Why do we expect the solution of a linear ode to have the same shape as the input function?

Question

I just started learning about control theory and there is something fundamental that I do not quite understand.

For a linear time-invariant system modelled by a linear ode $Ly(t) = x(t)$, if the characteristic roots of the differential operator $L$ all have negative real parts, we know that the homogeneous solution $y_c(t)$ converges to $0$ as $t\to\infty$. Which means that the total response of the system $y(t)=y_c(t) + y_p(t)$ will eventually, in a sense, converge to its particular integral solution "the steady state" after some period of time "the transient state".

We then find the steady state error by subtracting the output function, at the steady state, from the input function.

So here is my question, why do we even expect the solution of the linear ode to have the same shape as the input, at the steady state at least, in the first place ?. I do understand that if we a have process variable (output), we can compare it with a reference or set point (input), but again, why does the linear ode model a system where we input some signal to the system and expect the output to eventually reach that input at the steady state, it just seems unjustified to me. Is this only a characteristic of linear time-invariant systems?

Answer 1

There are two questions here. The first is why an output of a linear ODE is similar to the input. And the second question is why do we consider the difference between the input and the output.

For the second, we consider this difference, that is the tracking error, because we want the output to be as the input, and we change our system (regulate it) to ensure this tracking, as good as we can or need.

Regarding the first question, you can think of it in the following way.

1. All good enough references can be seen as their Fourier representations, i.e., a composition of sinusoidal waves. Slow references consist mainly of low-frequency components, and fast-changing inputs correspond to high frequencies.
2. For a linear ODE, we know that the particular solution for a sinusoidal input is also a sinusoidal function of the same frequency but with different magnitude and phase.
3. In the Control Systems context, the vast majority of linear control systems (plants) are low-pass filters, i.e., they have small magnitude distortions for low-frequency components and pass them almost as they are, maybe with a delay.
4. Thus, for linear control systems, outputs to slow-varying (low-frequency) inputs have mainly the same shape as these inputs; they mainly consist of low-frequency components that are not distorted.