## Linearization Techniques for Control Design

Learn how to use Simulink® and Simscape™ tools together to model physical processes and design control systems. Linearizing electrical models is an important step in a wider control design workflow. You can use different techniques to linearize electrical models and tune controllers.

### Benefits of Linearization

When you linearize a model, you create a linear approximation of a linear or nonlinear system. This approximation is valid in a small region around a particular operating or trim point, a steady-state condition in which all model states are constant. You need to use linearization to evaluate or design control systems using classical design techniques. By linearizing models, you can:

• Use tools intended for linear controller design.

• Determine the stability of control systems using Bode plots and other frequency analysis plots.

• Create reduced-order models, which have lower computational requirements and run faster than their corresponding nonlinear models.

After you generate a linear model, you can optimize controllers for electrical systems so that the controller:

• Responds faster.

• Minimizes overshoot.

• Reduces oscillations.

• Minimizes reactive power consumed.

• Maximizes efficiency.

• Increases disturbance rejection.

• Is more robust. For example, a controller that has better gain and phase margins.

#### Determine Stability of Control Systems Using Bode Analysis

You can use Bode plots to assess the stability of a control system. A Bode plot describes the frequency response of a linear, time-invariant system.

To generate Bode plots and achieve your desired closed-loop system performance by graphically shaping the open-loop frequency response, you can use these functions and apps:

You can also calculate the magnitude and phase from the state-space representations and generate Bode plots by running a series of MATLAB® commands. To learn how you can use these tools for Bode analysis, see Examples.

### Linearization Techniques

Some models are continuous, meaning that they have no abrupt discontinuities such as saturations or dead-bands. You can linearize continuous models easily. For example, you can use the steady-state solver to find an operating point and the `linmod` function to extract a linear model. To learn how you can linearize continuous models, see Examples.

Note

The `linmod ` function provides basic linearization functionality. For full linearization functionality, use Simulink Control Design™ software. For more information about the tools you can use for linearization, see Choose Linearization Tools (Simulink Control Design).

Discontinuities are common in electrical networks because switches have a binary, on or off, state. For example, circuits driven by pulse-width modulation usually have high-frequency switching components, such as the metal-oxide-semiconductor field-effect transistor (MOSFET). You cannot linearize models with discontinuities exactly.

To find a linear approximation of models with discontinuities, you first need to approximate the discontinuous signal with a continuous approximation. If your model contains a converter block, you can use the `Averaged Switch` option to simplify your model so that you can linearize it. For more information about converter blocks which support the `Averaged Switch` option, see Linearize Models with Converters Using Averaged Switching.

For more complex models with discontinuities, you can estimate a linear model from the results of a dynamic simulation. For more information, see Estimate Linear Models from Simulation Results.

### Examples

These examples explore the linearization and analysis techniques that you can apply to models without discontinuities. These techniques include using the `linmod` function and the Model Linearizer app to extract a linear model.

### Tuning Controllers

MathWorks® provides tools that you can use to tune your controller after you generate a linear model. These examples show two different techniques for tuning the current and velocity loops in a linear electric actuator with saturation limits:

You can also automatically tune the gains of a proportional-integral-derivative (PID) controller for a single-input-single-output (SISO) plant to achieve a balance between performance and robustness, using the PID Tuner (Control System Toolbox) app. If the PID Tuner app cannot linearize the plant at the operating point you want to use for tuning, you can use the Frequency Response Based PID Tuner app. The Frequency Response Based PID Tuner app finds a linear approximation of your model and tunes your controller by running a dynamic simulation. You can apply this method to models with discontinuities. For more information about the Frequency Response Based PID Tuner app, see Frequency-Response Based Tuning (Simulink Control Design). For more information about advanced techniques for tuning controllers that you can apply to models with discontinuities, see Tune Controllers by Running Dynamic Simulation.