Balanced truncation for linear switched systems

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Resumé

In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems from Shaker and Wisniewski (2011, 2009) and . This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems (Wood et al., 1996) [3]. Specifically in this paper, we provide a bound on the approximation error in the L2L2 norm for continuous-time and the l2l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of balanced truncation depends only on the input–output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2L2 and l2l2 gains depend only on the input–output map.
OriginalsprogEngelsk
TidsskriftNonlinear Analysis: Hybrid Systems
Vol/bind10
Sider (fra-til)4-20
ISSN1751-570X
DOI
StatusUdgivet - 2013

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Balanced Truncation
Switched Linear Systems
Controllability
Observability
Discrete-time
Quadratic Stability
Linear Parameter-varying Systems
Norm
State-space Representation
Switched Systems
Model Reduction
Approximation Error
Singular Values
Reachability
Continuous Time
Theoretical Analysis
Estimate

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title = "Balanced truncation for linear switched systems",
abstract = "In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems from Shaker and Wisniewski (2011, 2009) and . This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems (Wood et al., 1996) [3]. Specifically in this paper, we provide a bound on the approximation error in the L2L2 norm for continuous-time and the l2l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of balanced truncation depends only on the input–output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2L2 and l2l2 gains depend only on the input–output map.",
keywords = "Switched systems, Model reduction, Balanced truncation, Realization theory",
author = "Mihaly Petreczky and Rafal Wisniewski and John-Josef Leth",
year = "2013",
doi = "10.1016/j.nahs.2013.03.007",
language = "English",
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pages = "4--20",
journal = "Nonlinear Analysis: Hybrid Systems",
issn = "1751-570X",
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Balanced truncation for linear switched systems. / Petreczky, Mihaly; Wisniewski, Rafal; Leth, John-Josef.

I: Nonlinear Analysis: Hybrid Systems, Bind 10, 2013, s. 4-20.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Balanced truncation for linear switched systems

AU - Petreczky, Mihaly

AU - Wisniewski, Rafal

AU - Leth, John-Josef

PY - 2013

Y1 - 2013

N2 - In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems from Shaker and Wisniewski (2011, 2009) and . This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems (Wood et al., 1996) [3]. Specifically in this paper, we provide a bound on the approximation error in the L2L2 norm for continuous-time and the l2l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of balanced truncation depends only on the input–output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2L2 and l2l2 gains depend only on the input–output map.

AB - In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems from Shaker and Wisniewski (2011, 2009) and . This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems (Wood et al., 1996) [3]. Specifically in this paper, we provide a bound on the approximation error in the L2L2 norm for continuous-time and the l2l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of balanced truncation depends only on the input–output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2L2 and l2l2 gains depend only on the input–output map.

KW - Switched systems

KW - Model reduction

KW - Balanced truncation

KW - Realization theory

U2 - 10.1016/j.nahs.2013.03.007

DO - 10.1016/j.nahs.2013.03.007

M3 - Journal article

VL - 10

SP - 4

EP - 20

JO - Nonlinear Analysis: Hybrid Systems

JF - Nonlinear Analysis: Hybrid Systems

SN - 1751-570X

ER -