Hierarchical Steady State Control of Complex Systems1

Copyright © IFAC New Trends in Design of Control Systems, Smolenice, Slovak Republic, 1997

HIERARCHICAL STEADY STATE CONTROL OF COMPLEX SYSTEMS

Ladislav Harsanyi*, Zdenka Kralova*, Vojtech Vesely*, Khalil S. Hindi**

*Department of Automatic Control Systems, SIovak University of Technology in Bratislava, Ilkovicova 3, 812 19 Bratislava, SIovak Republic! **Department of Manufacturing Systems, Brunei University, Uxbridge, Middlesex, UK

Abstract: This paper presents the basic principles and main properties of new modified feasible methods that can be applied in steady-state optimization of complex systems. The essential features of the proposed structure are both using the feedback from the controlled system and the linearity of a mathematical model for rmding the optimal solution on the coordinator level and on the subsystem one. These properties are important from a practical point of view. An example of optimal steady state control of a complex system is given. Keywords : hierarchical systems, steady states, optimization

- mixed method. 1. INTRODUCTION

2) Techniques where the feedback from real processes is used - interaction balance method with feedback, IBMF, - modified feasible method, MFM, proposed in this paper.

The problem of complex system optimization for industrial processes is becoming more and more important because of increasing raw material and energy costs. To handle a steady-state optimization problem of a complex system which consists of N subsystems, the hierarchical control structure, composed of local decision making units, local control systems and a coordinator can be used (Mesarovic, et aI. , 1970, Findeisen, et aI. , 1978 , Singh and Titli, 1978). There are two principal modes by which the coordinator could intervene into the local problem:

3) Integrated system optimization and parameter estimation, ISOPE. . Later modifications have been made to these methods which extend their utility and reduce some of their disadvantages by incorporating several types of approaches according to their availability (Tatjewski, 1985, Sidaou, et aI. , 1991, Tatjewski, et aI. , 1990, Roberts and Lin, 1991 , BrdiS, et aI. , 1987, Vesely, et al., 1992, etc.).

-action on the objective functions of the subproblems (goal cordination), -interaction prediction among subproblems (model coordination).

In this paper the modified feasible method, MFM is presented. Its main advantage is that the linear algebraic system equations carry out on the coordinator level.

Generally, the steady-state optimising technique can be divided as follows.

The work is organised as follows . First, the problem statement and some preliminary results are given in Section 2. The main results are given in Section 3 and an example in Section 4.

1) Methods that do not use the feedback from real processes, - goal coordination or non-feasible method, - model coordination or feasible method,

I

The research work has been partially supported by th e Slovak Scientific Grant Agency, grant No. 114231/97. 439

2. PROBLEM STATEMENT

Generally, we assume that the mappings Tt (ui' zi) ,

Consider an industrial process consisting of N interconnected subsystems. Each subsystem is represented by the following model (Fig. 1).

Qi (ui' zi) and gi (ui' zi) : Rf1I; xp; are C 2 functions, continuously differentiable and Qi (ui' zi) is convex fori EN . Based on the above, a process optimising control problem may be formulated as follows:

.;, z,

Yi

•

min J(u,z)

Si

subject to z = Cy U;

(5)

y=T(u,z) g(u, z)

5,

0

Fig. 1. Representation of the ith subsystem where ui

E

Rf1I; , Yi

E

RI; ,

zi E

where

RP; and ~i are the

T

= [T Uj , ... ,UN T] E Rm ,

zT

T] E RP, = [T zl , .. . , Z N

YT

I = [T YI , .. ·,Y N T] ER.

U

control input, output (controller set points in many cases), input and disturbances, respectively. It is assumed that external disturbances ~i affecting the ith subsystem are constant over the considered time interval and therefore disturbances can be omitted from the system equations or the system equations described by the mean value of process variables. The input-output steady-state behaviour of the ith subsystem is described by the following functions

are control input, input and output of the complex system. Different coordination methods could be used for fmding out the solution of (5) (Findeisen, 1978, Singh and Titli, 1978, Roberts, 1978, Tatjewski, 1985, and references therein).

(1)

With this optimization problem we can associate the Lagrangian:

The interconnections between the subsystems are identified as the physical connection between the inputs of a subsystem and the outputs of the other subsystems. This can be described by N linear coupling equations

N

L=

I

{Qi(Ui,Zi)+A/[Tt(Ui,Zi)- Y;]+

i=1

N

z· /

= .I . Clj.. y·/ J

(2)

=/

where Ai' 9 i are Lagrange multiplier vectors and po xl

where C!J ER'

J

.

Pi is the Kuhn-Tucker multiplier which have been

is the interconnection matrIx.

introduced to take into account the equalities (1), (2) and inequality (4) as constraints. In the feasible or model coordination method (Singh and Titli, 1978), the coordinator level fixes Yj and provides it for the

Generally, the purpose of process operations optimization is to achieve the optimum performance of the process in the sense that the production process yields the maximum profit for the minimum cost, Findeisen et al. (1978), Roberts and Lin (1991).

lower level. At the subsystem level, the Lagrangian (6) can be written in the following separable form

The global performance index in hierarchical situation is specified as a summation of the performed indices of each individual subsystem.

N

L=ILi i=1

N

J

= I

Qi(ui,zi)

(3)

i= 1

In order to ensure that the process always operates within the region specified by standards and capacity limits it is necessary to impose constraints on the process optimization. Such constraints are usually identified as the local constraints. The local constraints are (4) gi(ui,zi)5,O, iEN

The ith subproblem is min Qi(ui ,zi)

440

(7)

where

N

(8)

zi=LCijYj

iJIjl

iJIjl

iJIj

ail

aipi

ai

iJIj,,

ilJ.[I,

ail

aip,

j=l

for a given Y j' j EN. For fixed Y j' j EN the model constraints (4) and interconnections constraints are always satisfied. The gradient type coordination algorithm for model coordination is obtained as follows Y

(n)

=y

(n-I)

0L

oy

i EN etc.

For the ith subsystem the number of equations and the number of searched parameters are equal to mi + 2(Pi + li) . In the modified feasible method the following procedure is proposed:

(9)

±a-

ER' " xP ,

1. On the local level for the each subsystem the input control variables ui' i EN have to be chosen. The

where a > 0, + for the maximum and - for the minimum.

chosen ui T would be adjusted to the real process and the output variables Yi of all subsystems will have to be measured.

Thus the coordination algorithm will improve the value of the performance index (3) at each iteration and the results could be applied in "real time" since all the constraints are satisfied at each iteration. Generally, in the subsystem level, the nonlinear equations (8) have to be solved. On the gradient coordinator level the calculations are simpler. For the solution of complex optimization problems the designer needs N subsystems with complicated microprocessors, and for the coordinator just a simple one. This approach may not be economical from a practical point of view, which is one of important drawbacks of the proposed coordination method (Findeisen, et aI. , 1978, Singh and Titli, 1978, Roberts, 1978, Tatjewski, 1985).

2. The input variables of

Zi'

i EN are determined on

the coordinator level from the last equation of (10). 3. Substituting the third equation for the second one N

of (10) we obtain N.

LI

j

linear equations in the

j=1 form -A 1' +"C N JI" L., j=l

T

CQj

orj

J

J

1

[a+ . []T a-. ..1.=0 J ' iEN

3. MODIFIED FEASIBLE METHOD

N

= oL = CQi at . 1

at . 1

Lagrangian multipliers

Ai' i EN. The coordinator carries out the solution

for the system of linear algebraic equations (11) with respect to Ai' and the results would be sent to the local level. 4. Local control systems carry out the solution Lu ,

+[iJIj]T A. = 0 at

CQ

=__ + [ 1

ati

T

ilJ.. _I

ati

]

Ai' i EN

(12)

1

1

If for some iEN 1 and ci>O is ILuil
LYi

unknown

j

j=l

The optimal solution of (8) must satisfy the stationary conditions of the Lagrangian

u,

LI

N.

with

Hence, in the feasible method the constraints are satisfied at each iteration. If gi (ui' zi) exists in (7) then its constraints should be included in the cost function (3) as a penalty function, Singh and TitIi (1978). Let us assume that output variables of all subsystems Yi' i E N (I) can be measured.

L

(11)

oL

N

VYI

j=1

= -;:J, . = -A 1' + "C L., J1··

T

= ui k ± aLui '

.

I E

N2

(13)

where N = NI uN 2 , sign +( -) valid for maximum

,9 J. = 0

(10)

(minimum) of L with respect to ui' i EN . Remark I. Substituting Ai' i E N from the third equation of (10) to the second one we can obtain a system of linear algebraic equations with respect to ,9i,iEN .

i EN

441

The first difference of Lyapunov function is given

Remark 2. The most practical situations in a real process: the input and output variables Zi and Yi' i EN can be measured. In this case the second step of the above proposed procedure can be omitted.

!:lV(t)

where

Because of (10), Lq

i EN to be the

where

i=1,2, ... ,N

!:lV(t)

(15)

are the input-output mappings of the real

Thus, it is sufficient that both the respective outputs and the derivatives with respect to the input controls of the model and system should match the solution to the model based problem at the optimum. Convergence of the proposed coordination algorithm is given by the following theorem.

The gradient type coordinator with proposed steps 14 is asymptotically stable with respect to the optimal solution of the system (10) if there exists a positive coefficient a < a * such that the following inequality holds p(!:lu(t), !:lq(t»

a*

for each iteration process,

= -aLu T (t)Lu(t) + p(!:lu(t), !:lq(t»

(20)

In general, a process model may not be a true representation of the real system and this usually results in a non-optimum solution. To overcome this disadvantage an Integrated System Optimization and Parameter Estimation method has been proposed by Roberts (1979). An important feature of the ISOPE method is that it is capable of producing a true optimum in the presence of model reality differences. This is achieved by introducing a modifier into the model based optimization problem so that the error caused by model reality differences is compensated for when the algorithm converges. Using the ISOPE techniques the following Lagrangian forms:

Theorem I .

u >

=0

Summarizing the main properties of the proposed modified feasible method we can conclude that the situation for carrying out the optimal solution is easier than in Findeisen, et al. (1978), Singh and Titli (1978). On the local "coordinator" level the proposed method uses the very simple equation (13). On the above level the control system carries out the solution of the system of linear algebraic equations, which is independent on linearity or non linearity of the mathematical model of each subsystems. The above mentioned properties are the main advantage of the proposed method.

YiR is the output of the real system.

TL

the

Equation (20) implies the sufficient stability condition given by (16) . This completes the proof.

systems,

Lu

is

then using (17) we obtain

Oui

1iR

=0

remaining part of the Taylor series expansion.

solution of the system optimization problem (10) are given by

Oui

p(!:lu(t), !:lq(t»

lim tlu(t),~(t)~O

may be slightly different from those given by (I).

or orR _, IU=U*=_'_lu=u*,

= M(t)

M = Lu T (t)!:lu(t) + Lq T !:lq(t) + p(!:lu(t), !:lq(t» (19)

(14)

Ui'

L(t)

Using the Taylor series expansion of Lagrangian L (7)

Remark 3. If the process has disturbances, thus it is not noise free, the input-output mappings of the ith subsystem (1) for the real process

The sufficient conditions for

= L(t + 1) -

(16)

where p(!:lu(t), !:lq(t» is given later. Proof. Let q be the set of variables given as follows:

q T = [T Z ,Y T,A.T, 9 T] , where A. T = [A. / ' ... , A. NT] , 9 T = [9 / ' ... , 9 NT] The local control systems equation can be written in the form of difference equation (13) (21) u(t + I) = u(t) - aLu

(17)

,

where t = 1,2 ... , n is the discrete sample time .

where Yi (v) ER

We seek the minimum of L by a gradient procedure.

of ith subsystem

i

be the optimal solution, and let the Lyapunov Let function be chosen as (Singh and Titli, 1978) V(t)

= L(t) -

L

/. I

is the measured output variable

9 i ,r i' A. i , 'li are the Lagrangean multiplier vectors

(18) 442

Vi, wi are the vector variables which associate with identification process of process model

ai V

are the unknown parameters of the model process

1

o[

T= [T vI "",vN T]

xQ. + N L &. I .

J= I

I

The necessity conditions for optimal solution of (21) are as follows.

T .9 . = 0 r.I T C·· JI J '

iEN

(23)

Equations (23) are a system of linear algebraic equations with respect to .9 i, i E N which can be discovered on the coordinator level. The solutions of .9 j , i E N have to be sent to the local level. 3. On the local level the vectors

TJ j, A i

are

calculated from (22e) and (22c). The local level i EN and if for some carries out the solution of Lu, I i EN and for a positive small number

L v; =k+[Oy/(V)]TJ I 6V. I

c)

-~[T/]TJ' =O 6V . I

I

d) L W ,

I

I

ILujl <

&i

then stop the operation of those local control systems, else

I

Ui

= Y -~(T/)TJ ' =0 av. I

&i

(k+l)

=ui(k)±aLuj

(22)

I

I

4. STEADY STATE OPTIMIZATION OF CHEMICAL TECHNOLOGY PROCESSES SIMULA TION EXAMPLE

. =~[Q' +~T/CJ.9 · l-~(T:)TJ ' =O & . L.. & .

L GI

e)

I

I

JI

J

I

j=1

I

I

I

Let us consider the following nonlinear three-subsystem example. The subsystem output mappings are

I

t)

LIJ;

=Yi (v)-T(vi,wi , ai)=O

= ajuj 2 +bjUi + TJizi,

Yj

i

= 1,2,3

(25)

N

L(};

g)

= ICij7i

-Zi

with interconnection constraints (2)

=0

j=1

=Yj -1 for i=I,2,3

Zj

h) Lr j = w·I - ul· ' L /I.; , = v I· - uI· = 0

where Yo = cons! > O.

The following procedure for fmding the solution of (12) is proposed.

The local constraints set is

Yj min :::; Yj :::; Yj max for i = 1,2,3

I. On the local level for each subsystem the input control variables The chosen

Vi

and local performance indices are (3)

E RfrI; , i EN have to be chosen.

would be adjusted on the real process

Vi

(26)

I

and the output variables of the ith subsystem Yi (v) (and zi) have to be measured. If

Zj

where aj,bj,TJj , qj,Pj fori=I,2,3 are supposed to

could not be

be known parameters.

measured on the subsystem level, the output variables

With this optimization problem we can associate the Lagrangian (6):

I

Yj (v) have to be sent to coordinator level using the equations (22g) and variables

Zj,

i EN

can be

3

calculated. From (22t) the unknown parameter aj , i EN is calculated. The derivatives of

L

2 2 = "(q L.. I.u I + p I' u·I + k I [Y I' _(a I.u I 2

I

Oyi (v) / 6V j , i EN , could be determined as

In

Pi(Yi - Yd 2 ) +

Roberts and Lin (1991).

o[ Q. +

Oz. I

I

.

J

N L

r.I T CJI·· T .9.J

=1

L .9 j (Yj - Zj+d j=1

2. Substituting (22d) and (22e) to (22b) we obtain

-

+b·u· + I I + TJ 'I z.)] I

j=1

The optimal solution of (27) must satisfy the following conditions:

1-.9. + I

Lu;

=

ilL Ou =2qi Uj +Pj -Aj(2ajuj +bj)=O I

443

(27)

(28) REFERENCES

L Vj = y j LZk

- Z j+1

=0

Brdis, M. , N. Abdullah and P.D. Roberts (1987). Augmented model based double iterative loop techniques for integrated system optimisation and parameter estimation of large-scale industrial processes. Proceedings of 10th lFAC World Congress, 7, 97-102.

j = 1,2

=-Ak7]ki- 9 k-1 =0

k=2,3

The parameters of both models with constraints (25) and performances indices (26) are as follows.

Findeisen, W ., M. Brdis, K. Malinowski, P. Tatjewski and A. Wozniak (1978). On-Line Hierarchical Control for Steady-State Systems. IEEE Trans. on AC, 13, No. 3, 189-208.

Pi -1.4958

19.6062

2

0.2467

13.9693

3

0.2739

-8 .0270

1.1626

6.6759

0.8

4.3525

-3 .8628

0.8

0.0085

14.7900

Mesarovic, M .D., D. Macko and Y.Takahara (1970). Theory of Hierarchical Multi/evel Systems. Academic Press, New York. Roberts, P.D., (1979). An Algorithm for Steady-State System Optimisation and Parameter Estimation, Int. Journal of System Science, 10, 719-734.

and PI = 0.2, P2 = 1.5, P3 = I.

Roberts, P.D. and J. Lin (1991). Potential for Hierarchical Optimising Control in the Process Industry, Proceedings ECC 91 , Grenoble, France, 213-217.

For the first step the following input control variables have been chosen: UI = 1.4, u2 =2.7, u3 =5, Yo =20

On the coordinator level the following algebraic equations have been obtained (11):

Sidaoui, H. , Z.Binder and R. Perret (1991). Tracking Approach in Hierarchical Optimization of Large Scale Systems. Proceedings ECC 91, Grenoble, 2376-2379.

linear

AI-A27]2 +2PI(YI-Ylz)=0

Singh, M.G. and A. Titli (1978). Decomposition, Optimisation and Control. Pergamon Press, Oxford.

A2 - A3 7]3 + 2P2 (Y2 - Y2z) = 0 A3 +2P3(Y3 - Y3z)

=0

Tatjewski, P., (1985). On Hierachical Control of Steady-State System Using IBMF. Large Scale Systems, 1, No. 1.

The optimal solution of (28) is given as follows:

uI = 1.5638,

YI = 46.9863 , Al = 0.69

u2 = 2.4662 ,

Y2 = 73.6084 , A2 = 1.8539

u3 = 5.2266 ,

y 3 = 24.450 I , A3 = -2.9005

and

with

Tatjewski, P., N. Abdullah and P.D. Roberts (1990) . Comparative Study of Development of Integrated Optimisation and Parameter Estimation Algorithms for Hierarchical SteadyState Control.IJC, 51, No. 2, 421-443 .

9 1 =-1.4831 , 9 2 =2.3204

Q=

n

z:

Vesely, V. et al. (1992) . Design of Complex Systems Control. ALF A, Bratislava (in Slovak).

Qi = 133.7053

i = I

5. CONCLUSION The paper presents the main properties of a new modified feasible method . The essential features of the proposed method are both in the procedure for finding the optimal solution for the steady state optimization problems on the coordinator level and on the local one, only the linear algebraic equations are used. In the process of calculation of optimal solution the feedback variables from the real process are needed. The above mentioned properties of the proposed method are important from a practical implementation point of view. A nonlinear threesubsystem example is given.

444