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93: Systems theory; control

Introduction

Loosely speaking, this is the mathematical study of complex dynamic structures in engineering. One can attempt a mathematical or statistical test for system identification, that is, to deduce the laws of evolution which govern the system. One can attempt system control, that is, to determine appropriate inputs (e.g. initial conditions for the differential equation) so that the system demonstrates desired outputs; this (and kinematics -- section 70) is used in the field of "robotics". One can study system stability, that is, the tendency to achieve a steady-state configuration. Since systems of interest in applications are subject to noise and imprecision, this area includes the study of stochastic systems as well; control is usually achieved using filters (e.g. the Kalman filter) to make best estimates of a system's condition.

Clearly systems analysis and control requires tools from differential equations and functional analysis, statistics, and (in the case of Optimal Control -- section 49) differential geometry.

Norbert Weiner coined the phrase "cybernetics" to mean, roughly, the mathematical study of the processes by which complex systems control their development using information-sharing between parts of the system. Arguably this is part of section 93 (or section 94, or 68, or 92, or...) The phrase is used much more frequently in the Russian literature.

Applications and related fields

For optimal control, See 49-XX

Subfields

• 93A: General
• 93B: Controllability, observability, and system structure
• 93C: Control systems, guided systems
• 93D: Stability
• 93E: Stochastic systems and control

This is among the larger areas in the Math Reviews database. Indeed, subfield 93B is one of the largest of the three-digit areas, but the other four subfields are also quite large.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Of the very many texts in this area, a fairly accessible but comprehensive text is Jacobs, O. L. R., "Introduction to control theory", Oxford University Press, Oxford, 1993. 390 pp. ISBN 0-19-856249-7, MR 95g:49001

Guilbaud, G. T.: "What is cybernetics?", Grove Press, Inc., New York 1960 126 pp. MR23#B1039

Klir, G. J.: "Systems science: a guided tour." J. Biol. Systems 1 (1993), no. 1, 27--58. CMP1231861

Top Ten Research Problems in Nonlinear Control[Richard Murray]

Newsgroups sci.engr.control, sci.systems.

For your amusement: Glasser, William, "Staying together : the control theory guide to a lasting marriage", HarperCollins Publishers, New York, 1995, 133 p. ISBN: 0-060-17247-9. (No, it's not really relevant!-- just a chance encounter through a library database. Similarly: "Mathematicians in Control", Bull Inst Math Appl)

Software and tables

Benchmarks for Systems and Control Theory.

Here is a wide variety of software for use in civil engineering