scispace - formally typeset

Journal ArticleDOI

A switching algorithm for global exponential stabilization of uncertain chained systems

07 Oct 2003-IEEE Transactions on Automatic Control (IEEE)-Vol. 48, Iss: 10, pp 1793-1798

TL;DR: A novel switching control strategy is proposed involving the use of input/state scaling and integrator backstepping and the ability to achieve Lyapunov stability, exponential convergence, and robustness to a set of uncertain drift terms is proposed.

AbstractThis 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 Kexponentially 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.

Topics: Backstepping (59%), Exponential stability (59%), Robust control (57%), Lyapunov stability (56%), Robustness (computer science) (53%)

Summary (2 min read)

I. INTRODUCTION

  • As explained and illustrated in [11] , [18] , and the references therein, many nonlinear mechanical systems with nonholonomic constraints on velocities can be transformed, either locally or globally, to chained form systems via coordinates and state-feedback transformation.
  • 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 exponential regulation of nonholonomic systems in chained form [2] , [10] , [16] , and [20] .
  • Finally, some conclusions are given in Section V.

II. PROBLEM FORMULATION

  • Throughout this note, the following assumptions will be required.
  • The authors first recall a notion of K-exponential stability from [20] , which reduces to exponential stability in the Lyapunov sense when the function is a linear function.
  • Notice that the forward invariance property proved in Lemma 1 will be used in the controller design and stability analysis in the next section.

A. Input-State Scaling and Backstepping Design

  • P2) is only convenient for the control design.
  • In the remainder of this section, the authors focus on designing the control input u provided that Assumptions 1-3 are satisfied.
  • 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] .
  • Therefore, the following theorem can be obtained.

B. Switching Scheme

  • The purpose of this section is to answer this question by proposing a globally exponentially stabilizing static state feedback.
  • To prevent this phenomenon from happening, the following switching control strategy for both control inputs u 0 and u is proposed.
  • Then, the following static discontinuous feedback law globally K-exponentially stabilizes the uncertain chained form system (1).
  • Proof: Remark 1: It should be emphasized that a feedback controller may become excessively large even for small states.
  • Note that along the trajectories of the closed-loop system, when u 0 tends to zero, the state does the same.

IV. EXAMPLE

  • A tricycle-type mobile robot with nonholonomic constraints on the linear velocity has often been used as a benchmark example in the recent literature on nonholonomic control systems design [8] , [10] .
  • The convergence rate is not exponential but asymptotic.
  • The closed-loop system is not Lyapunov stable although the convergence rate is exponential.
  • The authors will design a robust state-feedback controller to drive the states of (21) to the origin with exponential convergence and Lyapunov stability.
  • So, the following controller can be obtained.

V. CONCLUSION

  • The problem of global K-exponential stabilization is considered for a class of nonholonomic chained systems with strongly nonlinear input/state driven disturbances and drifts.
  • Using input-state scaling and backstepping techniques, a globally exponentially convergent state-feedback control law is designed.
  • Using a switching scheme dependent on the initial condition, Lyapunov stability and exponential convergence are guaranteed for the closed-loop system.
  • The simulations results in a wheeled mobile robot have demonstrated the effectiveness of the proposed control design approach.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

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
)
jj
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
)

Citations
More filters

Journal ArticleDOI
TL;DR: A novel switching controller is proposed with guaranteed robustness to orientation error and unknown parameters in mobile robots and a generalized version of Krasovskii-LaSalle theorem in time-varying switched systems is proposed.
Abstract: This paper is concerned with the study of, both local and global, uniform asymptotic stability for general nonlinear and time-varying switched systems. Two concepts of Lyapunov functions are introduced and used to establish uniform Lyapunov stability and uniform global stability. With the help of output functions, an almost bounded output energy condition and an output persistent excitation condition are then proposed and employed to guarantee uniform local and global asymptotic stability. Based on this result, a generalized version of Krasovskii-LaSalle theorem in time-varying switched systems is proposed. For switched systems with persistent dwell-time, the output persistent excitation condition is guaranteed to hold under a zero-state observability condition. It is shown that several existing results in past literature can be covered as special cases using the proposed criteria. Interestingly, as opposed to previous work, the main results of this paper are applicable to the situation where some switching systems are not asymptotically stable at the origin. The robust practical regulation problem of nonholonomic mobile robots is studied as a way of demonstrating the power of the proposed new criteria. A novel switching controller is proposed with guaranteed robustness to orientation error and unknown parameters in mobile robots.

165 citations


Cites background from "A switching algorithm for global ex..."

  • ...Recently, many interesting results were presented to provide a solid foundation for the performance analysis of hybrid control [2], [4], [5], [7], [9]–[12], [24], [27]–[29], [31], [33]....

    [...]

  • ...Other interesting directions include the study of the limiting systems of switched systems as done in [22] with applications to the design of switching-based stabilizing and tracking controllers for physical systems; see [33] for preliminary results....

    [...]


Journal ArticleDOI
TL;DR: A novel switching control strategy with help of homogeneity, time-rescaling, and Lyapunov-based method is proposed to design a nonsmooth state feedback law such that the controlled chained form system is both LyAPunov stable and finite-time convergent within any given settling time.
Abstract: This note considers finite time stabilization of uncertain chained form systems. The objective is to design a nonsmooth state feedback law such that the controlled chained form system is both Lyapunov stable and finite-time convergent within any given settling time. We propose a novel switching control strategy with help of homogeneity, time-rescaling, and Lyapunov-based method. Also, the simulation results show the effectiveness of the proposed control design approach.

114 citations


Journal ArticleDOI
TL;DR: A switching control strategy is employed to get around the smooth stabilization issue (difficulty) associated with nonholonomic systems when the initial state of system is known and a dynamic output feedback controller is developed with a filter of observer gain.
Abstract: This paper deals with chained form systems with strongly nonlinear disturbances and drift terms. The objective is to design robust nonlinear output feedback laws such that the closed-loop systems are globally exponentially stable. The systematic strategy combines the input-state-scaling technique with the so-called backstepping procedure. A dynamic output feedback controller for general case of uncertain chained system is developed with a filter of observer gain. Furthermore, two special cases are considered which do not use the observer gain filter. In particular, a switching control strategy is employed to get around the smooth stabilization issue (difficulty) associated with nonholonomic systems when the initial state of system is known.

111 citations


Cites background or methods from "A switching algorithm for global ex..."

  • ...Step i (2pipn 2): As in [17,23,26], consider the Lyapunov function candidate Vi 1⁄4 V i 1ðt;P; e; z1; ....

    [...]

  • ...Then the design of the control input u will be obtained using the standard backstepping method shown in [17,18,23,26] to the transformed system (9), i....

    [...]

  • ...We recently proposed a switching scheme to achieve Lyapunov stability and exponential convergence for uncertain chained form systems using state feedback in [23]....

    [...]

  • ...As in [26,17,18,23], it is easy to know that some suitable smooth functions jðn 1Þjðt;P; x0;z1; ....

    [...]


Journal ArticleDOI
Abstract: This paper investigates the problem of output-feedback adaptive stabilization control design for non-holonomic chained systems with strong non-linear drifts, including modelled non-linear dynamics, unmodelled dynamics, and those modelled but with unknown parameters. An observer and an estimator are introduced for state and parameter estimates, respectively. By using the integrator backstepping approach and based on the observer and parameter estimator, a constructive design procedure for output-feedback adaptive stabilization control is given. It is shown that, under some conditions, the control design ensures the closed-loop system is globally asymptotically stable when there is no non-linear drift in the first subsystem, and semiglobally asymptotically stable, otherwise. An example is given to show the effectiveness of the theory.

73 citations


Cites background from "A switching algorithm for global ex..."

  • ...In Xi et al. (2003), it is assumed not only that ’i0 ¼ 0 in the inequality (7), but also that the modelled dynamics fi and the dynamics with unknown parameters Ti ð yÞ do not exist, i.e., fi¼ 0 and Ti ð yÞ ¼ 0, although the gains of u0, x2u0, . . . , xn 1u0 and u on the right hand sides of the…...

    [...]


Journal ArticleDOI
TL;DR: For a group of linear systems, each under a saturated linear, not necessarily stabilizing, feedback law, the problem of designing such a switching scheme as a constrained optimization problem with the objective of maximizing an estimate of the domain of attraction is solved.
Abstract: For a group of linear systems, each under a saturated linear, not necessarily stabilizing, feedback law, we design a switching scheme such that the resulting switched system is locally asymptotically stable at the origin with a large domain of attraction. By expressing each saturated linear feedback in a convex hull of a group of auxiliary linear feedbacks, we formulate and solve the problem of designing such a switching scheme as a constrained optimization problem with the objective of maximizing an estimate of the domain of attraction. Simulation results indicate that the resulting domain of attraction extends well beyond the linear regions of the actuators.

70 citations


References
More filters

Book
01 Jan 1995
Abstract: From the Publisher: Using a pedagogical style along with detailed proofs and illustrative examples, this book opens a view to the largely unexplored area of nonlinear systems with uncertainties. The focus is on adaptive nonlinear control results introduced with the new recursive design methodology--adaptive backstepping. Describes basic tools for nonadaptive backstepping design with state and output feedbacks.

6,912 citations


Journal ArticleDOI
Abstract: We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

2,864 citations


"A switching algorithm for global ex..." refers background in this paper

  • ...Examples of these approaches are open-loop periodic steering control, either smooth or continuous time-varying control, and discontinuous feedback control; see, for example, [1]–[3], [6], [7], [9], [10], [13], and [15]–[20]....

    [...]


01 Jan 1983
Abstract: We consider the loca1 behavior of control problems described by (* = dx/dr) i=f(x,u) ; f(x,0)=0 o and more specifically, the question of determining when there exists a smooth function u(x) such that x = xo is an equilibrium point which is asymPtotically stable. Our main results are formulated in Theorems 1 and 2 be1ow. Whereas it night have been suspected that controllability would insure the exlstence of a stabilizing control law, Theorem I uses a degree-theoretic argument to show this is far from being the case. The positive result of Theorem 2 can be thought of as providing an application of high gain feedback in a nonlinear setting. 1. Introduction In this paper we establish general theorems which are strong enough to irnply, among other things, that a) there is a continuous control law (u,v) = (u(xry, z) rv(x,y,z)) which makes the origin asympEoticatly stable for x=u y=v z=xy and that b) there exists no continuous control law (urv) = (u(xryrz), v(x,y,z)) which makes the origin asymptotically stable for 181

2,831 citations


Journal ArticleDOI
TL;DR: Methods for steering systems with nonholonomic c.onstraints between arbitrary configurations are investigated and suboptimal trajectories are derived for systems that are not in canonical form.
Abstract: Methods for steering systems with nonholonomic c.onstraints between arbitrary configurations are investigated. Suboptimal trajectories are derived for systems that are not in canonical form. Systems in which it takes more than one level of bracketing to achieve controllability are considered. The trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level. A class of systems that can be steered using sinusoids (claimed systems) is defined. Conditions under which a class of two-input systems can be converted into this form are given. >

1,726 citations


Journal ArticleDOI
Abstract: Provides a summary of recent developments in control of nonholonomic systems. The published literature has grown enormously during the last six years, and it is now possible to give a tutorial presentation of many of these developments. The objective of this article is to provide a unified and accessible presentation, placing the various models, problem formulations, approaches, and results into a proper context. It is hoped that this overview will provide a good introduction to the subject for nonspecialists in the field, while perhaps providing specialists with a better perspective of the field as a whole. The paper is organized as follows: introduction to nonholonomic control systems and where they arise in applications, classification of models of nonholonomic control systems, control problem formulations, motion planning results, stabilization results, and current and future research topics.

1,233 citations


"A switching algorithm for global ex..." refers background in this paper

  • ...INTRODUCTION Over the past decade, the control and stabilization of nonholonomic systems has formed an active area within the nonlinear control community; see, for example, the recent survey papers [5], [11], and the references cited therein for an interesting introduction to this quickly expanding area....

    [...]

  • ...As explained and illustrated in [11], [18], and the references therein, many nonlinear mechanical systems with nonholonomic constraints on velocities can be transformed, either locally or globally, to chained form systems via coordinates and state-feedback transformation....

    [...]