Simulink® Control Design™

What Is Linearization?

Linearization Background

A linearized model is an approximation to a nonlinear system, which is valid in a small region around the operating point of the system. Engineers often use linearization in the design and analysis of control systems and physical models.

The following figure shows a visual representation of a nonlinear system as a block diagram. The diagram consists of an external input signal, u(t), a measured output signal, y(t), and the nonlinear system that describes the system's states and its dynamic behavior, P.

You can also express a nonlinear system in terms of the state space equations

where x(t) represents the system's states, u(t) represents the inputs, and y(t) represents the outputs. In these equations, the variables vary continuously with time. Discrete-time and multi-rate models are discussed in Linearization of Discrete-Time Models in the online documentation. A linear time-invariant approximation to this nonlinear system is valid in a region around the operating point at t=t₀, x(t₀)=x₀, and u(t₀)=u₀. If the values of the system's states, x(t) and inputs, u(t) are close enough to the operating point, the system will behave approximately linearly.

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Linearization of Simulink^® Models

Simulink^® uses a series of connected blocks to model physical systems and control systems. Input and output signals connect the blocks, which represent mathematical operations. The nonlinear system, P, in the previous figure, represents a series of connected Simulink blocks.

The Simulink^® Control Design™ software linearizes both continuous and discrete-time nonlinear systems by computing the state-space matrices of the linearized model, A, B, C, and D, using one of the linearization algorithms described in Understanding Analysis in the Simulink^® Control Design™ Software in the Simulink Control Design User's Guide.

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Linearization Process Overview

The main steps to linearize a model using the Simulink Control Design GUI are as follows:

1. Create or open a model. See Creating or Opening a Simulink^® Model.

2. Create a new linearization task on the Control and Estimation Tools Manager. See Beginning a New Linearization Task.

3. Specify an operating point for the model. See Specifying Operating Points.

4. Insert linearization input and output points in the model. See Configuring Inputs and Outputs for the Linearized Model.

5. Linearize the model. See Linearizing the Model.

6. Inspect and validate the linearization. See Viewing Linearization Results.

7. Save your project and export the results to the MATLAB^® Workspace. See Saving and Exporting Your Work.

The first three steps in this process were completed in the previous chapters. This chapter continues the magball model example to give a detailed discussion of the remaining steps.

You can also use the Simulink Control Design command-line functions to linearize a model. Linearizing Models Using Functions in the Simulink Control Design User's Guide gives detailed information on these functions. For a discussion of the advantages and disadvantages of the GUI versus the command-line interface, refer to Using the GUI Versus Command-Line Functions.

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Tutorial — Linearizing a Plant in a Single-Loop Control System Using the Command Line

Configuring Inputs and Outputs for the Linearized Model