Linearization of nonlinear models in Simulink - Mini Project: Build & Apply

Linearization of Nonlinear Models

📖 Scenario: You are working with a nonlinear system model in Simulink. To analyze and design controllers easily, you want to create a linear approximation of this nonlinear model around a specific operating point.

🎯 Goal: Build a simple nonlinear Simulink model, specify an operating point, linearize the model at that point, and display the linearized system matrices.

📋 What You'll Learn

Create a nonlinear Simulink model with a Gain and a Saturation block

Define an operating point structure for linearization

Use the linearize function to get the linear model

Display the linearized system matrices (A, B, C, D)

💡 Why This Matters

🌍 Real World

Linearization helps engineers analyze and control complex nonlinear systems by approximating them as simpler linear systems near specific operating points.

💼 Career

Understanding model linearization is essential for control engineers, system analysts, and data scientists working with dynamic system models in industries like automotive, aerospace, and robotics.

1

Create a nonlinear Simulink model

Create a Simulink model named nonlinear_model that contains a Gain block with gain value 2 connected to a Saturation block with limits -1 and 1. Save the model.

Simulink

% Create a Simulink model named nonlinear_model with a Gain block of 2 and a Saturation block with limits -1 and 1
% Your code here

Hint

Use new_system to create the model, add_block to add Gain and Saturation blocks, set their parameters, connect them with add_line, and save the model with save_system.

2

Define the operating point for linearization

Create an operating point object named op for the model nonlinear_model using findop with initial guess at input 0.

Simulink

open_system(new_system('nonlinear_model'));
add_block('simulink/Commonly Used Blocks/Gain', 'nonlinear_model/Gain', 'Gain', '2');
add_block('simulink/Commonly Used Blocks/Saturation', 'nonlinear_model/Saturation', 'UpperLimit', '1', 'LowerLimit', '-1');
add_line('nonlinear_model', 'Gain/1', 'Saturation/1');
save_system('nonlinear_model');

% Create an operating point object named op for nonlinear_model with input 0
% Your code here

Hint

Use getlinio to get linearization I/O points and findop to find the operating point with input 0.

3

Linearize the nonlinear model

Use the linearize function with the model nonlinear_model, the operating point op, and the linearization I/O points io to create a linear model named lin_sys.

Simulink

open_system(new_system('nonlinear_model'));
add_block('simulink/Commonly Used Blocks/Gain', 'nonlinear_model/Gain', 'Gain', '2');
add_block('simulink/Commonly Used Blocks/Saturation', 'nonlinear_model/Saturation', 'UpperLimit', '1', 'LowerLimit', '-1');
add_line('nonlinear_model', 'Gain/1', 'Saturation/1');
save_system('nonlinear_model');

io = getlinio('nonlinear_model');
op = findop('nonlinear_model', struct('Inputs', 0));

% Linearize the model and save it as lin_sys
% Your code here

Hint

Use linearize with the model name, operating point, and I/O points to get the linear system.

4

Display the linearized system matrices

Print the A, B, C, and D matrices of the linearized system lin_sys using disp.

Simulink

open_system(new_system('nonlinear_model'));
add_block('simulink/Commonly Used Blocks/Gain', 'nonlinear_model/Gain', 'Gain', '2');
add_block('simulink/Commonly Used Blocks/Saturation', 'nonlinear_model/Saturation', 'UpperLimit', '1', 'LowerLimit', '-1');
add_line('nonlinear_model', 'Gain/1', 'Saturation/1');
save_system('nonlinear_model');

io = getlinio('nonlinear_model');
op = findop('nonlinear_model', struct('Inputs', 0));

lin_sys = linearize('nonlinear_model', op, io);

% Display the A, B, C, D matrices of lin_sys
% Your code here

Hint

Use disp to print each matrix property of lin_sys.

Practice

1. What is the main purpose of linearizing a nonlinear model in Simulink?

easy

A. To make the model run faster in simulation

B. To simplify the model around an operating point for easier analysis

C. To convert the model into a discrete-time system

D. To add noise to the model for robustness testing

Solution

1. Step 1: Understand linearization concept

   Linearization approximates a nonlinear system by a linear one near a specific operating point.

2. Step 2: Identify the purpose in Simulink

   This simplification helps in easier analysis and controller design by using linear system tools.

3. Final Answer:

   To simplify the model around an operating point for easier analysis -> Option B

4. Quick Check:

   Linearization = Simplify near operating point [OK]

Hint: Linearization means simplify near a point [OK]

Common Mistakes:

• Thinking linearization speeds up simulation
• Confusing linearization with discretization
• Assuming linearization adds noise

2. Which Simulink function is used to linearize a nonlinear model at a specified operating point?

easy

A. linmod

B. ode45

C. step

D. sim

Solution

1. Step 1: Recall Simulink linearization functions

   Simulink provides linmod and linearize functions to get linear models from nonlinear ones.

2. Step 2: Identify the correct function

   linmod is a common function used to linearize models at operating points.

3. Final Answer:

   linmod -> Option A

4. Quick Check:

   Linearize function = linmod [OK]

Hint: Use linmod to linearize nonlinear models [OK]

Common Mistakes:

• Choosing sim which runs simulation
• Selecting step which plots step response
• Using ode45 which solves ODEs

3. Given the following MATLAB code snippet for linearizing a Simulink model:

sys = 'myModel';
op = operpoint(sys);
[A,B,C,D] = linmod(sys, op);
disp(size(A));

What will be the output of disp(size(A)) if the system has 4 states?

medium

A. [4 1]

B. [1 4]

C. [4 4]

D. [1 1]

Solution

1. Step 1: Understand linmod output matrices

   linmod returns state-space matrices A, B, C, D. A is the state matrix with size (number_of_states x number_of_states).

2. Step 2: Apply to given system

   Since the system has 4 states, A will be a 4x4 matrix.

3. Final Answer:

   [4 4] -> Option C

4. Quick Check:

   State matrix A size = (states x states) = [4 4] [OK]

Hint: A matrix size is states by states [OK]

Common Mistakes:

• Confusing A size with input or output dimensions
• Assuming A is a vector instead of a matrix
• Mixing up B or C matrix sizes

4. You tried to linearize a nonlinear Simulink model using linmod('modelName') but got an error. What is the most likely cause?

medium

A. The MATLAB path is incorrect

B. The model has no inputs defined

C. The model is already linear

D. The model is not loaded or compiled before linearization

Solution

1. Step 1: Understand linmod requirements

   linmod requires the model to be loaded and compiled to access its operating point and states.

2. Step 2: Identify common error cause

   If the model is not loaded or compiled, linmod throws an error.

3. Final Answer:

   The model is not loaded or compiled before linearization -> Option D

4. Quick Check:

   Model must be loaded before linmod [OK]

Hint: Load and compile model before linmod [OK]

Common Mistakes:

• Assuming no inputs cause error
• Thinking linear model can't be linearized
• Blaming MATLAB path without checking model status

5. You want to design a controller using a linearized model of a nonlinear Simulink system. Which step is essential before controller design?

hard

A. Linearize the nonlinear model at the desired operating point

B. Convert the nonlinear model to a discrete-time model directly

C. Simulate the nonlinear model without linearization

D. Add random noise to the nonlinear model

Solution

1. Step 1: Understand controller design needs

   Controller design often requires a linear model to apply linear control techniques.

2. Step 2: Identify necessary preparation

   Linearizing the nonlinear model at the operating point provides the linear approximation needed.

3. Step 3: Recognize incorrect options

   Direct discretization or simulating nonlinear model alone does not prepare for linear controller design.

4. Final Answer:

   Linearize the nonlinear model at the desired operating point -> Option A

5. Quick Check:

   Linearize before controller design [OK]

Hint: Always linearize model before designing controller [OK]

Common Mistakes:

• Skipping linearization and designing controller on nonlinear model
• Trying to discretize nonlinear model first
• Adding noise instead of linearizing