Computational complexity of a problem arising in fixed order output feedback design

Minyue Fu; Zhi Quan Luo

doi:10.1016/S0167-6911(97)00014-5

Computational complexity of a problem arising in fixed order output feedback design

Minyue Fu, Zhi Quan Luo

Electrical and Computer Engineering

Research output: Contribution to journal › Article › peer-review

71 Scopus citations

Abstract

This paper is concerned with a matrix inequality problem which arises in fixed order output feedback control design. This problem involves finding two symmetric and positive definitive matrices X and Y such that each satisfies a linear matrix inequality and that XY=I. It is well-known that many control problems such as fixed order output feedback stabilization, H_∞ control, guaranteed H₂ control, and mixed H₂/H_∞ control can all be converted into the matrix inequality problem above, including static output feedback problems as a special case. We show, however, that this matrix inequality problem is NP-hard.

Original language	English (US)
Pages (from-to)	209-215
Number of pages	7
Journal	Systems and Control Letters
Volume	30
Issue number	5
DOIs	https://doi.org/10.1016/S0167-6911(97)00014-5
State	Published - Jun 1997

Bibliographical note

Funding Information:
The motivation of the problem above stems from the fact that several important fixed order output feedback control problems, which include static output feedback control problems as a special case, can be converted into the above. Examples of these problem are such as fixed order output feedback stabilization, H~ control, guaranteed H2 control, and mixed H2/Hoo control. See \[3 10, 12\] for references and Section 2 for discussions. It is generally believed that the problem in (1) is difficult to solve. Nevertheless, several recent approaches fcr solving the fixed order output feedback control problems rely on this conversion and certain iterative algorithms for solving (1). For example, the following iterative algorithm is proposed * Corresponding author. E-mail: [email protected]. 1T he research of the first author is supported by Australian Research Council, while the research of the second author is supported by the Natural Science and Engineering Research Council of Canada under grant No. OPG0090301.

Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

Keywords

Computational complexity
H control
Linear matrix inequality
Output feedback control
Static output feedback

Publisher link

10.1016/S0167-6911(97)00014-5

Find It

Find It at U of M Libraries

Cite this

@article{a04ad6b670fd4cd18eef0fb499ff1c95,

title = "Computational complexity of a problem arising in fixed order output feedback design",

abstract = "This paper is concerned with a matrix inequality problem which arises in fixed order output feedback control design. This problem involves finding two symmetric and positive definitive matrices X and Y such that each satisfies a linear matrix inequality and that XY=I. It is well-known that many control problems such as fixed order output feedback stabilization, H∞ control, guaranteed H2 control, and mixed H2/H∞ control can all be converted into the matrix inequality problem above, including static output feedback problems as a special case. We show, however, that this matrix inequality problem is NP-hard.",

keywords = "Computational complexity, H control, Linear matrix inequality, Output feedback control, Static output feedback",

author = "Minyue Fu and Luo, \{Zhi Quan\}",

note = "Funding Information: The motivation of the problem above stems from the fact that several important fixed order output feedback control problems, which include static output feedback control problems as a special case, can be converted into the above. Examples of these problem are such as fixed order output feedback stabilization, H\textasciitilde{} control, guaranteed H2 control, and mixed H2/Hoo control. See \textbackslash{}[3 10, 12\textbackslash{}] for references and Section 2 for discussions. It is generally believed that the problem in (1) is difficult to solve. Nevertheless, several recent approaches fcr solving the fixed order output feedback control problems rely on this conversion and certain iterative algorithms for solving (1). For example, the following iterative algorithm is proposed * Corresponding author. E-mail: [email protected]. 1T he research of the first author is supported by Australian Research Council, while the research of the second author is supported by the Natural Science and Engineering Research Council of Canada under grant No. OPG0090301. Copyright: Copyright 2017 Elsevier B.V., All rights reserved.",

year = "1997",

month = jun,

doi = "10.1016/S0167-6911(97)00014-5",

language = "English (US)",

volume = "30",

pages = "209--215",

journal = "Systems and Control Letters",

issn = "0167-6911",

publisher = "Elsevier B.V.",

number = "5",

}

TY - JOUR

T1 - Computational complexity of a problem arising in fixed order output feedback design

AU - Fu, Minyue

AU - Luo, Zhi Quan

N1 - Funding Information: The motivation of the problem above stems from the fact that several important fixed order output feedback control problems, which include static output feedback control problems as a special case, can be converted into the above. Examples of these problem are such as fixed order output feedback stabilization, H~ control, guaranteed H2 control, and mixed H2/Hoo control. See \[3 10, 12\] for references and Section 2 for discussions. It is generally believed that the problem in (1) is difficult to solve. Nevertheless, several recent approaches fcr solving the fixed order output feedback control problems rely on this conversion and certain iterative algorithms for solving (1). For example, the following iterative algorithm is proposed * Corresponding author. E-mail: [email protected]. 1T he research of the first author is supported by Australian Research Council, while the research of the second author is supported by the Natural Science and Engineering Research Council of Canada under grant No. OPG0090301. Copyright: Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1997/6

Y1 - 1997/6

N2 - This paper is concerned with a matrix inequality problem which arises in fixed order output feedback control design. This problem involves finding two symmetric and positive definitive matrices X and Y such that each satisfies a linear matrix inequality and that XY=I. It is well-known that many control problems such as fixed order output feedback stabilization, H∞ control, guaranteed H2 control, and mixed H2/H∞ control can all be converted into the matrix inequality problem above, including static output feedback problems as a special case. We show, however, that this matrix inequality problem is NP-hard.

AB - This paper is concerned with a matrix inequality problem which arises in fixed order output feedback control design. This problem involves finding two symmetric and positive definitive matrices X and Y such that each satisfies a linear matrix inequality and that XY=I. It is well-known that many control problems such as fixed order output feedback stabilization, H∞ control, guaranteed H2 control, and mixed H2/H∞ control can all be converted into the matrix inequality problem above, including static output feedback problems as a special case. We show, however, that this matrix inequality problem is NP-hard.

KW - Computational complexity

KW - H control

KW - Linear matrix inequality

KW - Output feedback control

KW - Static output feedback

UR - http://www.scopus.com/inward/record.url?scp=0010024237&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010024237&partnerID=8YFLogxK

U2 - 10.1016/S0167-6911(97)00014-5

DO - 10.1016/S0167-6911(97)00014-5

M3 - Article

AN - SCOPUS:0010024237

SN - 0167-6911

VL - 30

SP - 209

EP - 215

JO - Systems and Control Letters

JF - Systems and Control Letters

IS - 5

ER -

Computational complexity of a problem arising in fixed order output feedback design

Abstract

Bibliographical note

Keywords

Publisher link

Find It

Other files and links

Fingerprint

Cite this