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TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 1,

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1993

G. Kreissilmeier and R. Steinheuser, Application of vector per-

formance optimization to a robust control loop design for a fighter

aircraft,

Int. J . Contr.,

vol. 37, pp: 251-284, 1983.

A. Graham,

Kronecker Products and Matrix Calculus with Applica-

tions.

New York: Halsted Press, 1981.

A. Ben-Israel and T. N. E. Greville,

Generalized Inuerses: Theory

and Application.

W.

J.

Vetter, Vector structure and solutions

of

linear matrix

equations,

Linear Algebra IfsAppl. ,

vol. 10, pp. 181-188, 1975.

A. E. Bryson and Y. C. Ho,

Applied Optimal Control.

New York:

Wiley, 1975.

New York: Wiley, 1974.

Exact Feedback Linearization and Control

of

Space

Station Using CMG

Sahjendra N. Singh and Theodore C. Bossart

Abstract-Based on feedback linearization theory, a new approach to

attitude control of the space station using control momentgyros CMGs)

is presented. A linearizing transformation is derived to obtain a simple

linear representation of the nonlinear pitch axis dynamics. A feedback

control law for trajectory tracking

is

derived. Extension of this approach

to linearization of the coupled yaw and roll axis dynamics and control is

presented.

I.

INTRODUCTION

Attitude control of space vehicles employing control moment

gyro (CMG) is an interesting problem. The equations of motion

of the space station are described by nonlinear differential

equations. Often, attitude control system design using linear

control theory [1]-[3] is obtained. For large changes in orienta-

tion of space vehicles employing momentum exchange devices,

nonlinear controllers have been designed in literature [4]-[9].

The controller of [7] is based on the inversion of a nonlinear

input-output map.

A n

adaptive control design has been pre-

sented in [9].

In this note, we present a new approach to attitude control

system design of the space station employing control moment

gyros. Using feedback linearization theory [lo], [ll],

a

linear

representation of the nonlinear dynamics of the space station is

derived. In the new state space, a feedback control law is easily

derived for the control of pitch, yaw, and roll angles.

11. MATHEMATICALODEL ND CONTROL PROBLEM

It is assumed that the space station is in a circular orbit. An

orbital frame of reference (LVLH axis) with its origin at the

center of the mass of the space station is chosen. The axis of the

reference frame is chosen such that the roll axis is in the flight

direction, the pitch

axis

is perpendicular to the orbital plane,

and the yaw axis points toward the earth. The orientation of the

space station with respect to the reference frame is obtained by

a roll-pitch-yaw

(e, - 8,

13,) sequence of rotations, where e,,

0 2 , and

8,

are the roll, pitch, and yaw angles. The nonlinear

Manuscript received September 21, 1990; revised November

8,

1991.

This work was supported by the United States Army Research Office

under Gran t DAA L, 03-87-G-0004.

The authors are with the Department

of

Electrical and Computer

Engineering, U niversity of Nevada, Las V egas, NV 89154.

IEE E Log Number 9203091.

equations of motion can be written as [2]:

I ; = -610

+

3n2EIc U

(1)

-s ine,

o

,

(2)

(3)

where n,

= (0,

n,

0);

the orbital angular velocity is n = .0011

rad/s;

h

= (h, ,h,, hJ T ;

h,

is the body-axis component of CMG

momentum; U =

U ~ , U , , U , ) ~

is the control torque vector; the

inertia matrix I = ( I L , ) ,

j

= 1,2,3,; w = (wl, 2 ,w, l T; w , is

the body axis component of angular velocity; c = (c l , c2, , ) ~ ,

c1 = -sin e, cos e,, c2

=

cos 8 , sin 0, sin 8, sin 0, cos e,, c ,

=

-sin 0 , sin 9, sin

0,

+ cos 0, cos@,, and for any vector m =

( m l ,

m

JT, f i is defined as

os e, -cos 8 , sin 0, sin 0, sin

e,

0

sin e l cos

0,

cos O1 cos3

[ j

=

+[

case,

e 2

h + L h = u

0

-m3

m 2

m =

2,

,,

-71.

For certain configuration of the space station, one requires a

large pitch.

In

this note, we shall treat the question of control of

the space station for this configuration. The complete equations

of motion for this configuration have been derived in the litera-

ture [2]. These are:

I,i, 1

+

3 ~ 0 s ,)n2(1,

3)o l

- n I , - 1,+ Z 3 ) i 3

3(1, - 13)n2(sin , cos e, )@, = - u l

I,i, 3n2(1, ,)sin

0,

cos B, = -U ,

I,@,

+ (1 3sin2 B , ) ~ ~ ( Z , l)e,

+

n ( I , -

,

+ ~ ~ i ~

3(12 l)n2(sin

0,

cos

0 , ) 4 =

- u 3

h1

-

nh,

=

u l , h,

= U,, h3 + nh, = U,.

(4)

Equation 4) is derived from (l), (2), and (3) assuming that 0, is

large but the roll, and yaw attitude errors are small and also the

products

of

inertia are small. Here I ,

We are interested in designing a control system such that the

attitude angles of the space station can be controlled using

CMGs.

I l l ; = 1,2,3.

111.

LINEARIZINGRANSFORMATION

OR

PITCH DYNAMICS

The pitch dynamics which include the CMG momentum equa-

tion, differ from the dynamics of a pendulum. For the pitch axis

control system design, we consider the third order system:

k,

= -(1.5n2(1,

-

3)sin2e,

+ u 2 ) / 1 ,

The system(5) differs from the dynamics of a pendulum, since it

includes the CMG momentum equation. We are interested in

obtaining a nonlinear transformation such that ( 5 ) has a linear

representation. We shall obtain a linearizing transformation

using the result of [lo], [ll].

h,

= U,

( 5 )

We write 5 ) in a compact notation

=

f x ) +

gu,

(6)

where b, = (l/l,),

i =

1,2,3,

x = (e,,

e, h,)T,

u2

= 1.5n2 1,

[, /I,,

o

=

- U sin20,, f x)

=

x,, U , sin202,0) and

g

=

(0,

-b2,

1)

To this end, we introduce certain definitions [12],

[13]. The Lie bracket of

two

vector fields f and

g ,

denoted as

[f ] or

u d f g ,

is a vector field given by u d f g = [f ] =

( d g / d x ) f d f / d x ) g .

Repeated Lie brackets are denoted by

udyg =

g ,

ud g = [f,

ad f k - l g ] ,

k > 0.

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185

Definition:

A set of

1

vectors

f I ( x ) ; . . , r ( x )

is involutive if

there exist smooth functions

y i j k ( x )

uch that

I

[ f i , f / I ( x )= y l j k ( x ) f k ( x ) , V i , , k

=

I;..,

1.

k = 1

Now we investigate the existence of l inearizing transformation

on an open set

U

containing the origin

x

= 0. We can state the

following results of [lo],

[I11

Lemma 1: The dynamics of the nonlinear system (6) are

locally equivalent to the dynamics of a controllable l inear system

by change of state and input coordinates and state feedback if

and only if (1) the distribution, span

{ g ,

a d f g ,

a d j g }

has dimen-

sion 3 on

U,

and (2) the distribution, span

{ g , a d f g }

s involutive

on U .

In the following, we shall verify conditions 1 and 2. Computing

the Lie brackets, gives

a d f g

= [ f , g l = ( b 2 , 0 , 0 ) '

( 7 )

(8)

d f g

=

[ f , a d f g ]

=

= ( o , ~ u ~ ~ ~ c o s ~ ~ ~ , o ~ .

We note that the vectors

g , a d f g ,

and

a d ; g ( x )

are independent

at

x

= 0. Therefore, these vector fields are independent on an

open set U containing the origin x =

0.

The vector fields g , a d f g

are involutive

on U

since [ g ,

a d f g ]

= 0 E span

{ g , a d f g }

and

this verifies conditions 1 and 2.

According to Lemma 1, a linearizing transformation exists. In

order to find this transformation, one needs to solve a system

of

linear partial differential equations. This can be accomplished by

solving the following system of equations

d u A

ds

( X ) , x ( 0 )

= 0 (9)

(10)

du

g , x s , t , , o >

=

x ( s , , )

(11)

dt2

where f i x )

=

a d j ( g ) or f i x ) is any vector field that is linearly

independent of g and [ , 81.

For simplicity, we choose

f i x )

=

(0,

1,OIT. The set

of

equa-

tions (9)-(11) are solved sequentially. First solving (9) with the

initial condition

x 0 )

=

0,

gives

x ( s )

=

(0,

s , 0lT. Now we solve

(10) with initial condition

x ( s ,

0) = (0, s, O T to yields

x ( s ,

l )

=

(b2t l ,

,OlT. Finally, solving (11) with initial condition

x ( s ,

t , , 0)

= ( b , t , ,

s , O) one gets

(12)

Solving (12) for s, gives s

=

x 2 b , x , . We set s = tl.Then

the linearizing transformation is given by [ll]

5 =

(t , , 2 ,3>

where 5, = tl 6,

=

t2 ,nd

[ = ( x ,

+

b 2 x 3 ,

a2s in2 02, -2a2cos28,8,)

.

(13)

+ (14)

~ ( s ,, , 2 = ( b z t l , - b2 t2

+ s,

2 l T .

. T

A linear representation of (6) is given by

where e ,

=

(O,O, 1) and

0 1 2 x 2

=

[ o o,,,].

(15)

Here 0 and 1 denote null and identity matrices of indicated

dimensions, and

u2 = [4a2sin28,8,2 a;sin40,] + ( ~ u , ~ , c o s ~ ~ ) u ,

We notice that a choice of Foordinate transformation (13) and a

feedback control U, =

b; ( x ) [ - a z

U , ] results in a control-

A

a ; ( x ) +

b;u,.

lable linear system representation (14) of the nonlinear pitch

dynamics. It is interesting

to

note that the linear system (14) is

the exact representation of the third-order nonlinear system (5)

for all

x E R

in which b z - ' , exists, where

R

= { x

E

R3:

8,

f

+ ~ / 4 ) .A controller easily designed using (14) and a linear

control theory is valid in the region

R .

It is well known that the

a regulator designed using a linearized model obtained by the

Taylor series expansion of nonlinear functions about a chosen

operating point is effective only in a small region about the

operating point.

IV. PITCHAxis CONTROL

To this end, we introduce new state vector z1 which is integral

of 6,. The introduction

of

integral term leads to robustness in

the control system to uncertainty in system parameters. Thus

z1 = O2 z , and is= b 2 ~ 3 .Define a new state vector z =

(zl, z,, z,, 2,) =

(zl, E R 4

given by

z =

8, +

zs, x 2

b , x , ,

-a2 sin28,, -2a2cos262~,)T.One can easily verify that

i =

Az

+

bu,

(16)

where b =

(0, 0,

0,l) and

03x1 13x3

A = 0 OIX3 ]

Suppose that it is desired to track the reference trajectory zC1of

a command generator

~ , ( S ) Z , ~

A, w , , ~ , , ~ z , * ~where

n,(s>

= (s

A,)n~=, s2

25,,~,, ,s w;,,) , s

=

( d / d t ) and z,*1 is

the terminal value of z,, . Define

f

=

z , ,, , i

=

1,2,3, 4 and

z,(,+ 1)

=

i

=

1,2,3,4. Then one has from (16)

i

=

Af b ( u ,

, ~ ) . (17)

Now we choose a control law

U , = of, - p i 5 2 - ~ 2 f 3

p 3 Z 4

+ 2 s. (18)

In the closed-loop system 17) and

(18),

one has =

i

where

2 s obtained from

A

by replacing the bottomfow of

A

by

( - p o , -pl, - p 2 , - p 3 ) .

We choose

p ,

such that

A

is a Hunvitz

matrix. It is interesting to note that Z l satisfies a linear differen-

tial equation given by

n,(s)il=

0 where n,(s)= (s4 +p 3 s 3

p 2 s 2+

pl s

+p o ) .The parameters p , are chosen by equating the

polynomials

Il, s)

=

n:=,(s2

25,,0,,,s

+ w:,,)

where

leJ

0,

w,,

>

0

are appropriately chosen parameters. Since

A

is

Hunvitz, the tracking error

Z1( t )+

0, as

t

+ W.

v,

YAW

AND

ROLLDYNAMICSINEARIZATION

ND

CONTROL ESIGN

The yaw and roll dynamics are given by

A -

=

( Z , 0,) + g,u, + g,u,

where

U

=

[u , ,u31 x - OI,

l

8

e

hi, hJ , Zlfo,

=

-(I

+

3cos2 e,)n2(z2

- z3)o1

+ n ( ~ , 1,

+ Z -

31,

- 1,)n2

sin

28,8,; and

Z,fo,

= -(1 + 3sin2

8 , )n2 (1 , - Z I I 6 3

+ n(Zl

2

+ z3)el

-

3(1,

-

1,)n2 sin 20~ 0, .

In the following, we shall obtain a linearizing transformation

for the system (19). To this end, we introduce certain operators

which will be useful in the sequel. The Lie derivative of a scalar

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function h along a vector field f , denoted as L f h ( x ) s given by

L f h ( x )

=

[ ( d h ( x ) / d x ) ] f ( x )where x

= (Z',

8 2 , i2

t,) , f =

fT, e 2 ,

ez - b,u2, u , x , t))' (The readers should not be con-

fused with the vectors

x

and

f

used in Section 111). Let L h ( x )

= L f ( L f h X x )and Lg,Lf(x) =

L,,(LfhXx).

To this end, o ne can follow the procedure of [lo] for the exact

linearization of

(19)

which requires a solution of a set of differ-

ential equations. Instead, motivated by the transformation of

Section

111,

we shall choose coordinate transformation and a

feedback control law such that in the closed-loop system exact

linearization of (19) is obtained. Let

us

consider a transforma-

tion

@

= 411,

4 1 2 ,

413, 431,

3 2 ,

4 3 3 l T

(20)

where the coordinates

c ~ ~ ,

~ ~re chosen as

.

h , n u ,

-11

11

1 1

h3 n u 3 - 1 2 )

13

13

Cp = 81 +

8 3

81

,, = e +

(21)

The coordinates

4,1

and 431 are similar to 6 = 6 2 + ( h z / I z )

in (13), however they also include linear functions of 8 , , and 83.

These additional functions are introduced so that the control

inputs appear for the first time in the third derivatives of

411

and

431.

t is interesting to see that with this choice

of

d~~~

and

31,

one has

LgkLf+,,,.=, i E

{1,3};

j =

0,.1; k

=

{1,3h 412

=

4 = Lf4li' 4 1 3

= 4 1 2

= L 411,

4 3 2 = 4 3 1 = L f 4 3 1 ,

4 3 3

=

32 = Lf432

= L74317

and

a * ( x , u , ( x ,

t ) ) +

B* X)U

where 4:;)=

( d 3 4 , , / d t 3 ) .

We note that L?4 , , , and L3f43l in-

clude input U,, which has been already derived in Section 111

and IV. We choose a linearizing feedback control law

(23)

B * -

[

--* ( X , U 2 )

+

a l .

Then, linear representation of th e system

(19)

is

6, A,@ . ,

e,fi, ,

i

= 1 , 3

(24)

= (411, 12,13),T> a3

=

4 3 2 ,

c J,,)', and = ( i J l , c 3 ) T . We notice that, similar to the pitch

dynamics, this linear representation of the roll and yaw dynamics

is obtained by the nonlinear feedback (22) and the nonlinear

coordinate transformation (20).

Thus, the coupled yaw and roll dynamics decompose into two

third-order linear subsystems. This system is similar to the linear

system treated in Section I11 and IV. We introduce the integral

of 4,1, nd as additional state variables for robustness and

carry out the design as done for controlling pitch angle.

where @ =

(@:,@TI',

VI. SIMULAT~ONESULTS

In this section, for simplicity, results for only pitch axis control

are presented. Space station parameters are: (inertia slug-ft

2 ,

I , ,

= 50.23E6,

I,,

= 10.80E6, and I,, =

58.57E6.

The con-

troller parameters are:

A,

=

.0005,

,

=

1.0,

o,,,

.0005,

i

=

1,2,

o,,,=

.0015, le,= 1.0,

i

= 1,2. The initial conditions were

chosen to be

8,(0) =

30, i 2 O > =

.005 /s.,

h,(O)

=

0, and z , =

0. The initial conditions for the command generator were chosen

such that z,,(0) = z ,(0) , i = 1,2,3,4, and

z,,(O)

= 0. Notice that

with this choise of

z,,(O),

one has

f , ( O )

= 0. Selected responses

are shown in Fig.

1.

We observe that the pitch angle O2 and

h2

ID

Y

m

ZW

0.00E.

lwI

.00

Y

:::

3.00

i-

1.00

.:::I

-,

Loo

-3.00

0 00

200.00 400.w

600110

Time. .Xinuces

C)

Fig. 1.

Attitude control. (a) Pitch angle.

b)

Control torque. (c)

CMG

momentum.

converges to zero in about 500 minutes for the chosen command

trajectory zCl(t). The maximum values of the torque input u 2

and the angular momentum h, were less than 7 ft-lb and 5900

ft-lb .

s,

respectively, which are well within permissible limits. It

is interesting to note that the tracking error

Z,( t )

=

0, for all

t

as

predicted.

CONCLUSION

Based on feedback linearization, a new attitude control system

was designbed for controlling the orientation of the space sta-

tion using

CMG's.

Feedback linearization gave rise to three

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decoupled controllable linear systems which describe the nonlin-

ear, coupled pitch, yaw, and roll dynamics. Control law is rela-

tively easily derived based

on

the linear system representation in

the new state space.

REFERENCES

New

York

Sparton, 1970.

A. L. Greensite,

Analysis and Design of Space Vehicle Flight Control

Systems,

vol. 11.

B. Wie,

K.

W. Bynn, V. W. Warren, D. Geller, D. Long, and J.

Sunkel, New approach to attitude/momentum control

for

the

space station, J .

Guidance, Contr. Dynam.,

vol. 12, pp. 714-722,

Sept.-Oct. 1989.

A. Iyer and

S.

N. Singh, MFDs of spinning satellite and attitude

control using gyrotorquers,

IEEE Trans. Aerosp. Electron. Syst.,

vol. 25, pp. 611-620, Sep t. 1989.

S. B. Skaar and L. G. Kraige, Large-angle spacecraft attitude

maneuvers using an optimal reaction wheel power criterion,

J .

Astronaut. Sci.,

vol. 32, no. 1, pp. 47-61, Jan .-Mar. 1984.

S .

R. Vadali and

J.

L. Junkins, Spacecraft large angle rotational

maneuvers with optimal m omentum transfer,

AIAA/A AS Astro-

dynam. Conf.,

San Diego, CA, Aug. 9-11, 1982.

T.

E.

Dabbous and N. U. Ahmed, Nonlinear optimal feedback

regulation of satellite angular moments,

IEEE Trans. Aerosp.

Electron. Syst.,

vol. A E S - 1 8 , no. 1,

pp. 2-10,

Jan. 1982.

S. N.

Singh and A. D. de Arahjo, Asymptotic reproducibility in

nonlinear systems and attitude control of gyrostat,

IEEE Trans.

Aerosp. Electron. Syst.,

vol.

AES-20,

no. 2, pp. 94-103, Mar. 1984.

T. A. W. Dwyer, 111 and A. L. Batten, Exact spacecraft detum-

bling and reorientation maneuvers with gimbaled thrusters and

reaction wheels,

1.

Astronaut. Sci.,

vol. 33, no.

2,

pp. 217-232,

Apr.-June 1983.

S. N. Singh, Attitude control of a three rotor gyrostat in the

presence of uncertainty, J .

Astronaut. Sci.,

vol. 35, pp. 329-345,

July-Sept. 1987.

L. R . Hunt,

R. Su,

and G. Meyer, Design for multiinput nonlinear

system,

Differential Geometric Control Theoy,

R. W. Brockett,

R. S. Millman, and H. Sussamann, Eds. New

York:

Birkhauser,

B. Jakubczyk and W. Respondek, On linearization

of

control

systems, Bull. Academy Polon. Sci. Str. Sci. Math, vol. 28, pp.

517-522, 1980.

A. Isidori,

Nonlinear Control Systems: An Introduction.

New

York

Springer-Verlag, 1989.

H. Nijmeijer and A. J. V. der Schaft,

Nonlinear Dynamical Control

Systems.

New York: Springer-Verlag, 1990.

J.-J. E. Slotine and W. Li,

Applied Nonlinear Control.

NJ: Pren-

tice-Hall, 1991.

pp. 268-298, 1983.

On

the Asymptote

of

the Optimal Routing Policy

for

Two

Service Stations

Susan

H.

Xu and Hong Chen

Abstract-Hajek 111 among others, proved that the optimal routing

control of a two-station Markovian network with linear cost is described

by a monotone switching curve. With the discounted cost objective

Manuscript received O ctober 12, 1990; revised Septem ber 6, 1991 and

January 31, 1992. This work was supported in part by the National

Science Foundation unde r Gran t DDM -89-09972 and in part by the

Natural Sciences and Engineering Research Council of Canada.

S . H . Xu is with the College of Business Administration, Pennsylvania

State University, University Park, PA 16802.

H. Chen is with the Faculty of Commerce and Business Administra-

tion, University of British Columbia, Vancouver, Cana da an d is also with

the Department of Mechanical and Industrial Engineering, New Jersey

Institute of Technology, New ark, NJ 07102.

IEE E Log Num ber 9203968.

function, we prove that the optimal switching curve has

a

finite asymp-

totic limit when

c 1 f c 2 ,

where

ci

is the unit inventory cost at station

i.

Whereas, for the case with c 1 = c2 , as well as the case with the long-run

average objective function, the switching curve does not have a finite

asymptote.

I.

INTRODUCTION

We consider a special case of Hajeks model [l], where upon

their arrivals jobs are to be routed to one of the two exponential

service stations. The arrival process is a Poisson process with

rate A, and the exponential service rates at stations 1 and 2 are

pI nd p 2 , espectively. First we focus on the discounted objec-

tive function, i.e., to route the arrived jobs in order to minimize

where c,, c 2 and

(Y

are given positive scalars,

X, t)

denotes the

number of

jobs

in station

i

at time

t , =

1,2,

x =

(Xl(0),X,(O))

is the initial state, and U is a control (routing) policy. For

simplicity, we will not make the dependency of the probability

and hence the expectation on the initial state x and the control

U

explicit in future presentation.

It was shown in [l ] that the optimal policy is characterized by

a nondecreasing function

s

from

Z ,

to 2 , such that a job that

arrives when the queue lengths are x 1 and x 2 is sent to station 2

if x 2 (xl) and to station 1 otherwise. One might think that

s xl) is finite when x1 s and s x , ) approaches infinity when

x1

does, i.e., a new job will inevitably be routed to another queue

when one of the queue lengths is sufficiently large. However,

this is true only in a very special case when

c 1 = c , .

In fact, the

switching curve s ( x , ) approaches a finite asymptote as x1 goes

to infinity if

c1 < c , ;

symmetrically, if

c1 >

c 2 , there exists a

finite asymptote

xT

such that s x , ) goes to infinity as x1+ xT.

We will prove the first case.

Theorem1:

Suppose c1 < c . Let

s .)

be the optimal switching

curve. Then

s(nl)

converges to a finite asymptote as x1 + m.

Proof Let V ( x ) be the minimum expected cost given the

initial state x (the existence of such I/was proved in [l]). Then it

was proved in [l ] that it is optimal to route a job to station 1 if

(1)

and to station 2 otherwise, where x = x , , x,) is the state of the

queue length process at the arrival epoch of the job. Therefore,

it suffices to show that there exists a finite xg such that inequal-

ity (1) holds for all x1 2

0

and x 2

2

x g .

Let X = {(Xl( t) , J t ) ) , ) and

Y = ( (Y J t ) , ,(t>),

2 0)

denote the queue length processes with initial states xl, 2 1)

and

x,

+ 1,x2) , respectively. We couple the two processes as

follows: Let process Y follow the optimal policy for process X

(such that they have the same arrival processes to both stations)

and assume that a given job has the same service time realiza-

tion in both processes; we further let the job x, 1 at station i

i = 1,2 , be served only when

no

other jobs are waiting at station

i and be preempted by the new job arrived to station

i

during

the service of job

x,

+ 1.Due t o the memoryless property of the

exponential distribut ion, the expected costs of the processes

subject to these shuffling and preemptions are identical to

those of the processes without the shufflings and preemption.

We reckon the cost difference of the coupled processes. First

consider the cost difference inducted by station 1.Let CT, be the

first time station 1 in process

Y

becomes empty, i.e., CT,

=

inf

( T :

Yl(t>

O}. Note that

X , ( t )

must also equal 0 at time vl. Clearly,

Y,(t)=

X l ( t )

1

for t E O

CT,),

and

Y,( t)

= X , ( t ) for

t E

Vh ,

+

1) V Xl + 1, x , ) 2 0

0018-9286/93 03.00 0 1993 IEEE