Time-limited <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.gif" overflow="scroll"><mml:msub><mml:mi mathvariant="bold-script">H</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>-optimal model order reduction

Goyal, Pawan; Redmann, Martin

doi:10.1016/j.amc.2019.02.065

Cited by 23 publications

(39 citation statements)

References 14 publications

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“…In [54], a projection‐based algorithm, i.e. the time‐limited IRKA (TLIRKA), is presented that tends to achieve optimality conditions (28)–(30).…”

Section: Preliminariesmentioning

confidence: 99%

“…Let

false({\hat{A}}_{r}, {\hat{B}}_{r}, {\hat{C}}_{r} false)

be the initial guess of the ROM, and let

S = Q {\hat{A}}_{r} Q^{- 1}

scriptB = Q {\hat{B}}_{r}

, and

scriptC = {\hat{C}}_{r} Q^{- 1}

where

Q = (][, \begin{matrix} q_{1} & \dots & q_{r} \end{matrix})

and

q_{i}

is the eigenvector of

λ_{i} false({\hat{A}}_{r} false)

. The algorithm in [54] heuristically suggests replacing

B B^{*}

and

C^{*} scriptC

in IRKA with

B B^{*} - e^{A t} B B^{*} e^{S t}

and

C^{*} scriptC - e^{A^{*} t} C^{*} scriptC e^{S t}

, respectively, for the calculation of the reduction subspaces [see (32)–(33)]. The input and output subspaces in TLIRKA [54] are computed as

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A {\overset{false^}{V}}_{r, t} + {\overset{false^}{V}}_{r, t} S + B {scriptB}^{*} - e^{A} \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…The algorithm in [54] heuristically suggests replacing

B B^{*}

and

C^{*} scriptC

in IRKA with

B B^{*} - e^{A t} B B^{*} e^{S t}

and

C^{*} scriptC - e^{A^{*} t} C^{*} scriptC e^{S t}

, respectively, for the calculation of the reduction subspaces [see (32)–(33)]. The input and output subspaces in TLIRKA [54] are computed as

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A {\overset{false^}{V}}_{r, t} + {\overset{false^}{V}}_{r, t} S + B {scriptB}^{*} - e^{A t} B {scriptB}^{*} e^{S t} = 0, \end{matrix}

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A^{*} {\overset{false^}{W}}_{r, t} + {\overset{false^}{W}}_{r, t} S + C^{*} C - e^{A^{*} t} C^{*} C e^{S t} = 0 . \end{matrix}

{\hat{V}}_{r, t}

and

{\hat{W}}_{r, t}

is set to

orth false({\hat{V}}_{r, t} false)

and

…”

Section: Preliminariesmentioning

confidence: 99%

“…Further, TLBT is extended to unstable systems in [53]. In [54], IRKA is generalised to time‐limited scenario to approximately achieve the first‐order optimality conditions for the time‐limited

H_{2}

‐MOR problem. The first‐order optimality conditions for the time‐limited

H_{2}

‐MOR problem are expressed as bi‐tangential Hermite interpolation conditions in [55], and a descent‐based iterative algorithm is presented for the SISO systems.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we present a time‐limited MOR algorithm which satisfies a subset of the first‐order optimality conditions for the time‐limited

H_{2}

‐MOR problem (as derived in [54, 55]). The algorithm is iteration free, and it guarantees the stability of ROM.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Time‐limited pseudo‐optimal ‐model order reduction

Zulfiqar

Sreeram

2020

IET Control Theory & Appl

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“…In [54], a projection‐based algorithm, i.e. the time‐limited IRKA (TLIRKA), is presented that tends to achieve optimality conditions (28)–(30).…”

Section: Preliminariesmentioning

confidence: 99%

“…Let

false({\hat{A}}_{r}, {\hat{B}}_{r}, {\hat{C}}_{r} false)

be the initial guess of the ROM, and let

S = Q {\hat{A}}_{r} Q^{- 1}

scriptB = Q {\hat{B}}_{r}

, and

scriptC = {\hat{C}}_{r} Q^{- 1}

where

Q = (][, \begin{matrix} q_{1} & \dots & q_{r} \end{matrix})

and

q_{i}

is the eigenvector of

λ_{i} false({\hat{A}}_{r} false)

. The algorithm in [54] heuristically suggests replacing

B B^{*}

and

C^{*} scriptC

in IRKA with

B B^{*} - e^{A t} B B^{*} e^{S t}

and

C^{*} scriptC - e^{A^{*} t} C^{*} scriptC e^{S t}

, respectively, for the calculation of the reduction subspaces [see (32)–(33)]. The input and output subspaces in TLIRKA [54] are computed as

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A {\overset{false^}{V}}_{r, t} + {\overset{false^}{V}}_{r, t} S + B {scriptB}^{*} - e^{A} \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…The algorithm in [54] heuristically suggests replacing

B B^{*}

and

C^{*} scriptC

in IRKA with

B B^{*} - e^{A t} B B^{*} e^{S t}

and

C^{*} scriptC - e^{A^{*} t} C^{*} scriptC e^{S t}

, respectively, for the calculation of the reduction subspaces [see (32)–(33)]. The input and output subspaces in TLIRKA [54] are computed as

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A {\overset{false^}{V}}_{r, t} + {\overset{false^}{V}}_{r, t} S + B {scriptB}^{*} - e^{A t} B {scriptB}^{*} e^{S t} = 0, \end{matrix}

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ A^{*} {\overset{false^}{W}}_{r, t} + {\overset{false^}{W}}_{r, t} S + C^{*} C - e^{A^{*} t} C^{*} C e^{S t} = 0 . \end{matrix}

{\hat{V}}_{r, t}

and

{\hat{W}}_{r, t}

is set to

orth false({\hat{V}}_{r, t} false)

and

…”

Section: Preliminariesmentioning

confidence: 99%

H_{2}

‐MOR problem. The first‐order optimality conditions for the time‐limited

H_{2}

‐MOR problem are expressed as bi‐tangential Hermite interpolation conditions in [55], and a descent‐based iterative algorithm is presented for the SISO systems.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we present a time‐limited MOR algorithm which satisfies a subset of the first‐order optimality conditions for the time‐limited

H_{2}

‐MOR problem (as derived in [54, 55]). The algorithm is iteration free, and it guarantees the stability of ROM.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Time‐limited pseudo‐optimal ‐model order reduction

Zulfiqar

Sreeram

2020

IET Control Theory & Appl

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Balanced truncation reduced models of quadratic‐bilinear systems in time interval

Zhang

Jiang

2023

Asian Journal of Control

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Accumulative error along with time impedes the application of many computational methods on high‐precision simulation; therefore, time‐interval methods emerge. In this paper, a new model reduction method of quadratic‐bilinear (QB) systems based on time‐interval Gramians (TIGs) is presented. The conditions for generalized Lyapunov equations whose solutions are exactly TIGs to be solvable are derived. Lyapunov stability and error bound are discussed to demonstrate the succession and advance of time‐interval balanced truncation (TIBT). The accuracy and robustness are improved, as is illustrated in numerical results.

show abstract

Mixed control of vibrational systems

Nakić¹,

Tomljanović²,

Truhar³

2019

Z Angew Math Mech

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We consider new performance measures for vibrational systems based on the H2 norm of linear time invariant systems. New measures will be used as an optimization criterion for the optimal damping of vibrational systems. We consider both theoretical and concrete cases in order to show how new measures stack up against the standard measures. The quality and advantages of new measures as well as the behaviour of optimal damping positions and corresponding damping viscosities are illustrated in numerical experiments.

show abstract

Time-limited H2-optimal model order reduction

Cited by 23 publications

References 14 publications

Time‐limited pseudo‐optimal ‐model order reduction

Time‐limited pseudo‐optimal ‐model order reduction

Balanced truncation reduced models of quadratic‐bilinear systems in time interval

Mixed control of vibrational systems

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