Second-order system simulation in Simulink

Introduction

We use second-order system simulation to understand how systems like springs or circuits respond over time. It helps us see how they react to changes or inputs.

To study how a car's suspension reacts to bumps on the road.

To analyze how a speaker cone moves when playing sound.

To model how a thermostat controls room temperature.

To simulate how an electrical circuit responds to voltage changes.

Syntax

Simulink

Use Simulink blocks to build the system:
- Use 'Transfer Function' block with numerator and denominator coefficients.
- Set numerator as [0 0 1] for a standard second-order system.
- Set denominator as [1 2*zeta*wn wn^2], where zeta is damping ratio and wn is natural frequency.
- Connect input and output blocks to simulate response.

The denominator defines the system's behavior: damping and frequency.

Simulink uses blocks connected by lines to represent system equations visually.

Examples

This sets a system with damping ratio 0.25 and natural frequency 2 rad/s.

Simulink

Numerator: [0 0 1]
Denominator: [1 1 4]

This models a system with damping ratio 0.5 and natural frequency 3 rad/s.

Simulink

Numerator: [0 0 1]
Denominator: [1 3 9]

Sample Program

This code defines a second-order system with damping 0.7 and natural frequency 5 rad/s. It then plots the step response showing how the system reacts to a sudden input.

Simulink

% MATLAB script to simulate a second-order system
% Define parameters
zeta = 0.7; % damping ratio
wn = 5; % natural frequency (rad/s)

% Create transfer function numerator and denominator
num = [0 0 1];
den = [1 2*zeta*wn wn^2];

% Create transfer function system
sys = tf(num, den);

% Simulate step response
step(sys)
title('Step Response of Second-Order System')
grid on

Important Notes

Higher damping ratio means less oscillation and faster settling.

Natural frequency controls how fast the system responds.

Simulink allows visual block diagrams, but MATLAB code can also define and simulate systems.

Summary

Second-order system simulation helps us see how systems respond over time.

Use transfer function blocks with numerator and denominator coefficients in Simulink.

Damping ratio and natural frequency shape the system's behavior.

Practice

1. What does the damping ratio (ζ) affect in a second-order system simulation in Simulink?

easy

A. The color of the simulation plot

B. The speed and overshoot of the system's response

C. The size of the Simulink blocks

D. The input signal amplitude

Solution

1. Step 1: Understand damping ratio role

   The damping ratio controls how oscillations decay in the system response.

2. Step 2: Connect damping ratio to response behavior

   A higher damping ratio reduces overshoot and speeds settling time, affecting speed and overshoot.

3. Final Answer:

   The speed and overshoot of the system's response -> Option B

4. Quick Check:

   Damping ratio = speed and overshoot control [OK]

Hint: Damping ratio controls oscillation and overshoot [OK]

Common Mistakes:

• Confusing damping ratio with input amplitude
• Thinking damping ratio changes block size
• Assuming damping ratio affects plot color

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

easy

A. Numerator: [1 0 0], Denominator: [1 2*zeta*wn wn]

B. Numerator: [1 2*zeta*wn wn^2], Denominator: [1]

C. Numerator: [1], Denominator: [1 2*zeta*wn wn^2]

D. Numerator: [wn^2], Denominator: [1 2*zeta*wn 1]

Solution

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

   The standard form is numerator = [1], denominator = [1 2*zeta*wn wn^2].

2. Step 2: Match options to standard form

   Numerator: [1], Denominator: [1 2*zeta*wn wn^2] matches the correct numerator and denominator coefficients for a second-order system.

3. Final Answer:

   Numerator: [1], Denominator: [1 2*zeta*wn wn^2] -> Option C

4. Quick Check:

   Standard form = Numerator: [1], Denominator: [1 2*zeta*wn wn^2] [OK]

Hint: Denominator has 1, 2*zeta*wn, wn squared [OK]

Common Mistakes:

• Swapping numerator and denominator
• Using wrong powers for wn
• Missing damping term 2*zeta*wn

3. Given the Simulink transfer function block with numerator [1] and denominator [1 4 4], what is the natural frequency (wn) and damping ratio (ζ) of the system?

medium

A. wn=2, ζ=1

B. wn=4, ζ=2

C. wn=1, ζ=4

D. wn=2, ζ=0.5

Solution

1. Step 1: Identify denominator coefficients

   Denominator is [1 4 4], which matches [1 2*zeta*wn wn^2].

2. Step 2: Calculate wn and ζ from coefficients

   wn^2 = 4 -> wn = 2; 2*zeta*wn = 4 -> ζ = 4/(2*2) = 1.

3. Final Answer:

   wn=2, ζ=1 -> Option A

4. Quick Check:

   wn=2, ζ=1 from denominator [OK]

Hint: wn is sqrt of last denominator term [OK]

Common Mistakes:

• Confusing wn^2 with wn
• Dividing damping term incorrectly
• Ignoring coefficient order

4. You created a second-order system in Simulink with numerator [1] and denominator [1 0 4], but the simulation shows no oscillations. What is the likely error?

medium

A. The damping ratio is zero, causing no oscillations

B. The input signal is missing

C. The numerator should be [1 0 4]

D. The denominator coefficients are incorrect for oscillations

Solution

1. Step 1: Analyze denominator coefficients

   Denominator [1 0 4] means 2*zeta*wn = 0, so damping ratio ζ = 0.

2. Step 2: Understand effect on oscillations

   Zero damping ratio means undamped oscillations, but the system denominator form is incorrect for natural frequency 2 (since wn^2=4).

3. Step 3: Identify correct denominator form

   Correct denominator should be [1 0 4] only if system is undamped, but zero middle term can cause simulation issues.

4. Final Answer:

   The denominator coefficients are incorrect for oscillations -> Option D

5. Quick Check:

   Incorrect denominator = no oscillations [OK]

Hint: Check denominator matches 1, 2*zeta*wn, wn^2 [OK]

Common Mistakes:

• Assuming zero damping means no oscillations
• Swapping numerator and denominator
• Ignoring input signal presence

5. You want to simulate a second-order system with a natural frequency of 5 rad/s and a damping ratio of 0.3 in Simulink. What should be the denominator coefficients for the Transfer Function block?

hard

A. [1 3 25]

B. [1 3 5]

C. [1 3 15]

D. [1 3 30]

Solution

1. Step 1: Calculate denominator coefficients

   Given wn=5, ζ=0.3, calculate 2*ζ*wn = 2*0.3*5 = 3 and wn^2 = 25.

2. Step 2: Form denominator array

   Denominator coefficients are [1, 3, 25].

3. Final Answer:

   [1 3 25] -> Option A

4. Quick Check:

   2*zeta*wn=3, wn^2=25 [OK]

Hint: Multiply 2*damping*wn and square wn for denominator [OK]

Common Mistakes:

• Using wn instead of wn squared
• Mixing damping ratio with wn
• Incorrect multiplication order