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Approximate Models for Nonlinear Deterministic Systems

  • Juan I. Yuz
  • Graham C. Goodwin
Part of the Communications and Control Engineering book series (CCE)

Abstract

This chapter extends the ideas presented in Chap.  8 to develop discrete-time models for nonlinear deterministic systems. In particular, we develop sampled models based on up-sampling and on truncated Taylor series expansion of the normal form. The approximation errors are precisely characterized.

Further Reading

A general introduction to the accuracy of numerical integration schemes is given in

  1. Butcher JC (2008) Numerical methods for ordinary differential equations, 2nd edn. Wiley, New York CrossRefzbMATHGoogle Scholar

Normal forms for nonlinear systems are described in detail in

  1. Isidori A (1995) Nonlinear control systems, 3rd edn. Springer, Berlin CrossRefzbMATHGoogle Scholar

Discrete-time normal forms are discussed in

  1. Califano C, Monaco S, Normand-Cyrot D (1998) On the discrete-time normal form. IEEE Trans Autom Control 43(11):1654–1658 MathSciNetCrossRefzbMATHGoogle Scholar

State-space models for nonlinear systems, in both continuous time and discrete time, are discussed in

  1. Belikov J, Kotta U, Tonso M (2012) State-space realization of nonlinear control systems: unification and extension via pseudo-linear algebra. Kybernetika 48(6):1100–1113 MathSciNetzbMATHGoogle Scholar

Approximate models for nonlinear systems based on variable truncated Taylor series were first described in

  1. Yuz JI, Goodwin GC (2005) On sampled-data models for nonlinear systems. IEEE Trans Autom Control 50(10):1477–1489 MathSciNetCrossRefGoogle Scholar

The definitions of vector truncation errors and an associated accuracy study of truncated Taylor series models were first presented in

  1. Carrasco DS, Goodwin GC, Yuz JI (2013) Vector measures of accuracy for sampled data models of nonlinear systems. IEEE Trans Autom Control 58(1):224–230 MathSciNetCrossRefGoogle Scholar

Sampled-data models for Hamiltonian systems have been proposed in

  1. Laila D, Astolfi A (2006) Construction of discrete-time models for port-controlled Hamiltonian systems with applications. Syst Control Lett 55:673–680 MathSciNetCrossRefzbMATHGoogle Scholar
  2. Monaco S, Normand-Cyrot D, Tiefensee F (2011) Sampled-data stabilization; a PBC approach. IEEE Trans Autom Control 56(4):907–912 MathSciNetCrossRefGoogle Scholar

The problem of sampled-data control for nonlinear systems has been extensively investigated. See, for example

  1. Laila D, Nesic D, Astolfi A (2006) Sampled-data control of nonlinear systems. In: Advanced topics in control systems theory II, pp 91–137 CrossRefGoogle Scholar
  2. Nesic D, Teel AR (2004) A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models. IEEE Trans Autom Control 49(7):1103–1122 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Juan I. Yuz
    • 1
  • Graham C. Goodwin
    • 2
  1. 1.Departamento de ElectrónicaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.School of Electrical Engineering & Computer ScienceUniversity of NewcastleCallaghanAustralia

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