Interpolatory Projection Techniques for H2 Optimal Structure-Preserving Model Order Reduction of Second-Order Systems

In this paper, we explore the Riccati-based boundary feedback stabilization of the incompressible Navier–Stokes flow via the Krylov subspace techniques. Since the volume of data derived from the original models is gigantic, the feedback stabilization process through the Riccati equation is always infeasible. We apply a ℋ 2 optimal model-order reduction scheme for reduced-order modeling, preserving the sparsity of the system. An extended form of the Krylov subspace-based two-sided iterative algorithm (TSIA) is implemented, where the computation of an equivalent Sylvester equation is included for minimizing the computation time and enhancing the stability of the reduced-order models with satisfying the Wilson conditions. Inverse projection approaches are applied to get the optimal feedback matrix from the reduced-order models. To validate the efficiency of the proposed techniques, transient behaviors of the target systems are observed incorporating the tabular and figurative comparisons with MATLAB simulations. Finally, to reveal the advancement of the proposed techniques, we compare our work with some existing works.

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“…Authors in reference [49] derived the way of estimating H 2 norm of the error system (31) as follows:…”

Section: H 2 Norm Of the Error Systemmentioning

confidence: 99%

Sparsity-Preserving Two-Sided Iterative Algorithm for Riccati-Based Boundary Feedback Stabilization of the Incompressible Navier–Stokes Flow

Islam

Uddin

et al. 2022

Mathematical Problems in Engineering

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“…For the error system (22), the Authors in [5] explored an efficient approach to approximate the ℋ 2 -norm as…”

Section: Computing the Optimal Feedback Matrix From Rommentioning

confidence: 99%

“…The implications of LTI continuous-time systems are inescapable in the branches of engineering fields with the applications of applied mathematics, for example, system and control theory, mechatronics, power electronics [3]- [5]. Continuous-time Algebraic Riccati Equation (CARE) plays a premier role in engineering applications, such as the systems that originated from mechanical and electrical fields [6]- [7] .…”

Section: Introductionmentioning

confidence: 99%

SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models

2021

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We propose an efficient sparsity-preserving reduced-order modelling approach for index-1 descriptor systems extracted from large-scale power system models through two-sided projection techniques. The projectors are configured by utilizing Gramian based singular value decomposition (SVD) and Krylov subspace-based reduced-order modelling. The left projector is attained from the observability Gramian of the system by the low-rank alternating direction implicit (LR-ADI) technique and the right projector is attained by the iterative rational Krylov algorithm (IRKA). The classical LR-ADI technique is not suitable for solving Riccati equations and it demands high computation time for convergence. Besides, in most of the cases, reduced-order models achieved by the basic IRKA are not stable and the Riccati equations connected to them have no finite solution. Moreover, the conventional LR-ADI and IRKA approach not preserves the sparse form of the index-1 descriptor systems, which is an essential requirement for feasible simulations. To overcome those drawbacks, the fitting of LR-ADI and IRKA based projectors from left and right sides, respectively, desired reduced-order systems attained. So that, finite solution of low-rank Riccati equations, and corresponding feedback matrix can be executed. Using the mechanism of inverse projection, the Riccati-based optimal feedback matrix can be computed to stabilize the unstable power system models. The proposed approach will maintain minimized ℌ2 -norm of the error system for reduced-order models of the target models.

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“…The optimal feedback matrix for the system (1) via X can be achieved in plenty of ways. When reduced-order matrices must be stored and used for subsequent manipulations in some of them, optimal feedback matrices can be approximated from the reduced-order feedback matrix using the inverse projection scheme or any other counter approach, such as the Singular-Value Decomposition (SVD)-based Balanced Truncation (BT) [15][16][17][18], the Krylov subspace-based Iterative Rational Krylov Algorithm (IRKA) [19][20][21][22], and a recently developed hybrid approach Iterative SVD-Krylov Algorithm [23][24][25][26]. In those methods, storing the reduced-order matrices claims redundant memory allocation and delays the convergence of the simulation.…”

Section: Introductionmentioning

confidence: 99%

Memory-Efficient Interpolatory Projection Techniques for the Stabilization of Incompressible Navier-Stokes Flows

Uddin,

Khan

2024

Contemp. Math.

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In this article, we are proposing an updated form of Krylov subspace-based interpolatory projection techniques for the stabilization of incompressible Navier-Stokes flows. In the proposed techniques, we utilize the reduced-order modelling approach implicitly, where reduced-order matrices need not be gathered explicitly. To estimate the aimed optimal feedback matrix, only the factored solution of the desired continuous-time algebraic Riccati equation (CARE) needs to be stored through the classical eigenvalue decomposition. The sparse structure of the target systems will remain invariant within the matrix-vector operations for generating the bases of the projector matrices through Krylov subspace techniques, where a cohesive projection scheme will be incorporated to ensure the potency of the projector matrices. So, the proposed techniques will be feasible for memory allocation and enhance the rapid convergence of the simulation. Analysis of the target systems’ transient characteristics, such as eigenvalues and step-responses, will be used to ascertain the competence and reliability of the proposed techniques. Necessary computation will be done numerically through MATLAB. Stabilization of the transient behaviors and minimization of the simulation time are the prime concerns in this work. Eventually, by comparing with the contemporary techniques the advancement of the proposed techniques will be confirmed.

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Interpolatory Projection Techniques for H2 Optimal Structure-Preserving Model Order Reduction of Second-Order Systems

Cited by 8 publications

References 20 publications

Sparsity-Preserving Two-Sided Iterative Algorithm for Riccati-Based Boundary Feedback Stabilization of the Incompressible Navier–Stokes Flow

Sparsity-Preserving Two-Sided Iterative Algorithm for Riccati-Based Boundary Feedback Stabilization of the Incompressible Navier–Stokes Flow

SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models

Memory-Efficient Interpolatory Projection Techniques for the Stabilization of Incompressible Navier-Stokes Flows

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