Exponential observer design for nonlinear systems with real parametric uncertainty
type: Item
instance of: scholarly article
title: Exponential observer design for nonlinear systems with real parametric uncertainty
zbMATH Open document ID: 1029.93034
DOI: 10.1016/S0895-7177(03)80013-4
published in: Mathematical and Computer Modelling
publication date: 2003-10-15T00:00:00Z
review text: The aim of this paper is to present a characterization and construction procedure of local exponential observers for nonlinear systems with real parametric uncertainty of the form \[ \dot x= f(x,\omega),\qquad y= h(x,\omega),\tag{1} \] where \(x\in \mathbb{R}^n\) is the state, \(y\in \mathbb{R}^p\) the output, \(\omega\in \mathbb{R}^m\) the real parametric uncertainty, and \(f(0,0)= 0\), \(h(0,0)= 0\). Since \(\omega\) is not known, it is convenient to consider \(\omega\) as an additional state variable, i.e., the plant (1) is considered in the form \[ {\dot x\brack\dot\omega}= {f(x,\omega)\brack 0},\qquad y= h(x,\omega).\tag{2} \] First, a necessary condition for the existence of a local asymptotic observer for the plant (2) is obtained, namely, that the plant (2) is asymptotically detectable, i.e., any state trajectory \((x(t),\omega(t))\) of (2) with small initial condition \((x_0,\omega_0)\) satisfying \[ h(x(t),\omega(t))\equiv 0 \] must be such that \[ \|(x(t), \omega(t))\|\to 0,\quad\text{as }t\to\infty. \] Using this result it is proved that there is no local asymptotic observer for the plant (2) if the equilibrium \(x=0\) does not change with the disturbance, i.e., \(f(0,\omega)\equiv 0\) and if the output function \(y\) is purely a function of \(x\), i.e., \(h(x,\omega)\equiv \psi(x)\). The main result states that if the plant dynamics in (2) is Lyapunov stable at \((x,\omega)= (0,0)\), then a local exponential observer for (2) described by \[ {\dot z\brack\dot\mu}= {g(z,\mu,y)\brack j(z,\mu,y)}, \] with \(g,j\in C^1\), \(g(0,0,0)= 0\), \(j(0,0,0)= 0\) exists if and only if (a) the submanifold defined via \(e=0\) is invariant under the flow of the composite system \[ \begin{aligned}\dot x &= f(x,\omega),\\ \dot\omega &=0,\\ \dot e_1 &= g(x+ e_1, \omega+ e_2, h(x,\omega))- f(x,\omega),\\ \dot e_2 &= j(x+ e_1,\omega+ e_2, h(x,\omega)),\end{aligned} \] (b) the dynamics \[ \begin{aligned} \dot e_1 &= g(e_1, e_2,0),\\ \dot e_2 &= j(e_1, e_2,0)\end{aligned} \] h…
reviewed by: Marian Ivanovici
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93B07
Mathematics Subject Classification ID: 93B51
Mathematics Subject Classification ID: 93C73
zbMATH DE Number: 1993253
zbMATH Keywords: exponential observers
zbMATH Keywords: real parametric uncertainty
zbMATH Keywords: exogenous disturbance
zbMATH Keywords: nonlinear systems
author: Sundarapandian Vaidyanathan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1993253