A first-order plant P(s) = 1/(s+1) is connected in unity feedback configuration with a PI controller. If the controller gains are K_i =2 and K_p =1, then which of the following statements are true?

A) The closed loop system has zero steady-state error for a step input.

B) The closed loop system has complex conjugate poles at  s=-1 \pm j  .

C) The closed loop system is a first-order system.

D) The closed loop system has a zero in the LHP.

Question 1Select one:

a.
Statements A, B, C and D are all correct.

b.
Only statement C is correct.

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

d.
Only statement D is correct.

e.
Only statement A is correct.

f.
Only statement B is correct.

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.**

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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.

Explore the closed-loop transfer function of a unity feedback system with a PI controller and the plant P(s) = 1/(s+1). Understand how the controller gains affect system stability, pole placement, and steady-state error. Detailed analysis and solutions provided.

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