A multi-objective synthesis of optimal control system bythe network operator method

1

A.I. Diveev

Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS, Moscow, Russia (e-mail: aidiveev@mail.ru).

Classical problem of optimal control 2

The mathematical model of object of control is set

uxfx ,Initial values

nTnxx R 00

100 xx

Functional is criterion optimization

min0

0 dtttfJft

ux ,

A solution is optimal control

mRU u

xgu

thu ~

,Rnx

Tfn

fff xxt 1xx

Terminal condition

ftt ,0

Synthesizing function

2121 xxyu ,~

0 0 ,2

0 0 ,1

0 0 ,2

0 0 ,1

21if 22

1

21if

21if 22

1

21if

21

xxx

x

xx-

xxx

x

xx

xxy

,

,

,

,

,

21 xx

ux 2 min ftJ

11,U u

3Synthesis of optimal control

1x

2x

иначе0

0 если ,1

,

aa

Problem statement 4

uxfx ,

Initial values are in closed bounded domain

nTnxx RX 000

100 xx

min001

00

0

dtdxdxttfJ n

t

ii

f

ux ,,X

It is necessary to find the admissible control that satisfies the restrictions

mRU u

xgu qxgu ,

nRx

Ni ,1

or

min001

1

22

0

dtdxdxxtxJ n

n

i

fifi

X

Tpqq 1q is vector of parameterswhere

Pareto set is considered to be the solution of problem

Kii ,~P~

1: xg

U xg P~~ xg i xgJxgJ i~

Theorem 1. Assume

5

0X is a finite denumerable set

Suppose P~

is the solution of the multi-objective synthesis problem.

and

P~

P̂ Then

P̂

M,, ,,X 0100 xx

is the solution of the same problem, but with the set of

initial condition 00 XX̂

The network operator 6

nxx ,,X 1

pqq ,,Q 1

zzzz W ,,,O 211

zzzzzz V ,,,,,,O 1102

The set of variables

The set of parameters

The set of unary operations

The set of binary operations

zzzz ii ,,

zzzzzz iiii ,,,,

zezze iiii ,,

Commutative

Associative

Unit element

Унарные операции 7

Binary operations 8

The network operator 9

Network operator is a directed graph with following properties:

a) graph should be circuit free;

b) there should be at least one edge from the source node to any non-source node;

c) there should be at least one edge from any non-source node to sink node;

d) every source node corresponds to the item of set of variables or of parameters ;

e) every non-source node corresponds to the item ofbinary operations set ;

f) every edge corresponds to the item of unary operations set .

XQ

2O

1O

10

Definition 1. Program notation of mathematical equation is a notation of equation with the help of elements of constructive sets X,

Q,O1,O2.

Definition 2. Graphic notation of mathematical equations is the notation of program notation that fulfills the following conditions:a) binary operation can have unary operations orunit element of this binary operation as its arguments;b) unary operation can have binary operation, parameteror variable as its argument;c) binary operation cannot have unary operationswith equal constants or variables as its arguments.

Theorem 2. Any program notation can be transformedin graphic notation.

zzlk , zz mlk ,

zzzlk ,, 1 mlmk ez ,

qmqlk eaa ,, 1

11

An example

3 22

21

311

xx

xqy

22120155114111 xxxqy ,,,

Program notation

Graphic notation

,, 11411111 xqy 221201505 0 xx ,,

Network operator

12

Network operator matrix (NOM) 13

Definition 4 Network operator matrix (NOM) is an integer upper-triangular matrix that has as its diagonal elements numbers of binary operations and non-diagonal elements are zeros or numbers of unary operations, besides if we replace diagonal elements with zeros andnonzero non-diagonal elements with ones we shall get an vertex incident matrix of the graph that satisfies conditions a-c of network operator definition..

1000000100000001000001001000001000000110000001000

A

NOM vertex incidence matrix

100000050000000150000010010000020000002140000001000

Ψ

Network operator matrix (NOM) 14

vector of numbers of nodes for input variables Tnbb 1b

vector of numbers of nodes for parameters Tpss 1s

vector of numbers of nodes for output variables Tmdd 1d

otherwise

if

if0

1 , ,

1 , ,

,,

,

iie

pjsiq

nkbix

z jj

kk

i

a node vector TLzz 1z

Li ,1

11 i

iij

ij zzz

ijjj,

11 Li , Lij ,1

0ijif

Theorem 3. If we have the NOM ijΨ Lji ,, 1

Tnbb 1bnumbers of nodes for variables Tpss 1s Tmdd 1d

parameters

and outputs

the mathematical expression is described by NOM

100000050000000150000010010000020000002140000001000

Ψ

T32b T1s T7d

Txxq 10012110 z

141 , 0111

14 1 zz ,

1442 , 0214

141

24 zzz ,

252 , 0220

25 0 zz ,

253 , 032

250

35 zzz ,

174 , 2411

47 1 zz ,

1565 , 35150

56 0 zz ,

576 , 565

471

67 zzz ,

and vectors of

then it is sufficient to calculate

15

16Small variations of network operator: 0 - replacement of unary operation by the edge of the graph; 1 - replacement of binary operation in the node; 2 - addition of the edge with unary operation; 3 - removal of unary operation with the edge of the graph.

Vector of variations Twwww 4321 w

specifies the number of variation,

is the number of node that the edge comes out,

is the number of node that the edge comes in,

is number of unary or binary operation.

1w

2w

3w

4w

T3 6 2 2w

13 2

221

311

1xxx

xqy

The principle of basic solution 17

When solving optimization problems, initially we set the basic solution that is one of admissible solutions, then define small variations of basic solution and create search algorithm that searches for the optimal solution on the set of small variations

10000005000000015000001001000002000002140000001000

0

Ψ

10000005000000015000001001000002000002140000001000

3

Ψw

3 22

21

311

xx

xqy

The genetic algorithm 18

00ijΨThe basic solution is a basic NOM Lji ,, 1

ilii ww ,,W 1 Hi ,1A structural part of chromosome

01 ΨwwΨ ilii Hi ,1NOM for chromosome i

TiM

ii yy 1yA parametrical part of chromosome 10,ijy Mj ,1 Hi ,1

RMMM 21

1M is number of bit for integer part 2M is number of bit for fractional part

TiR

ii cc 1cVector of parameters TiM

ii yy 1yGrey code <===>

21

211

11

2MM

j

iMMkj

jMik qc

else

0 mod 1 if ,

1

21

,ij

ij

iji

jqy

MMjyq

Rk ,1 211 MMRj ,

21 xx

m

Su

xx

xRgx z 0

2323

21

12

02

1

zRxx

exxx

23

212

4222

43 xx

m

Su

xx

xRgx z 0

2323

21

32

04

1

zRxx

exxx

23

212

4224

001 x

An example

002 cos0 Vx 03 0 hRx z

,004 sin0 Vx

000

19

20

Pareto set 21

00000000000000001000000000000000010000000000000001511000000000000000100000000000000001000000000000001000000000000000000100000000000300301100000009010010100000000900010000000000000000040000000000000000010000000000000000300000000080000100000000000000001000000

Ψ

the solution

otherif

if

,,

,

yuyu

uyu

u

12103 1210

2

2

2

2

1

1

1

1zz

e

ezz

e

ey

v

v

v

v

112211 1 qvvqqv

0

23

21

1 h

Rxxv

z

1arctg

1

3

10

x

xR

L

hhz

f

f /

4132

23

214321

2 arctgxxxxR

xxxxxxv

z

3

1

23

21

arctgarctg

xx

RL

HRxx

zf

fz

22

gF

c ,t

23

00000000000000001000000000000000010000000000000001501000000000000000100000000000000001000000000000001000000000000000000100000000000300301100000009010910100000000900010000000000000000040000000000000000010000000000000000300000000080004100000000000000001000000

Ψ

3 1210

2

2

1

1zz

e

ey

v

v

221110 vqqvz

112211 1 qvvqqv

322sgn qvv

2211 vqqv

11412 qvqz

24