Systems theory (engineering)

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In engineering and mathematics, systems theory or mathematical systems theory refers broadly to the mathematical study of (complex) systems in engineering and interconnections among such systems. Systems in this context refers to dynamical systems.

The fundamental principle in systems theory is mathematical modelling of systems. This means that the dynamics of the system or systems under study should be describable or representable by a set of mathematical equations. For example, it may be that the appropriate equations can be derived from first principles by using the laws of physics or empirically derived by system identification. A mathematical model is considered essential for a quantitative study of systems and may be used for purposes such as:

• Analysis of the behaviour of systems to make quantitative predictions about a system's long term behavior. This can be seen, for example, in the use of mathematical models in the study of population dynamics [1].
• Design of another system that can be interconnected with an existing system so that the resulting interconnected system has certain desired properties. This is typically the type of problem studied in control engineering.

Although systems theory was originally developed in the context of engineering and technological systems, its general principles and tools can in principle be adapted and applied to other types of systems, such as economic and biological systems, as long as mathematical models are available or can be developed. For example, the theory of electrical networks can also be applied to mechanical networks, by analogy, if the equations representing the latter network are analogous to the equations for the former. More recently, there has been significant interest and activity in extending the tools of systems theory for application to complex biological systems within the rapidly expanding field known as systems biology[2] (see, for example, The Hamilton Institute Systems Biology Group's homepage (retrieved on 2007-09-19) and the paper [3]), bringing biologists, control engineers and applied mathematicians together to study some important and difficult problems in the life sciences.

Systems theory in the above sense is also recognized as a branch of mathematics and is currently assigned the American Mathematical Society's mathematical subject classification (MSC) of 93-xx (Systems theory; control) [4][5].

References

1. F. Bauer and C. Castillo-Chavez, Mathematical models in Population Biology and Epidemiology, ser. Text in Applied Mathematics 40, New York: Springer-Verlag, 2001
2. Drack, Manfred., Apfalter, Wilfried., Pouvreau, David. (2007) On the Making of a System Theory of Life: Paul A. Weiss and Ludwig von Bertalanffy's Conceptual Connection. The Quarterly Review of Biology, Volume 82, No 4, The University of Chicago, Chicago

Introduction In terms of publications, “systems biology” is currently an exploding field (see, e.g., Chong and Ray 2002; Grant 2003). Note, however, that system thinking is not at all new in biology. In the sciences of life, it can be traced back at least to the German “teleo-mechanicist” tradition inaugurated by Immanuel Kant’s Critics of Judgement, of which Johannes Mu¨ ller and Karl von Baer were the main proponents (Kant 1789; Lenoir 1982). More generally, integrative (or “holistic”) thinking is not a “discovery” of the 20th century. It has a rather long history going back to ancient Greek thinkers: Aristotle already claimed that “the whole is of necessity prior to the part” (Aristotle 1966:1253a20)

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3. P. Wellstead, "The role of control and system theory in systems biology," edited text of a Plenary Lecture presented at the 10th International Federation of Automatic Control (IFAC) International Symposium on Computer Applications in Biotechnology and the 8th IFAC Symposium on Dynamics and the Control of Process Systems, The Hamilton Institute, 2007, Online: http://www.hamilton.ie/systemsbiology/files/2007/ControlAndSystemsBiology.pdf (retrieved on 2007-9-18).
4. 2000 Mathematical Subject Classification, American Mathematical Society, Online: http://www.ams.org/msc/ (retrieved on 2007-08-21).
5. D. Rusin, The Mathematical Atlas (93: Systems Theory;Control), Online: http://www.math.niu.edu/~rusin/known-math/index/93-XX.html (retrieved on 2007-09-19).