Multivariable feedback: a personal reminiscence

Abstract

My involvement in the development of linear multivariable feedback theory and its applications is described from a highly personal point of view, concentrating on the origin of key ideas and their development in association with many colleagues. Detailed references are given to all my papers and books, and explanatory notes provided where further material may be found.

Keywords:

• feedback theory
• control systems design

Acknowledgements

I am most grateful to Peter Wellstead, Jan Maciejowski and Nick Karcanias for very helpful comments and suggestions made during the preparation of this account. It gives me great pleasure that this festschrift is appearing in the International Journal of Control which I had the privilege of editing for some 17 years.

Notes

Notes

1. When the wartime MIT Radiation Laboratory closed, a far-sighted decision was made to document the huge amount of work done on radar. Between 1947 and 1953 a 28-volume series was produced, edited by Louis N. Ridenour and published by McGraw-Hill.

2. Bode's book was published by Van Nostrand in 1945.

3. The names come from the Rainbow Code which was used to identify parts of the then secret integrated air defence system, now declassified. Like everything else, nowadays there is a lot of information available on the Internet.

4. It was presented to a meeting of the Mathematics section of the British Association in Dundee in August 1978, in a paper entitled Dynamics and the Designer (MacFarlane 1968c). A full treatment is given in Chapter 4 of Dynamical System Models(MacFarlane 1970).

5. Functionals of Dynamical System Performance and Their Use in Electrical Network Theory, A.G.J. MacFarlane, Ph.D. Thesis, University of London, 1962.

6. G. Birkhoff and S. MacLane: A Survey of Modern Algebra, Macmillan, New York, 1941.

7. Best illustrated by what I came to regard as nearly-orthogonal questions and answers. Such as when Howard decided not to attend the 1972 IFAC World Congress. Realising that almost all of the people I would meet there would ask me why he was not present, I decided to find out. So, going over to CSC, I asked ‘why are you not going to Paris?’, and he replied ‘I've been there’.

8. G. Springer: Introduction to Riemann Surfaces, Addison-Wesley, Reading, MA, 1957.

9. F. Klein: On Riemann's Theory of Algebraic Functions and their Integrals,Dover reprint, 1963, of 1881 original.

10. I chaired SERC's major computing committees for many years. Highlights included a visit to Cray's headquarters in Minneapolis, and to Mitsubishi and Fujitsu in Japan. I was invited to lecture at Mitsubishi on multivariable control and join a group of their senior engineers on a weekend retreat near Mount Fuji. The mountain remained shrouded in mist. I hoped that multivariable control did not remain shrouded in mystery.

11. A Generalised Nyquist/Root-Locus Theory for Multi-loop Feedback Systems, M.C. Smith, Ph.D. thesis, University of Cambridge, 1982. This is the definitive treatment of the algebraic function theory used in linear multivariable feedback.

12. A.G.J. MacFarlane: Notes on the Vector Frequency Response Approach to Multivariable Feedback Systems, UMIST Report, February 1973.

13. N.P. Karcanias: Phase Compensation of General Multivariable Feedback Systems, M.Sc. Dissertation, UMIST, October 1973.

14. J.M. Maciejowski, Multivariable Feedback Design, Addison-Wesley, Reading, MA, 1989.

15. S. Skogestad and Ian Postlethwaite, Multivariable Feedback Control-Analysis and Design, Wiley, New York, 1996.

16. Jacques Hadamard, the famous French mathematician, said that the shortest path between truths in the real domain passes through the complex domain. Sadly he left no advice on how to find it. The textbook which best conveys the beauty of complex analysis is Visual Complex Analysis by Tristan Needham, Clarendon Press, Oxford, 1997. He makes an impassioned plea for visualisation (as opposed to formal deduction). Kron would have loved the pictures. Unfortunately he does not deal with Riemann surfaces, although giving all the necessary background. A method for constructing Riemann surfaces is described in Appendix 4 of (Postlethwaite and MacFarlane 1979), and in (MacFarlane 1980a, p. 123). A Riemann surface is simply a compact 2-dimensional manifold, like a sphere or a doughnut, (or sometimes more complicated, like a sphere with handles). Once you know how to construct them the abstract descriptions given in advanced texts become comprehensible. Chapter 8 of the splendid The Road to Realityby Roger Penrose (Vintage 2005) deals briefly with Riemann surfaces. Finding out good ways of visualising representations involving several complex variables would be a formidable problem, but I remain convinced that complex variable theory still has a lot to offer in the development of feedback theory.

17. In searching the literature on functions of several complex variables (see references in (Hung and MacFarlane 1982a)), Sam Hung and I discovered the useful concept of pseudopolynomials (Hung 1982, Appendix A2.1). These are polynomials in one complex variable whose coefficients are polynomials in another complex variable. I had used the thermodynamic concept of adiabatic (i.e. slow) variation when modelling aircraft gas turbines, and the idea of an adiabatic variation of parameters in a complex function dynamical model remains an attractive one.