Dynamic Models of Comprehensive Scheduling for Ground-Based Facilities Communication with Navigation Spacecrafts

ELSEVIER

Copyright @ IFAC Automatic Control in Aerospace. Saint-Petersburg. Russia, 2004

IFAC PUBLICATIONS www.elsevier.c:omllocatelifac:

DYNAMIC MODELS OF COMPREHENSIVE SCHEDULING FOR GROUND-BASED FACILITIES COMMUNI CATION WITH NA VIGA TION SPACECRAFTS B.V. Sokolov

St.-Petersburg Institute for Informatics and Automation. 39. 14th Line. SPIIRAN. St. Petersburg, 199178, Russia, [email protected]

Abstract: A multiple-model description of interaction between a ground-based control complex (GCC) and orbital system (OrS) of navigation spacecrafts (NS) is presented. A dynamic interpretation of operations and control processes is implemented. The proposed approach lets use fundamental scientific results of the modem control theory for new applied problems. In particular, in this case, a scheduling problem for GCC ground-based technical facilities was reduced to a boundary problem with the help of the local section method. Scheduling problems of the considered class are usually solved via methods of discrete programming, but when the dimensionality is high, the optimal solution is not provided and heuristic algorithms are needed. This paper introduces an original approach, based on models and methods of optimal control theory, to scheduling problems of high dimensionality'. Copyright © 2004 IFAC Keywords : boundary conditions, dynamic models, large-scale systems, scheduling algorithms, satellite control, planning, qualitative analysis.

automatic planning, for SF CS design, maintenance, and improvement remains a very important problem.

1. INTRODUCTION Modem control systems (CS) of space facilities (SF) belong under a class of complex technicalorganizational systems (CTOS). Control systems include orbital and ground-based SF (OrSF and GSF). In their turn, OrSF and GSF interact with each other and combine into orbital systems (OrS) of SF, in other words, OrS of spacecrafts and into ground-based control complexes (GCC) (Raab, et al., 1977; Sokolov, and Kalinin, 1995). General organization and performance of tasks in the above mentioned control systems and complexes are being carried out by personnel of different categories. The specialists take charge of stable and reliable functioning of SF CS.

Planning, in the broad sense, is a purposeful, organized, and continuous process including examination of complex technical system (CT)S elements, analysis of their current state and interaction, forecasting of their development for some period, forming of mission-oriented programs and schedules. Three planning approaches (concepts, philosophies) emerged by now (Ackoff, 1978): satisfactory (incremental), formal, and system (comprehensive) planning. Formal planning concentrates on prediction of situation in terms of mathematical models, satisfactory planning consider CTS reactions to external impacts, system planning supports CTS interaction with the environment. System planning implies problem resolution and redefmition through leaming process, rather than problem solving. This

Nowadays the development of computer-aided decision-making procedures, as well as procedures of , The work was supported by the Russian Foundation for Basic Research under the project 02-07-90463.

263

2. VERBAL DESCRIPTION OF A SCHEDULING PROBLEM

lets interpret planning not as discrete operations, but as continuous adaptive process. In (Ackoff, 1978) this was called adaptation planning. A posteriori, current, and a priori infonnation can be used for plan adaptation (adaptation to the "past", "present", or "future"). Various planning theories realizing the abovementioned concepts for different applications were developed by now (Ackoff, 1978; Klir, 1985).

Let A = {A;, i e N}, N= {l, ... ,n} be a set of navigation spacecrafts (NS).The spacecrafts are elements of an Orbital System (OrS). The main function of Ors is to provide users, in other words, client objects (CO) with navigation messages. Let li = {lit, k E K}, K= {1, ... ,m} be a set of CO. Navigation support for CO requires special configuration of Ors which should be designed and

The scope of this paper does not allow describing advantages and disadvantages ofall existing approaches to planning. Let us consider development of programs (plans, schedules) for operation of ground-based technical facilities embodied in GCe. Various practical planning and resource-allocation problems for some Gce elements and subsystems are elaborated. This concerns planning of operations for measuring-andcontrol radio communication facilities, for computing and telecommunication facilities, and for personnel (Raab, el al., 1977). Mathematical models (first of all, models of mathematical programming and simulation models), logic-algebraic, and logic-linguistic models were used for planning automation (Ackoff, 1978; Klir, 1985). However, failure to comply with principles of system approach resulted in discordance of Gce elements operation, in high peak data traffic, in ineffective work of cs. Besides, nowadays tendency of integration in Ors CS, and advantages of CS structuredynamics control (Sokolov, and Kalinin, 1995) were not taken into account.

kept up in order that each CO B t , K= {I, ... m} have a given number ofNS A; (i = I, ... ,n) in its interaction zone (lZ) over a fixed period. It is assumed that the location of CO B k is evaluated without request through measuring of pseudodistance and pseudo-velocity for each NS A; in IZ of

CO lit at time I E (/0, IJ1 (/0 is the beginning and It is the end of a planning period). In this case the navigation equations bounding measured and computed motion parameters of NS A;, can be solved if the following infonnation is available: motion state vector

x;g) for each NS B at j

every time; the current space-time location of the whole OrS; the current time-and-frequency corrections for each NS. Let us name the mentioned data respectively navigation infonnation of the first type (NI- i), navigation infonnation of the second type (NI-2), and navigational infonnation of the third type (NI-3). Modem navigation aids of CO provide simultaneous processing of navigational infonnation from each NS (NI-I, NI-2, NI-3) and measuring of pseudo-distances and pseudo-velocities. These data let CO to calculate its real-time and exact location. Accurate operation of OrS depends on pennanent corrections of on-board navigational infonnation for all NS. These corrections are perfonned by a groundbased control complex (GeC) consisting of distributed measuring and guidance stations, for short, interaction stations (IS) and of a control station (stations) (eST) coordinating the work ofIS.

The approach proposed in this paper helps to realize conception of comprehensive plannirig. The developed unified dynamic models of Gce facilities functioning can be used at the phases of SF abilities analysis and planning of SF operation (long-range and operational planning), as well as at the phase of plan implementation. Then, the general technology of Gce operational planning should include the following stages: - structural and parametric adaptation of planning models, and algorithms to the past and present states of SF and to the past and present states of the environment; - structure-functional synthesis of the main Gce elements and subsystems;

Let us introduce a set B = {Bj , ) E M}, M= {l, ... ,m} of IS and eST. All the stations use unified communication facilities and computer aids therefore they can act for each other. In this connection, let us defme one more set, namely, the set of Gce facilities, in other words, the set of channels

- GCe scheduling, simulation of possible scenarios of Gce functioning according to the schedule, structural and parametric adaptation of the schedule, models, and algorithms to future GCe states and environment states (predicted via simulation models).

C(J)

In this paper, mainly is considered the only one stage of the described technology, namely scheduling of GCe facilities operations with navigational spacecrafts. Possible approaches to other stages are considered in (Hanziverov and Ostrouhov, 1989; Malishev, el al., 1989).

= {CiJ) ,A E A j}'

Aj = {I, ... l

j}.

Generation

and correction of NI-I, NI-2, and NI-3 are perfonned as the components of a general computer aided control technology implemented for NS and OrS as well as for IS and eST. Let us introduce two more sets: a set of interaction

. D{i) -- {D(i) operatIOns z ' re

264

E

} ,

~ = {I , ••. ,S; } an d

'V

a set of flows P

(,) -_{(I) -} -_ Pp ,p ER, R - {l, ...,1t,} .

aD

These sets will be used in formal statement of the considered scheduling problem.

~p

,

D, =

Execution of technological operations implies transitions of NS CS elements and subsystems to one of possible designed (or off-design, abnormal) macrostates.

The following notations were used: I is a macrooperation directly concerned with the main mission of an element; 11 is a macro-operation being performed by a reserve element; III is a macro-operation of element maintenance or repair; 11 ~ Ill; III ~ 11; I ~ Ill; III ~ I; I ~ 11; 11 ~ I are macro-operations of intermediate macro-states. Is assumed that planning is performed when technical characteristics of NS CS main elements and subsystems are known and timespace, technical and technological constraints for OrS and GCC functioning are determined. Besides, is assumed that interaction between NS, IS, and CTS is performed in a centralized control mode. The fulfilled analysis confirmed that the accuracy and quickness of CO localization principally depends on the precision of estimate of NS A, motion state vector and on the time of updated navigational information delivery aboard NS. This information includes ephemeris, system almanac, and time-frequency corrections. Thus, for example, accuracy of navigation field for stationary CO interacting with three NS can be evaluated according to the formulas (Malishev, et al., 1989; Raab, et ai., 1977):

D,

kx, _X p )2 +(y, - yp)2 +(z, _Zp)2 J1/2;

K"

=b~ + 2bobl (t -t,) + bl2 (t - t,)2;

(2)

K e,

= Ko, +Kli (t-t,)+K 2,(t-t,)2;

(3)

=Ilx:g) y,(g) z:g)

Ir

is a motion state

Greenwich coordinate system; bo, b l are given coefficients characterizing onboard time scale accuracy of NS; Ko" K li , K2, are given correlation matrices characterizing ephemeris accuracy; K" is a correlation function of time scale errors for NS A;; K e, is a correlation matrix of NS A; location errors in Greenwich coordinate system; t; is a beginning of a period the navigational information (NI-I, NI-2, NI3) is computed for. Is used the expressions (2) and (3) in the statement of a scheduling problem to define the quality of IS and CST functioning. The presented considerations permit the following verbal description of the scheduling problem: it is necessary to fmd such allowable program (a plan of functioning) for activities of ground-based technical Facilities (GTF) that all operations of NS technological control cycles (TCC) are executed in time and completely, and the quality of CO navigation support meets the requirements. In addition, if several allowable programs of GTF control are available, the best one would be selected according to the optimality criteria.

3. DYNAMIC MODELS OF OPERATIONS PLANNING FOR GROUND-BASED TECHNICAL FACILITIES The formal statement of the scheduling problem will be produced, as it was noted in the introduction, via dynamic interpretation of operation execution processes (Athans, and Falb, 1966; Sokolov, and Kalinin, 1985; Zimin, and Ivanilov, 1971). Models for program control o{interaction operations and channels a) Models of processes

,=1

hT 1

T

p

D,

x~g) =11 X pYpz p liT is a vector of CO location in

=±(c2 K,,'ii,'ii; +'ii,h,TKe,b,'ii;), = b2

z(g)-z p _'

vector for NS A, in Greenwich coordinate system;

Fig. I. Diagram of transitions from NS CS macro-states

Bo

D,

time t; x,(g)

------e

'ii, = (B~Borl h,;

y(g) -y P _'

I

where Kz is a correlation matrix for CO location errors; c is the velocity of radio wave transmission; 'ii, ,Bo are auxiliary vectors and matrices; D; is a distance from CO to NS A; measured (computed) at

Fig.l shows a diagram of transitions from aggregated macro-states for the main elements and subsystems of NSCS.

Kz

I x(g) -x

E.T = _ , = _

(I)

~~)

b3

T

m

I,

=L ~>ij(t)e'Zj).u~J)., i = l, ... ,n;

re = l, ... ,s;;

j=I).=1

(4)

265

n

S

(k,l) ~ ~ (0,1) . ;t jA. =~LUilCjA.,I-

1, ... ,m,. A-I /. - 'OO"j'

be executed at time t E (to, tJ1 through some active (J) Do ( ) • Ch anne I C J.. an d ~;aeJJ. t = O I'f not; x p(k,l) .. IS a

(5)

i=IIC=1

n =~ ~ fl u(o~l) ~L IlCjA.· S

;t(k)

j

variable equal to total duration of the channel

(6)

xY) is a variable equal to joint

activity by the time t;

;=IIC=IA.=I

CiJ )

activity duration of BJ (IS or eST) channels by the time t; U~]A. (t) is a control input that is equal to I if the operation D~) involving NS A; is being executed

(7)

via the channel

ci

J ) , that is set on

Bj (IS or eST)

and equal to 0 if not. The constraints (7) set a sequence of interaction operations (10) of NS Tee. These equations imply

L L u~JA. (t) ~ 1 ; ViE N, V re E <1>;

blocking of 10 D~) until the previous operation

(8)

jeM I A.EAI

n

s·

~~

(0)

~LU;zjA. ;=1 se=1

~

(t) <

~

~ u(o) (t)

~ ~ ~ ZEll>i jEM z A.EA z

n

Sj

(I).

_CjA.'

IZjA.

(/) IS. D(1lC-1)

d' (0) (0) d th execute , I.e., X,(IlC_I) - a,(IlC_I) an e

required

amount

of

(processed, transmitted):

information

is

received

LLx~~2-I)jA.p

= a~~2_I)p

A.

j

< C<2) . ViE N - I' ,

(9)

E

(0)

a;(IlC_I) an

d

(n)

a;(IlC_I)p are

kn

own

(preset) values, x~~_I)jA.P is a variable that is equal to

";

L L ~:U~JA.V;ZP ~ HW ;V j

- . Here R

for V pE

M 2, V A E A 2; (10)

1=1 se=lp=1

actual amount of information that belongs under the

u~JA.(t)E{O,I}; V M=M1 uM2, V A=A 1uA2 · (11)

class

c) End conditions For the initial time t = to:

from (to) NS Ai during the operation D~) through

x(o)

IIlC

(t ) = 0

=d(k,I) . jllC

d(o). X(k,l)

jA.

11lC'

x(k) (t ) 0

j

'

0

(12)

= dlj·(k)

..

To simplify the expressions describing NS Tee (see Fig.5) it was assumed that the operations of Tee are strictly ordered and are executed one after another [see expression (7)].

j

t (a~) - X~)

=! i 2

The expressions (8), (11) are actual for non-separable operations. These constraints mean that at a fixed time each NS Ai can participate in at most one 10

2 )

(14)

D~) through the channel C~j) of IS (eST) Bj and

;=1 IC=I

n

m

s,

If

J~O) = LLLL

and was received (processed, transmitted)

the channel C~j) of Bj .

(t ) =

For the end point t = If X(O)(t )=a(O). X(k,I)(t ) X(k)(t )eRI (13) IIlC f 11lC' jA. f' j f . d) Quality measures of schedule for ground-based facilities operation J1(0)

pp(i)

vice versa the channel

If

fY;IlC('t)U}:}IJt)d1:;

Ci(J)

can execute at most one

10 D~j).

(15)

;=1 j=1 z=1 A.=I 10

JY)

=! 'II 2

f

t~(x~~I)(tf)-xj~,1)(tf») ,(16)

The constraints (9), (10), and (11) are actual for operations that share resources (the number of operations being executed by a channel at the same

q=1 j=q+1 A.=I '1=1

where x~) is a variable characterizing the state of

time is defmed by a constant c;~, the number of

operation D~) relating to interaction between NS Ai

resources used by interaction operation D~)

and IS BJ through the channel C~j), here is interpreted

defmed by a constant

the state of operation as an actual amount of work done by a specified time; EiP) is a preset matrix time function of time-spatial constraints for interaction between NS Ai and Bj (IS or eST), here Eilt) = 1 if Ai falls within the interaction zone of BJ, Eilt) = 0 if not; SiaeJA.(t) is a preset matrix time function that characterizes technical abilities of operation execution,

is

c?» .

The constraints (10) specify the maximum allowable amount of information to be processed by the channel

C~J) at the current time. I addition, the following values are known: memory capacity H;~ of the channel C~j} and amount Viaep of memory used by

here SiaeJA.(t) = 1 if the interaction operation D~) can

266

D~)

type

(c) J iz'

infonnation. End conditions (12), (13) specify the values of variables at the beginning and the end of

+lr

operation

·

for

. d H

p Iannmg peno.

processing

d(o)

ere

'z'

d(t ,l)

jz

of

'

Pp(i)

d(t) j ,

(0)

'T'

functions y;.,{'t) defme the time of beginning (tennination) of operations. The quality measure (16) helps to estimate the unifonnity of channels use by the end point t = tf of planning period.

(c,2)

re -- I ,... ,Si,.

'

• (0,1) _ (0). Z;ZjA - U iZjA '

'(0,3) _ ZiZjA -

(0)

W iZjA •

ej

~ ~ U(c,I)(t)

L..L.. "Sv


Vi EN

-,

,

,Ilv

,11;

,Sv'

[

lie

lie

(0) a iz -

(27)

.

j

L LI m

{O I} . ,

(0,1) ZiZj..t

being executes ,

(28)

'

E

U(e ,2)(t)

U(c,l)

(0) Wizj..t

_ X(O») = 0

1

= 0,

operation' D~)

(29)

,

channel

w~J..t (t) E {O,I} . (30)

w(o) IZjA

j=I..t=1

0

IOV

IOV

-- O',

Z(o,3)(t ) izj..t 0 -

0

'/Z

O'

(0 ,2) ( )_ ,Z/Zj..t 10 - .

(31)

0

For the end point I = tf x(e,l) (t IDV

)=

f

a(e,I). IDV

32

'

(e,2) ( ) (0,1)( ) (0,2)( ) (0)( ) x iz If ,Zizj..t t f ,ZiZj..t If ,ZiZj..t If E

RI ' (

5

=.!.2 L.. ~ L.. ~ L.. ~ (a(e,l) _ x(c,l) \2 1 ., ~ ,,5v ) IDV

1=lo=lv=1

(33)

1=1/

7

. ,

(35)

i=lz'eD(;) (e) J il!

=

b2 0

+ 2b 0 b I (C,2) XiI!

\2

interaction

,

otherwise

u(e,l) IlIv

(t)

=0 '.

(C,2») - Xi(li!+I)

+

J)

U(c,2)(t) IZ

=0 ifnot·'

w(o) (t)=1 IZjA

is

if the

that was executed through the of IS (CST) Rj is completed, and

(t) =0 if not . The values

d(e ,l) d(e,2) a(e ,l) IlIv , IZ , IlIv

(Xi~,2) - x;(ti~I»)

are equal to interval

Now the scheduling problem can be fonnulated as a following problem of dynamic system (4)-(6), (23)(25) program control: it is necessary to fmd an allowable control ii(t) , t E (to, t.J that meets the requirements (7)-(11), (26)-(30) and guides the dynamic system (4)-(6), (23)-(25) from the initial state (13), (32) to the specified fmal state (12), (31).

n

= ~ ~ f c) L- L..J lte'

last

size of NS Ai on-board TFC. Similarly to (34), the functional (35) denotes decrease of navigation field, but here the reason is late correction of on-board ephemerid infonnation. In the expression (37) the symbol Tr denotes a spur of an appropriate correlation matrix.

(34) J(e)

ci

differences

)

d) Quality measures ofprogram control J(e)

the

are known right-hand members of end constraints. The measure (33) of scheduling quality expresses the accuracy of end conditions accomplishment or let minimize the losses caused by failure to implement end conditions. The functionals (34)-(37) are directly linked with and influence upon the quality measures (1)-(3) of navigation field. For example the measure (34) characterizes total decrease of OrS NS navigation field accuracy as caused by tardy timeand-frequency corrections (TFC). Here the

c) End conditions For the initial time t = to: x(e,l) (t ) = d(e,I). x(e,2) (t ) = _ d(e,2). (0,1) ( ) _ iz ,Zizj..t to -

when

U~,2) (t) =1 if the interaction operation D~)

O',

Ill ;

U(e .2) (a(O) le ~ tse

U~~I) (t) = 1

x(O)(t) = a(O» IS; IS;

D=lv=1

U(C,I)(a(O) _x(O»)=

(c,2) (c,2) X iz ' -xi(z'+I»)

2;

operation D;~) of NS TCC is completed (when (26)

'

(37)

\2

(K )(

Here

(25)

b) Constraints hi

+

'1' . bl es Z;ZjA (0,1) (t) (0,2) ( ) (0,3) ( ) auxllary vana ' ZiZjA t , ZiZjA t are used for operations with interrupt prohibition (Sokolov, and Kalinin, 1985). The constraint (26) states that at a given time each NS should be in a single macro-state and occupy a single "place". Similar constraint can be fonned for macro-states of IS and CST. The constraints (27)-(30) specify start time for macro-operations and auxiliary operations.

(24)

i(o,2) _ Z(o,l). iZjA iZjA'

(C,2») -Xi(z'+I)

time passed since the operation D~) completion; the

Model of program control for macro-operation operations. a) Models ofprocesses • (c,l) _ (c,l) . - 1 . s:: I h I ( X;lIv - U;lIv ' I , ... ,n, u = , ... , ;; V = , ... ,ei; 23) u;z

'T' (K )( (c,2) +lr I; X iz '

state the spacecraft Ai is in while the operation is executed, v is a ''place'' number of NS Ai in the macro-state (for example, the number of this NS within a group of spacecrafts with a common orbital plane); X~,2) is an auxiliary variable that is equal to

time gaps for execution of 10 D~). In this case the

_ -

0;

macro-operation D~~; 8 is a number of a macro-

given values, RI = [0,00). The measure (14) of program control quality characterizes the accuracy of end conditions accomplishment or expresses extent of losses caused by misclosures. The functional (15) lets specify preferable

(c,2) x· iZ

(K )

where X~~I) is a variable characterizing a state of

are

a;z

'T'

= lr

(36)

+b2(x(C,2) _x(c,2) I IZ l(z+I»)

267

If there are several allowable control (schedules) then the best one (optimal) should be selected in order to maximize (minimize) the quality measures of program control (14)-(16), (33)-(37). The components of the program-control vector ii(t) possess Boolean values and specify time intervals for works of the appropriate ground-based facilities with navigation spacecraft.

CONCLUSION In this paper, is considered a scheduling problem for ground-based control facilities communicating with an orbital system of navigational spacecrafts. Is used a dynamic interpretation of NS TCC operations for formal statement of the problem. This approach resulted in essential reduction of a problem dimensionality. The dimensionality is determined by the number of independent paths in a network diagram of GCC operations and by current spatio-temporal, technical, and technological constraints. In its turn, the degree

algorithmic connectivity depends on a dimensionality of the main and the conjugate state vectors x(t'),

iji(t'), where t'

E

(to,

t.n,

t' is a point the solving

process is being interrupted (Sokolov, and Kalinin, 1985). If the vectors are known then the schedule calculation may be resumed after removal of appropriate constraints. The dynamic interpretation of GCC functioning also lets gear the results of NS Ors use (characteristics of navigation field) to control technology. This opens up possibilities of comprehensive analysis and synthesis of effective control technologies for NS OrS. For example, the qualitative analysis based on the control theory as applied to the dynamic systems (4)-(6), (23)-(25) provides the results listed in the table 1. The table also presents possible directions of practical implementation (interpretation) for these results in real scheduling of communication with NS.

Table I Possible directions of practical interpretation the main results of qualitative analysis The main results of qualitative analysis of space facilities control processes Analysis of solution existence in the problems of SF control Conditions of controllability and attainability in the problems of SF control Uniqueness condition for optimal program controls in scheduling problems Necessary and sufficient conditions of optimality in SF control problems Conditions of reliability and sensitivity in SF control problems

The directions of practical implementation of the results Adequacy analysis of the SF control processes description in control models Analysis SF control technology realizability on the planning interval. Detection of main factors of NS OrS goal and information-technology abilities. Analysis of possibility of optimal schedules obtaining for SF functioning Preliminary analysis of optimal control structure, obtaining of main expressions for SF scheduling algorithms Evaluation of reliability and sensitivity of SF control processes with respect to perturbation impacts and to alteration of input data contents and structure

search and rescue and remote tracking. Navigation, Vol. 24, N!! 3, pp. 216-222 (in USA). Sokolov, B.V. and V.N Kalinin (1985). A Dynamic Model and an Optimal Scheduling Algorithm for Activities with Bans of Interrupts. Automation and Telechanics, N 1, pp. 106-1 14 (in Russian). Sokolov, B.V. and V.N Kalinin (1995). Multi-model Approach to the Description of the Air-Space Facilities Control Process. Control Theory and process, NI, pp.149-156 (in Russian). Zimin, I.N. and Yu.,P. Ivanilov (1971). Solving of network planning problems via a reduction to optimal control problems. Journal of Calculus Mathematics and Mathematical Physics, Vol. 11, N 3, pp.632-631 (in Russian).

REFERENCES Ackoff, R.L. (1978). The Art of Problem Solving. WileyInterscience, New York. Athaus, M. and P.L. Falb (1966). Optimal control: An Introduction to the Theory and Its Applications. McGrow-Hill Book Coinpany. New York, San Francisco, Sidney. Hanziverov, F.R. and V.V.Ostrouhov (1989). Simulation of Space Systems for Natural Resources Research. Mashinostroenie, Moscow. Klir, G.J. (1985). Architecture of Systems Problem Solving. Plenum Press, New York. Malishev, V.V., M.N. Krasilshikov and V.I. Karlov (1989). Optimal Monitoring and Control of Spacecrafts. Mashinostroenie, Moscow. Raab F.H, G.W. Board and S.D. Arling et al. (1977). An application of the Global Positioning System to

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