- Email: [email protected]
A Structure for Time-Horizon Decomposition
A STRUCTURE FOR TIME-HORIZON DECOMPOSITION M.R.Javdan Electn'cal Engineenng Department, Arya-Mehr University of Technology, P. O. Box 3406, Tehran, Iran
INTRODUCTION Although much theoretical progress has been made towards the solution of optimal control problems, with the Linear-Quadratic case being virtually solved completely, there are still immense problems associated with the numerical solutions in cases where the dimensionality is large and when there are widely differing time constants in the system (giving rise to 'stiffness' in the problem). Large-scale decision problems are, however, being solved every day in industrial, management, and biological systems_ In such systems there appears to be a division of the overall problem into levels which are concerned with different time horizons, which tends to alleviate both the problems of dimensionality and 'stiffness'. A characteristic feature of these systems seems to be that long-time-scale problems are not directly affected by the short-time dynamics. For example, the decisions of the management regarding the required rate of production of a product from a plant are normally based on a model of the market conditions (which are slow varying) and not on the detailed dynamics of the plant. On the other hand, the short-timescale problems are directly influenced by the solution of the long-time-scale problems. In the example, the operation of a plant is directly dependent on the amount of product required from it. In this paper a preliminary, and in parts heuristic, study is made of a time-horizon decomposition procedure based on the structures which seem to exist in the systems mentioned above. The analysis is based mainly on the second order linear-quadratic case and it is intended as a starting point for a more general investigation. The class of system models considered has been studied, in a different context, by Singh and Drew (see Singh 1973; Singh, Drew and Coales 1975) for serially connected systems. 1_
The Proposed Structure
index (1) are slowly changing variables, and those with index (2) are fast changing variables. Suppose that the above problem is solved in two stages in the following way:
A possible model of the control problem which leads to the multi-time-horizon structure considered is the problem !t~
Hin
0
s. t. where A=
R
~
-
{I~I~ A~
Stage 1 (Problem A) + II!)I!} dt
sf' :,] ['~I) .:,J [, (I)J
[" :,] A3
B3
~=
'
ER
n1
n
~ (2)£R 2 ,1!)(I)ER
[Q(I) Q=
and
~(2)
~ (1)
(1 )
Solve
+ BI!)
o
1
0
s . t.
~ (1)
1
1
= A ~ ( 1) +B I!) ( 1)
Q(2)
.. ("(Il]
Stage 2 (Problem B)
1!)(2) PI
and 1!)(2)ER
P2
Solve
.
The matrices Q and R are positive definite and the system is controllable and observable.
Hin 1!)(2)
6~
2
2
{1~(2)IQ(2)+11!)(2)IR(2)} dt (2B)
s.t. ~(2)=A2~(2)+B21!)(2)+A3~*(I)+B31!)*(I)
It is assumed that the variables with
83
84
A structure for time-horizon decomposition
Note that the solution of problem (2A), i.e. {/(I), m"' (I)} , acts as deterministic dist~rbance-to problem (2B) . Thus we have that the slowly varying problem (A) is independent of the fast varying parameters; and the fast changing problem (B) is dependent on the slowly varying parameters of problem (A) . We shall now consider the merits of the problem in the form of (2) over the integrated form of (I). The question of the equivalence of the problems (I) and (2) is investigated in the following section. Suppose that the problem (I) is to be solved in an on-l ine situation in the presence of plant uncertainty, where the optimal input sequence m* is appl ied to the system, the response of-the system measured, new estimates of the model parameters made, and a new optimal input sequence computed based on the updated parameters of the problem. In this case a two-level structure, based on problems (2A) and (2B), can be synthesised as shown in Fig. I . On the higher level the problem (lA) is solved initially for the whole interval t=O to t f . The solution {~*(I), ~* (I)}t=O to t l ,where t~ < tf' is f
sent to the lower level. On the lower level the problem (2B) is solved for the interval hI ' 1arger t= O to t fI (were t .IS severa ltimes f than the largest time-scale of the variables of problem (2B». There is little loss in optimality resulting from shortening the I time-horizon of problem (2B) from t to t f f since the variables on this level are fast changing. The optimal input sequence{~*} t=O to T where T(t~ is now appl ied to the system. The response of the system is measured and the parameters of the lower-level problem updated. The problem on this level is now solved for the interval t = T to t 1 and the f procedure is repeated until the timet=t~ is reached . At this time the measurement of the variables {z(I), m(I)} 0 1 is sent to t= to t f the upper level. The parameters of the upper level updated and the upper level problem is now solved for the interval t=l} to t . f Again the solution {~* (1), ~*(I)}t=tl t02t1 f
f
is sent to the lower level and the same cycle of events is repeated . In the structure of Fig 1 the wide spread in the time-constants has been incorporated in a natural way. The "stiffness" problem has been eliminated since the time-constants of the variable~ on each level are similar.
There is also considerable saving in computation time. During a time period of t~, where problem (2A) is solved once and problem (2B) t~/T times, the processing time for the two-level solution is of the order of t 1 ,I, ,1jJ2 f '''' I {n
xr+n
2
l
}
'1jJ where n 2 is of the order of the processing 2 time of the problem (2B) on the lower level, and
~i l
is of the order of the process i ng
time of the problem (2A) on the higher level. 1jJ l =1jJ2( >I) since, although the time-horizon t f for problem (2A) is larger than the timehorizon t~ of problem (2B), larger sample time can be used in the numerical solution of problem (2A), and thus the two problems involve similar computational effort. (The sampling-time is proportional to the smallest time-constant present in the prob lem.) The corresponding processing time of the integrated solution is of the order of ,
,
t
1jJ
r
(n +n ) 0 x
1
2
1
f
where 1jJo>1jJl since, although this problem and problem (2A) have the same time-horizons, longer sampl ing-time can be used for the latter problem. Also 1jJ >1jJ since the integrated solution anS t~e problem (2B) have similar sampl ing-time but the timehorizon of the integrated problem is longer. 1
Suppose that "1="2=8, 1jJo=4, 1jJl=1jJ2=3 and
t f /T=5. then the ratio of the processing
times for the integrated solution to the decentralised one is
- 120
which is a substantial improvement. Notice that these estimates of the processing times are independent of the 1 inearily of the system equations. 2.
The Equivalence of the Decomposed and the Integrated Structures
The solution of problem (I) is given by the two-point boundary value problem
[:] {: -"-"] [:J -A'
{~}t=O =~ -1 B_ \ ~ -R
0
and Y A
e-
{Pt:.tf=Q
(3)
M. R. Javdan
One way of assessing whether the two problems (I) and (2) have similar solutios is to determine whether the two matrices Z e and Za have similar eigenvalues and eigen-
where "e:R(n l +n 2) is the adjoint variable. Define-the matrix
~
Z
e
-BR- IB
1
-A'
J
[ A
_Q
85
vectors, and the matrices Ye and Y have a similar elements . Note that the two methods give identical solutions if A3=0 and B3=0 . This, however, is a trivial result as it is the case where the two problems (2A) and (2B) are decoupled.
For a given ZO the solut ion of equation (3) is uniquely determined by the eigenvalues and the eigenvectors of the matrix Ze' The solution of problem (2B) is given 2 I -B R-
b:(2)ll A2 [ ~(2) = -Q(2)
~(2~
-A
. {~(2)}t=0=~O(2) ,
~(2)
(2)B2J(~(2~ .F3~t1)+B3~t1)1
1
{~(2)}t=tf =Q
Q
J
3.
The scalar case Let us consider the case when n =n =1 l 2
and PI=P2=1. Z and Z become e 2 a b
(4B)
o
-R-I(2)B2'~(2)
l
where z * (I) and m* (I) are the solutions of problem (2A) given by
[
~(1)l=rA' -BIR-I(~)BI'] ~(l) ~(1)
l-Q(l)
~(1)
_AI
o (4A)
{~(I)}t=O = ~o(l) , {~(I)}t=tf = Q ~(1)
and
= -R- I (l)BI'~(1)
The solution obtained when the integrated problem (1) is solved in the two-level structure of problem (2) is given by the combination of equations (4A) and (4B),
z (1) ~(2)
Z
;'(1)
a
[~(2)
o
o Z a
o
[z (I)j
o
~(2) ,,(1) ~(2)
Jl~(I)]=-~5)[~(I~
I
r:(1)]/ =[-R-I(I)B ' 0 l~(2) 0 -R- I (2) B2'
~ (2)
The matrices Ye and Y are also given by a
~ r
a ~ (2)1
Ye =
with the same boundary conditions as in (3) . The matrix Za is given by :-BI(R(l)-IBI' A2 :-B 3 (R(l)-I BI'
- 1 ' - - - - - - - --
-Q(1)
0
:
-A
o
-Q(2)
I
o
0 2' -A
[
l
o
and
o
+
1
Ya =
[: :~l
Direct inspection of the matrices give the first two conditions for the equivalence of the solutions. For the similarity of Y e and Y a sufficient condition is that a « (C1) 1 I bll 3 For the matrices Z and Z we have the first e a condition 2 2 b b (C2) 2 -.l. « r r 2 l
Ib
the matrix Z can be written as e
86
A structure for time-horizon decomposition
Let us consider the eigenvalues of the matrices Z and Z which are the solutions of e a
and
IAI
o
IAI
o
For ~e,;~a we need 3 3 a
2 ql 2 «b 2 ql -+a 2 r2 3 2 q2
which can be simpl ified to the following sufficient condition
Now A 2 A2 2 A2 2 q2 \A I-Z I=(a - A )(a - A )+b e I 2 2 r 2
(C4) Furthermore, we have
and I ~ I-Z
a
1 =(a2_ ~ 2) (a2_ ~ 2)+b2.:2(a2_ ~ 2)+b2~(a2_ ~ 2) I
2
2r
2
I
Ir
l
2
and assuming (C4) is satisfied,
+ bLb2 qlq2
Ae I z I = -a
I 2 r r l 2
The two polynomials are similar except for the constant term. Assuming that conditions (Cl) and (C2) hold, we have a further condition
2 b2 b I b3 Ae A Ae { - + - - z - (a - A)Z } r r 3 2 2 2 l 3
In summary , a set of dufficient conditions for the similarity of problems ( I ) and ( 2 ) a re i)
2 2 q2 q2 {l a 3 b l ~ - 2a l a b l b ~I } 3 3 Since all the terms on the left hand side are positive this condition can be reduced to the following sufficient condition, obtained by considering only the term 2 2 qlq2 b b -I 2 r Ir2 '
«
la3(2alb3-a3bl)l «
blb~
b~b~ ~: ql
(C3)
r2
The eigenvectors corresponding to each A are given by ( ~ I-Z )~e
A
where
e Aa
( A I-Z)~
are Ae
~
i i i)
iV)
1-2ala3blb3+a;b~ 1 i.e.
i i)
Q 0
Aa 4 and Z EC .
. Ae Aa . Taking z4=z4=1, by elementar y manipulations we get 2 ql ql 2 A2 2 b2 + (a 2 - A )+a 3
r; q;-
These conditions make quantitative the intuitive notions about the conditions for equivalence of problems (I) and (2) given in Singh (1973) and Singh, Drew, Coales (1975). These conditions essentially state that the two problems (I) and (2) have similar solutions if, either the couplings between the two stages of problem (2) are weak (i .e . a ,b +O), or when the weightings 3 3 on the state and input vectors z(I),m(l) are much greater than those on the ~ectors z(2), 1lI(2). It is expected that the condition~ for (~1+~2»2 will not be significantly different
from the conditions derived here for the
(~1+~2)=2 case . A numerical example Consider a system with A= [-0.1
o. I
0] ' B=[o.5I
-I
~J
'R=[ ~
:]'"d Q-G
87
M. R. Javdan
This system satisfies the four conditions i) to iv) which ensure that its integrated solution (through problem 1) and the decomposed solut ion (through problem 2) are similar . To verify the conditions for th i s system form the corresponding Ze and Za
4.
A prel iminary investigation of a new structure for time-horizon decomposition has been given. The treatment is centred on linear systems, although the concepts equally apply to non-linear systems with the same basic structure as in problem (1). It is i ntuitively clear that similar conditions to those derived in section 3 for 1 inear systems will also apply to the nonlinea r case, i.e. for the equivalence of the integrated and the decomposed structures we either need weak coupling between the two stages or require that much more priority be given to the long-time-horizon problem than to the short-time-hor izon one.
matrices and evaluate their eigenvalues and eigenvectors: eigenvalue
corresponding eigenvector
~ 1=-11 .180
(0.914
0.011
0.0405
0.000)
~ 1=-11 .180
(0.914
0.011
0.405
0 . 000)
~ 2= 11.180
(-0.911 -0.024 0.411
0.002)
~ 2= 11.180 a ~ 3= 1.415
(-0.911 -0 . 024 0.411
0.002)
(-0.013 -0.383 -0.017
0.923)
~ 3=
(0.000
0.000
0.923)
(-0 . 010
0.923 -0.008
0 . 382)
(-0 . 000
0 . 923 -0.000
0 . 382)
e
a e
e
1.414 a ,4 A = -1.415 e ,4 A = -1.414 a
Concluding Remarks
References -0 . 382
1.
Singh, M.G. "Some Applications of Hierarchical Control for Dynamical Systems", Ph . D. Thesis, 1973, Cambridge Univers i ty.
2.
Singh, M.G., Drew, S.A . W. and Coales, J . F. "Comparison of Practical Hierarchical Control Methods for Interconnected Dynam ical Systems", Automatica , 1975, Vol. 11.
The eigenvalues are identical to within two place s of decimal and the eigenvectors are the same to within one place of decimal.
Sol v e Pr o blerl (l A) t
"
!I t
~
fo r . the
in t er v al
to t f
Solution
, ( ' I · . m{ ' I
c( ' 1
So l v e Pr o b l l"'" (2 B) (o r t = (."
t: . .
'T 1 to (.
SO l u tion
! (2 )
: ()
,
t '"
t n l"
in terv.)1
+,) ~ t
. ,:- ( 2 )
Lt
t. ( ••
:+;, Tl I
( • t : + (.. ' +,
t~+ l, lT) t O ( L +I ) t :
to
1TIr -----I.I._---,
Proc e ss
The Time-Horizon-Decomposition Structure
INTRODUCTION Although much theoretical progress has been made towards the solution of optimal control problems, with the Linear-Quadratic case being virtually solved completely, there are still immense problems associated with the numerical solutions in cases where the dimensionality is large and when there are widely differing time constants in the system (giving rise to 'stiffness' in the problem). Large-scale decision problems are, however, being solved every day in industrial, management, and biological systems_ In such systems there appears to be a division of the overall problem into levels which are concerned with different time horizons, which tends to alleviate both the problems of dimensionality and 'stiffness'. A characteristic feature of these systems seems to be that long-time-scale problems are not directly affected by the short-time dynamics. For example, the decisions of the management regarding the required rate of production of a product from a plant are normally based on a model of the market conditions (which are slow varying) and not on the detailed dynamics of the plant. On the other hand, the short-timescale problems are directly influenced by the solution of the long-time-scale problems. In the example, the operation of a plant is directly dependent on the amount of product required from it. In this paper a preliminary, and in parts heuristic, study is made of a time-horizon decomposition procedure based on the structures which seem to exist in the systems mentioned above. The analysis is based mainly on the second order linear-quadratic case and it is intended as a starting point for a more general investigation. The class of system models considered has been studied, in a different context, by Singh and Drew (see Singh 1973; Singh, Drew and Coales 1975) for serially connected systems. 1_
The Proposed Structure
index (1) are slowly changing variables, and those with index (2) are fast changing variables. Suppose that the above problem is solved in two stages in the following way:
A possible model of the control problem which leads to the multi-time-horizon structure considered is the problem !t~
Hin
0
s. t. where A=
R
~
-
{I~I~ A~
Stage 1 (Problem A) + II!)I!} dt
sf' :,] ['~I) .:,J [, (I)J
[" :,] A3
B3
~=
'
ER
n1
n
~ (2)£R 2 ,1!)(I)ER
[Q(I) Q=
and
~(2)
~ (1)
(1 )
Solve
+ BI!)
o
1
0
s . t.
~ (1)
1
1
= A ~ ( 1) +B I!) ( 1)
Q(2)
.. ("(Il]
Stage 2 (Problem B)
1!)(2) PI
and 1!)(2)ER
P2
Solve
.
The matrices Q and R are positive definite and the system is controllable and observable.
Hin 1!)(2)
6~
2
2
{1~(2)IQ(2)+11!)(2)IR(2)} dt (2B)
s.t. ~(2)=A2~(2)+B21!)(2)+A3~*(I)+B31!)*(I)
It is assumed that the variables with
83
84
A structure for time-horizon decomposition
Note that the solution of problem (2A), i.e. {/(I), m"' (I)} , acts as deterministic dist~rbance-to problem (2B) . Thus we have that the slowly varying problem (A) is independent of the fast varying parameters; and the fast changing problem (B) is dependent on the slowly varying parameters of problem (A) . We shall now consider the merits of the problem in the form of (2) over the integrated form of (I). The question of the equivalence of the problems (I) and (2) is investigated in the following section. Suppose that the problem (I) is to be solved in an on-l ine situation in the presence of plant uncertainty, where the optimal input sequence m* is appl ied to the system, the response of-the system measured, new estimates of the model parameters made, and a new optimal input sequence computed based on the updated parameters of the problem. In this case a two-level structure, based on problems (2A) and (2B), can be synthesised as shown in Fig. I . On the higher level the problem (lA) is solved initially for the whole interval t=O to t f . The solution {~*(I), ~* (I)}t=O to t l ,where t~ < tf' is f
sent to the lower level. On the lower level the problem (2B) is solved for the interval hI ' 1arger t= O to t fI (were t .IS severa ltimes f than the largest time-scale of the variables of problem (2B». There is little loss in optimality resulting from shortening the I time-horizon of problem (2B) from t to t f f since the variables on this level are fast changing. The optimal input sequence{~*} t=O to T where T(t~ is now appl ied to the system. The response of the system is measured and the parameters of the lower-level problem updated. The problem on this level is now solved for the interval t = T to t 1 and the f procedure is repeated until the timet=t~ is reached . At this time the measurement of the variables {z(I), m(I)} 0 1 is sent to t= to t f the upper level. The parameters of the upper level updated and the upper level problem is now solved for the interval t=l} to t . f Again the solution {~* (1), ~*(I)}t=tl t02t1 f
f
is sent to the lower level and the same cycle of events is repeated . In the structure of Fig 1 the wide spread in the time-constants has been incorporated in a natural way. The "stiffness" problem has been eliminated since the time-constants of the variable~ on each level are similar.
There is also considerable saving in computation time. During a time period of t~, where problem (2A) is solved once and problem (2B) t~/T times, the processing time for the two-level solution is of the order of t 1 ,I, ,1jJ2 f '''' I {n
xr+n
2
l
}
'1jJ where n 2 is of the order of the processing 2 time of the problem (2B) on the lower level, and
~i l
is of the order of the process i ng
time of the problem (2A) on the higher level. 1jJ l =1jJ2( >I) since, although the time-horizon t f for problem (2A) is larger than the timehorizon t~ of problem (2B), larger sample time can be used in the numerical solution of problem (2A), and thus the two problems involve similar computational effort. (The sampling-time is proportional to the smallest time-constant present in the prob lem.) The corresponding processing time of the integrated solution is of the order of ,
,
t
1jJ
r
(n +n ) 0 x
1
2
1
f
where 1jJo>1jJl since, although this problem and problem (2A) have the same time-horizons, longer sampl ing-time can be used for the latter problem. Also 1jJ >1jJ since the integrated solution anS t~e problem (2B) have similar sampl ing-time but the timehorizon of the integrated problem is longer. 1
Suppose that "1="2=8, 1jJo=4, 1jJl=1jJ2=3 and
t f /T=5. then the ratio of the processing
times for the integrated solution to the decentralised one is
- 120
which is a substantial improvement. Notice that these estimates of the processing times are independent of the 1 inearily of the system equations. 2.
The Equivalence of the Decomposed and the Integrated Structures
The solution of problem (I) is given by the two-point boundary value problem
[:] {: -"-"] [:J -A'
{~}t=O =~ -1 B_ \ ~ -R
0
and Y A
e-
{Pt:.tf=Q
(3)
M. R. Javdan
One way of assessing whether the two problems (I) and (2) have similar solutios is to determine whether the two matrices Z e and Za have similar eigenvalues and eigen-
where "e:R(n l +n 2) is the adjoint variable. Define-the matrix
~
Z
e
-BR- IB
1
-A'
J
[ A
_Q
85
vectors, and the matrices Ye and Y have a similar elements . Note that the two methods give identical solutions if A3=0 and B3=0 . This, however, is a trivial result as it is the case where the two problems (2A) and (2B) are decoupled.
For a given ZO the solut ion of equation (3) is uniquely determined by the eigenvalues and the eigenvectors of the matrix Ze' The solution of problem (2B) is given 2 I -B R-
b:(2)ll A2 [ ~(2) = -Q(2)
~(2~
-A
. {~(2)}t=0=~O(2) ,
~(2)
(2)B2J(~(2~ .F3~t1)+B3~t1)1
1
{~(2)}t=tf =Q
Q
J
3.
The scalar case Let us consider the case when n =n =1 l 2
and PI=P2=1. Z and Z become e 2 a b
(4B)
o
-R-I(2)B2'~(2)
l
where z * (I) and m* (I) are the solutions of problem (2A) given by
[
~(1)l=rA' -BIR-I(~)BI'] ~(l) ~(1)
l-Q(l)
~(1)
_AI
o (4A)
{~(I)}t=O = ~o(l) , {~(I)}t=tf = Q ~(1)
and
= -R- I (l)BI'~(1)
The solution obtained when the integrated problem (1) is solved in the two-level structure of problem (2) is given by the combination of equations (4A) and (4B),
z (1) ~(2)
Z
;'(1)
a
[~(2)
o
o Z a
o
[z (I)j
o
~(2) ,,(1) ~(2)
Jl~(I)]=-~5)[~(I~
I
r:(1)]/ =[-R-I(I)B ' 0 l~(2) 0 -R- I (2) B2'
~ (2)
The matrices Ye and Y are also given by a
~ r
a ~ (2)1
Ye =
with the same boundary conditions as in (3) . The matrix Za is given by :-BI(R(l)-IBI' A2 :-B 3 (R(l)-I BI'
- 1 ' - - - - - - - --
-Q(1)
0
:
-A
o
-Q(2)
I
o
0 2' -A
[
l
o
and
o
+
1
Ya =
[: :~l
Direct inspection of the matrices give the first two conditions for the equivalence of the solutions. For the similarity of Y e and Y a sufficient condition is that a « (C1) 1 I bll 3 For the matrices Z and Z we have the first e a condition 2 2 b b (C2) 2 -.l. « r r 2 l
Ib
the matrix Z can be written as e
86
A structure for time-horizon decomposition
Let us consider the eigenvalues of the matrices Z and Z which are the solutions of e a
and
IAI
o
IAI
o
For ~e,;~a we need 3 3 a
2 ql 2 «b 2 ql -+a 2 r2 3 2 q2
which can be simpl ified to the following sufficient condition
Now A 2 A2 2 A2 2 q2 \A I-Z I=(a - A )(a - A )+b e I 2 2 r 2
(C4) Furthermore, we have
and I ~ I-Z
a
1 =(a2_ ~ 2) (a2_ ~ 2)+b2.:2(a2_ ~ 2)+b2~(a2_ ~ 2) I
2
2r
2
I
Ir
l
2
and assuming (C4) is satisfied,
+ bLb2 qlq2
Ae I z I = -a
I 2 r r l 2
The two polynomials are similar except for the constant term. Assuming that conditions (Cl) and (C2) hold, we have a further condition
2 b2 b I b3 Ae A Ae { - + - - z - (a - A)Z } r r 3 2 2 2 l 3
In summary , a set of dufficient conditions for the similarity of problems ( I ) and ( 2 ) a re i)
2 2 q2 q2 {l a 3 b l ~ - 2a l a b l b ~I } 3 3 Since all the terms on the left hand side are positive this condition can be reduced to the following sufficient condition, obtained by considering only the term 2 2 qlq2 b b -I 2 r Ir2 '
«
la3(2alb3-a3bl)l «
blb~
b~b~ ~: ql
(C3)
r2
The eigenvectors corresponding to each A are given by ( ~ I-Z )~e
A
where
e Aa
( A I-Z)~
are Ae
~
i i i)
iV)
1-2ala3blb3+a;b~ 1 i.e.
i i)
Q 0
Aa 4 and Z EC .
. Ae Aa . Taking z4=z4=1, by elementar y manipulations we get 2 ql ql 2 A2 2 b2 + (a 2 - A )+a 3
r; q;-
These conditions make quantitative the intuitive notions about the conditions for equivalence of problems (I) and (2) given in Singh (1973) and Singh, Drew, Coales (1975). These conditions essentially state that the two problems (I) and (2) have similar solutions if, either the couplings between the two stages of problem (2) are weak (i .e . a ,b +O), or when the weightings 3 3 on the state and input vectors z(I),m(l) are much greater than those on the ~ectors z(2), 1lI(2). It is expected that the condition~ for (~1+~2»2 will not be significantly different
from the conditions derived here for the
(~1+~2)=2 case . A numerical example Consider a system with A= [-0.1
o. I
0] ' B=[o.5I
-I
~J
'R=[ ~
:]'"d Q-G
87
M. R. Javdan
This system satisfies the four conditions i) to iv) which ensure that its integrated solution (through problem 1) and the decomposed solut ion (through problem 2) are similar . To verify the conditions for th i s system form the corresponding Ze and Za
4.
A prel iminary investigation of a new structure for time-horizon decomposition has been given. The treatment is centred on linear systems, although the concepts equally apply to non-linear systems with the same basic structure as in problem (1). It is i ntuitively clear that similar conditions to those derived in section 3 for 1 inear systems will also apply to the nonlinea r case, i.e. for the equivalence of the integrated and the decomposed structures we either need weak coupling between the two stages or require that much more priority be given to the long-time-horizon problem than to the short-time-hor izon one.
matrices and evaluate their eigenvalues and eigenvectors: eigenvalue
corresponding eigenvector
~ 1=-11 .180
(0.914
0.011
0.0405
0.000)
~ 1=-11 .180
(0.914
0.011
0.405
0 . 000)
~ 2= 11.180
(-0.911 -0.024 0.411
0.002)
~ 2= 11.180 a ~ 3= 1.415
(-0.911 -0 . 024 0.411
0.002)
(-0.013 -0.383 -0.017
0.923)
~ 3=
(0.000
0.000
0.923)
(-0 . 010
0.923 -0.008
0 . 382)
(-0 . 000
0 . 923 -0.000
0 . 382)
e
a e
e
1.414 a ,4 A = -1.415 e ,4 A = -1.414 a
Concluding Remarks
References -0 . 382
1.
Singh, M.G. "Some Applications of Hierarchical Control for Dynamical Systems", Ph . D. Thesis, 1973, Cambridge Univers i ty.
2.
Singh, M.G., Drew, S.A . W. and Coales, J . F. "Comparison of Practical Hierarchical Control Methods for Interconnected Dynam ical Systems", Automatica , 1975, Vol. 11.
The eigenvalues are identical to within two place s of decimal and the eigenvectors are the same to within one place of decimal.
Sol v e Pr o blerl (l A) t
"
!I t
~
fo r . the
in t er v al
to t f
Solution
, ( ' I · . m{ ' I
c( ' 1
So l v e Pr o b l l"'" (2 B) (o r t = (."
t: . .
'T 1 to (.
SO l u tion
! (2 )
: ()
,
t '"
t n l"
in terv.)1
+,) ~ t
. ,:- ( 2 )
Lt
t. ( ••
:+;, Tl I
( • t : + (.. ' +,
t~+ l, lT) t O ( L +I ) t :
to
1TIr -----I.I._---,
Proc e ss
The Time-Horizon-Decomposition Structure