# Control theory - Differential equations and state variable description

I'm a 3rd-year bachelor student and I may be mistaken but from what I have understood: In control theory we assume that the studied systems have a State Variable Description. Then from there we can derive that for such systems the transfer function must be proper. But, to have a State Variable Description means that the system can be described by a simple differential equation or equivalently by a system of differential equations. I am not really convinced that all physical systems can be described by such equations. Although I don't know what kind of other description there could be, I would like to find some strong argument. I guess it must have to do with the equations of fluid or Newtonian dynamics but if someone could be more specific I would appreciate it.

• Partial differential equations also describe some systems, but control theory for such systems is far more complicated in comparison to regular ODE's.
– Paul
May 14 '16 at 20:24
• Aside from systems that can only be described realistically using quantum mechanics, you could in principle describe any physical system using Newtonian or relativistic mechanics, plus Maxwell's equations for EM radiation. So almost every "real world" application of control theory can be described by a system of partial differential equations. Of course the "clever part" of real-world control theory is to find a small and useful description of the system's behaviour - a 10 DOF model is a lot more manageable than a large mechanical or fluid dynamics model with say 1,000,000 DOFs. May 14 '16 at 20:58

A professor of mine, long ago, once quipped:

"Give me a word... any word at all... and I'll show you how the root of that word is a PDE!"

A touch of Philosophy
All physical systems are modeled by some set of mathematical equations, be they differential, algebraic, integral, stochastic, etc. The simplest models of physical systems only consider variations in the time variable (i.e. ordinary differential equations). While these models tend to be very simplified compared to the "true physics", they are often just "good enough" for specific engineering applications because we, as engineers, often only care about the input-output behavior of the system over time.