Journal Article10.1109/tac.2012.2210836
Open Stochastic Systems
Jan C. Willems
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TL;DR: Open stochastic systems are defined as probability triples and are characterized by their events, linearity, and interconnection.
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Abstract: The problem of providing an adequate definition of a stochastic system is addressed and motivated using examples. A stochastic system is defined as a probability triple. The specification of the set of events is an essential part of a stochastic model and it is argued that for phenomena with as outcome space a finite dimensional vector space, the framework of classical random vectors with the Borel sigma-algebra as events is inadequate even for elementary applications. Models very often require a coarse event sigma-algebra. A stochastic system is linear if the events are cylinders with fibers parallel to a linear subspace of a vector space. We address interconnection of stochastic systems. Two stochastic systems can be interconnected if they are complementary. We discuss aspects of the identification problem from this vantage point. A notion that emerges is constrained probability, a concept that is reminiscent but distinct from conditional probability. We end up with a comparison of open stochastic systems with probability kernels.
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References
The Behavioral Approach to Open and Interconnected Systems
TL;DR: This article has presented an approach to the mathematical description of dynamical systems, and described a methodology for modeling interconnected systems, called tearing, zooming, and linking, that is much better adapted to the physics of interconnected systems than input/output-modeling procedures such as Simulink.
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Terminals and ports
Willems
- 01 Jan 2010
TL;DR: In order to formalize the architecture of an electrical circuit it is convenient to use a digraph with leaves, which correspond to the external terminals of a circuit, the edges to the circuit elements, and the vertices to the connections.
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