Closed form expressions for linear MIMO system responses and solutions of the Lyapunov equation

Anna Soffía Hauksdóttir; Sven Th Sigurosson; Sigurour Örn Aoalgeirsson; Gísli Herjólfsson

doi:10.1109/CDC.2007.4434159

Closed form expressions for linear MIMO system responses and solutions of the Lyapunov equation

Anna Soffía Hauksdóttir^*, Sven Th Sigurosson, Sigurour Örn Aoalgeirsson, Gísli Herjólfsson

^*Corresponding author for this work

Faculty of Electrical and Computer Engineering

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

3 Citations (Scopus)

Abstract

In this paper, closed form expressions of linear MIMO system responses are presented. These expressions make use of matrix polynomial formulations of e^tA. They depend on the eigenvalues of A, the characteristic polynomial of A, as well as the partial fraction expansion coefficients of the corresponding unity numerator transfer function. The general partial fraction coefficients may be calculated using a computationally efficient recursive formula that has been derived in the context of SISO system responses. The final expressions are presented in a form that enhances the understanding of linear systems as such and emphasizes efficient computational implementation and the resulting time complexity. Finally, a closed form solution of the Lypunov equation, the Gramian, is presented.

Original language	English
Title of host publication	Proceedings of the 46th IEEE Conference on Decision and Control 2007, CDC
Pages	2797-2802
Number of pages	6
DOIs	https://doi.org/10.1109/CDC.2007.4434159
Publication status	Published - 2007
Event	46th IEEE Conference on Decision and Control 2007, CDC - New Orleans, LA, United States Duration: 12 Dec 2007 → 14 Dec 2007

Publication series

Name	Proceedings of the IEEE Conference on Decision and Control
ISSN (Print)	0191-2216

Conference

Conference	46th IEEE Conference on Decision and Control 2007, CDC
Country/Territory	United States
City	New Orleans, LA
Period	12/12/07 → 14/12/07

Other keywords

Gramian
Linear continuous time MIMO system responses
Lyapunov equation
Matrix exponential
Partial fraction expansion coefficients

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10.1109/CDC.2007.4434159

Cite this

Hauksdóttir, A. S., Sigurosson, S. T., Aoalgeirsson, S. Ö., & Herjólfsson, G. (2007). Closed form expressions for linear MIMO system responses and solutions of the Lyapunov equation. In Proceedings of the 46th IEEE Conference on Decision and Control 2007, CDC (pp. 2797-2802). [4434159] (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2007.4434159

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title = "Closed form expressions for linear MIMO system responses and solutions of the Lyapunov equation",

abstract = "In this paper, closed form expressions of linear MIMO system responses are presented. These expressions make use of matrix polynomial formulations of etA. They depend on the eigenvalues of A, the characteristic polynomial of A, as well as the partial fraction expansion coefficients of the corresponding unity numerator transfer function. The general partial fraction coefficients may be calculated using a computationally efficient recursive formula that has been derived in the context of SISO system responses. The final expressions are presented in a form that enhances the understanding of linear systems as such and emphasizes efficient computational implementation and the resulting time complexity. Finally, a closed form solution of the Lypunov equation, the Gramian, is presented.",

keywords = "Gramian, Linear continuous time MIMO system responses, Lyapunov equation, Matrix exponential, Partial fraction expansion coefficients",

author = "Hauksd{\'o}ttir, {Anna Soff{\'i}a} and Sigurosson, {Sven Th} and Aoalgeirsson, {Sigurour {\"O}rn} and G{\'i}sli Herj{\'o}lfsson",

year = "2007",

doi = "10.1109/CDC.2007.4434159",

language = "English",

isbn = "1424414989",

series = "Proceedings of the IEEE Conference on Decision and Control",

pages = "2797--2802",

booktitle = "Proceedings of the 46th IEEE Conference on Decision and Control 2007, CDC",

note = "46th IEEE Conference on Decision and Control 2007, CDC ; Conference date: 12-12-2007 Through 14-12-2007",

}

Hauksdóttir, AS, Sigurosson, ST, Aoalgeirsson, SÖ & Herjólfsson, G 2007, Closed form expressions for linear MIMO system responses and solutions of the Lyapunov equation. in Proceedings of the 46th IEEE Conference on Decision and Control 2007, CDC., 4434159, Proceedings of the IEEE Conference on Decision and Control, pp. 2797-2802, 46th IEEE Conference on Decision and Control 2007, CDC, New Orleans, LA, United States, 12/12/07. https://doi.org/10.1109/CDC.2007.4434159

Closed form expressions for linear MIMO system responses and solutions of the Lyapunov equation. / Hauksdóttir, Anna Soffía; Sigurosson, Sven Th; Aoalgeirsson, Sigurour Örn et al.

Proceedings of the 46th IEEE Conference on Decision and Control 2007, CDC. 2007. p. 2797-2802 4434159 (Proceedings of the IEEE Conference on Decision and Control).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Sigurosson, Sven Th

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AU - Herjólfsson, Gísli

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N2 - In this paper, closed form expressions of linear MIMO system responses are presented. These expressions make use of matrix polynomial formulations of etA. They depend on the eigenvalues of A, the characteristic polynomial of A, as well as the partial fraction expansion coefficients of the corresponding unity numerator transfer function. The general partial fraction coefficients may be calculated using a computationally efficient recursive formula that has been derived in the context of SISO system responses. The final expressions are presented in a form that enhances the understanding of linear systems as such and emphasizes efficient computational implementation and the resulting time complexity. Finally, a closed form solution of the Lypunov equation, the Gramian, is presented.

AB - In this paper, closed form expressions of linear MIMO system responses are presented. These expressions make use of matrix polynomial formulations of etA. They depend on the eigenvalues of A, the characteristic polynomial of A, as well as the partial fraction expansion coefficients of the corresponding unity numerator transfer function. The general partial fraction coefficients may be calculated using a computationally efficient recursive formula that has been derived in the context of SISO system responses. The final expressions are presented in a form that enhances the understanding of linear systems as such and emphasizes efficient computational implementation and the resulting time complexity. Finally, a closed form solution of the Lypunov equation, the Gramian, is presented.

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KW - Matrix exponential

KW - Partial fraction expansion coefficients

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Closed form expressions for linear MIMO system responses and solutions of the Lyapunov equation

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