A CSD approach to H-infinity controller synthesis

Mi-Ching Tsai, Da Wei Gu

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

H optimal control, which minimizes the H -norm of a closed-loop system, has been developed in the last 30 years and been applied in various domains. The original H optimal control problem involves an equivalent model matching problem, which can be transformed into a four-block distance problem. By applying spectral factorizations, the four-block distance problem can be reduced to a Nehari problem, and Hankel norm approximation can be considered [2–4, 6]. Operator theory approach is very mathematics involved, and numerical solution procedures are difficult to be developed for general form problems. Notable progress was made in finding suboptimal solutions of a general control synthesis problem by solving two algebraic Riccati equations (AREs) [2, 3, 10, 13]. However, even with such solution procedures, questions of “why?” and “how?” often arise from students and engineers who want to understand and use them. An alternative development based on the framework of J-lossless coprime factorizations was proposed by Green in which the solutions can be characterized in terms of transfer function matrices [5]. A similar framework based on a single chain scattering description (CSD) was initially proposed by Kimura [10]. As described in Chap. 8, the general four-block problem can be solved by augmenting with some fictitious signals. Furthermore, since the transformation from LFT to CSD does not guarantee stability of the resulting CSD matrix, the J-lossless factorization with an outer matrix cannot be found directly by the coprime factorization-based method. In this book, the proposed H CSD solution framework involves constructing two coupled (right and left) CSD matrices by solving two J-lossless coprime factorizations and is fairly straightforward. The method is generally valid and does not need to introduce any fictitious signals for matrix augmentation. Based on Green’s approach of J-lossless coprime factorizations, the proposed CSDs framework is significantly different from Kimura’s approach.

Original languageEnglish
Title of host publicationAdvances in Industrial Control
PublisherSpringer International Publishing
Pages267-302
Number of pages36
Edition9781447162568
DOIs
Publication statusPublished - 2014 Jan 1

Publication series

NameAdvances in Industrial Control
Number9781447162568
ISSN (Print)1430-9491
ISSN (Electronic)2193-1577

Fingerprint

Factorization
Scattering
Controllers
Riccati equations
Closed loop systems
Transfer functions
Mathematical operators
Students
Engineers

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Automotive Engineering
  • Aerospace Engineering
  • Industrial and Manufacturing Engineering

Cite this

Tsai, M-C., & Gu, D. W. (2014). A CSD approach to H-infinity controller synthesis. In Advances in Industrial Control (9781447162568 ed., pp. 267-302). (Advances in Industrial Control; No. 9781447162568). Springer International Publishing. https://doi.org/10.1007/978-1-4471-6257-5_9
Tsai, Mi-Ching ; Gu, Da Wei. / A CSD approach to H-infinity controller synthesis. Advances in Industrial Control. 9781447162568. ed. Springer International Publishing, 2014. pp. 267-302 (Advances in Industrial Control; 9781447162568).
@inbook{826080f25d404c298e34869c4aa2321b,
title = "A CSD approach to H-infinity controller synthesis",
abstract = "H ∞ optimal control, which minimizes the H ∞-norm of a closed-loop system, has been developed in the last 30 years and been applied in various domains. The original H ∞ optimal control problem involves an equivalent model matching problem, which can be transformed into a four-block distance problem. By applying spectral factorizations, the four-block distance problem can be reduced to a Nehari problem, and Hankel norm approximation can be considered [2–4, 6]. Operator theory approach is very mathematics involved, and numerical solution procedures are difficult to be developed for general form problems. Notable progress was made in finding suboptimal solutions of a general control synthesis problem by solving two algebraic Riccati equations (AREs) [2, 3, 10, 13]. However, even with such solution procedures, questions of “why?” and “how?” often arise from students and engineers who want to understand and use them. An alternative development based on the framework of J-lossless coprime factorizations was proposed by Green in which the solutions can be characterized in terms of transfer function matrices [5]. A similar framework based on a single chain scattering description (CSD) was initially proposed by Kimura [10]. As described in Chap. 8, the general four-block problem can be solved by augmenting with some fictitious signals. Furthermore, since the transformation from LFT to CSD does not guarantee stability of the resulting CSD matrix, the J-lossless factorization with an outer matrix cannot be found directly by the coprime factorization-based method. In this book, the proposed H ∞ CSD solution framework involves constructing two coupled (right and left) CSD matrices by solving two J-lossless coprime factorizations and is fairly straightforward. The method is generally valid and does not need to introduce any fictitious signals for matrix augmentation. Based on Green’s approach of J-lossless coprime factorizations, the proposed CSDs framework is significantly different from Kimura’s approach.",
author = "Mi-Ching Tsai and Gu, {Da Wei}",
year = "2014",
month = "1",
day = "1",
doi = "10.1007/978-1-4471-6257-5_9",
language = "English",
series = "Advances in Industrial Control",
publisher = "Springer International Publishing",
number = "9781447162568",
pages = "267--302",
booktitle = "Advances in Industrial Control",
edition = "9781447162568",

}

Tsai, M-C & Gu, DW 2014, A CSD approach to H-infinity controller synthesis. in Advances in Industrial Control. 9781447162568 edn, Advances in Industrial Control, no. 9781447162568, Springer International Publishing, pp. 267-302. https://doi.org/10.1007/978-1-4471-6257-5_9

A CSD approach to H-infinity controller synthesis. / Tsai, Mi-Ching; Gu, Da Wei.

Advances in Industrial Control. 9781447162568. ed. Springer International Publishing, 2014. p. 267-302 (Advances in Industrial Control; No. 9781447162568).

Research output: Chapter in Book/Report/Conference proceedingChapter

TY - CHAP

T1 - A CSD approach to H-infinity controller synthesis

AU - Tsai, Mi-Ching

AU - Gu, Da Wei

PY - 2014/1/1

Y1 - 2014/1/1

N2 - H ∞ optimal control, which minimizes the H ∞-norm of a closed-loop system, has been developed in the last 30 years and been applied in various domains. The original H ∞ optimal control problem involves an equivalent model matching problem, which can be transformed into a four-block distance problem. By applying spectral factorizations, the four-block distance problem can be reduced to a Nehari problem, and Hankel norm approximation can be considered [2–4, 6]. Operator theory approach is very mathematics involved, and numerical solution procedures are difficult to be developed for general form problems. Notable progress was made in finding suboptimal solutions of a general control synthesis problem by solving two algebraic Riccati equations (AREs) [2, 3, 10, 13]. However, even with such solution procedures, questions of “why?” and “how?” often arise from students and engineers who want to understand and use them. An alternative development based on the framework of J-lossless coprime factorizations was proposed by Green in which the solutions can be characterized in terms of transfer function matrices [5]. A similar framework based on a single chain scattering description (CSD) was initially proposed by Kimura [10]. As described in Chap. 8, the general four-block problem can be solved by augmenting with some fictitious signals. Furthermore, since the transformation from LFT to CSD does not guarantee stability of the resulting CSD matrix, the J-lossless factorization with an outer matrix cannot be found directly by the coprime factorization-based method. In this book, the proposed H ∞ CSD solution framework involves constructing two coupled (right and left) CSD matrices by solving two J-lossless coprime factorizations and is fairly straightforward. The method is generally valid and does not need to introduce any fictitious signals for matrix augmentation. Based on Green’s approach of J-lossless coprime factorizations, the proposed CSDs framework is significantly different from Kimura’s approach.

AB - H ∞ optimal control, which minimizes the H ∞-norm of a closed-loop system, has been developed in the last 30 years and been applied in various domains. The original H ∞ optimal control problem involves an equivalent model matching problem, which can be transformed into a four-block distance problem. By applying spectral factorizations, the four-block distance problem can be reduced to a Nehari problem, and Hankel norm approximation can be considered [2–4, 6]. Operator theory approach is very mathematics involved, and numerical solution procedures are difficult to be developed for general form problems. Notable progress was made in finding suboptimal solutions of a general control synthesis problem by solving two algebraic Riccati equations (AREs) [2, 3, 10, 13]. However, even with such solution procedures, questions of “why?” and “how?” often arise from students and engineers who want to understand and use them. An alternative development based on the framework of J-lossless coprime factorizations was proposed by Green in which the solutions can be characterized in terms of transfer function matrices [5]. A similar framework based on a single chain scattering description (CSD) was initially proposed by Kimura [10]. As described in Chap. 8, the general four-block problem can be solved by augmenting with some fictitious signals. Furthermore, since the transformation from LFT to CSD does not guarantee stability of the resulting CSD matrix, the J-lossless factorization with an outer matrix cannot be found directly by the coprime factorization-based method. In this book, the proposed H ∞ CSD solution framework involves constructing two coupled (right and left) CSD matrices by solving two J-lossless coprime factorizations and is fairly straightforward. The method is generally valid and does not need to introduce any fictitious signals for matrix augmentation. Based on Green’s approach of J-lossless coprime factorizations, the proposed CSDs framework is significantly different from Kimura’s approach.

UR - http://www.scopus.com/inward/record.url?scp=85021161566&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021161566&partnerID=8YFLogxK

U2 - 10.1007/978-1-4471-6257-5_9

DO - 10.1007/978-1-4471-6257-5_9

M3 - Chapter

T3 - Advances in Industrial Control

SP - 267

EP - 302

BT - Advances in Industrial Control

PB - Springer International Publishing

ER -

Tsai M-C, Gu DW. A CSD approach to H-infinity controller synthesis. In Advances in Industrial Control. 9781447162568 ed. Springer International Publishing. 2014. p. 267-302. (Advances in Industrial Control; 9781447162568). https://doi.org/10.1007/978-1-4471-6257-5_9