[A whole new spiral arm of the galaxy]
ChatGPT says:
an C. Willems’ behavioral systems theory offers a paradigm shift in understanding dynamical systems by focusing on their ‘behavior’—the set of all possible trajectories—without predefining inputs and outputs. This contrasts with traditional approaches that emphasize input-output relationships.
In this framework, a dynamical system is defined as a triple (T, W, B), where T represents the time set, W denotes the signal space, and B is the behavior comprising all signals that satisfy the system’s laws. This perspective treats all variables equally, allowing for multiple interpretations of cause and effect.
A key advantage of the behavioral approach is its flexibility in modeling complex systems, especially when distinguishing between inputs and outputs is challenging or arbitrary. It has led to new insights in areas such as controllability, where the focus is on the ability to transition between trajectories, and system interconnection, viewed as the sharing of variables among subsystems.
For a comprehensive introduction to this approach, consider the book “Introduction to Mathematical Systems Theory: A Behavioral Approach” by J.W. Polderman and J.C. Willems.
what space of ‘systems theory’ does this belong to? Why, in over ten years of documenting everything I find on www.syscoi.com, have I never come across this before?
ChatGPT said:
Jan C. Willems’ behavioral systems theory is a distinctive approach within systems theory, emphasizing the set of all possible trajectories of a system—its “behavior”—without predefining inputs and outputs. This perspective contrasts with traditional methods that prioritize input-output relationships.
The behavioral approach has been particularly influential in control theory and dynamical systems analysis. It offers a framework that accommodates complex systems where traditional input-output distinctions may be ambiguous or limiting. This methodology has led to advancements in understanding system interconnections and controllability.
Regarding its relative obscurity, several factors might contribute:
- Niche Focus: The behavioral approach is primarily applied in specific areas of control theory and mathematical systems analysis. If your research or documentation has concentrated on other aspects of systems theory, it’s possible this particular approach didn’t surface prominently.
- Terminology Variations: Differences in terminology can lead to oversight. Willems’ work might be referenced under different terms or within contexts that don’t explicitly label it as “behavioral systems theory.”
- Publication Channels: Willems’ contributions are often found in specialized journals and conferences focused on control theory and applied mathematics. If your sources haven’t included these specific outlets, his work might not have been as visible.
To delve deeper into this approach, you might explore Willems’ publications and related literature in control theory and mathematical systems. This could provide a more comprehensive understanding of the behavioral perspective within systems theory.
The Behavioral Approach to Modeling and Control of Dynamical Systems
Jan C. Willems
Dept. of Electrical Engineering University of Leuven Belgium and Mathematics Institute University of Groningen The Netherlands
Abstract
The behavioral approach provides a mathematical framework for modeling, analysis, and synthesis of dynamical systems. The main difference from the classical view is that it does not the input/output partition as its starting point. In this setting, control is viewed as interconnection.
https://skoge.folk.ntnu.no/prost/proceedings/cpc6-jan2002/willems.pdf
BEHAVIORAL SYSTEMS THEORY: A SURVEY
EVA ZERZ Lehrstuhl D für Mathematik RWTHAachen University, 52062 Aachen, Germany e-mail: eva.zerz@math.rwth-aachen.de
We survey the so-called behavioral approach to systems and control theory, which was founded by J. C. Willems and his school. The central idea of behavioral systems theory is to put the focus on the set of trajectories of a dynamical system rather than on a specific set of equations modelling the underlying phenomenon. Moreover, all signal components are treated on an equal footing at first, and their partition into inputs and outputs is derived from the system law, in a way that admits several valid cause-effect interpretations, in general.
https://intapi.sciendo.com/pdf/10.2478/v10006-008-0024-9
The Behavioral Approach to Open and Interconnected Systems
JAN C. WILLEMS
MODELING BY TEARING, ZOOMING, AND LINKING
During the opening lecture of the 16th IFAC World Congress in Prague on July 4, 2005, Rudy Kalman articulated a principle that resonated very well with me. He put forward the following paradigm for research domains that combine models and mathematics: 1) Get the physics right. 2) The rest is mathematics. Did we, system theorists, get the physics right? Do our basic model structures adequately translate physical reality? Does the way in which we view interconnections respect the physics? These questions, in a nutshell, are the theme of this article. The motivation for the behavioral approach stems from the observation that classical systemtheoretic thinking is unsuitable for dealing on an appropriately general level with the basic tenets at which system theory aims, namely, open and interconnected systems. By an open system, we mean a system that interacts with its environment, for example, by exchanging matter, energy, or information. By an interconnected system, we mean a system that consists of interacting subsystems. Classical system theory introduces inputs, outputs, and signal-flow graphs ab initio. Inputs serve to capture the influence of the environment on the system, outputs serve to capture the influence of the system on the environment, while output-to-input assignments, such as series and feedback connection, serve to capture interconnections. A system is thus viewed as transmitting and transforming signals from the input channel to the output channel, and interconnections are viewed as pathways through which outputs of one system are imposed as inputs to another system.
The Behavioral Approach to Systems Theory | The n-Category Café
June 15, 2018The Behavioral Approach to Systems TheoryPosted by John Baezguest post by Eliana Lorch and Joshua TanAs part of the Applied Category Theory seminar, we discussed an article commonly cited as an inspiration by many papers1 taking a categorical approach to systems theory, The Behavioral Approach to Open and Interconnected Systems. In this sprawling monograph for the IEEE Control Systems Magazine, legendary control theorist Jan Willems poses and answers foundational questions like how to define the very concept of mathematical model, gives fully-worked examples of his approach to modeling from physical first principles, provides various arguments in favor of his framework versus others, and finally proves several theorems about the special case of linear time-invariant differential systems.
The Behavioral Approach to Systems Theory | The n-Category Café
Systems Community of Inquiry 