On the robustness of linear systems with nonlinear uncertain parameters
type: Item
instance of: scholarly article
title: On the robustness of linear systems with nonlinear uncertain parameters
zbMATH Open document ID: 0951.93024
DOI: 10.1016/S0005-1098(98)00040-5
author: V. Pereyra
published in: Automatica
publication date: 2001-01-02T00:00:00Z
review text: The authors study the stability robustness of linear uncertain state space systems of the form (after the control law is given by \(u=Kx\)): \[ x'= (I +\Delta_m)[A +\Delta_a]x,\tag{L} \] where \(x\) denotes the state vector, \(x'\) is the derivative of \(x\), \(I\) is an identity matrix, and \[ \Delta_m=\sum^m_{j=1}k_jG_j,\quad \Delta_a=\sum^l_{i=1}r_iE_i,\quad \underline{k}_j\leq k_j\leq \overline{k}_j,\quad \underline{r}_i\leq r_i\leq \overline{r}_i, \] where \(k_j\), \(r_i\) are uncertain parameters, \(G_j\), \(E_i\) are known constant metrices, \((\overline *)\) and \((\underline{*})\) represent the upper and the lower bounds of \((*)\), respectively. It is supposed that the nominal system of (L) with \(\Delta_m = 0\), \(\Delta_a = 0\), is asymptotically stable. It is shown in the paper that the above model has a strong practical background. By applying the Lyapunov theorem, a new robustness criterion is established for System (L), which yields three different sufficient conditions for the existence of a positive-definite matrix \(P\) such that \(V = x^T Px\) is the desired Lyapunov function. The results obtained extended the existing approaches on stability robustness analysis to those systems with a class of second-order nonlinear uncertain parameters. An example is also given to convey the advantages of the proposed method.
Mathematics Subject Classification ID: 93B35
Mathematics Subject Classification ID: 93D09
zbMATH DE Number: 1325622
zbMATH Keywords: Lyapunov theorem
zbMATH Keywords: second-order nonlinear uncertain parameters
zbMATH Keywords: stability robustness
zbMATH Keywords: linear uncertain state space systems
MaRDI profile type: Publication
full work available at URL: s0005-1098(98)00040-5
OpenAlex ID: W2041797390
description: scientific article; zbMATH DE number 1325622