## 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

In **Control System Designer**, you can specify multiple models for any
plant or sensor in the current control architecture using an array of LTI models
(see Model 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 the

`tf`

,`ss`

,`zpk`

, or`frd`

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

Create an array of LTI models using the

`stack`

command.% Create individual LTI models. G1 = tf(1, [1 1 8]); G2 = tf(1, [1 1 9]); G3 = tf(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 example Reference Tracking of DC Motor with Parameter Variations (Simulink Control Design).

Sample an uncertain state-space (

`uss`

) model using`usample`

(Robust Control Toolbox).Compute a

`uss`

model from a Simulink^{®}model. Then use`usubs`

(Robust Control Toolbox) or`usample`

(Robust Control Toolbox) to create an array of LTI models. See Obtain Uncertain State-Space Model from Simulink Model (Robust Control Toolbox).Specify a core Simulink block to linearize to a

`uss`

(Robust Control Toolbox) or`ufrd`

(Robust Control Toolbox) model. See Specify 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 opening
**Control System Designer** from the MATLAB^{®} command line. For more information, see **Control System Designer**.

You can also import model arrays into **Control System Designer** when
configuring the control architecture. In the Edit Architecture dialog
box:

In the

**Value**text 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?

The *nominal model* is a representative model in the
array of LTI models that you use to design the controller in **Control 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, in **Control System
Designer**, click **Multimodel Configuration**. Then,
in the Multimodel Configuration dialog box, select a **Nominal model
index**. The default index is `1`

.

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:

if *G* and *H* are both three-element arrays
and the nominal model index is `2`

, the software uses the
second element in both the arrays to compute the nominal model:

The nominal response from *r* to *y*
is:

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

The app also computes and plots the responses showing the effect of
*C* on the remaining pairs of plant and sensor models
—
*G _{1}*

*H*

_{1}and

*G*

_{3}

*H*

_{3}.

If only *G* is 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 from *r* to
*y* is:

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

The app also computes and plots the responses showing the effect of
*C* on the remaining pairs of plant and sensor model
— *G*_{1}*H* and
*G*_{3}*H*.

### Frequency Grid

The frequency response of a system is computed at a series of frequency values,
called a *frequency grid*. By default, **Control System
Designer** computes 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, see `logspace`

.

**Note**

Modifying the frequency grid does not affect the frequency response
computation for the nominal model. The app always uses the **Auto
select** option 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 using
**Control System Designer**.

**Create Array of Plant Models**Create an array of LTI plant models using the

`stack`

command.`% 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**Open

**Control System Designer**, and import the plant and sensor model arrays.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, in

**Control System Designer**, click on the small dropdown arrow on the**IOTransfer_r2y: step**plot and then select`Maximize`

.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 select

**Multimodel Display**>**Individual Responses**.**Note**To view an envelope of all model responses, right-click the plot area, and select

**Multimodel Display**>**Bounds**The plot updates to display the responses for the other models.

**Select Nominal Model**On the

**Control System**tab, click**Multimodel Configuration**.In the Multimodel Configuration dialog box, specify a

**Nominal Model Index**of`2`

.Click

**Close**.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 supported Control 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 to

`0.4`

. For more information, see Edit 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%.