Give step-by-step solution with explanation and final answer:A heater process has the transfer function GP (s) = Gp(s) = Te .Itis regulated by a controller
with transfer function G(s) = 3(1 + 2 The s

Question

Give step-by-step solution with explanation and final answer:A heater process has the transfer function GP (s) = Gp(s) = Te .Itis regulated by a controller
with transfer function G(s) = 3(1 + 2 The sensor used to feed back the output signal has the
transfer function Gs(s) = 1.
a. Construct the system in Simulink.
b. Simulate a unit step input signal. Measure maximum overshoot, settling time and rise time.
Also check control input (signal from controller to process).
c. Simulate a unit step disturbance. Measure overshoot, settling time, rise time and check control
input.
d. ‘What type of controller is used?
e. Perform the same simulation as in task b above, but in Matlab without using Simulink. This
can be done using the following commands:
1. Define the transfer function variable s: s=tf{('s")
2. Define the transfer function GP: GP = 1/(1 + 3*s)"2
3. Define the transfer function GC: GC = 3*(1 + 1/(2*s))
4. Calculate the feed forward transfer function, GCGP: GCGP = GC * GP
5. Calculate the transfer function of the entire feedback loop, GTOT: GTOT = feedback(GCGP, 1)
6. Plot the step response: step(GTOT)
7. Calculate step response properties: stepinfo(GTOT)

VIPSolutions · Accepted Answer

Let's solve the problem step by step:

---

## **Given Transfer Functions**

- **Process (Plant):**  
  $$
  G_P(s) = \frac{1}{(1 + 3s)^2}
  $$

- **Controller:**  
  $$
  G_C(s) = 3 \left(1 + \frac{1}{2s}\right)
  $$

- **Sensor:**  
  $$
  G_S(s) = 1
  $$

---

## **a. Construct the System in Simulink**

**Blocks to Use:**
- **Sum** (for feedback)
- **Transfer Function** (for $ G_P(s) $)
- **Transfer Function** (for $ G_C(s) $)
- **Step Input** (to provide step input)
- **Scope** (to view outputs)
- **Gain** (if needed)
- **(Optional) Disturbance Input**

**Configuration:**
1. Step input to the **sum block** (+ and - for feedback).
2. **Sum block** output to **controller** ($ G_C(s) $).
3. **Controller output** to **process** ($ G_P(s) $).
4. **Process output** to **scope** and back to **sum block** (feedback).
5. To measure control input, connect a scope after $ G_C(s) $.

---

## **b. Simulate Unit Step Input (in Simulink)**

**Steps:**
1. Run the simulation with a **step input**.
2. Measure:
   - **Maximum overshoot**
   - **Settling time**
   - **Rise time**
   - **Control input** (output of $ G_C(s) $ block)

---

## **c. Simulate Unit Step Disturbance**

**Steps:**
1. Add a **step disturbance** at the process input.
2. Run the simulation.
3. Measure:
   - **Maximum overshoot**
   - **Settling time**
   - **Rise time**
   - **Control input**

---

## **d. What Type of Controller is Used?**

Expand $ G_C(s) $:

$$
G_C(s) = 3\left(1 + \frac{1}{2s}\right) = 3 + \frac{3}{2s}
$$

This is a **Proportional-Integral (PI) Controller**:
- **Proportional term:** $ 3 $
- **Integral term:** $ \frac{3}{2s} $

---

## **e. Simulate in MATLAB (Without Simulink)**

### **MATLAB Code Steps**

```matlab
% 1. Define s as the Laplace variable
s = tf('s');

% 2. Define the transfer function of the process
GP = 1 / (1 + 3*s)^2;

% 3. Define the controller transfer function
GC = 3 * (1 + 1/(2*s));

% 4. Calculate the open-loop transfer function
GCGP = GC * GP;

% 5. Calculate the closed-loop transfer function
GTOT = feedback(GCGP, 1);

% 6. Plot the step response
step(GTOT);

% 7. Calculate step response properties
S = stepinfo(GTOT);
disp(S);
```

### **Explanation of Each Step**

1. **Laplace Variable:**  
   `s = tf('s');`  
   This creates a Laplace variable for transfer functions.

2. **Process Transfer Function:**  
   `GP = 1 / (1 + 3*s)^2;`

3. **Controller Transfer Function:**  
   `GC = 3 * (1 + 1/(2*s));`

4. **Open-Loop Transfer Function:**  
   `GCGP = GC * GP;`

5. **Closed-Loop System:**  
   `GTOT = feedback(GCGP, 1);`  
   This creates a unity feedback system.

6. **Plot Step Response:**  
   `step(GTOT);`

7. **Calculate Step Response Info:**  
   `stepinfo(GTOT);`  
   This gives overshoot, settling time, rise time, etc.

---

### **Summary Table**

| Task   | Step(s) | Output/Result |
|--------|---------|---------------|
| a      | Simulink construction as per above block diagram | System built |
| b      | Step input, measure response | Overshoot, rise time, settling time, control input |
| c      | Step disturbance, measure response | Overshoot, rise time, settling time, control input |
| d      | Controller Type | PI Controller |
| e      | MATLAB code as above | Step response plot and metrics |

---

## **Final Answers**

- **Controller Type:** PI Controller ($ 3 + \frac{3}{2s} $)
- **Simulation in Simulink:** Follow the block diagram as described.
- **Simulation in MATLAB:** Use the code above.
- **Measured Quantities:** Use `stepinfo` in MATLAB for:
  - Maximum overshoot
  - Settling time
  - Rise time

### **(You will get the numerical values for overshoot, settling time, rise time after running the MATLAB code).**

---

Let me know if you'd like help constructing the Simulink diagram or interpreting MATLAB output!

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