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The behavioral approach to
systems theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...
and
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
was initiated in the late-1970s by J. C. Willems as a result of resolving inconsistencies present in classical approaches based on state-space, transfer function, and convolution representations. This approach is also motivated by the aim of obtaining a general framework for system analysis and control that respects the underlying
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. The main object in the behavioral setting is the behavior – the set of all signals compatible with the system. An important feature of the behavioral approach is that it does not distinguish a priority between input and output variables. Apart from putting system theory and control on a rigorous basis, the behavioral approach unified the existing approaches and brought new results on controllability for nD systems, control via interconnection,J.C. Willems On interconnections, control, and feedback IEEE Transactions on Automatic Control Volume 42, pages 326-339, 1997 Available online http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/1997.4.pdf and system identification. I. Markovsky, J. C. Willems, B. De Moor, and S. Van Huffel. Exact and approximate modeling of linear systems: A behavioral approach. Monograph 13 in “Mathematical Modeling and Computation”, SIAM, 2006. Available online http://homepages.vub.ac.be/~imarkovs/siam-book.pdf


Dynamical system as a set of signals

In the behavioral setting, a dynamical system is a triple :\Sigma=(\mathbb,\mathbb,\mathcal) where * \mathbb\subseteq\mathbb is the "time set" – the time instances over which the system evolves, * \mathbb is the "signal space" – the set in which the variables whose time evolution is modeled take on their values, and * \mathcal\subseteq \mathbb^\mathbb the "behavior" – the set of signals that are compatible with the laws of the system :(\mathbb^\mathbb denotes the set of all signals, i.e., functions from \mathbb into \mathbb). w\in\mathcal means that w is a trajectory of the system, while w\notin\mathcal means that the laws of the system forbid the trajectory w to happen. Before the phenomenon is modeled, every signal in \mathbb^\mathbb is deemed possible, while after modeling, only the outcomes in \mathcal remain as possibilities. Special cases: * \mathbb=\mathbb – continuous-time systems * \mathbb=\mathbb – discrete-time systems * \mathbb = \mathbb^q – most physical systems * \mathbb a finite set – discrete event systems


Linear time-invariant differential systems

System properties are defined in terms of the behavior. The system \Sigma=(\mathbb,\mathbb,\mathcal) is said to be * "linear" if \mathbb is a vector space and \mathcal is a linear subspace of \mathbb^\mathbb, * "time-invariant" if the time set consists of the real or natural numbers and :\sigma^t\mathcal \subseteq\mathcal for all t\in\mathbb, where \sigma^t denotes the t-shift, defined by :\sigma^t(f)(t'):=f(t'+t). In these definitions linearity articulates the superposition law, while time-invariance articulates that the time-shift of a legal trajectory is in its turn a legal trajectory. A "linear time-invariant differential system" is a dynamical system \Sigma=(\mathbb,\mathbb^q,\mathcal) whose behavior \mathcal is the solution set of a system of constant coefficient linear ordinary differential equations R(d/dt) w=0, where R is a matrix of polynomials with real coefficients. The coefficients of R are the parameters of the model. In order to define the corresponding behavior, we need to specify when we consider a signal w:\mathbb\rightarrow\mathbb^q to be a solution of R(d/dt) w=0. For ease of exposition, often infinite differentiable solutions are considered. There are other possibilities, as taking distributional solutions, or solutions in \mathcal^(\mathbb,\mathbb^q), and with the ordinary differential equations interpreted in the sense of distributions. The behavior defined is :\mathcal = \. This particular way of representing the system is called "kernel representation" of the corresponding dynamical system. There are many other useful representations of the same behavior, including transfer function, state space, and convolution. For accessible sources regarding the behavioral approach, see J. Polderman and J. C. Willems. "Introduction to the Mathematical Theory of Systems and Control". Springer-Verlag, New York, 1998, xxii + 434 pp. Available online http://wwwhome.math.utwente.nl/~poldermanjw/onderwijs/DISC/mathmod/book.pdf. . J. C. Willems. The behavioral approach to open and interconnected systems: Modeling by tearing, zooming, and linking. "Control Systems Magazine", 27:46–99, 2007. Available online http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/2007.1.pdf.


Observability of latent variables

A key question of the behavioral approach is whether a quantity w1 can be deduced given an observed quantity w2 and a
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
. If w1 can be deduced given w2 and the model, w2 is said to be
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
. In terms of mathematical modeling, the to-be-deduced quantity or
variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
is often referred to as the
latent variable In statistics, latent variables (from Latin: present participle of ''lateo'', “lie hidden”) are variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or me ...
and the observed variable is the manifest variable. Such a system is then called an observable (latent variable) system.


References

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Additional sources

*Paolo Rapisarda and Jan C.Willems, 2006
''Recent Developments in Behavioral System Theory''
July 24–28, 2006, MTNS 2006, Kyoto, Japan *J.C. Willems
Terminals and ports.
IEEE Circuits and Systems Magazine Volume 10, issue 4, pages 8–16, December 2010 *J.C. Willems and H.L. Trentelman
On quadratic differential forms.
SIAM Journal on Control and Optimization Volume 36, pages 1702-1749, 1998 *J.C. Willems
Paradigms and puzzles in the theory of dynamical systems.
IEEE Transactions on Automatic Control Volume 36, pages 259-294, 1991 *J.C. Willems
Models for dynamics.
Dynamics Reported Volume 2, pages 171-269, 1989 Systems theory Dynamical systems