Hierarchical Optimisation and Control of Large Scale Systems with Dynamical Interconnection System

HIERARCHICAL OPTIMISATION AND CONTROL OF LARGE SCALE SYSTEMS WITH DYNAMICAL INTERCONNECTION SYSTEM

J.

L. Calvet and A. Titli

Laboratot"re d'Automatt"que et d'Analyse des Systemes du C.N.R.S. 7, Avenue du Colonel Roche, 31400 Toulouse, France

Abstract. A hierarchical and decentralised approach is proposed for optimisation of dynamical interconnected systems. After a quickly review of the well known decomposition-coordination methods using a classical formulation with static interconnection between dynamical subsystems, we introduce dynamical interconnection equations. The new formulation on interconnected systems corresponds to some practical problems (e.g. the control of power system). Limiting our formulation to the linear quadratic case, we suggest a relaxed algorithm based on interaction prediction. The convergence is discussed and promising examples are given. The use of a mini-computer network to implement such a hierarchical and decentralised algorithm is also investigated. Finally, for an infinite horizon time and under the realistic hypothesis that the modes of interconnection system are slower than the modes of subsystems, a time space decomposition is derived from singular perturbation theory. We thus develop a decentralised control (closed loop control on each subsystem and on the interconnection system) using the state observation of each system.An application on a power system gives satisfactory results even with fairly coupled subsystems. Keywords. Large scale systems ; optimal control hierarchical systems ; power system control ·

INTRODUCTION

trol u of the overall system belongs to ~ = XI x ••• x ~ and U = UI x ••• x UN respecti-

The automatic control theory is now widely concerned \li th the op timisa tion of comp lex systems. Among these systems, the large scale systems viewed as interconnected systems have been receiving a great amount of emphasis during the past few years (Mahmoud, 1977).

vely, whilst vi and zi would belong to a socalled interaction space V. We will start with the non linear model dx.

In optimisation of dynamic systems, which are obviously of major importance for control theory, two approaches seem of a great interest

(I )

z.

(2)

h.(z., u.) j = I, ... ,N (3) ~ J J n· m· where x· ElRn=nl +. · · +nN) , u·~ Em, ~ . ~ E IR ~ (xq. v. EIRP~ , z. € m. ~ and where the functions f~, e., h.~ 6elong to class C2 in all argu~ ments. v.

~

This paper deals with these two kinds of approaches and is concerned with

4

It should be emphasized at this point that by substituting Eq. 2 into Eq. 3, the inter-

- the time space separation of interconnected systems - the decomposition and the orJer reduction involved in the calculating task.

connection system can be designed by the static relation

For this study, we will consider a dynamic state variable model for N interactive subsystems. Such a subsystem representation is ~iven in Fig. I., where xi is the local set of state variables, ui the local set of controls'Yi the local set of outputs and vi(zi) the input (output) coupling vectors. Letting xi~X;, u.€U., then the state x and the CQn1

dt~ ~ xi = fi(xi,ui,v i ) ~

- the model simplification approach - the hierarchical approach

.L

system order reduction

z = g (x, u, z)

(4)

where z E. IRq. Optimisation of such interconnected systems using decomposition-coordination methods has been depthly investigated in the literature, see for example the works by (Takahara, 1965;

~

117

118

J. L. Calvet and A. Titli

Pearson, 1971 ; Singh and Titli, 1978 ; Cohen, 1975, ••• ).

s. t.

x.=A.x.+B.u.+C.v.

Remind that the proposed decomposition techniques are based upon some hypothesis on the interaction space, which may be satisfied by the convergence of an iterative coordination process. This yields a hierarchical algorithm which consists of fixing at the upper level some coordination parameters in order to obtain a separable form of the Lagrangian function. In the fi~st section of the paper, we quickly review the "Prediction Method" developped by Takahara (1965) to the linear quadratic case. In the second part we propose to introduce a dynamical interconnection system : z = g (x, u, z)

(5)

It is believed that no particular attention has been devoted to such a model, however we will see on various examples that many interconnected systems of practical interest fit this description. Restricting to the linear quadratic case, we propose in this second section a hierarchical and decentralized algorithm based upon the "Interaction Prediction Principle" (Mesarovic, Macko and Takahara, 1970). The point of this is threefold - it decomposes the overall system into N+l "subsystems" (N subsystems + interconnection system) - it involves a double task of optimisation-coordination for both levels - the coordination is performed via a mutual prediction of coupling variables (from both levels). Moreover an attempt is made to prove the convergence and two power system examples are simulated. Finally, in the last section we consider the regulator problem and we analyse the hypothetic separation of the time scales of subsystems and interconnection system. Here, this realistic hypothesis yields a complete separation of the N+l "subsystems". Thus, referring to the application of "singular perturbations" to large scale systems for a time space decomposition (Titli, 1979), we develop a decentralised control using the state observation of each subsystem and of the interconnection system. An illustrative application is made on a

power system. PREDICTION METHOD

t

N

{u E. i

tR""'}

1

(f

2

~"2)~ i= 1

Q.)O 1/'

2

(!Ix. HQ +Bu. HR ) dt to 1 i 1 i R.

1

>0

Z. 1

(7) (8)

D. x. 1

1

N

v. 1

N

L.

L

H.. z.

j=1

1J

j=l

J

L .. x. 1J

(9)

J

~ x~Q. x. and (xi' ui' v· ,z.) , 1 i 111 1 1 (Ai' Bi,Ci,D.,H,L) are respectively vectors 1 and matrices 0f appropriate dimension.

where \I x. \\ Q2

For given coupling inputs vi and Lagrange parameters ~i associated to the coupling constraint (9), chosen as coordination parameters, the variables zi may be dropped and the Lagrangian1r of the overall problem is separable into N independent minimisation ~ro­ blems. That yields the following two-level calculation. Higher Level : coordination level The problem here is to set the values of coordination parameters ; i.e. to solve the stationnarity conditions :

f

@~=

ty

dV"

f3_ C 'tfJ::: o j

~~:::v - Lx =- 0 }-J

(10)

where'+' is the cos tate vec tor. That enables one to predict v and 0, at stage n+l, by an iterative process of finding fixed point of a contraction mapping (Takahara, 1965), i.e. v

n+l

=

L x

n

{ 11 )

Remind that the convergence of this coordinating algorithm depends closely on the choice of the weighting matrices Q, R and of the horizon time : t f - to. Lower Level: optimisation level The problem is here of solving for given v and ~ (-+ v*, 0*), N independent optimisation problems. Figure 2 gives the complete algorithm where asterisks denote the fixed variables. Notice that such a method giving the possibility of a parallel resolution of N subproblems, provides in fact a decentralised iterative calculation, since the higher level only transfers informations between these subproblems and doesn't execute any calculating task. Thus, the task of the upper level is to ensure coordination while the lower level ensures optimality. EXTENSION TO DYNAMICAL INTERCONNECTION SYSTEM

Let us consider the linear-quadratic formulation of the overall problem : Min.

xi(tO)=x iO

1111111

i=I, ••• ,N

(6)

Statement of the Problem and Assumptions Suppose now the interconnection system S' is designed by the dynamic model (5). For more generality, we also select amoung the control variables, the local control ui and the proper control u' of the system S'. Hence, we can consider the following linear quadratic

Hierarchical Optimisation and Control formulation of the overall problem. t

11

"2

Min

t

,

[11 z\lQ2 ,+\\u ,2 HR'

0 m}

mi

tu.EG<.

f

, u'E.eR

2 +

1

Coordination N

2

+ ~ (\\xiIl Q. + 1=1

1

]

\\uiH .) dt R

(12)

1

s. t.

I~i

A.x. + B.u. + C.v. 1 1 1 1 1 1 i ;: 1 to N

L

(14 )

v. = H.z + G.u' 111

A'z + B'u' + C' w ; z(t O) N H!x.+ G~u. H'x + G'u = 1 1 1 1 i=l

z

zo

L.

w

(15) ( 16)

a.

Define now ltJ. , as cos ta te variab les and (3. ) f1..1 1 . 1 J~ as Lagrange parameters assoc1ated to the state equations (13) (15) and the coupling constraints (14),(16) respectively. Hence, we are in position to provide a two level decomposition-coordination method of solving the overall problem by respective prediction of (3 i ' vi at the lower level ~', w at the upper level. Hierarchical and Decentralised Algorithm Associated with the problem (12)-(16) we can define the lagrangian : 1[ 2

LO

2

\="2 llzIlQ,+\\u'\\R'+

~ 2 2 1 [:-1 (\\XiIlQi+lluiIIRi) j

T {\. (~-A'z-B'u'-C'w)+

)T

\J

+

(w-H'x-G'u) +

N

~r~~ (~.-A. x .-B. \.l. -C. v.)+ (l.. ~ (v. -H. z-G.1J) ] '-IL 1 1 1 1 1 1 1 1 ,v 1 1 1

(17)

1-

This lagrangian takes a separable additive form choosing 0' and (3. as coordination variables and, a fortiori 10 " 0i' wand v· in order to use prediction algorithm. 1 This decomposition allows one to define the following sub-problems and the corresponding coordination task:

Subproblem nO i (i = 1 to N) (t f [ 1

min J u. t 1 0

"2

-0'~T(H~x.+ 1 1

2 2 'T* x (\lx·H Q +Hu.H R )+~ w 1 . 1 . 1 1 ] G~u.) dt 1

s.t. Xi = Aix

i

+ Biu

i

+ Civ

1

i

(18)

t

u

to

s.t.

i:

[t (lIz\\~,+lIu'II~,)+f-l*~v:l:-HZ-GU')]dt (20)

A'z + B'u'

+ C'w

~~l= ~·V\

tt~. lfw

L-

~

W =

= 0

;>

0·1

= 0

;> ;>

0

0

v. 1

f>'

H'x+G'u

C~

lP.

1 1 H. z + G.u' 1 1 C'T A.

(22) (23) (24) (25 )

Usine the particular structure of the coordination equations and distributing them between subproblems one obtains the decomposition coordination algorithm shown in the figure 3. (where n is the iteration number). In this algorithm the task of coordination is performed at each level where the variables ,B i , vi and (3], w can be respectively predicted from the stationnarity conditions on ~. The task of optimisation is also carried out by the two levels : N independent optimisation problems are solved at the lower level whilst an optimisation problem on the interconnection system is solved at the higher level. Remarks (i) This algorithm is a proper "relaxed algorithm", see (Cohen, 1979). At each step of coordination, it involves a parallel decomposition into N subproblems at the lower level, combined with a sequential decomposition into the N subproblems and the optimisation on the interconnection system at the upper level. Let us note that so a particular decomposition can have some similitude with the one investigated in (Findeisen, 1968) and yielding a two level relaxed algorithm derived from a "primal coordination method". (ii) The whole decomposition uses a particular form of the overall system matrix A which is composed of N independent blocks connected to a coupling system. (figure 4). We"ll see later that such a structure is met in many physical examples. (iii) By seeking a solution for ~i and ~ : 'Pi = Ki x i + qi and ~ = Kz + q, and under the assumption of stabilizability and detectability for each "subsystem", one can easily design for each of them, a decentralised closed loop control with an open loop control using informations on the coordination parameters to improve iteratively the local feedbacks (in a periodic system). Convergence

;x (to)=x iO (19) i

Subprob1em (N+l) tf

M~n f

119

;z(tO)=zO

(21)

It can be proved by a ~eneralization to this kind of decomposition of the Pearson approach (1971),the following tneorem : TheoreM (1) : There always exists an horizon time [to' tfJ; t ~ over which the decomposition coordinaEion algorithm (Fig.3) achieves its convergence to the optimal solution. t""

tf

J. L. Calvet and A. Titli

120 Illustratives Examples

In this section, an attempt is made to apply the previous theory to obtain an optimal controller improving the dynamic response of a power system consisting of two (three) coupled hydraulic machines.'

Note that fj,Ptie.l represents the change of the total area power export mesured positive out from the 1st area. If we choose diagonal matrices Q and R for the objective function (26), Q

d i ag (q 1l' q 22' q 33

=

i I

q 44' q 55' q 66 : q)

These generators are assumed to be in constant steady state whose value was obtained by a standard load flow analysis.

(32) (33)

Here we consider the following trajectory optimisation problem for the deterministic case, where each variable represents a deviation from its steady state after small disturbances. T

Min uE(Rm

i- 5
dt

Q~ 0

0

Then the formulation of the opt1m1sation problem (26) to (33) will be obviously recognized as equivalent to the formulation (12) to (16) So we can apply the two level algorithm defined in Fig. 3.

>(1)

R

Fig. 5 gives the combined and detailed structure for this example.

(26)

txample 1 : two-area system The system under investigation consists of two hydraulic machines connected via a power line ("tie line"). The dynamic model in state-variable form, for small disturbances about the given operating point, can be found in (Elgerd, 1971). If we rewrite as follows the state vector x and the control vector u from their own definition in the reference above : XI

~

x

~xEI

x x ~

x x x z

u T 6== I

U

I'

U

2

AP

3 4 5

6

fI

=

GI (28)

IJ £2 Ax

E2 ~PG2

Tpi = 20 p.u ; ~i = 120 p.u. ; T = 0.08p.u. Gi I/Ri = 0.417 p.u. ; TTi = 0.333 p.u. : i=I,2 ; = - 1 ; T~2 = 0.0866 p.u. a l2 We have also chosen for Q, R Q = diag (2,1,1,2,1,1,2) ; R

-

~pc2l

diag (1,1)

and for Xo the initial value to which a disturbance on the first generator 6.P = - 0.01 (6P = 0,) took the system. D1 D2 The two-level algorithm was programmed on the IBM 370/165 digital computer of LAAS Toulouse. The initial guess for z(t) and~(t) was chosen to be O. throughout. The test for termination was taken to be : C A c -

C1 P tie. I

2 J =LaP C1 '

Simulation and results . This problem was solved with numerical values taken from (Elgerd, 1971), i.e. we have considered two identical machines

f T [ (zn+ 1-z n) 2+ ( xn+ 1-xn) 2+ (xn+ 1-xn) 2 0

1

1

4

4

+ (~n+l_ A n) 2 + (~n+l _A n) 2 +( d n+l_ A'n )2 I'" 1 I'" 1 2 J 2 }.J 1 ,.. . 1 J

(29)

+

then the system matrix A and the input distribution matrix B can be set as

(~I~+I_~1 ~)2

<

] dt

10- 4

where n is the iteration number.

-K

pl

/T pl

0 0

- A

o

T

B

=[

_0

o

I_/T_G_1_ _0 {}

-I/Tp2

0

-I/R T Z G2 0

-I/T

-ZI\T

0 I2

I/T

GZ

TZ


+-

0

-aI2~2/Tp2 0

-I/T

0

T2

I/T

G2

(30)

0

0

0

-

Kp2 /T pZ 0

0_ _

]

(31 )

Hierarchical Optimisation and Control

Table 1 shows the convergence obtained for different values of the horizon time T. Note that in these calculations, the period of optimisation is long enough for the system to be sufficiently close to steady state. Furthermore, we have compared our approach with the Takahara's one where the interconnection system will appear as a (N+l)th subsystem in parallel with the other N subsystems. In Table I, we also have indicated into brackets the error value En for this algorithm, that shows on this example the interest of our approach.

121

and Vl,v2,w'fi~,~lbe coordination parameters, then the overall problem breaks-up into three optimisation problems which can be coordinated as in Fig. 3. Fig. 7. shows this particular two-level hierarchical structure. Simulation and Results The values are globally the same as in the previous example, except the local weighting matrices Qi' Ri given as Qi = diag (2,0,0,1) ; Ri = 1.

: i=1,2,3

and

Example 2 : three-area system The optimal control problem discussed above for a two-area system can be intuitively extended to an n-area system. The three-machine configuration studied here is defined in (Calvet and Hurteau, 1979) and is shown in Fig.6 • This yields directly a global system matrix A consisting of three independent blocks for each generator model, and a coupling system mode led by the dynamical model of the two tie lines expressing the change of the total area power export mesured positive out from the 1st and 3rd areas : ~Ptie.l, ~Ptie.3.

Note that it is interesting to use a multiprocessor computational system, since parallel computation is available in such a method. This was investigated on three mini-computers of type MITRA 105 (16 K words memory), connected as in Fig.7. Table 2 shows the fairly convergence which has been obtained for an initial value of x corresponding to a disturbance ~PDl=-O.012; ~ PD2 = ~ PD3 = 0 (E. being defined as previously, from the coordination variables).

Denoting :

SEPARATION OF TIME SCALES

xT~[~fl ~~1' ~PGl' l:!Jf 2 ,!:tx E2 ,

{JP G2 ,

Lif , llx , !:l P , APtie. I, ~Ptie.3] E3 G3 3 (34) (35)

then it is straightfon~ard to obtain an analogous formulation to the problem (26) to (33) with

Regulator Problem Let us consider now the optimisation problem (12) to (16) with an infinite horizon time. For the sake of simplicity, we also suppose there is no control u' on the interconnection system.

~Ptie. 1

21TT

~f2)

(36)

Letting "A, "B, 1\ Q, AR be the block diagonal matrices with blocks Ai' Bi , Qi' Ri : ~=l, ••• ,~ then the overall problem can be rewr1tten as follows

t.Ptie.3

21l'T~2( tlf 3- ~f2)

(37)

Min

o I2

( Af t

00

Thus, choosing the system (36) (37) as an interconnection system, the optimality conditions may be arranged in a similar hierarchical structure as in Fig. 5 with three subproblems solved at the 1st level. However, we can obviously consider an other decomposition where the interconnection system is the generator G2 itself, see Fig. 6 Here, we denote :

=

(~Ptie.2

~Ptie.l+L\Ptie.3)(38)

X~~[~fl' ~xEI' ~PGI' APtie. 1]

(39)

X~I~[t\f3' l'ixE3 ,

(40)

LlP

G3

, flPtie. 3 ]

fi'

Letting ~1' (1:J:z ' be Lagrangian parameters for the coup11ng constraints : VI

A =:

1\

uf

2

v

2

~

6.f

2

w

T

J u E. 0<."' s. t. ..

1\

1'\

J:J..2J(1\z\\2,+\\x\'~ +nuu~R Q Q

) dt (42)

o

x=Ax+Bu+Dz ; x(O)=xO (D@CH)

(43)

z=A'z+D'x+D"u; z(O)=zO (D'~C'H', D"~C'G') (44) It would be interesting to consider subsystems and interconnection system governed by different dynamics. It is obvious that many practical examples fall under this assumption. Several order reduction technics as singular perturbations, can be motivated by such an eventual two-time scale property. For this study, we'll apply the complete separation technique derived from the latter theory for linear models (Chow and Kokotovic, 1976).

Then, we suppose that the modes of the interconnection system are slower than the modes of subsystems. This realis tic hypothes is enables one to derive

J. L. Calvet and A. Titli

122

a "slow" system

z

A'z

s

G'

+ D'x + D"u

s

s

with 1\

!,_

o

I

+ Bu

= Ax

z(t)

~

+ Dz

s

(46)

s

z (t)

(47)

s

and with the performance index :

foe>(H z

1

J = s 2

1\

0

2

-

s Q'

2 Q

2

+11 x HI'. +\1 u 11 A) s R

X

=

f

f

+

BU

where (49)

; xf(O)=xO-x(O)

f

ote that here, these symetrical results are due to the fact we have considered two identical generators and a transfer function a = -1. 12 For comparison, we calculated tte optimal regulator of the global system:

indepen-

-

~

-1.373

(48)

dt

and a "fast" system consisting of dent subsystems : A AX

G-

with

1

f ="2

foco(lI x o

f

2

2

Q

R

l\/\ +\\uf\ll\) dt

(51 )

Then, the space decomposition into N+l "subsystems" is achieved by the complete suboptimal time decomposition. It should be noted at this point, that the initial trajectory optimisation problem (12) to (16) with a fixed horizon time can also be solved by such an order reduction technique (Kokotovic, O'Malley and Sannuti, 1976). Let us also note that in an other way using a multimodel formulation, a more general situation with weakly coupled fast subsystems was investigated by Khalil and Kokotovic (1978) by means of Pareto optimal strategies.

'"G

, the gain matrices If we call now Gs and f associated to the problem (45) to (48) and (49) to (51) respectively, then we can design a near optimum decentralised regulator using the local information xi and the infonnation z on the interconnection system (with the requirements of detectability and stabilizability on each "subsystem" as formulated by Chow and Kokotovic (1976))

=

Us + u

= f

"

Gl~ + G x

(52)

where G' = [G

s

+

~f

1- 1

(D +

~

G )

s

J

(53)

and 1\

I\.

(54)

G = G f

Illustrative Example We considered the regulator problem of the previous two-area system. In that case, the hypothesis of two-time scale separation is realistic since the system matrix A' of the interconnection system is null. Thus, we have defined the decentralised control visualized in Fig. 8. umerical results. Using the same values as previously defined, we got : G

1

= G 2 =[ -

1.031

0.4xI0-

4

O.27xI0-

3

: -1.002

-0.7005

-1.672

~

1.374

J

-1.374

(50)

and with the performance index

u

[_-~.~~2_~~~o~~ _~~~7~ __ L=o~~ _~._4~O~~ ~.~7~1~-~ 1 -0.03

x(t) ~ x(t) + xf(t)

J

=[ 1.373-j

(45)

- 0.7004

-1.671

J

This result shows the near optimality of the previous decentralized regulator. Moreover, the relative importance of the gains related to the interconnection system allows us to conclude that the two generators are fairly coupled. Remarks : The stabilizing solutions K of the Riccati equations involved in the resolution of the N+l optimisation problems (45) to (51) has been computed by a Runge-Kutta-4 integration routine. The test for termination was choosen to be

£~I\K(t

+ CH) - K(t)

\\~ ~

10-

10

where

1I.1l112~~ L .L

j

[k .. (t+ t,t)-k .. (t)J ~J

~J

2

;

ll.t=0.02

Convergence of the Riccati solutions for each generator was achieved when t=8.62s whereas convergence of the solution to the overall problem required backward integration to 17.06 s. This shows the damping effects of the interconnection system on the global system behaviour. CONCLUSIO S In this paper a hierarchic scheme for the optimal control of dynamical interconnected systems is proposed, whereby optimisation is carried out on both the subsystems and the interaction system assumed to be modeled by dynamical equations. The present results appreciatly differ from the classical approach where the dynamical subsystems are interconnected via static relations. They concern a particular structure of coupled systems ; however, it is admitted that many systems fall under this class, as power systems, hierarchical systems, articulated mechanical systems, ... Restricting the study to the linear quadratic case, we showed how the optimal control can be computed using a hierarchical and decen-

Hierarchical Optimisation and Control tralized algorithm which is indeed an extension of the "Prediction Method" by Takahara. However, as mentionned above, it would be noted that here, both the two levels assume a double task of optimisation and coordination. It must be also emphasized that the algorithm is well suited to be implemented on a multiprocessor network structured in a masterslaves configuration. Various examples have been experimented giving promising results. Finally, the assumption of the separation of subsystem and interconnection modes seems to be a realistic hypothesis and would concern a wide class of control systems. The resulting decentralised regulator presents two mainly advantages : (i) the order reduction and the decomposition provided in the calculation procedure (ii) the local regulator implementation using reduced information from both the subsystem and the interconnection system. Although a particular structure was considered, our main conclusion is that more general formalism, but also others specific cases, can possibly be handled, contributing to time space decomposition of complex systems.

REFERENCES Calvet, J.L. and Hurteau, R. (1979.l "Vers une commande decentralisee et hierarchisee des systemes dynamiques interconnectes". Rapport technique EP79-R-I0, Ecole Poly technique de Montreal. Chow, J.H. and Kokotovic, P.V. (1976). "A decomposition of near-optimum regulators for systems with slow and fast modes". IEEE Trans. Automat. Contr., vol. AC-21, 701705. Cohen, G. (1975). "Contribution a la theorie de la commande decentralisee et a la coordination en ligne des systemes dynamiques" These Dr. Ingenieur, Univ. Paris Sud. Cohen, G. (1979). "Dne approche unifiee des algorithmes d'optimisation par decomposition-coordination". Chap. VIII in "Analyse et Connnande des Systemes Complexes" Titli et al. Monographie AFCET, Cepadues Editions Elgerd, 0.1. (1971). "Electric energy systems theory: an introduction". Mc Graw Hill. Findeisen, W.(1968). "Parametric optimization by primal method in rnultilevel systems" IEEE Trans. Syst. Sciences Cybern., vol. SSC-4 , nO 2, 155-164. Khalil, H.K. and Kokotovic,P.V. (1978). "Control strategies for decision makers using different models of the same system". IEEE Trans. Automat. Contr., vol. AC-23, nO 2, 289-297. Kokotovic, P.V., O'Malley, R.E. Jr and Sannuti, P. (1976). "Singular perturbations and order reduction in control theory: an overview". Automatica, vol. 12, 123-132.

123

Mahmoud, M.S. (1977). "Multilevel systems control and applications: a survey". IEEE Trans. Syst. Man and Cybern., vol. SMC-7, nO 3, 125-143. Mesarovic, M.D., Macko, D. and Takahara, Y. (1970). "Two coordination principles and their application in large scale systems control". Automatica, vol. 6, 261-270. Pearson, J.D. (1971). "Dynamic decomposition techniques" in "Optimization methods for large scale systems •.• with applications". Wismer Editor. Singh, M.G. and Titli, A. (1978). "Systems: Decomposition, Optimization, and Control" Pergamon Press. Takahara, Y. (1965). "A nultilevel structure for a class of dynamical optimization problems". M.S. Thesis, Case Western Reserve Univ., Cleveland. Titli, A. (1979). "Partitioning and time decomposition for the control of interconnected systems". Symposium Optimization Methods : Applied Aspects. Varna, Oct. 15- I8, 1979.

124

J. L. Calvet and A. Titli

u. 1.

-C>

Yi=x

ss.1.

i

(for simplicity)

x. z. 1. Fig.l. Subsystem representation

......... x i

l{J

=

V

~~=

,,¥i

0

==)

0

~

(3 n+l = CT v

n+l

~

'f*n

"'llIIlIIII

~n =L x

.!" [z

Min

m1.

uiE

fR

s.t.

1

xj '

v·,ft

vi,j3

'r.J

~ ,J 2

x .\lQ

1.

N

i

- L

+\11\2)

u i R. 1.

to

j=1

. x

x = A.1. x.1. + B.1. u.1. + C.1. v.1. i

*T

(3.

Optimisation on

L ..

J1. Xi]dt

J

--- -

; xi(t ) = x O iO

the subsystem j

Fig. 2. Decomposition-coordination algorithm

x.,u. 1. 1.

s.t.

,8.1.

--

.z =

A'z + B' u' + C' w

tf w

= 0 ~

~f.>'=

0

===>

r:

n+l n+l

w

; z(t ) = zo O ~n+l

=

C,T

=

H' x

~n+l

;

,

+ G

u

xn+l

z, u'

,p' ~I~

~I' Min

Subproblem

m. 1. u.f~ 1.

s.t. x. = A.x. + B.u. + C.v. 1. 1. 1. 1. 1. 1. 1.

Lf~.= l

0

~

n+l v. 1.

=

H. z~n + G. u,~n ; 1. 1.

Fig. 3. Decomposition-coordination algorithm

j

125

Hierarchical Optimisation and Control

Fig. 4. System matrix A

....

z(O) ; ;

. •

xI(O)

X

A

~

s

= -

w 2 x4/R2TG2-xs/TG2 + '5/ r 2TG2

~6 = (x s -x 6 )/TT2 ; xrr(O) =

= x ro

·
12

•

2

(x 2-x 3 )/TTI ;

TO

= 2 \T

x 4 = - x 4 /T p2 + ~2(x6-aI2z )/T p2

x 2 = - xl/RITGl-x2/TGl+~2/ r l TG1

·3 =

~(T) = 0

.

~

Xl = - xl/T pl + Kpl (x -z )/T pl 3

x

~}

'·2

zo

=

-

)~

01

~4

= Q44 x 4+


0/2 = Q22 x 2 + \fJ2/ TG1- ~3/TT 1 ; yJr (T) =0

~s = qssx S+ 'PS/TG2 -

~

~6

= Q33x3 - K p1

u I =~+\/ r l TGI

~I/Tpl

;01

+\f/TTI

=-~ICYl/Tpl

=

Q66 x 6 -

X

1rO

+ If's/R 2TG2 -

'Y6 /T T2

~2 If'3/ Tp2

~~

; \frr(T)=0

+ lP6/ TT2

u 2 = yJS/ r 2TG2 ; (32=-a I2 ~2 ty 4/ Tp2

Fig.S. Decomposition-coordination algorithm

Fig.6. Three-machine configuration

J. L. Calvet and A. Titli

126

-...

.......

-.

.....

..--~

'~I

MASTER

~

Optimisation on G 2

~f2,~1

~f2' ~'

~I'

~f3'

~I'

SLAVE 1

SLAVE 2

Optimisation on G 1

Optimisation on G 3

Fig.7. Two-level computer network structure

z

Fig.8. Decentralised control scheme

~ T = I

n = 5

n = 10

n = 15

3 1.58 x 10(7.82 x 10- 3 )

6 8.77 x 104 (6.56 x 10- )

8 6.68 x 105 (3.14 x 10- )

---- -- -- ----------T=I.5

2 1.60 x 102 (2.38 x 10- )

4.12 x 10- 3 2 (1.21 x 10- )

4 9.97 x 103 (2.20 x 10- )

T = 2

2 8.74 x 102 (5.79 x 10- )

1.64 x 10- 1 2 (5.88 x 10- )

1 2.69 x 102) 10(7.85 x

TABLE I

ysn

n

= 5

n = 10

T = 3

4 2.41 x 10-

6 1.74 x 10-

T = 6

3 1.26 x 10-

2.89 x IO-f.

TABLE 2

CONVERGENCE

DIVERGENCE