reducespec

Create model order reduction specifications

Since R2023b

Syntax

R = reducespec(sys,method)

Description

The reducespec function is the entry point for model order reduction workflows in Control System Toolbox™ and Robust Control Toolbox™ software. Use this function to create a model order reduction (MOR) task based on the model type and selected method.

For information on configuring model order reduction tasks, see the object corresponding to your model type and selected algorithm. For details on how to select orders and obtain reduced-order models, see the corresponding view and getrom functions.

Algorithm	Supported Models	Objects	Object Functions
Balanced truncation	Nonsparse and sparse LTI models	BalancedTruncation SparseBalancedTruncation	process view (balanced) getrom (balanced)
Balanced truncation of normalized coprime factors (NCF)	Nonsparse LTI models	NCFBalancedTruncation	process view (ncf) getrom (ncf)
Modal truncation	Nonsparse and sparse LTI models	ModalTruncation SparseModalTruncation	process view (modal) getrom (modal)
Proper orthogonal decomposition (since R2024b)	Nonsparse and sparse LTI models	ProperOrthogonalDecomposition	process view (pod) getrom (pod)

This function creates only the MOR specification object and does not perform any computation. This allows you to properly configure options before you run the MOR algorithm, which can be computationally expensive in the case of sparse models.

Tip

For the full workflow, see Task-Based Model Order Reduction Workflow.

R = reducespec(sys,method) creates a model order reduction (MOR) specification object for a dense or sparse linear time-invariant (LTI) model sys. method is the model order reduction algorithm, specified as one of the following:

• "balanced"

• "ncf" (Requires Robust Control Toolbox software)

• "modal"

• "pod"

Examples

Model Order Reduction Using Task-Based Workflow

This example shows the basic workflow for obtaining reduced-order models using the task-based model order reduction workflow.

Generate a random discrete-time state-space model with 40 states.

rng(0)
sys = drss(40);

Create a balanced truncation model-order reduction task using reducespec.

R = reducespec(sys,"balanced");

The function returns a BalancedTruncation object.

View the graphical information to help decide the reduced order using the view function.

view(R)

For balanced truncation, view plots the state contributions in terms of the Hankel singular values.

Obtain the reduced order model using getrom. For this example, select an order 15.

rsys = getrom(R,Order=15);

Compare the Bode response of both models.

bode(sys,rsys,'r--')
legend("Original","Order 15")

Input Arguments

sys — Dynamic system model

dynamic system model

Dynamic system model, specified as an ordinary or sparse LTI model.

This model must have a valid state-space representation, such as a tf, ss, sparss, or mechss model. For generalized or uncertain state-space models (genss, uss), the function uses the current value of the model. For identified models (idss), the function uses the identified value.

method — Model order reduction algorithm

"balanced" | "ncf" | "modal" | "pod" (since R2024b)

Model order reduction algorithm, specified as one of these strings:

• "balanced" — Use the balanced truncation algorithm. This method quantifies the state contributions to the I/O transfer and lets you discard states with weak contribution.

• "ncf" — Use the balanced truncation algorithm applied to the normalized coprime factors (NCF) of sys. This method requires Robust Control Toolbox software and is available only for ordinary LTI models.

• "modal" — Use the modal truncation algorithm. This method computes the system modes (poles) and lets you discard modes based on their location or DC contribution.

• "pod" — Use the proper orthogonal decomposition algorithm. This method uses simulation data to compute and extract the dominant modes (principal components) of the state vector. (since R2024b)

Output Arguments

R — Model order reduction specification object

BalancedTruncation object | SparseBalancedTruncation object | NCFBalancedTruncation object | ModalTruncation object | SparseModalTruncation object | ProperOrthogonalDecomposition object

Model order reduction specification object, returned as one of the objects in this table. The type of object depends on the model type and algorithm.

Model Type (sys)	Algorithm (method)	Object (R)
LTI	"balanced"	BalancedTruncation
	"ncf"	NCFBalancedTruncation
	"modal"	ModalTruncation
	"pod"	ProperOrthogonalDecomposition
Sparse LTI	"balanced"	SparseBalancedTruncation
	"modal"	SparseModalTruncation
	"pod"	ProperOrthogonalDecomposition

Limitations

Modal Truncation

• Sparse modal truncation in discrete-time is only applicable if A+E is definite.

• Performs poorly when DC contributions of modes have varying signs and cancellations occur (extreme case is models with zero DC gain). Modal approximation will then tend to have much higher DC gain, which is not acceptable.

• Performs poorly for models whose dynamics are packed in a narrow band. Discarded modes must be reasonably separated from retained modes.

Proper Orthogonal Decomposition

• POD method is applicable only to stable LTI models.

• The Galerkin algorithm is recommended specifically for symmetric positive definite problems with A = A^T, E = E^T ≥ 0 or M = M^T ≥ 0, C = C^T, K = K^T. The algorithm may perform poorly for asymmetric problems because it only takes into account the input-to-state map.

Tips

• Use findop to compute matching steady-state initial conditions for the reduced-order model.

References

[1] Benner, Peter, Jing-Rebecca Li, and Thilo Penzl. “Numerical Solution of Large-Scale Lyapunov Equations, Riccati Equations, and Linear-Quadratic Optimal Control Problems.” Numerical Linear Algebra with Applications 15, no. 9 (November 2008): 755–77. https://doi.org/10.1002/nla.622.

[2] Benner, Peter, Martin Köhler, and Jens Saak. “Matrix Equations, Sparse Solvers: M-M.E.S.S.-2.0.1—Philosophy, Features, and Application for (Parametric) Model Order Reduction.” In Model Reduction of Complex Dynamical Systems, edited by Peter Benner, Tobias Breiten, Heike Faßbender, Michael Hinze, Tatjana Stykel, and Ralf Zimmermann, 171:369–92. Cham: Springer International Publishing, 2021. https://doi.org/10.1007/978-3-030-72983-7_18.

[3] Varga, A. “Balancing Free Square-Root Algorithm for Computing Singular Perturbation Approximations.” In [1991] Proceedings of the 30th IEEE Conference on Decision and Control, 1062–65. Brighton, UK: IEEE, 1991. https://doi.org/10.1109/CDC.1991.261486.

[4] Green, M. “A Relative Error Bound for Balanced Stochastic Truncation.” IEEE Transactions on Automatic Control 33, no. 10 (October 1988): 961–65. https://doi.org/10.1109/9.7255.

Version History

Introduced in R2023b

R2024b: Compute low-order models using Proper Orthogonal Decomposition method

You can now compute reduced-order models 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.

To perform simulations, extract state-snapshot data, and approximate the controllability and observability Gramians, the software provides two types of built-in excitation signals (impulse and chirp). By default, the software uses an impulse excitation signal. Alternatively, you can provide:

• State-snapshot data obtained from Simulink^® or a third-party software.

• Custom POD data generated from a simulation with lsim and incrementalPOD.

R2024b: Support for modal truncation of nonsymmetric sparse models

You can now perform modal truncation of nonsymmetric sparss and mechss models. Previously, sparse modal truncation was limited to first-order models with A = A^T and E = E^T definite, or second-order models with K = K^T, M = M^T definite, and Rayleigh-type damping.

R2024b: Support for balanced truncation of discrete-time sparse models

You can now perform balanced truncation of sparss and mechss models in discrete-time.

R2024a: Support for modal truncation of discrete-time sparse models

You can now perform modal truncation of sparss and mechss models in discrete-time. Sparse modal truncation in discrete-time is only applicable if A+E is definite.