How does the intuitive notion of causality fit in with control systems?

Question

Edit: By causality, in this question, I do not mean the traditional mathematical definition in the theory o signals and systems; I mean causality as in an intuitive 'what's moving/pushing what notion'...

I'm having trouble understanding how causality fits into our intuitive ideas of inputting our output back into the system as feedback. What does it mean physically that y, the output, is dependent on the previous y values?

I understand the control/feedback engineering intuition, but once I get into the math, what I see are neat differential equations that govern how every single component should vary under such constraints. But, I don't see how we get to talk about how a certain component's behavior is caused by others.

Take exponential growth, for example, the output's behavior is caused by the behaviors of its rate of change and of the input force. Is this causality statement really the case, or is it just nature doing its thing, or am I just mixing ideas and being way too pedantic?

Either way, I want to understand this concept to be able to fully grasp what it really means when, for example, a servomotor gives feedback information about its rotation, and how this relates to the mathematical/physical differential equation framework.

Answer 1

As near as we can tell by experiment, causality is nature's way of doing its thing.

Causality says that if you have a system $y(t) = h\left(x(t), t\right)$, and it is causal, then $y(t_0)$ is dependent only on values of $x(t)$ for $t < t_0$*.

Causality doesn't say you can't combine systems into larger systems (as a rather pertinent example, you can certainly combine systems in a way that makes a control loop).

Causality doesn't imply stability, or instability. There's a rich set of behaviors that causality allows -- it just doesn't allow the output to follow future inputs.

So in your exponential growth example, if you have a system described by $\dot x = a x + u$, and $a > 0$, the output will tend to grow exponentially (unless $u$ magically or through feedback damps the tendency). We can say that the growth is caused by $x$ feeding back on itself -- but that causation still happens without violating causality. Any change in $x$ or $u$ at time $t_0$ will affect the derivative of $x$ at time $t_0$, but (in the absence of infinities) it won't change the immediate value of $x$ -- so, the system is still causal.

* I may be overly strict here -- I'm ruling out $x = u$ as a causal system. I'm not sure what the popular definition is in control system terms, but I'm lazy so I'm going to let it stand.