Transfer Function:
The given transfer function of the system is G(s) = 100/(s+1)(s+100).

Step Response:
To find the settling time for a unit step input, we need to analyze the step response of the system. The step response is the output of the system when a unit step input is applied.

Settling Time:
The settling time is defined as the time it takes for the system output to reach and stay within a certain percentage (usually 2%) of the final value.

2% Criterion:
In this case, the 2% criterion means that the output must reach and stay within 2% of the final value.

Calculation:
To find the settling time, we need to calculate the time it takes for the output to reach and stay within 2% of the final value.

Step 1:
Convert the transfer function to its time-domain representation. We can use partial fraction expansion to do this.

G(s) = 100/[(s+1)(s+100)] = A/(s+1) + B/(s+100)

Multiply both sides by (s+1)(s+100) to get rid of the denominators.

100 = A(s+100) + B(s+1)

Step 2:
Now, we can equate the coefficients of the powers of s on both sides of the equation.

For s^0 term: 100 = 100A + B
For s^1 term: 0 = 100A + A + 100B

Solving these equations, we get A = 1/99 and B = -1/99.

Step 3:
Now, we can write the time-domain representation of the transfer function.

G(s) = (1/99)/(s+1) - (1/99)/(s+100)

Step 4:
To calculate the settling time, we can find the time it takes for the output to reach and stay within 2% of the final value.

Let's assume the final value is 1 (since it's a unit step input).

2% of 1 is 0.02.

So, we need to find the time it takes for the output to reach and stay within 0.02 of 1.

Step 5:
We can use the inverse Laplace transform to find the time-domain representation of the transfer function.

Taking the inverse Laplace transform of G(s), we get g(t) = (1/99)e^(-t) - (1/99)e^(-100t).

Step 6:
Now, we need to find the time it takes for the output to reach and stay within 0.02 of 1.

Setting g(t) = 1 - 0.02 = 0.98, we can solve for t.

0.98 = (1/99)e^(-t) - (1/99)e^(-100t)

Solving this equation, we get t ≈ 4 seconds.

Therefore, the approximate settling time for a unit step input with a 2% criterion is 4 seconds.