Behavioral modeling

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.
Dynamical system as a set of signals.
In the behavioral setting, a dynamical system is a triple 
where
formula_8 means that formula_9 is a trajectory of the system, while formula_10 means that the laws of the system forbid the trajectory formula_9 to happen. Before the phenomenon is modeled, every signal in formula_5 is deemed possible, while after modeling, only the outcomes in formula_13 remain as possibilities.
Special cases:
Linear time-invariant differential systems.
System properties are defined in terms of the behavior. The system formula_1 is said to be 
where formula_24 denotes the formula_25-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.
A "linear time-invariant differential system" is a dynamical system formula_27 whose behavior formula_13 is the solution set of a system of constant coefficient linear ordinary differential equations formula_29, where formula_30 is a matrix of polynomials with real coefficients. The coefficients of formula_30 are the parameters of the model. In order to define the corresponding behavior, we need to specify when we consider a signal formula_32 to be a solution of formula_29. For ease of exposition, often infinite differentiable solutions are considered. There are other possibilities, as taking distributional solutions, or solutions in formula_34, and with the ordinary differential equations interpreted in the sense of distributions. The behavior defined is
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 
Observability of latent variables.
A key question of the behavioral approach is whether a quantity w1 can be deduced given an observed quantity w2 and a model. If w1 can be deduced given w2 and the model, w2 is said to be observable. In terms of mathematical modeling, the to-be-deduced quantity or variable is often referred to as the latent variable and the observed variable is the manifest variable. Such a system is then called an observable (latent variable) system.