arXiv subjects

Zhi-Hua Xiao

Publications and source records attributed to Zhi-Hua Xiao.

7 recordsLinked to original sources

arXiv2026

Data-driven Implementations of Various Generalizations of Balanced Truncation

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 require transfer function samples in the right half of the $s$-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.

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arXiv2025

$\mathcal{H}_2$-optimal Model Reduction of Linear Quadratic Output Systems in Finite Frequency Range

In frequency-limited model order reduction, the objective is to maintain the frequency response of the original system within a specified frequency range in the reduced-order model. In this paper, a mathematical expression for the frequency-limited $\mathcal{H}_2$ norm is derived, which quantifies the error within the desired frequency interval. Subsequently, the necessary conditions for a local optimum of the frequency-limited $\mathcal{H}_2$ norm of the error are derived. The inherent difficulty in satisfying these conditions within a Petrov-Galerkin projection framework is also discussed. Using the optimality conditions and the Petrov-Galerkin projection, a stationary point iteration algorithm is proposed, which approximately satisfies these optimality conditions upon convergence. The main computational effort in the proposed algorithm involves solving sparse-dense Sylvester equations. These equations are frequently encountered in $\mathcal{H}_2$ model order reduction algorithms and can be solved efficiently. Moreover, the algorithm bypasses the requirement of matrix logarithm computation, which is typically necessary for most frequency-limited reduction methods and can be computationally demanding for high-order systems. An illustrative example is provided to numerically validate the developed theory. The proposed algorithm's effectiveness in accurately approximating the original high-order model within the specified frequency range is demonstrated through the reduction of an advection-diffusion equation-based model, commonly used in model reduction literature for testing algorithms. Additionally, the algorithm's computational efficiency is highlighted by successfully reducing a flexible space structure model of order one million.

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arXiv2025

Balanced Truncation via Tangential Interpolation

This paper examines the construction of rth-order truncated balanced realizations via tangential interpolation at r specified interpolation points. It is demonstrated that when the truncated Hankel singular values are negligible-that is, when the discarded states are nearly uncontrollable and unobservable-balanced truncation simplifies to a bi-tangential Hermite interpolation problem at r interpolation points. In such cases, the resulting truncated balanced realization is nearly H2-optimal and thus interpolates the original model at the mirror images of its poles along its residual directions. Like standard H2-optimal model reduction, where the interpolation points and tangential directions that yield a local optimum are not known, in balanced truncation as well, the interpolation points and tangential directions required to produce a truncated balanced realization remain unknown. To address this, we propose an iterative tangential interpolation-based algorithm for balanced truncation. Upon convergence, the algorithm yields a low-rank truncated balanced realization that accurately preserves the r largest Hankel singular values of the original system. An adaptive scheme to automatically select the order r of the reduced model is also proposed. The algorithm is fully automatic, choosing both the interpolation data and the model order without user intervention. Additionally, an adaptive low-rank solver for Lyapunov equations based on tangential interpolation is proposed, automatically selecting both the interpolation data and the rank without user intervention. The performance of the proposed algorithms is evaluated on benchmark models, confirming their efficacy.

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arXiv2024

Time-limited H2-optimal Model Order Reduction of Linear Systems with Quadratic Outputs

An important class of dynamical systems with several practical applications is linear systems with quadratic outputs. These models have the same state equation as standard linear time-invariant systems but differ in their output equations, which are nonlinear quadratic functions of the system states. When dealing with models of exceptionally high order, the computational demands for simulation and analysis can become overwhelming. In such cases, model order reduction proves to be a useful technique, as it allows for constructing a reduced-order model that accurately represents the essential characteristics of the original high-order system while significantly simplifying its complexity. In time-limited model order reduction, the main goal is to maintain the output response of the original system within a specific time range in the reduced-order model. To assess the error within this time interval, a mathematical expression for the time-limited $\mathcal{H}_2$-norm is derived in this paper. This norm acts as a measure of the accuracy of the reduced-order model within the specified time range. Subsequently, the necessary conditions for achieving a local optimum of the time-limited $\mathcal{H}_2$ norm error are derived. The inherent inability to satisfy these optimality conditions within the Petrov-Galerkin projection framework is also discussed. After that, a stationary point iteration algorithm based on the optimality conditions and Petrov-Galerkin projection is proposed. Upon convergence, this algorithm fulfills three of the four optimality conditions. To demonstrate the effectiveness of the proposed algorithm, a numerical example is provided that showcases its ability to effectively approximate the original high-order model within the desired time interval.

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arXiv2024

Balanced Truncation of Linear Systems with Quadratic Outputs in Limited Time and Frequency Intervals

Model order reduction involves constructing a reduced-order approximation of a high-order model while retaining its essential characteristics. This reduced-order model serves as a substitute for the original one in various applications such as simulation, analysis, and design. Often, there's a need to maintain high accuracy within a specific time or frequency interval, while errors beyond this limit can be tolerated. This paper addresses time-limited and frequency-limited model order reduction scenarios for linear systems with quadratic outputs, by generalizing the recently introduced structure-preserving balanced truncation algorithm. To that end, limited interval system Gramians are defined, and the corresponding generalized Lyapunov equations governing their computation are derived. Additionally, low-rank solutions for these equations are investigated. Next, balanced truncation algorithms are proposed for time-limited and frequency-limited scenarios, each utilizing its corresponding limited-interval system Gramians. The proposed algorithms ensure accurate results within specified time and frequency intervals while preserving the quadratic-output structure. Two benchmark numerical examples are presented to demonstrate the effectiveness of the algorithms, showcasing their ability to achieve superior accuracy within the desired time or frequency interval.

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arXiv2023

Frequency-limited H$_2$ Model Order Reduction Based on Relative Error

Frequency-limited model order reduction aims to approximate a high-order model with a reduced-order model that maintains high fidelity within a specific frequency range. Beyond this range, a decrease in accuracy is acceptable due to the nature of the problem. The quality of the reduced-order model is typically evaluated using absolute or relative measures of approximation error. Relative error, which represents the percentage error, becomes particularly relevant when reducing a plant model for the purpose of designing a reduced-order controller. This paper derives the necessary conditions for achieving a local optimum of the frequency-limited H2 norm for the relative error system. Based on these optimality conditions, an oblique projection algorithm is proposed to ensure a small relative error within the desired frequency interval. Unlike existing algorithms, the proposed approach does not necessitate solving large-scale Lyapunov and Ricatti equations. Instead, the proposed algorithm relies on solving sparse-dense Sylvester equations, which typically emerge in the majority of H2 model order reduction algorithms, but can be efficiently solved. To evaluate the performance of the proposed algorithm, a comparison is conducted with three existing techniques: frequency-limited balanced truncation, frequency-limited balanced stochastic truncation, and frequency-limited iterative Rational Krylov algorithm. The comparative analysis focuses on designing reduced-order controllers for high-order plants. Numerical results confirm that the reduced-order controllers obtained using the proposed algorithm ensure superior robust closed-loop stability.

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arXiv2022

Relative Error-based Time-limited H2 Model Order Reduction via Oblique Projection

In time-limited model order reduction, a reduced-order approximation of the original high-order model is obtained that accurately approximates the original model within the desired limited time interval. Accuracy outside that time interval is not that important. The error incurred when a reduced-order model is used as a surrogate for the original model can be quantified in absolute or relative terms to access the performance of the model reduction algorithm. The relative error is generally more meaningful than an absolute error because if the original and reduced systems' responses are of small magnitude, the absolute error is small in magnitude as well. However, this does not necessarily mean that the reduced model is accurate. The relative error in such scenarios is useful and meaningful as it quantifies percentage error irrespective of the magnitude of the system's response. In this paper, the necessary conditions for a local optimum of the time-limited H2 norm of the relative error system are derived. Inspired by these conditions, an oblique projection algorithm is proposed that ensures small H2-norm relative error within the desired time interval. Unlike the existing relative error-based model reduction algorithms, the proposed algorithm does not require solutions of large-scale Lyapunov and Riccati equations. The proposed algorithm is compared with time-limited balanced truncation, time-limited balanced stochastic truncation, and time-limited iterative Rational Krylov algorithm. Numerical results confirm the superiority of the proposed algorithm over these existing algorithms.

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