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2017 | Book

Singular Perturbation Methods and Time-Scale Techniques Read chapter Read first chapter

Finite Frequency Analysis and Synthesis for Singularly Perturbed Systems

Authors: Chenxiao Cai, Zidong Wang, Jing Xu, Yun Zou

Publisher: Springer International Publishing

Book Series : Studies in Systems, Decision and Control

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

This book is a self-contained collection of recent research findings providing a comprehensive and systematic unified framework for both analysis and synthesis for singularly perturbed systems. It paves the way for the gap between frequency-domain-transfer-function-based results and time-domain-state-space-based results to be bridged.

It is divided into three parts focusing on: fundamental background of singular perturbation; general singular perturbation methodologies and time-scale techniques and the theoretical foundation of finite-frequency control; the analysis and synthesis of singularly perturbed systems; and real-world engineering applications implementing the results developed in systems like wind turbines and autonomous-aerial-vehicle hovering.

It also presents solutions to analysis and design problems in terms of linear matrix inequalities. Lastly, it provides valuable reference material for researchers who wish to explore the design of controllers for such systems.

Table of Contents

Frontmatter

Preliminaries

Frontmatter
Chapter 1. Singular Perturbation Methods and Time-Scale Techniques
Abstract
For control engineering, typical tasks can generally be classified into three main categories: optimal regulation, tracking and guidance. To overcome the external disturbances, parameter variations and other uncertainties, control systems should possess a sufficient degree of robustness or insensitivity to extraneous effects.
Chenxiao Cai, Zidong Wang, Jing Xu, Yun Zou
Chapter 2. Theoretical Foundation of Finite Frequency Control
Abstract
Finite frequency control strategy has been proven to be an important method for modern control system. Combined with the particular frequency characteristics of the plant, many control specifications in the full frequency domain can be simplified into finite frequency ones. Commonly used tools in the frequency division are the weighting function and general Kalman-Yakubovich-Popov (GKYP) Lemma. In this chapter, some background information and useful lemmas in the field of finite frequency control have been investigated in detail.
Chenxiao Cai, Zidong Wang, Jing Xu, Yun Zou

Singularly Perturbed Systems Analysis and Synthesis

Frontmatter
Chapter 3. Stabilization of Singularly Perturbed Systems
Abstract
Stability analysis and controller design are significant problems of dynamic systems in theory and practice that have attracted the interest of many investigators [4]. In recent decades, researchers have focused on the problem of stability analysis and stabilization for SPSs, and these approaches can improve the control precision of system. In this chapter, we first discuss the concept of stability in general, and then present four techniques for assessing the stability of a system: (1) introducing Lyapunov functions; (2) finding the eigenvalues for state-space notation; (3) finding the location of the poles in the complex frequency plane of the closed-loop TF; and (4) providing a descriptor-system method to stabilize the SPSs. Note that the stability of the system should be guaranteed in the entire frequency range, while the related control system specifications should be specified in the finite frequency ranges to reduce their conservatism.
Chenxiao Cai, Zidong Wang, Jing Xu, Yun Zou
Chapter 4. Finite Frequency Control for Singularly Perturbed Systems
Abstract
In this chapter, we consider the finite requency \(H_\infty \) methodologies of SPSs based on a dual version of GKYP lemma approach, which is more suitable for feedback synthesis. Two different methods are mainly utilized, namely the classical slow-fast decomposition method and the descriptor-system method for SPSs. The classical slow-fast decomposition method aims at applying this dual GKYP lemma in slow and fast subsystems, respectively, to achieve the desired performance characteristics for the closed-loop SPSs.
Chenxiao Cai, Zidong Wang, Jing Xu, Yun Zou
Chapter 5. Finite Frequency Positive Real Control for Singularly Perturbed Systems
Abstract
With the development of high-performance mechatronics, the feedback control system is often required to exhibit some particular control system specifications such as high control bandwidth and FF disturbance suppression capability. Meanwhile, the integrated design of both mechanical plant and controller design is carried out to achieve performance specifications. In this chapter, a GKYP lemma-based algorithm is proposed for simultaneous finite frequency design of a mechanical SPS plant and controller through a convex separable parametrization. In Sect. 5.1, basic definitions and some related engineering background of passivity and positive real property are introduced. The design procedure using the classical slow-fast decomposition is represented in Sect. 5.2. To extend the results to nonstandard SPS, a descriptor-system method for SPSs are demonstrated in Sect. 5.3.
Chenxiao Cai, Zidong Wang, Jing Xu, Yun Zou
Chapter 6. The Sensitivity-Shaping Problem for Singularly Perturbed Systems
Abstract
In this chapter, a design technique is carried out by applying robustness criteria to obtain stability and satisfy some performances. A loop-shaping technique has been researched by selecting a suitable open-loop TF and then the robust controller is constructed. This chapter is arranged as follows: the basic definitions are presented in Sect. 6.1. The loop-shaping design procedure for SISO SPSs via using the finite frequency strategy is demonstrated in Sect. 6.2. Such method has been extended to be applied in MIMO SPSs in Sect. 6.3. Using observer-based controllers, the fault detection (FD) issue for SPSs based on finite frequency methods has been investigated in Sect. 6.4. Simulation examples are given respectively to show the validity and effectiveness of the design procedure.
Chenxiao Cai, Zidong Wang, Jing Xu, Yun Zou

Applications

Frontmatter
Chapter 7. Applications
Abstract
The theory of SPM originated from fluid dynamics and applied widely in the area of aerospace systems (Ramnath, Multiple Scales Theory and Aerospace Applications (2010) [110]). In (Stoica, Berbente, Condurache Rev Roum des Sci Tech Serie de Mecani Appl, 45, 277–293 (2002) [134]), a design method was proposed for the manual flight control system corresponding to the lateral motion of the aircraft based on SPaTSs
Chenxiao Cai, Zidong Wang, Jing Xu, Yun Zou
Backmatter
Metadata
Title
Finite Frequency Analysis and Synthesis for Singularly Perturbed Systems
Authors
Chenxiao Cai
Zidong Wang
Jing Xu
Yun Zou
Copyright Year
2017
Electronic ISBN
978-3-319-45405-4
Print ISBN
978-3-319-45404-7
DOI
https://doi.org/10.1007/978-3-319-45405-4