scholarly journals Solution formulas for differential Sylvester and Lyapunov equations

CALCOLO ◽  
2019 ◽  
Vol 56 (4) ◽  
Author(s):  
Maximilian Behr ◽  
Peter Benner ◽  
Jan Heiland
Keyword(s):  
Large Scale ◽  
Krylov Subspace ◽  
Applied Mathematics ◽  
Order Reduction ◽  
Lyapunov Equation ◽  
Spectral Theorem ◽  
Lyapunov Equations ◽  
Normal Operators ◽  
Symmetric Version ◽  
Induced Operator

AbstractThe differential Sylvester equation and its symmetric version, the differential Lyapunov equation, appear in different fields of applied mathematics like control theory, system theory, and model order reduction. The few available straight-forward numerical approaches when applied to large-scale systems come with prohibitively large storage requirements. This shortage motivates us to summarize and explore existing solution formulas for these equations. We develop a unifying approach based on the spectral theorem for normal operators like the Sylvester operator $${\mathcal {S}}(X)=AX+XB$$S(X)=AX+XB and derive a formula for its norm using an induced operator norm based on the spectrum of A and B. In view of numerical approximations, we propose an algorithm that identifies a suitable Krylov subspace using Taylor series and use a projection to approximate the solution. Numerical results for large-scale differential Lyapunov equations are presented in the last sections.

The extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations

2021 ◽  
Vol 8 (3) ◽  
pp. 526-536
Author(s):  
L. Sadek ◽  
◽  
H. Talibi Alaoui ◽  
Keyword(s):  
Matrix Equation ◽  
Large Scale ◽  
Krylov Subspace ◽  
Low Rank ◽  
Lyapunov Equations ◽  
Lanczos Algorithm ◽  
Lyapunov Matrix ◽  
Block Lanczos ◽  
Low Dimensional ◽  
Differential Lyapunov Equations

In this paper, we present a new approach for solving large-scale differential Lyapunov equations. The proposed approach is based on projection of the initial problem onto an extended block Krylov subspace by using extended nonsymmetric block Lanczos algorithm then, we get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix equation is solved by the Backward Differentiation Formula method (BDF) or Rosenbrock method (ROS), the obtained solution allows to build a low-rank approximate solution of the original problem. Moreover, we also give some theoretical results. The numerical results demonstrate the performance of our approach.

scholarly journals AN ITERATIVE MODEL ORDER REDUCTION METHOD FOR LARGE-SCALE DYNAMICAL SYSTEMS

2017 ◽  
Vol 59 (1) ◽  
pp. 115-133
Author(s):  
K. MOHAMED ◽  
A. MEHDI ◽  
M. ABDELKADER
Keyword(s):  
Dynamical Systems ◽  
Model Order Reduction ◽  
Large Scale ◽  
Reduction Method ◽  
Linear Time ◽  
Order Reduction ◽  
Lyapunov Equation ◽  
Model Order ◽  
Iterative Model ◽  
Order Reduction Method

We present a new iterative model order reduction method for large-scale linear time-invariant dynamical systems, based on a combined singular value decomposition–adaptive-order rational Arnoldi (SVD-AORA) approach. This method is an extension of the SVD-rational Krylov method. It is based on two-sided projections: the SVD side depends on the observability Gramian by the resolution of the Lyapunov equation, and the Krylov side is generated by the adaptive-order rational Arnoldi based on moment matching. The use of the SVD provides stability for the reduced system, and the use of the AORA method provides numerical efficiency and a relative lower computation complexity. The reduced model obtained is asymptotically stable and minimizes the error ($H_{2}$and$H_{\infty }$) between the original and the reduced system. Two examples are given to study the performance of the proposed approach.

scholarly journals A Broadband Enhanced Structure-Preserving Reduced-Order Interconnect Macromodeling Method for Large-Scale Equation Sets of Transient Interconnect Circuit Problems

Energies ◽  
2020 ◽  
Vol 13 (21) ◽  
pp. 5746
Author(s):  
Ning Wang ◽  
Huifang Wang ◽  
Shiyou Yang
Keyword(s):  
Large Scale ◽  
Krylov Subspace ◽  
Equivalent Circuit Model ◽  
Order Reduction ◽  
Wide Band ◽  
Circuit Model ◽  
Band Frequency ◽  
Reduced Order ◽  
Structure Preserving ◽  
Expansion Point

In the transient analysis of an engineering power electronics device, the order of its equivalent circuit model is excessive large. To eliminate this issue, some model order reduction (MOR) methods are proposed in the literature. Compared to other MOR methods, the structure-preserving reduced-order interconnect macromodeling (SPRIM) based on Krylov subspaces will achieve a higher reduction radio and precision for large multi-port Resistor-Capacitor-Inductor (RCL) circuits. However, for very wide band frequency transients, the performance of a Krylov subspace-based MOR method is not satisfactory. Moreover, the selection of the expansion point in this method has not been comprehensively studied in the literature. From this point of view, a broadband enhanced structure-preserving reduced-order interconnect macromodeling (SPRIM) method is proposed to reduce the order of equation sets of a transient interconnect circuit model. In addition, a method is introduced to determine the optimal expansion point at each frequency in the proposed method. The proposed method is validated by the numerical results on a transient problem of an insulated-gate bipolar transistor (IGBT)-based inverter busbar under different exciting conditions.

scholarly journals Basic principles and aims of model order reduction in compliant mechanisms

2011 ◽  
Vol 2 (2) ◽  
pp. 197-204 ◽  
Author(s):  
M. Rösner ◽  
R. Lammering
Keyword(s):  
Model Order Reduction ◽  
Large Scale ◽  
Krylov Subspace ◽  
Compliant Mechanisms ◽  
Linear Time ◽  
Order Reduction ◽  
Research Direction ◽  
Future Research ◽  
Model Order ◽  
Advantages And Disadvantages

Abstract. Model order reduction appears to be beneficial for the synthesis and simulation of compliant mechanisms due to computational costs. Model order reduction is an established method in many technical fields for the approximation of large-scale linear time-invariant dynamical systems described by ordinary differential equations. Based on system theory, underlying representations of the dynamical system are introduced from which the general reduced order model is derived by projection. During the last years, numerous new procedures were published and investigated appropriate to simulation, optimization and control. Singular value decomposition, condensation-based and Krylov subspace methods representing three order reduction methods are reviewed and their advantages and disadvantages are outlined in this paper. The convenience of applying model order reduction in compliant mechanisms is quoted. Moreover, the requested attributes for order reduction as a future research direction meeting the characteristics of compliant mechanisms are commented.

scholarly journals Model order reduction of linear time invariant systems

2008 ◽  
Vol 6 ◽  
pp. 129-132 ◽  
Author(s):  
Lj. Radić-Weissenfeld ◽  
S. Ludwig ◽  
W. Mathis ◽  
W. John
Keyword(s):  
Krylov Subspace ◽  
Linear Time ◽  
Order Reduction ◽  
Efficient Solutions ◽  
Lyapunov Equations ◽  
Linear Time Invariant ◽  
Weak Part ◽  
Multipoint Approximation ◽  
Linear Time Invariant Systems ◽  
Order Reduction Method

Abstract. This paper addresses issues related to the order reduction of systems with multiple input/output ports. The order reduction is divided up into two steps. The first step is the standard order reduction method based on the multipoint approximation of system matrices by applying Krylov subspace. The second step is based on the rejection of the weak part of a system. To recognise the weak system part, Lyapunov equations are used. Thus, this paper introduces efficient solutions of the Lyapunov equations for port to port subsystems.

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