Exponential observer design for discrete-time nonlinear systems with real parametric uncertainty.
type: Item
instance of: scholarly article
title: Exponential observer design for discrete-time nonlinear systems with real parametric uncertainty.
zbMATH Open document ID: 1055.93012
DOI: 10.1016/S0895-7177(03)80014-6
published in: Mathematical and Computer Modelling
publication date: 2003-10-15T00:00:00Z
review text: The author considers discrete-time nonlinear plants of the form \[ \left[\begin{matrix} x(k+1)\\ \omega(k+ 1)\end{matrix}\right]= \left[\begin{matrix} f(x(k), \omega(k))\\ \omega(k)\end{matrix}\right],\quad y(k)= h(x(k), \omega(k)),\tag{1} \] where \(x\in \mathbb{R}^n\) is the state, \(y\in\mathbb{R}^p\) the output of the plant, and \(\omega\in\mathbb{R}^m\) the real parametric uncertainty. The problem of finding local exponential observers for (1) is investigated. A discrete-time nonlinear system described by \[ \left[\begin{matrix} z(k+1)\\ \mu(k+ 1)\end{matrix}\right]= G(z(k), \mu(k), y(k))= \left[\begin{matrix} g(z(k),\, \mu(k),\, y(k))\\ j(z(k),\,\mu(k),\,y(k))\end{matrix}\right],\tag{2} \] where \(z\in\mathbb{R}^n\), \(\mu\in\mathbb{R}^m\) and \(g,j\in C^1\) maps in a neighborhood of the origin with \(g(0,0,0)= 0\), \(j(0,0,0)= 0\), is called a local asymptotic (respectively, exponential) observer for plant (1) if the composite system (1)--(2) satisfies the following two conditions: I: If \({x(0)\brack \omega(0)}= {z(0)\brack \mu(0)}\), then \({x(k)\brack \omega(k)}= {z(k)\brack \mu(k)}\) for all \(k\geq 0\). II: There exists a neighborhood \(V\) of the origin of \(\mathbb{R}^n\times \mathbb{R}^m\) such that for all \({z(0)\brack \mu(0)}- {x(0)\brack \omega(0)}\) in \(V\), the measurement error \(\|{z(k)- x(k)\brack \mu(k)- \omega(k)}\|\) tends to zero asymptotically (respectively, exponentially) as \(k\to\infty\). First, the author shows that for the classical case of problems, when the state equilibrium \(x= 0\) does not change with the disturbance, i.e. \(f(0,\omega)\equiv 0\), and when the output function \(y\) is purely a function of \(x\), i.e. \(h(x,\omega)\equiv \psi(x)\), there does not exist even a local asymptotic observer for the plant. In sharp contrast to this case, for the general case of systems where we allow the equilibrium \(x= 0\) to change with the disturbance \(\omega\), it is shown that typically there exist local exponential observers e…
reviewed by: Marian Ivanovici
Mathematics Subject Classification ID: 93B07
Mathematics Subject Classification ID: 93B51
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 93C73
zbMATH DE Number: 1993254
zbMATH Keywords: Discrete-time systems
zbMATH Keywords: Exponential observers
zbMATH Keywords: Real parametric uncertainty
zbMATH Keywords: Exogenous disturbance
zbMATH Keywords: Invariant submanifold
author: Sundarapandian Vaidyanathan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1993254