Linearization of nonlinear models in Simulink

Introduction

Linearization helps us understand and work with complex nonlinear systems by making them simpler and easier to analyze.

When you want to predict how a system behaves near a specific operating point.

When designing controllers that work best with simple linear models.

When you need to simplify a complex model to save time in simulations.

When analyzing stability of a system around a steady state.

When comparing different system designs using a common simple model.

Syntax

Simulink

linmod('model_name')

% or using linearize function
linearize('model_name', operating_point)

linmod is a basic function to get linear models from Simulink models.

linearize is more flexible and works with operating points and Simulink Control Design toolbox.

Examples

This command linearizes the Simulink model named 'my_nonlinear_model' at its default operating point.

Simulink

sys = linmod('my_nonlinear_model')

First, find the operating point of the model, then linearize the model at that point for more accurate results.

Simulink

op = operpoint('my_nonlinear_model');
lin_sys = linearize('my_nonlinear_model', op);

Sample Program

This code loads a sample nonlinear Simulink model, finds its steady operating point, linearizes it there, and shows the linear system matrices.

Simulink

% Load example Simulink model
load_system('sldemo_nonlinear')

% Find operating point at steady state
op = operpoint('sldemo_nonlinear');

% Linearize the model at this operating point
lin_sys = linearize('sldemo_nonlinear', op);

% Display the linear system matrices
disp('State-space matrices of the linearized model:');
disp(lin_sys);

Important Notes

Linearization only approximates the model near the chosen operating point.

Results may not be accurate far from this point.

Make sure the model is configured correctly before linearizing.

Summary

Linearization simplifies nonlinear models around a point.

Use linmod or linearize functions in Simulink.

It helps in analysis and controller design.

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