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How behavior defines a system: Are your signals compliant?
In the late 1970s, J. C. Willems proposed the systems theory and control theory of the behavioral approach, which was intended to resolve the inconsistencies existing in the classical approach. These classical methods rely on concepts such as state spaces, transfer functions, and convolutional representations. Willems' behavioral approach aims to establish a general framework for system analysis and control that respects fundamental physics. The main object of this approach is behavior, that is, the set of all signals that the system is compatible with.
In behavioral settings, the main component of the system is behavior—the collection of signals that are legally compatible with the system.
An important characteristic of the behavioral method is that it does not set a priority order for input and output variables. This also means that when we analyze the signals of the system, no matter which signals they are, they are an important part of the system behavior. This approach not only provides a rigorous foundation for system theory and control, but also unifies existing approaches and introduces a new framework for controllability of nD systems, new control consequences, and system identification.
Dynamic systems as sets of signals
In the behavioral approach, the dynamic system can be expressed as a triplet Σ = (T, W, B), where:
Trepresents a time set, which is a collection of time instances of system evolution.Wis the signal space, which is the range of system variable values.Bis behavior, that is, the set of signals that is consistent with the laws of the system.
The key to this model is that, of course, signals may be considered legal before they are established, but after modeling, only those signals that are in the set B can be regarded as legal system behavior. .
If
w ∈ B, thenwis a trajectory of the system; ifw ∉ B, the system law prohibits the existence of this trajectory .
Linear time-invariant differential system
The characteristics of a system lie in its behavior. The system Σ = (T, W, B) is said to be "linear" if W is a vector space and B is W A linear subspace of ^T; said to be "time-invariant" if the time set consists of real or natural numbers, and σ^tB ⊆ for all , where t ∈ T Bσ^t represents the displacement of t.
A "linear time-invariant differential system" is a dynamic system Σ = (R, R^q, B) whose behavior B is a set of constant coefficients Set of solutions to linear ordinary differential equations.
The behavior is defined as
B={w ∈ C∞(R, R^q) | R( d/dt )w(t) = 0, ∀ t ∈ R}.
This type of representation is called a "kernel representation" of the corresponding dynamic system. There are many other useful expressions, such as transfer functions, state spaces, convolutions, etc.
Observability of latent variables
A key question in behavioral approaches is whether it is possible to derive another quantity w1 based on an observed variable w2 and a model. If w1 can be derived, then w2 is called an observable. In mathematical modeling, latent variables are often called variables to be derived, while observed variables are called manifest variables.
Such systems are called observable (latent variable) systems.
The proposal of behavioral methods provides a new entry point for in-depth analysis of system theory. This is not only a theoretical exploration, but how to build these systems in practical applications is also an important topic for future research. As technology develops, can we build more powerful models so that these behaviors are not just theoretical but can be applied in a variety of fields?
