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SparseBalancedTruncationOptions

Options for sparse model order reduction with balanced truncation method

Since R2023b

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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

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

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 damping, specified as a vector of form [ωn,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 ωn. For best results, pick ωn 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.

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.

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

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.

Relative tolerance for rank decisions, specified as a positive scalar. Increasing this value reduces the ranks of Lr and Lo 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.

An accumulation of pole-zero cancellations near the stability boundary (common in DAEs discretized with Tustin transform) can introduce directions with large gain but low contribution in Lr or Lo. With the default RankTol value, this can result in a low-rank Lr (Lo) that only includes cancelling directions, leading to poor approximations. If Lr has low rank and large norm compared to Lo (or Lo has low rank and large norm compared to Lr), try reducing RankTol to increase the rank and improve the quality of 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.

Version History

Introduced in R2023b

See Also

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