Reduction of triangular controlled systems to linear systems without changing the control
type: Item
instance of: scholarly article
title: Reduction of triangular controlled systems to linear systems without changing the control
zbMATH Open document ID: 1050.93034
DOI: 10.1023/A:1014895307896
author: E. V. Sklyar
published in: Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: Single-input, triangular control systems of the form \[ \dot x_i= f_i(x_1,\dots, x_{i+1}),\quad\text{for }i= 1,2,\dots, n-1,\quad \dot x_n= f_n(x_1,\dots, x_n, u),\tag{\(*\)} \] with bounded controls \(| u|\leq 1\), are examined. The main result of this paper (Theorem 3) states and proves necessary and sufficient conditions for making the system \((*)\) equivalent by a state space diffeomorphism to the normal form \[ \dot z_i= z_{i+1},\quad i= 1,2,\dots, n-1,\quad \dot z_n= a_1 z_1+\cdots+ a_n z_n+ u.\tag{\(**\)} \] These conditions impose some constraints on the functions \(f_i\), \(i= 1,\dots, n\), and provide an explicit definition of the state space diifeomorphism in terms of differential operators \[ A_i= f_1(x_1, x_2){\partial\over\partial x_1}+\cdots+ f_i(x_1,\dots, x_{i+1}){\partial\over\partial x_i},\quad i= 1,2,\dots, n-1. \] In particular, these conditions allow one to decide when \((**)\) takes the form \(\dot z_i= z_{i+1}\), \(i= 1,2,\dots, n-1\), \(\dot z_n= u\), which is useful in optimal control.
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93B10
Mathematics Subject Classification ID: 93B17
zbMATH DE Number: 1916005
zbMATH Keywords: nonliner control system
zbMATH Keywords: triangular form
zbMATH Keywords: state space of diffeomorphism
zbMATH Keywords: normal form
reviewed by: Krzysztof Tchoń
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1916005