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Concept
Behavioral modeling
Summary
The behavioral approach to systems theory and control theory 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.
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, and system identification.
In the behavioral setting, a dynamical system is a triple
where
is the "time set" – the time instances over which the system evolves,
is the "signal space" – the set in which the variables whose time evolution is modeled take on their values, and
the "behavior" – the set of signals that are compatible with the laws of the system
( denotes the set of all signals, i.e., functions from into ).
means that is a trajectory of the system, while means that the laws of the system forbid the trajectory to happen. Before the phenomenon is modeled, every signal in is deemed possible, while after modeling, only the outcomes in remain as possibilities.
Special cases:
– continuous-time systems
– discrete-time systems
– most physical systems
a finite set – discrete event systems
System properties are defined in terms of the behavior. The system is said to be
"linear" if is a vector space and is a linear subspace of ,
"time-invariant" if the time set consists of the real or natural numbers and
for all ,
where denotes the -shift, defined by
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.
Official source
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