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,[1] and system identification.
[2] In the behavioral setting, a dynamical system is a triple where
means that the laws of the system forbid the trajectory
is deemed possible, while after modeling, only the outcomes in
Special cases: System properties are defined in terms of the behavior.
-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.
is the solution set of a system of constant coefficient linear ordinary differential equations
is a matrix of polynomials with real coefficients.
In order to define the corresponding behavior, we need to specify when we consider a signal
For ease of exposition, often infinite differentiable solutions are considered.
, and with the ordinary differential equations interpreted in the sense of distributions.
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 [3] .