# Is System Stability of LTI System Proven?

• the_amateur
In summary: The system is LTI as\frac{d}{dt} (a_1x_1(t) + a_1x_1(t)) = a_1 \frac{d}{dt} x_1(t) + a_2 \frac{d}{dt} x_2(t) = a_1y_1(t) + a_2y_2(t)and\frac{d}{dt} x(t-\tau) = y(t-\tau).In summary, the system is stable if its impulse response is absolutely integrable.
the_amateur
Is the following system stable. If so how.

y(t)= $\frac{d}{dt}$ x(t)I have tried the following proof but i think it is wrong.

PROOF:

1. The System is LINEAR
2. The system is time invariant

So on applying the stability criterion for LTI systems

ie . $\int^{\infty}_{-\infty} h(t) dt$ < $\infty$ --------- 1 For the above system h(t) = $\delta^{'}(t)$

so on applying h(t) = $\delta^{'}(t)$ in eq. 1
$\int^{\infty}_{-\infty} \delta^{'}(t) dt$ = $\delta(0)$

So the system is not stable.
I think the above proof is way off the mark.
please provide the correct proof. thanks

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The system (BIBO) stable if the impulse response is absolutely integrable, that is,

$\int^{\infty}_{-\infty} |h(t)| dt < \infty$

For the differentiator, we have (as you have found in your question)

$h(t) = \delta (t)'$.

Since

$\int^{\infty}_{-\infty} | \delta (t)' | dt= \infty$,

the system is not stable.The system is LTI as

$\frac{d}{dt} (a_1x_1(t) + a_1x_1(t)) = a_1 \frac{d}{dt} x_1(t) + a_2 \frac{d}{dt} x_2(t) = a_1y_1(t) + a_2y_2(t)$

and

$\frac{d}{dt} x(t-\tau) = y(t-\tau)$.

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Thanks for your reply but i am trying to find if it is stable or not. i think my derivation is wrong.

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here let x(t) be input that is bounded. As you let t-> inifinity, y(t) remains real and bounded, therefore the system is Stable.

if we had y(t) = tx(t), then if t->infinity x(t) is still bounded, but it is being multiplied by a factor of infinity so your output y(t) -> infinity therefore is unstable.

@Larrytsai

you mentioned the following

Larrytsai said:
here let x(t) be input that is bounded. As you let t-> inifinity, y(t) remains real and bounded, therefore the system is Stable.

if we had y(t) = tx(t), then if t->infinity x(t) is still bounded, but it is being multiplied by a factor of infinity so your output y(t) -> infinity therefore is unstable.

where x(t) can be any input signal. You say that in the system that i have posted the system is stable since x(t) is stable but i find by intuition that this is not the case if the input signal has discontinuities. At the point of discontinuity the output would then be a DIRAC delta $\delta(t)$ whose value is infinite at t = 0 (http://en.wikipedia.org/wiki/Dirac_delta_function" ). So by intuition the system is not stable.

The problem is I am unable to provide a rigorous proof to back my intuition. I believe my proof is wrong.

Any help would be appreciated. thanks!

Last edited by a moderator:

## 1. What is an LTI system?

An LTI (linear time-invariant) system is a type of mathematical model used to describe the behavior of a system that is both linear (follows the principle of superposition) and time-invariant (its behavior does not change over time).

## 2. How is stability defined in the context of LTI systems?

In the context of LTI systems, stability refers to the ability of the system to maintain a steady response, without growing or decaying, when subjected to a specific input. This can be mathematically proven by analyzing the system's transfer function or impulse response.

## 3. Can system stability of an LTI system be proven?

Yes, the stability of an LTI system can be proven by analyzing its transfer function or impulse response. If the system's transfer function is bounded, or its impulse response decays over time, then the system is stable.

## 4. What is the significance of proving system stability in LTI systems?

Proving system stability is important because it ensures that the system will not exhibit unpredictable or uncontrollable behavior, which can have serious consequences in real-world applications. It also allows for accurate prediction and control of the system's response.

## 5. Are there any limitations to proving system stability in LTI systems?

While proving system stability in LTI systems is a rigorous and reliable method, it does have some limitations. It assumes that the system is linear and time-invariant, which may not always be the case in real-world systems. Additionally, it requires accurate modeling of the system, which can be challenging in complex systems.

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