Polytopic approximations of reachable sets applied to linear dynamic games and a class of nonlinear systems

Inseok Hwang, Dušan M. Stipanović, Claire J. Tomlin

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

This chapter presents applications of polytopic approximation methods for reachable set computation using dynamic optimization. The problem of computing exact reachable sets can be formulated in terms of a Hamilton-Jacobi partial differential equation (PDE). Numerical solutions which provide convergent approximations of this PDE have computational complexity which is exponential in the continuous variable dimension. Using dynamic optimization and polytopic approximation, computationally efficient algorithms for overapproximative reachability analysis have been developed for linear dynamical systems tikya[1]. In this chapter, we extend these to feedback linearizable nonlinear systems, linear dynamic games, and norm-bounded nonlinear systems. Three illustrative examples are presented.

Original languageEnglish (US)
Title of host publicationSystems and Control
Subtitle of host publicationFoundations and Applications
PublisherBirkhauser
Pages3-19
Number of pages17
Edition9780817643850
DOIs
StatePublished - Jan 1 2005

Publication series

NameSystems and Control: Foundations and Applications
Number9780817643850
ISSN (Print)2324-9749
ISSN (Electronic)2324-9757

Fingerprint

Reachable Set
Dynamic Games
Dynamic Optimization
Nonlinear systems
Partial differential equation
Nonlinear Systems
Reachability Analysis
Linear Dynamical Systems
Partial differential equations
Hamilton-Jacobi
Continuous Variables
Feedback Systems
Approximation
Approximation Methods
Approximation Algorithms
Computational Complexity
Efficient Algorithms
Numerical Solution
Norm
Computing

Keywords

  • Convex Polytope
  • Disturbance Input
  • Hybrid System
  • Linear Dynamical System
  • Protected Zone

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Computer Science Applications
  • Control and Optimization
  • Computational Mathematics

Cite this

Hwang, I., Stipanović, D. M., & Tomlin, C. J. (2005). Polytopic approximations of reachable sets applied to linear dynamic games and a class of nonlinear systems. In Systems and Control: Foundations and Applications (9780817643850 ed., pp. 3-19). (Systems and Control: Foundations and Applications; No. 9780817643850). Birkhauser. https://doi.org/10.1007/0-8176-4409-1_1

Polytopic approximations of reachable sets applied to linear dynamic games and a class of nonlinear systems. / Hwang, Inseok; Stipanović, Dušan M.; Tomlin, Claire J.

Systems and Control: Foundations and Applications. 9780817643850. ed. Birkhauser, 2005. p. 3-19 (Systems and Control: Foundations and Applications; No. 9780817643850).

Research output: Chapter in Book/Report/Conference proceedingChapter

Hwang, I, Stipanović, DM & Tomlin, CJ 2005, Polytopic approximations of reachable sets applied to linear dynamic games and a class of nonlinear systems. in Systems and Control: Foundations and Applications. 9780817643850 edn, Systems and Control: Foundations and Applications, no. 9780817643850, Birkhauser, pp. 3-19. https://doi.org/10.1007/0-8176-4409-1_1
Hwang I, Stipanović DM, Tomlin CJ. Polytopic approximations of reachable sets applied to linear dynamic games and a class of nonlinear systems. In Systems and Control: Foundations and Applications. 9780817643850 ed. Birkhauser. 2005. p. 3-19. (Systems and Control: Foundations and Applications; 9780817643850). https://doi.org/10.1007/0-8176-4409-1_1
Hwang, Inseok ; Stipanović, Dušan M. ; Tomlin, Claire J. / Polytopic approximations of reachable sets applied to linear dynamic games and a class of nonlinear systems. Systems and Control: Foundations and Applications. 9780817643850. ed. Birkhauser, 2005. pp. 3-19 (Systems and Control: Foundations and Applications; 9780817643850).
@inbook{cea07f70b3894b4c8c07afbd8381fd3e,
title = "Polytopic approximations of reachable sets applied to linear dynamic games and a class of nonlinear systems",
abstract = "This chapter presents applications of polytopic approximation methods for reachable set computation using dynamic optimization. The problem of computing exact reachable sets can be formulated in terms of a Hamilton-Jacobi partial differential equation (PDE). Numerical solutions which provide convergent approximations of this PDE have computational complexity which is exponential in the continuous variable dimension. Using dynamic optimization and polytopic approximation, computationally efficient algorithms for overapproximative reachability analysis have been developed for linear dynamical systems tikya[1]. In this chapter, we extend these to feedback linearizable nonlinear systems, linear dynamic games, and norm-bounded nonlinear systems. Three illustrative examples are presented.",
keywords = "Convex Polytope, Disturbance Input, Hybrid System, Linear Dynamical System, Protected Zone",
author = "Inseok Hwang and Stipanović, {Dušan M.} and Tomlin, {Claire J.}",
year = "2005",
month = "1",
day = "1",
doi = "10.1007/0-8176-4409-1_1",
language = "English (US)",
series = "Systems and Control: Foundations and Applications",
publisher = "Birkhauser",
number = "9780817643850",
pages = "3--19",
booktitle = "Systems and Control",
edition = "9780817643850",

}

TY - CHAP

T1 - Polytopic approximations of reachable sets applied to linear dynamic games and a class of nonlinear systems

AU - Hwang, Inseok

AU - Stipanović, Dušan M.

AU - Tomlin, Claire J.

PY - 2005/1/1

Y1 - 2005/1/1

N2 - This chapter presents applications of polytopic approximation methods for reachable set computation using dynamic optimization. The problem of computing exact reachable sets can be formulated in terms of a Hamilton-Jacobi partial differential equation (PDE). Numerical solutions which provide convergent approximations of this PDE have computational complexity which is exponential in the continuous variable dimension. Using dynamic optimization and polytopic approximation, computationally efficient algorithms for overapproximative reachability analysis have been developed for linear dynamical systems tikya[1]. In this chapter, we extend these to feedback linearizable nonlinear systems, linear dynamic games, and norm-bounded nonlinear systems. Three illustrative examples are presented.

AB - This chapter presents applications of polytopic approximation methods for reachable set computation using dynamic optimization. The problem of computing exact reachable sets can be formulated in terms of a Hamilton-Jacobi partial differential equation (PDE). Numerical solutions which provide convergent approximations of this PDE have computational complexity which is exponential in the continuous variable dimension. Using dynamic optimization and polytopic approximation, computationally efficient algorithms for overapproximative reachability analysis have been developed for linear dynamical systems tikya[1]. In this chapter, we extend these to feedback linearizable nonlinear systems, linear dynamic games, and norm-bounded nonlinear systems. Three illustrative examples are presented.

KW - Convex Polytope

KW - Disturbance Input

KW - Hybrid System

KW - Linear Dynamical System

KW - Protected Zone

UR - http://www.scopus.com/inward/record.url?scp=84959089048&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84959089048&partnerID=8YFLogxK

U2 - 10.1007/0-8176-4409-1_1

DO - 10.1007/0-8176-4409-1_1

M3 - Chapter

AN - SCOPUS:84959089048

T3 - Systems and Control: Foundations and Applications

SP - 3

EP - 19

BT - Systems and Control

PB - Birkhauser

ER -