To determine the response of the system with unity feedback to an input, we need to analyze the closed-loop transfer function and apply it to the input.

Given:

The open-loop transfer function is:

$G(s) = \frac{9}{s^2 + 4s}$

Since the system has unity feedback, the closed-loop transfer function $T(s)$ is given by:

$T(s) = \frac{G(s)}{1 + G(s)}$

Step 1: Determine the closed-loop transfer function

Substitute $G(s)$ into the formula for $T(s)$:

$T(s) = \frac{\frac{9}{s^2 + 4s}}{1 + \frac{9}{s^2 + 4s}} = \frac{9}{s^2 + 4s + 9}$

Thus, the closed-loop transfer function is:

$T(s) = \frac{9}{s^2 + 4s + 9}$

Step 2: Apply the input to the system

Let the input to the system be $R(s)$. The output $Y(s)$ of the system is given by:

$Y(s) = T(s) \cdot R(s) = \frac{9}{s^2 + 4s + 9} \cdot R(s)$

To determine the system's time-domain response, you would need to know the specific form of the input $R(s)$. Common inputs are:

1. Step input: $R(s) = \frac{1}{s}$
2. Impulse input: $R(s) = 1$
3. Sinusoidal input: $R(s) = \frac{1}{s^2 + \omega^2}$, for a sinusoid of frequency $\omega$.

Would you like me to proceed with one of these inputs, or do you have a specific input in mind? Let me know, and I’ll continue from there!

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Related Questions:

1. What is the time-domain response of a system with a step input?
2. How can we find the frequency response of the system?
3. How would the response differ if the input were a sinusoidal signal?
4. What is the stability of this system based on its transfer function?
5. How do we calculate the system's poles and zeros?

Tip:

To find the time-domain response of a system, it's often useful to perform the inverse Laplace transform of the closed-loop transfer function multiplied by the input's Laplace transform.