<p>Quadrature-based approximation of Gramians in standard balanced truncation yields a non-intrusive, data-driven implementation that requires only transfer function samples on the imaginary axis, which can be measured experimentally. This idea has recently been extended to several generalizations of balanced truncation, including positive-real balanced truncation, bounded-real balanced truncation, and balanced stochastic truncation. However, these extensions require samples of some spectral factorizations on the imaginary axis, and no practical method exists to obtain such data experimentally. As a result, these non-intrusive implementations are mainly of theoretical interest at present. This paper shows that if the Gramians in these generalizations are approximated via rational interpolation rather than numerical integration, the resulting non-intrusive implementations do not require spectral factorization samples. Instead, they rely only on transfer function samples. Based on this idea, non-intrusive implementations are first developed for several variants of balanced truncation, wherein the Gramians are approximated implicitly using low-rank Alternating Direction Implicit (ADI) methods for Lyapunov and Riccati equations. These formulations include Linear Quadratic Gaussian (LQG) balanced truncation, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {H}_\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> balanced truncation, positive-real balanced truncation, bounded-real balanced truncation, self-weighted balanced truncation, and balanced stochastic truncation. They require transfer function samples in the right half of the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s$</EquationSource> <EquationSource Format="MATHML"><math> <mi>s</mi> </math></EquationSource> </InlineEquation>-plane, which cannot be measured experimentally. Next, building on these results, novel data-driven non-intrusive implementations are proposed that require only transfer function samples on the imaginary axis. Hence, unlike the quadrature-based and ADI-based approaches, these non-intrusive formulations can be implemented using practically measurable data. Numerical results are presented for benchmark problems in model order reduction, which show that the proposed non-intrusive implementations achieve accuracy comparable to their intrusive counterparts.</p>

Data-driven implementations of various generalizations of balanced truncation

期刊论文
发表日期：

2026年5月11日

查看全文

Umair Zulfiqar,
Qiu-Yan Song,
Zhi-Hua Xiao,
Victor Sreeram

摘要

Quadrature-based approximation of Gramians in standard balanced truncation yields a non-intrusive, data-driven implementation that requires only transfer function samples on the imaginary axis, which can be measured experimentally. This idea has recently been extended to several generalizations of balanced truncation, including positive-real balanced truncation, bounded-real balanced truncation, and balanced stochastic truncation. However, these extensions require samples of some spectral factorizations on the imaginary axis, and no practical method exists to obtain such data experimentally. As a result, these non-intrusive implementations are mainly of theoretical interest at present. This paper shows that if the Gramians in these generalizations are approximated via rational interpolation rather than numerical integration, the resulting non-intrusive implementations do not require spectral factorization samples. Instead, they rely only on transfer function samples. Based on this idea, non-intrusive implementations are first developed for several variants of balanced truncation, wherein the Gramians are approximated implicitly using low-rank Alternating Direction Implicit (ADI) methods for Lyapunov and Riccati equations. These formulations include Linear Quadratic Gaussian (LQG) balanced truncation, $\mathcal {H}_\infty $ $H_{\infty}$ balanced truncation, positive-real balanced truncation, bounded-real balanced truncation, self-weighted balanced truncation, and balanced stochastic truncation. They require transfer function samples in the right half of the $s$ $s$ -plane, which cannot be measured experimentally. Next, building on these results, novel data-driven non-intrusive implementations are proposed that require only transfer function samples on the imaginary axis. Hence, unlike the quadrature-based and ADI-based approaches, these non-intrusive formulations can be implemented using practically measurable data. Numerical results are presented for benchmark problems in model order reduction, which show that the proposed non-intrusive implementations achieve accuracy comparable to their intrusive counterparts.