Analysis of an iterative scheme for approximate regulation for nonlinear systems

E. Aulisa, D. S. Gilliam, T. W. Pathiranage

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

This paper concerns the analysis of an iterative scheme delivering approximate control laws for the tracking regulation problems for nonlinear systems. The procedure can be applied to finite- and infinite-dimensional systems, and the underlying methodology derives from the geometric methods, which have been developed for both linear and nonlinear systems. In the nonlinear case, the main tool is the center manifold theorem. Indeed, in the geometric methodology, under the assumption that the signals to be tracked are generated by a finite-dimensional exo-system, the desired control is obtained by solving a pair of operator equations called the regulator equations. In this paper, we extend the concept of regulator equations to what we refer to as the dynamic regulator equations. Just as it is generally quite difficult to solve the regulator equations, it can be equally difficult to solve the dynamic regulator equations. As the authors have already shown in the linear case, a straightforward attempt to solve the dynamic regulator equations leads to a singular system, which can be regularized to obtain an iterative scheme that provides approximate control laws that provide accurate tracking with very a small tracking error after only a couple of iterations. In this paper, we generalize the iterative scheme to nonlinear systems and provide error estimates for the first 3 iterations. Both finite- and infinite-dimensional examples are given to validate the estimates. We comment that the method has also been applied to a wide range of nonlinear distributed parameter examples described in the references.

Original languageEnglish
Pages (from-to)3140-3173
Number of pages34
JournalInternational Journal of Robust and Nonlinear Control
Volume28
Issue number8
DOIs
StatePublished - May 25 2018

Keywords

  • iterative scheme
  • lumped and distributed parameter systems
  • nonlinear control
  • tracking regulation

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