Passivity of nonlinearity incremental systems: application to PI stabilization of nonlinear RLC circuits
type: Item
instance of: scholarly article
title: Passivity of nonlinearity incremental systems: application to PI stabilization of nonlinear RLC circuits
zbMATH Open document ID: 1231.94081
author: Bayu Jayawardhana
author: Eloísa García-Canseco
author: Fernando Castaños
author: Romeo Ortega
published in: Systems \& Control Letters
publication date: 2009-02-09T00:00:00Z
review text: In the paper, the authors investigate a class of nonlinear systems described by the set of vector nonlinear equations \(dx/dt=f(x) + g\cdot u\), \(y=h(x)\). Here, \(x\), \(y\) and \(u\) are column vectors containing functions of time \(t\), \(f\) and \(h\) are nonlinear vector mappings and \(g\) is a constant matrix of full rank. All matrices have adequate sizes. If \(x_0\) is a fixed non-zero equilibrium point, one can associate to the initial system a new system where the evolution is described in terms of the deviations from the equilibrium point, i.e. \(x-x_0\). This last system is called the incremental system (IS). It is worth pointing out that the behavior of the IS is of importance in nonlinear circuit theory. A standard result states that in the class of linear time-invariant systems (LTIS), if the initial system is passive, then its associated IS has the same property. Then the following nontrivial question can be asked: Under what conditions is the implication which is true for the class LTIS, also true for nonlinear systems? Some important results pertaining to this question are given in the present paper for the class described by the aforementioned system. The main quantity which here is under attention is precisely the storage function, denoted by \(H(x)\). For the class of LTIS such a function exists and is non-negative. The results obtained in the paper are expressed in three propositions. In Proposition 1, the canonic field taken into account is assumed to be passive, to have a non-negative storage function and to satisfy a certain inequality involving the variables of the associated IS. Then its IS is also passive and the form of its \(H(x)\) is given. Proposition 2 gives conditions under which the storage function has a unique global minimum. It is then shown that, under the conditions contained in Proposition 1, the system can be globally stabilized with a linear PI control. In order to avoid some difficult calculations of \(H(x)\) for an RLC ne…
reviewed by: Dumitru Stanomir
Mathematics Subject Classification ID: 94C05
Mathematics Subject Classification ID: 93D05
Mathematics Subject Classification ID: 93D15
zbMATH DE Number: 5504340
zbMATH Keywords: nonlinear systems
zbMATH Keywords: RLC circuits
zbMATH Keywords: passivity conditions
zbMATH Keywords: global stability
MaRDI profile type: Publication
full work available at URL: j.sysconle.2007.03.011
OpenAlex ID: W2174497162
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DOI: 10.1016/J.SYSCONLE.2007.03.011
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description: scientific article; zbMATH DE number 5504340