# Transients in marginally stable system

I know that if a system is asymptotically stable then after infinite time its output contains only characteristic of input signal and it is also known as steady state and there are no transient component left in output, and if system is asymptotically unstable then whatever will be input is its output will become infinite after infinite time so we can say that it doesn't reach at steady state and it's transient components become infinitely large,

but what about transient and steady state in marginally stable systems?

Because in marginally stable systems even after infinite time it contains both inputs characteristics and systems characteristics so what exactly transient and steady state components will be?

I don't know whether I understand transient and steady state conditions properly (whatever I explained above) or it is just poorly defined in books?

## 1 Answer

I think your understanding is correct. The steady state response of an LTI system is the part of the response that is caused by a steady excitation at the input, like a sinusoidal signal, or any other periodic signal. The transient response is caused by changes at the input, like switching on a signal, or changing the parameters of a periodic input signal, such as amplitude, phase or frequency.

As an example of a marginally stable system, take an ideal integrator with impulse response $$h(t)=u(t)$$, where $$u(t)$$ is the unit step function. You can show that the response to a switched sinusoid at the input

$$x(t)=\sin(\omega_0t)u(t)\tag{1}$$

is

$$y(t)=\frac{1}{\omega_0}u(t)-\frac{1}{\omega_0}\cos(\omega_0t)u(t)\tag{2}$$

The first term on the right-hand side of $$(2)$$ is a non-decaying contribution caused by switching on the sinusoid at the input at $$t=0$$, and the second part is the steady state response to the sinusoidal input. The non-sinusoidal part of the output signal is what would be a transient if the system were a stable system, in which case it would decay. Since for this example it is non-decaying due to marginal stability, the term transient would of course be a misnomer.

• Thanks for such a great insight, as in your example, by observing output we can guess which component Of output might behave as a transient and which one as steady state but what about more general example like sinusoidal input and sinusoidal impulse response (I.e L-C circuit) where output is implicit function, then how to know which component behave as transient and which as steady state? – user215805 Jul 16 '20 at 13:12
• @user215805: The part of the output that has the same frequency as the input is the steady state response, and the other part is only determined by the system's impulse response. So for an ideal resonator you'll get a contribution at the resonator's frequency, which is unrelated to the frequency of the input signal. – Matt L. Jul 16 '20 at 13:15