SparseBalancedTruncationOptions

Options for sparse model order reduction with balanced truncation method

Since R2023b

Description

This object contains model order reduction options of sparse linear time-invariant models, and is contained in the Options property of a SparseBalancedTruncation object R created using reducespec. To configure these options, use dot notation, for example, R.Options.MaxRank = 1500.

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

Properties

Focus — Dynamic range

[0 Inf] (default) | vector

Dynamic range of interest, specified as a vector of form [fmix,fmax]. Use this option if you know all poles are in this range.

Offset — Spectral offset

0 (default) | positive scalar

Spectral offset, specified as a positive scalar.

Sparse balanced truncation is only supported for stable systems. For first-order models (sparss) with integral action, you can use this option to implicitly shift poles to enforce stability. The algorithm shifts poles as follows.

• Continuous time — p to p-Offset

• Discrete time — p to (1-Offset)p

This option is not supported for mechss models.

Rayleigh — Rayleigh damping

[] (default) | vector

Rayleigh damping, specified as a vector of form [ω_n,ζ].

Sparse balanced truncation is supported only for stable systems. To enforce stability for undamped second-order models (mechss), you can use this option to implicitly add Rayleigh damping with minimum damping ζ at the frequency ω_n. For best results, pick ω_n close to the dominant mode and ζ in the range [0.001,0.1].

For an example, see Add Rayleigh Damping to Enforce Stability for Sparse Balanced Truncation.

This option is not supported for sparss models.

CustomShift — Shifts to accelerate convergence

[] (default) | vector

Custom shifts to accelerate the convergence of the algorithm, specified as an n-by-1 vector.

Specify shifts based on prior knowledge of pole locations. The algorithm applies these shifts in addition to the default shifts. See [1] and [2] for details.

MaxRank — Maximum rank of Cholesky factors

5000 (default) | positive integer

Maximum rank of Cholesky factors, specified as a positive integer. The algorithm terminates when the column size of the low-rank Gramian factors L_r and L_o reaches this limit.

LyapTol — Relative tolerance for Lyapunov residuals

1e-8 (default) | positive scalar

Relative tolerance for Lyapunov residuals, specified as a positive scalar. Increasing LyapTol helps speed up computation at the expense of reduced-order model accuracy. Decrease this value to capture more Hankel singular values.

RankTol — Relative tolerance for rank decisions

1e-8 (default) | positive scalar

Relative tolerance for rank decisions, specified as a positive scalar. Increasing this value reduces the ranks of L_r and L_o and results in less accurate reduced-order models. Decreasing this value helps compute small Hankel singular values more accurately and obtain more accurate reduced-order models.

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.

Version History

Introduced in R2023b