Control Systems

Step 1: Analyzing the transfer function.
The transfer function represents a second-order system. The step response of such a system depends on the nature of its poles.

Step 2: Explanation of options.
- (A) The characteristic equation indicates underdamped behavior, leading to a damped oscillatory response. - (B) Overdamped systems do not exhibit oscillatory behavior. - (C) The response does not have a non-zero slope at the origin. - (D) The system is stable because its poles have negative real parts.

Final Answer:

Step 1: Understanding the first-order system.
The amplitude ratio for a first-order system subjected to sinusoidal input at the frequency equal to is , which corresponds to the frequency at which the system's response is reduced by 3 dB.

Step 2: Explanation of options.
- (B) is the correct amplitude ratio at . - (A) and (C) These values are incorrect for the specified frequency. - (D) is incorrect because the amplitude ratio does not go to infinity.

Final Answer:

Step 1: Identify damping ratio.
Standard 2nd-order denominator: . Here, , .

Step 2: Formula for percent overshoot.
Substituting :

Final Answer:

Step 1: Understanding the Bode diagram.
A Bode diagram is a graphical representation of a system's frequency response. The amplitude ratio (AR) and phase angle ( ) are plotted against the frequency ( ).

Step 2: Explanation of the Bode plot.
- (A) log AR vs log and vs log : This is the standard form of a Bode plot.
- (B) log AR vs and log vs log : This is incorrect because both AR and are plotted against log .
- (C) AR vs log and log vs log: This is incorrect because should be plotted against log .
- (D) AR vs and vs : This is incorrect as Bode plots are typically log-log plots.

Step 3: Conclusion.
The correct option is (A), as it correctly represents the standard Bode diagram plotting both amplitude and phase against the logarithm of frequency.

Step 1: Use the steady-state error formula for a control system.
For a unit step change in the input, the steady-state error can be found by applying the final value theorem or evaluating the system's response at steady state.

Step 2: Conclusion.
The steady-state error for the given system is 0.4, so the correct answer is (C).

Step 1: Apply the Routh-Hurwitz criterion.
The Routh-Hurwitz criterion is applied to determine the stability of the system. We form the Routh array using the coefficients of the characteristic equation. For stability, the number of sign changes in the first column determines the value of that will keep the system stable.

Step 2: Conclusion.
Using the Routh test, we find that the correct value of that keeps the system stable is (C) 13.5.