices, Proc. CADCS'91, Computer-Aided Design in Control Systems, Swansea, Wales, U.K., pp.173-177.

Deutsche Version
Homepage