In the derivation of expression for peak percent overshoot, Mp = exp – πξ √1-52 ×100% Which one %3D of the following conditions is NOT required? (a) System is linear and time invariant (b) The system transfer function has a pair of complex conjugate poles and no zeroes. (c) There is no transportation delay in the system. (d) The system has zero initial conditions
The Correct Answer and Explanation is :
The correct answer is (d) The system has zero initial conditions.
Explanation:
The derivation of the expression for peak percent overshoot in a second-order system primarily depends on the system’s transfer function and its dynamic characteristics. To understand why option (d) is not required, let’s break down the conditions:
- (a) System is linear and time-invariant:
This condition is necessary because the analysis for overshoot applies to linear time-invariant (LTI) systems. The peak percent overshoot and other transient characteristics are derived assuming the system follows the principles of linearity and time-invariance. Nonlinear systems or systems that change over time (time-varying systems) would have different transient behaviors, making the usual expressions for overshoot invalid. - (b) The system transfer function has a pair of complex conjugate poles and no zeroes:
This is a critical condition for a second-order system to exhibit oscillatory behavior, which is a key characteristic for the peak percent overshoot. If the system has complex conjugate poles, the response is oscillatory (underdamped), leading to overshoot in response to a step input. Zeroes do not impact the presence of overshoot in the standard second-order system, but complex poles are directly related to the overshoot. - (c) There is no transportation delay in the system:
This is also a necessary condition because transportation delay (also known as time delay) causes a phase shift in the system’s response. This delay can significantly affect the overshoot and transient response. Without delay, the system’s overshoot can be analyzed with standard second-order approximations. A system with a delay would need additional considerations for accurate overshoot estimation. - (d) The system has zero initial conditions:
This condition is not strictly required for the derivation of peak percent overshoot. The derivation of overshoot generally assumes that the system is responding to a step input and starts from rest, which implicitly means zero initial conditions. However, systems with non-zero initial conditions can also exhibit a peak overshoot, but the formula for overshoot typically assumes zero initial conditions. If initial conditions are non-zero, the transient response would be influenced by these conditions, but they do not affect the derivation of the peak overshoot formula for a typical second-order system.
Thus, the only condition that is not essential for the derivation of peak percent overshoot is (d).