Question

An LTI system is shown in the figure where \[ G(s) = \frac{100}{s^2 + 0.1s + 10} \] The steady state output of the system, to the input \( r(t) \), is given as \[ y(t) = a + b \sin(10t + \theta) \] The values of 'a' and 'b' will be

• A. ( a = 1, b = 10 \)
• B. ( a = 10, b = 1 \)
• C. ( a = 1, b = 100 \)
• D. ( a = 100, b = 1 \)

Hint:

In LTI systems, the steady-state output to a sinusoidal input can be found by evaluating the magnitude of the frequency response at the input frequency.

Answer 1

The Correct Option is A

Step 1: Understand the system's transfer function.
The transfer function \( G(s) = \frac{100}{s^2 + 0.1s + 10} \) describes a second-order system. The steady-state output for a sinusoidal input is given by the form \( y(t) = a + b \sin(10t + \theta) \), where \( b \) is the amplitude and \( a \) is the offset. Step 2: Calculate the values of 'a' and 'b'.
At steady state, the response of an LTI system to a sinusoidal input depends on the frequency of the input. The value of \( b \) is related to the magnitude of the system's frequency response at the given input frequency, which is 10 rad/s. The magnitude of the frequency response is \( |G(j\omega)| = \frac{100}{\sqrt{(10)^2 + (0.1 \cdot 10)^2}} = 10 \), so \( b = 10 \). The value of \( a \), which represents the DC offset, is 1. Step 3: Conclusion.
The correct answer is (A), where \( a = 1 \) and \( b = 10 \).