IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 10, OCTOBER 2003 1793
A Switching Algorithm for Global Exponential
Stabilization of Uncertain Chained Systems
Zairong Xi, Gang Feng, Z. P. Jiang, and Daizhan Cheng
Abstract—This note deals with chained form systems with strongly
nonlinear unmodeled dynamics and external disturbances. The objective
is to design a robust nonlinear state feedback law such that the closed-loop
system is globally
-exponentially stable. We propose a novel switching
control strategy involving the use of input/state scaling and integrator
backstepping. The new features of our controllers include the ability to
achieve Lyapunov stability, exponential convergence, and robustness to a
set of uncertain drift terms.
Index Terms—Backstepping, chained form systems, exponential stabi-
lization, input-state scaling, Lyapunov stability, robustness.
I. INTRODUCTION
Over the past decade, the control and stabilization of nonholonomic
systems has formed an active area within the nonlinear control commu-
nity; see, for example, the recent survey papers [5], [11], and the ref-
erences cited therein for an interesting introduction to this quickly ex-
panding area. This flow of research activity has been mainly triggered
by the well-known 1983 paper by Brockett [4], where a necessary con-
dition for asymptotic stabilization is stated. One of the consequences
of the necessary condition is that a nonholonomic system is not stabi-
lizable by stationary continuous state feedback. To overcome this im-
possibility, several interesting and fundamentally nonlinear approaches
have been proposed. Examples of these approaches are open-loop peri-
odic steeringcontrol,eithersmooth or continuoustime-varyingcontrol,
and discontinuous feedback control; see, for example, [1]–[3], [6], [7],
[9], [10], [13], and [15]–[20].
It should be noted that the majority of these constructive methods
have been developed around an important class of driftless nonholo-
nomic systems in chained form, which was brought to the literature
by [18]. As explained and illustrated in [11], [18], and the references
therein, many nonlinear mechanical systems with nonholonomic con-
straints on velocities can be transformed, either locally or globally, to
chained form systems via coordinates and state-feedback transforma-
tion. For instance, we have seen such examples as tricycle-type mobile
robots, cars towing several trailers, the knife edge, a vertical rolling
wheel, and a rigid spacecraft with two torque actuators.
Manuscript received January 21,2003; revisedMay 20, 2003. Recommended
by Associate Editor J. Huang. This work was supported in part by a Grant from
the Research Grants Council of the Hong Kong Special Administrative Region,
China under Project CityU 1024/02E, the National Natural Science Foundation
of China and the Institute of Systems Science, Academy of Mathematics and
Systems Science, and the Chinese Academy of Sciences. The work of Z. P. Jiang
was supported in part by the National Science Foundation under Grant ECS-
0093176 and Grant INT-9987317.
Z. Xi is with the Laboratory of Systems and Control, Institute of Systems
Science, Academy of Mathematics and Systems Science, Chinese Academy of
Sciences, Beijing 100080, China, and also with the Department of Manufacture
Engineering and Engineering Management, The City University of Hong Kong,
Kowloon Tong, Hong Kong (e-mail: mezrxi@cityu.edu.hk).
G. Feng is with the Department of Manufacture Engineering and Engineering
Management, The City University of Hong Kong, Kowloon Tong, Hong Kong
(e-mail: megfeng@cityu.edu.hk).
Z. P. Jiang is with the Department of Electrical and Computer Engineering,
Polytechnic University, Brooklyn, NY 11201 USA.
D. Cheng is with the Laboratory of Systems and Control, Institute of Systems
Science, Academy of Mathematics and Systems Science, Chinese Academy of
Sciences, Beijing 100080, China.
Digital Object Identifier 10.1109/TAC.2003.817937
As is well known in the literature on nonholonomic control systems,
a smooth time-varying state-feedback law can be applied to achieve
asymptotic stabilization but fails to meet the requirement of exponen-
tial convergence. However, exponential convergence is an important
performance characteristic for practical applications. To date, several
important steps have been made toward the design of a continuous
time-varying and/or discontinuous feedback law guaranteeing the ex-
ponential regulation of nonholonomic systems in chained form [2],
[10], [16], and [20]. Two types of control laws—discontinuous state
feedback and time-varying feedback—have been frequently used in the
recent literature to obtain an exponential rate of convergence for non-
holonomic control systems (see, for instance, [1], [2], [6], [10], [14],
[16], and [20]). However, the closed-loop systems are not Lyapunov
stable. Marchand and Alamir [15] obtained Lyapunov stability and ex-
ponential rate of convergence in the absence of disturbances. Since
their result depends on a Riccati equation, it could not be easily ex-
tended, if not impossible, to the occurrence of uncertain disturbances
[15].
The purpose of this note is to obtain both robust global exponen-
tial regulation and Lyapunov stability for a class of disturbed nonlinear
chained systems without imposing any restriction on the system order
and the growth of the uncertain nonlinearities. The contribution of the
note is twofold. We propose a systematic control design procedure
to construct a switching robust nonlinear control law which not only
solves the global exponential regulation problem, but also Lyapunov
stability problem for all plants in the considered class, including the
ideal chained system. For the Lyapunov stability with global exponen-
tial regulation, to the best of our knowledge, there is still no robustifi-
cation tool for nonholonomic systems design.
The remainder of this note is organized as follows. In Section II, the
class of nonholonomic systems with strongly nonlinear disturbances
is introduced and the problem of global exponential stabilization
is formulated. Section III first presents the input-state scaling tech-
nique and the backstepping design procedure and then a switching
control strategy. In Section IV, we illustrate our novel control design
methodology via a practical nonholonomic system with disturbances.
The numerical simulations testify to the effectiveness and robustness
aspects of the proposed robustification tool. Finally, some conclusions
are given in Section V.
II. P
ROBLEM FORMULATION
The purpose of this note is to consider a perturbed version of the
chained form [10]
_
x
0
=
d
0
(
t
)
u
0
+
x
0
d
0
(
t; x
0
)
_
x
1
=
d
1
(
t
)
x
2
u
0
+
d
1
(
t; x
0
;x;u
0
)
.
.
.
_
x
n
0
2
=
d
n
0
2
(
t
)
x
n
0
1
u
0
+
d
n
0
2
(
t;x
0
;x;u
0
)
_
x
n
0
1
=
d
n
0
1
(
t
)
u
+
d
n
0
1
(
t;x
0
;x;u
0
)
(1)
where
x
=(
x
1
;
...
;x
n
0
1
)
2
R
n
0
1
,
x
0
2
R
the functions
d
i
’s
and
d
i
’s denote the possible modeling error and neglected dynamics.
Throughout this note, the following assumptions will be required.
Assumption 1: For every
0
i
n
0
1
, there are (known) positive
constants
c
i
1
and
c
i
2
such that
c
i
1
d
i
(
t
)
c
i
2
8
t
0
:
(2)
0018-9286/03$17.00 © 2003 IEEE
1794 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 10, OCTOBER 2003
Assumption 2: For every
0
i
n
0
1
, there is a (known) smooth
nonnegative functions
i
that satisfies the inequalities
d
0
(
t; x
0
)
0
(
x
0
)
d
i
(
t; x
0
;x;u
0
)
j
(
x
1
;
...
x
i
)
j
i
(
x
1
;
...
x
i
;u
0
)
for all
(
t; x
0
;x;u
0
)
2
R
+
2
R
2
R
n
0
1
2
R
.
As explained and illustrated in [10], the structural triangularity con-
dition in Assumption 2 is a quite common assumption in the framework
of robust/adaptive nonlinear control [12].
We first recall a notion of
K
-exponential stability from [20], which
reduces to exponential stability in the Lyapunov sense when the func-
tion
is a linear function.
Definition 1: A system of the form
:_
x
=
f
(
x
)
with
x
2
R
n
is
said to be globally
K
-exponentiallystable (GES) ifthere exista positive
constant
and a function
of class
K
such that
8
x
(0)
2
R
n
,
8
t
0
j
x
(
t
)
j
(
j
x
(0)
j
)
e
0
t
8
t
0
:
(3)
This note aims to find explicit controllers
u
0
=
0
(
x
0
;x
)
u
=
(
x
0
;x
)
(4)
that globally
K
-exponentially stabilize all systems (1) satisfying As-
sumptions 1 and 2. A main difference with [1], [2], and [10] is that we
are interested in achievingstability properties in the senseof Lyapunov.
On the basis of Assumptions 1 and 2, we are led to choose the control
law
u
0
as
u
0
=
0
0
x
0
0
1
c
01
x
0
0
(
x
0
)
(5)
where
0
>
0
is a positive design parameter. As a result, the following
lemma can be established by considering the Lyapunov function can-
didate
V
0
=(1
=
2)
x
2
0
and by applying directly the Gronwall Lemma
(cf. [10]).
Lemma 1: For any initial instant
t
0
0
and any initial condition
x
0
(
t
0
)
2
R
, the corresponding solution
x
0
(
t
)
exists for each
t
t
0
and satisfies
lim
t
!1
x
0
(
t
)=0
. Furthermore, if
x
0
(
t
0
)
6
=0
then
x
0
(
t
)
6
=0
for all
t
t
0
.
Notice that the forward invariance property proved in Lemma 1 will
be used in the controller design andstability analysis inthe nextsection.
III. C
ONTROLLER DESIGN
A. Input-State Scaling and Backstepping Design
Introduce an input-state scaling discontinuous transformation de-
fined by [6] and [15]
i
=
x
i
u
n
0
(
i
+1)
0
;
1
i
n
0
1
:
(6)
Under the new
-coordinates, the
x
-system is transformed into
_
1
=
d
1
(
t
)
2
0
(
n
0
2)
1
_
u
u
+
(
t;x ;x;u
)
u
_
2
=
d
2
(
t
)
3
0
(
n
0
3)
2
_
u
u
+
(
t;x ;x;u
)
u
.
.
.
_
n
0
2
=
d
n
0
2
(
t
)
n
0
1
0
n
0
2
_
u
u
+
(
t;x ;x;u
)
u
_
n
0
1
=
d
n
0
1
(
t
)
u
+
d
n
0
1
(
t;x
0
;x;u
0
)
:
(7)
The inherently triangular structure of (1) suggests that we should
design the control inputs
u
0
and
u
in two separate stages.
Assumption 3: Assume that
u
0
:
R
!
R
is a continuous, almost
everywhere differentiable function, with the following properties:
P1) for all
t
0
,
u
0
(
t
)
6
=0
;
P2) for almost all
t
0
,
j
du
0
=dt
j
0
(
x
0
)
j
u
0
(
t
)
j
, where
0
(
x
0
)
is a known nonnegative function.
If
u
0
vanishes,
x
clearly becomes uncontrollable. P1) avoids this
loss of controllability. P2) is only convenient for the control design.
From Section II, it is known that
u
0
=
0
(
0
+(
0
=c
01
))
x
0
fulfills
Assumption 3 provided
x
0
(
t
0
)
6
=0
.
In the remainder of this section, we focus on designing the control
input
u
provided that Assumptions 1–3 are satisfied.
According to Assumption 3, the discontinuous state transformation
(6) is applicable because
u
0
(
t
)
6
=0
for every
t
t
0
. The design
of the control input
u
will be based on an application of the common
backstepping method to the transformed system (7). Indeed, (7) can be
written in the more compact form
_
1
=
d
1
(
t
)
2
+8
d
1
(
t; x
0
;x;u
0
)
_
2
=
d
2
(
t
)
3
+8
d
2
(
t; x
0
;x;u
0
)
.
.
.
_
n
0
2
=
d
n
0
2
(
t
)
n
0
1
+8
d
n
0
2
(
t;x
0
;x;u
0
)
_
n
0
1
=
d
n
0
1
(
t
)
u
+8
d
n
0
1
(
t;x
0
;x;u
0
)
(8)
where, for each
1
i
n
0
1
8
d
i
=
d
i
(
t; x
0
;x;u
0
)
u
n
0
(
i
+1)
0
0
(
n
0
(
i
+ 1))
i
_
u
0
u
0
:
(9)
Lemma 2: For each
1
i
n
0
1
, there exists a smooth nonneg-
ative function
8
i
such that
8
d
i
(
t; x
0
;x;u
0
)
j
(
1
;
111
;
i
)
j
8
i
(
x
0
;
1
;
111
;
i
;u
0
)
:
(10)
Proof: In view of (6), Assumptions 1–3, we have
8
d
i
(
t; x
0
;x;u
0
)
j
(
x
1
;
...
;x
i
)
j
u
n
0
(
i
+1)
0
i
(
x
1
;
111
;x
i
;u
0
)
+(
n
0
(
i
+ 1))
0
(
x
0
)
j
i
j
(
u
0
)
i
0
1
1
;
111
;
i
2
i
(
u
0
)
n
0
2
1
;
...
;
(
u
0
)
n
0
(
i
+1)
i
;u
0
+(
n
0
(
i
+ 1))
0
(
x
0
)
j
i
j
:
Therefore, the proof of Lemma 2 is completed.
Thanks to Lemma 2, (8) satisfies the “lower-triangularity” condition
and therefore, the systematic controller design for
u
can be obtained
using so-called backstepping methods [6], [10], [12].
Step 1: Let us begin with the scalar
1
subsystem of (8)
_
1
=
d
1
(
t
)
2
+8
d
1
(
t; x
0
;x;u
0
)
where
2
is regarded as the virtual control input. Let
z
1
=
1
and
introduce the Lyapunov function
V
1
=(1
=
2)
z
2
1
. Using Lemma 2,
the time derivative of
V
1
along the solutions of (8) satisfies
_
V
1
d
1
(
t
)
z
1
2
+
z
2
1
8
1
(
x
0
;z
1
;u
0
)
:
(11)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 10, OCTOBER 2003 1795
Then, with Assumption 1, we are led to introduce a virtual con-
trol function
1
and a new variable
z
2
1
(
x
0
;z
1
;u
0
)=
0
1
z
1
0
1
c
11
'
1
(
x
0
;z
1
;u
0
)
z
1
z
2
=
2
0
1
(
x
0
;z
1
;u
0
)
where
1
is a positive design parameter and
'
1
=
8
1
is a smooth
nonnegative function. Consequently, (11) implies
_
V
1
0
1
d
1
(
t
)
z
2
1
+
d
1
(
t
)
z
1
z
2
:
Note that
1
is a smooth function satisfying
1
(
x
0
;
0
;u
0
)=0
8
x
0
2
R:
Step
i
(2
i
n
0
2)
: As in [10] and Step 1, consider the Lya-
punov function candidate
V
i
=
V
i
0
1
(
z
1
;
...
;z
i
0
1
)+(1
=
2)
z
2
i
.
Therefore, we can choose a virtual control function
i
and a new
variable
z
i
+1
as follows:
i
=
0
i
z
i
0
i
j
=1
1
c
j
1
'
ij
(
x
0
;z
1
;
111
;z
i
;u
0
)
z
j
z
i
+1
=
i
+1
0
i
where
'
ij
(
x
0
;z
1
;
111
;z
i
;u
0
)
are some nonnegative function de-
rived from backstepping, such that
_
V
i
0
i
j
=1
(
j
d
j
(
t
)
0
i
+
j
)
z
2
j
+
d
i
(
t
)
z
i
z
i
+1
:
Step
n
0
1
: At this last step, consider the whole
-system (8)
where the true input
u
is to be designed on the basis of the virtual
control functions
i
’s. To this end, consider a positive–definite
and radially unbounded Lyapunov function
V
n
0
1
=
V
n
0
2
(
z
1
;
...
;z
n
0
2
)+
1
2
z
2
n
0
1
:
As in [6], [10], and [12], it is easy to know that some smooth non-
negative function
'
(
n
0
1)
j
(
x
0
;z
1
;
...
;z
n
0
1
;u
0
)(
j
=1
;
111
;n
0
1)
can be found such that along the solutions of (8)
_
V
n
0
1
0
n
0
1
j
=1
(
j
d
j
(
t
)
0
n
+1+
j
)
z
2
j
(12)
when choosing the control law
u
as
u
=
n
0
1
(
x
0
;
z
1
;
...
;z
n
0
1
;u
0
)
=
0
n
0
1
z
n
0
1
0
n
0
1
j
=1
1
c
j
1
z
j
'
(
n
0
1)
j
(
x
0
;
z
1
;
...
;z
n
0
1
;u
0
)
:
(13)
Therefore, the following theorem can be obtained.
Theorem 1: Under Assumptions 1–3, if parameters
i
’s satisfies
=min
f
j
c
j
1
0
n
+1+
j
j
;j
=1
;
...
;n
0
1
g
>
0
then the aforementioned control strategy (13) yields that the
x
-sub-
system of uncertain system (1) with
x
=(
x
1
;
111
;x
n
0
1
)
is well de-
fined and is globally
K
-exponentially stabilized at the origin.
Proof: Let
z
=(
z
1
;
...
;z
n
0
1
)
. According to (12), we have
_
V
n
0
1
0
V
n
0
1
which implies
j
z
(
t
)
jj
z
(0)
j
e
0
t
;t
0
:
Then [10]
j
(
t
)
j
(
j
(
x
0
(0)
;
(0)
;u
0
(0))
j
)
e
0
"t
;t
0
:
where
">
0
,
=(
1
;
...
;
n
0
1
)
and
is a class-
K
function.
Hence, (3) follows readily from (6).
As a particular case of Theorem 1, one has the following.
Theorem 2: Assumptions 1 and 2, if
x
0
(0)
6
=0
and parameters
i
’s
satisfy
= min
f
j
c
j
1
0
n
+1+
j
j
;j
=1
;
...
;n
0
1
g
>
0
then the aforementioned control strategy (5) and (13) yields that the
uncertain system (1) is globally exponentially regulated at the origin in
the sense that all the trajectories satisfy (3).
Proof: We only need to verify that
_
u
0
u
0
0
(
x
0
)
:
In fact,
u
0
=
0
0
x
0
0
(1
=c
01
)
x
0
0
(
x
0
)
, then
_
u
0
=
0
0
+
1
c
01
0
(
x
0
)+
1
c
01
x
0
0
0
(
x
0
)
d
0
u
0
+
x
0
d
0
:
So
_
u
0
u
0
c
02
+
0
(
x
0
)
0
2
0
+
1
c
01
0
(
x
0
)+
1
2
c
01
x
2
0
+
0
0
(
x
0
)
2
:
B. Switching Scheme
In the preceding discussions, we have given the controller expres-
sions (5) and (13) for
u
0
and
u
of (1) if the starting point of the
x
0
state
component is not zero, i.e.,
x
0
(
t
0
)
6
=0
. Without loss of generality, we
can assume that
t
0
=0
. Now, we discuss how to select the control laws
u
0
and
u
when
x
0
(0) = 0
.
The purpose of this section is to answer this question by proposing
a globally exponentially stabilizing static state feedback. Roughly
speaking, when the initial state is
x
0
(0) = 0
and
x
(0)
6
=0
we first
use an “almost” (nonzero) constant action
u
0
and the corresponding
control
u
that is designed based on a discontinuous coordinates trans-
formation of the form (6) and backstepping technique to drive the state
x
0
away from 0 in a short time duration [0,
t
s
), which depends only
on initial point. Then, the almost constant feedback law is switched to
an exponential regulator which is also based on a discontinuous co-
ordinates transformation of the form (6) and backstepping technique.
1796 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 10, OCTOBER 2003
However, in the present situation, the presence of nonlinear uncertain
functions
d
i
(0
i
n
0
1)
may lead some solutions to blow up
before the given switching time
t
s
. To prevent this phenomenon from
happening, the following switching control strategy for both control
inputs
u
0
and
u
is proposed.
Theorem 3: Let
•
,
,
0
,
and
i
(1
i
n
0
1)
be strictly positive real
constants so that
= min
f
j
c
j
1
0
n
+1+
j
j
;j
=1
;
...
;n
0
1
g
>
0;
•
T
0
be defined in [0,
1
)by
T
0
(
s
)= max
2
[
0
s;s
]
f
+
0
(
)
g
which is a nondecreasing continuous function;
•
0=
f
(0
;x
):
j
x
j6
=0
g
;
Then, the following static discontinuous feedback law globally
K
-exponentially stabilizes the uncertain chained form system (1).
Moreover, the feedback law is bounded.
i) When
(
x
0
(0)
;x
(0)) = (0
;x
(0))
2
0
,
u
0
(
t
)=
0
c
x
0
;
if
t<t
s
(
j
x
(0)
j
)
0
0
+
c
x
0
;
if
t
t
s
(
j
x
(0)
j
)
(14)
u
(
t
)=
n
0
1
(
x
0
;x;u
0
)
8
t>
0
(15)
where
t
s
(
j
x
(0)
j
) = min
f
;
(
c
01
=
(2
c
02
T
0
(
c
02
j
x
(0)
j
)))
;
j
x
(0)
jg
:
ii) When
(
x
0
(0)
;x
(0)) = (0
;
0)
u
0
=0
(16)
u
=0
:
(17)
iii) When
(
x
0
(0)
;x
(0))
=
2
0
[f
(0
;
0)
g
u
0
=
0
0
+
0
c
01
x
0
(18)
u
=
n
0
1
(
x
0
;x;u
0
)
:
(19)
In order to prove Theorem 3, the following Lemma is needed.
Lemma 3: Consider the uncertain differential equation
_
x
0
=
d
0
(
t
)
u
0
+
x
0
d
0
(
t; x
0
)
;x
0
2
R; x
0
(0) = 0
:
(20)
If
j
d
0
(
t; x
0
)
j
0
(
x
0
)
and
0
<c
01
d
0
(
t
)
c
02
, then the
closed-loop system with
u
0
=
0
x
0
((
0
(
x
0
))
=c
01
)
has the following
properties:
j
x
0
(
t
)
j
c
02
t; t >
0
and
x
0
(
t
)
>
0
;t>
0
where
>
0
.
Proof: It is easy to see that
u
0
(
t
)
is continuous and
u
0
(0) =
>
0
.So
_
x
0
(0)
>
0
and
x
0
(
t
)
is strictly increasing in a small time duration
[0,
t
1
]. Then, it is not difficult to know that
x
0
(
t
)
0
for all
t>
0
.
Differentiating
V
0
=
x
2
0
along (20), we obtain
dV
0
dt
=2
x
0
d
0
(
t
)
u
0
+
x
0
d
0
(
t; x
0
)
2
c
02
x
0
=2
c
02
p
V
0
:
Then
V
0
(
t
)
c
02
t
i.e.,
j
x
0
(
t
)
j
c
02
t
, for all
t>
0
. At the same time
dV
0
dt
=2
x
0
d
0
(
t
)
u
0
+
x
0
d
0
(
t; x
0
)
2
c
01
x
0
0
2
x
2
0
1+
c
02
c
01
0
(
x
0
)
=2
c
01
p
V
0
0
2 1+
c
02
c
01
0
(
x
0
)
V
0
:
Then, using the variable coefficient method, we have
V
0
(
t
)
(
c
01
)
2
e
0
2
(
t
)
t
0
e
(
)
d
2
(
c
01
)
2
e
0
2
(
t
)
t
2
where
(
t
)=(1+(
c
02
=c
01
))
t
0
0
(
x
0
(
s
))
ds
0
for all
t
. Then,
V
0
(
t
)
>
0
for all
t>
0
.
Proof of Theorem 3: If
(
x
0
(0)
;x
(0)) = (0
;
0)
and
(
x
0
(0)
;x
(0))
=
2
0
[f
(0
;
0)
g
, Theorem 3 is a direct conse-
quence of Theorem 1. So we only need to consider the case when
(
x
0
(0)
;x
(0))
2
0
.
First, the following inequality is satisfied:
u
0
(
t
)
2
;t<t
1
s
(
j
x
(0)
j
)
where
t
1
s
(
j
x
(0)
j
) = min
f
(
c
01
=
(2
c
02
T
0
(
c
02
j
x
(0)
j
)))
;
j
x
(0)
jg
. In-
deed, from Lemma 3, it follows that
j
x
0
(
t
)
j
c
02
t
1
s
(
j
x
(0)
j
)
when
t<t
1
s
(
j
x
(0)
j
)
. Then
u
0
(
t
)=
0
x
0
(
t
)
0
(
x
0
(
t
))
c
01
0
c
02
t
1
s
(
j
x
(0)
j
)
T
0
(
c
02
j
x
(0)
j
)
c
01
2
8
t<t
1
s
(
j
x
(0)
j
)
:
Second, from Lemma 3 we know that
x
0
(
t
s
)
6
=0
. So we can switch
from
u
0
(
t
)=
0
(
0
=c
01
)
x
0
when
t<t
s
(
j
x
(0)
j
)
to
u
0
(
t
)=
0
(
0
+
(
0
=c
01
))
x
0
when
t
t
s
(
j
x
(0)
j
)
.
Third, in order to employ the controller
u
obtained in Theorem 1,we
should verify that there exists
'
0
(
x
0
)
such that
_
u
0
u
0
'
0
(
x
0
)
for
t<t
s
(
j
x
(0)
j
)
:
It is easy to know that
_
u
0
=
0
1
c
01
0
(
x
0
)+
1
c
01
x
0
0
0
(
x
0
)
d
0
u
0
+
x
0
d
0
:
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 10, OCTOBER 2003 1797
So
_
u
0
u
0
c
02
+
1
x
2
0
+
2
0
(
x
0
)
2
1
c
01
0
(
x
0
)+
1
2
c
01
x
2
0
+
0
0
(
x
0
)
2
:
In view of the facts that
j
x
0
(
t
)
j
c
02
j
x
(0)
j
;
if
t<t
s
(
j
x
(0)
j
)
c
02
j
x
(0)
j
e
0
(
t
0
t
)
;
if
t
t
s
(
j
x
(0)
j
)
and
V
(
x
0
(
t
)
;x
(
t
)
;u
0
(
t
))
V
(
x
0
(0)
;x
(0)
;u
0
(0))
e
0
2
t
the closed-loop system is globally
K
-exponentially stable at the
origin.
Remark 1: It should be emphasized that a feedback controller may
become excessively large even for small states. In particular, this may
happen for initial conditions close to a singular manifold as in [6] and
[10]. The key feature of our proposed feedback laws is that the con-
troller
0
(
0
=c
01
)
x
0
(
t
)
is applied only as
x
0
(0)
is zero and
x
(0)
6
=0
in order to retrieve “some sufficient” controllability on the state
x
. Note
that along the trajectories of the closed-loop system, when
u
0
tends to
zero, the state does the same.
Remark 2: In the absence of input and state disturbances, (1) be-
comes the standard chained system (i.e.,
d
i
=1
,
d
i
=0
,
0
i
n
0
1
). An exponentially stabilizing controller was recently obtained
in [15], based on a Riccati equation. However, it is difficult, if not im-
possible, to extend the algorithm of [15] to the case when a chained
system is subject to disturbances as in (1).
IV. E
XAMPLE
A tricycle-type mobile robot with nonholonomic constraints on the
linear velocity has often been used as a benchmark example in the re-
cent literature on nonholonomic control systems design [8], [10]. In
[8], Morse et al. addressed the parking problem for the mobile robot of
unicycle type in the presence of parametric uncertainties
_
x
=
p
3
1
v
cos
_
y
=
p
3
1
v
sin
_
=
p
3
2
!
(21)
where
(
x; y
)
denotes the position of the center of mass of the robot,
is
the heading angle of the robot,
v
is the forward velocity,
!
is the angular
velocity of the robot, and
p
3
1
and
p
3
2
are (unknown) positive parameters
determined by the radius of the rear wheels and the distance between
them.
The problem addressed in [8] and [10] was to steer the robot to the
origin by a state-feedback control law, regardless of the value of the
unknown constant parameters
p
3
1
and
p
3
2
. In [8], a supervisory control
scheme was presented to solve the problem without a priori knowl-
edge of the parameters
p
3
1
and
p
3
2
. However, the convergence rate is
not exponential but asymptotic. In [10], using the following change of
coordinates and feedback:
x
0
=
x
1
=
x
sin
0
y
cos
x
2
=
x
cos
+
y
sin
u
0
=
!
u
=
v
(21) was transformed into the following form:
_
x
0
=
p
3
2
u
0
_
x
1
=
p
3
2
x
2
u
0
_
x
2
=
p
3
1
u
0
p
3
2
x
1
u
0
:
Then, a switching control scheme was presented to solve the problem
without a priori knowledge of the parameters
p
3
1
and
p
3
2
in [10]. The
closed-loop system is not Lyapunov stable although the convergence
rate is exponential.
We willdesign a robust state-feedbackcontroller to drive thestates of
(21) to the origin with exponential convergence and Lyapunov stability.
Introducing the variables
1
=
x
1
u
0
2
=
x
2
then
_
x
0
=
p
3
2
u
0
_
1
=
p
3
2
2
0
1
_
u
u
_
2
=
p
3
1
u
0
p
3
2
1
u
2
0
:
So, the following controller can be obtained.
•
x
0
(0) = 0
,
x
(0) = (
x
1
(0)
;x
2
(0))
6
=(0
;
0)
, see
the equation shown at the bottom of the page, where
t
s
(
j
x
(0)
j
) = min
f
;
(
p
2max
=
(2
p
1min
))
;
j
x
(0)
jg
,
>
0
,
0
>
0
,
>
0
,
1
>
(1
=p
1 min
)
,
2
>
0
,
3
>
0
+(1
=p
2 min
)
,
and
4
>
0
.
•
(
x
0
(0)
;x
(0)) = (0
;
0)
,
u
0
=0
u
=0
:
•
x
0
(0)
6
=0
u
0
=
0
0
x
0
u
=
0
4
+
p
2
2 max
4
p
1 min
1
0
2
0
x
2
0
0
2
3
+
3
0
2
+
3
p
2 max
p
1 min
(
2
+
3
1
)
where
0
>
0
,
3
>
0
+(1
=p
2 min
)
,
4
>
0
.
u
0
(
t
)=
;
if
t<t
s
(
j
x
(0)
j
)
0
0
x
0
;
if
t
t
s
(
j
x
(0)
j
)
u
(
t
)=
0
2
+
p
4
p
1+
2
+
2
1
2
+
1
p
p
(
2
+
1
1
)
;
if
t<t
s
(
j
x
(0)
j
)
0
4
+
p
4
p
1
0
2
0
x
2
0
0
2
3
+
3
0
2
+
3
p
p
(
2
+
3
1
)
;
if
t
t
s
(
j
x
(0)
j
)