To analyze the given system and determine which of the statements are true, we will first derive the transfer function of the closed-loop system with the given PI controller.

Step 1: Define the System

• The plant is $P(s) = \frac{1}{s+1}$.
• The PI controller is given by $C(s) = K_p + \frac{K_i}{s} = 1 + \frac{2}{s} = \frac{s + 2}{s}$.

Step 2: Open-Loop Transfer Function

The open-loop transfer function is: $G_{\text{OL}}(s) = C(s) \cdot P(s) = \frac{s + 2}{s} \cdot \frac{1}{s+1} = \frac{s + 2}{s(s + 1)}$

Step 3: Closed-Loop Transfer Function

The closed-loop transfer function is: $G_{\text{CL}}(s) = \frac{G_{\text{OL}}(s)}{1 + G_{\text{OL}}(s)} = \frac{\frac{s + 2}{s(s + 1)}}{1 + \frac{s + 2}{s(s + 1)}} = \frac{s + 2}{s^2 + 2s + 2}$

Step 4: Analyze the Statements

Statement A: Zero steady-state error for a step input

• A PI controller generally ensures zero steady-state error for a step input because it introduces an integrator, making the system type 1.
• True.

Statement B: Complex conjugate poles at $s = -1 \pm j$

• To find the poles, solve the characteristic equation $s^2 + 2s + 2 = 0$.
• The roots are $s = -1 \pm j$.
• True.

Statement C: The closed-loop system is a first-order system

• The characteristic equation is quadratic ($s^2 + 2s + 2$), indicating a second-order system.
• False.

Statement D: The closed-loop system has a zero in the left-half plane (LHP)

• The numerator of $G_{\text{CL}}(s)$ is $s + 2$, which gives a zero at $s = -2$.
• This zero is in the LHP.
• True.

Conclusion

The correct answer is:

c. Statements A, B, and D are correct, and C is incorrect.

---

Do you want details or have any questions?

Here are 5 related questions:

1. How does the presence of a PI controller affect the stability of a system?
2. What is the significance of the system poles being complex conjugates?
3. Why does a PI controller ensure zero steady-state error for a step input?
4. How does the location of a zero in the left-half plane affect system behavior?
5. Can a system with a second-order characteristic equation be considered first-order under any conditions?

Tip: Understanding the pole-zero configuration of a system helps in predicting its time-domain and frequency-domain behavior.