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.

• Proper orthogonal decomposition (POD) — Use simulation data to compute and extract the dominant modes (principal components) of the state vector and perform an approximate balanced truncation. (since R2025a)

• Frequency response fitting — Fit low-order models to the frequency response data of large sparse state-space models. (since R2026a)

• Zero-pole truncation — Obtain a zero-pole-gain approximation of sparse state-space models in a desired low-frequency band. (since R2026a)

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.

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

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.

Balanced Truncation and Proper Orthogonal Decomposition (since R2025a) Tabs

Reduction criteria — Model order reduction criteria

Order | Maximum error | Minimum energy | Maximum loss

Specify the model order reduction criteria, specified as one of the following:

• Order — Desired order of the reduced-order model, specified as a nonnegative scalar or vector.

• Maximum error — Maximum approximation error, specified as a nonnegative scalar or vector. The app selects the order using the error bound $${\Vert G_s-G_r\Vert }_{\infty }$$ < R.Error(r+1) = $$2{\displaystyle \sum _{j=r+1}^n{\sigma }_j}$$ which is not exact and not always tight. As a result, smaller orders may still meet the specified Maximum error, so you can try smaller orders when seeking the lowest possible order compatible with Maximum error.

• Minimum energy — Minimum bound on the normalized energy, specified as a nonnegative scalar or vector. The app discards all the states that have normalized energies lower than the specified value.

• Maximum loss — Maximum energy loss as a fraction of total energy, specified as a nonnegative scalar or vector. The app selects the lowest order with loss below the specified value.

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 scalar value, Model Reducer computes and displays the responses of a model of that order. If you specify a vector, 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.

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 $${\Vert G^{-1}\left(G-G_r\right)\Vert }_{\infty }$$.

Modal Truncation Tab

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 canceling fewer poles and zeros. Moving the slider to the right, or entering a larger value, simplifies the model more by canceling 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 $${\Vert G^{-1}\left(G-G_r\right)\Vert }_{\infty }$$.

Frequency Response Fitting Tab

Since R2026a

Enforce fit stability — Return stable state-space model

"on" (default) | "off"

Enable this option to return a stable state-space model. Stability is encoded in the frequency response, but coarse frequency sampling may lose this information and the software may produce unstable fits. When you enable this option, the software enforces stability on the model, but this can degrade the fit quality. You can rectify this by increasing the frequency grid density near the underdamped poles (resonances in the frequency response). However, the more frequency points you have, the longer it takes to evaluate the frequency response.

The software ignores this option if the fit is improper.

Minimize fit order — Minimize fit order while maintaining desired accuracy

"off" (default) | "on"

When you enable this option, the software applies balanced truncation on the initial fit to eliminate redundant dynamics. By default, the software is not guaranteed produce a minimal order fit especially for multi-input multi-output models. Therefore, enable this option to minimize the fit order.

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 $${\Vert G^{-1}\left(G-G_r\right)\Vert }_{\infty }$$.

Zero-Pole Truncation Tab

Since R2026a

Roll-off slope — Roll-off slope

nonpositive scalar | matrix

Minimum roll-off slope, specified as a nonpositive scalar or matrix.

• Use a scalar value for SISO models or when the slope is uniform for all input-output pairs for MIMO models.

• Use a matrix when the slope is different for each input-output pair for MIMO models.

For example, Roll-off slope of -2 ensures the gain rolls off at a rate of at least –40 dB/decade (the roll-off rate of 1/s²).

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 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) 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) opens the app and imports the specified models.

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.

More About

Choosing Model Reduction Method

Method	Description	When to Use	Algorithms
Balanced Truncation	Remove states with relatively small energy contributions.	When you want to obtain lower-order approximation of your model by neglecting states that have relatively low effect on the overall model response. This method measures state contributions by Hankel singular values and discards states with smaller values.	BalancedTruncation Algorithms NCFBalancedTruncation Algorithms SparseBalancedTruncation Algorithms
Modal Truncation	Discard poles based on their locations or DC contributions.	When you want to obtain low-order approximations by discarding poles based on locations, damping, or DC contributions. This method is useful when you want to focus your analysis on a particular subset of system dynamics.	ModalTruncation Algorithms SparseModalTruncation Algorithms
Pole-Zero Simplification	Eliminate canceling or near-canceling pole-zero pairs.	When you want to reduce model order exactly by canceling pole-zero pairs or eliminating states that have no effect on the overall model response. This method is applicable to only nonsparse LTI models.	Pole-zero cancellation is a straightforward search through the poles and zeros looking for matches that are within tolerance. Transfer functions are first converted to zero-pole-gain form.
Proper Orthogonal Decomposition (POD) (since R2025a)	Use simulation data to compute and extract the dominant modes (principal components) of the state vector and perform an approximate balanced truncation.	When you want to obtain low-order approximation of stable LTI models using simulation data. This method takes snapshots of the state vector during simulation and uses principal component analysis (PCA) to obtain principal directions. Since POD is a simulation-based technique, it is more efficient when reducing models with smaller number of inputs and outputs.	ProperOrthogonalDecomposition Algorithms
Frequency Response Fitting (since R2026a)	Fit low-order models to the frequency response data of large sparse state-space models.	When you want to obtain low-order approximations of sparse LTI models in the frequency band of interest. The frequency response fitting method is applicable to all types of sparse models with fewer limitations than Balanced Truncation or POD. In particular, it is effective at reducing models with unstable or undamped poles. Frequency response fitting is also applicable to non-sparse models but is usually less effective than Balanced Truncation for such models.	FrequencyResponseFitting Algorithms
Zero-Pole Truncation (since R2026a)	Obtain a zero-pole-gain approximation of sparse state-space models in a desired low-frequency band.	When you want to obtain low-order zero-pole-gain approximations of sparse state-space models. This method computes a subset of the zeros and poles of sparse models, typically in a specific low-frequency band [0 fmax]. It can yield better low-frequency approximations than modal truncation at the expense of more computation. This method also provides direct control over the roll-off slope past the frequency fmax. Because this method calculates zeros for each input-output pair, it is most suitable for models with small input-output sizes	SparseZeroPoleTruncation Algorithms

Version History

Introduced in R2016a

R2026a: Compute Truncated Zero-Pole-Gain Approximation of Sparse Models

You can use the new zero-pole truncation method to obtain low-order zero-pole-gain approximations of sparse state-space models. This method computes a subset of the zeros and poles of sparse models, typically in a specific low-frequency band [0 f_max]. It can yield better low-frequency approximations than modal truncation at the expense of more computation. This method also provides direct control over the roll-off slope past the frequency f_max. Because this method calculates zeros for each input-output pair, it is most suitable for models with small input-output sizes. Additionally, this method is applicable only to models with a valid sparss representation.

R2026a: Fit low-order models to frequency-response data of large sparse models

You can now use the new frequency response fitting method to fit low-order models to the frequency response data of large sparse state-space models. The frequency response fitting method is applicable to all types of sparse models. In particular, it is useful when you want to reduce unstable or damped models. This method uses the AAA algorithm, which is an interpolation algorithm to perform rational fitting of the frequency response data.

R2025a: Compute reduced order models using Proper Orthogonal Decomposition method

You can now compute reduced-order models for sparse and ordinary LTI models interactively using the proper orthogonal decomposition (POD) method. POD is a simulation-based technique to extract dominant state directions and perform an approximate balanced truncation. This method takes snapshots of the state vector during simulation and uses principal component analysis (PCA) to obtain principal directions.

R2025a: Balanced Truncation: App displays analysis plot based on reduction criteria

Model Reducer now automatically selects the analysis plot type based on the model reduction criteria for balanced truncation and proper orthogonal decomposition methods. For example, when you set Reduction criteria to Minimum energy the app automatically shows the plot for normalized state energies.

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.