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Stabilization of Nonholonomic Systems 209

5.5 DISCONTINUOUS CONTROL OF CHAINED SYSTEMS
To begin with, we transform system (5.22) through the σ process

ξ 1 = x 1
ξ 2 = x 2
.
.
.
(5.24)
x n−1
ξ n−1 = (n−3)
x
1
x n
ξ n = (n−2)
x
1
yielding a discontinuous system described by equations of the form
˙ ξ 1 = u 1
˙ ξ 2 = u 2
.
.
.
(5.25)
ξ n−2 − (n − 3)ξ n−1
˙ ξ n−1 = u 1
ξ 1
ξ n−1 − (n − 2)ξ n
˙ ξ n = u 1 .
ξ 1
Remark 5.8 The σ process (5.24) is a special case of (5.17) with α i = 1 for
all i = 1, ... , n − 1 and β i = i − 1 for all i = 1, ... , n − 1. Observe that such
β i fulfill the conditions (5.19).

Consider now the system (5.25) and apply the control u 1 =−kξ 1 , with
k > 0. A simple computation shows that the resulting system, described by
equations of the form

˙ ξ = Aξ + b 2 u 2 (5.26)

where ξ =[ξ 1 , ξ 2 , ... , ξ n ] ,
 
−k 0 0 0 ... 0
0 0 0 0 ... 0
 
 
0 −k k 0 ... 0
 
 
0 0 −k 2k ... 0 (5.27)
A =  

 . . . . . . 
 . . . . . . . . . . . . 

0 0 0 0 ··· (n − 2)k
© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 209 — #23