Application of Bernstein expansion to the solution of control problems
type: Item
instance of: scholarly article
title: Application of Bernstein expansion to the solution of control problems
zbMATH Open document ID: 0965.65095
DOI: 10.1023/A:1009934614393
author: Jürgen Garloff
published in: Reliable Computing
publication date: 2001-08-02T00:00:00Z
review text: The author provides an informative survey of the use of Bernstein expansion in control theory. In particular, he describes reliable methods for solving stability problems. Consider a family of polynomials \[ f(s,p)= a_0(p) s^m+\cdots+ a_{m-1}(p) s+ a_m(p), \] where the coefficients \(a_k(p)\) depend polynomially on the parameter vector \(p= (p_1,p_2,\dots, p_n)\). The parameter vector \(p\) is known to belong to a given box \(B\subseteq \mathbb{R}^n\). For \(p\in B\), the polynomial \(f(.,p)\) is \({\mathcal D}\)-stable, if all zeros of \(f(.,p)\) are inside \({\mathcal D}\), where \({\mathcal D}\) is a subset of the complex plane. The family is robust \({\mathcal D}\)-stable, if \(f(.,p)\) is \({\mathcal D}\)-stable for every \(p\in B\). Special choices of \({\mathcal D}\) lead to Hurwitz-stability, to Schur-stability, and to damping. Also covered is a related topic: Solving systems of strict polynomial inequalities over a given box \(B\subseteq \mathbb{R}^n\) with Bernstein expansion. Numerical examples using interval arithmetic illustrate the described methods.
reviewed by: Herbert Fischer
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 93B35
Mathematics Subject Classification ID: 93D09
Mathematics Subject Classification ID: 65G40
zbMATH DE Number: 1513373
zbMATH Keywords: sensitivity
zbMATH Keywords: interval analysis
zbMATH Keywords: numerical examples
zbMATH Keywords: Bernstein expansion
zbMATH Keywords: control theory
zbMATH Keywords: stability
zbMATH Keywords: Hurwitz-stability
zbMATH Keywords: Schur-stability
zbMATH Keywords: damping
zbMATH Keywords: strict polynomial inequalities
zbMATH Keywords: interval arithmetic
MaRDI profile type: Publication
full work available at URL: a:1009934614393
OpenAlex ID: W1572998018
description: scientific article; zbMATH DE number 1513373