The proposed work integrates the Arnoldi algorithm within the Krylov Subspace Method (<Emphasis Type="BoldItalic">KSM</Emphasis>) framework to construct the reduced-order model (<Emphasis Type="BoldItalic">ROM</Emphasis>) of a commensurate-order fractional-order system. First, the original commensurate fractional-order model (<Emphasis Type="BoldItalic">FOM</Emphasis>) is transformed into an equivalent integer-order system (<Emphasis Type="BoldItalic">IOS</Emphasis>) through a suitable domain transformation. This conversion enables the use of well-established linear system reduction techniques. On the transformed <Emphasis Type="BoldItalic">IOS</Emphasis>, a Krylov subspace is generated to approximate the system’s dominant dynamics. The Arnoldi algorithm is then employed to construct an orthonormal basis for this Krylov subspace. By iteratively building a Hessenberg matrix representation of the large system matrix, Arnoldi efficiently captures the most influential spectral properties while preserving numerical stability. Thus, Arnoldi complements <Emphasis Type="BoldItalic">KSM</Emphasis> by ensuring stable and computationally efficient basis construction, while <Emphasis Type="BoldItalic">KSM</Emphasis> provides the theoretical foundation for projection-based model reduction. After reduction, the <Emphasis Type="BoldItalic">IOS</Emphasis>-based <Emphasis Type="BoldItalic">ROM</Emphasis> is transformed back into fractional-order form using the inverse transformation, yielding a compact fractional-order <Emphasis Type="BoldItalic">ROM</Emphasis>. The main contribution of the work is to propose an Arnoldi-enhanced Krylov subspace framework tailored for commensurate-order fractional-order systems via domain transformation. The method achieves improved approximation accuracy in both time and frequency domains, as validated through performance indices such as Integral Absolute Error (<Emphasis Type="BoldItalic">IAE</Emphasis>), Integral of Time Weighted Absolute Error (<Emphasis Type="BoldItalic">ITAE</Emphasis>), Integral Time Square Error (<Emphasis Type="BoldItalic">ITSE</Emphasis>), Integral Square Error (<Emphasis Type="BoldItalic">ISE</Emphasis>), and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {H}_{\infty }\)</EquationSource> </InlineEquation> norm. Comparative studies demonstrate that the proposed Arnoldi–<Emphasis Type="BoldItalic">KSM</Emphasis> approach produces reduced models that closely replicate the dynamics of high-order fractional systems, outperforming existing <Emphasis Type="BoldItalic">ROM</Emphasis> techniques in terms of accuracy and compactness. To test the effectiveness of the proposed method, two examples of higher order <Emphasis Type="BoldItalic">FOM</Emphasis> are simulated in MATLAB.

Reduced order modeling of commensurate fractional order systems using Arnoldi–Krylov subspace optimization technique

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发表日期：

2026年4月13日

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Anirban Bose,
Arindam Mondal,
Chandan Das

摘要

The proposed work integrates the Arnoldi algorithm within the Krylov Subspace Method (KSM) framework to construct the reduced-order model (ROM) of a commensurate-order fractional-order system. First, the original commensurate fractional-order model (FOM) is transformed into an equivalent integer-order system (IOS) through a suitable domain transformation. This conversion enables the use of well-established linear system reduction techniques. On the transformed IOS, a Krylov subspace is generated to approximate the system’s dominant dynamics. The Arnoldi algorithm is then employed to construct an orthonormal basis for this Krylov subspace. By iteratively building a Hessenberg matrix representation of the large system matrix, Arnoldi efficiently captures the most influential spectral properties while preserving numerical stability. Thus, Arnoldi complements KSM by ensuring stable and computationally efficient basis construction, while KSM provides the theoretical foundation for projection-based model reduction. After reduction, the IOS-based ROM is transformed back into fractional-order form using the inverse transformation, yielding a compact fractional-order ROM. The main contribution of the work is to propose an Arnoldi-enhanced Krylov subspace framework tailored for commensurate-order fractional-order systems via domain transformation. The method achieves improved approximation accuracy in both time and frequency domains, as validated through performance indices such as Integral Absolute Error (IAE), Integral of Time Weighted Absolute Error (ITAE), Integral Time Square Error (ITSE), Integral Square Error (ISE), and \(\mathcal {H}_{\infty }\) norm. Comparative studies demonstrate that the proposed Arnoldi–KSM approach produces reduced models that closely replicate the dynamics of high-order fractional systems, outperforming existing ROM techniques in terms of accuracy and compactness. To test the effectiveness of the proposed method, two examples of higher order FOM are simulated in MATLAB.