Are steady-state values in LTI systems differentially influenced by the rate an input is introduced

signal processing

I have a general question about LTI systems I was hoping someone could clarify.
I'm currently learning about steady state behavior in LTI systems, and how you can evaluate them with different inputs. So here's my question-say we have an LTI system, and let's consider 2 different inputs: a step input with a long duration (say maybe 150 time steps), let's call this A. And another input that gradually increases from 0 to 1 (maybe this gradual increase is logarithmic or linear-I wonder if it matters?), in 135 time steps, and then remains at that value for an additional 15 steps-let's call this B. So that means it takes B 135 steps to reach A's value, but it does maintain that value for a little while at least. Are there general features of LTI systems that could help you predict the behavior of the system in response to the abrupt vs. gradual introduction of an input? Will they reach the same steady state values?

My intuition is that they should both give for the same steady state, but that abruptly introducing an input should reach steady state faster. Although I wonder if the gradual case would allow for a more stable system? Does anyone have any thoughts? I'd appreciate it, this has been bothering me for a while now haha

Best Answer

Whenever the inputs to an LTI system stay constant for a time that is greater than the length of the impulse response, the system will reach the steady state for these inputs, regardless of history.

Whenever that is not the case, history matters. How the system reacts to different inputs is completely dependent on the system -- if you look at discrete time systems, [ 5 -5 0 0 0] will react differently than [ 1 1 1 1 1].

The first system will reach steady state after just two ticks, but produces fairly violent spikes on every input change:

[ 1  1  1  1  1  6  6  6  6  6 11 11 11 11 11] -> [ 5  0  0  0  0 25  0  0  0  0 25 ...]
[ 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15] -> [ 5  5  5  5  5  5  5  5  5  5  5 ...]

The second takes five ticks, and smooths out steps of length 5:

[ 1  1  1  1  1  6  6  6  6  6 11 11 11 11 11] -> [ 1  2  3  4  5 10 15 20 25 30 35 ...]
[ 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15] -> [ 1  3  6 10 15 20 25 30 35 40 45 ...]

You can see that apart from the start the outputs are the same (a ramp increasing by 5 each step).

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