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1680 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER 2007

Event-Triggered Real-Time Scheduling of Stabilizing

Control Tasks

Paulo Tabuada, Senior Member, IEEE

AbstractIn this note, we revisit the problem of scheduling stabilizing

control tasks on embedded processors. We start from the paradigm that areal-time scheduler could be regarded as a feedback controller that decides

which task is executed at anygiven instant. This controller hasfor objectiveguaranteeing that (control unrelated) software tasks meet their deadlinesand that stabilizing control tasks asymptotically stabilize the plant. We in-vestigate a simple event-triggered scheduler based on this feedback para-digm and show how it leads to guaranteed performance thus relaxing themore traditional periodic execution requirements.

Index TermsEvent-triggered, input-to-state stability, real-time sched-uling.

I. INTRODUCTION

Small embedded microprocessors are quickly becoming an essen-

tial part of the most diverse applications. A particularly interesting ex-

ample are physically distributed sensor/actuator networks responsiblefor collecting and processing information, and to react to this infor-

mation through actuation. The embedded microprocessors forming the

computational core of these networks are required to execute a variety

of tasks comprising the relay of information packets in multihop com-

munication schemes, monitoring physical quantities, and computation

of feedback control laws. Since we are dealing with resource-limited

microprocessors, it becomes important to assess to what extent we can

increase the functionality of these embedded devices through novel

real-time scheduling algorithms based on event-triggered rather than

time-triggered execution of control tasks.

We investigate in this note a very simple event-triggered scheduling

algorithm that preempts running tasks to execute thecontrol task when-

ever a certain error becomes large when compared with the state norm.

This idea is an adaptation to the scheduling context of several tech-

niques used to study problems of control under communication con-

straints [6], [11], [18]. We take explicitly into account the response

time of the control task and show that the proposed scheduling policy

guarantees semi-global asymptotical stability. We also show existence

of a minimal time between consecutive executions of the control task

under the proposed event-triggered scheduling policy. This time, when

regarded as a lower bound on release times of the control task, suggests

that periodic implementations of control tasks can be relaxed to more

flexible aperiodic or event-triggered real-time implementations while

guaranteeing desired levels of performance. The proposed approach is

illustrated with simulation results.

Real-time scheduling of control tasks has received renewed interest

from the academic community in the past years [2], [5], [8][10], [14],[22]. Common to all these approaches, that followed from the initial

work of Seto and co-authors [23], is the underlying principle that better

control performanceis achievedby providingmore CPUtimeto control

tasks. This can be accomplished in two different ways: letting control

Manuscript received May 30, 2006; revised May 17, 2007. Recommendedby Associate Editor J. Hespanha. This research was partially supported by theNational Science Foundation EHS Award 0712502.

The author is with the Department of Electrical Engineering at the Uni-versity of California at Los Angeles, Los Angeles, CA 90095 USA (e-mail:[email protected]; URL: http://www.ee.ucla.edu/~ tabuada).

Color versions of one or more of the figures in this note are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2007.904277

tasks run for longer amounts of time using anytime implementations or

model predictive controllers; or by scheduling control tasks more fre-

quently. All these approaches assume the existence of a performance

criterion for the control task such as a cost function used to design an

optimal linear quadratic regulator. Scheduling strategies are then ob-

tained through optimization algorithms seeking to determine schedules

maximizing the performance criterion. The work presented in this note

does not resort to optimization and does not require a performance cri-terion. Instead, the decision to execute the control task is determined

by a feedback mechanism based on the state of the plant.

The above described work focused on optimizing schedules and

sampling rates while leaving controllers unchanged. A further gener-

alization of such ideas appears in [20], [21] where the optimization

process also changes the feedback controllers in order to enforce

stability of the controlled processes and schedulability. Close at the

technical level, although addressing very different problems, is the

recent work on stabilization under communication constraints [6],

[7], [11], [15], [18], [19]. All these approaches are concerned with

the stabilization of continuous systems under reduced communication

and the employed techniques share with some of the techniques

described in this note a common ancestor: the perturbation approach

to stability analysis of control systems, described, for example, in [13].

Similar techniques have also been used in [16] and [27] to show how

sample-and-hold implementations of stabilizing controllers guarantee

stability under sufficiently fast periodic executions. Unfortunately,

these results are either nonconstructive or rely on assumptions much

stronger than those used in this note. Finally, we would like to refer

the reader to [1] where some advantages of event-driven control over

time-driven control are presented in a stochastic setting. A preliminary

version of the results presented in this note was reported in [25].

II. NOTATION AND PROBLEM STATEMENT

A. Notation

We shall use the notation j x j to denote the Euclidean norm of an

element x 2 I R n . Given matrices A and B , [ A j B ] denotes the matrix

formed by the columns of matrix A followed by the columns of matrix

B . A function f : I R n ! I R m is said to be Lipschitz continuous on

compacts if for every compact set S I R n there exists a constant

L > 0 such that j f ( x ) 0 f ( y ) j L j x 0 y j for every x ; y 2 S . A

continuous function : [ 0 ; a [ ! I R +0

, a > 0 , is said to be of class K

if it is strictly increasing and ( 0 ) = 0 . It is said to be of class K1

if

a = 1 and ( r ) ! 1 as r ! 1 .

B. Problem Statement

We consider a control system

_x = f ( x ; u ) ; x 2 I R

n

; u 2 I R

m

(1)

for which a feedback controller

u = k ( x )

(2)

has been designed rendering the closed-loop system

_x = f ( x ; k ( x + e ) ) : (3)

Input-to-State Stable (ISS) with respect to measurement errors e 2

I R

n . We shall not need the definition 1 of ISS in this note but rather

the following characterization.

1

See, for example, [24] for an introduction to ISS and related notions.

0018-9286/$25.00 2007 IEEE

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER 2007 1681

Definition II.1: A smooth function V : I R n ! I R +0

is said to be

an ISS Lyapunov function for the closed-loop system (3) if there exist

class K1

functions ; ; and satisfying

( j x j ) V ( x ) ( j x j ) (4)

@ V

@ x

f ( x ; k ( x + e ) ) 0 ( j x j ) + ( j e j ) : (5)

Closed-loop system (3) is said to be ISS with respect to measurement

errorse I R

n if there exists an ISS Lyapunov function for (3).

The implementation of the feedback law (2) on an embedded pro-

cessor is typically done by sampling the state at time instants

t

0

; t

1

; t

2

; t

3

; t

4

. . .

computing u ( ti

) = k ( x ( t

i

) ) and updating the actuator values at time

instants

t

0

+ 1 ; t

1

+ 1 ; t

2

+ 1 ; t

3

+ 1 ; t

4

+ 1 ; . . .

where 1 0 represents the time required to read the state from the

sensors, compute the control law and update the actuators. This means

that to a sequence of measurements

x ( t

0

) ; x ( t

1

) ; x ( t

2

) ; x ( t

3

) ; x ( t

4

) ; . . .

there corresponds a sequence of actuation updates

u ( t

0

+ 1 ) ; u ( t

1

+ 1 ) ; u ( t

2

+ 1 ) ; u ( t

3

+ 1 ) ; u ( t

4

+ 1 ) ; . . . :

Between actuator updates the input valueu

is held constant according

to

t [ t

i

+ 1 ; t

i + 1

+ 1 [ = ) u ( t ) = u ( t

i

+ 1 ) : (6)

Furthermore, the sequence of times t0

; t

1

; t

2

; t

3

; t

4

; . . . is typically pe-

riodic meaning thatt

i + 1

0 t

i

= T

where T > 0 is the period. We can

thus regard the execution of the control task implementing the con-

trol law (2) as being time-triggered. In this note we consider instead

event-triggered executions where the sequence t0

; t

1

; t

2

; t

3

; t

4

. . . of

execution times is no longer periodic neither specified in advance but

rather implicitly defined by an execution rule based on the state of the

plant. To introduce this execution rule we define the measurement error

e to be

t [ t

i

+ 1 ; t

i + 1

+ 1 [ = ) e ( t ) = x ( t

i

) 0 x ( t ) : (7)

We can thus describe the evolution of (1) under the implementation (6)

of control law (2) by

_x = f ( x ; k ( x + e ) )

with e I R n as defined in (7).

Let us first consider the hypothetical case 1 = 0 with the single

purpose of explaining the execution rule. From (5) we see that if we

restrict the error to satisfy

( j e j ) ( j x j ) ; > 0 (8)

the dynamics of V is bounded by

@ V

@ x

f ( x ; k ( x + e ) ) ( 0 ) ( j x j )

thus guaranteeing that V decreases provided that 0 , the control task needs to be executed before the

equality ( j e j ) = ( j x j )

is satisfied in order to account for the delay

1

between measuring the state and updating the actuators.

Although the simple execution rule (9) guarantees global asymptot-

ical stability by construction, there are three important questions that

need to be answered in order to assess the feasibility of this schedulingpolicy.

1) Since theexecution times areonly implicitly defined, canwe guar-

antee that they will not become arbitrarily close resulting in an ac-

cumulation2 point?

2) In the absence of accumulation points, can we compute an esti-

mate of the time elapsed between consecutive executions of the

control task?

3) How can we use the execution rule (9) when there are more tasks

competing for processor time and still guarantee that no deadlines

are missed?

Answering the above questions is the objective of Sections III VI.

III. EXISTENCE OF A LOWER BOUND FOR INTEREXECUTION TIMES

We start immediately with one of the main contributions of this note.

Theorem III.1: Let _x = f ( x ; u ) be a control system and let u =

k ( x ) be a control law renderingthe closed-loopsystem ISSwith respect

to measurement errors. If the following assumptions are satisfied:

1) f : I R n 2 I R m ! I R n is Lipschitz continuous on compacts;

2) k : I R n ! I R m is Lipschitz continuous on compacts;

3) there exists an ISS Lyapunov function V for the closed-loop

system satisfying (5) with 0 1 and Lipschitz continuous on

compacts,

then, for any compact set S I R n containing the origin, there exists

an " > 0 such that for all response times 1 [ 0 ; " ] there exists a time

I R

+ such that for any initial condition in S the inter-execution

timesf t

i

+ 1

0 t

i

g

i 2 I N implicitly defined by the execution rule (9) with 0 is large enough so that S R . Such

always exists since by (4), V is proper or radially unbounded. Set

R is forward invariant for the closed-loop system since the execution

rule (9) guarantees _V 0 . We now define another compact set E by

all the points e I R n satisfying j e j 0 1 ( ( j x j ) ) for all x

R . Since 0 1 and are Lipschitz continuous on compacts, then so is

0 1

( ( j r j ) = )

. LetP

be the Lipschitz constant for the compact setE

so that j 0 1 ( ( j r j ) = ) 0 0 1 ( ( j s j ) = ) j P j r 0 s j . Ifr = e and s =

0 we obtain 0 1 ( ( j e j ) = ) P j e j . Note that by enforcing the more

conservative inequality P j e j j x j we guarantee 0 1 ( ( j e j ) = ) j x j

(which is (8)) so that it suffices to show that the inter-execution times

are bounded for the execution rule P j e j = j x j . As a first step towardsshowing boundedness we note that it follows from Lipschitz continuity

on compacts of f ( x ; u ) and k ( x ) that f ( x ; k ( x + e ) ) is also Lipschitz

continuous on compacts, that is

j f ( r ; k ( r + s ) ) 0 f ( r

; k ( r

+ s

) ) j L j ( r ; s ) 0 ( r

; s

) j

by taking r = x , s = e and r = 0 = e we obtain

j f ( x ; k ( x + e ) ) j L j ( x ; e ) j L j x j + L j e j (10)

when ( x ; e ) R 2 E . Note that R 2 E is forward invariant for the

state as well as the error dynamics. We can now bound the interevent

2In the context of hybrid systems this corresponds to another example of theinfamous Zeno behavior [4].

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times by looking at the dynamics of j e j = j x j

d

d t

j e j

j x j

=

d

d t

( e

T

e )

1 = 2

( x

T

x )

1 = 2

=

( e

T

e )

0 1 = 2

e

T

_e ( x

T

x )

1 = 2

0 ( x

T

x )

0 1 = 2

x

T

_x ( e

T

e )

1 = 2

x

T

x

= 0

e

T

_x

j e k x j

0

x

T

_x

j x k x j

j e j

j x j

j e k _x j

j e k x j

+

j x k _x j

j x k x j

j e j

j x j

= +

j e j

j x j

j _x j

j x j

+

j e j

j x j

L j x j + L j e j

j x j

b y ( 0 )

= L +

j e j

j x j

+

j e j

j x j

:

(11)

If we denotej e j = j x j

byy

we have the estimate_y L ( + y )

2 andwe conclude that y ( t ) ( t ;

0

) where ( t ; 0

) is the solution of_

= L ( + )

2

satisfying ( 0 ; 0 ) = 0 .Assume now that 1 = 0 . Then, the inter-execution times are

bounded by the time it takes for to evolve from 0 to = P , thatis, the inter-execution times are bounded by the solution 2 I R +

of ( ; 0 ) = = P . Since ( ; 0 ) = L = ( L 0 ) we obtain = = ( L + L P ) .

For 1 > 0 we need a more detailed analysis. First, pick satisfying <

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where the inequality follows from @ = @ 0 > 0 , we have

( 1 ) 0 : 4 4 b = j K

T

B

T

P + P B K j = 8

and m

(

Q

)

is the smallest eigenvalue ofQ

. Since b

needs to besmaller that 0.44, we select

= 0 : 0 5

and obtain the execution rule

j e j = 0 : 0 5 j x j . For 1 = 0 : 0 0 5 s and according to Corollary 4.1 we can

select any 0 satisfying 0 : 0 2 3 5 0 0 : 0 4 0 5 . The theoretical value

for the interexecution time corresponding to 0 = 0 : 0 4 is

1 + = 0 : 0 0 5 s + 0 : 0 0 9 s = 0 : 0 4 s

while simulation results for the initial conditions

( x

1

( 0 ) ; x

2

( 0 ) ) = 0 c o s

2

3 0

; 0 s i n

2

3 0

= ; . . . ; 3 0

provided a lower bound of 0 : 0 2 3 7 s .

In Fig. 1 we can see how the error norm never reaches j x j even

though it goes beyond 0 j x j which is used as execution rule. Although

forsmall values of time there is a large gapbetween the maximum value

reached by j e j and j x j , this gap decreases as the state approaches the

origin.

For different values of 0 we obtained the interexecution time es-

timates reported in Table I. We can see that the estimated values, al-

though conservative, do not overestimate the values obtained through

simulation by more than a factor of 3.

VI. COSCHEDULABILITY OF STABILIZING CONTROL TASKS

The lower bound for interexecution times of the control task, pre-

sented in Theorem 3.1, can be used to analyze schedulability of a set

of tasks T = f Ti

g

i 2 I

. We shall assume a preemptive scheduler inwhich the control task has the highest priority and thus cannot be pre-

Fig. 1. Evolution of j e j , j x j and j x j for 1 = 0 : 0 0 5 , = 0 : 0 5 and =0 : 0 4 .

empted by anyother task and is executedwithout delayswhen ( j e j ) =

( j x j ) . Note that timing overheads associated with context switching

can be captured in the proposed framework by suitably enlarging 1 .

We regard execution of the control task as a timing overhead imposed

on the tasksT

i

by the scheduler in the sense thatT

i

may need to be

interrupted to execute the control task. When a set of software tasksT

can be scheduled despite the overhead associated with the control task

we say that T is coschedulable with the control task. Coschedulability

is now ensured by the following sufficient condition where d r e denotes

the smallest integer greater thanr 2 I R

.

Proposition VI.1: Let _x = f ( x ; u ) be a control system, let u =

k ( x ) be a control law renderingthe closed-loopsystem ISSwith respect

to measurement errors and let T = f Ti

g

i 2 I

be a set of tasks with

execution timesf 1

i

g

i 2 I

. If the set of tasksT

is schedulable with new

execution times 1 0i

given by

1

0

i

= 1

i

+

1

i

1

then T is coschedulable with the control task under the execution rule(9).

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TABLE ISIMULATED AND ESTIMATED INTER-EXECUTION TIMES FOR SEVERAL VALUES

OF

Proof: Since the time between executions of the control task

is lower bounded by 1 + , each task Ti

can be interrupted at

most d 1i

= e times. Each interruption will delay the execution of

T

i

by1

units of time resulting in a combined execution time of

1

i

+ d 1

i

= e 1 .

Coschedulability is ensured by the existence of a schedule where the

duration 1i

of each task Ti

has been inflated to 1 0i

in order to accom-

modate the timing overhead imposed by the control task. This result

is a very simple illustration of how the interexecution time can be

used for schedulability analysis of the proposed event-triggered policy.

Much less conservative results can be obtained when more structure is

assumed on the set of tasks T (periodicity for example) and on the

scheduler.

VII. DISCUSSION

A. Real-Time Scheduling Policies for Nonpreemptible Tasks

The simple preemptive scheduling strategy presented in Section VI

relied on the possibility to preempt all but the control task. If there are

other non-premptible tasks and its execution times are small compared

with1

, then the same scheduling strategy can still be used by suit-ably enlarging 1 to account for execution delays. In many, if not most,

situations this assumption may not hold and more elaborate scheduling

strategies are necessary. The results presented in this note are also rele-

vant in this more general context since the lower bound on the interex-

ecution times can be used to construct a timed-automaton model for

the control task. This model can then be composed with models of the

remaining tasks and a scheduler, regarded as a supervisor, can be syn-

thesized by resorting to control and/or game theoretic techniques for

timed automata [3], [12].

B. ISS With Respect to Actuation Errors and Networked Control

Systems

The results presented in this note are based on the assumption thatthe controller to be implemented renders the closed-loop system ISS

with respect to measurement errors. From a theoretical point of view

it is not difficult to see that Theorem 3.1 still holds, mutatis mutantis,

if ISS with respect to actuation errors is assumed in place of ISS with

respect to measurement errors. This is an assumption that can be made

without loss of generality for control affine systems. In this case, the

execution rule would be based on the error e ( t ) = k ( x ( ti

) ) 0 k ( x ( t ) )

which makes this approach uninteresting from the practical point of

view as k ( x ( t ) ) needs to be computed to determine if the control task

computing k ( x ( t ) ) should be executed! Under the ISS with respect to

measurement errors assumption used in this note we only need to com-

putej x ( t

i

) 0 x ( t ) j

andj x ( t ) j

to determine if the control task should

be executed and this can be done in hardware without requiring pro-

cessor time. However, the ISS with respect to actuation errors, doesmake sense in a networked control setting where the shared resource to

be scheduled is not processor time but rather the transmission medium.

Similar ideas have been more or less explicitly explored in [17], [19],

[26].

ACKNOWLEDGMENT

The author would like to thank his former colleague Mike Lemmon

at the University of Notre Dame for insightful discussions about someof the issues addressed in this note.

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Positive Controllability Test for Continuous-Time

Linear Systems

Hiroshi Yoshida, Member, IEEE, and Tetsuro Tanaka, Member, IEEE

AbstractThis note presents a method to test the positive controllabilityof a continuous-time linear system. Based on the Jordan canonicalform, thecontrollability of a given system can be reduced to those of its subsystemswith real eigenvalues. Because the dimension of the subsystem is smallerthan that of the given system, the controllability test can be simplified. It is

pointed out that the conditions for a continuous-time system to be positivecontrollable have almost the same expressions as those for a discrete-timesystem.

Index TermsContinuous-time linear system, controllability test,Jordan canonical form, positive control.

I. INTRODUCTION

The concept of controllability of systems was introduced by Kalman

[1], which is a significant and fundamental subject in control system

theory. As for the controllability under positive input, necessary and

sufficient conditions for continuous-time linear systems were obtained

which arise frequently in the practical problems, such as antivibration

control of pendulums system [2], optimal control of economic system

[3], and others [4], [5]. It is very important to test the controllability in

designing systems. But unfortunately the subject of testing the positive

controllability of continuous-time linear systems has not been investi-

gated completely as yet.

On the other hand, the controllability of discrete-time linear systems

with input constraints was also discussed in [6][10]. A necessary and

sufficient condition for single input discrete-time linear systems withpositive input was obtained by Evans et al. [7]. The authors presented

a necessary and sufficient condition for multiple input discrete-time

linear systems to be positive controllable, where a method to test the

positive controllabilitywas proposed [10]. Thepurpose of this note is to

present a method to test the positive controllability of continuous-time

linear systems based on the results of discrete-time linear systems [10].

By the way, the controllability of positive discrete-time linear systems

Manuscript received October 25, 2005; revised July 19, 2006 and May 7,2007. Recommended by Associate Editor A. Hansson.

The authors are with the Department of Electrical and ElectronicsEngineering, Kagoshima University, Kagoshima 890, Japan (e-mail:[email protected]).

Digital Object Identifier 10.1109/TAC.2007.904278

was discussed recently [11], [12]. But the controllability of positive

systems is different from that of ordinary systems with positive controls

because both of inputs and states are restricted to be nonnegative in

positive systems.

This note is constructed as follows. In Section II, some definitions

are given for preliminaries. In Section III, a necessary and sufficient

condition is given for a continuous-time linear system to be positive

controllable based on the Jordan canonical form and a method is pro-posed to test the positive controllability. In Section IV, an example is

presented to illustrate the proposed method. Finally in Section V, con-

cluding remarks are stated.

II. PRELIMINARIES

Consider a multiple input continuous-time linear system described

by

S : _xxx ( t ) = A xA xA x ( t ) + B uB uB u ( t ) (1)

wherexxx ( t ) 2

n ,uuu ( t ) 2

m ,AAA 2

n 2 n ,BBB 2

n 2 m . The control

input that satisfies uuu ( k ) 2m

i = 1

[ 0 ; 1 ) is called a positive control.

Definition 1: Let xxx0

and xxxf

be any initial and final states, respec-

tively. Then system S is calledpositive controllableif there exist a finitetime

and a positive controluuu ( t ) 0

;0 t

, which will bring

the system from xxx ( 0 ) = xxx0

to xxx ( ) = xxxf

.

Definition 2: If xi

> 0 or xi

0 for all i = 1 ; 2 ; . . . ; n , then

xxx > 0 or xxx 0 , respectively, where xxx [ x1

; x

2

; . . . ; x

n

]

T with T

denoting the transposition. Further we use IIIn

2

n 2 n as the identity

matrix and the following notations:

C ( AAA ; B B B ) BBB ; A A A BBB ; . . . ; AAA

n 0 1

BBB 2

n 2 n m

EEE

m

[ 1 ; 1 ; . . . ; 1 ]

T

2

m

; 0

n

[ 0 ; 0 ; . . . ; 0 ]

T

2

n

eee

m 1

[ 1 ; 0 ; 0 ; . . . ; 0 ]

T

2

m

; eee

m 2

[ 0 ; 1 ; 0 ; . . . ; 0 ]

T

2

m

; . . . :

Definition 3: Let f k1

; k

2

; . . . ; k

m

g be one of the rearrangements

of f 1 ; 2 ; . . . ; m g . Then WWW [ eeem k

; eee

m k

; . . . ; eee

m k

] 2

m 2 m is

called a permutation matrix.

Definition 4: By a nonsingular real transformation of the state vari-

able, ~xxx ( t ) = VVV 0 1 xxx ( t ) , system S described by (1) is transformed into

~

S :

_

~xxx ( t ) =

~

AAA ~xxx ( t ) +

~

BBB uuu ( t ) ; ~xxx ( t ) 2

n

(2)

where

~

AAA VVV

0 1

AAA VVV ;

~

BBB VVV

0 1

BBB ; V V V 2

n 2 n

; d e t ( VVV ) 6= 0 : (3)

If the following holds:

~

AAA VVV

0 1

AAA VVV

~

AAA

1 1

0

~

AAA

2 1

~

AAA

2 2

~

BBB VVV

0 1

BBB

~

BBB

1

~

BBB

2

; VVV

0 1

xxx ( t ) ~xxx ( t )

~xxx

1

( t )

~xxx

2

( t )

~

AAA

1 1

2

~n 2 ~n

;

~

AAA

2 1

2

~n 2 ~n

;

~

AAA

2 2

2

~n 2 ~n

;

~

BBB

1

2

~n 2 m

~

BBB

2

2

~n 2 m

; n = ~n

1

+ ~n

2

(4)

then system ~S1

described by

~

S

1

:

_

~xxx

1

( t ) =

~

AAA

1 1

~xxx

1

( t ) +

~

BBB

1

uuu ( t ) ; ~xxx

1

( t ) 2

~n (5)

is called a subsystem of S .

Note that in obtaining the triangular form ~AAA in (4), VVV can be calcu-

lated based on anAAA

-invariant subspace [13].0018-9286/$25.00 2007 IEEE

04303247

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