Optimal induced \(l^1\)-norm state feedback control
type: Item
instance of: scholarly article
title: Optimal induced \(l^1\)-norm state feedback control
zbMATH Open document ID: 0946.93016
DOI: 10.1016/S0005-1098(98)00214-3
author: Jie Yu
author: A. Sideris
published in: Automatica
publication date: 1999-12-05T00:00:00Z
review text: The authors discuss first the advantages of \(\ell^2\), \(\ell^\infty\), and \(\ell^1\) norms in closed loop feedback systems synthesis. Amplification in \(\ell^2\) generally refers to amplification of some form of energy, the \(\ell^\infty\) norm amplifies some component of the system, while the \(\ell^1\) norm distributes the effects of amplification among the components. Frequently it is used to take care of the worst possible case. It is known that closed loop robustness is easily handled within the \(\ell^1\) framework. Because of the duality between \(\ell^\infty\), and \(\ell^1\) norms, in linear systems with linear controls, one can manipulate the control of a system and the one of the transposed (adjoint) system. However, in nonlinear systems these are distinct, separate problems. The following (standard) linear discrete time system is considered: \[ x(k+ 1)= Ax(k)+ B_1w(k)+ B_2u(k),\tag{1} \] \[ z(k)= C_1x(k)+ D_{11}w(k)+ D_{12}u(k),\quad y(k)= x(k), \] where \(x\in\mathbb{R}^n\), \(w\in \mathbb{R}^{n_1}\), \(u\in \mathbb{R}^{n_2}\), \(z\in\mathbb{R}^n\), and \(B_1\) is a full rank vector. Substituting a given control law \(u= u(x)\) the closed loop system reduces to \[ x(k+ 1)= f[x(k)]+ B_1w(k),\quad z(k)= g[x(k)]+ D_{11}w(k),\tag{2} \] with \(f(x)= Ax+ B_2u(k)\), and \(g(x)= C_1x+ D_{12}u(x)\). The transfer function \(G\) maps the input \(w\) into the output \(z\). A definition of the ``analysis problem'' as interpreted by the reviewer consists in finding the solution (the state of the system: \(x(k)\)) at all times \(k= 0,1,2,\dots\) when all the other quantities (and in particular \(G\)) are known, and the transfer function \(G\) does not exceed some a priori given \(\ell^1\) norm: \(\|G\|_{\ell^1}= \sup\|z\|_1/\|w\|_1\), if \(\|w\|_1\neq 0\). The authors' definition differs from that one, which is certainly admissible, since it is consistent. The synthesis (or control) problem consists of finding a control law (it is not necessarily unique) \(u= u(x)\…
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 93B36
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 93B40
zbMATH DE Number: 1325752
zbMATH Keywords: \(\ell^1\) synthesis
zbMATH Keywords: generalized bounded real inequality
zbMATH Keywords: \(\ell^1\) norms
zbMATH Keywords: discrete time system
zbMATH Keywords: \(\lambda\)-contractive set
MaRDI profile type: Publication
full work available at URL: s0005-1098(98)00214-3
OpenAlex ID: W2181236131
description: scientific article; zbMATH DE number 1325752