Question

Adding a Zero to the transfer function generally:

• A. Increases the overshoot and improves the speed of response
• B. Decreases the overshoot
• C. Makes the system unstable
• D. Has no effect on the transient response

Hint:

Adding a Zero \( \implies \) Derivative action \( \implies \) Faster response, shorter rise time, but higher peak overshoot.
Adding a Pole \( \implies \) Integral action \( \implies \) Slower response, longer rise time, but reduced peak overshoot.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the general effect of adding a zero to the open-loop or closed-loop transfer function on the system's transient response characteristics.
In control systems, the addition of poles and zeros is used to shape the performance and stability profile of the feedback loop.

Step 2: Key Formula or Approach:
A zero adds a derivative-like action to the system.
Consider a second-order system with an added zero at \( s = -z \):
\[ T_z(s) = \frac{\omega_n^2 (s/z + 1)}{s^2 + 2\zeta\omega_n s + \omega_n^2} = T(s) + \frac{1}{z} s T(s) \] Where \( T(s) \) is the standard second-order transfer function.
The term \( s T(s) \) represents the derivative of the output response. This derivative component introduces a predictive nature to the system.

Step 3: Detailed Explanation:
Let us analyze the impact of this zero on the step response:

• Impact on Speed of Response:
- The derivative action increases the initial rate of rise of the output.
- This reduces the rise time (\( t_r \)) and peak time (\( t_p \)), making the system respond much faster.
- Therefore, the speed of the response is significantly improved.

• Impact on Overshoot:
- Because the system responds faster and with higher initial acceleration, the output tends to overshoot the steady-state target more than it would without the zero.
- Consequently, the maximum peak overshoot (\( M_p \)) increases.
- The closer the zero is located to the origin in the Left Half of the s-plane, the more pronounced this increase in overshoot and speed becomes.

• Comparison with adding a Pole:
- Adding a pole acts like an integrator, which increases the rise time (slowing down the system) and decreases the overshoot, making the system more sluggish.

Step 4: Final Answer:
Adding a zero to the transfer function generally increases the overshoot and improves the speed of response.