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I am working through some example problems about stability in systems and came across this one that I can't find the answer to. I'm wondering whether I am misunderstanding something about the definition of stability in this context or how it relates to solutions of the system equation.

The example question asks which of these statements are guaranteed to be true when an LTI system modeled by the equation $m\ddot{x} + b\dot{x} + kx = b \dot{y}$ is stable, given that m, b, and k are positive. y is the input signal and x is the system response (output). The input signal is sinusoidal.

All solutions decay to 0 as 𝑡 t approaches infinity. All homogeneous solutions decay to 0 as 𝑡 t approaches infinity. All solutions are sinusoidal. All solutions are asymptotically the same as 𝑡 t approaches infinity. All solutions are overdamped and do not oscillate. Initial conditions don't have a significant long-term effect on the solutions. The unforced system always returns to equilibrium.

I think that in general, if you plot the homogenous solutions of a system, they should all be asymptotic, not getting larger and larger as t approaches infinity, but they don't necessarily all need to have the same asymptote or all approach 0 specifically (I think). Does the fact that m, b, and k are all positive change that? If not, it makes me think that all 3 of the options that mention t approaching infinity are not guaranteed.

I also don't think all the solutions have to be sinusoidal because if they are real solutions not complex solutions, I don't think they will be sinusoidal.

I also don't think all solutions will be overdamped, because I looked that up and it seems like that is true when $b^2 > 4mk$, but we don't know that for this particular system.

The last two options sound like they could be true, since a stable system should stay in a fairly steady state over time and intuitively it seems like it should return to equilibrium if it's stable, but I'm not sure if there is a subtlety that I'm missing there.

Can anyone see where I am going wrong in my reasoning above? None of the combinations of answers I thought made sense were correct so far, and I don't know what the correct set of answers is. Please let me know if anything needs to be clarified further.

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I assume that we're talking about BIBO stability, and hence we exclude marginally stable systems. In that case, all poles of the system's rational transfer function lie in the left half plane, and, consequently, all eigenmodes decay exponentially. So yes, all homogeneous solutions decay to zero as $t\to\infty$ (statement #2). This is equivalent to saying that the effect of initial conditions decays with time, i.e., transients die out (statement #6).

So for a sinusoidal input, the output converges to a sinusoid of the same frequency with its amplitude and phase determined by the constants in the differential equation. But the parameters of this sinusoid are independent of the initial conditions (statement #4).

Statement #7 basically says the same, namely that the output stabilizes with zero input. In the case of BIBO-stability, "stabilize" means "converge to zero".

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  • $\begingroup$ Thanks. Then would you say the second and last statements are correct, but none of the others? The autograder doesn't mark that combination as correct, but I'm wondering if there's just an issue with the autograder. $\endgroup$
    – Keara
    Commented Jul 19 at 1:19
  • $\begingroup$ No, try 2, 4, 6 and 7. If that works I can add an explanation. $\endgroup$
    – Matt L.
    Commented Jul 19 at 7:19
  • $\begingroup$ thanks, that did work - why is #4 correct? $\endgroup$
    – Keara
    Commented Jul 22 at 2:42
  • $\begingroup$ @Keara: I've edited my answer to specifically point out why the different statements are correct. $\endgroup$
    – Matt L.
    Commented Jul 22 at 7:52

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