PREDICTIVE COORDINATION IN TWO LEVEL
HIERARCHICAL SYSTEMS
K.Stoilova*, T.Stoilov*, R.Yoshinov**
*Institute of Computer and Communication Systems,** Telematics Laboratory
Bulgarian Academy of Sciences , 1113 Sofia, Acad.G.Bontchev str. BL.2, BULGARIA
Abstract: - Noniterative coordination strategy is extended to the case of coordination with prediction. It is applied for
optimal resource allocation in a two level system. Appropriate model for steady state optimization is derived.
Key-Words: - Noniterative coordination, multilevel systems, on-line operation, optimization, complex systems.
1 Introduction
The development of the multilevel theory has been
strongly influenced by the experience of the human
beings in coordinating their efforts reaching an overall
goal which can not be achieved individually. The control
of hierarchical systems is described by the solution of an
optimization problem, decomposed to low order
subproblems, solved by the subsystems and coordinated
appropriately. Hence instead of direct solution of a high
order optimization problem, the multilevel theory
manages and coordinates low order subproblems [3].
Two main coordination strategies have been worked out:
goal and predictive. The goal coordination influences the
local goals of the subproblems. The predictive
coordination assumes constant values for the global
arguments or constraints. The coordination influences
are performed iteratively and insist multiple data
transmissions between the levels. This spends time for
data communications and prevents the real time control.
Generally the multilevel approach is used for off-line
solution of large scale optimization problems [5]. In [2]
the multilevel theory is applied for nonlinear time delay
problem. Hierarchical methods were used for joint
coordination in steady state control and identification
[1]. The gaps between the iterative computations and the
real time requirements is addressed by the noniterative
coordination [4]. It applies a ‘preposition - correction’
protocol. The local subsystems solve and send to the
coordinator their prepositions x(0) found with lack of the
global constraints. Using x(0) the coordinator modifies
them towards the global optimal solution xopt and
transmits it to the subsystems for implementation. The
noniterative coordination founds on the explicit
analytical description and approximation of the dual
Lagrange problem.
2 Noniterative Coordination
The coordination in a two level system consists of
iterative data transfer between the levels, Fig.1. The
coordinator influences with the parameters  by
defining, the optimization subproblems Zi(), i=1,n.
Coordinator


x1()
Z1()

x2()
Z2()
...
xn()
Zn()
Fig.1. Two level hierarchical system
The solutions xi(), i=1,n are sent to the coordinator.
The last improves m  to * , *=*(xi()), i=1,n. Next
* is returned to the subsystem for implementation. Thus
an iterative communication sequence is performed till
finding the optimal coordination opt, which results in
optimal solutions xiopt(opt),i=1,n. The iterative
coordination yields delays in the control process. They
have been reduced by the noniterative coordination, to
an information transfer between the system levels in the
steps: the coordinator sends to the subsystem initial 0;
using 0 the subsystems solve their problems and
evaluate the prepositions x(0); the coordinator corrects
x(0) to the global optimal xopt or evaluate the optimal
coordination opt without iterative computations; the
subsystems evaluate/implement x(opt).
This noniterative concept is applied for resource
allocation between the subsystems as follows
1. By y is denoted the common resource, which the
coordinator must allocate per subsystems. The
subsystems assuming lack of resources, y=0, evaluate its
control x(0)=x(y=0), which are sent to the coordinator.
2. Assessing the solutions x(0), the coordinator evaluates
the optimal resource allocation per subsystems
yopt=(y1opt, y2opt, ..., ynopt), n - number of subsystem.
3. Using the given resource yiopt, the subsystem i
evaluates its optimal management xiopt = xi(yopt).
3 Noniterative Resource Allocation Model
A global optimization problem,
hierarchical system is stated as
solved
by
the
min F ( x)   f i ( x i )
d
d 2 f dx d 2 g T ( N ) dx dg T d
(
A
)




 0 or
dyT
dxT dx dyT dxT dx
dyT
dx dy T
g 1 ( x1 )  y1  0.....g n ( x n )  y n  0
 d2 f
d 2 g T ( N )  dx dg T d
(6)  T  T   T 
0
dx dy T
 dx dx dx dx
 dy
d
dg dx dg d
( B)  T T 
 Im  0 .
T
dy
dx dy
d dy T
dg
 0 and
As g doesn’t rely explicitly on ,
d
dg dx
(7)
 I m ,Im - identity matrix m x m .
dx T dy T
N
(1)
x
i 1
(2) y1  y 2  ...  y n  T
x i  ( x i1 , x i 2 ,..., x in i ); y i  ( y1 ,..., y m )
g i  ( g i1 , g i 2 ,..., g im ); T  (T1 ,..., Tm )
The yi are resources, allocated to subsystem i and the
total amount of resources is limited to T. If an allocation
*
*
y*=(y ,...,y ) is performed, the global problem is
1
n
decomposed to N optimization subproblems,
(3)
min f i ( x i )
g i ( x i )  y i  0.
The solution xi(yi) of (3) is influenced by the allocated
resources yi The coordination task is to find the optimal
opt
opt
allocation yopt=(y
,....,y
) .Thus the subproblems
1
N
give the solutions x(yopt), which are also the optimal
solution of the global problem (1). Applying the
noniterative coordination, the sequence is:
- the subsystem assumes lack of resources, yi = 0,
i=1,...,N, and solves (3) and found x(0t),
- receiving xi(0), the coordinator evaluates the optimal
allocation
y opt  ( y1opt ,....., y Nopt )  y ( xi (0)) ;
opt
- using y
, the subsystem solves (3) and finds the
i
global solution x opt  ( x1 ( y1opt ),...... x N ( y Nopt )).
To be able to perform the noniterative coordination, the
relations xi(yi), i=1,N, and y = y(xi(0)) must be found
in analytical way. It can be linearly approximated
(4)
xi ( y i )  xi ( y i )  xi 0 
dxi
dyiT
xi 0
yi ,
where xi(0) is found from (3) .The unknown matrix
dxi
dyiT
is found from the
From (6) it follows
1
xi
Right and Dual Lagrange
problem of (3) (the index i is skipped for simplicity)
dL( x( y ),  ( y ) df ( x( y )) dg T ( x( y ))
(5)(A)


( y)  0
dx
dx
dxT
dL( x( y ),  ( y ))
 g ( x( y ))  y  0 .
(B)
d
The nonlinear system (5) has (n+m) unknown (x,)
and (n+m) equations. The vectors df , dg ,g are explicit
functions of x. The unknown matrices
dx d
,
are
dy T dy T
found from (A)-(B) by differentiating to y:
 d2 f
dx
d 2 g T ( N )  dg T d
(8) T    T  T  
.
T
dy
 dx dx dx dx
 dx dy
By substituting (8) in (7) it follows
1
 d2 f
 dg T d
dg
d 2gT
 Im
(9) T ( 1)  T  T ( N ) 
T
dx
 dx dx dx dx
 dx dy
d
dy T
To be kept (8) true, the matrix
corresponding
must be a
inverse
matrix,
 dg  d 2 f
d
d 2 g T ( N )  dg T 



 T T

dy T
dx T dx  dx 
 dx  dx dx

1
(10)
Substituting (10) in (8), the explicit relations
dx i  d 2 f i
d 2 g iT
(N ) 





i
dy iT  dx Ti dxi dxiT dxi

(11)
 dg
 Ti
 dxi
1
1
dx
dy T
is
dg iT
dxi
1
.
 d fi
d g
dg 
(N ) 
 T

 T i i 
 dxi dxi dxi dxi
 dxi 
Hence the subsystems evaluate xi0 and i0 with a lack of
2
1
2
T
i
resources. The coordination problem is derived here in
explicit way. Substituting xi(y0) in fi(xi) according to (4)
it becomes a function of yi f i ( xi ( yi ))  wi ( yi ) and
N
min
(12)

f i ( xi ( yi ))  min
i 1
N
 yi  T
i 1
N
w (y )
i
i
i 1
N
y
i
T
i 1
f i ( xi ( yi ))  wi ( yi ) is not explicit,
As
approximation to quadratic form is done.
an
f i ( xi )  f i ( xi 0 ) 
(13)
df i
dxiT
xi 0
( xi  xi 0 ) 
T ( 2)
0
x
.
d2 f
1 T
( xi  xiT0 ) T i
( xi  xi 0 )
2
dxi dxi xi 0
Using (4) the coordination problem is explicitly defined
1 dxT d 2 f i
(16) min  yiT i
dyi dxiT dxi
i 1 2
N
dxi
df
yi  Ti
T
xi 0 dyi
dxi
dxi
yi
xi 0 dyi
AI . y  T , where AI  I m  I m  I m .

m
yopt and duals l0 is
From [4] the solution of
(17) y
opt
i
li0  (ai

2
 
min f i x (i ) 
x(i )
1 ( i )T
x qi  x (i )  riT x (i )
2
Gi ( x i )  0  a ( i ) x ( i )  y i
x (1)T  ( x1 , x 2 , x 3 ),
x ( 2 )T  ( x 4 , x 5 , x 6 ) ,
y1T  ( y11 , y12 ), y 2T  ( y 21 , y 22 ), q1  diag (3,1,4),
q 2  diag (1,5,2), r1T   2,4,3 ,
2 3 4
1 1  5
,
a2 
r2T   1,6,5 ,
5 1 6
2 3 4
The resources yi satisfy y1  y 2  T 
.
8
. Following
1
(4), the optimal subproblems solutions are
x (1) ( y1 )  x0(1) 
T (1)
0
x
dx (1)
dx ( 2 )
y1 , x ( 2 ) ( y2 )  x0( 2 ) 
y2
dy1
dy2
 0.0156
min f1 ( x (1) ) 
 ( x10 , x20 , x30 )  arg 
  0.049
G1 ( x(1) )  0
0.0041
1
2 y
i 1
T
i
qqi y i  rriT y i ,
0.0747 0.0263
,
0.0263 0.1411
rr1 
qq2 
1 0 1 0 y1
8

0 1 0 1 y2  1
0.0999  0.1548
 0.15483 0.3126
 1.2084
 0.8162
, rr2 
 0.37
0.6570
The solution of (20) can be evaluated according to (17)
y1opt  8.6719 1.3022 y 2opt   0.6719 2.3022 .
Applying y iopt in (20), the optimal solutions of (18) are

 x1  x 2  5 x3  2 x 4  3 x5  4 x6  1  0
Two subproblems are derived
a1 
The coordination problem, applying (16) , is
qq1 
The global optimization problem is stated like
1
2
2
2
2
2
2 
 3 x1  x 2  4 x3  x 4  5 x5  2 x6 

min  2
x 


2
x

4
x

3
x

x

6
x

5
x
1
2
3
4
5
6 

(18)
2 x1  3 x 2  4 x3  5 x 4  x5  6 x6  8  0
 0.6570.
0.0410  0.0295
0.1896  0.1491
dx (1)
dx ( 2)
 0.2502 0.2199 , T  0.1129  0.2185 .
dy1T
dy 2
0.0418  0.1501
 0.0101 0.1606
qi1 aiT ) 1 ( ai
4 Numerical Example
0.37 , l 20T  0.8162
According to (11) it holds
(20) min
Thus if f is approximated to a quadratic and x is a linear
series at xi0, , where xi0 solve (3), the optimal resources
are found from. Substituting (17) in (4) the suboptimal
solution of the poblem (1) is found.
(19)
l10T  1.2084
N
 N

  q 1i ri  qi1  ( qi1 )1   ( qi1ri )  T 
 i 1
  i 1

qi1 ri ) , i  1,2 .
 1.767
min f 2 ( x( 2) ) 
 ( x40 , x50 , x60 )  arg 
  0.6425
G2 ( x ( 2) )  0
1.3654
x (1) opt
x1opt 0.3013
x 4opt  1.5512
 x 2opt  2.461 x ( 2 ) opt  x 5opt  1.0698
x 3opt 0.1714
x 6opt
1.0024
5 Conclusions
The noniterative prediction coordination has been
applied for resource allocation in a hierarchical system.
The coordination allows the multilevel system to
perform the resource allocation without iterative
computations, which speeds up the control of the system.
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Tammer,K.
Two-level
optimization
with
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