Foundations and Trends® in Systems and Control > Vol 9 > Issue 2-3

Finite-Time Stability Tools for Control and Estimation

By Denis Efimov, Inria, Lille, France, denis.efimov@inria.fr | Andrey Polyakov, Inria, Lille, France, andrey.polyakov@inria.fr

 
Suggested Citation
Denis Efimov and Andrey Polyakov (2021), "Finite-Time Stability Tools for Control and Estimation", Foundations and TrendsĀ® in Systems and Control: Vol. 9: No. 2-3, pp 171-364. http://dx.doi.org/10.1561/2600000026

Publication Date: 09 Dec 2021
© 2021 D. Efimov and A. Polyakov
 
Subjects
Control of Nonlinear Systems,  Systems Theory
 

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In this article:
1. Introduction and motivation
2. Notions of interest
3. Homogeneity
4. Control Design
5. State Estimation
6. Implementation and discretization
7. Extensions
Appendix
References

Abstract

This monograph presents some existing and new results on analysis and design of finite-time and fixed-time converging systems. Two main groups of approaches for analysis/ synthesis of this kind of convergence are considered: based on Lyapunov functions and the theory of homogeneous systems. The focus is put on the dynamics described by ordinary differential equations, time-delay models and partial differential equations. Some popular control and estimation algorithms, which possess accelerated converge rates, are reviewed. The issues of discretization of finite-/fixed-time converging systems are discussed.

DOI:10.1561/2600000026
ISBN: 978-1-68083-926-5
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ISBN: 978-1-68083-927-2
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Table of contents:
1. Introduction and motivation
2. Notions of interest
3. Homogeneity
4. Control Design
5. State Estimation
6. Implementation and discretization
7. Extensions
Appendix
References

Finite-Time Stability Tools for Control and Estimation

This monograph presents some existing and new results on analysis and design of finite-time and fixed-time converging systems. Two main groups of approaches for analysis/synthesis of this kind of convergence, Lyapunov functions and the theory of homogeneous systems, are considered.

The authors focus on the dynamics described by ordinary differential equations, time-delay models and partial differential equations. Some popular control and estimation algorithms, which possess accelerated converge rates, are also reviewed. Finally, the issues of discretization of finite-/fixed-time converging systems are discussed.

Divided into 3 parts, this monograph provides the reader with a complete and accessible review of the topic. In the first part, the definitions of the different finite-/fixed-time stability properties are given together with their characterizations via the Lyapunov function approach. In the second part, several stabilization algorithms for linear and nonlinear systems are formalized, which are based on the implicit Lyapunov function approach. In the third part, the issues of discretization of finite-/fixed-time stable systems are discussed, with a special attention to the solutions obtained with the implicit Lyapunov function method. Finally, the accelerated converge concepts are presented for systems described by time-delay and partial differential equations.

This monograph is an excellent introduction to the complex field of Finite-Time Stability Tools. It enables the reader to synthesize the important concepts and further their own research in the area.

 
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