Numerical Methods for Linear Control Systems By Biswa Datta
Remarkable progress has been made in both theory and applications of all important areas of control theory. Theory is rich and sophisticated. Some beautiful applications of control theory are presently being made in aerospace, biomedical engineering, industrial engineering, robotics, economics, power systems, etc. Unfortunately, the same assessment of progress does not hold in general for computations in control theory.
Many of the methods described in earlier control and systems theory text books were developed before the computer era and were based on approaches that are not numerically sound. Most of these methods, for example, require reduction of the system matrices to some condensed forms, such as a companion form or the Jordan canonical form, and it is well-known that these forms cannot, in general, be achieved in a numerically stable way.
The situation is, however, changing quite fast. In the last 20 years or so, numerically viable algorithms have been developed for many of the common linear control problems. Softwares based on these methods have been developed and are still being built.
Unfortunately, these methods and softwares do not seem to be widely known and easily accessible to broad groups of applied mathematicians, control theorists, and practicing control engineers. They are still largely confined in reprints and preprints (in this context it is noted that a reprint book on “Numerical Linear Algebra Techniques for Systems and Control” edited by R.V. Patel, A. Laub, and E Vandooren containing a large number of important published papers in this area has recently been published by IEEE/CRC Press). The primary reason for the inaccessibility of these algorithms and the softwares, in my opinion, is that an understanding, efficient implementations, and making possible modifications of these methods needed for some applications of special interests, require an interdisciplinary knowledge of linear algebra, numerical linear algebra, control theory, and computer science; and such a combined expertise is hard to find.
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