First-order system simulation in Simulink - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a first-order system in control theory?

A first-order system is a system whose behavior can be described by a first-order differential equation. It has one energy storage element and its output depends on the current input and its own previous state.

beginner

What is the general form of the transfer function for a first-order system?

The transfer function is usually written as G(s) = \frac{K}{\tau s + 1}, where K is the system gain and \tau is the time constant.

beginner

In Simulink, which block is commonly used to simulate a first-order system?

The Transfer Function block is commonly used. You enter the numerator and denominator coefficients of the system's transfer function to simulate its behavior.

beginner

What does the time constant \tau represent in a first-order system?

The time constant \tau represents the time it takes for the system's response to reach about 63% of its final value after a step input.

beginner

How can you verify the simulation results of a first-order system in Simulink?

You can use the Scope block to visualize the output response and compare it with the expected theoretical response, such as the exponential rise or decay.

What is the order of a system described by the differential equation \( \tau \frac{dy}{dt} + y = Ku \)?

A First-order

B Second-order

C Zero-order

D Third-order

The equation has a first derivative and no higher derivatives, so it is a first-order system.

In the transfer function \( G(s) = \frac{5}{2s + 1} \), what is the time constant \( \tau \)?

A 5

B 2

C 0.5

D 1

The denominator is \( 2s + 1 \), so \( \tau = 2 \).

Which Simulink block is best to simulate a first-order transfer function?

A Gain block

B Integrator block

C Sum block

D Transfer Function block

The Transfer Function block allows you to enter numerator and denominator coefficients to simulate the system.

What does the output of a first-order system look like after a step input?

A Exponential rise or decay to final value

B Oscillations around final value

C Instant jump to final value

D No change

First-order systems respond with an exponential rise or decay to the new steady state.

How can you observe the output of your first-order system simulation in Simulink?

A Using the Step block

B Using the Constant block

C Using the Scope block

D Using the Gain block

The Scope block displays the simulation output as a graph.

Explain how to set up a first-order system simulation in Simulink starting from the transfer function.

Think about the blocks needed and how to connect them.

Describe the meaning of the time constant in a first-order system and how it affects the system response.

Relate it to how quickly the system reacts to changes.

Practice

1. What does a first-order system simulation in Simulink typically show?

easy

A. How the system output changes over time after an input

B. The exact solution of differential equations analytically

C. The internal code of Simulink blocks

D. The hardware specifications of the computer running Simulink

Solution

1. Step 1: Understand the purpose of first-order system simulation

   It models how the output of a simple system changes over time when given an input.

2. Step 2: Identify what simulation shows in Simulink

   Simulink simulates the system response graphically, showing output behavior over time.

3. Final Answer:

   How the system output changes over time after an input -> Option A

4. Quick Check:

   Simulation = output change over time [OK]

Hint: Simulation shows output change over time [OK]

Common Mistakes:

• Confusing simulation with code details
• Thinking simulation shows hardware specs
• Assuming simulation gives exact analytic solutions

2. Which of the following is the correct way to represent a first-order transfer function in Simulink's Transfer Function block?

easy

A. Numerator: [5], Denominator: [1 0]

B. Numerator: [1 5], Denominator: [1]

C. Numerator: [1], Denominator: [1 5]

D. Numerator: [1], Denominator: [5 1]

Solution

1. Step 1: Recall first-order transfer function form

   A first-order system has denominator of form s + a, so denominator coefficients are [1 a].

2. Step 2: Match numerator and denominator arrays

   Numerator is usually constant [1], denominator [1 5] means s + 5, correct for first-order.

3. Final Answer:

   Numerator: [1], Denominator: [1 5] -> Option C

4. Quick Check:

   Denominator = [1 a] for first-order [OK]

Hint: Denominator starts with 1 for s term [OK]

Common Mistakes:

• Swapping numerator and denominator
• Using wrong order of coefficients
• Using zero in denominator leading coefficient

3. Given the Simulink Transfer Function block with Numerator = [2] and Denominator = [1 3], and a step input of magnitude 1, what is the steady-state output value?

medium

A. 2

B. 0.67

C. 0.5

D. 3

Solution

1. Step 1: Calculate steady-state gain from transfer function

   Steady-state gain = numerator constant / denominator constant = 2 / 3 ≈ 0.67.

2. Step 2: Apply step input magnitude

   Step input is 1, so output steady-state = 0.67 * 1 = 0.67.

3. Final Answer:

   0.67 -> Option B

4. Quick Check:

   Gain = 2/3 = 0.67 [OK]

Hint: Steady-state = numerator/denominator constant term [OK]

Common Mistakes:

• Using numerator over denominator reversed
• Ignoring step input magnitude
• Confusing transient with steady-state

4. You created a first-order system in Simulink with Numerator = [1] and Denominator = [1 0], but the simulation shows no response. What is the likely error?

medium

A. Denominator [1 0] represents an integrator, not a first-order system

B. Numerator should be zero for first-order systems

C. Step input magnitude is missing

D. Scope block is not connected

Solution

1. Step 1: Analyze denominator [1 0]

   Denominator [1 0] means s + 0 = s, which is an integrator, not a first-order system with constant term.

2. Step 2: Understand simulation behavior

   Integrator causes output to ramp, but if input is zero or setup wrong, no visible response occurs.

3. Final Answer:

   Denominator [1 0] represents an integrator, not a first-order system -> Option A

4. Quick Check:

   Denominator must have constant term for first-order [OK]

Hint: Denominator must have constant term, not zero [OK]

Common Mistakes:

• Assuming zero constant term is valid
• Ignoring block connections
• Forgetting to set step input

5. You want to simulate a first-order system with time constant 4 seconds and gain 3 in Simulink. Which numerator and denominator coefficients should you use in the Transfer Function block?

hard

A. Numerator: [1], Denominator: [1 0.25]

B. Numerator: [3], Denominator: [4 1]

C. Numerator: [3], Denominator: [1 0.25]

D. Numerator: [3], Denominator: [1 4]

Solution

1. Step 1: Write transfer function formula for first-order system

   Transfer function = gain / (time_constant * s + 1) = 3 / (4s + 1).

2. Step 2: Convert to Simulink coefficients

   Numerator [3], Denominator [1 4] for s + 4.

3. Final Answer:

   Numerator: [3], Denominator: [1 4] -> Option D

4. Quick Check:

   Denominator = [1 tau] for first-order [OK]

Hint: Denominator coefficients = [1 tau] [OK]

Common Mistakes:

• Swapping numerator and denominator
• Using time constant as denominator constant
• Not normalizing denominator leading coefficient to 1