Hierarchical on-line control with applications to regional water distribution and agricultural systems

HIE~RCH~CAL ON-LINE CONTROLPiITR A~P~~~AT~O~STO REOIONAL %ATER DISTRIB~~~~~ AND AGR~C~LT~L Reinisch,

K, , Institute

of

SYSTR%S

Technology

of Ilmenata, P&R’

Abstasc t In the fixst part of the paper hierarchical methods for on-line optlmization and control wi3.l be presented. They are cleseified according to the meaeurtzbility and pred~~t8bility of the main d~~t~ban~ee influencing the optimality of the proceee. In the second part some ~~~~~~&~~~n~ in the field of control of regional water distribution and of greenhouse climate control will be demonstrated. Keywords Rierarchical Control, on-line control, muIti~eveI~ult~l~yer control, water distribution system, ~reenhouse-~I~m~te control.

1. Introduction.

problem

formulation

The appearance of processes of increasing complexity in nearly all fields of human activities connected with demands on high efficiency and limited consumption of energy snd material and on protection of environment require the determination and realization of (nearly) optimal strategies for the development and control of such processes. For elabgrating the control scheme, the type of the control task, the cheratter, meseurabilfty and predictability of the external influences and the availability and temporal (in-Ivariance of a process model have special importance. Three classes of such externel inputs axe of particular interests Zf : ~o~-~requen~~

d~~t~r~~e~* usually of larger ~pI~tude expansion and with approximate predictability

and wider

22: Medium-frequency disturbances often of medium amplitude sion which can be predicted only for a shorter horiaon

and expan-

spatial

23: High-frequency dieturbsnces usually of smaller emplitudee and locally restricted which oennot be followed by the opttimizstion. While 21 should be taken into consideration by d$~ermi~i~g the (lone termjstrategy, the influence of 23 should be suppressed by feedback control. In case of Z2 8 contfnuous or discrete-time correction of the strategy by suitable tactical measures will improve the performance. It

1) Prof, Dr. so. tech& K. Reinisch, Technische Hochschule IlmenRu, Sektion Technische und Biomed.Kybernet~k, DDR-63 Ilmenau, Ehrenberg

is

adviwble

to a%%ign the proceasing

of 21,

22 and 23 to a %trategic

(high), % tactical (medium) and CLbasic (low) control layer, rasp. Thie results in the hZerarchica1 multilavez structure of Fig. f, Clr,~’ the rofig-fmediura-lahort-term intervention intervals) T% h designate

Tm’

CeaeroZ

objectlw%:

eaonomIca1, eaologicel,

tachaologioal, *. l

1 Strategic

Zf --Long-term prediction%

t

Elaboretion (off-line

leyer:

of (user-) optimal stretegla6 to prepsrstory phase) ference

tr%jecturie%

Tl Facticel lsyert Repelitfve correofioa% of %tr%t%g (on-line in operating p&m*)

t Controle

4

T

Eh

Controlled

t

prooere

&r*me 1 iBTlUOoiZO%

Fig.

92

1.

Hierarchicel

multilayer

control

0%

eh

outputs

It

should

be emphasized

the process long-term

that

to the direct strategy,

and tactical

however,

once in the preparatory

the on4ine

control control

can be determined

phase or repeatedly

needs a feedback layers off-line

with large

from

@V.g. 1).

The

usually

only

time intervala.

In many applicationa, it will be advioabll? not to exclude the human In a computer-aided operator during the operating phaee. Thie reeulte decision making by man (operative oontrol, Fig. 1) and allowe to introI duce further information and criteria (multi-criteria problems) In the deoision making. In thie paper, we will preeume that the different original objectives may be reprelented by a scalar mathematical eubstitute criterion the free parameters of which will, on the basis of simulation run8, be choosenaccording to the original objectives. In the case of large-scale proceraee consisting of several interaotiw aubprooeaaea, the elaboration of the strategy in the preparatory phare au well a8 the determination of the tactical operations during the operational phase will favourably be solved by hierarchical multilevel methods. (Theoe are oharacterized by a pyramidal structure with a coordinator at the top of the pyramid,) In this way, the on-line optimization and control of large.0scale proceaaee will result in a hierarchical multilevelwmultilarer system fsee below Pig. 3). On the basis of the characteristics dfecurraed above, groups of problems can be classified of which the following are, in general, of a p8rtfcuLar

Pit

fnterestt

Continuous on-line tracking of the optimum state of a dynamic aystem in oaae of not predicted disturbances. The actual main diaturbancea 21 and 22 should be a) measurable, b) not directly meaeurable.

IQ: Optimum on-line tely

predictable

control inputs

of a dynamic process

in c&Be of approxima-

2t and 22.

P3: Transition processes along a trajeotory optimally defined on the be&s of known 21, with 22 being neglectable and Z3 being suppressed. In case of P3, the strategy is determined off-line and not corrected on-line. Therefwe, this problem is not considered here. For a qualiof a procees fied eolution of problems P3 and P2, the availability model must be preeupposed. In the following chapter@, we will at firet dfacuee methods for solving problem P1 and then problem P2. The last two chaptera are concerned with applications. The methods and applicationtr presented have been developed and inveetigated and realized, reap., by the author’s

93

research

group.

2. Multilayer control In ca8e of mearurable available process model

main disturbances

Suppose that the original objectives may be substituted wing quadratic criteriona t+T min I = ‘I(r)dr 1 T very large, finite i J 1 U

and

by the follo-

(2.la)

t

-

y,)To(yk)

The process should observable) r i(z)

= Ax(r)

+

.-

yd)

+

&u@)

be a disturbed

Buk)

+ Be(z),

linear

”

u,)TRh(r)

-

(2.lb)

u,)

one (controllable

and

Y(T) - Cx(V)

(2.2)

the y(r), x(r), u(c), z(t’) are the vectors of the outputs, the states, controls and the disturbances reap. (all being elements of finite and COntrOle dimensional vector spaces); yd, ud are desired OUtpUt8 semidef.) and A, B, C, E are matrices of reap. ; R(pos. def. 1, Q (pee. -1 shall exist (this may be assured by corresponding dimensions! A special

measures).

Supposing that the main disturbances can be measured, but cannot or will not be predicted. Then, the assumption that the disturbance z measured at the time t remains unchanged will be reasonable. At the time t + dt, optimization will be repeated ‘with an eventually modified disturbance z (t+dt), but with the same assumption. This approach corresponds to problem (2.1), (2.2) with z(t) instead of z(T’). Under theee conditions, the optimal centralized control (using full information on process and disturbance model and state) can be decomposed into the following twolayer structure, Hozietulski /5/r Superordinate optimization layerr depending on disturbance z:

with

the static

the stationary

process y,

eouation

Static

(from

(2.2)

setting

optimization

g(r)=

0; 7, c are

u>:

z(t)

= P,;(t)

+ Pzz(t),

Pu = -A”B,

y(t)

= s,uw

+ Szz(t),

5, =

94

feedforward

CP,,

P, = -A-'E sz = CP,

(2.4)

From the stdltianarity follow

oonditione

the staticmwthm

far

the appertaining

Lagrangian,

there

ixmtrolt

f2.5) and the etlstic-optimum = X$(s(W

%‘tW)

state

+ fzz(t)

(2.6)

Subordfnete dirslot ooatrol.lavert Optimum tranoition into the instantaneous static-optimum state. Using the reeulta of optimum etate-feedback control one gets (M Riccati-Matrlx)

Therefore, aided

the multilager

control

u*(t)

Ffg.

2. Two-layer

u?t,

= &zw stet~~~~ti~ feedforward

control

acoording

control:

control

to

static

hw

Fig.

reetite

in 8 X#tatiC=-ODtimdlg

2;

opttially

aided

control

- KXfXW - %CW>J optimum feedback of the state deviation (suppreseing Z3)

(2.8)

Thie control is optimal. under the presuppoaitione cited above {whloh correspond to disturbances of atepfunction type). In contrast to P central&ted solution it can heuristlcslly be extended to nonlinear cpt~i~at~on at the upper dynamio processes. In this case, the atetic layer must be effected by a search procedure of nonlinear programmfng performed on a static prooees model.

Investigations

processes

of nonlinear

of Rakmersteln-type

by means of

computer simulations have confirmed the geod qualitier of the multllayer algorithm and its euperlorlty to centralloed solutions (with linearieation in the actual state) If the state feedback K,e,(t) ia optimally determined for the process being linearlsed in the actual static-optimum state, Kurplnaki /6/, K, Is purposefully computed in the preparatory phase for olaeees of dieturbanoe raluea to be expeated and stored up for being used in the operating phase.

3. Multilayer oontrol in case using a process model

of non-measurable

main dirturbaneee

If the main disturbances 21 and 22 are not directly measurable, then the solution of the control problem may be reduoed to the above one by conetructing a dieturbanoe observer or by current/repetitive adaptation of a process model which parametrically depends on the actual dlrturbance. observation of dleturbanoee and optlmlIn oase of linear proaesses, zation can be combined In a favourable way and realized by feedback of the error between the output yR and its reference value yRd of a static process observer which are defined a8 follows (S,, Q, R, see (2.3)(2.4)h y,(t)

= s$y(t)

+

Ru(t)i

YBd

Using (2.4), one oan showr If disturbance) converges to Its u(t) converges to the desired

4. Rultllevel-multilayer

i~(Ui,Y$(Up+5i))

i vi = My (M coupling

96

direotly

on the process

and dlsturbanaes, one can only search for stationary state (corresponding to the are preeuppoeed to vary slowly) directly on thie is favourably done by using eyutems, this structure of Fig. 3. For eimplicity, here only with regard to the stat lonary

that the global static local ones (Tilt

%u,y,d = r

(3.1)

the observer output yR (depending on the referenoe value ygd, then the oontrol etatlo-optimum value “u(s(t)) /5/.

control

Having no model of proeese the Instantaneous optimum actual disturbances which the process. For large-scale the multilevel-multllayer problem will be discussed behaviour. Presuppose posed into

+Y, + Rud

=

T Equ. (2.1)

criterion

= r

iibi,Y,Ci)

i matrix);

ui = (uoi,

uyi)

may be deoom-

(4.1)

r;lOptimiferOi

I L-dud1

1

i

1

1 Optimifzer 02 b

. Ydr1

0%

Basic control layer (Decentralized co$roll

Fig. 3. Multilevel-multilayer control directly on the process Yi’ zi denote the controls, outputs and disturbance8 of the i-th subsystem. Uelng the direct coordination method (Findeisen /3/J, the outputs yi are aerving as coordinating variables. They are set by the coordinator, the desired values ydi being followed and maintained by with dim u = dimydi= dim yi. The means of the oontrola u Yi Yi remaining controls uoi, with dim uoi = dim ui - dim yi > 0, are used by the optimizer Oi for optimizing the local criterion yi under the Ui’

condition Y = yda L! ;:" Ii(Ui' YdC zi) - Ii(Yd, zi)

(4.2)

This is realized by means of the references udi and the feedback denote the deviationa). The tack of the cocontrollers Ri l(e Oi' eYi the sum of the coAditioMlly optimum ordinator consists in opt iI@ziAg local objective functions Ii* min{?(yd,z) Yd

fi(Yd'"i)} - Z*(z)

= z

(4.3)

i

Coordinator and controllers are acting alternately.

5. Yultilevel-multilayer control by repetitive optimization based on undated ztate8 and WediCtiOm In case of approximately predictable environmental inputo, a repetitive correction of the etrategy according to Pig. 1 i8 advantageous. For

97

disorete-time I&problems

with inaative inequality constraints, a +mp-

le control algorithm with fixed Initial and final values and finite horisan may be derived by advancing the three-level method of Tamura /ll/ in the following way (ThUmmler, Relniech /12/ /g/)1 The state and output equations of aubproeesa i should read xi(k+l) = A;xi(k) + Bjui(k) + Cbi(k) yi(k) E D;xi(k), i = l(l)H,

+ ei(k)

(5.1)

k = O(l)K-1

and the coupling relation between input vi and outputs y vi(k) =

j

z? '~j yj(k) = M~jDjxj(_k) .SL j-l(fi) j=l(+i)

(5.2)

The global objective functional I shouldbe decomposable N 1(X,&V)

= z

(5.3a)

li(xi*ui#vi) 1

K Ii( ) = $ z{(lxi(k) k-0

- xdi(k)i2 + ((ui(k)-udi(k)/i2+ I[vi(k)'vdi(k)12} RI Qi gI. (5.3b)

with Xdi' Udit Vdi being reference trajectories (from strategic layer) and Cl!, Ri, Si diagonal weighting matrices, K horizon. Introducing vectors which deecribe the whole trajectory of a subsystem for fixed initial and final value6 x.

1

vi

(K-l))T,

E

(x,(1

e

(vi(0),...,vi(K-lUT

>

****,Xi

ui

-

(u,(O),...,U,(K-~))~,

(5.4)

equations (5.1) . .. (5.3) may be contracted to Ai~i + Bi~i

+ Civi + hi * Oi vi -

)

MijDjxj - ci = O

(5.5)

j=l(fi)

1i(x~#ui9vi)

5

4

+

lj"i'-udii/2 + +di(i} Ri

Combining once again the vectors of the subsystems to those deeorlbing the whole system spatially and temporally: x = (x1 ,...,

xN)

T

, u -

(u,, .

..)

UN)

T )

v - (v

,,...(

VN, T

(5.7)

Equs. (5.5) for i = l(l)N may be combined to Ax+Bu+Cv+h=O,v-RDx-c=O

(5.8)

Criterion

with

(5.3e),

a static OP which

and of

fi I(x,u,v)

the subproblems N

) = I@pp

Lit

substituted

is equivalent

solved by determining problem L(x,u,v,y,h)

(5.3b)

vi )

by (5.6),

to the original

the saddlepoints

of

and (5.8)

dynamical

the Lagrenglans

one. of

+ ~T[Ax+Bu~Cv+h) + ~T(v-c-MRx) with given

describe It

is

the whole

(5.9)

x = 3,

+$(Aixi + Biui

+

civi

+ hi>

(5.10)

N +

XpyQIl - > j=l (fi)

AT j”j iDPi

The results are optimum prices pi) of the subsystems i = 1(1 )Ns

im

V;‘(PTF’

\

-

end $($I

and optimum controls

(Axd + Bud + Cvd + h) + c + MDXd)

-(P%% + S-l + fifDQ-'(f~%D)~; P Y

vr= -(AQ~‘A~

+ BR”‘R~ + cs”‘cT),

n

ci($,

(5.11)

AQ-‘(m)T - cs-’

v = diag vri Y

The desired feedback controller followe l'romEqu. (5.13) with the simplification that only the first subvector ui(O) of ui is calculated and implemented on the process. In the following time, calculation is repeated. In this way, the current ui(k) reeulte from the actual ~~(0). except for h, c, In Equs. (5.11) . . . (5.13) all parameters (matrloea) hi9 ci may be determined only once in the preparatory phaee. Only hi = hi(xifk), Xi(k+K), ti(k)

l

** Zi(k+K-lf)i ci P c~(Xi(k),xi(k~))

(5.14) the wanted final values and the predicdepending on the actual states, ted disturbancea have to be updated on-line at the discrete timen k. This results in the three-level algorithm of Pig. 4$ which can be implemented on a distributed microoomputer syetem.

99

3dlevel

f h - Cocrdinatcr 1

.zNbK

&k)

;

-1

pdi& -r-

) 2d levet

1x,(k)

lSt level

11 I

(W

i.+(k) I,

I

1

vl(kf

c Subprocess pl (state xl 1 z,(k) w

Process level

j

Fig.

4.

Three-level

discrete-time

and predicted The h-coordinator processes of

the

via

balances prices

preceding

states

of

the

feedback

control

with

x,

the

interactions

the /i-coordinator

time-intervals folIowing

between

of

intervals

the

cares for

subsystem

the

i equalling

by means of

prices

Subprocesses transfer

Pi:

values

transfer

and predict

si[k,

CaIculate

(6)

At time

ri

In case

of

calculated values. tors

i(k)

put

minor

transfer

to A -coordinator o,(k)...cK(k)),

di(k)

k := k+l,

violations

of

= /+(!(k)),

= ui($(k)

go back existing

to

transfer V l,(k)

),

of

and controls

coordinate

to

be iteratively

(optimixers)

in turn

apply to Pi

restrictions,

ci* s may be (suboptimally) serious violations of constraints, have

to Ci

(1).

unconstrained

search.

The algorithm

K+K] I

E X(hl(k)...hK(k),

,$%(k)

Calculate

and ~:ontrolIess

strained)

100

Ci:

In c6se

prices

hi(k),

Calculate

k+l,

states initial

-coordinators

(4) pr-coordinators: Controller

xi(k)

Calculate

to

(5)

Measure

final

sub-

to ,,+,-coordinator

f2) /Qi -coordinators: (3) h-coordinator:

horlson

different the

pi*

reeds: (1)

finite

disturbances

determined using

the

reduced to feasible the optimum by the

procedures

of

ccordina(ccn-

6. Application control

of

hierarchical

of a regional

The method presented

methods to etructural

water distribution In Sect.

expansion

and

network

5 has been applied

to the structural

ex-

pansion and control of a large-ecale regional water eupply system where the water reaches the consumers almost exclusively in free fall via a pipe-line network including Borne 100 reservoirs (elevated tanks), Ropfgarten /4/, Reiniech /8/. The input is from a large hydrodam reaervoir and from local eupplies (deep wells, springs). The mean hourly demands of the different municipal, industrial and agricultural consumer8 have a periodicity of one day. As the water transport ie effected by gravity and controlled by means of valve6 at the inputs of the tanks and of eubordinate feedback controllers, the inflows of the tanks will be regarded aa controls of the euperordinate layer (Ci in Fig. 4). This proceee can be described by linear difference equation (5.1) with conetant constraints of coordinatesi x,_(k) states (content8 of reeervoirs) at time k; ui(k) controlled inflowe; vi(k) inflow6 from other subeystems into subeystem 2 and zi(.k) consumption8 and local supplies d subayatem i, all in interval [k, k+l). Regarding responsibilities of different authorities and principles of control and data transmission, the system haa been decomposed into 7 subsystems. The structure of subey@tem 3 (again decompoeed in 4 lower subsystems)’ is shown by Fig.5.

-pipeline Fig.

5. Subsystem of

a--deep

the regional

well wpring

water eupply

+c consumer

syetem investigated

101

The objectives

are the fulfilment

of

the demand8 expeoted

for the next

two decades, the avoidance of overflow8 in the tanks and the smoothing of the water flow, especially in the water treatment plant at the input from the hydrodam reservoir (for stabilizing water quefity). This should be realized with minimal investmente for tanks and pipes. For solving the structural Wcpanaion problem, the substitute control from the preeent structure problem (5.3) has been solved - starting for different structural variants (sites and capacities of tanks and pipe-lines) which seemed to be candidates for an optimum solution (their number is limited by the topography of the area) with the following specifications being made: Control interval fk,k+l) = 1 hour; optimization horizon K = 24 houre from 4a, m to 4 a,m, with x(O) = of the mean hourly X(24) = 60 % xmax (because of one-day periodioity consumption8 and low fluctuations during the night). The zi(k), maximum hourly consumption@ in the k = 0, ... . 23, are the predicted years e1985, 1990 and 2000. The references xdi(k) have been for the moat udi(k) and vdi(k) the part Set xdi = 60 % Tax = conat. Aa references daily

mean values

iii and vi ,re8p.l have been taken.

Some of these parameters as well. as the weighting matricee Qi, Ri, Si have been chosen after Borne simulation run8 according to the original criteria. The results of the structural varianta have been compared. This comparison has been repeated for the years 1985* 1990 and 2000, As result of the expansion strategy derived, Borne expenefve investmenta which had already been planned could be saved. The Fontrol problem has been solved using the results (5. $1). . . (5.13) and the algorithm cited there-after with the following specifications: The reference trajectories Xdi(k), udi(k), vdi(k.) are determined on the strategic layer (Fig. 1) once in the preparatory phase by solving the (5.3) with the zi(k) being the mean consumptlons in the OP (5.1)... different hours of the day. For the repetitive optimization on the tactical layer, a control interval and a repetition period of 1 hour (2 hours) have been chosen. The zi(k) are the real or simulated consumptione of the different users. The optimization horizon has been taken a) as shortening horizon hk, K], with K being 4 a.m. and b) aa shifted horizon [k, k+K], with K being 24 hours. The final values are fixed on the reference trajectories. In this way, the long-term experiences (by the reference trajectorims) as well as the aotual information (actual states and ahort-term demand predictions) are taken into consideration. By controlling all tanks, the change8 in the water treatment plant could be reduced to lese than 3 % of the past valuea. A decentralized microcomputer system, consisting of looal unite - fulfilling the tasks of a

102

pi-coordinator

and a Ci-controlleririg. 4) and of B subordinate

follow-

up controller - and superordinate X-ooordinatora for the different systems (and for the whole system) is under construction 121.

sub-

7. Multilayer

in

optimiration

and control

of

the plant

growth factors

a greenhouse Hierarchical methods have also been applied to greenhouse climate eontrol for maximizing the yield of cucumbers with special regard to the earliness of the harvest and to the limitation of the energy consumpth The strategy h8s been determined by optimization on the basis of a model of the growing cucumber, A plant-phvsioloaical model. has been developed for that purpose by describing and combining a large number of physiological subproceasea, a8 photoayntheais, reepi.ration,oeeimilation, distrlbution, growth and adaptation, by llugustin /I/ from the hcademy of Agricultural Sciences. This has been simulated on a computer and approximated (by identification) by a simplified nonlinear long-term inputoutput model of Hammerstein and Wiener type, reap., by Schmidt /~/,/ICI/, The linear dynamic part is being described by a difference equation and the nonlinear static part by polynomiala of jd degree with bilinear interactiona, all parameters being time-variable. The controls are the C02-concentration u,(k), the night-time temperature u2(k) (during darknesa) and the day-time basic temperature IQ(~) (the real temperature result8 from the additional influence exerted by radiation), thg output is the cumulative yield Ax(k), all variables holding in a 9-daya interval (k-l ,k]. For simplicity, the radiation has been apprpxlmated by mean trajectories depending on the day-time and the date. For determining the long-term strategy, the profits from the yields Cy(k, Ax(u,k)) = 1, (k)Ax(k) minus the costs for heating Ch(k,u) = PH. f2fu2(k),

u3(k) T and for

been optimized __ K T(u) Ir;[Cy(k,u) k=-Q

CC2-input

CCC2(k,u)

- CCC (k,u) - C#w~ 2

= PCC2 f,(u,fk))

' min =

have

(7.1)

U

the latter being higher for early yields. The *HI Py are prices, energy demand f2 for wanted temperature6 u2,u3 is determined by pi speradiation and wind as uncontrolled cial model with outside temperature, inputs, @ time period before harvesting, K time of last harvest.

‘CO$

The dynamic OF has been solved by transforming it into an equivalent static one and by determining the global saddle-point of the appertaining Lagrangian. Because of the nonlinearities, an augmented Lagrangian haa been taken, Schmidt /lo/. By using the input-output model, the computing time could be reduced to < 1 %, the main aforrge to

103

model (which ie unsuitable fm 5 $b of that needed for the physiological optimization). When larger deviations from the computed trajectorlee occur at time ki (86 a result of longer lasting deviation6 in the climate from the mean ourves), the etrategy will be updated by repetitive optimization with shortening horizon [ki,K] and initial states x(ki), the final states x(K) being free. The strategy will be transferred to a subordinate tactical laver, where on the basis of short-term models and the actual outside climate (radiation, temperature) tactical meaauree may be taken in shorter time intervals. The final reference control trajectories are realized by the lowest, the direct feedback control layer. Special control algorithms using short-term predictions of the outer climate have been implemented on microprocessorbased controllers for that task, Mahlendorf et.al./?/. First tests in the greenhouse have confirmed the results from the simulation rune which show yield improvements of about 20 $ as against the situation at the beginning of these investigations,

Literetur A model of yield formation as bssis for optimal control of growth factors in the culture of greenhouse cucumber, 1st Intern. Symp. on System Analysis and Simulation, Berlin 1980 Bonitz, F.: Implementation of a multilevel-multilayer algorithm on /2/ 8 distributed microprocessor system for controlling a regional water supply network. 7th IFAC-Conference on Digital Computer Applications to Frocess Control, Vienna 1985 F. N.; Brdys, M.1 Malinowski, K.; Tetjewaki, 131 Findeisen, W.; Bailey, P.; Wozniak, A.: Control and Coordination in Hierarchical Systems, r).Wiley, New York-Toronto 1980 /4/ Hopfgarten, H.; Puts, H.; Reiniach, K.8 ThUmmler, Chr.: Application of hierarchical methods to the structural expansion and control of a regional water distribution network. IFAC/IFORS Symposium on LargeScale Systems, Warsaw 1983 151 Kozietulski, M.; Kurpinski, K.; Liniger, W.; Reinisch, K.: On-line multilevel-multilayer control of stationary and transient behaviour of disturbed dynamic processes. VIII. IFAC World Congress, Kyoto 1981 161 Kurpinski, K,; Jlihrig,U.8 Linfger, W.; Reiniach, K.r Multilayer algorithms for on-line control of nonlinear disturbed dynamic processes and their implementetion on a multimicroprocessor system, IFAC/IFORS Symposium on Large Scale Systems, Wereew 1983 R.; Engztann,U.; Diezemann, M.; Laudant U.; Schmeil, H,: /?/ ivlahlendorf, Control strategy for a microcomputer-controlled greenhouse. IX. IFACWorld Congress, Budapest 1984 /81 Reinlsch, K.; ThQmmler, Chr.; Hopfgarten, S.: Hierarchical on-line control algorithm for repetitive optimization with predicted environ ment and its application to water management problems. Syst. Anal. BBodel.Simul. lfl984)4, 263-280, Akademie-Verlag Berlin /9/ Schmidt, BB.;Puta, H.5 Reinisch, K.: Contribution to the optimization of the plant growth factors in the greenhouse end the open land. IX.IFAC World Congress, Budapest 1984 /lO/Schmidt, M.: Modelbuilding and optimization of the plant growth process of the greenhouse cucumber. 2nd Internat.Symp.on Syst. Anal.

111 Augustin, P.; Schmidt, 1.:

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