03-twotanksystem

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Approximately Time Optimal FuzzyControl of a Two Tank SystemThomas Heckenthaler and Sebastian Engell

his article deals with the level controls in a laboratory two-T ank system. The plant is strongly nonlinear due to the basicdynamic equations and the characteristics of the valves. Wedeveloped a nonlinear control law which achieves robust ap-proximately time-optimal control over the full range of op erationconditions. The developm ent is based on ideas from fuzzy con -trol, but in contrast to usual fuzzy controller designs , most of therules are not derived from h euristics but rather are mathem aticalformulas which, together with the standard fuzzy quantization ofthe systems variables, approximate the time-optimal control law.This approximation is improved by heuristic rules which weregained from the observation of the behavior of the controlledplant. The resulting nonlinear control law exhibits a performancewhich is neither attainable with standard linear control nor w ithclassical time-optimal control.

New Approach to Fuzzy Control LawFuzzy control is at present being extensively explored as ameans to develop controllers for plants with complex and oftennot precisely known dynamics. In almost all cases, the basis ofa fuzzy control law is heuristic knowledge about a suitablecontrol strategy. This knowledge may be acquired by interview-ing operators who have a lot of experience in the m anual controlof the plant, or it may stem from basic common sense engi-neering reasoning. If the dynamics of the plant are complex,however, basic reasoning is often insufficient for the derivationof suitable controls. Furthermore, even if there are experiencedoperators available, the formalization of their knowledge is usu-ally difficult, and doubts about the quality of the solution mayremain even if the implemented controller behaves reasonably.

Therefore, we decided to take another approach to the deri-vation of a fuzzy control law. Our starting point was to assumethat there exists a - ossibly crude- athematical model ofthe plant dynamics. Then, instead of asking, What does theoperator do? we ask, How would the ideal operator perform?One an swer to this question, and in many cases a very reasonableone, is that large deviation s of the actua l state of the plant fromthe desired state should lead to control actions which fully usethe available range of actuator movem ents to bring the plant stateto the vicinity of the desired state as quickly as possible, whereasaround this desired state, a behavior similar to that of a linearcontroller or, a linear controller with a dead zone, is appropriate.Presented at the Second IEEE Con ference on Control Applications,Vancouvei;B.C., Canada, September 13-16 ,1993 .The authors arewith the Department of Chemical Engineering. University of Dort-mund, 4422 Dortmund, Germany. This researchwas upported bythe Deutsche F ~ ~ r s c h u i ~ g s ~ e ~ ~ e i n s c ~ ~nder Grant DRG En 15218-1.

In other w ords, we w ant almost time-optimal control for largedeviations and robust, not too hectic control, for small ones.Thus as a first step, time-optimal controls are derived andsimulated for a plant model. Then the quantitative system vari-ables are classified into qualitative values and for each of thesevalues, a formula for the control law is computed. The classifi-cation is fuzzy with significant overlap, so that several controllaws are usually activated. The resulting controller output is theweighted mean of the outputs of these local control laws. Fuzz-ification is used here to provide a smooth interpolation of thelocal control laws.The controller which we thus o btained contains eight rules forthe control of the level of one tank. It approximated the time-op-timal controller very well, including its unattractive features,particularly its high sensitivity to noise in the m easurements andto m odeling errors. At this point, after the theoretically foundedmethods had been exhausted and not before, we brought inheuristics. The trajectories of the system were observed andadditional rules were formulated to improve the robustness. Oneobvious and simple measure was to introduce a linear controlband for small regulation errors. In our case, the d ecisive stepwas to change the rules with respect to one input in order tomodify the time-optimal control law depending on the state ofthe system. The implementation of the com plete control law inthe form of rules provides a flexible and uniform representationand allows for the easy implementation of further heuristicknowledge, constraints, etc.This method proved to be successful in approaching thecontrol of the cou pled system. The strictly time-optimal controllaw for the fourth o rder nonlinear system is not only hard tocompute, but it is also undesirable from a practical point of view(see below). In co ntrast, the fuzzy controllers for the decoupledtanks were suitable to integrate further heuristic knowledge inorder to handle the couplings. By means of only four simpleadditional rules we obtained a performance at the real plant whichis superior both to conv entional linear control and strictly time-optimal control.

Description of the Wo Tank SystemThe plant we want to control consists of two tanks T1 and T2connected as shown in Fig. 1. Their levels h i and h2 are controlledby the actuators V1 and V2. The control valves can only beopened and closed rather slowly. It takes about 80 s for a completemovement from fully open to closed and vice versa. Due to thisspeed limitation, a three-term controller is applied, so that thevalves are either opened or closed at maximal speed or remainin their present position.The following mathematical model results from the massbalance equations for the two tanks:

24 0272-1708/94/ 04.0001994IEEE IEEE Control Systems


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I ~2 outletFig. I Laboratoiy two-tank system.

hi = ( V , V12 ) / A I and h2 = (V12 - V a ) / A 2 . ( I )A1 and A2 are the cross-sectional areas, and 5 e is the volumetricinflow of water into T I . The outgoing volum etric flows 5/12and Va are determined by Toricelli's law:

P1.2 = [O 801 (3)

P1,2 I = IS - ' = const.The acceleration in the tubes can be neglected due to the sm allmass of water in the tubes. The coefficients K1 and K2 aredetermined by the respective valve positions Pi and P2; Fig. 2shows the relation between the coefficient K1 and the valveposition for various ingoing volumetric flow rates, as determinedexperimentally. The resulting curve was determined by interpo-lation and is significantly inaccurate at the extreme positions ofthe valve. If this curve is used to calculate mod el-based con trolstrategies, this inaccuracy can lead to unsatisfactory systemperformance, especially when sensitive time-optimal control is

applied. In addition, none of the flows (Ve, 5/12, Va ) are meas-ured online. The inlet flow is estimated on the basis of theinterpolated valve characteristic by evaluating ( 1 ) and can beused as an (inacc urate) controller input.

+ 340 I h% 240 I hx 140 I h

, \

a0 20 40 60Valve position PIFig. 2. Valve low characteristic.

June 1994

The plant exhibits a strongly nonlinear behavior because ofthe n onlinear flow characteristics, Toricelli's outflow equation,actuator saturation, and the three-term controller. The controlobjective for the plant is to get reasonable transients for (simul-taneous) setpoint changes in both tanks over the entire operationrange. The com plexity of this seem ingly simple plant becameobvious as we first tried to contro l it manu ally. The time nee dedfor manual control experiments was very long becau se of theslow system dynamics. For this reason, the development ofheuristics based on exhaustive experiments which could eventu-ally be used in a fuzzy rule base did not seem promising.Columns T1 and T have different diameters ( d n = 0.12 mand d n=0.05m). Because the time constant of tank 1 is six timeslarger than that of tank 2 the level in TI is more difficult tocontrol. Therefore we started with the development of a con trol-ler for the leve l in this tank.

Conventional Controllers for Decoupled Tank TILinear ControllerAs one might expect, a fixed-gain linear controller cannotadequately cope with the nonlinearities of this system. AP D-con-troller must be used because of the double integrator which ispresent in the plant. With the regulation error el the rate time

7 6 12 s, and the sampling interval Ts = 1 s. the PD -controllerwas formulated as follows:u ~ > D B close valveU,,


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second orde r (nonlinear) system, the valve motion has only to beswitched onc e, since the linearized system always has real eigen-values and the valve characteristic is monotonously decreasing.Supp osing that the regulating error is negative, the valve VI hasto be driven in such a way as to release the water above the desiredset point level as fast as possible. The valve h as to be opened firstand then must be closed at a critical time so that the level doesnot oversh oot. It is therefore clear that more switching opera tionsincrease the settling time.Th e time optimal control action can be found by computingthe valve position Poptwhich allow s the set point to be reachedas fast as possible without overshooting. This can be doneassuming that the inlet flow Ve is constant over time and that themathematical m odel as described in the second section is valid.The optimal valve position P o p can be found by backwardintegration of equations (1)-(3), starting from the set point atsteady state. The control law can then be stated as:

if P ( t )>Popthen close valveif P ( t )< P o p hen open valve.Fig. 4shows the simulated system using the time-optimalcontroller. Although this controller performs well in situationswithout measurement noise, it is not a good choice at the realplant. Modeling inaccuracies and measurement noise lead toinefficient valve movem ents, and this causes settling times whichare not only non-optimal, but worse than those obtained with PDcontrol (see Fig. 5 ) .

Approximation of Time-Optimal C ontrol Lawby a Fuzzy ControllerMotivationThe application of fuzzy log ic in a controller typically resultsin a controller which consists of three compone nts. In the fuzzi-fication compon ent, the controller inputs are classified into cer-tain fuzzy variables. i.e., they are converted from real numbers

80

5 60g 400

.

0u

00 100 200 3000 0 1 200 300TII IK 1 Time [ \ ]Fig. 4. Time-optimal coilfro1 of level (.\imulntion).

Time[s] e IO0 200 300Time1sFig 5. Timc~-q ~ t ima lontrol ofle\zel Lit the real plunt.

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into qualitative variables with fuzzy mem bership functions. Thisis followed by the evaluation of the rule base in the inferencecomp onent. Here the actuator move ment is determined by meansof linguistic rules. The results of these are combined into aunique , but still fuzzy, output using fuzzy logic. Sinc e the finaloutput of the controller has to be a crisp real number or one of anumber of alternative discrete outputs, retranslation by the de-fuzzification comp onent is necessary. W hereas the fuzzificationof the input variables and the qualitative rules in the rule base usuallycan be understood more or less easily, the application of one of thevarious possible inference and defuzzification methods giv es rise toa behavior of the controller which is in general not transparent to theuser. A graphical user interface which shows wh ich rules are activein special cases and the resulting control surface can be helpful. Still,the standard fuzzy controller design is characterized by trial anderror and not by clear systematics.

General ApproachOur approach differs from the one described above, in thatthe identification of the fuzzy rule base was automa ted and thatthe rules are formula s for comp utations with real variables. Themethod we used to determ ine an optimal fuzzy rule base was byand large the one proposed by S ugeno [I] . In this approach, theconclusions of the fuzzy rules take a special form. The output is

determined by a parametrized function of the controller inputsand not by fuzzy values. This means each rule returns a pair ofvalues truth value. which is determined by the truth values(membership degrees) of the variables in the precedent, and acrisp value for the control output according to the consequence.If the fuzzy sets of the qualitative input variables have sufficientoverlap, a soft interpolation results with the advantage that thenumber of implications is reduced, and a smooth transitionbetween the different control laws is guaranteed [ I]-[3]. WithA ; being the linguistic values, the format of the fuzzy rule baseis:

Rule I : IF SI is A \ , x is A: , ..., xm is AfnT H E N j1 p : + p t . X I + p i . ~2 + ... + p fi . ~ m .

The consequent functions are thus characterized and deter-mined by the parameter vectorp;. The final output y* is computedas the weighted average of the individual outputs 4;; of the rules:

p A : l l , ~ , , l )s the degree of membership of the (crisp) input xm tothe linguistic variable. With

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x . m + l ) = [P ' ,...,b I p1 . x1, . . . p x 1 I I p ' . x m .....p X n 1 ]1

P , I n + 1 ,1 = [ p:,,...,pa I pi ,...,p ; I . . . I P I ...., ; j Tequation ( 5 )can be arranged into a sy stem of linear equations:

y * = x p . (7)The matrix Xrepresen ts the fuzzification and the values of theinputs. and the vector P contains the unknown param eters of theconsequences. By using singletons as output values in the con-sequences, the complicated but normally required defuzzifica-tion step is no longer needed. In order to approximate thetime-optimal control strategy for the full range of operating condi-tions, simulations were performed over the entire feasible region ofthe state space of the process. These computations yield, for eachpoint considered, a pair of controller inputs and the desired time-op-

timal control output. Fork ordered sets, 7) becomes:

The ma trix X, on the right-hand side is given by the controllerinputs sL nd their respective fuzzy values A:, . The parameter-vector P is unknown. For k=n m+l) ordered pairs, the solutionof (8) is uniquely determined. But we used far more points atwhich the optimal control is compu ted than there are free parame-ters. Thus, (8) is an overdetermined linear system of equations.The parameter vector P can be calculated using the pseudoin-verse of X, such that the root mean square error F

is minimized. Thus

The magnitude of the minimal value of F is determined by thestructure of the fuzzy rule base. T he important factors here arethe numb er of rules and the choice of the membership functions.For a g iven number of rules, the mem bership functions can beoptimized so that the error Fmin is minim ized. Usually, o ne willstart with the sma llest reasonable n umb er of fuzzy partitions andincrease it until the error is sufficiently sma ll. In Fig. 6, a fuzzymodel of sampled pairs of .x ,f x)= ) is identified. We usedfuzzy sets with triangular shape because they only have fewparameters and are therefore easy to represent and to optimize.The effect of optimizing the fuzzy values can be seen in Fig. 6(b).The optimization can be accom plished by finding the best hori-zontal and vertical position of the intersection of neighboringfuzzy sets. We used a simple downhill simplex method from

10 80 60.40 2

x

0 0 2 0.4 0 6 0 8 10 2 0 4 0 6 0 8 1X X

Gg. 6 . Fuzzy approximation (a)not optimized (b )with optimize(fuzzi fcatio n (d ots represent the values which are approximated).Nelder and Mead [4] or this multidimensional optimizationproblem to minimize the resulting error F. It has to be noted thatthis method only finds local minima. Our experience show ed thatthe best fuzzification is usually given with a significant overlapof the fuzz y sets. Starting with an initial overlap as sh own in Fig.6(a), the error F is very likely to reach the global minimum. T henecessary accu racy of the approxim ation, Le., the magnitude ofFmlrmust be determined by simulations or tests of the resultingcontroller. Finding the optimal rule base is an iterative processwhere step by step the number of rules is increased, and theoptimal fuzzification is computed.

Application to the Control of LevelNow we show how our approach was applied to the tanksystem described in the previous section. The controller inputs

considered in the premises of the fuzzy rules have to be deter-mined first. These inputs are the same as the inputs of thetime-optimal controller. In the case of the decoupled first tank,the inputs are the regulation error el the ingoing volumetric flowrate Ve,and the actual level hi.Next, the state sp ace of the processhas to be discretized into physically reasonable partitions. Therange of values for the input was dete rmined by the plant limits( V e : 150. 350 j L/h and h : r0.1, 0.91 m). The range of theregulation error el depends on the level hi. In discrete steps of50 L/h, 0.1 m, and 0.015 m for V, , hi and el respectively, weget 3339 pairs of inputs, for each of which a time-op timal valveswitch has to be calculated. In Fig. 7, a switching surface forconstant inlet flow is shown.

i nn >

Height h, [m ]Fig. 7. Time-o p t ima l va lve p o s i t io n P o p = f ( h i , e l ) forV? o n , = 200 L.K1.

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One obvious way to reduce the number of pairs is to notconsider the values P o p beyond o r at the hard limits. Since thefuzzy contro ller is able to extrapolate according to the fuzzifica -tion, values bigger or smaller than the constraints can be cut out.It suffices that the fuzzy c ontroller yields values which are equ alto, larger, or smaller than the constraints. This reduces the numberof pairs to only 966. In addition, it improves the quality of theapproximation n the range of interest for a given number of rules,because one nonlinearity has been removed.Several fuzzy rule bases with different numbers of rules wereidentified. The minimal error Fmin which is a measure of thedeviation from the time-optimal valve position, decreases withan increasing number of rules as one would expect. With a fuzzyrulebase of eight rules for approximating he given 966 switchingpoints of the time -optimal controller, the resulting error is only

F m i n = 1.4 s, which is in the order of magn itude of the samplingtime T s = 1 s.It is know n that time-optimal control tends to be hectic andineffective around the set point. Therefore after the computationof the basic fuzzy co ntroller with eight rules, three more rule s fortiny regulation errors were added on a heuristic basis whichrepresent switching the strategy to an insensitive PD-controllerwith a deadzone DZ. T he final rule base is then:Rule 1:IF Vc, is small AND hi is small AND el is posit iveTHEN P&t = ph + p i . Ve+ p i , hi + p i , el

Rule 8: IF Ve s big AND hi is big AND el is negativeTHEN P&t = p f j+ p ? .Ve+ p g . h i+ p 3 . e lRule 9: IF e is tiny AND PD> DZ THEN P&t = 80Rule 10: IF el is tiny AND PD-DZTHEN P&t = P ( t )

with

and P ( t ) the actual valve position. The mem bership functionscorresponding to rules 1-8 were optimized as described above.The membership function of tiny was adjusted based on simula-tions. The final mem bership functions are shown in Fig. 8.Fig.9 shows that the behavior of the fuzzy controller insimulation s is very close to a tim e-optima l controller. The time-

Fig.8. Fuzii fcation of controller inputs.

0 10 3d 1 200 300 0 100 2 3Time[s] Time[s]iFig. 9. Control of level by the fu z z y controller with 8 rules(simulation).

0.7 ~~~~~~E 0.6

.5-....> Y

0 40.3

0 100 200 300 0 100 200 300Time[s] Time[slFig.10 Control of level I by the 6iz zy controller at the real plant.

optimal strategy can indeed be approximated by a fuzzy control-ler with a relatively small number of ru les.Next, the controller was tested for robustness at the actualsystem. While the simulations so far were made under the as-sumption that the existing model is correct, now noise andmodeling errors are present. It cannot be expected that a fuzzycontroller which is determined as an approximation of the idealtime-optimal controller would provide a significant improve-ment. Despite frequent claims by advocates of fuzzy control,fuzzy control is not m ore robust p er se . Indeed, the fuzzy con-troller exhibits the same sensitivity to measurement noise andmodeling errors. But observations of the transients gave addi-tional insight which was then represen ted by a dditional rules. Inorder to suppress overshooting, it is obvious that the ingoingvolumetric flow rate can be manipulated. For positive levelerrors, the volum etric rate is increas ed, and for negative errors itis decreased. Manipulating the estimated inflow rate in thismanner leads to earlier movements of the valve to its finalsteady-state position. Experiments have shown that an overshootcan be avoided by a 20% change in both directions. This manipu-lation can be directly considered in the rulebase by chan ging the

respective parameters p\ . An almost time-optimal system per-forman ce with the real plant is now achieved over the full rangeof operation cond itions (see Fig. IO) .Control of the Coupled System

In the previous sec tions, only the first tank withou t couplingto the second one was considered, Le., the case h2


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- 8E2 0 6

0 4

--W-

20 10 100 200 300 400Time[ s ]

ig. 11. PD-control of the coupled system.For the same reasons as discussed above, we want to achievea time-op timal behavior, Le., the smalle st possible settling timewithout overshoot for both levels. However, as can be seen fromthe simulation of the globally time-optimal controller inFig. 12,to merely spe cify the smallest possible settling time of the overallsystem produces trajectories which would not be consideredacceptable from a practical point of view. This is a commonfeature of standard time-optimal control laws for multivariablesystems with different maximal speeds of the subsystems [SI. Asthe settling time of level 1 is always much longer than the settlingtime of level 2 , and hence determines the overall settling timewhich is to be minimized, the globally time-optimal controller

manipulates level 2 in a manner which yields the maximalsupport for level 1 (Le., reduces the settling time of level 1before level 2 itself approaches the desired reference value. Thisindeed produces a minimal overall settling time but at the ex-pense of large deviation s of level 2 from the desire d value in thetransient period.The global time-optimal control was determined by backwardintegration of the system dynamics as in the case of level 1 alone.

0 50 1Time[slFig. 12. Globally time-optim al control ojboth evels (simulation).Because of the coupling and the constraints of the state variables,it is not possible to determine the necessary number of switchingoperations. Therefore the time-optimal strategy was found bynumerical search based on simulations. This observation led usto the conclusion that globally time-optimal behavior might bemathem atically nice but of little practical interest. Another dis-advantage is that the mathematical effort to calculate globaltime-optimal controls for this strongly nonlinear system is huge.This is also true in principle if one would try to approximate theglobal time optimality by a fuzzy model of the format describedabove, because seven inputs enter into the computation of thetime optimal control law. If each input is partitioned into onlytwo fuzzy sets, 2=128 rules with 1024 parameters have to beidentified.

Our idea then was to restrict the time-optimal behavior toisolated subsystems. We first identified a fuzzy controller (10rules) for tank 2 in the same manner as for tank 1. The fuzzy-con-troller for level 1 was expanded by one reasonable rule which isdue to the backwards cou pling of level 2 into tank 1 fhz>H thenreduce input hi by h2-H). From Fig. 13, we can see that theperformance of the fuzzy-control of level 1 is alm ost unaffectedin the coupled case. On the other hand, the performance of thefuzzy controller for level 2 is unsatisfactory. This is due to thelack of the input V12 ,which is not a m easured variable. Thus theinlet flow V c in tank 1 is used as an input in fuzzy controller 2as well. This means that the strongly variable flow into tank 2due to the actions of the controller for level 1 is neglected. Theconsequence is seen in Fig. 13which exhibits significant over-shoot of level 2.

0 0 100 2 300 400 500Time[s]

.... I100 200 300 400 500Time[s]

Fig. 13. Control of the coupled system with 2 independentfuzzycontrollers (simulation).A s one examines the transients it can be seen that the simul-taneous control in tanks 1 and 2 can be classified into 3 cases.These are determined by com binations of the signs of the regu-

lation errors el and e2. According to these combinations, one isable to formu late rules to take the effect of the actuator movementof valve VI into acco unt for tank 2 by m eans of a correction ofthe volumetric flow into tank 2:

2) if e10 then AVi2 = K 2 . el3)if e1>0 and e2


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F 0 8 I

O

60

40

20

-e..i

al-100 2 300 400Time[s]

Tig. 14. Behavior o the coupled system with expunded f u z z ycontrollers at real plant.

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IFig. 15. System behavior of the coupled system with expundedj1z:ycontrollers (simulation).In Fig. 15the reference steps from Fig. 12 are repeated withthe expande d fuzzy controllers. It can be seen that in comparisonto global time-optimal control, the (fuzzy) rule-based approachonly marginally increases the settling time of level 1 and leadsto very reasonable transients of both controlled variables. An-other benefit is the much smaller computational effort for thefuzzy controllers compared to the global time-optimal control,which m akes real-time application much easier.

Superior ControllersWith this contribution, we want to support two main statem ents:1 . With unconven tional methods, it is possible to obtain control-lers which are superior both to conventional fixed time-invariantcontrollers (PI, PD, PID) and to classical optim al controllers. Wehave dem onstrated this for the control of the levels in a tank systemwhich has co nsiderable nonlinearities and poses a nontrivial, but stillrelatively transparent, control problem. Our controller combine s thelow sen sitivity of the standard control algorithm s with the perform-ance of the time-optima l control law.2.The standa rd approach to fuzzy control, the implemen tationof heuristic knowledge, is in many situations of limited value.For plants with complex dynamics, heuristic knowledge is hard

to acquire and eq ually hard to assess with respect to the resultingquality of control. Stability and performance of the resultingcontrol laws have to be tested by extensive simulations or bytrials at the real system . In mo st cases, there will be som e model

of the plant dynamics available. It therefore makes much moresense to use the av ailable analytical tools first and then eve ntuallymodify and improve the resulting controller using heuristicknowled ge gained by expe riments or further physical insight. Inthis process it is beneficial o represent the control law in the formof rules with analytical formulae as conclusions. The parametersof the rules can be determined by optimization. This repre-sentation provides a basis for the additional representation ofheuristic knowledge and for an efficient implementatio n of com-plex nonlinear control laws which might be impossible to com-pute on-line , as in the case of time-optim al control of high-orderlinear or nonlinear systems [ 5 ] .Further work in this area will concentrate on the systematicsimplification of the rule base in the case of a la rge numbe r ofvariables which en ter into the rules, as in the case of the couple dtank system, to allow for easy modification and implementationof the controller.

References[ I ] T. Takagi and M. Sugeno, Fuzzy identification of systems and itsapplications to modelling and control, / T v m s . Sysf . , Man, Cybevn., vol.SMC-15, no. I Feb. 1985. [ ] S.M. Smith and D.J. Comer, Automatedcalibration of a fuzzy logic controller using a cell state space algorithm.IEEE Control SJIsf.Mug.. vol. 11, no. 5, pp. 18-27, Aug. 1991.[3] M . Sugeno , An introductory survey of fuzzy control, Info. Sei. , vol. 36.pp. 56-83, 1985.4 J.A. Nelder and R . Mead, A simplex method for function minimization,

Cotnput..I.OI 7, pp. 308-313.[ 5 ] M.H. Kim, New algorithms for time-optimal control of discrete-timelinear systems. Ph.D. dissertation. Univ. of Dortmund, 1993 (in German).

Thomas Heckenthaler w a s born in 1965 inLathen, Germany. He received the Diplom-In-genieur degree in chemical engineering in 1992from the University of Dortm und, Germany. Sincethen he has been working as a Research Assistanttoward his doctoral degree in the process controlgroup of the chemical engineering department atthat same university. His current research focuse son fuzzy systems. fuzzy logic control lers , andtime-optimal control.

Sebastian Engell \cas born in Dusqeldorf, Ger-many, in 1954 He obtain ed the Dip1 -1ng degree inelectncal engineering troin the Unikersity of Bo-chum in 1978. the Dr -In& degree from the Depart-ment of Mechanical Engineering at the Universityof Duisburg in I98 and the Venia Legendi in thesame department in 1987 He Lisited the ControlS)stems Centre at UMIST, Mancheqter in 1978 andspent one year at McCill Universit], Montreal in19841985 From 1986 to 1990 he worked for the

Fraunhofer-Institute IITB, Karlsruhe, hedding R&D groups in the area ofprocess automation and factory control Since 1990, he has been Professorof Procecs Control in the Department of Chemical Engineering at the Uni-versit) of Dortmund He i s chairman of the German Working Group onControl Theory, Co-Editor of the IEEE T v c ~ ~ ~ a c t i o n sn ConfvolS> t e imTechnolog\. and was elected to the bodrd of the German A utomation Soc iety(GM A) or the period 1994-1996 His re\ear ch interest? are controller de9ignand on-line production xheduling

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