TPI - 3:30

NONLINEAR CONTROL OF AN EXOTHERMIC CSTR

A. Cinar, K. Rigopoulos # S.M. Meerkov +

Dept. of Chemical EngineeringIllinois Institute of TechnologyChicago, IL 60616

Dept. of Electrical Engineeringand Computer ScienceThe University of MichiganAnn Arbor, MI 48109

Dept. of Electrical EngineeringIllinois Institute of TechnologyChicago, IL 60616

ABSTRACT

Nonlinear control strategies are designedfor an adiabatic CSTR with an exothermic reac-

tion. Conventional feedback and nonlinear feed-back strategies are implemented using the feedflow rate as the manipulated variable. Vibra-tional control is implemented using either vibra-tions in feed flow rate or vibrations in bothfeed flow rate and feed concentration. Experi-mental results illustrating the performance ofvarious strategies are presented.

INTRODUCTION

Control of nonlinear chemical processes isusually carried out using conventional feedbackstrategies. These strategies are sometimes ade-quate for successful control but occasionallypractical limitations may prevent the implementa-tion of the linear control strategies. Use ofnonlinear control strategies may circumvent theselimitations. In this study, two nonlinear con-

trol strategies, the nonlinear feedback (push/pull) control and the vibrational control will beapplied to a nonlinear system.

The adiabatic CSTR is controlled in order tomaintain the reactor states in the vicinity ofunstable steady states. To achieve stabilizedoperation at an unstable state a conventionalfeedback controller was proposed and tested ex-

perimentally 11i. The reactor temperature was

used as the measured variable and cooling waterflow rate was manipulated to regulate the reactortemperature. In this study reactant flow rates andreactant inlet concentrations are used as manipu-

lated variables. This choice is stimulated bytwo reasons. If successful control can beachieved by manipulating the properties of thereactor feed stream the cooling system my beeliminated. Consequently, both capital and oper-ating expenses vould be reduced. Furthermore,the constraints on the cooling system (heat ex-

change area available, maximum coolant flow rate)are such tighter than the constraints on the feedstream. Also, variations in the inlet streamproperties are more effective on the reactor

Supported by the U. S. Department of Energyunder Grant DE-FW02-84ER132O5.Supported by the U.S. Departmnt of Energy

under Grant DE-FGO2-85ER13315.

states than the variations in coolant flow rate.Consequently, the use of the feed stream proper-ties would result in a more flexible controlsystem under emergency situations.

CONTROL STRATEGIES

Three control strategies are implemented forstabilizing reactor operation in the vicinity ofan unstable steady state. Conventional feedbackcontrol with feed flow rate as the manipulatedvariable is used as a reference case. Althoughthis configuration is not used often in practiceit enables a good basis for comparison. Non-linear feedback control and vibrational controlare discussed in the following sections.

Nonlinear Feedback Control

Nonlinear feedback control is based on theuse of a relay with hysteresis as the controlelement E2,3]. The closed-loop system is de-signed to ensure a stable, small amplitude limitcycle in the vicinity of the desired unstablesteady state operating point. For the design ofthe relay, techniques such as Describing Functionanalysis or Tsypkin's method can be used. The

necessary and sufficient local stability condi-tions and solutions for oscillatory states can bedeveloped t2,31.

Vibrational Control

Vibrational control is a method for modifi-cation of the dynamic properties of linear andnonlinear systems by introducing fast, zero-av-

erage oscillations in the parameters of the sys-

tem t4-63. Using the method of averaging C71 theconditions of vibrational stability and vibra-tional controllability have been derived [6].Since a detailed summary of the method of analy-sis is given in C8] only a brief description willbe provided here.

The analysis of a vibrationally controlledsystem is equivalent from the mathematical stand-point to the analysis of a dynamical system withparametric oscillations.

For a system of the form

i a XCx,ca) (1)

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X. Shu +

where x is the state, t is the dimensionless timeand 0 is a fast oscillating parameter defined by

-a=a + (a/c)f(ti) . (2)

Here r, is a constant, f(.) is a periodic orquasi-periodic zero-average vector, O<E <<I andO<ct are scalars. Assuming that (1) with cr as in(2) has the form

dx/dt 5 Xl(x)+(a/Jc)X.(t/fc,x), (3)

where X. (t/t_,.) is a periodic or quasi-periodicfunction.

Defining the fast time 0 t/t, (3) can be writtenas

dx/de 2 EXd(x) * aXe (8,x) (4)

The general solution of a generatingequation

dx/d6 a aXm (O,x) (5)

is denoted as

x(e) = h(e,c) , c = constant. (6)

When X (9, x) is differentiable with respect to xa substitution which reduces (4) to a standardform is x(e) a h8,y(G)). The reulting equationin the standard form is

dy/do a ccah/ayl-'X.(h(e,y)) * sY(8,y). (7)

To analyze (7) further, asymptotic techniques canbe employed. Introducing

IT Alim - J C3h/By-lX,Ch(8,y)3de Z(y) (8)

T-1z2 T 0

the averaged equation can be defined:

dz/de = eZ(z) - (9)

If the assumptions stated in (83 are satisfiedwhenever (9) has a locally asymptotically stableequilibrium point zs, (7 ) has an asymptoticallystable periodic solution y*(O) and for positiveS as small as desired (ee C81)

II X (y*).-.j(zs) II <6 (10)

Here x5(y*) is the average steady state of (4)and x5(z) is the approximated average steadystate of (4).

THE REACTOR SYSTEM

The homogenecus liquid phase reactionbetween sodium thiosulfate (Na,S.C3) and hydrogenperoxide (H1O10) is used in this study. For thereactant concentrations used in the feed streamthe stoichioaetric equation for this second orderexothermic reaction is

2Na.SM ,SU + 4H O. -> Nae St O + Ma, SO* + 4H, 0.

The experimental system (Fig. 1) is con-structed such that feed flow rate and/or concen-tration vibrations can be accomplished withoutchanging the ratio of the reactant concentrationsin the feed. This asures that there are no sidereactions. Previously a two feed stream configu-ration was used instead of the three feed streamconfiguration, causing the formation of hydrogensulfide. The current configuration eliminatessuch side reactions but the injection of a purewater stream necessitates the use of more concen-trated reactant solutions. Since the Na,S.Ossolution started to crystallize in the feed lineswhen its concentration was increased beyond 2.9mol/l, the magnitude of the amplitude variationswas severely constrained for this test reaction.For the experiments presented here the coolingcoil was removed from the reactor.

The Reactor Model

The material and energy balances describingthe CSTR (83 reduce to the following set ofdimensionles equations for adiabatic reactoroperation:dx1/dt s axI1+a(1-xI )(-x) exp [x](14.zJY)]dx2/dt a -x24pDa(1-xl )(1I-x2) exptx3/(1+xjy)1dx3/dt a -Xx3+XBDa(1-xl)(1-x2)exp(xy'(1+x3/)M]Here b and p denote the ratio of the stoichiomet-ric coefficients and of the feed concentrations,respectively, c and c denote the H&C6 andNa.Se03 concentrations, T is the reactor tempera-ture and the dimnsionless variables are:

x aCC - cA)/C X 2 (C3 C3)/C39

y a E/RTf. X3 = (T - Tf)/TfI y, r=V/F

Oa a C3f k0 exp(-y)r, X s l/(l4mfrCrIQCpV)

B a (-A)CAf /p cp TfThe temperature dependent exponential term issimplified using Q, a 1-x.I;4x.'/r' z i/(ix./yX)for the remainder of this work since x./jc<<.

VIBRATIONAL CONTROL OF THE CSTR

Vibrational control of the reactor was con-sidered using forced oscillations in one or twoinput variables. When a single input variableis vibrated, forced oscillations can be intro-duced in the feed flow rate, in the feed concen-tration or in the coolant flow rate. When twoinput variables are vibrated any two of thesevariables may be used.

In this work, vibrations in feed flow rateand vibrations in feed flow rate and concentra-tion are studied. The formulation of the modelequations for feed flow rate vibrations and theapplication of the method of asymptotic analysisfor this case is presented in (Sb. The model ofthe system with two input vibrations and theaveraged equations will be presented here.

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When two input variables are vibrated theamplitude and frequency of forced oscillation maybe selected. If a common frequency ia used thephase shift between the two oscillations providea degree of freedom. In this study, zero-averagenonsyuwtric rectangular waveform (Fig. 2) areused. Denoting the duty fraction by oG and thephase shift by a, the three alternatives areoO>O, 0aOQ and a>-U>G. For this reactor systemthe mt promising results are obtained using thethird alternative.

Oscilationa in feed flow rate and concentra-tion are represented as

F F j o (1 +AFf(wt)) -

CAF=C4F (1+ACf(ft))a

IFMFm

nT<tc(n+a)T(n+a)T<t<(n+1 )T

CAFM nT<t<(n+a)TlAFm (n+a)T<t<(n+l)1

where the subscript o denotes the average value,and f and m denote the maximum and sinimumvalue respectively. Defining the variousaverage values

CAF aCAFM + (1a)CAFm F0o * aFm + (1-a)Fn0~~~~~~~~~4 i

Rc CAFM/CAFrmRF a fff/Fm

CAF = I1TF00

CkF n (aRC+lI-a)cAFfm

T

so cAF F dt

CAF KFC cAF0 0

KrFC [(aRF+1 a)(aRc+l-a)I/[(a+r)RcRF-(Rc+R)+1 + C+a]

The dimensionless equations for the reactorsystem with vibrating feed flow rate and feedconcentration are

dxI/dtn - x1 l+AFf(ct)]+f1 (wt)

+ Da(l-x1)(1-x2)exp(F0(x3)]

dx2/dta-x21[-+AeFfx(t)]+f(xt)+ bpDa(I-x1)(l-x2)exp[F,,(Y3]

dx3/dtnX{-x331 +AFf (ct) ]+ 80a(1-x1)(1-x2)exp [F0(x3)] }

with

fl(tt)= (l-KFC)AF Z Bkcos(kwt) KFCAc

m

L 'kCosk-

[k(wt-WCt)F-K AcAF U [as iin (kwt)+8cos (kwt)]k=1

The coefficient Si is the same for all cases buta" and S vary for each case3

k = (2/k-r)sin ak-r

all= (1/ kTr)(- (1 -a) cos (a+2a) kr+cosakTr-acos (2a-a) k7r}

8-(1/kr){(1 -a)s in (a+2a)kr+C1( -2a)si nakrv+asi n(2a-a) kTr

The resulting averaged equations are

1'- Zk + S + Da expFa1 + Ya C(AF 1)2 + A2F2] + C7(F 2 + F2) -(z1+z2)(1 + X'4 (F1 F)]

+ z1z2(1 + 92 ((Xfl + 1)2 + X2 F2)] + C7(F 2 + F)1

*2; {1Z + S + bp Da exp F {I + y2 ((F - 1)2 + A2FJ] + C (F2 + F) -(z+Z)[l + A22 (F2 + F)]2 z2+SbDaxF a 1 2 7 1 2 1 2 a'1 2

+ z1z2(1 + y9 ((xr1 + 1)2 + A2F2)] + C 2(F2 + F2)1

A{- z + BDa exp Fo [I +Ya 2((F1 1)2 + F2) + C7x2(F12 + F2j3 3

+Ba x F 1+ x + + 2' +-(Z + Z )(1 + Y9 (X2F2 + (x - 1 x )2)) + z1z2(I + y2(X2F2 + (x-2-XF1)2) + C7(2-X)2(F2 + F2 1}

with

S (1 - Krc)Ar (1/8) aF BB2k B - KFC A0 ((1/8) ar2

BkB2k cos k2ira + (1/4)aF2 Si'

k-l c k=l

(81+ S2)cos 2Ta + (1/2)aF U Bk8k sin k 2rra} - KFCACAF ((aFt2)Y ak Bk + (4F/8) U kknl k=l k=l

the material balance of A gives

VdCA/dt'n - VkOC;ACB exp(- E/RT)

Flo (1+AFf (w 't' ) ) {KFcCAFo (1 +AC f lw' (t'oT) )-CA}

and4

f n=o /(_Y)n=o

a = (2/rrk 2) sin akr

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4F1 = Z (n+l)znT /C-r)n

n=o

4F2 E (n+1)2 Z3 n'l /(_,)n

n=o

y2 = 1/4 a2CLFn=l

6k2

Discussion of Theoretical Results

The effect of vibrations in feed flow rateusing nonsymmetric rectangular waveforms arepresented in Fig. 3. Curve 1 is for fixed feedflow rate, Curves 2 to 4 show the average steadystates for various frequencies of vibrationsranging from 1 to 3 respectively. As the vibra-tion frequency is increased the average steadystate curve approaches the curve for stationaryinputs. Various properties of vibrations in oneinput variable are presented in CJ].

Vibrations in two input variables providemore freedom in shaping the characteristics ofthe response. The ratio of the maximum to mini-mum flow rate, the ratio of maximum to minimumconcentration, the oscillation frequency, and thephase shift between the two oscillations areamong the parameters that can be modified toimprove the behavior of the state variables. Theeffect of these parameters on the average steadystate curves are presented in Figures 4-8. Inall these figures the dimensionless conversion isrepresented with a solid line while the dimn-sionless temperature is shown with a broken line.The effect of frequency is shown in the absence(Fig. 4) and in the presence of phase shift (Fig.5). As it was observed with a single variableoscillations an increase in frequency (curve 2-4correspond to frequencies of 8, 6 and 4, respec-tively) shifts the average steady state curvecloser to the steady state curve for fixed inputs(curve 1). The presence of phase shift causesdramatic changes (Fig. 5). The effect of thefrequency variat±ons are much more pronounced.

The effect of varying the flow rate ratio isgiven shown in Fig. 6. As the ratio R. isincreased (Curve 2-4 corresponding to R.= 4, 8,12, respectively) the average steady state currvemoves away from the fixed input curve (Curve 1).In Fig. 7 the effect of phase shift ( Curve 1-6corresponding to r-O, -0.05, -0.1, -0.15, -0.2,-0.25) is presented. Of the three types of phaseshifts only Type 3 (t>-U>O) gave promisingresults for this reactor system.

Simulation studies using the reactor modelindicates two major advantages of vibration oftwo input variables. Figure 8 illustrates a casewhere large oscillations in concentration andtemperature are observed. The conversion achievedin this case is substantially larger (0.88) thanthe conversion corresponding to the average reac-tor temperature (0.81). Hence at the expense ofan increase in oscillation amplitude a separationof the conversion curve from the temperaturecurve is achieved. Figure 8 shows the other al-

ternative. Here the conversion is equal to thatindicated by the corresponding temperature butthe amplitude of the tempersture swings is dras-tically reduced to about 106C. Hence a tradeoffexists between an increase in conversion over andabove to that corresponding to the reactor tea-perature and the amplitude of the temperatureswings in the reactor.

Another achievement with vibrations in twoinput variables is the intersection of thenegative slope part of the steady state curvewith fixed inputs (Curve 1 in Fig. 10) by theaverage steady state curve ( curve 2). Since atthe intersection point the average steady statecurve corresponds to stabilized operation,stabilized operation is achieved at the sameresidence time as that of the fixed operation.Not only the states (concentration and tempera-ture) but also the residence times match.Therefore, operation at a closer vicinity of thedesired unstable steady state can be achieved.

EXPENIMENTAL RESULTS

Experiments with conventional and nonlinearfeedback, with vibrations of single or multipleinput variables have been conducted. Properlytuned P1 controllers were successful in enablingoperation in the vicinity of the unstable point.A limit cycle always existed when feed flow ratewas used as the manipulated variable (Fig. 11*.The P1 controller was able to handle both setpoint variations and disturbances. The existenceof a variable dynamic process gain is shown asthe set-point is changed C Fig. 11). The manipu-lated variable oscillates between the two extremevalues (Fig. 12), the frequency of oscillationdepends on the set-point value.

Nonlinear feedback control was implementedby specifying an upper and a lover temperaturefor switching the feed flow rate from its maxi-mum to its minimum value. The controller perform-ed well (Fig. 13) at various set-point values andunder set-point changes and disturbances. Theamplitude of the temperature swings depended onthe set-point location (at higher temperaturesthe amplitude increased) and on the temperaturedifference between the upper and lover switchingpoints. As the difference is reduced the magni-tude of the swings first reduced then becameconstant. For our system the amplitude of thereactor temperature variations stayed constantfor switching point differences less than 50C.Only syumetric switching point settings aroundthe set-point temperature were tested.

Vibrational control experiments supportedthe results of the analytical studies and simu-lations. The transient and steady state behaviorfor vibrations in a single input are given inCS]. The reactor temperatures measured in theexperiments vith vibrations in two inputs confirsthe theoretical results. The conversion seasure-ments gave promising results but the measurementprocedure needs further refinement. A summary ofthe stabilized reactor behavior for various con-trol approaches is given in Fig. 14. Further

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a-F £AF

experimntal results will be presented at the Air (meeting. SuppLy u t

CONCLUSIONS

All control strategies tested were success-ful in enabling stabilized reactor operation in N2 2 H2the vicinity of an unstable steady state. When nHt;Sthe input flow rate is used as the manipulatedvariable limit cycles develop for all strategies a bused. If the amplitude of temperature swings areacceptable all three approaches would provide pupsuccessful controllers. Otherwise, a coolingsystem must be installed and the cooling waterflow rate must be used as the manipulated bath |variable, resulting in an increase in capital and 2N23 bathoperating costs.

In vibrational control, use of vibrations intwo input variables enable achievement of stabi-lized behavior with small swings in the control-led variables or with higher conversions than the drum {conversions corresponding to the stabilized reac-

tor temperature. Since an open-loop controller ControLis used, vibrational control is attractive for veLv|scases when measurements are too difficult or 4 L

involve considerable delays. 1 1

REFERENCES Adi aboti c

Cl3 Chang M. and R. A. Schmitz (1975). Feedback J product streaControl of unstable states in a laboratoryreactor. Chem. Ena. Sci. 30 837-846. Figure 1 Schatic Diagram of CSTR Network

E2] Bruns D.D. and J.E. Bailey (1975). Processoperation near an unstable steady state usingnonlinear feedback control. Chem. Ena, ScT. _l T30 755-762.

(3] Bruns D.D. and J.E. Bailey (1977). Nonlinear C ocTfeedback control for operating a nonisother- CAF Amal CSTR near an unstable steady state. Chem.Enc.Sci. 32 257-264. FM F

(4] Meerkov S.M. (1980). Principle of VibrationalControl: Theory and applications. IEEE Trans. C.- __Autom. Cntrl AC-25 755-762.

(5] Meerkov S.M. (1982). Condition of Vibrational Fo F--- - - CFmstabilizability for a class of nonlinear sys- [tea. IEEE Trans. Autom. Cntrl AC-27 485-487. T ME

CS] Bellman R., J. Bentsman and S.M. Meerkov (cx+)T l dEt(1983). Nonlinear systems with fast paramet- Fig. 2. Nonsymmetric rectangular pulse.ric oscillations. J. Math. Anal. And1. 97572-589. 1 90

t7] Bogoliubov N.M. and Yu. A. Mitropolsky b 1(1961). Asnvmtotic Methods in the Theory of 44Nonlinear Oscillations. Gordon & Breach, New 3 |

York, MY.23£8] Cinar A, K. Rigopoulos, S.h. Meerkov and X.

thermic CSTR. Proc. ACC Seattle, WA.

0nonsymetric . nonsynetricZJ rectangular t rectangular

S 10 15 $ 10 15

Residence time T (sec)

Fig. 3. Stabilized response to vibrations in feedflow rate.

1021

Fig 4. Effect of frequency (f--O) Fig 5. Effect of frequencyand phase shift.

Fig 7. Effect of phase shiftFig a. Transient response.

Separation of 7 and c.

Fig 6. Effect of feed flowratio.

Fig 9. Transient response.small 7 swings.

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LiLU0

U)IC

I.-cccLUILxLliI.-

Fig 10. Intersection of sta-bilized states.

win

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TIM~U SEC. 3Fic

Fig ii. Experimental response.Feedback (PI)

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12. Experimental response.Manipulated variable.

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Lu

LiSL

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1001

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Fig 13. Experimental response.

Nonlinear feedback.

LuI

~ ~

LLI50&4*rimpits ~ ~ ~ Silulatiorts

v04

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Fig 14. Summary of experimeental responses. Vibra-

tional control with feed flow rate.

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