Continuous Stabilizing of First Order Single Input Nonlinear Systems

doi:10.4236/ica.2011.23022

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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

E-mail: [email protected]

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:

 

fx gxu

 (1)

where n

R is state space vector, is control

input vector and

uR

 

, n

xgx R.

Definition [1]: A differentiable positive definite and

radially unbounded function is

called a CLF for the system (1), if for each ,



Vx RR

0x

 

LV xLV x  (2)

If there exist nonzero points where



0Vx



, then

is sometimes referred to as Weak Control

Lyapunov Function (WCLF) [2,3].



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



,ukxx0





Vx is a

CLF for the system (1), then a particular stabilizing con-

trol law





, 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







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:

 



 



 



(())

()

0 0

ffgg T

sgg

LVxLVxLVx LVxLV xLVx

ux LV xLV x

LV x















(3)

A. SHAHMANSOORIAN

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,



xx 

u (5)

The equation has a root at



0gx1

, 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,



xx 

u (6)

Although the equation has a root at



0gx1



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,



xx

u (7)

The equation has a root at



0gx1

 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,



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





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,

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,

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



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,





xx xu

 (12)

The equation





0gx



has the root . The un-

forced system is unstable, but with the initial states

0x





10x0





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



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



. 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,







12 1121

2122

xxxxx

xxxxu

 

 



(13)

It can be proved that the function 2

is a

WCLF for the system. With all initial states interior the

circle



Vxx x

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,



21 12

πarctan543

221 25

xx x





 







u

(14)

In [2] the function:



Vxxx

is used as a CLF in the Sonag’s formula. It can be veri-

fied the function:



π2

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

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.





fx gxu

 (15)

where



is state variable, is control input

vector and

uR







and



xR are smooth. As-

sume the unforced system,



fx

 (16)

is locally asymptotically stable and its domain of attrac-

tion is









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



Vx x as a WCLF for the system,

the Sontag’s formula gives:





 



 



() ,0

0, 0

fxSgnxfxx gxgx





















(17)

It can be shown that this control law globally asymp-

totically stabilizes the system (15). Assume





0gx



has a nonzero root



such that . Accord-

ing to the assumption of lemma

acb





0,xf x







,, 0.xabx





Thus we have:

 



 



 



lim lim

lim 0

xc xc

xgx

ux fxSignxfxxgx

xgx

xf xxf xxgx















This proves the continuity of the control law (17).

When the equation





xR has root(s) at 0.x



Then it is clear that:



lim



is equal to zero or infinity (when



x and





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:

 



 



lim lim

xgx

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

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.

7. References

[1] R. Sepulchre, M. Jankovic and P. V. Kokotovic, “Con-

structive Nonlinear Control,” Springer Verlag, London,

1997.

[2] J. A. Primbs, V. Nevistic and J. C. Doyle, “A Receding

Horizon Generalization of Pointwise Min Norm Control-

lers,” IEEE Transactions on Automatic Control, Vol. 45,

No. 5, May 2000, pp. 898-909. doi:org/10.1109/9.855550

[3] M. Krstic, I. Anellakopoulos and P. V. Kokotovic,

“Nonlinear and Adaptive Control Design,” John Wiley &

Sons, New York, 1995.

[4] E. D. Sontag, “A ‘Universal’ Construction of Artstein’s

Theorem on Nonlinear Stabilization,” System & Control

letters, Vol. 13, No. 2, 1989, pp. 117-123.

doi:10.1016/0167-6911(89)90028-5

[5] F. Ceragioli, “Some Remarks on Stabilization by Means

of Discontinuous Feedbacks,” System & Control Letters,

Vol. 45, No. 4, 2002, pp. 271-281.

doi:org/10.1016/S0167-6911(01)00185-2

[6] F. A. C. C. Fontes, “Discontinuoud Feedbacks, Discon-

tinuous Optimal Controls, and Continuous-Time Model

Predictive Control,” International Journal of Robust Con-

trol and Nonlinear Control, Vol. 13, No. 3-4, 2003, pp.

191-209.

[7] R. W. Brockett, R. S. Millman and H. S. Sussmann,

“Differential Geometric Control Theory,” Birkhouser,

Boston, 1983.

[8] A. Bacciotti and L. Rosier, “Liapunov Functions and

Stability in Control Theory,” Springer-Verlag, London,

2001.