A. SHAHMANSOORIAN

Copyright © 2011 SciRes. ICA

183

fx gxu

(4)

where

R is the state space vector, is the

control input vector and

uR

,

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

xx

u (5)

The equation has a root at

0gx1

, 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

xx

u (6)

Although the equation has a root at

0gx1

,

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

xx

u (7)

The equation has a root at

0gx1

but the

unforced system solutions with initial states

00x

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

xx

u (8)

The equation

0gx has roots at 1

and

. For the root

1x 1

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

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

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

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

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

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,