Download The network operator

1
A multi-objective synthesis of optimal control system by
the network operator method
A.I. Diveev
Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS,
Moscow, Russia (e-mail: [email protected]).
Classical problem of optimal control
The mathematical model of object of control is set
x  f x, u x  Rn ,
Initial values
x0  x 
0

u  U  Rm
 R
0
0T
x1  xn
Terminal condition

 
x t f  x f  x1f  xnf
n

T
Functional is criterion optimization
tf
J
 f0 xt ,ut dt  min
0
A solution is optimal control
~  ht 
u
Synthesizing function
u  gx

t  0,t f

2
x1  x2
x2  u
3
Synthesis of optimal control
u U  1,1
J  t f  min
u~  1  2  yx1 , x2 
1, если a  0
 a   
0 , иначе
1, if x1  0 , x2  0

2
x
 x1  2 , if x1  0 , x2  0

2
y x1 , x2   
-1, if x1  0 , x2  0


x22
 x1  2 , if x1  0 , x2  0
x2
x1
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Problem statement
x  f x, u x  R n u  U  Rm
Initial values are in closed bounded domain
x0  x 
0
Ji    
X0
tf


 X
0
0T
x1  xn
n

R
0
f 0 ,i xt , ut dx10  dxn0dt  min
0
J2    
X0
 xi t f  xi
n
i 1
i  1, N
 dx dx dt  min
f 2
0
1
0
n
It is necessary to find the admissible control that satisfies the restrictions


T
q

q

q


u

g
x
,
q
where
is vector of parameters
u  gx or
1
p


Pareto set is considered to be the solution of problem
~ ~i
P  g x : i  1, K
~
gxU ~
g i x  Jgx
g i x  P J ~
 
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Theorem 1. Assume
X0
is a finite denumerable set

X 0  x 0 ,1 , , x 0 ,M
Suppose
and
P̂
~
P
is the solution of the multi-objective synthesis problem.
is the solution of the same problem, but with the set of
initial condition
Then

X̂ 0  X 0
~
P̂  P
The network operator
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The set of variables
X  x1 , , xn 
The set of parameters

Q  q1 , , q p

The set of unary operations
O1  1z   z , 2 z , , W z 
The set of binary operations
O2  0 z  , z ,1 z  , z , ,V 1 z  , z 
Commutative
i z  , z   i z  , z 
Associative

Unit element

i i z , z, z  i z ,i z , z
i ei , z   i z ,ei   z
Унарные операции
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Binary operations
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The network operator
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 nonsource 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
of parameters Q ;
X
e) every non-source node corresponds to the item of
binary operations set O2 ;
f) every edge corresponds to the item of unary operations set
O1 .
or
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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 or
unit element of this binary operation as its arguments;
b) unary operation can have binary operation, parameter
or variable as its argument;
c) binary operation cannot have unary operations
with equal constants or variables as its arguments.
Theorem 2. Any program notation can be transformed
in graphic notation.
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 k  l z  , z 
 k  l z ,  m z 
 k 1  l z  , z , z 

k  m l z , em 

 k l a, 1  q  m a, eq

An example
y
q1 x13
3 x2  x2
1
2
Program notation
y  1 1 q1 , 14 x1 , 5 15  0  2 x1 ,  2 x2 
Graphic notation
y  1 1 1 1 q1 ,14 x1 ,5  0 0 , 15  0  2 x1 ,  2 x2 
Network operator
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Network operator matrix (NOM)
Definition 4 Network operator matrix (NOM) is an integer uppertriangular 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 and
nonzero non-diagonal elements with ones we shall get an vertex
incident matrix of the graph that satisfies conditions a-c of network
operator definition..
vertex incidence matrix
NOM
0
0
0
Ψ  0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
14
0
1
0
0
0
0
2
2
0
0
0
0
0
0
0
0
15
0
0
0
0
0
1
0
5
1
0
0
0
A  0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
1
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Network operator matrix (NOM)
T
vector of numbers of nodes for input variables b  b1  bn 


vector of numbers of nodes for parameters s  s1  s p T
vector of numbers of nodes for output variables d  d1  d m T
a node vector
z  z1  z L T
 xk , if i  bk , k  1, n

zi0   q j , if i  s j , j  1, p
e , otherwise
 ii
if  ij  0
i  1, L
 
z ji    jj  z ji 1 ,  zii 1 
ij


i  1, L  1
j  i  1, L
 
i , j  1, L and vectors of
Theorem 3. If we have the NOM Ψ  
ij
numbers of nodes for variables b  b1  bn T parameters s  s1  s p T
T
and outputs d  d1  d m  then it is sufficient to calculate


the mathematical expression is described by NOM
  
 14 z     z   ,  z   
 2 z     0 ,  z   
2
z     z   ,  z   
1
z     1,  z   
 15 z     0 ,  z   
 5 z     z   ,  z   
 1,4  1
 2 ,4
 2 ,5
 3 ,5
 4 ,7
 5 ,6
 6 ,7
z41  1 1, 1 z10
2
4
2
5
3
5
4
7
5
6
6
7
1
1 4
0
2
0 5
1
0
4
1 7
0
14 2
0
2 2
0
2 3
2
1 4
3
15 5
5
5 6
0
0
0
Ψ  0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
14
0
1
0
0
0
0
2
2
0
0
0
0
0
0
0
0
15
0
0
0
0
0
1
0
5
1
b  2 3T s  1T d  7T
z 0  q1 x1 x2 1 0 0 1T
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Small 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
w  w1 w2 w3 w4 T
w1
specifies the number of variation,
w2
is the number of node that the edge comes out,
w3
is the number of node that the edge comes in,
w4
is number of unary or binary operation.
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The principle of basic solution
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
w  2 2 6 3T
0
0
0
Ψ  0
0
0
0
0
0
0
0
0
0
0
y
0
0
0
0
0
0
0
1
14
0
1
0
0
0
0
2
2
0
0
0
0
0
0
0
0
15
0
0
q1 x13
3 x2  x2
1
2
0
0
0
1
0
5
1
0
0
0
w  Ψ  0
0
0
0
y1 
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
14
0
1
0
0
0
0
2
2
0
0
0
0
0
3
0
0
15
0
0
0
0
0
1
0
5
1
q1x13
3 x2  x2
1
2
 x1
The genetic algorithm
 
Ψ 0   ij0 i , j  1, L
The basic solution is a basic NOM


i  1, H
Ψ i  w i1    w il  Ψ 0
i  1, H
W i  w i1 , , w il
A structural part of chromosome
NOM for chromosome i
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i
A parametrical part of chromosome y 


i
i T
y1  yM
y ij  0,1 j  1, M i  1, H
M  M1  M 2 R
M1 is number of bit for integer part


M 2 is number of bit for fractional part
i


i
i T
y1  yM
<===> Grey code y 
 y i , if  j  1 mod M  M   0
M1  M 2
 j
1
2
i
M1  j i
i
q


j
i
i
ck   2
q j  k 1M  M 
y

q
j

j 1 , else
1
2

Vector of parameters c i 
j 1
k  1, R
i
i T
c1 c R
j  1, RM 1  M 2 
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An example
x1  x2

x10  0
x2 0  V0cos0
x3 0  Rz  h0
x4 0  V0sin0 ,
0  0  0
x 2  
g 0 R z2 x1
 x 2  x 2 
3
 1
32

1  u S0 
m


  x12  x32  Rz 

 x2 x22  x42 e 
x3  x4
x 4  
g 0 R z2 x3
 x 2  x 2 
3
 1
32

1  u S0 
m


  x12  x32  Rz 

 x4 x22  x42 e 
20
Pareto set
21
the solution
22
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 0 0 0 8 0 0 0 
u  , if y  u 
0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 
 
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 

u

u
,
if
y

u

0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 
 y , other
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 9 

0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 9 
 v2
 v2
1

e
1

e
Ψ  0 0 0 0 0 0 0 1 1 0 3 0 0 3 0 0 
y3
z z 
z z 
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 
 v2 10 12
 v2 10 12
1 e
1 e
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 
 v1  q1  q2v2  1  v1  q1 
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 1 15 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 
x12  x32  R z   h f / h0  1 
x
1
 R arctg
 1
  
z
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 v1 


Lf
x3
h0




0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


 2


2
2
2 
x

x

R

H
 x1 x 2  x3 x 4  x1  x3 
1
3
z
f 
v 2  arctg
  arctg 
R z x 2 x3  x1 x 4 

x1 




 L f  Rz arctg x

3 

23
Fg
t, c
24
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 4 0 0 0 8 0 0 0 
1  e  v2
0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 
3
z10 z12 
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0  y 
1  e  v2
0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 9 
0 0 0 0 0 0 0 0 1 0 1 9 0 1 0 9  v1  q1  q2v2  1   v1  q1 
Ψ  0 0 0 0 0 0 0 1 1 0 3 0 0 3 0 0 
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0  z  v  q  q v 
1
1
2 2
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 10
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 
 sgn v2  v2  q3
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 015 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 z12  q4   v1  q1  
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
v q q v
1
1
2 2