Intelligent Control and Automation, 2011, 2, 182-185
doi:10.4236/ica.2011.23022 Published Online August 2011 (http://www.SciRP.org/journal/ica)
Continuous Stabilizing of First Order Single Input
Nonlinear Systems
Aref Shahmansoorian
Department of Electrical Engineering, Imam Khomeini International University, Qazvin, Iran
Received October 20, 2010; revised May 14, 2011; accepted May 21, 2011
Abstract
In this paper, stabilizability of first order nonlinear systems by a smooth control law is investigated. The
main results are presented by the examples and finally summarized in a lemma. The proof for the lemma is
according to Sontag’s formula. In addition, it is explained that using weak control Lyapunov functions in
Sontag’s formula generates (possibly nonsmooth) the control law, which globally stabilizes the system-
globally asymptotic stability needs more investigation.
Keywords: Control Lyapunov Function, Inverse Optimality, Sontag’s Formula
1. Introduction
Consider the following nonlinear system:
 
x
fx gxu
 (1)
where n
x
R is state space vector, is control
input vector and
m
uR
 
, n
f
xgx R.
Definition [1]: A differentiable positive definite and
radially unbounded function is
called a CLF for the system (1), if for each ,

:
n
Vx RR
0x
0
0
 
0
gf
LV xLV x  (2)
If there exist nonzero points where

0Vx

, then
is sometimes referred to as Weak Control
Lyapunov Function (WCLF) [2,3].

Vx
Assume that is a CLF for the system (1). It is
known that the existence of a CLF for the system (1) is
equivalent to the existence of a globally asymptotic sta-
bilizing control law which is continuous
everywhere except possibly at [2]. If

Vx

,ukxx0


Vx is a
CLF for the system (1), then a particular stabilizing con-
trol law


s
ux
, smooth for all , is given by Son- 0x
tag’s formula: Equation (3) [3,4]
It is often desirable to guarantee at least Lipschitz con-
tinuity of the control law at in addition to its
smoothness elsewhere [1]. A further characterization of a
stabilizing control law for (1) with a given
0x

x

s
u


Vx is continuous at 0x

if and only if the CLF
satisfies the small control property [3]. It is well known
there is a class of nonlinear systems that can not be stabi-
lized by a continuous time-invariant feedback. Examples
of systems which do not admit continuous stabilizing
feedback laws are systems which do not satisfy brokett’s
necessary condition for continuous stabilizability [5,6].
Stabilizability of nonlinear systems is studied in lit-
eratures [7,8]. In [3] Brockett defines a necessary condi-
tion for stabilizability of nonlinear systems by a con-
tinuous feedback. In this paper sufficient condition for
stabilizability of single input nonlinear systems by a con-
tinuous feedback is introduced.
2. Problem Formulation
Consider the following nonlinear system:
     

  


2
2
(())
()
0 0
T
ffgg T
gg
T
sgg
g
LVxLVxLVx LVxLV xLVx
ux LV xLV x
LV x







0
(3)

A. SHAHMANSOORIAN 183
 
x
fx gxu
 (4)
where
x
R is the state space vector, is the
control input vector and
uR
 
,
f
xgx R are smooth.
The question is, “when can the system (1) be stabi-
lized at by a smooth feedback control law?”
0x
It is clear that when the unforced system is GAS, then
the problem is solved. But when the unforced system is
unstable or locally stable, the problem depends on the
roots of the equation . In the next section all
possible situations by using numerical examples will be
presented.

0gx
3. Examples
Example 1:
Consider the following nonlinear system,

31
x
xx 
u (5)
The equation has a root at

0gx1
x
, but the
unforced system is stable. Thus this system can be stabi-
lized by a continuous control law.
Example 2:
Consider the following nonlinear system,

21
x
xx 
u (6)
Although the equation has a root at

0gx1
x

,
the unforced system solutions with initial states
converge to the origin.

00x
Hence this system can be stabilized by a continuous
control law.
Example 3:
Consider the following nonlinear system,

31
x
xx
u (7)
The equation has a root at

0gx1
x
 but the
unforced system solutions with initial states


00x
escape to infinity. For that reason the system can not be
stabilized by a continuous control law.
Example 4:
Consider the following nonlinear system,

22
1
x
xx 
u (8)
The equation


0gx has roots at 1
x

and
. For the root 1x 1
x
x the argument is as example 2,
but for the initial states

00

the unforced system
solutions escape to infinity. Because of that the system
can not be stabilized by a continuous control law.
Example 5:
Consider the following nonlinear system,
32
x
xxu
 (9)
The unforced system is unstable, and the equation


0gx

has the root 0x

. This system can be stabi-
lized by a smooth control law (i.e. ). Actually
when the equation
2u x


0gx

has only the root 0x

,
the system can be stabilized by a control law which is
smooth everywhere except possibly at
0x
Example 6:
Consider the following nonlinear system,
23
x
xxu 
 (10)
This system can not be stabilized by a smooth control
law. Nevertheless, this system can be stabilized by a
control law, which is continuous at every nonzero
x
and is right-continuous at . The control law,
0x

2
11
0
0 0
Sgn xxx
uxx






(11)
globally asymptotically stabilizes the system and this
control law is right-continuous at . 0x
The reason is that the unforced system solution with
initial states


00x is stable.
Example 7:
Consider the following nonlinear system,


2
14
x
xx xu
 (12)
The equation


0gx

has the root . The un-
forced system is unstable, but with the initial states
0x


10x0


0x
the unforced system solutions converge
to the origin. Therefore the system can be stabilized by a
control law, which is continuous at every nonzero x, and
at

is left continuous.
4. The Existence of WCLF
Actually the single input system (1) when the unforced
system is not stable and the equation has real
nonzero root(s) has not CLF and has only WCLF. The
existence of WCLF is not the sufficient condition for the
existence of a globally asymptotic stabilizing control law
which is continuous everywhere except possibly at

0gx
0x

. Furthermore using WCLF in Sontag’s formula
generates a (possibly nonsmooth) control law, which
guarantees asymptotic stability-globally asymptotic sta-
bility need to more investigation.
Example 8:
Consider the following second order nonlinear system,



32
12 1121
3
2122
2
x
xxxxx
xxxxu
 
 


u
2
(13)
It can be proved that the function 2 is a
WCLF for the system. With all initial states interior the
circle

2
1
Vxx x
22
12
1xx

 and a smooth stabilizing control law,

A. SHAHMANSOORIAN 184
the state trajectories converge to the origin. Globally
asymptotic stabilizing by smooth control law is not pos-
sible.
Example 9:
Consider the following second order nonlinear
system,


12
2
1
21 12
2
1
5 πarctan543
221 25
xx
x
x
xx x
x


 



u
(14)
In [2] the function:

22
12
π
2
Vxxx
is used as a CLF in the Sonag’s formula. It can be veri-
fied the function:

22
12
π2
2
Vxx x
is a WCLF for the system. Using this WCLF in the Son-
tag’s formula (Equation (3)) yields a discontinuous con-
trol law which does not globally asymptotically stabilize
the system. The state trajectory with this control law and
the initial state converges to the point
(2.946,0). In Figure 1 the state trajectory is shown.
 
032
T
x
5. The Main Results
From the above examples the following lemma can be
suggested.
Lemma 1: Consider the following single input first or-
der system,
Figure 1. State trajectory for the example 9.


x
fx gxu
 (15)
where
x
R

is state variable, is control input
vector and
uR


f
xR

and

g
xR are smooth. As-
sume the unforced system,

x
fx  (16)
is locally asymptotically stable and its domain of attrac-
tion is




0,
x
ab and the roots of the equation


0gx

belong to the interval The system can
be stabilized by a continuous control law.

,.ab
Proof: Using

2
1
2
Vx x as a WCLF for the system,
the Sontag’s formula gives:


  
 

4
22
() ,0
0, 0
ux
fxSgnxfxx gxgx
gx
gx











(17)
It can be shown that this control law globally asymp-
totically stabilizes the system (15). Assume


0gx

has a nonzero root
x
c

such that . Accord-
ing to the assumption of lemma
acb


0,xf x



,, 0.xabx


Thus we have:
 

    
 
 



3
2
24
2
3
3
24
2
lim lim
lim 0
xc xc
xc
xgx
ux fxSignxfxxgx
xgx
xf xxf xxgx







This proves the continuity of the control law (17).
When the equation


g
xR has root(s) at 0.x

Then it is clear that:

 
2
0
lim
x
f
x
g
x

is equal to zero or infinity (when

f
x and


g
x are
smooth and the system (16) is locally stable, this limit
can not be equal to a nonzero finite value). Using this
fact, it can be proved that:
 


 
 
 
2
2 00
2
22
lim lim
0
xx
xgx
ux
fx fx
Sign xx
gx gx









Remark 1: If the unforced system is unstable and the
unforced system solutions with initial states




00,xb0

,






0,0xa converge to the origin,

A. SHAHMANSOORIAN 185
then the system can be stabilized by a control law which
is continuous at every nonzero
x
and right/left-continuous
at . 0x
0
6. Conclusions
The stabilizability of affine single input first order sys-
tems by a continuous control law is investigated. It is
demonstrated that sometimes a stabilizing control law
can be defined that is right/left-continuous at the origin.
. In addition, using WCLF in Sontag’s formula
generates a (possibly nonsmooth) control law, which
globally stabilizes the system and globally asymptotic
stability needs more investigation.
x
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