For each of the following statements determine whether it is True or False. Providedetailed analysis and rationale with appropriate drawings and/or equations.(1+4) x 4= 20(a) A system with a transfer function ๐๐๐ ๐ +๐๐ can be stable if b < 0.(b) The system with transfer function ๐๐๐ ๐ 2 +๐๐๐ ๐ +๐๐ is a critically damped system, if ๐๐ = ๐๐ 24 .(c) Consider the feedback system in Figure 2, if controller C is chosen to be a proportionalcontroller, then the system output can never track the reference, i.e., |e| = |Yref โY| > 0.(d) For the feedback system in Figure 2, consider the controller C is a proportional-integralcontroller with proportional gain ๐พ๐พ๐๐ and integral gain ๐พ๐พ๐ผ๐ผ. If for the system a =1 andthe integral gain is chosen to be ๐พ๐พ๐ผ๐ผ > (๐๐+๐พ๐พ๐๐)24 , the system will possess underdampedbehaviour
Question
For each of the following statements determine whether it is True or False. Providedetailed analysis and rationale with appropriate drawings and/or equations.(1+4) x 4= 20(a) A system with a transfer function ๐๐๐ ๐ +๐๐ can be stable if b < 0.(b) The system with transfer function ๐๐๐ ๐ 2 +๐๐๐ ๐ +๐๐ is a critically damped system, if ๐๐ = ๐๐ 24 .(c) Consider the feedback system in Figure 2, if controller C is chosen to be a proportionalcontroller, then the system output can never track the reference, i.e., |e| = |Yref โY| > 0.(d) For the feedback system in Figure 2, consider the controller C is a proportional-integralcontroller with proportional gain ๐พ๐พ๐๐ and integral gain ๐พ๐พ๐ผ๐ผ. If for the system a =1 andthe integral gain is chosen to be ๐พ๐พ๐ผ๐ผ > (๐๐+๐พ๐พ๐๐)24 , the system will possess underdampedbehaviour
Solution
The question you've asked is quite complex and requires a detailed understanding of control systems. Here's a step-by-step analysis of each statement:
(a) False. A system with a transfer function as/(as+b) can be stable if b > 0. The stability of a system is determined by the roots of the denominator of the transfer function. If all roots have negative real parts, the system is stable. If b < 0, the root of the denominator is positive, making the system unstable.
(b) True. The system with transfer function cs^2 + bs + c is critically damped if c = b^2/4. This is because the damping ratio ฮถ = b/(2sqrt(c)). For a system to be critically damped, ฮถ = 1. If c = b^2/4, then ฮถ = b/(2sqrt(b^2/4)) = 1, so the system is critically damped.
(c) False. Even with a proportional controller, the system output can track the reference. The error e = Yref - Y can be reduced to zero if the controller gain is properly adjusted. However, the system may take some time to reach the reference value and may exhibit overshoot and oscillations.
(d) True. For the feedback system with a proportional-integral controller, if a = 1 and the integral gain Ki > (b+Kp)/4, the system will exhibit underdamped behavior. This is because the roots of the characteristic equation will have negative real parts and non-zero imaginary parts, leading to oscillatory response.
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