Balanced SP: Low-Frequency Model Reduction

• Balanced SP is a model reduction technique for asymptotically stable LTI systems that enhances low-frequency and near steady-state approximation compared to Balanced Truncation.
• It employs a reciprocal transformation to recast BT operations, enabling low-rank and data-driven implementations with computational efficiency on large-scale systems.
• The method guarantees stability and an a priori H∞ error bound, making it a reliable and practical approach in modern model order reduction pipelines.

Searching arXiv for the target paper and closely related model reduction work. Balanced SP, also termed Balanced Singular Perturbation Approximation (SPA), is a model order reduction method for asymptotically stable linear time-invariant systems that preserves asymptotic stability and admits an a priori $\mathcal H_\infty$ error bound, in close analogy with Balanced Truncation (BT). Its distinguishing approximation profile is that the reduced models obtained by SPA generally introduce better approximation in the lower frequency range and near steady-states, whereas BT is better suited for the higher frequency range. Modern work on Balanced SP has focused on closing the practical gap with BT by deriving low-rank and data-driven implementations whose computational complexity is comparable to established BT counterparts (Liljegren-Sailer et al., 2023).

1. Formal setting and reduction objective

Balanced SP is formulated for the asymptotically stable LTI system

$$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$

with $A\in\mathbb R^{n\times n}$, $B\in\mathbb R^{n\times m}$, $C\in\mathbb R^{p\times n}$, and $D\in\mathbb R^{p\times m}$. Its transfer function is

$H(s)=C\,(sI-A)^{-1}B+D.$

The objective is to construct a reduced-order model of order $r\ll n$,

$$\dot x_r(t)=A_r\,x_r(t)+B_r\,u(t), \qquad y_r(t)=C_r\,x_r(t)+D_r\,u(t),$$

such that $y_r\approx y$ in a system norm such as the $$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$0 norm (Liljegren-Sailer et al., 2023).

In Balanced SP, one first transforms the full model into a balanced realization and partitions the balanced state as $$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$1, with $$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$2 and $$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$3. The method then replaces the fast subsystem by $$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$4, yielding the reduced system

$$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$5

$$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$6

$$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$7

$$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$8

This singular-perturbation form is the defining reduction step of SPA (Liljegren-Sailer et al., 2023).

A central practical motivation for Balanced SP is its frequency-selective behavior. The reported comparison with BT is not that one uniformly dominates the other, but that SPA is particularly aligned with low-frequency and near-steady-state fidelity, while BT remains more often used in practice because efficient algorithmic realizations have historically been more mature (Liljegren-Sailer et al., 2023).

2. Balanced realizations, Gramians, and error guarantees

The theoretical basis of Balanced SP is the balanced realization of the original LTI system. The controllability Gramian $$\dot x(t)=A\,x(t)+B\,u(t), \qquad y(t)=C\,x(t)+D\,u(t),$$9 and observability Gramian $A\in\mathbb R^{n\times n}$0 are the unique symmetric positive-definite solutions of the Lyapunov equations

$A\in\mathbb R^{n\times n}$1

$A\in\mathbb R^{n\times n}$2

The Hankel singular values are defined by

$A\in\mathbb R^{n\times n}$3

ordered as $A\in\mathbb R^{n\times n}$4 (Liljegren-Sailer et al., 2023).

A balanced realization is any realization in which

$A\in\mathbb R^{n\times n}$5

In that coordinate system, the state partition used by Balanced SP is aligned with the decay of the Hankel singular values. This is the same balancing premise underlying BT, but the reduced model is formed by singular perturbation rather than direct truncation (Liljegren-Sailer et al., 2023).

Balanced SP inherits two key guarantees stated in the source material. First, the SPA reduced system is stable. Second, it satisfies exactly the same a priori $A\in\mathbb R^{n\times n}$6 error bound as Balanced Truncation:

$A\in\mathbb R^{n\times n}$7

The coexistence of stability preservation and a certified error bound is one of the main reasons Balanced SP is treated as a principled MOR method rather than a purely heuristic low-frequency approximation (Liljegren-Sailer et al., 2023).

A plausible implication is that Balanced SP occupies a structurally similar theoretical niche to BT while targeting a different approximation regime. The source text makes this comparison explicit by pairing equal error-certification guarantees with different empirical frequency preferences (Liljegren-Sailer et al., 2023).

3. Reciprocal transformation and the reinterpretation of SPA

A central conceptual device in modern implementations of Balanced SP is the reciprocal transformation. Given

$A\in\mathbb R^{n\times n}$8

the reciprocal system is defined by replacing

$A\in\mathbb R^{n\times n}$9

Equivalently,

$B\in\mathbb R^{n\times m}$0

In the frequency domain, this reciprocal system satisfies $B\in\mathbb R^{n\times m}$1, and applying the reciprocal transformation twice recovers the original system (Liljegren-Sailer et al., 2023).

The crucial identity is the classical reinterpretation

$B\in\mathbb R^{n\times m}$2

That is, if BT of order $B\in\mathbb R^{n\times m}$3 is applied to the reciprocal system and the reciprocal is then taken again, the result is exactly the SPA reduced model. The 2023 work identifies this “reciprocal–BT–reciprocal” view as the main tool for deriving practical low-rank and data-driven algorithms, and argues that its significance for practical realization had been overlooked in the literature (Liljegren-Sailer et al., 2023).

This reinterpretation changes implementation strategy. Instead of constructing SPA directly in a dense balanced coordinate system, one can leverage established BT machinery on a transformed problem. This suggests why the paper presents Balanced SP not only as a classical reduction principle but also as a method whose algorithmic bottlenecks can be reformulated into BT-type subroutines (Liljegren-Sailer et al., 2023).

4. Low-rank implementation for large-scale systems

For large-scale systems, the paper derives a low-rank SPA algorithm designed to parallel the standard square-root or low-rank BT workflow. The starting point is to compute low-rank approximate Cholesky factors

$B\in\mathbb R^{n\times m}$4

with $B\in\mathbb R^{n\times m}$5 and $B\in\mathbb R^{n\times m}$6. These factors may be obtained by low-rank ADI, sign-function methods, or related techniques. One then forms the skinny SVD

$B\in\mathbb R^{n\times m}$7

and uses it to define the BT-style projection bases

$B\in\mathbb R^{n\times m}$8

The standard projected BT realization is

$B\in\mathbb R^{n\times m}$9

These ingredients are then repurposed through the reciprocal viewpoint to obtain SPA (Liljegren-Sailer et al., 2023).

The low-rank SPA algorithm proceeds as follows. After computing the low-rank Lyapunov factors and the dominant singular triplets, it forms the balanced bases

$C\in\mathbb R^{p\times n}$0

It then solves the two reduced linear systems

$C\in\mathbb R^{p\times n}$1

and builds an intermediate reciprocal reduced-order model:

$C\in\mathbb R^{p\times n}$2

$C\in\mathbb R^{p\times n}$3

$C\in\mathbb R^{p\times n}$4

$C\in\mathbb R^{p\times n}$5

Finally, applying the reciprocal once more gives the SPA model:

$C\in\mathbb R^{p\times n}$6

The defining implementation feature is that the full inverse of $C\in\mathbb R^{p\times n}$7 is never formed (Liljegren-Sailer et al., 2023).

The total reported complexity is

$C\in\mathbb R^{p\times n}$8

which is stated to be commensurate with low-rank BT. This directly addresses the historical practical disadvantage of SPA relative to BT in the large-scale regime (Liljegren-Sailer et al., 2023).

5. Data-driven and realization-free QuadSPA

Balanced SP also admits a data-driven implementation when only input-output data or transfer-function samples are available. The source derives this formulation from quadrature-based Gramian approximations used for BT, but applies them to the reciprocal system while expressing the needed quantities again in terms of the original transfer function $C\in\mathbb R^{p\times n}$9 (Liljegren-Sailer et al., 2023).

The key object is an $D\in\mathbb R^{p\times m}$0 Loewner-style data matrix constructed from transfer-function samples. Given quadrature nodes $D\in\mathbb R^{p\times m}$1 and $D\in\mathbb R^{p\times m}$2, the matrix entries are

$D\in\mathbb R^{p\times m}$3

Two additional data matrices are defined as

$D\in\mathbb R^{p\times m}$4

The paper states that the $D\in\mathbb R^{p\times m}$5 largest singular triplets of $D\in\mathbb R^{p\times m}$6 give exactly the SPA projection for the reciprocal system (Liljegren-Sailer et al., 2023).

The resulting realization-free algorithm, termed QuadSPA in the detailed description, is:

1. Compute $D\in\mathbb R^{p\times m}$7 and $D\in\mathbb R^{p\times m}$8.
2. Form the data matrices $D\in\mathbb R^{p\times m}$9, $H(s)=C\,(sI-A)^{-1}B+D.$0, and $H(s)=C\,(sI-A)^{-1}B+D.$1.
3. Compute the rank-$H(s)=C\,(sI-A)^{-1}B+D.$2 SVD of $H(s)=C\,(sI-A)^{-1}B+D.$3.
4. Form the reduced intermediate reciprocal model

$H(s)=C\,(sI-A)^{-1}B+D.$4

$H(s)=C\,(sI-A)^{-1}B+D.$5

$H(s)=C\,(sI-A)^{-1}B+D.$6

$H(s)=C\,(sI-A)^{-1}B+D.$7

1. Apply the reciprocal formulas to obtain the final SPA model (Liljegren-Sailer et al., 2023).

Since these steps require only transfer-function samples and an $H(s)=C\,(sI-A)^{-1}B+D.$8 SVD, the construction is completely non-intrusive. This is the realization-free counterpart to intrusive large-scale SPA, and it places Balanced SP in the same computational ecosystem as quadrature-based, data-driven BT methods (Liljegren-Sailer et al., 2023).

6. Computational profile, numerical behavior, and relation to BT

The 2023 study frames both proposed implementations as efforts to make SPA practical in the same settings where BT is already standard. Low-rank SPA is reported to have essentially the same cost as low-rank BT, with the dominant operations being a small number of large-scale solves for Lyapunov factors together with two extra multi-right-hand-side solves for the reciprocal step. QuadSPA is dominated by constructing and computing the SVD of an $H(s)=C\,(sI-A)^{-1}B+D.$9 matrix, with complexity $r\ll n$0 (Liljegren-Sailer et al., 2023).

The numerical behavior summarized in the source has two parts. First, experiments on rail and ISS benchmarks indicate that for $r\ll n$1 up to $r\ll n$2, SPA can be computed in a few seconds, and this is described as orders of magnitude faster than dense solvers or Matlab’s balred. Second, QuadSPA is reported to accurately reproduce the SPA frequency-response curves, including the low-frequency advantage, once $r\ll n$3 is sufficiently large. In practice, the source states that $r\ll n$4 suffices for orders $r\ll n$5, keeping CPU times in the sub-second to second range and comparable to quadBT (Liljegren-Sailer et al., 2023).

The relation to BT is therefore dual. On the one hand, Balanced SP remains distinct in approximation behavior: SPA generally introduces better approximation in the lower frequency range and near steady-states, whereas BT is better suited for the higher frequency range. On the other hand, the practical algorithms now deliberately mimic BT in structure, complexity, and implementation style. A plausible implication is that the main barrier to SPA adoption is no longer theoretical but infrastructural: once reciprocal-based formulations are used, SPA can be integrated into large-scale and data-driven MOR pipelines without sacrificing the stability and $r\ll n$6-bound guarantees that traditionally motivated balanced methods (Liljegren-Sailer et al., 2023).