- Email: [email protected]
Experimental Investigation of Selftuning Control of a Gas Phase Fixed Bed Catalytic Reactor with Multiple Inputs
Copyright © IFA C Control Science a nd T echnology (8th T riennial World Congress) Kyo to. Japa n. 198 1
EXPERIMENTAL INVESTIGATION OF SELFTUNING CONTROL OF A GAS PHASE FIXED BED CATALYTIC REACTOR WITH MULTIPLE INPUTS L. Hallager and S. Bay Jorgensen Department of Chemical Engineering, Technical University of Denmark, Building 229, 2800 Lyngby, Denmark
Abstract . An ap plication of a multiinput - mul tioutput (MIMO) self tuning regu lator to a fix ed - bed reactor is investigated exp e rimentally . The regulator consists of a linear MIMO time serie s model , which is ident ifi ed using a r e cursive l east s quar es method , combined wit h a lin e ar quadrati c optimal control strategy modified to permit the on - l ine so lution of the Ri cc ati e quati on . An appropriate model st ructure is selecte d a pr io ri u s ing qualita t i v e kn owl edge about t h e proc ess dynamic s . The self tunin g regula to r is tested experi men tal ly using s ing le and mul t iple upstream disturban ces . Th e r es ulting self tunin g regulato r perfo rms ve ry satisfactorily . The iden tifie d models confirm the a priori assumptio n s and t h e control performance is good . Keyw o rd s . Adaptive co ntrol ; optima l co n trol ; che mical r eact or cont rol; structured models ; distributed paramete r s yst ems ; ti me lag systems ; parameter estimation .
INTRODUCTION Dis tri buted syste ms with convective transport ar e widespread among c he mical unit operations and processes e . g . distillation co lumn s , tu bu l ar heat exchangers and chemical rea cto r s . An impo rtan t dynamical feature of the se systems i s that they are non - minimum phase , with the p oss ibili ty of e xhibi ting " wrong - way " be ha viour i . e . the immediate output- re spons e to an inpu t di stur ban ce i s o pposite to what is expe cte d from t h e stationary response . During the last few yea r s it has been d emon s trated experimentally that these syste ms often can b e satisfa ctorily cont r o lled by a pr ocess - co mp uter usin g v ariou s multi v ariable controller design techniques based up on o ff line i d entified mathematical models (Cleme nt , Jergensen and Se ren sen , 1980 ; Jutan and col leagues , 1977 ; Si l va , Wallman and Foss , 1979 ; Sere n sen , 19 77 ; Serensen , Jergensen and Cl ement , 1980 ; Tremblay and Wright , 1977 ; Wallman , Silva and Foss , 19 79) . In practice , however , i t i s ti me consuming to de ve l o p a mathematical mo del fr om basic physics and chemistry u s i ng conservation principles. Furth e rmor e it is often difficult and costly to ident i fy such a mod el , par tly because est i ma tes o f some paramet e r s may b e diffi cult to obtain from e xperim ents , and partly because the mode l often will be non - li ne ar in the parameter s . Finally the p arameters in chemical processes are usually t im e v arying , f or instance due to deactivation of cat alyst s in reacto rs o r depositions i n heat - e xchangers , thus necessitating more o r l ess regular up da ting o f parameters .
Therefore it may be much more e xp e di ent to use a selfadjusting appr oac h. The structure an d ord e r of a compar atively simple , linear mathematica l mod e l i s postulated and then a self tuning controlle r (see e . g . Astrem and Witt enmar k , 1973 ; Astrem , 1980 ; Borison , 1979) is used to est ima te t he parameters in t hi s model , and S imul ta neously cont r o l t h e process . Note that the postulated model c an be viewed as a reduced o rder lin ea riza t i o n of t h e " true " system model , th us t h e paramete rs in the lin ear mo d el d epen ds on t h e o p erating poin t as well a s on the physical paramete r s . The use o f a simplifi ed approach in a contro l cont e xt i s adequate in many cases . Astrem (1980) ha s si mulat ed t he appli catio n of a s elf tu n ing controlle r using pole - placement design to a fixed bed chemical reactor system s imi lar to the o ne investigated here . He uses single l oop cont r o l an d conclude s t hat the cont roller mode l can be conside rabl y simpler than t he system , st i ll achieving good control The contro l pr o blems in the chemical ind u st ry are usually r eg ula to ry i . e . h old ing a fixed ope r ating point in fa ce of di sturbances . A locally lineariz e d model has pr ove d to g i ve a reasonable description o f the fix ed - bed reac tor dynamic s ( Hansen and Jergensen , 1976) . Consequently i t is f easibl e to use a model which i s linear in the states and apply well known linear multi variabl e control t h eory. Thi s s implifi cat ion ~ ea d s to a mo del wh ich is li near in the unknown coe fficient s , thereby resul t ing in a linear ident if ication problem . However , a linear model is lik ely to b e ov erparam ete rized compared to a non - line ar mod e l
27 43
2744
L. Hallager a nd S. Bay
(especially when distributed processes are discretized spatially) . ;0 minimize this problem it is essential to use available qua litati ve a priori kn owledge in structuring the model . This "almost black-box " approach suffers from the disadvantage that it i s no t possible to estimate unmeasured quantities ( because quan titative a priori kn owled g e is not assumed available) , thus only oucput -feedbac k is p os sible and unmeasured quant i ties can not enter the control criterion .
e quidi scant temperature measurements along the reactor axis , composition measurements at the inl et and at one of 5 axially equ idistan t points through the re actor and a fl ow rate measurement . In industrial practice only a limited number of measurements are availab l e (2 - 5 temperature measurements and possibly a conce ntration measurement) . Hence t he te mpera ture profile is known on l y to a l i mited ex tent . The disturban ces are se lec te d by the operator and may be stochastic o r d ete rm inis tic. Th e purp ose of control may be d efi ned as : a.
to maintain all or part of the temperature profile , to reduce the thermal lo ad o n the bed .
b.
to maintain the exit cuncentration as a measure of product quality .
c.
a mixture of a and b.
THE PILOT PLANT The pilot plant reactor consists of a tubular fixed bed of catalyst partic le s , designed to be essent ially adiabatic and with negligible heat capacity of the reactor wall . It is de scribed in detail by Hansen and J0rgensen (1976) . In this ser i es of e xp eriments a stream of hydrogen containing app . 0 .2 mole% oxygen is fed to the packed cataly'st bed at a temper ature of approximately 82 °C . The adiabatic temperature rise during water formation is 169 °c pr . mole % reacted oxygen . Qualitative Dynamical Description In the gas phase fixed bed catalytic reactor , investigat ed here , the he at capacity of the gas is negligibl e compared to the heat capa city of the catalyst pellets , thus the t h ermal residence t ime is several h undred times great er than t h e gas residence time . Furthermore it i s known (Hanse n, 1973) that the mass transport processes are s ufficiently fast to make quasi stationary descriptions of the mass - balances r easonable . Finally there is almost thermal e q uilibrium between the catalyst pellets and the gas and within the pellets . A reasonable simplified mathematical model of the reactor therefore i s a pseudo homogeneous model consist in g of one dyn amic non - li near par tial differential e quati o n f o r the temperature and one quasistationary differential e q uation for the oxygen balance . Both including dispersive terms . The two important dynamical features of the reactor system are thus a.
b.
A fast propagation of concentration and flow rate changes . The mass balan ce is a quasi stationary functional of t he inlet cond itions and the temperature profile . A s low propagation of temperatu re changes. The thermal wa ve passes th r o ugh the reac to r in 11-25 minutes dependent on the flow rate .
The Control Problem The cont r ol problem conside r e d in th i s paper is control of the chemical reactor (see Fig . 1) . The plant has three inputs - temperature , composition and flow rate of the entering gas - in which di sturbances can enter and control actions be taken . The output co n s ist s of 11
J~rgensen
TlIEORY The problem of designing a r egulator for the chemical r eacto r sys t em without knowing the model parameters is approached by using the separation t heorem although it is not str i ctly valid . The design problem is thus divided in to two parts - one of identifying a suitable mode l and o n e of determining a control strategy . The ce rtain ty e q uivalence principle is furthermore used as an ad hoc d esign principle i . e . the control p art of the problem is han dled as a d e t erm in istic problems not taking the uncertainly of the parameters in to consideration . Identification It is assumed that th e input- output data from the chem i cal r eactor can b e de s cribed by a discrete - time multivariable time - se r ies model: x(t)=A(q
- 1
)x(t - l)+B(q
- 1
)u(t - d)+e(t)+c
wh e r e : x ( t)
(dim n) is a measurement vector
u(t)
(d im m) is an input \·ec tor
e( t)
(dim n) is a zero means , white noise sequence
c
( d im n) is a constant v ector i s the delay from input to output
d
A( q
A( q
B(q
- 1
-
1
) and B( q- ) are matrix polynomials of order a and b in t h e backward shi ft operator q-l i . e . 1 - 1 -a ) = AO + q Al + . . . + q Aa 1 -b )=B + + q Bb O
The dimensions (n , a , m, b) and the delay d are assumed known and the mat r ices A. , i = 0 , ... , a and B ., i = 0 , ... , b are assu~ed un known but co~stant or slowly varying .
Experime nta l
T
r--+<
,~
l'pstreMl { d [email protected]
l
2745
Investigation of Se l f - Tun i ng Control
0
!
C
c
~.
F
lC
.
,.
lC
X
lC
0
0
I(
lC
lC
0
0
J
I TC
Fc
Cs
l -- v----J
F L _
Co ___ _ __
Possible control signals
_
_ _~ _
Possible
x. (t)
1
temperature
o
concentrati on
sys~em
The loss function . which is minimize d by this algorithm is t
1
v . (t) 1
v,here :
T
T
T
{x ( l - l) . x ( t - 2) •.. . • x (t - a - 1) .
T
_ _ __ _---'
:'10. . ' rate
+ e. (t)
i
_
~easure~en ts :
Sketch of the reactor
The i'th state in equation (1) can be written :
_
I(
[!]
Fig . 1.
_
~
~
U
t- j
j =0
The algorit hm is starte d with initial values 1
0.
1
(superscript. i means the i ' th row o f the ma t. rix) The parameter ve c tor 0 i is estimated using a recurci v e least s quares algorithm with expo nential forgetting :
E.
( t+l)
J
1
thereby introducing exponential weighting of old data .
P ,( O) = a ' I
and
1
2( .)
E.
T
u (t - d) •. . .• u (t - d - b) . lJ
0. \ l h )
!---
0.
1
+ K. (t+llc (t+l) 1
1
x. ( t+l) - cpT(t+l)0. (t)
e1 (0)
= 0
where a is chosen sufficiently large to indi cate the lack of initial k n owledge of the parameters . I n general all paramete r s are assumed comple tely unk n own h ence it is reasonable to initi ate all covariance matrices at the same value and thus reduce computations considerably as it o nly ~il] be necessary to c alculate one estimator gain and one covariance matrix in each step (Bor i son . 1979) .
P . ( t+ll is t he n o rmalized cov a rian ce ma 1 trix of the estimate error l0 - 0 ( t +I ))
In the present case however extensive use is made of structural zeros . i . e . on the basis of a priori knowledge some parameters are set to zero . This may lead to differences be tween the different
Ki(l+l) is the corresponding gain vector
Structure selection
1
K. (t +l) 1
1
1
1
P . { t)CP. ( t+1) · 1
1
[cpT(l+l)P.(t+llCP.(t+l) + 1] - 1 1
I' . ( t + 1)
1
1
l/u[I - K. (t +1 ) ]cpT(t+llp (t)
1
1
1
1
\', here :
1
1
is an exponential forgetting fac tor I
is the unit matrix of appropriate dimension
The size of the model is defined when n . a . m. band d are fixed and it is decided which measurements and controls are to be used . If the model is no t further structured it would be necessary to estimate n(n ' (a+1) + m(b+l) +1) parameters ( including constant terms) . In the present case the number of parameters is reduced using the qualitative process
L. Hallager and S. Bay
2746 knowledge describe d earlier : a.
b.
The distributed disturbances (concentration and r l ow rate) inrl uences a ll meas ureme n ts with a delay or 1 sampling peri or (due to the sampling or the system) . The thermal wave as observed by a number or temperature measurements is described essentially as a delay rrom the temperature input to the rirst measurement and rrom one measurement to the next. This necessitates a proper selection or sampling time. The simplest approach is to select equi distant measurements, a sampling time equal to (or slightly larger than) the delay between two neighbouring measurements and set a=O , to describe convective propagat i on . A more e l aborat e approach is to select a s h orter sampling time but use additional old measurement ( i. e . a>O) , thus including a description or dispersive errects . Thi3 descrip t ion adds rlexibility in the selection of sampling time and allows ror larger rlow rate disturbances .
In this way the dominant thermal dynamics or the reactor and the inrluence or the inputs can be modelled with a reasonable number or parameters . Control Chemical processes are commonly subject to dr i rt. This is reflected in our model in the estimation or a constant term . It thererore seems reasonable to use some kind or integral action in the controller , which also eliminates orfset in case of constant disturbances . Consequently the identified model is rerormu lated to the rol l owing st ate space f orm (ror simplicity , d is assumed to be 1).
J~rgensen
and S(k) i s calculated by solving the Riccati equat i on : S(k) = 01 + A* T ' S(k+l) ' (A*-B* ' L(k)) sub ject to the condition: SIN) = 00 ' In the general selrtuning case this solution is not strictly usable because the system matrices A* and B* are not constant and their ruture variation is not known as the parame te r s are estimated on - line . Ir N is chosen as I , i . e. a one - step criterion is employed , the solution is applicab l e. In the p r esent case , howe ve r , the re is known to be a therma l time delay rrom inlet to outlet , which usual ly is 5- 10 times t he sampling time. It seems reasonable that t he control criterion should at least cover this delay in order to account ror the primary convective efrects or a given control input. Consequently the rollowing st rat egy is at tempted : N i s se t to inrinity indicating that the stationary solution is searched ror , the Riccati - equation is iterated (the ' wrong ' way) once at each sampling instant (sta r ti ng rrom the last solution) and the newe st value or S(k) is used in the calculation or the f ee dback gains . When the s y s t em parameters are constant the reedback gains converge to the stationary optimal gains . Thererore , if t h e parameters converge the reedback gains al so will con verge to the corresponding optimal stationary values . The purpose or this strategy is to approach the true optimal solution asymptotically and yet only use mod est amounts of com~uter time . It does not seem r easo nabl e to invest a lot of computer time in rinding the true station ary solution of the Ricca t i - equation as long as the parameters , which are the basis or the calculations , are only v arying , uncertain approximations .
wher e : T T {xT (t) , x (t - I) .... , x (t - a - 1) ,
z (t)
uT (t - I) , . . . , uT ( t - l - b) } T tlU(t) = {u(t) -
A consequence or us ing this st ra tegy is a transient period in which the process may have to work in open - loop if the reedback gains are rar rrom optimal.
u(t - l)} EXPERIMENTAL SET - GP
The matrices A* and B* are shown i n Table 1 . Derining the control criterion: T
z (N)OOz(N) + N- 1 ,E
[zT (i)Ol z(i) + AuT(i)02 Au (i) ]
l=O
Model Structure(s) In the presen t in v estigation t he fo l lo wing three model structures , dirrering in the number or control variables , were used: ./ .
the sampling time ~as c hos en as 4 min which is app . 1/5 or the mean thermal r esidence time , and t he delay d set to1 .
. /.
5 equidistant tempe ratur e measurements (T2 , T4 , T6 , T8 , 110) and one A matrix (i . e . (n , a) = (5 , O)) .
gives arter minimization the optimal control ler : u(k) where: L(k)
u(k - 1) - L(k)z(k)
2 74 7
Ex perime nt a l Investigation of Self-Tunin g Con tro l TABLE 1
System Matrices for the Control Problem (a+2)n Aa - Aa _ 1
- Aa
_._. . - _._. -_. -°
- 0 --
( a+2)n
- -0
0-
- I
0-
_ .°°
--0
0I
-°
0 '
°
.1. ,
(b+l)m
0._ ' I
° °
°
°
I
°
...J
m
B
n
°
(a +l ) n
O
BlI<
0
I
m
°
b·m
The A ma trix was selected as :
r X °X ° ° 0 1
and all other eleme n ts zero .
I
A
~ ~ ° ° °° °°° ~ I I
X X X X X X
°°°
In mod els 2 and 1 only the two first and first element s of Q2 we r e used .
j
whe r e X denotes a parameter to be est i mated. In ad d ition to the delay term from one meas urement to the next (- subdiagona l elements) , an autoregressive terlrt in eac h measurement i s included (- diagonal elements) . This expan s ion was employed to add some flexibility to the model .
./ .
1- 3 control inputs (i . e . u=(T ) in model 1 , u=(T , C ) in mod e l 2 and u~(T , C , F ) inmode{3 )c an d hJO B matrices (i~e .c b= I ) structured for model 3 as :
r~ ~ ~OoO}i
in
In models 1 and 2 the appropriate column s were deleted from the B matr i ces . The second B matrix (with onl y one non - z e ro ele ment) was in cluded to allow some de sc ription of thermal disper sion from t he te mp e rature input to t he f ir st meas ur ement . Control purp ose The pur p ose of control was select ed a s keep i ng a fixed te mp e ra ture profile . The weigh ts were , f or mode l 3 , chosen as: QO=Ql
=
Q2=diag . ( 1. 0 , 0 . 5 , 0 . 5)
di ag . (0 .2 , 0 . 2 , 0 . 2 , 0 . 2 , 0 . 2)
Proce dur e Initially all parameters were set to P, (0)
° and
= 10 ' [.
1
The fo rgett ing factor ~o =
~
was initiated as :
0.2
an d incremented acco rdin g to ~k+l = 0.8 ~k + 0 . 19
reaching the final value 0 . 95 in app . 2 5 samples . During the first period the r egulato r iden t i fied the process in open loop using independ e nt Ps eud o - Rand om Binary Sequences applied s imu ltaneously to each input . After app . 25 samples solut i on of the Riccati - equation was starte d and fe ed - back gains calculated . After a tota l of app . 4 0 s amples the ~R B S generati on was stopped , the loop c losed and the regulator allowed to br i ng the system to a steady s ta te . Then all parameters were stored on d i sk - f i l es , and a pulse dis turbance introduced . When the steady state was reache d again , the pa r ameters were restored from the di sk- files , and a new pulse disturbance introduced (etc . ) . The restoration of parameters was performed ,
L. Halla ge r and S. Ba y
274 8
because some changes of parameters occurred during pulse disturbances , and this procedure makes the regulator response to the individual disturbances independent of the regulator pre-history . RESULTS AND DISCUSSION Identification In Table 2 the identified system matrices of model 3 are shown . In the A matrix the diagonal elements all had relative standard devi ations larger than -90 % and the subdiagonal elements less than -30 % (based on 51 data points) . The variance of the prediction error ( € i ( t)) were from 8 to 23 % of the correspond ing measurement varianse . Judged on this basis we infer that this v ery s imple model is well - structured , but that a mo re elaborate model also describing disper sive effects probably will be able to reduce the error v ariance further. Note that the assumption of delay - term dominance is confirmed by the identified model . From the two last columns of B i t is seen that concentration and flow rat~ disturbances are distribu t ed with a maximum primary impact at the second measurement . However t h e two columns are c learly not linearly dependent , so the two di s turbances are indeed different .
Control Figures 2 - 4 shows the exit temperature (repre senting the profile) and control signals for the three different models when upstream pulse disturbances are applied . Figure 2 shows the response t o a pulse in the upstream temperature , when this is the only manipulated variable . The optimal action wi t h this measurement configuration is to cancel th e di s turbance by the control ( which is pos s ible due to the inclusion of integral action) , ~iLh a delay of 1 sampl e . Due t o the con~ective nature of the thermal disturbances , the controller can only handle the deterministic low frequency part of the disturbance . All the temperature control can do is to start a thermal wa v e to chase but ne ver reach the disturbance . Hence disturban ces with a duration of only a few samples will not be controlled very well in this case . :n the juxtaposed Fig . 3 the same disturbance is used , and inlet concentration is included as manipulated variable . Comparing the o u tlet temperature with the previous case , the pres e nt case is obviously the best with the smal lest excursions in exit temperature and con trol signals . The improvement stems from the utilization of the concentration control . Concentration changes immediately influences all of the reactor and is used in this case as a feed - forward - like action to late measure ments .
J~r g en se n
In F i g . 4 results are shown for model 3 whe r e upstream distur bances in both concentration and flow rate are introduced . The tempera ture control is practically inactive and the disturbances are very quickly , almost total ly compensated by the appropriate cont r ol sign als . This behaviour indicates that the controller i s actually able to dist i ngu i sh these very similar disturbances . CONCLUSION It is demonstrated that it is possible to use qualitative a priori knowledge about process dynam i cs to reduce the number of parameters to be estimated by MIMO self- tuning regula tors . An asymptotic stationary optimal con trol strategy is used where the dynamic Riccati equation is solved iteratively . The obtained control system is able to handle sing l e and simultaneous disturbances very well . REFERENCES Astrem , K. J . (1980) . Self tuning control of a fixed - bed chemical reactor system . Int . J . Control , 32 , 2 2 1- 256 . Astrem , K. J . and B. Wi t tenmark (L973) . On self-tuning regulators . Automatica , ~ , 185- 199 . Borison , V. (1979) . Self- tuning regulators for a class of multivariable systems . Automatica , 15 , 209- 215 . Clement , K., S . B. Jergensen and J .P . Serensen (1980) . Fixed - bed reactor Kalman filter ing and optimal control . 1 1 . Chem . Eng . Sci ., 35 , 1231 - 1236 . Hanscn , K. (lJ 7 3 ) . Simula t ion of the transi ent behaviour of a pilo t plant fixed - bed reactor . Chem . Eng. Sci ., 28 , 723 - 734 . Hansen , K. and S . B. Jergensen (L976) . Dynamic modelling of a gas phase catalytic fixe dbed r eactor . I - I1T. Chem . Eng . Sci ., l.!:., 3 1, 473 - 479 and 579- 598 . Jutan , A., J . P. Tremblay , J . F . MacGregor and J . D. Wright (l977) . Multivariable com puter control of a butane hydrogenolysis reactor . I - I l l . AIChE J ., 2 3 , 732-758 . Silva , J . M. , P . H. Wallman and A. S . Foss (1979) : Multi - bed reactor control systems : Con figuration development and experimental testing . I&EC Fundam ., 18 , 383- 391 . Serensen , J .P. (1977) . Experimental investi gation of the optimal control of a fixed bed reactor . Chem . Eng . Sct. , 32 , 763- 774. Serensen , J.P ., S . B. Jergensen and K. Clement (1980) . Fixed - bed reactor Kalman filter ing and optimal control . I . Chem . Eng . Sci ., 35 , 1223-1 230 . Tremblay , ~P . and J.D . Wright (1977) . Multi variable model reference adaptive control of a pilot scale packed tubular reactor . In Proc . IFAC/IFIP 5th Int . Conf . on Digi tal Computer Applications to Process Con trol , 513 - 521 . The Hague , The Netherlands . Wall man , P . H., J . M. Silva and A. S . Foss (1979). Mul ti variable integral controls f or fixed bed reac tors . I&EC Fundam ., 18 , 392 - 399.
Experimental Investigation of Self-Tuning Control I dentified S;tstem Matrices of Model 3
TABLE 2
r A
BO
B1
l'
0 . 004 1.051 0 0 0
0 - 0.066 0 . 804 0 0
0 0 0 . 119 0 . 911 0
r 0.805
0.07 2 0 . 336 0.31 3 0 . 277 0. 2 73
-0. 167l - 0. 2 11 - 0 . 136 , - 0 . 065 , 0.011
0 0 0 0 0
0 0 0 0 0
0 0 0 0 .- 0 . 2 53 0 0 0 0
0 0 0 0 . 098 0.891
.J
l'
out
l
0 0 0 0 0 . 09 1
o ut
cc F ig . 2 .
Out let temp e ra t Ul-e and co n t r ol f0 r mode l I , Disturbance : pul se of d ura t i on L2 min and he i g ht 5 un i t s i n up stream t e mpe r atu r e .
Fig . 3 .
Ou tl et te mperature an d controls f or mo de l 2 , Same di s turbance as Fig . 2 .
2749
L . Ha l lager and S . Bay
2750
J~rgensen
Explanation for Fig . 2 - 4 : Units : x- axis:
10 min
y - axis :
1
°c
( T , T t) or c ou .01 mol e % o xyge n (C) . 2 c 1 mg H /cm /sec (Fe)
or
2
outlet temper a tu r e temperature con trol signa l
~
~ ~ ~ ---- ---- ----.-..-.--.. -- - - - ------------.- I cc
\
~ '---- --- ---- -!
I I
I L, I L-l I - .. ----. . -..... - - . -_· __ ·_-- -- - - - -_·- - - --- 1 Fc ~~
I
l~---I
Fig . 4 .
Out l et tem p e r atu r e and control s for model 3 . Disturbance : pu l ses of dura t ion 12 min , + 5 units in up stream concentration and - 2 un i ts i n flow rate .
Discussio n to Paper 106.1 S. Ochiai (USA): Does any of Vm(n) (b ut not Va (n )) in Fig. 4 represent viscosity? Is an agitator motor power one of them? R. Strietzel (German Democratic Republic) : Simu lation tests have shown that the feedback of 8 equidistant sampling points of the viscosity profile (V (u)) is necessary to accomp lis h the demands oW the stability of the output viscosity . Using the motor power for calculating the correcting variable can be considered as a state variable feedback or profile feedback with fixed coefficients, because all parts of the fuse contribute to this power . The stirrer-driving power is often disturbed and for a precise control unsuitable. H. Unbehauen (Federal Republic of Germany) : Isn't it necessary to also adapt the observer , becal:se the system parameters are varying? If not , are the parameter variations of the system small and what is their range?
C
concentration control s ignal
F
flow rat e c ontrol si gnal
c c
R . Strietzel (German Democratic Republic) : The observe r contains the parameter sampling time, reactor length and model order. The model order is calculated with the filling mass and the rate of flow and therefore determined too. An adaption of the observer is not necessary . The parameter of the nonlinear part describe the polycondensation re action. Their variations depend o n the cata lysts and the g ly col concentration in the fuse. Their range is about 10 ... 20%. Discussion to Paper 106 . 2 A. Munack (Federal Republic of Germany) Does it have a significant effect on the results , when all elements of the A- matrix are allowed to be identified instead of the 9 parameters chosen here? L. Hallager (Denmark): In our preliminary work all elements of the A- matrix were estimated. The resulting A- matrix showed the same dominating sub-diagonal elements as in the present case . R . S . Crowder (USA) : Why are parameters restarted before each experiment? L . Hallager (Denmark): The pulse dis·tur bance used is not white - noise, therefore it leads to bias on the parameters. In order to make every disturbance - experiment independent of the immediate prehistory of the regulator, and thus facilitating comparison , parameters are restarted before each experiment.
EXPERIMENTAL INVESTIGATION OF SELFTUNING CONTROL OF A GAS PHASE FIXED BED CATALYTIC REACTOR WITH MULTIPLE INPUTS L. Hallager and S. Bay Jorgensen Department of Chemical Engineering, Technical University of Denmark, Building 229, 2800 Lyngby, Denmark
Abstract . An ap plication of a multiinput - mul tioutput (MIMO) self tuning regu lator to a fix ed - bed reactor is investigated exp e rimentally . The regulator consists of a linear MIMO time serie s model , which is ident ifi ed using a r e cursive l east s quar es method , combined wit h a lin e ar quadrati c optimal control strategy modified to permit the on - l ine so lution of the Ri cc ati e quati on . An appropriate model st ructure is selecte d a pr io ri u s ing qualita t i v e kn owl edge about t h e proc ess dynamic s . The self tunin g regula to r is tested experi men tal ly using s ing le and mul t iple upstream disturban ces . Th e r es ulting self tunin g regulato r perfo rms ve ry satisfactorily . The iden tifie d models confirm the a priori assumptio n s and t h e control performance is good . Keyw o rd s . Adaptive co ntrol ; optima l co n trol ; che mical r eact or cont rol; structured models ; distributed paramete r s yst ems ; ti me lag systems ; parameter estimation .
INTRODUCTION Dis tri buted syste ms with convective transport ar e widespread among c he mical unit operations and processes e . g . distillation co lumn s , tu bu l ar heat exchangers and chemical rea cto r s . An impo rtan t dynamical feature of the se systems i s that they are non - minimum phase , with the p oss ibili ty of e xhibi ting " wrong - way " be ha viour i . e . the immediate output- re spons e to an inpu t di stur ban ce i s o pposite to what is expe cte d from t h e stationary response . During the last few yea r s it has been d emon s trated experimentally that these syste ms often can b e satisfa ctorily cont r o lled by a pr ocess - co mp uter usin g v ariou s multi v ariable controller design techniques based up on o ff line i d entified mathematical models (Cleme nt , Jergensen and Se ren sen , 1980 ; Jutan and col leagues , 1977 ; Si l va , Wallman and Foss , 1979 ; Sere n sen , 19 77 ; Serensen , Jergensen and Cl ement , 1980 ; Tremblay and Wright , 1977 ; Wallman , Silva and Foss , 19 79) . In practice , however , i t i s ti me consuming to de ve l o p a mathematical mo del fr om basic physics and chemistry u s i ng conservation principles. Furth e rmor e it is often difficult and costly to ident i fy such a mod el , par tly because est i ma tes o f some paramet e r s may b e diffi cult to obtain from e xperim ents , and partly because the mode l often will be non - li ne ar in the parameter s . Finally the p arameters in chemical processes are usually t im e v arying , f or instance due to deactivation of cat alyst s in reacto rs o r depositions i n heat - e xchangers , thus necessitating more o r l ess regular up da ting o f parameters .
Therefore it may be much more e xp e di ent to use a selfadjusting appr oac h. The structure an d ord e r of a compar atively simple , linear mathematica l mod e l i s postulated and then a self tuning controlle r (see e . g . Astrem and Witt enmar k , 1973 ; Astrem , 1980 ; Borison , 1979) is used to est ima te t he parameters in t hi s model , and S imul ta neously cont r o l t h e process . Note that the postulated model c an be viewed as a reduced o rder lin ea riza t i o n of t h e " true " system model , th us t h e paramete rs in the lin ear mo d el d epen ds on t h e o p erating poin t as well a s on the physical paramete r s . The use o f a simplifi ed approach in a contro l cont e xt i s adequate in many cases . Astrem (1980) ha s si mulat ed t he appli catio n of a s elf tu n ing controlle r using pole - placement design to a fixed bed chemical reactor system s imi lar to the o ne investigated here . He uses single l oop cont r o l an d conclude s t hat the cont roller mode l can be conside rabl y simpler than t he system , st i ll achieving good control The contro l pr o blems in the chemical ind u st ry are usually r eg ula to ry i . e . h old ing a fixed ope r ating point in fa ce of di sturbances . A locally lineariz e d model has pr ove d to g i ve a reasonable description o f the fix ed - bed reac tor dynamic s ( Hansen and Jergensen , 1976) . Consequently i t is f easibl e to use a model which i s linear in the states and apply well known linear multi variabl e control t h eory. Thi s s implifi cat ion ~ ea d s to a mo del wh ich is li near in the unknown coe fficient s , thereby resul t ing in a linear ident if ication problem . However , a linear model is lik ely to b e ov erparam ete rized compared to a non - line ar mod e l
27 43
2744
L. Hallager a nd S. Bay
(especially when distributed processes are discretized spatially) . ;0 minimize this problem it is essential to use available qua litati ve a priori kn owledge in structuring the model . This "almost black-box " approach suffers from the disadvantage that it i s no t possible to estimate unmeasured quantities ( because quan titative a priori kn owled g e is not assumed available) , thus only oucput -feedbac k is p os sible and unmeasured quant i ties can not enter the control criterion .
e quidi scant temperature measurements along the reactor axis , composition measurements at the inl et and at one of 5 axially equ idistan t points through the re actor and a fl ow rate measurement . In industrial practice only a limited number of measurements are availab l e (2 - 5 temperature measurements and possibly a conce ntration measurement) . Hence t he te mpera ture profile is known on l y to a l i mited ex tent . The disturban ces are se lec te d by the operator and may be stochastic o r d ete rm inis tic. Th e purp ose of control may be d efi ned as : a.
to maintain all or part of the temperature profile , to reduce the thermal lo ad o n the bed .
b.
to maintain the exit cuncentration as a measure of product quality .
c.
a mixture of a and b.
THE PILOT PLANT The pilot plant reactor consists of a tubular fixed bed of catalyst partic le s , designed to be essent ially adiabatic and with negligible heat capacity of the reactor wall . It is de scribed in detail by Hansen and J0rgensen (1976) . In this ser i es of e xp eriments a stream of hydrogen containing app . 0 .2 mole% oxygen is fed to the packed cataly'st bed at a temper ature of approximately 82 °C . The adiabatic temperature rise during water formation is 169 °c pr . mole % reacted oxygen . Qualitative Dynamical Description In the gas phase fixed bed catalytic reactor , investigat ed here , the he at capacity of the gas is negligibl e compared to the heat capa city of the catalyst pellets , thus the t h ermal residence t ime is several h undred times great er than t h e gas residence time . Furthermore it i s known (Hanse n, 1973) that the mass transport processes are s ufficiently fast to make quasi stationary descriptions of the mass - balances r easonable . Finally there is almost thermal e q uilibrium between the catalyst pellets and the gas and within the pellets . A reasonable simplified mathematical model of the reactor therefore i s a pseudo homogeneous model consist in g of one dyn amic non - li near par tial differential e quati o n f o r the temperature and one quasistationary differential e q uation for the oxygen balance . Both including dispersive terms . The two important dynamical features of the reactor system are thus a.
b.
A fast propagation of concentration and flow rate changes . The mass balan ce is a quasi stationary functional of t he inlet cond itions and the temperature profile . A s low propagation of temperatu re changes. The thermal wa ve passes th r o ugh the reac to r in 11-25 minutes dependent on the flow rate .
The Control Problem The cont r ol problem conside r e d in th i s paper is control of the chemical reactor (see Fig . 1) . The plant has three inputs - temperature , composition and flow rate of the entering gas - in which di sturbances can enter and control actions be taken . The output co n s ist s of 11
J~rgensen
TlIEORY The problem of designing a r egulator for the chemical r eacto r sys t em without knowing the model parameters is approached by using the separation t heorem although it is not str i ctly valid . The design problem is thus divided in to two parts - one of identifying a suitable mode l and o n e of determining a control strategy . The ce rtain ty e q uivalence principle is furthermore used as an ad hoc d esign principle i . e . the control p art of the problem is han dled as a d e t erm in istic problems not taking the uncertainly of the parameters in to consideration . Identification It is assumed that th e input- output data from the chem i cal r eactor can b e de s cribed by a discrete - time multivariable time - se r ies model: x(t)=A(q
- 1
)x(t - l)+B(q
- 1
)u(t - d)+e(t)+c
wh e r e : x ( t)
(dim n) is a measurement vector
u(t)
(d im m) is an input \·ec tor
e( t)
(dim n) is a zero means , white noise sequence
c
( d im n) is a constant v ector i s the delay from input to output
d
A( q
A( q
B(q
- 1
-
1
) and B( q- ) are matrix polynomials of order a and b in t h e backward shi ft operator q-l i . e . 1 - 1 -a ) = AO + q Al + . . . + q Aa 1 -b )=B + + q Bb O
The dimensions (n , a , m, b) and the delay d are assumed known and the mat r ices A. , i = 0 , ... , a and B ., i = 0 , ... , b are assu~ed un known but co~stant or slowly varying .
Experime nta l
T
r--+<
,~
l'pstreMl { d [email protected]
l
2745
Investigation of Se l f - Tun i ng Control
0
!
C
c
~.
F
lC
.
,.
lC
X
lC
0
0
I(
lC
lC
0
0
J
I TC
Fc
Cs
l -- v----J
F L _
Co ___ _ __
Possible control signals
_
_ _~ _
Possible
x. (t)
1
temperature
o
concentrati on
sys~em
The loss function . which is minimize d by this algorithm is t
1
v . (t) 1
v,here :
T
T
T
{x ( l - l) . x ( t - 2) •.. . • x (t - a - 1) .
T
_ _ __ _---'
:'10. . ' rate
+ e. (t)
i
_
~easure~en ts :
Sketch of the reactor
The i'th state in equation (1) can be written :
_
I(
[!]
Fig . 1.
_
~
~
U
t- j
j =0
The algorit hm is starte d with initial values 1
0.
1
(superscript. i means the i ' th row o f the ma t. rix) The parameter ve c tor 0 i is estimated using a recurci v e least s quares algorithm with expo nential forgetting :
E.
( t+l)
J
1
thereby introducing exponential weighting of old data .
P ,( O) = a ' I
and
1
2( .)
E.
T
u (t - d) •. . .• u (t - d - b) . lJ
0. \ l h )
!---
0.
1
+ K. (t+llc (t+l) 1
1
x. ( t+l) - cpT(t+l)0. (t)
e1 (0)
= 0
where a is chosen sufficiently large to indi cate the lack of initial k n owledge of the parameters . I n general all paramete r s are assumed comple tely unk n own h ence it is reasonable to initi ate all covariance matrices at the same value and thus reduce computations considerably as it o nly ~il] be necessary to c alculate one estimator gain and one covariance matrix in each step (Bor i son . 1979) .
P . ( t+ll is t he n o rmalized cov a rian ce ma 1 trix of the estimate error l0 - 0 ( t +I ))
In the present case however extensive use is made of structural zeros . i . e . on the basis of a priori knowledge some parameters are set to zero . This may lead to differences be tween the different
Ki(l+l) is the corresponding gain vector
Structure selection
1
K. (t +l) 1
1
1
1
P . { t)CP. ( t+1) · 1
1
[cpT(l+l)P.(t+llCP.(t+l) + 1] - 1 1
I' . ( t + 1)
1
1
l/u[I - K. (t +1 ) ]cpT(t+llp (t)
1
1
1
1
\', here :
1
1
is an exponential forgetting fac tor I
is the unit matrix of appropriate dimension
The size of the model is defined when n . a . m. band d are fixed and it is decided which measurements and controls are to be used . If the model is no t further structured it would be necessary to estimate n(n ' (a+1) + m(b+l) +1) parameters ( including constant terms) . In the present case the number of parameters is reduced using the qualitative process
L. Hallager and S. Bay
2746 knowledge describe d earlier : a.
b.
The distributed disturbances (concentration and r l ow rate) inrl uences a ll meas ureme n ts with a delay or 1 sampling peri or (due to the sampling or the system) . The thermal wave as observed by a number or temperature measurements is described essentially as a delay rrom the temperature input to the rirst measurement and rrom one measurement to the next. This necessitates a proper selection or sampling time. The simplest approach is to select equi distant measurements, a sampling time equal to (or slightly larger than) the delay between two neighbouring measurements and set a=O , to describe convective propagat i on . A more e l aborat e approach is to select a s h orter sampling time but use additional old measurement ( i. e . a>O) , thus including a description or dispersive errects . Thi3 descrip t ion adds rlexibility in the selection of sampling time and allows ror larger rlow rate disturbances .
In this way the dominant thermal dynamics or the reactor and the inrluence or the inputs can be modelled with a reasonable number or parameters . Control Chemical processes are commonly subject to dr i rt. This is reflected in our model in the estimation or a constant term . It thererore seems reasonable to use some kind or integral action in the controller , which also eliminates orfset in case of constant disturbances . Consequently the identified model is rerormu lated to the rol l owing st ate space f orm (ror simplicity , d is assumed to be 1).
J~rgensen
and S(k) i s calculated by solving the Riccati equat i on : S(k) = 01 + A* T ' S(k+l) ' (A*-B* ' L(k)) sub ject to the condition: SIN) = 00 ' In the general selrtuning case this solution is not strictly usable because the system matrices A* and B* are not constant and their ruture variation is not known as the parame te r s are estimated on - line . Ir N is chosen as I , i . e. a one - step criterion is employed , the solution is applicab l e. In the p r esent case , howe ve r , the re is known to be a therma l time delay rrom inlet to outlet , which usual ly is 5- 10 times t he sampling time. It seems reasonable that t he control criterion should at least cover this delay in order to account ror the primary convective efrects or a given control input. Consequently the rollowing st rat egy is at tempted : N i s se t to inrinity indicating that the stationary solution is searched ror , the Riccati - equation is iterated (the ' wrong ' way) once at each sampling instant (sta r ti ng rrom the last solution) and the newe st value or S(k) is used in the calculation or the f ee dback gains . When the s y s t em parameters are constant the reedback gains converge to the stationary optimal gains . Thererore , if t h e parameters converge the reedback gains al so will con verge to the corresponding optimal stationary values . The purpose or this strategy is to approach the true optimal solution asymptotically and yet only use mod est amounts of com~uter time . It does not seem r easo nabl e to invest a lot of computer time in rinding the true station ary solution of the Ricca t i - equation as long as the parameters , which are the basis or the calculations , are only v arying , uncertain approximations .
wher e : T T {xT (t) , x (t - I) .... , x (t - a - 1) ,
z (t)
uT (t - I) , . . . , uT ( t - l - b) } T tlU(t) = {u(t) -
A consequence or us ing this st ra tegy is a transient period in which the process may have to work in open - loop if the reedback gains are rar rrom optimal.
u(t - l)} EXPERIMENTAL SET - GP
The matrices A* and B* are shown i n Table 1 . Derining the control criterion: T
z (N)OOz(N) + N- 1 ,E
[zT (i)Ol z(i) + AuT(i)02 Au (i) ]
l=O
Model Structure(s) In the presen t in v estigation t he fo l lo wing three model structures , dirrering in the number or control variables , were used: ./ .
the sampling time ~as c hos en as 4 min which is app . 1/5 or the mean thermal r esidence time , and t he delay d set to1 .
. /.
5 equidistant tempe ratur e measurements (T2 , T4 , T6 , T8 , 110) and one A matrix (i . e . (n , a) = (5 , O)) .
gives arter minimization the optimal control ler : u(k) where: L(k)
u(k - 1) - L(k)z(k)
2 74 7
Ex perime nt a l Investigation of Self-Tunin g Con tro l TABLE 1
System Matrices for the Control Problem (a+2)n Aa - Aa _ 1
- Aa
_._. . - _._. -_. -°
- 0 --
( a+2)n
- -0
0-
- I
0-
_ .°°
--0
0I
-°
0 '
°
.1. ,
(b+l)m
0._ ' I
° °
°
°
I
°
...J
m
B
n
°
(a +l ) n
O
BlI<
0
I
m
°
b·m
The A ma trix was selected as :
r X °X ° ° 0 1
and all other eleme n ts zero .
I
A
~ ~ ° ° °° °°° ~ I I
X X X X X X
°°°
In mod els 2 and 1 only the two first and first element s of Q2 we r e used .
j
whe r e X denotes a parameter to be est i mated. In ad d ition to the delay term from one meas urement to the next (- subdiagona l elements) , an autoregressive terlrt in eac h measurement i s included (- diagonal elements) . This expan s ion was employed to add some flexibility to the model .
./ .
1- 3 control inputs (i . e . u=(T ) in model 1 , u=(T , C ) in mod e l 2 and u~(T , C , F ) inmode{3 )c an d hJO B matrices (i~e .c b= I ) structured for model 3 as :
r~ ~ ~OoO}i
in
In models 1 and 2 the appropriate column s were deleted from the B matr i ces . The second B matrix (with onl y one non - z e ro ele ment) was in cluded to allow some de sc ription of thermal disper sion from t he te mp e rature input to t he f ir st meas ur ement . Control purp ose The pur p ose of control was select ed a s keep i ng a fixed te mp e ra ture profile . The weigh ts were , f or mode l 3 , chosen as: QO=Ql
=
Q2=diag . ( 1. 0 , 0 . 5 , 0 . 5)
di ag . (0 .2 , 0 . 2 , 0 . 2 , 0 . 2 , 0 . 2)
Proce dur e Initially all parameters were set to P, (0)
° and
= 10 ' [.
1
The fo rgett ing factor ~o =
~
was initiated as :
0.2
an d incremented acco rdin g to ~k+l = 0.8 ~k + 0 . 19
reaching the final value 0 . 95 in app . 2 5 samples . During the first period the r egulato r iden t i fied the process in open loop using independ e nt Ps eud o - Rand om Binary Sequences applied s imu ltaneously to each input . After app . 25 samples solut i on of the Riccati - equation was starte d and fe ed - back gains calculated . After a tota l of app . 4 0 s amples the ~R B S generati on was stopped , the loop c losed and the regulator allowed to br i ng the system to a steady s ta te . Then all parameters were stored on d i sk - f i l es , and a pulse dis turbance introduced . When the steady state was reache d again , the pa r ameters were restored from the di sk- files , and a new pulse disturbance introduced (etc . ) . The restoration of parameters was performed ,
L. Halla ge r and S. Ba y
274 8
because some changes of parameters occurred during pulse disturbances , and this procedure makes the regulator response to the individual disturbances independent of the regulator pre-history . RESULTS AND DISCUSSION Identification In Table 2 the identified system matrices of model 3 are shown . In the A matrix the diagonal elements all had relative standard devi ations larger than -90 % and the subdiagonal elements less than -30 % (based on 51 data points) . The variance of the prediction error ( € i ( t)) were from 8 to 23 % of the correspond ing measurement varianse . Judged on this basis we infer that this v ery s imple model is well - structured , but that a mo re elaborate model also describing disper sive effects probably will be able to reduce the error v ariance further. Note that the assumption of delay - term dominance is confirmed by the identified model . From the two last columns of B i t is seen that concentration and flow rat~ disturbances are distribu t ed with a maximum primary impact at the second measurement . However t h e two columns are c learly not linearly dependent , so the two di s turbances are indeed different .
Control Figures 2 - 4 shows the exit temperature (repre senting the profile) and control signals for the three different models when upstream pulse disturbances are applied . Figure 2 shows the response t o a pulse in the upstream temperature , when this is the only manipulated variable . The optimal action wi t h this measurement configuration is to cancel th e di s turbance by the control ( which is pos s ible due to the inclusion of integral action) , ~iLh a delay of 1 sampl e . Due t o the con~ective nature of the thermal disturbances , the controller can only handle the deterministic low frequency part of the disturbance . All the temperature control can do is to start a thermal wa v e to chase but ne ver reach the disturbance . Hence disturban ces with a duration of only a few samples will not be controlled very well in this case . :n the juxtaposed Fig . 3 the same disturbance is used , and inlet concentration is included as manipulated variable . Comparing the o u tlet temperature with the previous case , the pres e nt case is obviously the best with the smal lest excursions in exit temperature and con trol signals . The improvement stems from the utilization of the concentration control . Concentration changes immediately influences all of the reactor and is used in this case as a feed - forward - like action to late measure ments .
J~r g en se n
In F i g . 4 results are shown for model 3 whe r e upstream distur bances in both concentration and flow rate are introduced . The tempera ture control is practically inactive and the disturbances are very quickly , almost total ly compensated by the appropriate cont r ol sign als . This behaviour indicates that the controller i s actually able to dist i ngu i sh these very similar disturbances . CONCLUSION It is demonstrated that it is possible to use qualitative a priori knowledge about process dynam i cs to reduce the number of parameters to be estimated by MIMO self- tuning regula tors . An asymptotic stationary optimal con trol strategy is used where the dynamic Riccati equation is solved iteratively . The obtained control system is able to handle sing l e and simultaneous disturbances very well . REFERENCES Astrem , K. J . (1980) . Self tuning control of a fixed - bed chemical reactor system . Int . J . Control , 32 , 2 2 1- 256 . Astrem , K. J . and B. Wi t tenmark (L973) . On self-tuning regulators . Automatica , ~ , 185- 199 . Borison , V. (1979) . Self- tuning regulators for a class of multivariable systems . Automatica , 15 , 209- 215 . Clement , K., S . B. Jergensen and J .P . Serensen (1980) . Fixed - bed reactor Kalman filter ing and optimal control . 1 1 . Chem . Eng . Sci ., 35 , 1231 - 1236 . Hanscn , K. (lJ 7 3 ) . Simula t ion of the transi ent behaviour of a pilo t plant fixed - bed reactor . Chem . Eng. Sci ., 28 , 723 - 734 . Hansen , K. and S . B. Jergensen (L976) . Dynamic modelling of a gas phase catalytic fixe dbed r eactor . I - I1T. Chem . Eng . Sci ., l.!:., 3 1, 473 - 479 and 579- 598 . Jutan , A., J . P. Tremblay , J . F . MacGregor and J . D. Wright (l977) . Multivariable com puter control of a butane hydrogenolysis reactor . I - I l l . AIChE J ., 2 3 , 732-758 . Silva , J . M. , P . H. Wallman and A. S . Foss (1979) : Multi - bed reactor control systems : Con figuration development and experimental testing . I&EC Fundam ., 18 , 383- 391 . Serensen , J .P. (1977) . Experimental investi gation of the optimal control of a fixed bed reactor . Chem . Eng . Sct. , 32 , 763- 774. Serensen , J.P ., S . B. Jergensen and K. Clement (1980) . Fixed - bed reactor Kalman filter ing and optimal control . I . Chem . Eng . Sci ., 35 , 1223-1 230 . Tremblay , ~P . and J.D . Wright (1977) . Multi variable model reference adaptive control of a pilot scale packed tubular reactor . In Proc . IFAC/IFIP 5th Int . Conf . on Digi tal Computer Applications to Process Con trol , 513 - 521 . The Hague , The Netherlands . Wall man , P . H., J . M. Silva and A. S . Foss (1979). Mul ti variable integral controls f or fixed bed reac tors . I&EC Fundam ., 18 , 392 - 399.
Experimental Investigation of Self-Tuning Control I dentified S;tstem Matrices of Model 3
TABLE 2
r A
BO
B1
l'
0 . 004 1.051 0 0 0
0 - 0.066 0 . 804 0 0
0 0 0 . 119 0 . 911 0
r 0.805
0.07 2 0 . 336 0.31 3 0 . 277 0. 2 73
-0. 167l - 0. 2 11 - 0 . 136 , - 0 . 065 , 0.011
0 0 0 0 0
0 0 0 0 0
0 0 0 0 .- 0 . 2 53 0 0 0 0
0 0 0 0 . 098 0.891
.J
l'
out
l
0 0 0 0 0 . 09 1
o ut
cc F ig . 2 .
Out let temp e ra t Ul-e and co n t r ol f0 r mode l I , Disturbance : pul se of d ura t i on L2 min and he i g ht 5 un i t s i n up stream t e mpe r atu r e .
Fig . 3 .
Ou tl et te mperature an d controls f or mo de l 2 , Same di s turbance as Fig . 2 .
2749
L . Ha l lager and S . Bay
2750
J~rgensen
Explanation for Fig . 2 - 4 : Units : x- axis:
10 min
y - axis :
1
°c
( T , T t) or c ou .01 mol e % o xyge n (C) . 2 c 1 mg H /cm /sec (Fe)
or
2
outlet temper a tu r e temperature con trol signa l
~
~ ~ ~ ---- ---- ----.-..-.--.. -- - - - ------------.- I cc
\
~ '---- --- ---- -!
I I
I L, I L-l I - .. ----. . -..... - - . -_· __ ·_-- -- - - - -_·- - - --- 1 Fc ~~
I
l~---I
Fig . 4 .
Out l et tem p e r atu r e and control s for model 3 . Disturbance : pu l ses of dura t ion 12 min , + 5 units in up stream concentration and - 2 un i ts i n flow rate .
Discussio n to Paper 106.1 S. Ochiai (USA): Does any of Vm(n) (b ut not Va (n )) in Fig. 4 represent viscosity? Is an agitator motor power one of them? R. Strietzel (German Democratic Republic) : Simu lation tests have shown that the feedback of 8 equidistant sampling points of the viscosity profile (V (u)) is necessary to accomp lis h the demands oW the stability of the output viscosity . Using the motor power for calculating the correcting variable can be considered as a state variable feedback or profile feedback with fixed coefficients, because all parts of the fuse contribute to this power . The stirrer-driving power is often disturbed and for a precise control unsuitable. H. Unbehauen (Federal Republic of Germany) : Isn't it necessary to also adapt the observer , becal:se the system parameters are varying? If not , are the parameter variations of the system small and what is their range?
C
concentration control s ignal
F
flow rat e c ontrol si gnal
c c
R . Strietzel (German Democratic Republic) : The observe r contains the parameter sampling time, reactor length and model order. The model order is calculated with the filling mass and the rate of flow and therefore determined too. An adaption of the observer is not necessary . The parameter of the nonlinear part describe the polycondensation re action. Their variations depend o n the cata lysts and the g ly col concentration in the fuse. Their range is about 10 ... 20%. Discussion to Paper 106 . 2 A. Munack (Federal Republic of Germany) Does it have a significant effect on the results , when all elements of the A- matrix are allowed to be identified instead of the 9 parameters chosen here? L. Hallager (Denmark): In our preliminary work all elements of the A- matrix were estimated. The resulting A- matrix showed the same dominating sub-diagonal elements as in the present case . R . S . Crowder (USA) : Why are parameters restarted before each experiment? L . Hallager (Denmark): The pulse dis·tur bance used is not white - noise, therefore it leads to bias on the parameters. In order to make every disturbance - experiment independent of the immediate prehistory of the regulator, and thus facilitating comparison , parameters are restarted before each experiment.