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Distillation Control by Output Feedback Designed via Order Reduction
Cop,rigl" © 11'.\(: 10lh Trienni,,1 \\"orld Congress. \Iuni(h. FI{(: . 1" H7
DISTILLATION CONTROL BY OUTPUT FEEDBACK DESIGNED VIA ORDER REDUCTION S. K. Lehmann II U/'I/II! A(;. L\[R -P/al/lIl/g. TEP (1\ SO]). PO
HII.\
so
0] 20.
D-62 J(J FIIII/kjlnl!.\[ SO. FlU;
Abstract. mod~
In order to assess the feasibility of state space methods based on linear control system for a pilot plant of two coupled distillation columns was
designed. The columns, separating a mixture of three alcoholes , can be modelled as a nonlinear sys tem of order ]06. After derivation and verification of a linearized model, this model is analyzed using modal measures. These measures provide hints for the selection of a contro ller st ructure as well as for the succeeding modal order reduction. Using a reduced eighth-order model, a static output feedback of three mea sured temperature s on the four control input s is calculated using a numerical optimization sc heme. The performance of the obtained controller is discussed by means of data from simulations of the nonlinear model and from experiments with the plant itself. The results provide a base for the assessment of the methodology. Chemical variables control; distillation control; eigenvalues; large-scale sys tems ; linear systems: modal control; numerical methods; optimal sea rch techniques; order reduction; output feedback. K~ords .
INTRODUCTION
PLANT AND MODEL
Numerous theoretical results concern ing control system design by state space methods and based on lin ear models are available. As one contribution to the assessment of the application of these methods to real world problems, a control system for a pilot plant of two coup led distillation columns was designed and tested. The method employed is a conceptually s imple technique for the design of structural ly constrained controllers and \vas appl ied to a 1 inearized reduced model of The application oriented paper the plant. emphasizes the assessment of existing procedures rather than the derivation of new co ncepts. This in cl ude s the discussion of results from real plant experiments.
fhe plant, depicted in Fig. ], consists of two coupled distillation columns. The feed stream F contains three alcoholes. This mixture is to be separated into the (almost) pure components methano 1 (top of ma in column), ethanol (top of
Th l organization of the paper is as follows. After the description of the plant and the control ta sk. the derivation and verificatio n of a linear model is discussed. Subsequent sections treat the use of modal measures. the order reduction, the de sig n of a static output feedback and the pe"fonnance achieved by this control system. A summary of the relevant ,'esul ts concludes the pape,' .
*TThe-res'earchi'eportedTrlfhTs- paper waSCOil"=" ducted at the Institut fUr Regelungs· und Steuerungssysteme (Prof. Dr. rer. nat. Dr.-Ing. E. h. Otto Foll ingel') of the Uni ve l'sity Kar ', sruhe. The experiments were run at the Institut flil' Systeflldynamik und Regelungsteci1nik (Prof. D~.-Ing. Erns! Dieter Gille s) of the University Stuttgal't. Federal Republic of Germany.
Fig. ].
Structure of the pilot plant.
side column) and propanol (bottom of main co lumn). As the feedstream may vary in flow rate and composition. a control system i s necessary in order to keep constant the assigned compositions of the three outlet streams. The co ntrol system can influence the plant via four control inputs:
2S11
s.
K.
L C illll
The reflux ratios r, and r z of both co lumn s, the heating power Q. l of the evaporator and th e flow rate of the s ide s tream Os f rom the main to the si de column. The var iable s to be co ntrolled are the composition s of the outlet stre am s K" Kz and B,. As typical for di stillation columns, in s tead of the se concentrations only the temperature s on the tray s can be measured at acceptable cost.
Since the application of the powerful s tate space methods requires a lin ear model, the equations (1) have been linearized by expand ing the functions f and 9 to TAYLOR-~ell.<'n and neglecting all term s but the linear ones. Th e result is a 1 inear model given by ~
The (o nt ro 1 task is th erefore to fi nd a fe edba ck of suitab le measured temperat ure s to the co ntrol inputs that
; Ax
+
Bu
+
E~
,)'M ; fM ~
(2 )
The dynamic matrix A is a spar se matr i x with a s tructure reflecting t he excessive seri al structure of the plant. In Fig. 3 the behav i our of the linear and t he nonl inear mode l are compared by mean s of t he responses of the temperatures on
achieves an acceptable dynamic beha v· i our of the c l osed-loop sys tem , reduces th e impact of change s of feed flow rate and composit i on as much as possible. As th e evaporation enthalpies of the three alcoholes differ onl y s lightly , the energy balances of the tray s may be neglected . Th erefore, the plant can be modelled us ing only two material balance s for any tray. Th e model obtai ned is a non 1 inea r vector differential equatio n of order 106 , which reads
T34 (Nonlin. Mod .) T34 (Lln . Mod) 110
t:
--
:, 98 Q)
0
T34 (Nonlin. Mod) T34 (Lin Mod)
1------- ----
'-
Q)
0.
E Q)
(1)
86 .:;".-
..-:....- -
1-
n
Th e vec tor ~ co ntain s the co nce ntration s of me thanol and ethanol on all 53 trays, the vector l:! the four co ntrol inputs , the vec tor z th e three disturbance inputs, namel y the feed ffow r ate and two feed compos ition s. The temperatures on t he 53 trays form t he output vector IM' Wi th the model , t he behaviour of the open-loop system can be ana l yzed by I1wnell.(.caf ~<'mufaU(O,1. By these s i mu l atio ns, performed on a SIEMEN S 788! mainframe using a variable step s iz e RUN GEKUTTA-MERSON-Method, it cou l d be ver ifi ed that the mode l describe s the relevant properties of the system. One result of the simu lation s is s hown in Fi g. 2 below. Thi s figure s hows the var iati on of the temperature profi l e in the co lu mns ca used by a ste p change of the feed flow rate from 200 moles/ hour to 180 mole s/ hour.
F
=
0.9 Fo
n
, dd
(3 Vl Q) Q)
Q, Q)
9
90
~
I ~~,
,, '
{I
Q)
':J
0
-" , !I. ,...-;~>/
'-
Q)
0.
E Q)
1-
0.5 h 1.5 h
00 h 1.0 h
'" 1
... ~ ....
,<,> /,'/
::./
60 21
31
"Tray No. - Main Col. Fig. 2.
L 1 L3
53
Side Col .
Open -l oop mode l (s tep l'esponse) .
T'8 (Nonl in. Mod .) T'8 (Lin . Mod )
--- ----f--_ T'8 (Nonlln Mod) T'8 (Lin . Mod)
76,------,-----,-----,-----,-----,-----,
~
~
11.)
72
k
--------
~ --
"'-..._
~ 68 1-----1---~~~~~ =_~-r ~.-__---r-----r----~ -_~~-_-.:::
Cl)
1-
____ __
_~
__ _
641-----~-----r----~-----r----~----~
o
0.5
15
Time in Hours Fig. 3.
Nonl in ear and linearized mode l .
the trays 18 and 34 to step chan ges of the reflux rat i o r 1 of :t 2% of it s nominal value 0.84. Whil e the 1 inear mode l does not app,-oximate the sig n-dependable respon se of non 1 inear one. it doe s yield a good approximation. as long as the de via tion fro m the operating point. for whi ch the linearized mode l was calculated. i s suffi cie ntly sma ll. Since the co nt ro ll e r v/ill be desiqned so as to keep this deviation sma ll at all ti mes . the application of th is model for contro l des ign purposes is fea sible. To this end. at fir st a control s tru ct ure i s to be determined. This i s done using ,lil'dar
1)I(l ll~U't C ~ .
APPLICATION OF MODAL MEASURES
Modal meas ures eva luat e quantitati vely the mod al coordina tes of a system concerning propertie s like controllability or importance for the tran smiss ion frolll in puts to output s . During the last years seve ral moda l measure s ha ve been proposed. A re view of some of these can be found in LITZ ( 1983) .
DistillatioJl COJltrol 1)\· Output Feedback
The first question to be answered by modal measures is. on which trays the temperature should be measured. In prin ciple. it is possible to measure the temperature on all 53 trays. Such a structure. howe ver. is as expensive as unnece ssary. Therefore. it ha s to be de cided. on how many and on which trays the temperature i s to be mea sured. A first se le ction ca n be made us ing some heuristic chemical engineering rule s. which re comme nd to take the measurements on tray s. where the temperature varies considerab l y in time and from one tray to the other. In s ight of Fig. 2 above . measuring the temperatures on the tray s 18 and 34 (main co lumn) and 51 (s ide co lumn) appears feasible. After this tentative pla cement i s made. it has to be checked. if by this placement the properties of the sys tem ca n be corrected in the desired manner. This question can be answered by a model measure. The concept of the et9cl1\'a~ue bCI1;(t{\'(t,l. eigenvalue sensitivity wa s introdu ced by PORTER/ CROSSLEY (1972) and used for example by LITZ (198Ia). Eig enval ue sensit i vities are calculated fro m the sys tem transformed to JORDAN-canontcal 6o~m . '*
x
=
Y.. =
-,
"
TAT x
(3)
C ! ~.
In the se equations. ! denotes the medal' II1cttllix (th e matrix of the eigen vectors) of the sys tem (2). ~ contain s those ro\'l s of ~M that corres pond to the mea sured var i abl es. As the effect of a feedba ck of the measured output y on the control input s i s to be mea sured . the s en s iti vities are calculated us ing C T and T·t B only. The sen s itivity of an elgen-val ue equafs zero . if this eigenvalue ca nnot be shifted by the selected feedback struc ture. In order to achieve quantitatively correct va lues (higher sensitiv ity corre sponds to an eigen value that can be shifted more ea s ily) . inputs and outp ut s of the system (2) ha ve to be sc aled to maximal absolute values of approximately 1.0. With appropriate sc al ing factor s. the sensitivities of Table I are obtained. Th e ba s ic re sult of this tabl e is t hat all eigen values can be shifted by the co nt roller stru cture cons id ered. beca use all sen s iti viti es Esse ntial for th e are different from zero. fea sib ility of the control stru cture are the large se ns iti vitie s of the eigenvalue s clo se to the imaginary axi s . Th ese sensitivities i mply the pos s ibil i ty to shift the se e ig enva l ues to the left and thereby to speed up the s luggi sh beha viour of the col umn s . Therefore . the se l ected controller structure i s principally suitabl e for control purpo ses . Be s ide s the po ss ibility to shift an eigen va lue by feedba ck. th e eigenva lu e se ns iti vity assesses it s importance ( dC"1(1I,1I1C l') for the transmi ss ion from the input s to the output s (LITZ. 198Ia). Reviewing Table I , the f ollowing s tatement s regard i ng th e domi nan ce structure of the sys tem can be made: The eigen value s No. deci s i ve ly dominant.
l.
2 and 3 ar'e
The eigenvalue s No. 8. 9 and 10 po ss es an increased dominance. The dominance of the succeeding eige nva lu es declines co ntinuously.
TABLE 1
NO .
Sensitivity - illominance - ) Mea sures by Feedback of three Measured Temperatures EIGENVALUE REAL PART
IMAG . PART
SEN SIT IVITY/ STRUCTURE DOMINANCE MAXIMUM SUM
21
-=T2T.866
22 23 24 25
- 145 . 142 -149.292 - 178 . 414 - 178.414 -207.811 -207.8 11 -237.9 00 -237.900 -259.453
0.0 0.0 5.080 -5.080 7. 633 -7 .638 5.283 -5.283 0.0
16.47 64 . 00 5:21 0.04 0.33 0 . 60 0 .77 I. 42 0.55 0. 55 0.38 0.38 0. 33 0 . 33 0 . 00 0 . 52 0 . 61 0 .04 0.66 0.81 0.17 0.56 0.46 0.05 0.05 0.0 4 0 .04 0 .03 0.03 0.02
-856.002 -856.002
1.156 -1.156
0.22 0. 22
I
2 3 it 5 6 7 8
9
10 IT 12 13 14
15 16 17 18 19 20
26
27 28 29 30 66
67
-0.192 -0.33 0 -2.823 -3 . 302 -9.825 -18 . 433 -25.184 -30.066 -33.620 -33.620 -48.750 -48.750 -64.567 -64 . 567 -69.888 -85.929 -87.080 -105.114 - 117 .3 97
~.413
0.0
0:-0 0.0 0.0
0.0 0.0 0.0 0.0 o.s76 -0.576 T.558 -I. 558 2.895 -2.895 0.0 0.0 0.0 0.0 0.0
0.0 0.0
45 .95 263.16 l8':48
0.52 0.52
--0:14 I. 75 I. 22 I. 36
3.00
2.T2 2. 12 0.96 0. 96 0.88 0.88 0 . 02 I. 36 I. 54 0.15 2.13 2.27 0.26 I. 25
1.11 0.15 0.15 0. 10 0.10 0. 08 0.08 0. 03
39 additional eigenvalues all with dominance s smaller than 0.4
ORDER REDUCTION
After selecting a controller structure and checking it s feasibility. the co ntroller gains have to be determined. Despite the potential of nowaday's computer s, the use of a comp l ete model of the order 106 for thi s calculation yields an expen s i ve ta sk. Due to the do minan ce of only a few eigenvalues (cL Tabl e I). on the other hand. the I ineari zed model ca n be approximated by a model of considerab l y reduced ord er . Thi s l eads to the concept to ~et ermine a redu ced mode l and to use thi s model to compute the controller for the complete one . Many methods for the deri vation of a redu ced model ha ve been deve loped (for example LITZ. 1979: EITELBE RG. 1979 : ROTH. 1983 : POST. 1984) . The method of LIT Z (1979) comb ines numerical s implic ity with good or even excellent approximation qual ity of the reduced model (BONVIN/ MELLI CHAMP. 1982) and wa s therefore employed for the ca l culation of the reduced or'der model. The ba s i c idea of thi s procedure is to ca l culate a model containing only t he important (dominant) mode s of the original model and to op ti mall y appro ximate the impa ct of the negl ected mode s by a I i near' combinati on of the dominant one s (LITZ . 198Ib). As the red uced mode l \,ill be used for controller' design purpose s. primarily the responses of the three measured temperature s to the four contro l
S. h.
inputs must be approximated. Therefore and due to the equivalence of e i genva lue se ns iti vity and dominan ce (LITZ . 198Ia). th e domi nan t modes can be det ermin ed using Table I above. Due to this table. the use of a mode l of th ird order appears pos sible. Howeve r. so as to yie ld an approximatio n without steady state offset. th e order of the reduced mode l must at l east equal the number of input s, requiring a minimal order of seve n. Moreo ver . the approximation of the nondo minant modes requ i res an order exceed i ng the number of input s. For these reaso ns . the order of the reduced mode l was set to e ight. This cho ice corresponds to a dominance t hre shold of 2.0. The thereby obtained eight dominant eigenvalues are underlined in Ta bl e I. Du e to the in creased order of the red uced model . bes id e the three measured tempe ra ture s additional vari able s ha ve to be in co rporat ed into the reduced model. The se lect ed variab l es, name l y the outlet conce ntration s x, .
X 84
I.Clllll;!llll
mode l and therefore negl ible. Thus the re du ced mode l can be used t o design t he co ntrol la w u
( 5)
As the model (4) co nt a in s as stat e var iabl es not onl y the measured temperatures y. an <, u tpu t hee d b~cl is to be calculated. CONTROLLER DESIGN
The available so lution s for the output feedback problem have been recently reviewed by FbLLINGER (1986). The sc heme employed here i s Si mp l er than the majo r i ty of the method s covered in thi s survey. It re l ies on the numerical minim i zatio n of a sum of f our cr it eria
and
X 86 •
(6)
the co upling var iable s
and
X64
X'06
are of spe ci fic relevance for the dynamic beha viour of th e model. Th e resu lt of the order redu ction is an eig hth order model
by variation of the con t ro ll er gains. Th e criteria J E and J o eva luat e the clo sed-loop pole co nfi guration and are def in ed as n
JE
L
[max
(Re(~;
)
Re mi n + I
0)] 4.
i= l
(7) n
(4 )
Jo
L
1= 1
The dynamic behaviour of thi s mode l i s analyzed by s imulation s . In Fig . 4 the responses of the temperatures on tr ays 18 and 34 t o a st ep change of th e reflux ratio 1', from 0.84 to 0.823 are depicted for the nonl inear mode l . the comp l ete ---
T 34 (Lin. Mo d)
[max : Dl1li n
I Re( A; ) I + I I \; I
0)]3
with the sys tem order n. Th e de s i red d ~ 9~,' e ,'5 ~t(thUit j( Re m ;n and t he de s i red l'l i i/ , nl({{' «(WII).' i Il0 Dmi n have to be specif i ed by the designer. The criteri on
T34 (N onli n. Mod) (8)
T34 (R ed Mo d )
",
€:
as se ss es the ab sol ute values of the contro ll er gains with r mij as threshold s for the se gain s .
~ '04 :J
0 L
E
97
'"
f-
90
T,e (Nonl in Mo d )
T,e (L 'n Mod) T ' ,e (Red Mo d ) 75
t:
':J
74
2'
E
72
'"
f-
70
0
Fig. 4.
The main ad vantage s of th i s approach are:
0 .5
Tim e
Th e critel-ia J E • J o and J R aloe !.',· :/,, (' tlj ci"!lct i "iI ~. which penali ze violations of the bounds Rem ;n' Dm; n and r mij specified by the user. Th e differe nt exponent s of the criteria refle ct their different i mpor tance . Thu s during the minimizat i on of (6) fir st basically J E is mini mized (d ue to the exponent 4. J E ha s th e lal-g es t value for compara ble deviation s ) . When the de s ired degree of stability is achie ved. the contr ibu tion of J E equal s e ssentially zero. Then mainly J B (ex ponent 3) i s minimized. The minimization of J R (exponent 2) i s started. when the require ments for the dynamic beha vioul- of the clo sed- loop system are sati s fied. The contra st to usual penalt y function approaches (for exa mple POLAK. 1971) i s the use of instead of exponential funct ion s . polynomial Thereby . high nu mel-i cal robu s tne ss even fOl- large deviations fro m the minimum i s achi ev ed.
, r'
HO.Jrs
Nonlinear. 1 inearized and reduced model .
lin ear model and for the reduced mode l. The diffel'ences between the complete 1 inear and the reduced model are very small as compared to the di fferences between the nonl in eal' and the 1 inear
Any control stru c ture con straint s de centra l iz ed (output feedback. control) can be taken into ac count. Add i tional require me nt s can acc ounted fo r hy appending (6) suitable criteria .
be by
DistilLtti()11 COlJtrol 1)\ Olltpllt Feedback
As minimal steady -state offset i s the key requirement in this case, the norm of this offset due to ste p changes Zi of the disturbance inputs is added as the additional criterion Pz
L
[ I(~,
( 9)
i =l
with e i denoting the i-th co lumn of the distur and Pz the number of bance - input matrix fr disturbance input s. The application of this criterion is ba sed on the fact that the reduced model due to LITZ provides exact steady state values for all variables and due to all inputs. The effect of this criterion during the minimization of the sum (6) depends on it s value as compared to the other criteria. This value can be i nfl uen ced by the des i gner v i a the input of the Zi' It ha s proven effective to start the minimization with Zi=O and to increase Zi' when the threshold values Re min , Dmin and r",ij are met. The increase of the Zi i s continued, until one of the other specifications again fail s to meet the respective thresholds. The minimization of (6) i s done by a Quasi NEWTON -Algorithm that works using function va lue s onl.Y. For the model on hand , minimization of (6) requires le ss than five secon ds CPU-time on a SIEMENS 7881. With the final specificat ions
Dmin
=
r m13
1.3 mol h· 1 K" ,
r m2 1
r m23
1.0 J
r m31
r m33
0.3 K
r m41
r rn43
0.3 K
S
1
K·
20.0 mol h'
Z2
Z3 =
Re min Dmin
[\ T1800 [\ T3400 [\ T5100
De s ign Specific.
Reduced Model
Linear Model
-1.0 1.0 min min min
- 1.96 I. 00 0.61 0.24 -0.40
- 1.65 0.89 (I. 00) 0.62 0.24 -0.41
The stationary temperature offsets are calculated for a step change of the feed flow rate from 200 to 180 mole s/ hour. For the reduced model all de sig n specifications are met. The impact of the controller on the complete linear model i s quite s imilar - an expected result due to the sharp dominance structure of the system . The decrea sed damping 0.89 i s irrel evant as the corr'espo nding eigenvalue i s nondominant. The figure in brackets denotes the minimal damping of the dominant eigenvalues.
l
(11)
Zl
Table 2 Design Specifications and Propertie s of Reduced and Complete Linear Model
While the effect of the controller on the linear model can be investigated by eigenvalue calculation, for analyzing it s impa ct on the nonl inear model simulation is indi spe nsable. One typical s imulation re sult is shown in Fig. 5 below, namely the temperature profile s for a ste p change of the
1.0 .
r mll
Therefore, the achievement of the closed-loop specifications must be verified upon completion of the design. In Table 2 the design specifications are compared to the respective values of the controlled reduced and the controlled complete linear model.
F = 0.9 Fa
00 h 1.0 h
0.5 h 1.5 h
i ': . .,-------------------------- r -I
O. I
the following feedback matrix is obtained : 0.0496
0.7246
0.6712
-0.0211
-0.0275
0.0001
-0.0010
0.0001
~
(12 )
R
IU
i""
0.8010
~
11
70
I 50~------TI------'Ir------rIL------r11 I 1
11
-0.0308
-0 . 0187
-0.0003
TEST OF THE CONTROL SYSTEM
The used nonlinear minimization procedure permit s the de sig n of constrained co ntrollers and the use of a wide variety of design objectives. By minimization of the sum (6) a reasonable compromise of the - possibly I-ivalling - design objectives is found. Hovlever. as typical for nonl inear numerical optimization . neither the exi stence nor the achievement of the global minimum is guaranteed.
31
41
Tray No. - Ma in Col.
Fig. 5. Obviously. the absolute values of all gains are well below the bounds given b'y (Ill.
21
43
I i i 53
Side Co l .
Closed-loop mode l (step response).
feed flow rate from 200 to 180 moles / hoLII-. The sig nifi ca nt I-eduction of the impact of this di st ur ban ce on the system as compued to the open- loop system (cf. Fig. 2) is obvious. In ordel' to matc h real possible. the effect of
co ndition s as close as
meas urement noi se . time - lags at the control inputs. the finite sampl ing time of 10 s
S. K. Lehm
was tested by simulation . None of these caused any relevant deterioration of the closed-loop performance. As final step the controller was appl ied to the plant that is located at the Univer s ity Stuttgart. For this purpose , the controller was realized as a FORTRAN - subroutine and mapped into the proce s s control and data logging program that is implemented on a POP II and communicates with the plant via a Z 80 microprocessor. In Fig. 6 below the closed - loop response to a s tep change of the feed
cation of the order of the reduced model used and the properties of the closed-loop system is possible. The availability of methods for ~ ljmbL'Ric rl iS6rflcJl ti a ti oJl (BENNINGER/ KONIGORSKI, 1987) permit s the incorporation of the linearization step so as to achieve a complete system for controller design based on the complete nonlinear model. Subsequent re search will concentrate on the improvement of the non 1 i nea r mode 1, the use 0 f more complex control structures (multivariable PI - control , decentralized control) and the test of different design schemes.
1i,,1 . , - - - - - - ; - - - - - - ; - - - - - - , - - - - - - ,
TEl'f". ~2.
&J.
: T ~ --34
---
.......
,
: ...-.-~--
- - - - - - - - • - - - ., - • - - - - - - - - - - -
~
..--.--....
: - .-:----~
- -•- -- ---- - •-
,.. _. ...... ~. T~ .•_~L~~
~
.......
---
__....,. "_,i.______..~ __
~1. ~-:-:-:-::-::-~'f':-:-:-:-~::-:-:~ ~':-:-::':-::~'-~-:-:-:-:-'--:-:-:-:-:-:-'-:':-:-:' .
18
.
ACKNOWLEDGMENT
• - •• - - - - - - - - -
.
The research reported was funded Forschungsgemeinschaft under the and Gi 45/ 39. The support of RETZBACH was a prerequisite for experiment s with the distillation
by the Deutsche grants Fo 71 / 17 L. LANG and B. performing the columns.
REFERENCES
Fig. 6.
Closed-loop system (s tep response).
fl ow rate is depi cted . At t =O, the system is at steady s tate. Although no di s turbance is present, the temperatures on tray s 34 and 51 exhibit an offset. This offset is due to the mismatch between plant and nonl inear model. On the other hand , the s tep change acting at t =O, has only a minimal influence on the plant, as the measured temperatures vary only slightly.
CONCLUSION
The use of a linear reduced model enabled a controller de s ign for two coupled distillation columns. In spite of considerable mismatch between pl ant and nonl inear model the controller achieved (at least) a s ati s factory closed - loop performance. The use of a redu ced model calculated by the method of LITZ - was crucial for the des ign , since the low order of the redu ced model permitted the application of efficient numerical solution scheme s . Linearization and order reduction pro ved to be suitable tool s for the controller design (which is partly due to the exce s sive dominance stru cture of the system on hand), while the mi smatch between plant and nonlinear model limit s the performance achie vable by the control system.
BENNINGER , N.F. and U. KONIGORSKI (1987). Simula tion und Linearisierung ni chtlinearer Sy s teme im Rahmen de s Programmverbundes PILAR . Au tv l "Clt i ~ (.rl[w1g ~t<, c h Jli.l:, 35 , 72 · 78 . BONVIN, D. and D.A. MELLICHAMP (1982). A Unified Derivation and Critical Review of Modal Approaches to Model Reduct i on. III t e.flI1" ti.(' nCl ~ JuufllWJ! 00 COJ1tf\o J.', 35 , 829-848. EITELBERG , E. (1978) . Modelreduktion durch Minimi eren des Gl ei chungs feh 1ers. Re. geRU J1 <] 6 tr chl1 i/: , 26, 320-322. F6LLINGER, O. (1986). Entwurf konstanter Aus gang s ruckfuhrungen im Zustandsraum. Autult/CIU; «,flulIgMr.chll.{/' , 34, 5- 15. LITZ , L. (1979) . Rrdu ll t< OJ7 del( OfldJ1U119 J.'.iJl (, Cl I( ' 1( Zu; tClJld~ I[(lw)\Jno d c.eer
1[,," .
Kl'CfL'qu{'W!i " i\('-) .'~ l!d
As the large system s en countered require the appli c ation of computer s, a software package for simulation, modal analy s i s, order I-eduction and controller de s ign ha s been dev e loped. By mean s of thi s pa c kage a systematic interactive controller design ba sed on the 1 inear model and the s peci fi -
mU td'6
mt!rlCI~('1[
VCfl6ct h-
Hoch schul Verl ag , Stuttgart . LITZ , L. (1981a). Berechnung stabilisierender Ausgangs vektorruc kfuhrungen uber Po 1empfi ndlichkeiten. Rege~uJl9;tr. chJl i l: , 29 , 434 · 440. LITZ , L. (198Ib). Order reduction of linear state - space models via optimal approximation of the nondominant modes. LClI(9" SCClJ.' e Slf !> t em,\ , 2, 171 - 184. LITZ, L. (1983). Modale MaBe fur Steuerbarkeit , Beobachtbarkeit , Regelbarkeit und Domillanz Zusammenhange , Schwach s tellen , neue Wege. R"9l'eW'0.\t~chJ1{I:, 31 , 148 - 158. POLAK , E. (1971). Cc''''putllt ( c'JJ(lf l'lc t h('rI ~ { 11 c'pt i ",i:c1t{l':: . Academic Pre ss, Lond on. PORTER, B. an d R. CROSSLEY (1972). Alcd((f CUll t_~ ,;-J.' . Taylor Francis, London. POST. K. (1984). Alg ebrai sc he Ordnungsreduktion: Erweiterungen in sbe sondere zur Behandlung von MehrgroBensys te men. Bni (iJ t ~ /.1(\:J(( :(1)'1 2. -
dll ·'1.
DFG
Stl'ue't.i1 -
{I' ; .s c J. U t'L' '7.ru ;f !~tp'7.L'9'1. LlP11~1 R<'cll.'{' n \ ' l';1 SI' ~ tCIi;L'; i Pi i t
l:c' '''pfl'Xl' ~ St,~ul: tu~ " . Hanno ver. .' (1984). h:, ;'l'UI' ~ Vn S,liJ ,I' '' ;?u ./~J ~ ,~ edul~tt(' !1 unci ~l!~ire ''L (,J7tu.'U''1. 5
ROTH. H.
:1I .~ ~1U!'
cl,d de '7.
. Fort,c hl"itt · Berichte del' VDI -Nachri chten. Reihe 8 . NI'. 69 , DU s seldorf . Kni~
d(' ~
~ crlu:i"'t";j
.\/(,rI('(,f~
DISTILLATION CONTROL BY OUTPUT FEEDBACK DESIGNED VIA ORDER REDUCTION S. K. Lehmann II U/'I/II! A(;. L\[R -P/al/lIl/g. TEP (1\ SO]). PO
HII.\
so
0] 20.
D-62 J(J FIIII/kjlnl!.\[ SO. FlU;
Abstract. mod~
In order to assess the feasibility of state space methods based on linear control system for a pilot plant of two coupled distillation columns was
designed. The columns, separating a mixture of three alcoholes , can be modelled as a nonlinear sys tem of order ]06. After derivation and verification of a linearized model, this model is analyzed using modal measures. These measures provide hints for the selection of a contro ller st ructure as well as for the succeeding modal order reduction. Using a reduced eighth-order model, a static output feedback of three mea sured temperature s on the four control input s is calculated using a numerical optimization sc heme. The performance of the obtained controller is discussed by means of data from simulations of the nonlinear model and from experiments with the plant itself. The results provide a base for the assessment of the methodology. Chemical variables control; distillation control; eigenvalues; large-scale sys tems ; linear systems: modal control; numerical methods; optimal sea rch techniques; order reduction; output feedback. K~ords .
INTRODUCTION
PLANT AND MODEL
Numerous theoretical results concern ing control system design by state space methods and based on lin ear models are available. As one contribution to the assessment of the application of these methods to real world problems, a control system for a pilot plant of two coup led distillation columns was designed and tested. The method employed is a conceptually s imple technique for the design of structural ly constrained controllers and \vas appl ied to a 1 inearized reduced model of The application oriented paper the plant. emphasizes the assessment of existing procedures rather than the derivation of new co ncepts. This in cl ude s the discussion of results from real plant experiments.
fhe plant, depicted in Fig. ], consists of two coupled distillation columns. The feed stream F contains three alcoholes. This mixture is to be separated into the (almost) pure components methano 1 (top of ma in column), ethanol (top of
Th l organization of the paper is as follows. After the description of the plant and the control ta sk. the derivation and verificatio n of a linear model is discussed. Subsequent sections treat the use of modal measures. the order reduction, the de sig n of a static output feedback and the pe"fonnance achieved by this control system. A summary of the relevant ,'esul ts concludes the pape,' .
*TThe-res'earchi'eportedTrlfhTs- paper waSCOil"=" ducted at the Institut fUr Regelungs· und Steuerungssysteme (Prof. Dr. rer. nat. Dr.-Ing. E. h. Otto Foll ingel') of the Uni ve l'sity Kar ', sruhe. The experiments were run at the Institut flil' Systeflldynamik und Regelungsteci1nik (Prof. D~.-Ing. Erns! Dieter Gille s) of the University Stuttgal't. Federal Republic of Germany.
Fig. ].
Structure of the pilot plant.
side column) and propanol (bottom of main co lumn). As the feedstream may vary in flow rate and composition. a control system i s necessary in order to keep constant the assigned compositions of the three outlet streams. The co ntrol system can influence the plant via four control inputs:
2S11
s.
K.
L C illll
The reflux ratios r, and r z of both co lumn s, the heating power Q. l of the evaporator and th e flow rate of the s ide s tream Os f rom the main to the si de column. The var iable s to be co ntrolled are the composition s of the outlet stre am s K" Kz and B,. As typical for di stillation columns, in s tead of the se concentrations only the temperature s on the tray s can be measured at acceptable cost.
Since the application of the powerful s tate space methods requires a lin ear model, the equations (1) have been linearized by expand ing the functions f and 9 to TAYLOR-~ell.<'n and neglecting all term s but the linear ones. Th e result is a 1 inear model given by ~
The (o nt ro 1 task is th erefore to fi nd a fe edba ck of suitab le measured temperat ure s to the co ntrol inputs that
; Ax
+
Bu
+
E~
,)'M ; fM ~
(2 )
The dynamic matrix A is a spar se matr i x with a s tructure reflecting t he excessive seri al structure of the plant. In Fig. 3 the behav i our of the linear and t he nonl inear mode l are compared by mean s of t he responses of the temperatures on
achieves an acceptable dynamic beha v· i our of the c l osed-loop sys tem , reduces th e impact of change s of feed flow rate and composit i on as much as possible. As th e evaporation enthalpies of the three alcoholes differ onl y s lightly , the energy balances of the tray s may be neglected . Th erefore, the plant can be modelled us ing only two material balance s for any tray. Th e model obtai ned is a non 1 inea r vector differential equatio n of order 106 , which reads
T34 (Nonlin. Mod .) T34 (Lln . Mod) 110
t:
--
:, 98 Q)
0
T34 (Nonlin. Mod) T34 (Lin Mod)
1------- ----
'-
Q)
0.
E Q)
(1)
86 .:;".-
..-:....- -
1-
n
Th e vec tor ~ co ntain s the co nce ntration s of me thanol and ethanol on all 53 trays, the vector l:! the four co ntrol inputs , the vec tor z th e three disturbance inputs, namel y the feed ffow r ate and two feed compos ition s. The temperatures on t he 53 trays form t he output vector IM' Wi th the model , t he behaviour of the open-loop system can be ana l yzed by I1wnell.(.caf ~<'mufaU(O,1. By these s i mu l atio ns, performed on a SIEMEN S 788! mainframe using a variable step s iz e RUN GEKUTTA-MERSON-Method, it cou l d be ver ifi ed that the mode l describe s the relevant properties of the system. One result of the simu lation s is s hown in Fi g. 2 below. Thi s figure s hows the var iati on of the temperature profi l e in the co lu mns ca used by a ste p change of the feed flow rate from 200 moles/ hour to 180 mole s/ hour.
F
=
0.9 Fo
n
, dd
(3 Vl Q) Q)
Q, Q)
9
90
~
I ~~,
,, '
{I
Q)
':J
0
-" , !I. ,...-;~>/
'-
Q)
0.
E Q)
1-
0.5 h 1.5 h
00 h 1.0 h
'" 1
... ~ ....
,<,> /,'/
::./
60 21
31
"Tray No. - Main Col. Fig. 2.
L 1 L3
53
Side Col .
Open -l oop mode l (s tep l'esponse) .
T'8 (Nonl in. Mod .) T'8 (Lin . Mod )
--- ----f--_ T'8 (Nonlln Mod) T'8 (Lin . Mod)
76,------,-----,-----,-----,-----,-----,
~
~
11.)
72
k
--------
~ --
"'-..._
~ 68 1-----1---~~~~~ =_~-r ~.-__---r-----r----~ -_~~-_-.:::
Cl)
1-
____ __
_~
__ _
641-----~-----r----~-----r----~----~
o
0.5
15
Time in Hours Fig. 3.
Nonl in ear and linearized mode l .
the trays 18 and 34 to step chan ges of the reflux rat i o r 1 of :t 2% of it s nominal value 0.84. Whil e the 1 inear mode l does not app,-oximate the sig n-dependable respon se of non 1 inear one. it doe s yield a good approximation. as long as the de via tion fro m the operating point. for whi ch the linearized mode l was calculated. i s suffi cie ntly sma ll. Since the co nt ro ll e r v/ill be desiqned so as to keep this deviation sma ll at all ti mes . the application of th is model for contro l des ign purposes is fea sible. To this end. at fir st a control s tru ct ure i s to be determined. This i s done using ,lil'dar
1)I(l ll~U't C ~ .
APPLICATION OF MODAL MEASURES
Modal meas ures eva luat e quantitati vely the mod al coordina tes of a system concerning propertie s like controllability or importance for the tran smiss ion frolll in puts to output s . During the last years seve ral moda l measure s ha ve been proposed. A re view of some of these can be found in LITZ ( 1983) .
DistillatioJl COJltrol 1)\· Output Feedback
The first question to be answered by modal measures is. on which trays the temperature should be measured. In prin ciple. it is possible to measure the temperature on all 53 trays. Such a structure. howe ver. is as expensive as unnece ssary. Therefore. it ha s to be de cided. on how many and on which trays the temperature i s to be mea sured. A first se le ction ca n be made us ing some heuristic chemical engineering rule s. which re comme nd to take the measurements on tray s. where the temperature varies considerab l y in time and from one tray to the other. In s ight of Fig. 2 above . measuring the temperatures on the tray s 18 and 34 (main co lumn) and 51 (s ide co lumn) appears feasible. After this tentative pla cement i s made. it has to be checked. if by this placement the properties of the sys tem ca n be corrected in the desired manner. This question can be answered by a model measure. The concept of the et9cl1\'a~ue bCI1;(t{\'(t,l. eigenvalue sensitivity wa s introdu ced by PORTER/ CROSSLEY (1972) and used for example by LITZ (198Ia). Eig enval ue sensit i vities are calculated fro m the sys tem transformed to JORDAN-canontcal 6o~m . '*
x
=
Y.. =
-,
"
TAT x
(3)
C ! ~.
In the se equations. ! denotes the medal' II1cttllix (th e matrix of the eigen vectors) of the sys tem (2). ~ contain s those ro\'l s of ~M that corres pond to the mea sured var i abl es. As the effect of a feedba ck of the measured output y on the control input s i s to be mea sured . the s en s iti vities are calculated us ing C T and T·t B only. The sen s itivity of an elgen-val ue equafs zero . if this eigenvalue ca nnot be shifted by the selected feedback struc ture. In order to achieve quantitatively correct va lues (higher sensitiv ity corre sponds to an eigen value that can be shifted more ea s ily) . inputs and outp ut s of the system (2) ha ve to be sc aled to maximal absolute values of approximately 1.0. With appropriate sc al ing factor s. the sensitivities of Table I are obtained. Th e ba s ic re sult of this tabl e is t hat all eigen values can be shifted by the co nt roller stru cture cons id ered. beca use all sen s iti viti es Esse ntial for th e are different from zero. fea sib ility of the control stru cture are the large se ns iti vitie s of the eigenvalue s clo se to the imaginary axi s . Th ese sensitivities i mply the pos s ibil i ty to shift the se e ig enva l ues to the left and thereby to speed up the s luggi sh beha viour of the col umn s . Therefore . the se l ected controller structure i s principally suitabl e for control purpo ses . Be s ide s the po ss ibility to shift an eigen va lue by feedba ck. th e eigenva lu e se ns iti vity assesses it s importance ( dC"1(1I,1I1C l') for the transmi ss ion from the input s to the output s (LITZ. 198Ia). Reviewing Table I , the f ollowing s tatement s regard i ng th e domi nan ce structure of the sys tem can be made: The eigen value s No. deci s i ve ly dominant.
l.
2 and 3 ar'e
The eigenvalue s No. 8. 9 and 10 po ss es an increased dominance. The dominance of the succeeding eige nva lu es declines co ntinuously.
TABLE 1
NO .
Sensitivity - illominance - ) Mea sures by Feedback of three Measured Temperatures EIGENVALUE REAL PART
IMAG . PART
SEN SIT IVITY/ STRUCTURE DOMINANCE MAXIMUM SUM
21
-=T2T.866
22 23 24 25
- 145 . 142 -149.292 - 178 . 414 - 178.414 -207.811 -207.8 11 -237.9 00 -237.900 -259.453
0.0 0.0 5.080 -5.080 7. 633 -7 .638 5.283 -5.283 0.0
16.47 64 . 00 5:21 0.04 0.33 0 . 60 0 .77 I. 42 0.55 0. 55 0.38 0.38 0. 33 0 . 33 0 . 00 0 . 52 0 . 61 0 .04 0.66 0.81 0.17 0.56 0.46 0.05 0.05 0.0 4 0 .04 0 .03 0.03 0.02
-856.002 -856.002
1.156 -1.156
0.22 0. 22
I
2 3 it 5 6 7 8
9
10 IT 12 13 14
15 16 17 18 19 20
26
27 28 29 30 66
67
-0.192 -0.33 0 -2.823 -3 . 302 -9.825 -18 . 433 -25.184 -30.066 -33.620 -33.620 -48.750 -48.750 -64.567 -64 . 567 -69.888 -85.929 -87.080 -105.114 - 117 .3 97
~.413
0.0
0:-0 0.0 0.0
0.0 0.0 0.0 0.0 o.s76 -0.576 T.558 -I. 558 2.895 -2.895 0.0 0.0 0.0 0.0 0.0
0.0 0.0
45 .95 263.16 l8':48
0.52 0.52
--0:14 I. 75 I. 22 I. 36
3.00
2.T2 2. 12 0.96 0. 96 0.88 0.88 0 . 02 I. 36 I. 54 0.15 2.13 2.27 0.26 I. 25
1.11 0.15 0.15 0. 10 0.10 0. 08 0.08 0. 03
39 additional eigenvalues all with dominance s smaller than 0.4
ORDER REDUCTION
After selecting a controller structure and checking it s feasibility. the co ntroller gains have to be determined. Despite the potential of nowaday's computer s, the use of a comp l ete model of the order 106 for thi s calculation yields an expen s i ve ta sk. Due to the do minan ce of only a few eigenvalues (cL Tabl e I). on the other hand. the I ineari zed model ca n be approximated by a model of considerab l y reduced ord er . Thi s l eads to the concept to ~et ermine a redu ced mode l and to use thi s model to compute the controller for the complete one . Many methods for the deri vation of a redu ced model ha ve been deve loped (for example LITZ. 1979: EITELBE RG. 1979 : ROTH. 1983 : POST. 1984) . The method of LIT Z (1979) comb ines numerical s implic ity with good or even excellent approximation qual ity of the reduced model (BONVIN/ MELLI CHAMP. 1982) and wa s therefore employed for the ca l culation of the reduced or'der model. The ba s i c idea of thi s procedure is to ca l culate a model containing only t he important (dominant) mode s of the original model and to op ti mall y appro ximate the impa ct of the negl ected mode s by a I i near' combinati on of the dominant one s (LITZ . 198Ib). As the red uced mode l \,ill be used for controller' design purpose s. primarily the responses of the three measured temperature s to the four contro l
S. h.
inputs must be approximated. Therefore and due to the equivalence of e i genva lue se ns iti vity and dominan ce (LITZ . 198Ia). th e domi nan t modes can be det ermin ed using Table I above. Due to this table. the use of a mode l of th ird order appears pos sible. Howeve r. so as to yie ld an approximatio n without steady state offset. th e order of the reduced mode l must at l east equal the number of input s, requiring a minimal order of seve n. Moreo ver . the approximation of the nondo minant modes requ i res an order exceed i ng the number of input s. For these reaso ns . the order of the reduced mode l was set to e ight. This cho ice corresponds to a dominance t hre shold of 2.0. The thereby obtained eight dominant eigenvalues are underlined in Ta bl e I. Du e to the in creased order of the red uced model . bes id e the three measured tempe ra ture s additional vari able s ha ve to be in co rporat ed into the reduced model. The se lect ed variab l es, name l y the outlet conce ntration s x, .
X 84
I.Clllll;!llll
mode l and therefore negl ible. Thus the re du ced mode l can be used t o design t he co ntrol la w u
( 5)
As the model (4) co nt a in s as stat e var iabl es not onl y the measured temperatures y. an <, u tpu t hee d b~cl is to be calculated. CONTROLLER DESIGN
The available so lution s for the output feedback problem have been recently reviewed by FbLLINGER (1986). The sc heme employed here i s Si mp l er than the majo r i ty of the method s covered in thi s survey. It re l ies on the numerical minim i zatio n of a sum of f our cr it eria
and
X 86 •
(6)
the co upling var iable s
and
X64
X'06
are of spe ci fic relevance for the dynamic beha viour of th e model. Th e resu lt of the order redu ction is an eig hth order model
by variation of the con t ro ll er gains. Th e criteria J E and J o eva luat e the clo sed-loop pole co nfi guration and are def in ed as n
JE
L
[max
(Re(~;
)
Re mi n + I
0)] 4.
i= l
(7) n
(4 )
Jo
L
1= 1
The dynamic behaviour of thi s mode l i s analyzed by s imulation s . In Fig . 4 the responses of the temperatures on tr ays 18 and 34 t o a st ep change of th e reflux ratio 1', from 0.84 to 0.823 are depicted for the nonl inear mode l . the comp l ete ---
T 34 (Lin. Mo d)
[max : Dl1li n
I Re( A; ) I + I I \; I
0)]3
with the sys tem order n. Th e de s i red d ~ 9~,' e ,'5 ~t(thUit j( Re m ;n and t he de s i red l'l i i/ , nl({{' «(WII).' i Il0 Dmi n have to be specif i ed by the designer. The criteri on
T34 (N onli n. Mod) (8)
T34 (R ed Mo d )
",
€:
as se ss es the ab sol ute values of the contro ll er gains with r mij as threshold s for the se gain s .
~ '04 :J
0 L
E
97
'"
f-
90
T,e (Nonl in Mo d )
T,e (L 'n Mod) T ' ,e (Red Mo d ) 75
t:
':J
74
2'
E
72
'"
f-
70
0
Fig. 4.
The main ad vantage s of th i s approach are:
0 .5
Tim e
Th e critel-ia J E • J o and J R aloe !.',· :/,, (' tlj ci"!lct i "iI ~. which penali ze violations of the bounds Rem ;n' Dm; n and r mij specified by the user. Th e differe nt exponent s of the criteria refle ct their different i mpor tance . Thu s during the minimizat i on of (6) fir st basically J E is mini mized (d ue to the exponent 4. J E ha s th e lal-g es t value for compara ble deviation s ) . When the de s ired degree of stability is achie ved. the contr ibu tion of J E equal s e ssentially zero. Then mainly J B (ex ponent 3) i s minimized. The minimization of J R (exponent 2) i s started. when the require ments for the dynamic beha vioul- of the clo sed- loop system are sati s fied. The contra st to usual penalt y function approaches (for exa mple POLAK. 1971) i s the use of instead of exponential funct ion s . polynomial Thereby . high nu mel-i cal robu s tne ss even fOl- large deviations fro m the minimum i s achi ev ed.
, r'
HO.Jrs
Nonlinear. 1 inearized and reduced model .
lin ear model and for the reduced mode l. The diffel'ences between the complete 1 inear and the reduced model are very small as compared to the di fferences between the nonl in eal' and the 1 inear
Any control stru c ture con straint s de centra l iz ed (output feedback. control) can be taken into ac count. Add i tional require me nt s can acc ounted fo r hy appending (6) suitable criteria .
be by
DistilLtti()11 COlJtrol 1)\ Olltpllt Feedback
As minimal steady -state offset i s the key requirement in this case, the norm of this offset due to ste p changes Zi of the disturbance inputs is added as the additional criterion Pz
L
[ I(~,
( 9)
i =l
with e i denoting the i-th co lumn of the distur and Pz the number of bance - input matrix fr disturbance input s. The application of this criterion is ba sed on the fact that the reduced model due to LITZ provides exact steady state values for all variables and due to all inputs. The effect of this criterion during the minimization of the sum (6) depends on it s value as compared to the other criteria. This value can be i nfl uen ced by the des i gner v i a the input of the Zi' It ha s proven effective to start the minimization with Zi=O and to increase Zi' when the threshold values Re min , Dmin and r",ij are met. The increase of the Zi i s continued, until one of the other specifications again fail s to meet the respective thresholds. The minimization of (6) i s done by a Quasi NEWTON -Algorithm that works using function va lue s onl.Y. For the model on hand , minimization of (6) requires le ss than five secon ds CPU-time on a SIEMENS 7881. With the final specificat ions
Dmin
=
r m13
1.3 mol h· 1 K" ,
r m2 1
r m23
1.0 J
r m31
r m33
0.3 K
r m41
r rn43
0.3 K
S
1
K·
20.0 mol h'
Z2
Z3 =
Re min Dmin
[\ T1800 [\ T3400 [\ T5100
De s ign Specific.
Reduced Model
Linear Model
-1.0 1.0 min min min
- 1.96 I. 00 0.61 0.24 -0.40
- 1.65 0.89 (I. 00) 0.62 0.24 -0.41
The stationary temperature offsets are calculated for a step change of the feed flow rate from 200 to 180 mole s/ hour. For the reduced model all de sig n specifications are met. The impact of the controller on the complete linear model i s quite s imilar - an expected result due to the sharp dominance structure of the system . The decrea sed damping 0.89 i s irrel evant as the corr'espo nding eigenvalue i s nondominant. The figure in brackets denotes the minimal damping of the dominant eigenvalues.
l
(11)
Zl
Table 2 Design Specifications and Propertie s of Reduced and Complete Linear Model
While the effect of the controller on the linear model can be investigated by eigenvalue calculation, for analyzing it s impa ct on the nonl inear model simulation is indi spe nsable. One typical s imulation re sult is shown in Fig. 5 below, namely the temperature profile s for a ste p change of the
1.0 .
r mll
Therefore, the achievement of the closed-loop specifications must be verified upon completion of the design. In Table 2 the design specifications are compared to the respective values of the controlled reduced and the controlled complete linear model.
F = 0.9 Fa
00 h 1.0 h
0.5 h 1.5 h
i ': . .,-------------------------- r -I
O. I
the following feedback matrix is obtained : 0.0496
0.7246
0.6712
-0.0211
-0.0275
0.0001
-0.0010
0.0001
~
(12 )
R
IU
i""
0.8010
~
11
70
I 50~------TI------'Ir------rIL------r11 I 1
11
-0.0308
-0 . 0187
-0.0003
TEST OF THE CONTROL SYSTEM
The used nonlinear minimization procedure permit s the de sig n of constrained co ntrollers and the use of a wide variety of design objectives. By minimization of the sum (6) a reasonable compromise of the - possibly I-ivalling - design objectives is found. Hovlever. as typical for nonl inear numerical optimization . neither the exi stence nor the achievement of the global minimum is guaranteed.
31
41
Tray No. - Ma in Col.
Fig. 5. Obviously. the absolute values of all gains are well below the bounds given b'y (Ill.
21
43
I i i 53
Side Co l .
Closed-loop mode l (step response).
feed flow rate from 200 to 180 moles / hoLII-. The sig nifi ca nt I-eduction of the impact of this di st ur ban ce on the system as compued to the open- loop system (cf. Fig. 2) is obvious. In ordel' to matc h real possible. the effect of
co ndition s as close as
meas urement noi se . time - lags at the control inputs. the finite sampl ing time of 10 s
S. K. Lehm
was tested by simulation . None of these caused any relevant deterioration of the closed-loop performance. As final step the controller was appl ied to the plant that is located at the Univer s ity Stuttgart. For this purpose , the controller was realized as a FORTRAN - subroutine and mapped into the proce s s control and data logging program that is implemented on a POP II and communicates with the plant via a Z 80 microprocessor. In Fig. 6 below the closed - loop response to a s tep change of the feed
cation of the order of the reduced model used and the properties of the closed-loop system is possible. The availability of methods for ~ ljmbL'Ric rl iS6rflcJl ti a ti oJl (BENNINGER/ KONIGORSKI, 1987) permit s the incorporation of the linearization step so as to achieve a complete system for controller design based on the complete nonlinear model. Subsequent re search will concentrate on the improvement of the non 1 i nea r mode 1, the use 0 f more complex control structures (multivariable PI - control , decentralized control) and the test of different design schemes.
1i,,1 . , - - - - - - ; - - - - - - ; - - - - - - , - - - - - - ,
TEl'f". ~2.
&J.
: T ~ --34
---
.......
,
: ...-.-~--
- - - - - - - - • - - - ., - • - - - - - - - - - - -
~
..--.--....
: - .-:----~
- -•- -- ---- - •-
,.. _. ...... ~. T~ .•_~L~~
~
.......
---
__....,. "_,i.______..~ __
~1. ~-:-:-:-::-::-~'f':-:-:-:-~::-:-:~ ~':-:-::':-::~'-~-:-:-:-:-'--:-:-:-:-:-:-'-:':-:-:' .
18
.
ACKNOWLEDGMENT
• - •• - - - - - - - - -
.
The research reported was funded Forschungsgemeinschaft under the and Gi 45/ 39. The support of RETZBACH was a prerequisite for experiment s with the distillation
by the Deutsche grants Fo 71 / 17 L. LANG and B. performing the columns.
REFERENCES
Fig. 6.
Closed-loop system (s tep response).
fl ow rate is depi cted . At t =O, the system is at steady s tate. Although no di s turbance is present, the temperatures on tray s 34 and 51 exhibit an offset. This offset is due to the mismatch between plant and nonl inear model. On the other hand , the s tep change acting at t =O, has only a minimal influence on the plant, as the measured temperatures vary only slightly.
CONCLUSION
The use of a linear reduced model enabled a controller de s ign for two coupled distillation columns. In spite of considerable mismatch between pl ant and nonl inear model the controller achieved (at least) a s ati s factory closed - loop performance. The use of a redu ced model calculated by the method of LITZ - was crucial for the des ign , since the low order of the redu ced model permitted the application of efficient numerical solution scheme s . Linearization and order reduction pro ved to be suitable tool s for the controller design (which is partly due to the exce s sive dominance stru cture of the system on hand), while the mi smatch between plant and nonlinear model limit s the performance achie vable by the control system.
BENNINGER , N.F. and U. KONIGORSKI (1987). Simula tion und Linearisierung ni chtlinearer Sy s teme im Rahmen de s Programmverbundes PILAR . Au tv l "Clt i ~ (.rl[w1g ~t<, c h Jli.l:, 35 , 72 · 78 . BONVIN, D. and D.A. MELLICHAMP (1982). A Unified Derivation and Critical Review of Modal Approaches to Model Reduct i on. III t e.flI1" ti.(' nCl ~ JuufllWJ! 00 COJ1tf\o J.', 35 , 829-848. EITELBERG , E. (1978) . Modelreduktion durch Minimi eren des Gl ei chungs feh 1ers. Re. geRU J1 <] 6 tr chl1 i/: , 26, 320-322. F6LLINGER, O. (1986). Entwurf konstanter Aus gang s ruckfuhrungen im Zustandsraum. Autult/CIU; «,flulIgMr.chll.{/' , 34, 5- 15. LITZ , L. (1979) . Rrdu ll t< OJ7 del( OfldJ1U119 J.'.iJl (, Cl I( ' 1( Zu; tClJld~ I[(lw)\Jno d c.eer
1[,," .
Kl'CfL'qu{'W!i " i\('-) .'~ l!d
As the large system s en countered require the appli c ation of computer s, a software package for simulation, modal analy s i s, order I-eduction and controller de s ign ha s been dev e loped. By mean s of thi s pa c kage a systematic interactive controller design ba sed on the 1 inear model and the s peci fi -
mU td'6
mt!rlCI~('1[
VCfl6ct h-
Hoch schul Verl ag , Stuttgart . LITZ , L. (1981a). Berechnung stabilisierender Ausgangs vektorruc kfuhrungen uber Po 1empfi ndlichkeiten. Rege~uJl9;tr. chJl i l: , 29 , 434 · 440. LITZ , L. (198Ib). Order reduction of linear state - space models via optimal approximation of the nondominant modes. LClI(9" SCClJ.' e Slf !> t em,\ , 2, 171 - 184. LITZ, L. (1983). Modale MaBe fur Steuerbarkeit , Beobachtbarkeit , Regelbarkeit und Domillanz Zusammenhange , Schwach s tellen , neue Wege. R"9l'eW'0.\t~chJ1{I:, 31 , 148 - 158. POLAK , E. (1971). Cc''''putllt ( c'JJ(lf l'lc t h('rI ~ { 11 c'pt i ",i:c1t{l':: . Academic Pre ss, Lond on. PORTER, B. an d R. CROSSLEY (1972). Alcd((f CUll t_~ ,;-J.' . Taylor Francis, London. POST. K. (1984). Alg ebrai sc he Ordnungsreduktion: Erweiterungen in sbe sondere zur Behandlung von MehrgroBensys te men. Bni (iJ t ~ /.1(\:J(( :(1)'1 2. -
dll ·'1.
DFG
Stl'ue't.i1 -
{I' ; .s c J. U t'L' '7.ru ;f !~tp'7.L'9'1. LlP11~1 R<'cll.'{' n \ ' l';1 SI' ~ tCIi;L'; i Pi i t
l:c' '''pfl'Xl' ~ St,~ul: tu~ " . Hanno ver. .' (1984). h:, ;'l'UI' ~ Vn S,liJ ,I' '' ;?u ./~J ~ ,~ edul~tt(' !1 unci ~l!~ire ''L (,J7tu.'U''1. 5
ROTH. H.
:1I .~ ~1U!'
cl,d de '7.
. Fort,c hl"itt · Berichte del' VDI -Nachri chten. Reihe 8 . NI'. 69 , DU s seldorf . Kni~
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