Tarek Aissa, Christian Arnold, Steven Lambeck

Transcription

1 Preprints of the 9th orld Congress he International Federation of Automati Control Cape own, outh Afria. August 4-9, 04 Combined Approah of Fuzzy Deision Making and Preditive Funtional Control to Minimize Variations of Manipulated Variables in Proesses with Dead ime arek Aissa, Christian Arnold, teven Lambek Department of Eletrial Engineering and Information ehnology, University of Applied iene Fulda, Fulda, Germany, ( {tarek.aissa; hristian.arnold; Abstrat: Basially onventional ontrollers operate on a risp set point, even if the ontrol task does not require an eat value of the ontrolled variable. In fat the requirements for the ontrolled variable an often be desribed as intervals of aeptable or ideal ranges. In this paper we present an approah of a preditive funtional ontroller ombined with fuzzy deision making, whih leads to a ontroller that operates on omplete fuzzy goals. his approah is demonstrated for simple proesses with dead time and the performane is analyzed by omparison with onventional ontrollers.. IRODUCIO he aim of many ontrol appliations is to reah a speified range of the proess variable and therefore it is not neessary to hold the ontrolled variable at an eat set point w. he requirements of the ontrol task may then be defined by intervals of ideal, aeptable and inappropriate ranges, whereby the ontrolled variable should stay at least in the aeptable range, preferably in the ideal range. uh a linguistially formulated range an easily be desribed by formulating a fuzzy goal with the parameters of a trapezoidal membership funtion (figure ): f, f : 3 4, Fig.. Fuzzifiation of stationary requirements for ontrolled variable. It is obvious, that one requirement of the ontrol task is, to manipulate the proess in suh a way, that the ontrolled variable owns a high degree of membership to the stationary fuzzy goal represented in Fig.. Besides of reahing the stationary goal, the reduing of variations of the manipulated variable u is desirable. By reduing the variations of the manipulated variable u,it is possible to take are of the atuators durability. Hene, an additional fuzzy goal C for the dynami behaviour has to be formulated. herefore, initially the imum (fored) variation of the ontrolled variable an be formulated based on the imum gradients of the manipulated variable u : u f Using equation () and doing some simple transformations leads to the dynami fuzzy goal C. Regarding the atuators durability it is evident, that no variations of the manipulated variable u is ideal, whih is the ase, if there is no need for an adjustment of the ontrolled variable. Consequently we define the ore of the membership funtion only for 0. Hene we will get a resulting triangular membership funtion using the sample time with the optimal ase of 0 and the spread defined by the imum aepted variation : f 0 0, C : In many tehnial appliations that deal with IO-ystems simple ontinuous ontrollers (like PID) or disontinuous ontrollers (like two- or three point ontrollers) are used. For that type of ontrollers it is basially neessary to define a risp set point w and it is not possible to operate on a fuzzy goal like shown in figure. As one onsequene, the variations of the manipulated variable u an not be taken into aount. In ase of using a ontinuous ontroller it may be suitable to use a nonlinear transfer element at the input of the ontroller, whih operates as a dead zone element. For disontinuous ontrollers we an define a hysteresis h to redue the variations of the manipulated variable u. An approah for defining the parameters for hysteresis h or a dead zone, may be to use an alpha ut of the stationary fuzzy goal and to define the supremum and infimum of the resulting set as parameters of the set point interval w, ], as shown in figure. [ min w Copyright 04 IFAC 834

2 9th IFAC orld Congress Cape own, outh Afria. August 4-9, 04 Fig.. Defining the limiting set points out of a stationary membership funtion depended on the parameter of the alpha ut. ow it is possible to define the parameters of hysteresis h and dead zone as well as the desired value w of the ontrolled variable for the ontrollers by the following equations: w h,, w w, w,, w, min min Fig. 4. Approah for a fuzzy goal preditive funtional ontroller.. FUDAMEAL In this hapter we will introdue some basis of preditive funtional ontrol (PFC) and the theory of fuzzy deision making, whih is neessary to eplain the approah of the fuzzy goal preditive funtional ontroller (FGPFC) in the net setion.. Preditive Funtional Control he theory of preditive funtional ontrol was developed out of the Model based preditive ontrol (MPC), whih is a very ommon method of advaned ontrol strategies. he idea of MPC an easily be desribed by the following steps: initially the free proess behavior without ontrol is predited in a predition horizon n P. he predition is done by an internal proess model. he proess model requires some measurements or observations for the initial state. he future disturbanes of the proess an also be used if they are known. ith the predition of future proess states, it is possible to evaluate a ontrol sequene in suh a way, that the futures ontrol deviation (differene between set point and ontrolled proess variable) is minimized. Fig. 3. Adjusting the parameters for a disontinuous (above) and a ontinuous ontroller (below). he user has to define a ompromise out of stationary requirements and the variations in the manipulations. As we an see in figure 3 there are a lot of parameters to set up by the user, whih an lead to a onfusing tuning task. o avoid the defining of the parameters for ontrollers, hysteresis, dead zone, et., it may be suitable to use a ontroller whih operates on the omplete fuzzy goal, as shown in figure 4. In this paper we will desribe an approah of a fuzzy goal preditive funtional ontroller, whih operates on the omplete fuzzy goal. herefore, we will ombine the theories of preditive funtional ontrol and fuzzy deision making. e will use fuzzy deision making to replae the alulation of the referene trajetory for future proess states for the preditive funtional ontroller. First of all we will desribe some theoretial aspets in setion followed by the presentation of the approah of a fuzzy goal preditive funtional ontroller in setion 3. ome simulation results will be presented in setion 4. Finally, some onluding remarks and a short outlook will be given in setion 5. hen we have to deal with omple proesses, like nonlinear proesses or proesses with ontraints, it is often neessary to use a numerial solver with an (for eample iterative) optimization method to alulate the future ontrol sequene ((Adamy, 009), (Dittmar et al., 004)). Otherwise, for proesses that an be desribed as linear state spae systems without onstraints, there is an analyti way to alulate the ontrol sequene. In that ase, we are talking about preditive funtional ontrol (PFC), whih was presented first by Rihalet in (Rihalet et al., 978). As mentioned above, the the PFC has the advantage that a numerial optimization for alulating a ontrol sequene is not neessary. he eistene of an analyti solution leads to an easy implementation in pratie. here are many eamples whih demonstrate, that the PFC is getting more and more suessful applied in industrial appliations (see (Rihalet et al., 978), (Rihalet et al., 009), (Luft, 009)). Rihalet desribed PFC as easy to understand, to implement, to tune. ome important modifiations by Rihalet an simplify the optimization problem signifiant (Valenia-Palomo et.al., 0). On the one hand, there is the hoie of oinidene points, whih means that the ontrol error has to be minimized only at a random point and not in the omplete predition horizon (for eample by mean square error). his 834

3 9th IFAC orld Congress Cape own, outh Afria. August 4-9, 04 leads to an easier objetive funtion of the PFC. On the other hand, only basi funtions are used to desribe the behavior of the manipulated variable. Consequently, the solution spae for possible ontrol ations gets redued. he appliation of PFC will be demonstrated in this paper by the ontrol of a first-order lag element in ombination with a dead time. he results of using the proposed fuzzy goal PFC to ontrol a first order system without dead time are presented in (Arnold et.al., 03). he transfer funtion of the system regarded in this paper, whih is a suitable eample of a transport appliation, an be represented as F s K s t e s he aim is now to alulate the ontrol equation by using this transfer funtion as an internal model. herefore the dead time t is initially negleted and will be onsidered later. A time-disrete formulation of the resulting first order system using the sample time and a first order hold is then defined by using the parameter ep s : k e k K e uk k K uk Usually, the dynami behavior of the PFC is defined by a referene trajetory R ( k ). It is obvious that the requirement of the ontrol task is, that the internal model should follow the referene trajetory. Hene, we have the premise ˆ k R k. he future proess state an easily be alulated by the first order system (equation (6)) and the futures disturbane: ˆ k k K uk zk Assuming that the disturbane of the net sample z k will be the same as in the urrent one (in a simplified approah), we an alulate the disturbane by the urrent proess states in the following way: z k zk k ˆ k ome mathematial transformations of equations (6), (7) and (8) lead to the ontrol sequene equation: u k R k k ˆ k K he influene of the dead time on the proess is obviously given by the disrete formulation of the dead time d : t ˆ ( k d) ( ( k d) ˆ( Both equations an be ombined to: ( k d) ˆ( ˆ( k d) ( o ontrol the dead time system we have to replae ( in equation (9) by ( k d) of equation (). he struture of the PFC ontroller is shown in figure 5. It has to be mentioned that onstraints of the manipulation variable u '( may be taken into aount by using the onstrained value of the manipulation variable u ( as input of the proess and the inner model. Besides that, it is still not possible to operate on the omplete fuzzy goal, so that a risp set point is still neessary (figure 5). herefore, we replae the alulation referene trajetory R ( k ) by a fuzzy deision making approah. Fig. 5. truture of the PFC ontroller.. Fuzzy Goals and Fuzzy Deision Making As already mentioned, the requirements for the set points of some ontrol appliations are often formulated ineat or as intervals. hus, we often have to deal with linguistially formulated requirements. An opportunity to handle those requirements is to apply the fuzzy theory by using fuzzy numbers or fuzzy intervals. In that ase we an transform the knowledge of the user into a stationary fuzzy goal. By using a fuzzy formulation for the requirements of the ontrol task we will be able to take are of forbidden areas of the possible states. For reasons of simplifiation we will only use trapezoidal membership funtions as desribed in setion (see also figure ). In (Bellmann et.al., 970) the theory of fuzzy deision making was introdued. he idea is to find the optimal solution * out of a set of alternatives, whih represents the solution spae. hereby all requirements and onstraints have to be formulated as fuzzy sets. Hene we formulate an amount of fuzzy goals () (with i=, n) and fuzzy Gi ontraints () (with j=,,m). he best solution * has to Cj be alulated by an aggregation of all fuzzy goals () and onstraints (). here are several ways to find the Cj best solution. In this paper we fous on the approah presented in (Bellmann et.al., 970), where the fuzzy deision set (possible solutions for ) D is defined by the intersetion of all fuzzy goals and onstraints. One possibility to determine the optimal solution * out of the Gi 8343

4 9th IFAC orld Congress Cape own, outh Afria. August 4-9, 04 fuzzy deision set D is given by the imum value (Bandemer et.al., 970): D n m * Gi Cj i j D It has to be mentioned, that all onstraints an also be transformed into goals by a simple reversing of the set. hat s the reason why we will fous on fuzzy goals from now on (Bernard, 000). 3. COCEP OF A FUZZY GOAL PFC COROLLER In this hapter we will present our proposed approah of a fuzzy goal PFC. he advantage of the proposed ontroller is, that there is no need for a risp set point, like in a onventional PFC. he fuzzy formulation of the requirements an be used diretly at the input of the ontroller. he approah is to replae the referene trajetory with a fuzzy deision making approah, eplained shortly in setion. 3. Approah to a olution It is obvious that the optimal value of the ontrolled variable depends on the stationary and the dynami requirements. Hene, it is neessary to find a ompromise of both requirements. he stationary fuzzy goal is already formulated as a funtion of the ontrolled variable (see equation ()). Consequently the dynami requirements have to be transformed as a funtion of as well. e an do this by adding the urrent proess state to the aepted variations mentioned in equation (3): C f f k 0 0 k 0 0,, Fig. 6. Compromise of stationary and dynamial goal. Aording to equation () the optimal solution is given by the imum membership of the intersetion set of the stationary fuzzy goal and the dynami fuzzy goal C (also shown in figure 6):,, k D * C, Another benefit of using trapezoidal membership funtions is that the optimal deision an easily be alulated analytially by using a ase differentiation and the relations of table. able. Case differentiation for the optimal solution * k for k k k k k 4 4 k k k k 3 k k k k k 4 3. Closed loop dynami behaviour in aepted area For onveniene we fous on the lower part of the aepted area (seond ase of table ). It is evident, that the onsiderations are similar for the upper part of the aepted area (4th ase of table ). For the lower part we reeive the following solution for the net sample: k k k Using the z-ransformation and some mathematial operations leads to the following equation: L z e z z e L he resulting system represents a first order time delay system with a stati gain of one. he time onstants an be determined for the lower and the upper part of the aepted area L R ln 4 3 ln Hene, to adjust the dynami of the proposed fuzzy goal PFC it is only neessary to hoose the aepted variations. 4. IMULAIO everal simulations have been done to test and verify the approah of the fuzzy goal PFC. In this paper we will present the results of the ontrol of a transport system with a signifiant dead time. 8344

5 9th IFAC orld Congress Cape own, outh Afria. August 4-9, Model of the dead time system One of the lassi benhmark eamples for ontrol tasks with dead times are transport systems. he target of the ontrol task of transport systems is to ontrol the amount of material per time ( Q out) delivered to a speified plae by adjusting a valve at the inflow. Hene, the valve is regulating the supply of the material ( Q ). It may also be possible that the proess in is disturbed by an unknown variable funtion is shown in figure 7. Q z. he priniple Fig. 8. Controlled (above) and manipulated signal (below) during the simulation senario. Fig. 7. hemati representation of the proess. he dead time depends on the ratio of speed and length of the onveyor. he valve at the inflow an be desribed as a first order system. he resulting transfer funtion is given by: F K s s t s ( s) e s s he system investigated in this paper represents a first order lag element ( Ks 0., s s ) with dead time ( t 0s ), whih will be used as inner model of the ontroller. 4. imulation results All simulations were done with the simulation environment of (MALAB/imulin. A unified stohasti disturbane sequene Q z was used for all eeuted simulations. he stationary fuzzy goal was defined as: h f Q 0, 0,4 0,6 0,9 out,, h o verify the proposed approah we ompared the results of the fuzzy goal PFC with a simple PI ontroller and a smith preditor (Lunze, 004), whih is ommon pratie for systems with dead time delay. he adjustment was done by zero-pole ompensation. he performane of the mentioned ontrollers is shown in figure 8 in terms of a time plot of the ontrolled variable Q and the manipulated variable u Q out. in It is obvious, that the ontinuous ontrollers are foussed on the minimization of the ontrol error all the time, whih leads to higher variations of the manipulated variable, without ahieving muh better results regarding the stationary fuzzy goal. Another aspet to evaluate the performane of the ontroller is given by the mean fulfilment of the stationary fuzzy goal : hi i and the mean variation of the manipulated variable (both shown in figure 9): u i h i hi It is evident, that an adaption of the ontroller s gain leads to an improvement (higher fulfilment of the stationary goal) to a ertain degree. A higher gain for the PI ontroller leads to an osillating and unstable system and therefore high variations of the manipulated variable and less fulfilment of the stationary goal. For the fuzzy goal PFC is the parameter to adjust. he results of higher values for are similar to the results of the PI ontroller or smith preditor. Beause of the use of unertain set points it is to be seen that the fuzzy goal PFC solves the ontrol task by less variations of the manipulated variable. 8345

6 9th IFAC orld Congress Cape own, outh Afria. August 4-9, 04 onventional ontroller. e ould show that the proposed approah provides good results in simulative eperiments. Beause of the unertain set point the variations of the manipulated variable an be redued without loosing stationary auray. In the simulations we also found out, that the proposed ontroller is robust against hanging dead times. In future works we will try to transmit the proposed approah to MIMO-ystems and proesses with more than one inner state and we will verify the fuzzy goal PFC in pratial eperiments. REFERECE Fig. 9. Various ontrol strategies regarding the fulfillment of the stationary fuzzy goal and the mean variation of the manipulated variable depending on K p or. For a last test the influene of the dead time to the robustness of the ontrollers were investigated. herefore we analyzed what happens to the fulfilment of the stationary fuzzy goal for several dead times; the ontrollers parameters are not hanged. he results are shown in figure 0. Apparently the PI-Controller is not suitable to handle larger dead-times. he mith Preditor ahieves good results, although the performane of the fuzzy goal PFC beomes relatively better for larger dead times by less variations of the manipulated variable. Fig. 0. Various ontrol strategies regarding the fulfilment of the stationary dependent of the time delay t 5. COCLUIO AD OULOOK In this paper we presented a fuzzy goal PFC, whih operates on fuzzy formulated requirements. he way of doing this is to ombine the onventional PFC with the theory of fuzzy deision making. hereby, the alulation of a referene trajetory is replaed by a fuzzy approah. Hene, the requirements of the ontrol task an easily be formulated linguistially. he adjustment of the proposed ontroller will be muh easier to handle than finding several parameters of a Adamy, J. (009). ihtlineare Regelungen, pringer Verlag, Berlin, Germany. Arnold, C.,. Lambek, C. Ament (0). Multistage fuzzy ontrol using trapezoidal membership funtions and dynami programming. IEEE International Conferene on Fuzzy ystems (FUZZ-IEEE), Brisbane, Australia. Arnold, C.,. Aissa,. Lambek (03). Using Unertain et Points in Preitive Funtional Control Combining Preditive Funtional Control and Fuzzy Deision Making. IEEE International Conferene on Fuzzy ystems (FUZZ-IEEE), Hyderabad, India. Bandemer, H. and. Gottwald (993). Einführung in die Fuzzy-Methoden, Akademie Verlag, Berlin, Germany. Bellmann, R.E. and L.A. Zadeh (970). Deision Making in fuzzy enviroment. In: Management ienes, Vol. 7, pp. B4-B64. Bernard,. (000). Ein Beitrag zur gewihteten multikriteriellen Optimierung von Heizungs- und Lüftungsregelkreisen auf Grundlage des Fuzzy Deision Making, PhD-hesis, University Karlsruhe, Faulty for mehanial engineering. Dittmar, R. and B.M. Pfeiffer (004). Modellbasierte prädiktive Regelung, Oldenbourg issenshafts-verlag, Münhen, Germany. Luft, H. (009). Prozessführung mit PFC Anwendererfahrungen bei Evonik. PFC-orkshop, Colone, Germany. Lunze, J. (004). Regelungstehnik. pringer Verlag, Berlin / Heidelberg / ew York. Rihalet, J., A. Rault, J.L. estud, J. Papon (978). Model Preditive Heuristi Control: Appliations to industrial proesses. Automatia, Vol. 4. Rihalet, J. and D.O Donavan (009). Preditive Funtional Control Priniples and Industrial Appliations, Advanes in Industrial Control, pringer, London. Valenia-Palomo, G. and J.A.Rossiter (0). Comparison between an autotuned PI ontroller, a preditive ontroller and a preditive funtional ontroller in elementary dynami systems, online publiation, researh gate. 8346