Finite singularities of nonlinear systems. Output stabilization, observability, and observers
type: Item
instance of: scholarly article
title: Finite singularities of nonlinear systems. Output stabilization, observability, and observers
zbMATH Open document ID: 0944.93026
DOI: 10.1007/BF02259528
author: Philippe Jouan
author: Jean-Paul Gauthier
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: This paper presents a comprehensive treatment of observability, observers, and output-stabilization issues for observed dynamics and single-input multi-output control systems. The mathematical tools used in the paper come from differential geometry, algebraic geometry, and singularity theory. The main results of the paper can be summarized in the following way. 1. The observed dynamics are objects of the form \[ (\delta)\qquad\dot x= f(x),\quad y= y(x), \] where \(x\in X\subset R^m\), \(y\in R^p\), and all maps are analytic. Associated with the observed dynamics is a map \[ P_\delta: X\to C^\omega([0, \infty], R^p), \] defined as \[ P_\delta(s)(t)= y(\varphi_t(x)), \] and a family of observation maps, parameterized by \(n\); \[ \Phi^n_\delta: X\to R^{np}, \] such that \[ \Phi^n_\delta(x)= (y(x), y^{(1)}(x),\dots, y^{(n-1)}(x))= (y(x), \underline{y_{n-1}}(x)). \] The observed dynamics satisfy at a point \(x^0\) the \(\text{ACP}(N)\) condition (ascending chain property of order \(N\)), if the chain of rings of germs at \(x^0\) \[ A_n(x^0)= (\Phi^n_\delta)^* O(y^0) \] of analytic functions \(\varphi: X\to R\), where \(y^0= \Phi^n_\delta(x^0)\), \(O(y^0)=\) the ring of germs of analytic functions \(\varphi: R^{np}\to R\), \(^*\) denotes the pullback operation, saturates for \(n\geq N\), i.e. \[ A_1(x^0)\subset A_2(x^0)\subset\cdots\subset A_N(x^0)= A_{N+ 1}(x^0)=\cdots\;. \] Given a map \(\Phi^n_\delta\), the multiplicity of this map at \(x^0\) is equal to \(\dim(O(x^0)/(\Phi^n_\delta)^*(m(y^0))\cdot O(x^0))\), where \(m(y^0)\) stands for the maximum ideal in \(O(y^0)\). Results: \(\bullet\) If at \(x^0\) the multiplicity of \(\Phi^n_\delta\) is finite then the observed dynamics have \(\text{ACP}(N)\) for some \(N\geq n\). \(\bullet\) If \(X\) is compact, \(\Phi^n_\delta\) has finite multiplicity at every \(x^0\in X\) and \(\delta\) is observable (i.e., the map \(P_\delta\) is injective) then there exist an integer \(N\) and a smooth map \(\psi_N\) such that \(y^{(N)}= …
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93B07
Mathematics Subject Classification ID: 93B29
Mathematics Subject Classification ID: 93B27
Mathematics Subject Classification ID: 93C10
zbMATH DE Number: 1431812
zbMATH Keywords: semiglobal stabilization
zbMATH Keywords: ascending chain property
zbMATH Keywords: observability
zbMATH Keywords: observers
zbMATH Keywords: output-stabilization
zbMATH Keywords: singularity theory
zbMATH Keywords: dynamic extensions
zbMATH Keywords: dynamic output feedback
reviewed by: Krzysztof Tchoń
MaRDI profile type: Publication
cites work: Q5558469
cites work: Applications of centre manifold theory
cites work: High gain estimation for nonlinear systems
cites work: A Finiteness Criterion for Nonlinear Input–Output Differential Systems
cites work: Observability for any&lt;tex&gt;u(t)&lt;/tex&gt;of a class of nonlinear systems
cites work: A simple observer for nonlinear systems applications to bioreactors
cites work: Observability and Observers for Nonlinear Systems
cites work: Observability for systems with more outputs than inputs and asymptotic observers
cites work: Q4856506
cites work: Bilinearization up to output injection
cites work: Global Time-Varying Linearization up to Output Injection
cites work: Nonlinear controllability and observability
cites work: Observability of real analytic vector fields on compact manifolds
cites work: Linearization by output injection and nonlinear observers
cites work: Holomorphic functions of several variables. An introduction to the fundamental theory. With the assist. of Gottfried Barthel transl. by Michael Bridgland
cites work: Nonlinear Observers with Linearizable Error Dynamics
cites work: Contributions to stability theory
cites work: Introduction to the theory of analytic spaces
cites work: Single-input observability of continuous-time systems
cites work: Global stabilizability and observability imply semi-global stabilizability by output feedback
cites work: Stability of polymerization reactors using I/O linearization and a high- gain observer
cites work: A stable control structure for binary distillation columns
cites work: The structure of the level surfaces of a Lyapunov function
cites work: Q3249822
full work available at URL: https://doi.org/10.1007/bf02259528
OpenAlex ID: W2100781194
description: scientific article; zbMATH DE number 1431812