TRACKING CONTROL FOR UNCERTAIN DYNAMICAL SYSTEMS DESCRIBED BY DIFFERENTIAL INCLUSIONS WITH...

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TRACKING CONTROL FOR UNCERTAIN DYNAMICALSYSTEMS DESCRIBED BY DIFFERENTIAL INCLUSIONSWITH SET-VALUED CIRCULAR OBSERVATION MAPSJen-Fen Huang a & Jia-Wen Chen ba Department of Electronics Engineering , Wufeng Institute of Technology , Ming-Hsiung,Taiwan, 621, R.O.C.b Department of Applied Mathematics , National Chiayi University , 300 University Road,Chiayi, Taiwan, 600, R.O.C.Published online: 09 Feb 2010.

To cite this article: Jen-Fen Huang & Jia-Wen Chen (2007) TRACKING CONTROL FOR UNCERTAIN DYNAMICAL SYSTEMSDESCRIBED BY DIFFERENTIAL INCLUSIONS WITH SET-VALUED CIRCULAR OBSERVATION MAPS, Journal of the Chinese Instituteof Industrial Engineers, 24:1, 70-80, DOI: 10.1080/10170660709509023

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70 Journal of the Chinese Institute of Industrial Engineers, Vol. 24, No. 1, pp. 70-80 (2007)

TRACKING CONTROL FOR UNCERTAIN DYNAMICAL SYSTEMS DESCRIBED BY DIFFERENTIAL INCLUSIONS WITH SET-VALUED CIRCULAR OBSERVATION MAPS

Jen-Fen Huang Department of Electronics Engineering,

Wufeng Institute of Technology, Ming-Hsiung, Taiwan 621, R.O.C. Jia-Wen Chen*

Department of Applied Mathematics, National Chiayi University, 300 University Road, Chiayi, Taiwan 600, R.O.C.

ABSTRACT In this paper, we investigate the completely tracking control problem for a class of uncertain nonlinear dynamical systems described by differential inclusions. The goal is to construct a feedback control such that all tracking trajectories of the system are steered to the pre-specified set-valued observation map with an exponential convergence rate. An estimation of the distance from all tracking trajectories to the observation barycenter of observation map is given. Moreover, an estimation of the tracking time of the trajectory attaining the set-valued observation map is given. Finally, an example is given to illustrates the use of our main results. Keywords: tracking control, differential inclusion, feedback-controlled system, uncertain

dynamical systems, exponential asymptotic stability

*Corresponding author: [email protected]

1. INTRODUCTION

In the earlier work of tracking control for nonlinear uncertain systems, usually, it is described by ordinary differential equations (see, [4], [6-8]). But in many practical applications, it often meet discontinuous control. Hence we need consider more general types of uncertainties, and the traditional theory of ordinary differential equations is not applicable to both in analysis and synthesis for our framwork. Fortunately, the concept of differential inclusions is applicable to describe the uncertain nonlinear dynamical systems (cf. [5]) which we state as follows:

( )( )

,)(),(),()(,)(),()(

m

n

tutytxGtytytxFtx

&

& (1)

),(),(),( yxFyxfyxF + ,

[ ] ,)(),(),(),(),(),,(uFyxFyxQuyxQyxguyxG

+++

where ),0[ t is the time variable, ptu )( is the control inputs, and ntx )( , mty )( denote the states of the system. The set-valued maps

nyxF ),( , pyxF ),( , and

puF )( are

systems of uncertainty. The functions nmnf : , mmng : , and

pmmnQ : are single-valued continuous. Using the state feedback ),( yxuu = , the feedback-controlled system (1) becomes the closed-loop systems described by the differential inclusions as the form:

( )( )

,)(),()(,)(),()(

tytxGtytytxFtx

c&

& (2)

where

),(),(),( yxFyxfyxF + ,

( ) ( )( ))(),(),(),()(),( tytxutytxGtytxGc .

If }0{)(),(),( === uFyxFyxF for all nx , my and pu , we observe that the

original system (1) can be regarded as the model of no uncertainty nominal systems (3):

+==

.)())(),(())(),(()(,))(),(()(

tutytxQtytxgtytytxftx

&

& (3)

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Huang and Chen: Tracking Control for Uncertain Dynamical Systems Described by Differential Inclusions 71 with Set-Valued Circular Observation Maps

This nominal system (3) is a special case of the feedback-controlled system (1) with uncertainty.

Throughout this paper, let mnH a: be a set-valued map defined by ];0[)()( BxhxH += , regarded as a set-valued circular observation map, where )(h is a single-valued continuous differentiable Lipschitz map regarded as the observation barycenter of )(H , and

{ } xxB m];0[ is the closed ball centered at 0 with radius 0 in the Euclidean norm . In

this paper, we will consider the completely tracking control problem (see Definition 2.3 in the next section) of uncertain nonlinear dynamical systems described by differential inclusions with set-valued circular observation maps )(H . The goal is to find a feedback control ),( yxuu = such that for any initial state )(),( 00 HGraphyx , all solutions ( ))(),( yx of the system (2), starting from ),( 00 yx , will obey the trajectory ))(()( txHty for all 0t . Furthermore, if )(),( 00 HGraphyx , namely 0y is not traced by

)( 0xH . Under the feedback control, we will find a constant 0ft such that all solutions ( ))(),( yx of the system (2) satisfy ))(()( txHty for all ftt .

Moreover, the trajectory )(ty of the system (2) is steered by the set-valued observation map ))(( txH with an exponential convergence rate. An estimation of the tracking time T for the trajectory )(ty attaining the map ( ))(txH is given. 2. ASSUMPTIONS AND

DEFINITIONS OF COMPLETELY TRACKING CONTROL SYSTEMS

For convenience, denote as the Euclidean

norm or the corresponding induced norm of a matrix. Let yxF

xFy )(sup)(

, where F is a set-valued map.

For the existence of solutions to differential inclusions (2), in general case, ),( F and ),( cG need satisfy the assumption of upper semicontinuity. More precisely, if ),( F and ),( cG are upper semicontinuous with convex and compact values, for any initial state )(),( 00 HGraphyx , then there exist a positive T and a solution ( ))(),( yx defined on

],0[ T to the system (2) such that either =T or

72 Journal of the Chinese Institute of Industrial Engineers, Vol. 24, No. 1 (2007)

growth (see the Section 3). The Assumption 2.1 (A7) shows that all solutions )(ty of the system (2) are steered to the set-valued observation map ))(( txH with an exponential convergence rate (see the Section 4).

Let mnH a: be a set-valued map defined by ];0[))(())(( BtxhtxH + . We define an uncertain dynamical system which is completely viable controllable for H as follows. Definition 2.2. We say that the system (2) is tracking controllable for H if for all initial state

mnyx ),( 00 , there exist 0T , a feedback control ),( yxuu = and a solution ( ))(),( yx of the closed-loop differential inclusion (2) satisfying

))(()( txHty for all Tt . Definition 2.3. We say that the system (2) is completely tracking controllable for H if for all initial state mnyx ),( 00 , there exist 0T and a feedback control ),( yxuu = such that all solutions

( ))(),( yx of the closed-loop differential inclusions (2) satisfying ))(()( txHty for all Tt . Remark 2.2. Clearly, (from Definition 2.2 and Definition 2.3) the completely tracking controllable system for H is also tracking controllable for H. Remark 2.3. Since ];0[))(())(( BtxhtxH += , it is easy to see that the completely tracking controllable system for h is also completely tracking controllable for H. Hence, in this paper, we only consider the system (2) is completely tracking controllable for h. 3. DESIGN FEEDBACK

CONTROL INPUTS FOR TRACKING CONTROLLABLE SYSTEMS

Now, we consider the uncertain nonlinear

dynamical system (1) described by differential inclusions with control )(tu :

( ) ( ))(),()(),()( tytxutytxutu cn += , (4) ( ) [ ] ( )

( ) ,),()(,)(, )(

yxfxxh

yxgyyAuyxQ xHn

+

= (5)

( ) ( )( ) ,))(()()(),(),()(

txhtyMtytxQyxktu

T

c

= (6)

where )()()( xHyxH is called the projector of y

on )(xH , that is,

zyyyxHzxH

= )()(min)( , (7)

M is a positive definite symmetric mm matrix satisfying the following Lyapunov equation:

LMAMAT =+ , (8) L is an arbitrary positive definite symmetric mm matrix and A is an Hurwitz mm matrix, ),( yxk is a positive real-valued continuous function with linear growth satisfying

),(),( 0 yxkyxk , (9)

++ + nuyxkyxkx

xhyxQ

yxk

),(),(

)(),(

),(0

( ) 11 )1()()(

++

AQQQMM T

m

M ,

where )(Mm and )(MM are the minimum and the maximum eigenvalues of the real symmetric matrix M, respectively; +Q is the right inverse of Q, is any positive constant, and

( ) { }

=

0if10if

p (10)

is an upper semicontinuous function on p .

For the existence of solutions )(x defined on

),0[ to the closed-loop system (2), we need show that ),( yxF and ),( yxGc are Marchaud maps as follows. Lemma 3.1. The feedback-controlled system of (2) satisfies the assumptions (A1)-(A6), subject to the controller (4) with (5)-(10).Then ),( yxF and

),( yxGc in (2) are Marchaud maps. Proof. By (A1), we obtain that ),( yxF and

),( yxGc are upper semicontinuous with convex and compact values for all nx , my . In order to show such functions are Marchaud maps, we have only check that ),( yxF and ),( yxGc are dominated by any linear growth maps. By (A2), we have

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),(),(sup),(

),(yxFyxfzyxF

yxFz+

),(),( yxkyxf + .

Since ),( yxf and ),( yxk have linear growth, this implies that ),( yxF is linear growth.

By (A3)-(A6), we estimate that

zyxGyxGz

cc ),(

sup),(

uyxQyxg ),(),( +

[ ])(),(),( uFyxFyxQ ++ uyxQyxg ),(),( +

)(),(),(),( uFyxQyxFyxQ ++

uyxQyxg ),(),( +

uyxQryxkr ),(),( 12 ++

uyxQryxg ),()1(),( 1++

),(2 yxkr + , (11)

and

)(),(),( cn uuyxQuyxQ +=

[ ] [ ]),()(),()()( yxfxxh

yxgyyA xH

+

cuyxQ ),(+

( ) ),()( yxgxhyA ++ ),(),( 2 yxkryxfK h ++ . (12)

Combining (11) and (12), we obtain

zyxGyxGz

cc ),(

sup),(

),()2( 1 yxgr+

( )[ )()1( 1 xhyAr +++ ]),(),( 2 yxkryxfK h ++ ,

where 0hK is a Lipschitz constant of h. Hence

),( yxGc is also a linear growth map. 4. MAIN RESULTS

For convenience, we use , as the inner

product in m . We also define Sx, as the subset

{ }Sssx , of and define KSx , to denote Ksx , for all Ss , where s and

S is any subset of m .

In this paper, to obtain the main theorem, we need the following two lemmas. Lemma 4.1. Let ( ))(,)( tytx be any trajectories of the feedback-controlled systems (2) satisfying assumptions (A1)-(A7), subject to the controller (4) with (5)-(10). For any real number ),0[0 T with

( ))()( 00 TxhTy , there exists a positive real number ),( 0 TT such that all trajectories )(ty of the

feedback-controlled system (2) are steered to the map ( ))(txh with an exponential convergence rate on

],[ 0 TT and ( ))()( TxhTy = , where ( ))()( txhty for all ),[ 0 TTt . Proof. First, we show that T exists in ),( 0 T is the smallest positive real number such that

( ))()( TxhTy = . Suppose on the contrary that T does not exist, that is, T is infinite. Then ( ))()( txhty for all

0Tt . Let ))(()()( txhtyte = be the deviation of the state )(ty from the observation barycenter

))(( txh of ))(( txH . By ];0[))(())(( BtxhtxH += and the definition of )()( yxH , we obtain

))(()())(()( ];0[))(( tetetyty BtxH = .

For simplicity, we use the notations

( ))(,),( xhexfexf + , ( ))(,),( xhexgexg + , ( ))(,),( xhexFexF + , ( ))(,),( xhexFexF + ,

( ))(,),( xhexkexk + , ( ))(,),( xhexkexk + , ( ))(,),( xhexkexk + , ( ))(,),( 00 xhexkexk + , ( ))(,),( xhexQexQ + , ( ))(,),( xhexQexQ TT + , ( ))(,),( xhexQexQ + ++ ,

in terms of state x and error e. Then the closed-loop system (2) becomes

))(),(())(),(()( tetxFtetxftx +& , ))(()()( ],0[ teAtAete B &

)())(),(( tutetxQ c+ ))(),(())(),(( tetxFtetxQ+ ,

where

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))(())(),(())(),(( tuFtetxFtetxF +=

.))(),(()(

))(),(( tetxFxxh

tetxQ +

Let ( ) MeeeV T21)( = for all me . Then for all

( ))()( txhty , 0Tt , we have

( ) eMeeMeMeeeV TTT &&&& =+=21)(

c

B

uexQMe

eAMeAeMe

),(,

)(,, ],0[

+

+++ )(),(),(, cn uuFexFexQMe

+ ),(

)(),( exF

xxh

exQ

c

B

uexQMe

eAMeAeMe

),(,

)(,, ],0[

+

=

)(),(,),( cnT uuFexFMeexQ +++

),()(

),( exFxxh

exQ +

MeexQexk

eAMLee

T

MT

),(),(

)(21

+

),(),( exkuexk n ++

+

MeexQexkxxh

exQ T ),(),()(

),(

+ +

MeexQexk

eAMLee

T

MT

),(),()1(

)(21

+=

nuexk +

+ ),(

MeexQexkxxh

exQ T ),(),()(

),(

+ +

MeexQexk

eAMLee

T

MT

),(),()1(

)(21

0

+

nuexk +

+ ),(

MeexQexkxxh

exQ T ),(),()(

),(

+ +

eMexQeAMLee TMT ),()(

21 +=

( ) eMexQAQQQMM TT

m

M ),()()( 1

. (13)

Since ( ) ( ) )(21)(21 2 eVMeeeM Tm = , we have

MeQeQQQ

MeQQQQeeV

TT

TT

1

1

)(21

)(,21)(

=

MeQMeVQQQ T

m

T21

)()(2)(

21 1

.

Combining the above result and (13), for all

( ))()( txhty , 0Tt , we obtain

( )21

1)(

)(

)(2

)()(2

)(21)(

eVQQQ

M

MeV

AMLeeeV

T

m

mM

T

+

&

( )

( )( )2

1

1

1

)()(2

)()(

eVQQQ

MA

QQQMM

T

m

T

m

M

( ) 0)()(

)(221

21

1


( )( ) ( )( )

)()(

)(2

)()(2

0

21

21

21

021

TtQQQ

M

TeVteV

T

m

. (16)

Taking the limit t on the two sides of (16), it follows from (15) that

( )( )

( )( ) ( )( )

=

21

21

0

21

0

)()(2lim

)(2

teVTeV

TeV

t

=

t

QQQ

MT

m

t

21

21)(

)(2lim

.

This contradicts the fact that ( )( ) . Proof. Suppose on the contrary that there exists a positive real number ),(1 TT such that

( ))()( 11 TxhTy . By Lemma 4.1, there exists ),( 12 TT such that ( ))()( 22 TxhTy = and

( ))()( txhty for all ),[ 21 TTt . Without loss of generality, we can assume that there exists an interval

],[],[ 221 TTtt such that ( ))()( 11 txhty = , ( ))()( 22 txhty = and ( ))()( txhty for all ),( 21 ttt ,

where 21 tt < . By the same argument of Lemma 4.1,

(14) implies that

( )21

21)(

)()(2)( eV

QQQMeV

T

m

&

for all ),( 21 ttt , and so

( )

( )

2

1

2

1

21

21

)(

)(

21

)(

)(2)(

t

t T

mteV

teVdt

QQQ

MdVV

,

or

( ) ( )

)()(

)(2

))(())((2

12

21

21

21

121

2

ttQQQ

M

teVteV

T

m

,

( ) ( )

)()(

)(2

)0()0(20

12

21

21

21

21

ttQQQ

M

VV

T

m

=

.

This show that

12 tt = . This contradicts the fact that

12 tt > . Thus Lemma 4.2 is true. Remark 4.1. Let ( ))(,)( tytx be any trajectories of the feedback-controlled systems (2) satisfying assumptions (A1)-(A7), subject to the controller (4) with (5)-(10). By Lemma 4.2, if any initial state

( ))0()0( xhy = , then ( ))()( txhty = for all 0t .

Combining Lemma 4.1 and Lemma 4.2, we obtain the following theorem.

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Theorem 4.1. Let ( ))(),( tytx be any the trajectories of the feedback-controlled system (2) satisfying assumptions (A1)-(A7), subject to the controller (4) with (5)-(10). If the initial state ( ) )()0(,)0( HGraphyx , then the estimation of the tracking time

ft of all trajectories )(y attaining

H( ) is bounded by

,)0()()()(

1

e

MMQQQ

m

M

T

where >= ))0(()0()0( xhye denotes the

distance of the initial state to the observation barycenter )(h of )(H . Proof. Since ( ) )()0(),0( HGraphyx , we have

( ))0()0( xhy . From Lemma 4.1, there exists the tracking time T of the trajectory )(y attaining )(h , i.e., T is the smallest positive real number such that

( ))()( TxhTy = . Note that ( ) ( ))()( txHtxh . This implies that the tracking time

ft of the trajectory

)(y attaining )(H exists and Tt f . Now,

evaluate an upper bound of tracking time ft of the

trajectory )(y attaining )(H as follows. In Lemma 4.1, taking 00 =T , (13) induces that

( )21

1)(

)()(2

21)( eV

QQQM

LeeeVTmT

&

( )21

21)(

)(

)(2eV

QQQ

MT

m

for all ],0[ ftt .

To solve the above inequality, it is easy to get

( )

( )td

QQQMdVV

ff t

T

mteV

eV

21

21

0 21

)(

)0( )()(2)(

.

This implies that

( ) ( )

.)(

)(2

))0(())((2

21

21

21

21

fT

m

f

tQQQ

M

eVteV

Since ( ) ( ) 22 )(21)()(21 eMeVeM Mm , the above inequality implies that

,)()(21)0()(

212

)(

)(221

21

fmM

fT

m

teMeM

tQQQ

M

or

)()0()()()(

1

fm

M

T

f teeMMQQQ

t

.

Since

ft is the smallest positive real number such

that ))(()( ff txHty , that is, =)( fte , it follows

that

)0(

)()()( 1 e

MMQQQt

m

M

T

f.

Theorem 4.2. Let ( ))(),( tytx be any trajectories of the feedback-controlled system (2) satisfying assumptions (A1)-(A7), subject to the controller (4) with (5)-(10). Then all trajectories )(ty of the system (2) are steered to the pre-specified observation map

)(H with an exponential convergence rate, where

( ))0()0( xHy . Proof. Since ( ))0()0( xHy , we have

( ))0()0( xhy . In Lemma 4.1, taking 00 =T , then there exists 0>T such that ( ))()( txhty = for all

Tt , and by (17),

( )

( )

tML

xhyMM

txhty

M

m

m

M

)(2)(

exp)0()0()()(

)()(

for all ),0[ Tt . This shows that all trajectories )(ty of the feedback-controlled systems (2) are steered to the observation barycenter ( ))(txh of ( ))(txH with an exponential convergence rate. This also implies that all trajectories )(ty are steered to the observation map

( ))(txH with an exponential convergence rate. Corollary 4.1. In the preceding Theorem 4.2, we have proved that the estimation of the distance from

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)(y to the observation barycenter )(h of )(H is bounded by

( )

t

ML

xhyMM

M

m

m

M

)(2)(

exp)0()0()()(

,

that is,

( )

( ) .)(2

)(exp)0()0(

)()(

)()(

tML

xhyMM

txhty

M

m

m

M

Combining Theorem 4.1 and Theorem 4.2, we

show that the system (2) is completely tracking controllable for H as the following main theorem. Theorem 4.3. Let ( ))(),( tytx be any trajectories of the feedback-controlled system (2) satisfying assumptions (A1)-(A7). If for any initial state

)(),( 00 HGraphyx , namely 0y is not traced by

)( 0xH , then the controller (4) with (5)-(10) such that the feedback-controlled system (2) is completely tracking controllable for H and all trajectories )(ty of the system (2) are steered to the observation barycenter ( ))(txh of ))(( txH with an exponential convergence rate, More precisely, there exists a constant 0ft such that

))(()( txHty for all ftt

and ( )

( ))0()0()()(

)()(

xhyMM

txhty

m

M

0)(2

)(exp

t

ML

M

m

as t .

Moreover, an estimation of the tracking time

ft of

all trajectories )(y attaining )(H is bounded by

)0(

)()()(

1

eMMQQQ

m

M

T

,

where ))0(()0()0( xhye = denotes the distance

of the initial state to the observation map )(h . Corollary 4.2. Let ( ))(),( tytx be any trajectories of the feedback-controlled system (2) satisfying

assumptions (A1)-(A7). If for any initial state )(),( 00 HGraphyx , then the controller (4) with

(5)-(10), where

{ }0),(),( == yxkyxk ,

( ) +

AQQQMMyxk T

m

M 1

)()(:),( ,

such that the feedback-controlled system (3) is completely tracking controllable for H and all trajectories )(ty of the system (3) are steered to the observation barycenter ( ))(txh of ))(( txH with an exponential convergence rate. Proof. Note that the system (3) is a special case of the uncertain nonlinear dynamical system (2) subject to uncertainty, where ( ) { }0, =yxF , ( ) { }0, =yxF , and { }0)( =uF in the system (2), and the Assumption 2.1 is also satisfied for 0),( =yxk and

0),( =yxk . Taking

( )

+

AQQQMM

yxk Tm

M 1

)()(

),( ,

Theorem 4.1 and Theorem 4.2 also holds for the systems (3). This implies that Corollary 4.2 is true. 5. AN ILLUSTRATIVE

EXAMPLE

In order to illustrate our design procedure, an example is provided in the following. Example 5.1. Consider the uncertain dynamical system (2) described by differential inclusions as follows:

( )( )

,)(),()(,)(),()(

tytxGtytytxFtx

c&

&

where

( ) ( )( ))(),(),(),()(),( tytxutytxGtytxGc 2)cos(),( ++= xyyxyxf ,

4)sin(),( ++= xyxyyxg ,

( ) ( )22 cos)sin(1),( xxyyxQ ++= , ( ) )(sincos1),( xySIGNaxyyxayxF +++= ,

>=


)sin()( ucuuF = ,

11 a , 11 b , and 5.05.0 c . From Assumption (A2)-(A4), we have

1)1(),( 222 ++= xyxyxk ,

6),( ++= yxyxk , 5.0 = .

For example, for 1=a , 1=b and 5.0=c , some

typical phase trajectories of the uncontrolled system are depicted in Figure 1.

Figure 1. Typical phase trajectories of the

uncontrolled system.

Take 1 =A and 2=L . By (8), we obtain that 1=M . Furthermore, let xxh =)( ,

]1,1[]1;0[)()( +=+= xxBxhxH

and 5.0= in (9), then we can calculate the explicit form of the controller )(tu given by (4) with (5)-(10). They are shown as follows:

))(),(())(),(()( tytxutytxutu cn += ,

where

( ) 22]1,1[

)(cos)sin(1

)(),(

xxy

yyyxu xxn

++

= +

( ) 22 )(cos)sin(14)sin(

xxyxyxy++

++

( ) 22 )(cos)sin(12)cos(

xxyxyyx++

++ ,

( )= ),()( yxktuc , ( )[ +++= yxyxk 22),(

( ) ]5.025.06 ++++++ nuyx ( ) ( )[ ]22 )(cos)sin(1 xxyxy +++= .

By Theorem 4.3, all trajectories of the feedback-controlled system reach the observation map )(xh in a finite time and remain on )(xh thereafter. Some typical phase trajectories of the feedback-controlled system are depicted in Figure 2.

Figure 2. Typical phase trajectories of the feedback-

controlled system. 6. CONCLUSION

The completely tracking control property and the general results related to it appeared in this paper. The results dealing with the construction of trackers and observers, the decentralization property and hierarchical decomposition are taken from [5]. Theorems dealing with the existence of solutions

)(x defined on ),0[ to the closed-loop system (1.2) with linear growth to decomposable system of Marchaud maps are taken from the first part of the paper. In the case of Marchaud maps, the results was obtain in the section 3. The tracking property are the main topic of the paper. The goal is to construct a feedback control such that the nonlinear uncertain dynamical system satisfies the tracking property under the set-valued circular observation map. From the above results, conclusions are drawn up as follows:

1. A new design of feedback control inputs for trcking controllable systems is developed under the set-valued circular observation map.

2. All tracking trajectories of the system are steered to the circular set-valued observation map with an exponential convergence rate.

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3. A good estimation of the tracking time of the trajectory attaining the set-valued observation map is smaller than the result of Chen-Cheng-Hsieh (cf. [5; Theorem 2.2, p. 473]).

4. These results play important roles in the theory of uncertain dynamical systems about tracking control missiles.

5. These results play important roles in the theory of uncertain dynamical systems about viable control. As a illustration in this category, the trajectory of the medicinal carriers satisfying the uncertain dynamical system described by differential inclusions is guided to the mapping )(H in the nanomedicine system. Here, let x be the state of the medicinal carriers system and let )(H be the state of the arteries. In the nanomedicine system, the goal is to find a feedback control such that the closed-loop system is completely viable controllable for

)(H . This implies that the medicinal carriers )(x can be controlled to treate this cerebral embolism and cerebral thrombosis diseases in arteries )(H .

ACKNOWLEDGMENT

This work partially supported by the National

Science Council, Taiwan, R.O.C. under Grant NSC 91-2115-M-415-001.

REFERENCES

1. Aubin, J. P., Viability Theory, Birkhauser, Boston

(1991).

2. Aubin, J. P. and Cellina, A., Differential Inclusions, Springer-Verlag, Berlin (1984).

3. Aubin, J. P. and Frankowska, H., Set-valued Analysis, Birkhauser, Boston (1990).

4. Behtash, S., Robust output tracking for nonlinear systems, International Journal of Control, 51, 1381-1407 (1990).

5. Chen, J.W., J. S. Cheng and J. G. Hsieh, Tracking control for nonliear uncertain dynamical systems described by differential inclusions, Journal of Mathematical Analysis and Applications, 236, 463-479 (1999).

6. Fu, L.C. and T. L. Liao, Globally stable robust tracking of nonlinear systems using variable structure control and with an application to robotic manipulator, IEEE Transactions on Automatic Control, 35, 1345-1350 (1990).

7. Isidori, A. and C. I. Byrnes., Output regulation of nonlinear system, IEEE Transactions on Automatic Control, 35, 131-140 (1990).

8. Zhihong, M. and M. Palaniswami, Robust tracking control for rigid robotic manipulators, IEEE Transactions on Automatic Control, 39, 154-159 (1994).

ABOUT THE AUTHORS

Jen-Fen Huang is an assistant professor in the Department of Electronics Engineering at the Wufeng Institute of Technology, Taiwan. She received his B.S. (1987) and M.S. (1990) degrees in Physics from the Fu Jen Catholic University; and the Ph.D. (1998) in Electrical Engineering from the National Sun Yat-Sen University. Her research interests include RF and Microwave Components/Circuits, Antenna, RFID, and Video/Audio Processing. Jia-Wen Chen is a professor in the Department of Applied Mathematics at Chiayi University, Taiwan. He received his B.S. (1986) and M.S. (1988) degrees in Mathematics from the Fu Jen Catholic University; and the Ph.D. (1992) in Mathematics from the National Tsing Hua University. His research interests include Optimization, Nonsmooth Analysis, Dynamical Systems, Differential Inclusions, Tracking Control, Control Theory, and Viable Control. (Received November 2005; revised December 2005; accepted January 2006)

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(*[email protected])

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TRACKING CONTROL FOR UNCERTAIN DYNAMICAL SYSTEMS DESCRIBED BY DIFFERENTIAL INCLUSIONS WITH...

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