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.
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$\begingroup$ Partial differential equations also describe some systems, but control theory for such systems is far more complicated in comparison to regular ODE's. $\endgroup$– PaulMay 14, 2016 at 20:24
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$\begingroup$ 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. $\endgroup$– alephzeroMay 14, 2016 at 20:58
1 Answer
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.
Advantages of ODE Models:
Ignoring all variations with respect to other variables (especially space) simply makes things easier to both compute and analyze (in comparison to other types of differential equations). You can take advantage of a great deal of analysis tools, including impulse & frequency response, bode plots, etc... If your system of ODE's are linear or otherwise linearizable around a setpoint, you can often use standard control techniques like PID, LQR, and LQG.
Disadvantages of non-ODE models
I emphasize that ODE's are simpler to analyze insofar as they require the minimal mathematical complexity required of an engineer. There are, of course, more complicated models of physical systems (e.g. partial differential equations) that are likely more reflective of the true physics in comparison to an ODE model. However, the analysis tools that you learned for ODE's (a.k.a. classical control theory) no longer apply. Instead, you will likely have to use a set of tools that fall under the category of optimal control theory. For most problems of interest to engineers, optimal control requires fairly sophisticated numerical methods (usually adjoint-based optimization). The situation gets even more complicated if you include stochasticity in any of the variables or parameters.
In short, control of anything other than an ODE is both very complicated to do in practice and is often unnecessary to begin with. Non-ODE control is a very highly specialized field and is generally not taught as a part of core engineering classes. If you ever get the chance to control anything other than an ODE in practice, it will likely come as a part of state-of-the-art scientific research rather than run-of-the-mill engineering applications.
