Linearization of nonlinear models in Simulink - Step-by-Step Execution

Concept Flow - Linearization of nonlinear models

Start with nonlinear model

↓

Select operating point

↓

Calculate Jacobian matrices

↓

Form linear state-space model

↓

Analyze or simulate linear model

↓

End

Linearization takes a nonlinear model, picks a point to approximate around, calculates derivatives (Jacobian), and creates a simpler linear model for analysis.

Execution Sample

Simulink

1. Define nonlinear Simulink model
2. Choose operating point (e.g., steady state)
3. Compute Jacobian matrices A, B, C, D
4. Create linear state-space model
5. Simulate or analyze linear model

This sequence shows how Simulink linearizes a nonlinear model around a chosen operating point.

Execution Table

Step	Action	Input/State	Output/Result
1	Load nonlinear model	Nonlinear equations	Model ready for linearization
2	Select operating point	Initial guess or steady state	Operating point defined
3	Calculate Jacobian matrices	Nonlinear model + operating point	Matrices A, B, C, D computed
4	Form linear state-space model	Jacobian matrices	Linear model created
5	Simulate linear model	Linear model	Linear response output
6	End	Simulation complete	Linear approximation available

💡 Linear model formed and simulation done; linearization process complete

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	Final
Operating Point	None	Defined (e.g., steady state)	Same	Same	Same
Jacobian Matrices (A,B,C,D)	None	None	Computed	Used to form linear model	Same
Linear Model	None	None	None	Created	Simulated

Key Moments - 3 Insights

Why do we need to select an operating point before linearizing?

Because linearization approximates the nonlinear model only near that point, as shown in execution_table step 2 where the operating point is defined before Jacobians are calculated.

What do the Jacobian matrices represent in linearization?

They represent the derivatives of the nonlinear system at the operating point, capturing how small changes affect outputs, as seen in step 3 of the execution_table.

Is the linear model valid for all inputs and states?

No, it is only a good approximation near the operating point chosen, which is why step 5 simulates the linear model to check behavior close to that point.

Visual Quiz - 3 Questions

Look at the execution_table, at which step are the Jacobian matrices computed?

A Step 3

B Step 2

C Step 4

D Step 5

💡 Hint

Check the 'Action' column in execution_table for the step mentioning Jacobian matrices.

According to variable_tracker, when is the linear model created?

A After Step 2

B After Step 3

C After Step 4

D After Step 5

💡 Hint

Look at the 'Linear Model' row and see when it changes from None to Created.

If the operating point is not well chosen, what happens to the linear model's accuracy?

A It remains accurate everywhere

B It becomes less accurate away from the operating point

C It becomes more accurate

D It does not affect accuracy

💡 Hint

Refer to key_moments where the importance of the operating point is explained.

Concept Snapshot

Linearization of nonlinear models:
- Choose an operating point (steady state)
- Calculate Jacobian matrices (A,B,C,D) at that point
- Form linear state-space model
- Use linear model for analysis near operating point
- Valid only close to chosen point

Full Transcript

Linearization of nonlinear models in Simulink involves starting with a nonlinear system, selecting an operating point such as a steady state, then calculating the Jacobian matrices which are derivatives of the system at that point. These matrices form a linear state-space model that approximates the nonlinear system near the operating point. This linear model can then be simulated or analyzed. The process stops once the linear model is created and simulated. Variables like the operating point and Jacobian matrices change during the process, as shown in the variable tracker. Key points include the need to pick a good operating point and understanding that the linear model is only accurate near that point.

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