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The behavioral approach to systems and control theory was initiated in the late 70's by J. C. Willems as a result of resolving inconsistencies present in classical approaches based on state-space, transfer function, and convolution representations. The main object in the behavioral setting is the behavior --- the set of all signals compatible with the system. An important feature which distinguishes the behavioral approach from the classical approaches is that it does not distinguish a priori 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 .

Dynamical system as a set of signals

In the behavioral setting, a dynamical system is a triple

$\Sigma =(\mathbb {T} ,\mathbb {W} ,{\mathcal {B}})$

where

• $\mathbb {T} \subseteq \mathbb {R}$ is the "time set" --- the time instances over which the system evolves,
• $\mathbb {W}$ is the "signal space" --- the set in which the variables whose time evolution is modeled take on their values, and
• ${\mathcal {B}}\subseteq \mathbb {W} ^{\mathbb {T} }$ the "behavior" --- the set of signals that are compatible with the laws of the system

($\mathbb {W} ^{\mathbb {T} }$ denotes the set of all signals, i.e., functions from $\mathbb {T}$ into $\mathbb {W}$).

$w\in {\mathcal {B}}$ means that $w$ is a trajectory of the system, while $w\notin {\mathcal {B}}$ means that the laws of the system forbid the trajectory $w$ to happen. Before the phenomenon is modeled, every signal in $\mathbb {W} ^{\mathbb {T} }$ is deemed possible, while after modeling, only the outcomes in ${\mathcal {B}}$ remain as possibilities.

Special cases:

• $\mathbb {T} =\mathbb {R}$ --- continuous-time systems
• $\mathbb {T} =\mathbb {Z}$ --- discrete-time systems
• $\mathbb {W} =\mathbb {R} ^{q}$ --- most physical systems
• $\mathbb {W}$ a finite set --- discrete event systems

Linear time-invariant differential systems

System properties are defined in terms of the behavior. The system $\Sigma =(\mathbb {T} ,\mathbb {W} ,{\mathcal {B}})$ is said to be

• "linear" if $\mathbb {W}$ is a vector space and ${\mathcal {B}}$ is a linear subspace of $\mathbb {W} ^{\mathbb {T} }$,
• "time-invariant" if the time set consists of the real or natural numbers and

$\sigma ^{t}{\mathcal {B}}\subseteq {\mathcal {B}}$ for all $t\in \mathbb {T}$,

where $\sigma ^{t}$ denotes the $t$-shift, defined by

$\sigma ^{t}(f)(t'):=f(t'+t)$.

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.

A "linear time-invariant differential system" is a dynamical system $\Sigma =(\mathbb {R} ,\mathbb {R} ^{q},{\mathcal {B}})$ whose behavior ${\mathcal {B}}\,$ is the solution set of a system of constant coefficient linear ordinary differential equations $R(d/dt)w=0,$, where $R$ is a matrix of polynomials with real coefficients. The coefficients of $R$ are the parameters of the model. In order to define the corresponding behavior, we need to specify when we consider a signal $w:\mathbb {R} \rightarrow \mathbb {R} ^{q}$ to be a solution of $R(d/dt)w=0$. For ease of exposition, often infinite differentiable solutions are considered. There are other possibilities, as taking distributional solutions, or solutions in ${\mathcal {L}}^{\rm {local}}(\mathbb {R} ,\mathbb {R} ^{q})$, and with the ordinary differential equations interpreted in the sense of distributions. The behavior defined is

${\mathcal {B}}=\{w\in {\mathcal {C}}^{\infty }(\mathbb {R} ,\mathbb {R} ^{q})~|~R(d/dt)w(t)=0{\text{ for all }}t\in \mathbb {R} \}.$

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.

For accessible sources regarding the behavioral approach, see .

Additional sources

J.C. Willems. Terminals and ports. IEEE Circuits and Systems Magazine Volume 10, issue 4, pages 8-16, December 2010

J.C. Willems and H.L. Trentelman. On quadratic differential forms. SIAM Journal on Control and Optimization Volume 36, pages 1702-1749, 1998

J.C. Willems. Paradigms and puzzles in the theory of dynamical systems. IEEE Transactions on Automatic Control Volume 36, pages 259-294, 1991

J.C. Willems. Models for dynamics. Dynamics Reported Volume 2, pages 171-269, 1989