43:
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
1435:
35:
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
1440:
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.
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1532:
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
325:
802:
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668:
581:
292:
270:
169:
833:
1211:
1075:
1213:. For ease of exposition, often infinite differentiable solutions are considered. There are other possibilities, as taking distributional solutions, or solutions in
1519:
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
1500:. Exact and approximate modeling of linear systems: A behavioral approach. Monograph 13 in “Mathematical Modeling and Computation”, SIAM, 2006. Available online
1119:
1095:
853:
395:
345:
1430:{\displaystyle {\mathcal {B}}=\{w\in {\mathcal {C}}^{\infty }(\mathbb {R} ,\mathbb {R} ^{q})~|~R(d/dt)w(t)=0{\text{ for all }}t\in \mathbb {R} \}.}
1216:
1484:
J.C. Willems On interconnections, control, and feedback IEEE Transactions on
Automatic Control Volume 42, pages 326-339, 1997 Available online
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62:
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are the parameters of the model. In order to define the corresponding behavior, we need to specify when we consider a signal
1468:
and the observed variable is the manifest variable. Such a system is then called an observable (latent variable) system.
120:
1452:
A key question of the behavioral approach is whether a quantity w1 can be deduced given an observed quantity w2 and a
525:
1284:, and with the ordinary differential equations interpreted in the sense of distributions. The behavior defined is
942:, while time-invariance articulates that the time-shift of a legal trajectory is in its turn a legal trajectory.
861:
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is the "signal space" – the set in which the variables whose time evolution is modeled take on their values, and
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is the solution set of a system of constant coefficient linear ordinary differential equations
44:
basis, the behavioral approach unified the existing approaches and brought new results on
8:
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32:
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the "behavior" – the set of signals that are compatible with the laws of the system
1508:
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IEEE Circuits and
Systems Magazine Volume 10, issue 4, pages 8–16, December 2010
28:
24:
17:
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728:"time-invariant" if the time set consists of the real or natural numbers and
1534:
http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/2007.1.pdf
1521:
http://wwwhome.math.utwente.nl/~poldermanjw/onderwijs/DISC/mathmod/book.pdf
1486:
http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/1997.4.pdf
1277:{\displaystyle {\mathcal {L}}^{\rm {local}}(\mathbb {R} ,\mathbb {R} ^{q})}
1569:
SIAM Journal on
Control and Optimization Volume 36, pages 1702-1749, 1998
997:{\displaystyle \Sigma =(\mathbb {R} ,\mathbb {R} ^{q},{\mathcal {B}})}
1576:
IEEE Transactions on
Automatic Control Volume 36, pages 259-294, 1991
147:
is the "time set" – the time instances over which the system evolves,
945:
A "linear time-invariant differential system" is a dynamical system
1460:. In terms of mathematical modeling, the to-be-deduced quantity or
592:
System properties are defined in terms of the behavior. The system
207:{\displaystyle {\mathcal {B}}\subseteq \mathbb {W} ^{\mathbb {T} }}
637:{\displaystyle \Sigma =(\mathbb {T} ,\mathbb {W} ,{\mathcal {B}})}
104:{\displaystyle \Sigma =(\mathbb {T} ,\mathbb {W} ,{\mathcal {B}})}
1444:
For accessible sources regarding the behavioral approach, see .
769:{\displaystyle \sigma ^{t}{\mathcal {B}}\subseteq {\mathcal {B}}}
37:
1456:. If w1 can be deduced given w2 and the model, w2 is said to be
428:
is deemed possible, while after modeling, only the outcomes in
587:
397:
to happen. Before the phenomenon is modeled, every signal in
56:
In the behavioral setting, a dynamical system is a triple
1157:{\displaystyle w:\mathbb {R} \rightarrow \mathbb {R} ^{q}}
48:, control via interconnection, and system identification.
1574:
Paradigms and puzzles in the theory of dynamical systems.
377:
means that the laws of the system forbid the trajectory
51:
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denotes the set of all signals, i.e., functions from
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1502:http://homepages.vub.ac.be/~imarkovs/siam-book.pdf
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140:{\displaystyle \mathbb {T} \subseteq \mathbb {R} }
139:
103:
1588:
1583:Dynamics Reported Volume 2, pages 171-269, 1989
1552:Recent Developments in Behavioral System Theory
938:In these definitions linearity articulates the
1496:I. Markovsky, J. C. Willems, B. De Moor, and
552:{\displaystyle \mathbb {W} =\mathbb {R} ^{q}}
1421:
1304:
1101:with real coefficients. The coefficients of
1555:, July 24–28, 2006, MTNS 2006, Kyoto, Japan
927:{\displaystyle \sigma ^{t}(f)(t'):=f(t'+t)}
718:{\displaystyle \mathbb {W} ^{\mathbb {T} }}
421:{\displaystyle \mathbb {W} ^{\mathbb {T} }}
243:{\displaystyle \mathbb {W} ^{\mathbb {T} }}
588:Linear time-invariant differential systems
1549:Paolo Rapisarda and Jan C.Willems, 2006.
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513:{\displaystyle \mathbb {T} =\mathbb {Z} }
506:
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481:{\displaystyle \mathbb {T} =\mathbb {R} }
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125:
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370:{\displaystyle w\notin {\mathcal {B}}}
1543:
583:a finite set – discrete event systems
347:is a trajectory of the system, while
1490:
52:Dynamical system as a set of signals
1513:
320:{\displaystyle w\in {\mathcal {B}}}
31:was initiated in the late-1970s by
13:
1565:J.C. Willems and H.L. Trentelman.
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66:
14:
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1448:Observability of latent variables
797:{\displaystyle t\in \mathbb {T} }
1567:On quadratic differential forms.
1396:
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1021:{\displaystyle {\mathcal {B}}}
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687:{\displaystyle {\mathcal {B}}}
631:
605:
445:{\displaystyle {\mathcal {B}}}
98:
72:
46:controllability for nD systems
1:
1471:
1464:is often referred to as the
663:{\displaystyle \mathbb {W} }
576:{\displaystyle \mathbb {W} }
287:{\displaystyle \mathbb {W} }
265:{\displaystyle \mathbb {T} }
164:{\displaystyle \mathbb {W} }
7:
828:{\displaystyle \sigma ^{t}}
23:The behavioral approach to
10:
1618:
1206:{\displaystyle R(d/dt)w=0}
1070:{\displaystyle R(d/dt)w=0}
15:
488:– continuous-time systems
452:remain as possibilities.
694:is a linear subspace of
16:Not to be confused with
559:– most physical systems
520:– discrete-time systems
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670:is a vector space and
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1099:matrix of polynomials
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1581:Models for dynamics.
1560:Terminals and ports.
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1407: for all
1544:Additional sources
1507:2022-07-06 at the
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1602:Dynamical systems
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1114:{\displaystyle R}
1090:{\displaystyle R}
940:superposition law
848:{\displaystyle t}
390:{\displaystyle w}
340:{\displaystyle w}
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1509:Wayback Machine
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1466:latent variable
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455:Special cases:
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1597:Systems theory
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1579:J.C. Willems.
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1572:J.C. Willems.
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1558:J.C. Willems.
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29:control theory
25:systems theory
18:Behavior model
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33:J. C. Willems
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835:denotes the
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648:"linear" if
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327:means that
1591:Categories
1472:References
1458:observable
1414:∈
1323:∞
1311:∈
1140:→
953:Σ
867:σ
817:σ
787:∈
757:⊆
741:σ
600:Σ
358:∉
308:∈
188:⊆
130:⊆
67:Σ
1505:Archived
1462:variable
1077:, where
912:′
892:′
776:for all
38:physics
1362:
1354:
808:where
114:where
1454:model
1097:is a
272:into
27:and
1593::
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535:=
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70:=
20:.
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