Multimodel Control Design

Typically, the dynamics of a system are not known exactly and may vary. For example, system dynamics can vary because of:

Parameter value variations caused by manufacturing tolerances — For example, the resistance value of a resistor is typically within a range about the nominal value, 5 Ω +/– 5%.
Operating conditions — For example, aircraft dynamics change based on altitude and speed.

Any controller you design for such a system must satisfy the design requirements for all potential system dynamics.

Control Design Overview

To design a controller for a system with varying dynamics:

Sample the variations.
Create an LTI model for each sample.
Create an array of sampled LTI models.
Design a controller for a nominal representative model from the array.
Analyze the controller design for all models in the array.
If the controller design does not satisfy the requirements for all the models, specify a different nominal model and redesign the controller.

Model Arrays

InControl System Designer, you can specify multiple models for any plant or sensor in the current control architecture using an array of LTI models (seeModel Arrays). If you specify model arrays for more than one plant or sensor, the lengths of the arrays must match.

Create Model Arrays

To create arrays for multimodel control design, you can:

Create multiple LTI models using thetf,ss,zpk, orfrdcommands.

% Specify model parameters.m = 3; b = 0.5; k = 8:1:10; T = 0.1:.05:.2;% Create an array of LTI models.forct = 1:length(k); G(:,:,ct) = tf(1,[m,b,k(ct)]);end

Create an array of LTI models using thestackcommand.

% Create individual LTI models.G1 =特遣部队(1,(1 1 8));G2 =特遣部队(1,(1 1 9));G3 =特遣部队(1, [1 1 10]);% Combine models in an array.G = stack(1,G1,G2,G3);

Perform batch linearizations at multiple operating points. Then export the computed LTI models to create an array of LTI models. See the exampleReference Tracking of DC Motor with Parameter Variations(Simulink Control Design).
Sample an uncertain state-space (uss) model usingusample(Robust Control Toolbox).
Compute aussmodel from a Simulink^®model. Then useusubs(Robust Control Toolbox)orusample(Robust Control Toolbox)to create an array of LTI models. SeeObtain Uncertain State-Space Model from Simulink Model(Robust Control Toolbox).
Specify a core Simulink block to linearize to auss(Robust Control Toolbox)orufrd(Robust Control Toolbox)model. SeeSpecify Uncertain Linearization for Core or Custom Simulink Blocks(Robust Control Toolbox).

Import Model Arrays to Control System Designer

To import models as arrays, you can pass them as input arguments when openingControl System Designerfrom the MATLAB^®command line. For more information, seeControl System Designer.

You can also import model arrays intoControl System Designer当配置控制架构。在Edit Architecture dialog box:

在Valuetext box, specify the name of an LTI model from the MATLAB workspace.
To import block data from the MATLAB workspace or from a MAT-file in your current working directory, click.

Nominal Model

What Is a Nominal Model?

Thenominal modelis a representative model in the array of LTI models that you use to design the controller inControl System Designer. Use the editor and analysis plots to visualize and analyze the effect of the controller on the remaining plants in the array.

You can select any model in the array as your nominal model. For example, you can choose a model that:

Represents the expected nominal operating point of your system.
Is an average of the models in the array.
Represents a worst-case plant.
Lies closest to the stability point.

Tip

You can plot and analyze the open-loop dynamics of the system on a Bode plot to determine which model to choose as nominal.

Specify Nominal Model

To select a nominal model from the array of LTI models, inControl System Designer, clickMultimodel Configuration. Then, in the Multimodel Configuration dialog box, select aNominal model index. The default index is1.

For each plant or sensor that is defined as a model array, the app selects the model at the specified index as the nominal model. Otherwise, the app uses scalar expansion to apply the single LTI model for all model indices.

For example, for the following control architecture:

ifGandHare both three-element arrays and the nominal model index is2, the software uses the second element in both the arrays to compute the nominal model:

The nominal response fromrtoyis:

$T = \frac{C G_{2}}{1 + C G_{2} H_{2}}$

The app also computes and plots the responses showing the effect ofCon the remaining pairs of plant and sensor models —G₁H₁andG₃H₃.

If onlyGis an array of LTI models, and the specified nominal model is 2, then the control architecture for nominal response is:

In this case, the nominal response fromrtoyis:

$T = \frac{C G_{2}}{1 + C G_{2} H}$

The app also computes and plots the responses showing the effect ofCon the remaining pairs of plant and sensor model —G₁HandG₃H.

Frequency Grid

The frequency response of a system is computed at a series of frequency values, called afrequency grid. By default,Control System Designercomputes a logarithmically equally spaced grid based on the dynamic range of each model in the array.

Specify a custom frequency grid when:

The automatic grid has more points than you require. To improve computational efficiency, specify a less dense grid spacing.
The automatic grid is not sufficiently dense within a particular frequency range. For example, if the response does not capture the resonant peak dynamics of an underdamped system, specify a more dense grid around the corner frequency.
You are only interested in the response within specific frequency ranges. To improve computational efficiency, specify a grid that covers only the frequency ranges of interest.

For more information on specifying logarithmically spaced vectors, seelogspace.

Note

Modifying the frequency grid does not affect the frequency response computation for the nominal model. The app always uses theAuto selectoption to compute the nominal model frequency response.

Design Controller for Multiple Plant Models

This example shows how to design a compensator for a set of plant models usingControl System Designer.

Create Array of Plant Models

Create an array of LTI plant models using thestackcommand.

% Create an array of LTI models to model plant (G) variations.G1 = tf(1,[1 1 8]); G2 = tf(1,[1 1 9]); G3 = tf(1,[1 1 10]); G = stack(1,G1,G2,G3);

Create Array of Sensor Models

Similarly, create an array of sensor models.

H1 = tf(1,[1/0.1,1]); H2 = tf(1,[1/0.15,1]); H3 = tf(1,[1/0.2,1]); H = stack(1,H1,H2,H3);

Open Control System Designer
OpenControl System Designer,并导入植物和传感器阵列模型。
```
controlSystemDesigner(G,1,H)
```
The app opens and imports the plant and sensor model arrays.
Configure Analysis Plot
To view the closed-loop step response in a larger plot, inControl System Designer, click on the small dropdown arrow on theIOTransfer_r2y: stepplot and then selectMaximize.

By default the step response shows only the nominal response. To display the individual responses for the other model indices, right-click the plot area, and selectMultimodel Configuration>Individual Responses.

Note

To view an envelope of all model responses, right-click the plot area, and selectMultimodel Configuration>Bounds

The plot updates to display the responses for the other models.
Select Nominal Model
On theControl Systemtab, clickMultimodel Configuration.
在Multimodel Configuration dialog box, specify aNominal Model Indexof2.

ClickClose.

The selected nominal model corresponds to the average system response.
Design Compensator
To design a compensator using the nominal model, you can use any of the supportedControl System Designer Tuning Methods.
For this example, use the Compensator Editor to manually specify the compensator dynamics. Add an integrator to the compensator and set the compensator gain to0.4. For more information, seeEdit Compensator Dynamics.
Analyze Results
The tuned controller produces a step response with minimal overshoot for the nominal models and a worst-case overshoot less than 10%.