The Kalman-Yakubovich-Popov inequality for differential-algebraic systems
type: Item
instance of: scholarly article
title: The Kalman-Yakubovich-Popov inequality for differential-algebraic systems
zbMATH Open document ID: 1331.34018
author: Olaf Rendel
author: Matthias Voigt
author: Timo Reis
published in: Linear Algebra and its Applications
publication date: 2015-10-13T00:00:00Z
review text: In this well-written paper, the authors revisit the Kalman-Yakubovich-Popov inequality for differential-algebraic control systems of the form \[ E\dot x(t)=Ax(t)+Bu(t),\tag{1} \] where \(E,\,A\in \mathbb K^{n\times n}\), \(B\in \mathbb K^{n\times m}\), \(\mathbb K= \mathbb C\) or \(\mathbb R\), and the pencil \(sE-A\) is supposed to be regular. Unlike previous works on this topic, the authors neither assume impulse controllability nor make any index assumption on system (1). First, the authors point out the relation between the positive semi-definiteness of the Popov function on the imaginary axis and the solvability of the Kalman-Yakubovich-Popov inequality restricted to a certain subspace associated with system (1). Then, the authors characterize the solution set of the Lur'e equation associated with system (1) and discuss also its connection to the rank-minimizing solutions of the Kalman-Yakubovich-Popov inequality. Finally, consequences of these results for the linear-quadratic optimal control problem and comparison to other approaches are discussed.
reviewed by: Vu Hoang Linh
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 49N10
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 34H05
zbMATH DE Number: 6493724
zbMATH Keywords: differential-algebraic equations
zbMATH Keywords: Kalman-Yakubovich-Popov inequality
zbMATH Keywords: Lur'e equations
zbMATH Keywords: even matrix pencils
zbMATH Keywords: algebraic Riccati equations
zbMATH Keywords: linear-quadratic optimal control problems
MaRDI profile type: Publication
full work available at URL: j.laa.2015.06.021
OpenAlex ID: W1159463588
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DOI: 10.1016/J.LAA.2015.06.021
description: scientific article; zbMATH DE number 6493724