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I am a mathematics student, and am quite new to the whole control and system theory (haven't heard any lectures). I currently work with dynamic systems, but do not know the correct words for every attribute, yet. I would like to describe a system as follows:

A system is in "rest" until time $t=t_0$ for some $t_0\in\mathbb{R}$ if the system response $y(t<t_0)$ and the state $x(t<t_0)$ are equivalent to $0$ for zero input $u(t<t_0) \equiv 0$.

I am quite sure, that there is a word describing this scenario. I know the term causal in terms of systems, but I don't think this is the word I need.

EDIT: To maybe make it more clear what i want: I want to say "A system is in XXXX, when for zero input there is zero output." (~the system does nothing before any input is applied)

Lukas
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There are two things to consider here. First of all, a non-causal system can react to input that hasn't arrived yet. And, secondly, there could be non-zero initial conditions that would contribute to the output signal without any input signal.

So what you probably mean is a "causal system initially at rest" or "a causal system with zero initial conditions".

Matt L.
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  • Okay, so I need causality AND initially at rest. Thank you alot! Now I know what to google to go further! – Lukas Jan 12 '20 at 16:47
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This would be dead time. You could also call it delay, i think.

Max
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  • isn't *dead time* the time the system needs to react? So if for example the dead time is $\tau = 3$, and the input changes at time $t_0 = 10$ seconds, then the system response does not see react to the change of input until second $13$? i.e. the system response $y$ at time invteral $t_0$ to $t_0+\tau$ does not depend on the inputs made from $t_0$ to $t_0+\tau$. – Lukas Jan 12 '20 at 13:19
  • Yes, you are right, i misread your question. Matt L. has it right. – Max Jan 13 '20 at 08:27