Model Reducer

Reduce complexity of linear time-invariant (LTI) models

Description

The Model Reducer app lets you compute reduced-order approximations of high-order LTI and sparse LTI models. Working with lower-order models can simplify analysis and control design. Simpler models are also easier to understand and manipulate. You can reduce a plant model to focus on relevant dynamics before designing a controller for the plant. Or, you can use model reduction to simplify a full-order controller.

Using any of the following methods, Model Reducer helps you reduce model order while preserving model characteristics that are important to your application:

• Balanced truncation — Remove states with relatively small energy contributions.

• Modal truncation — Discard modes based on their locations or DC contributions.

• Pole-zero simplification — Eliminate canceling or near-canceling pole-zero pairs.

Model Reducer provides response plots and error plots to help ensure that the reduced-order model preserves important dynamics. For more information on model reduction and why it is useful, see Model Reduction Basics.

For an alternative to the Model Reducer app that lets you interactively perform model reduction and generate code for a live script, see the Reduce Model Order task in the Live Editor.

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Open the Model Reducer App

• MATLAB^® Toolstrip: On the Apps tab, under Control System Design and Analysis, click the app icon.

• MATLAB command prompt: Enter modelReducer.

Examples

• Model Reduction Basics
• Reduce Model Order Using Model Reducer App
• Balanced Truncation Model Reduction
• Pole-Zero Simplification
• Modal Truncation Model Reduction

Parameters

Balanced Truncation Tab

Model — Currently selected model for reduction

model name

Specify the model you want to reduce by selecting from the Model drop-down list. The list includes all models currently in the data browser. To get a model from the MATLAB workspace into the data browser, on the Model Reducer tab, click Import Model. You can import any:

• tf, ss, or zpk model that is proper. The model can be SISO or MIMO, and continuous or discrete.

  • Continuous-time models must not have time delays. To reduce a continuous-time model with time delays, first use pade to approximate the time delays as model dynamics.

  • Discrete-time models can have time delays. For the Balanced Truncation reduction method, the app uses absorbDelay to convert the delay into poles at z = 0 before reducing the model. The additional states are reflected in the response plot and Hankel singular-value plot.

• Generalized model such as a genss model. The Model Reducer app uses the current or nominal value of all control design blocks in model (see getValue).

• Sparse state-space models such as sparss and mechss models.

Note

Model Reducer assumes that the model time unit (specified in the TimeUnit property of the model) is seconds. If your model does not have TimeUnit = 'seconds', use chgTimeUnit to convert the model to seconds.

Reduction criteria — Number of states in reduced model

Order | Maximum error | Minimum energy

Specify the model order reduction criteria. Any value is permitted that falls between the number of unstable states in the model and the number of states in the original model. If you specify a single value, Model Reducer computes and displays the responses of a model of that order. If you specify multiple values, Model Reducer computes models of all specified orders and displays their responses on the same plot. To store reduced models in the data browser, click Save Reduced Model.

For more information, see Balanced Truncation Model Reduction.

Comparison plot — Type of comparison plot

Model response | Absolute error plot | Relative error plot

Specify the comparison plot type.

• Model response — Plot the model frequency response. This frequency response comparison is a Bode plot for SISO models, and a singular-value plot for MIMO models.

• Absolute error plot — Plot the frequency response of absolute error $${\Vert G-G_r\Vert }_{\infty }$$.

• Relative error plot — Plot the frequency response of relative error $${\Vert G^{-1}\left(G-G_r\right)\Vert }_{\infty }$$.

For more information, see Balanced Truncation Model Reduction.

Analysis plot — Type of analysis plot

Hankel singular values (default) | Energy

Specify the analysis plot type.

• Hankel singular values — Bar chart of Hankel singular values and associated error bounds.

• Energy — Bar chart of normalized state energies.

Modal Truncation Tab

Model — Currently selected model for reduction

model name

Specify the model you want to reduce by selecting from the Model drop-down list. The list includes all models currently in the data browser. To get a model from the MATLAB workspace into the data browser, on the Model Reducer tab, click Import Model. You can import any:

• tf, ss, or zpk model that is proper. The model can be SISO or MIMO, and continuous or discrete.

  • Continuous-time models must not have time delays. To reduce a continuous-time model with time delays, first use pade to approximate the time delays as model dynamics.

  • Discrete-time models can have time delays. For the Balanced Truncation reduction method, the app uses absorbDelay to convert the delay into poles at z = 0 before reducing the model. The additional states are reflected in the response plot and Hankel singular-value plot.

• Generalized model such as a genss model. The Model Reducer app uses the current or nominal value of all control design blocks in model (see getValue).

• Sparse state-space models such as sparss and mechss models.

For more information, see Modal Truncation Model Reduction.

Note

Reduce Model Order assumes that the model time unit (specified in the TimeUnit property of the model) is seconds. If your model does not have TimeUnit = 'seconds', use chgTimeUnit to convert the model to seconds.

Frequency Range — Frequency range

two-element vector

Frequency range of interest, specified as a vector of form [F_min,F_max]. The algorithm discards all the modes outside this range.

Damping Range — Damping range

two-element vector

Damping range of interest, specified as a vector of form [ζ_min,ζ_max]. The algorithm discards all the modes outside this range.

Minimum DC Contribution — Minimum DC contribution bound

nonnegative scalar

Minimum DC contribution bound for the reduced-order model, specified as a nonnegative scalar. The algorithm discards all the modes with normalized DC contributions smaller than this value.

Comparison plot — Type of comparison plot

Model response | Absolute error plot | Relative error plot | Mode locations

Specify the comparison plot type.

• Model response — Plot the model frequency response. This frequency response comparison is a Bode plot for SISO models, and a singular-value plot for MIMO models.

• Absolute error plot — Plot the frequency response of absolute error $${\Vert G-G_r\Vert }_{\infty }$$.

• Relative error plot — Plot the frequency response of relative error $${\Vert G^{-1}\left(G-G_r\right)\Vert }_{\infty }$$.

• Mode locations — Compare the pole locations of original and reduced models.

Analysis plot — Type of analysis plot

DC contribution (default) | Mode location | Mode damping and natural frequency

Specify the comparison plot type.

• DC contribution — Bar chart of normalized DC contributions.

• Mode location — Plot the location of the poles.

• Mode damping and natural frequency — Plot the damping and natural frequencies of the poles.

Pole/Zero Simplification Tab

Model — Currently selected model for reduction

model name

Specify the model you want to reduce by selecting from the Model drop-down list. The list includes all models currently in the data browser. To get a model from the MATLAB workspace into the data browser, on the Model Reducer tab, click Import Model. You can import any:

• tf, ss, or zpk model that is proper. The model can be SISO or MIMO, and continuous or discrete.

  • Continuous-time models must not have time delays. To reduce a continuous-time model with time delays, first use pade to approximate the time delays as model dynamics.

  • Discrete-time models can have time delays. For the Balanced Truncation reduction method, the app uses absorbDelay to convert the delay into poles at z = 0 before reducing the model. The additional states are reflected in the response plot and Hankel singular-value plot.

• Generalized model such as a genss model. The Model Reducer app uses the current or nominal value of all control design blocks in model (see getValue).

Simplification of Pole-Zero Pairs — Tolerance for pole-zero cancellation

positive scalar

Set the tolerance for pole-zero cancellation by using the slider or entering a value in the text box. The value determines how close together a pole and zero must be for Model Reducer to eliminate them from the reduced model. Moving the slider to the left or entering a smaller value in the text box simplifies the model less, by cancelling fewer poles and zeros. Moving the slider to the right, or entering a larger value, simplifies the model more by cancelling poles and zeros that are further apart.

For more information, see Pole-Zero Simplification.

Comparison plot — Type of comparison plot

Model response | Absolute error plot | Relative error plot

Specify the comparison plot type.

• Model response — Plot the model frequency response. This frequency response comparison is a Bode plot for SISO models, and a singular-value plot for MIMO models.

• Absolute error plot — Plot the frequency response of absolute error $${\Vert G-G_r\Vert }_{\infty }$$.

• Relative error plot — Plot the frequency response of relative error $${\Vert G^{-1}\left(G-G_r\right)\Vert }_{\infty }$$.

Programmatic Use

modelReducer

modelReducer opens the Model Reducer app with no models in the data browser. To import a model from the MATLAB workspace, click Import Model.

modelReducer(model)

modelReducer(model) opens app and imports the specified LTI model. model can be a:

• tf, ss, or zpk model that is proper. The model can be SISO or MIMO, and continuous or discrete.

  • Continuous-time models must not have time delays. To reduce a continuous-time model with time delays, first use pade to approximate the time delays as model dynamics.

  • Discrete-time models can have time delays. For the Balanced Truncation reduction method, the app uses absorbDelay to convert the delay into poles at z = 0 before reducing the model. The additional states are reflected in the response plot and Hankel singular-value plot.

• Generalized model such as a genss model. The Model Reducer app uses the current or nominal value of all control design blocks in model (see getValue).

• Sparse state-space models such as sparss and mechss models.

modelReducer(model1,...,modelN)

modelReducer(model1,...,modelN) opens the app and imports the specified models.

modelReducer(sessionFile)

modelReducer(sessionFile) opens the app and loads a previously saved session. sessionFile is the name of a session data file in the current working directory or on the MATLAB path.

To save session data to disk, in the Model Reducer app, on the Model Reducer tab, click Save Session. The saved session data includes the current plot configuration and all models in the data browser.

Version History

Introduced in R2016a

R2024a: Obtain reduced order models for sparss and mechss models

You can now compute reduced-order models of sparse state-space models interactively using the Model Reducer app. The software supports sparse model order reduction using these methods:

• Balanced truncation — Obtain low-order approximation by discarding states with low contribution.

• Modal truncation — Obtain low-order approximation by discarding undesired modes.

R2024a: Mode Selection method is now Modal Truncation

The Modal Truncation method replaces the Mode Selection method in the Model Reducer app. Modal truncation method provides better flexibility for choosing the criteria for discarding modes.

R2023b: Generates code using new model order reduction workflow

The Model Reducer app now generates code using the new model order reduction workflows. For example, this table describes the change in the model order reduction workflow in the generated code.

Method	Generated Code before R2023b	Generated Code in R2023b
Balanced Truncation	%% Reduce LTI model order using balanced truncation System = G; % Define System to reduce Order = 14; % Create option set for balred command Options = balredOptions(); % Offset for the stable/unstable boundary Options.Offset = 1e-05; % Compute reduced order approximation ReducedSystem = balred(System,Order,Options); % Create comparison plot bode(System,ReducedSystem);	%% Reduce LTI model order using balanced truncation System = G; % Define System to reduce % Compute reduced order approximation R = reducespec(System,'balanced'); % Set options for Balanced Truncation specification % Offset for the stable/unstable boundary R.Options.Offset = 1e-05; % Compute MOR data once R = process(R); % Get reduced-order model ReducedSystem = getrom(R,Order=14); % Create comparison plot bode(System,ReducedSystem);
Mode Selection	%% Reduce LTI model order using mode selection System = G; % Define System to reduce UpperCutoffFrequency = 100; LowerCutoffFrequency = 10; % Create option set for freqsep command Options = freqsepOptions(); % Accuracy loss factor for stable/unstable decomposition Options.SepTol = 100; % Select modes between lower and upper cutoff frequencies ReducedSystem = freqsep(System,... [LowerCutoffFrequency UpperCutoffFrequency],Options); % Create comparison plot bode(System,ReducedSystem);	%% Reduce LTI model order using mode selection System = G; % Define System to reduce % Select modes between lower and upper cutoff frequencies R = reducespec(System,'modal'); % Set options for Modal Truncation specification % Accuracy loss factor for stable/unstable decomposition R.Options.SepTol = 1e-11; % Compute MOR data once R = process(R); % Get reduced-order model ReducedSystem = getrom(R,Frequency=[10 100],Method='truncate'); % Create comparison plot bode(System,ReducedSystem);

For more information about the new workflow, see reducespec and Task-Based Model Order Reduction Workflow.