Fuzzy-PID Contolles vs. Fuzzy-PI Contolles M. Santos*, S. Domido**, J. M. de la Cuz* *Dpto. de Infomática y Automática. Facultad de Físicas. (UCM) **Dpto. de Infomática y Automática. Facultad de Ciencias. (UNED) Ciudad Univesitaia s/n. 28040-MADRID (Spain). FAX: (34)--3944687 e-mail: msantos@eucmax.sim.ucm.es Abstact The synthesis of a contol system includes both the contolle selection and the adjustment of its paametes. In some cases, the type of contolle might be moe complex o moe geneal, like PID instead PI o PD, to impove the contol system pefomance. In all cases, the tuning poblem must be satisfactoily solved. On the othe hand, Fuzzy Contol has made possible the establishment of intelligent contol. Howeve, Fuzzy Logic Contolles (FLC) ae only used in simple configuations and thei analytic knowledge is still poo. In this pape, a quantitative and qualitative study of fuzzy contolles is done fo the most complete case of a Fuzzy-PID. The FLC-PID analytic pefomance is summaized in tems of its thee input vaiables, which allows us to obtain initial values fo the FLC-PID scale factos in tems of the classical PID paametes. This initial tuning has been tested fo seveal systems and a qualitative tuning has also been established. The advantages of the deivative tem ae also examined.. Intoduction The need fo simple advanced contol altenatives especially aises in the Contol Pocess aea, whee most of the eal pocesses ae geneally complex and difficult to model [. The application of Fuzzy Logic to a wide ange of contol applications has made possible the establishment of intelligent contol in these aeas [4, [5. Its appeal, fom the Pocess Contol Theoy point of view, lies in the fact that this technique povides a good suppot fo tanslating the heuistic knowledge of the skilled opeato, expessed in linguistic tems, into compute algoithms. Fuzzy Contol solves eal poblems, peviously not tackled due to thei complexity o to lack of infomation [9. Howeve, Fuzzy Logic Contolles (FLC) ae usually applied with poo analytic knowledge of thei behavio and only in simple configuations. In fact, they nomally pefom like PI o PD. FLC-PI contolles ae quite simple, though they ae the most widely used in pactice and povide simila esults to conventional contolles. But in some applications it may be useful to employ moe geneal contolles, which make it easie to each the system specifications and impove thei pefomance, though they can be also moe difficult to tune. The complete study of fuzzy contolles should involve all the tems of conventional contolles. The thid contol action must be included so as to conside the FLC-PID case. Though the deivative tem is not commonly included -neithe in the conventional case-, this allows us to complete the development of Fuzzy contolles in a simila way that of the classical ones. It also makes it possible to obtain cetain conclusions about thei stability and specifications. But the main poblem in the synthesis of a contol system is not only the selection of a specific contolle but also the adjustment of its paametes, to veify cetain given specifications fo the contolled pocess. In this pape an analytic study of the FLC-PID is caied out in section 2, which allows us to establish an equivalence between the FLC-PID and a conventional PID; thus a tuning method is poposed fo these fuzzy contolles and is then evaluated. The qualitative analysis is done in section 3. The behavio of these contolles is compaed with the FLC-PI type in section 4 and some geneal conclusions ae summaized in the last section. 2. Analytic study of the FLC-PID contolle The aim of a contolle is to each o maintain a pocess in a specific state, by monitoing a set of vaiables and selecting the adequate contol actions. The Fuzzy-PID contolle pefoms like its classical homonym, but both the input vaiables and the contol action ae given in linguistic tems. The analytic development of fuzzy contolles allows us to explain the influence of each tuning paamete on the system
esponse, as well as to compae them to the conventional one to obtain geneal esults. The incopoation of the deivative tem povides a new contol action to the contolle. In the fuzzy case, inceasing the numbe of input vaiables causes a ise in the dimension of the ule table and, theefoe, in the complexity of the system; this makes its implementation moe complicated and can make difficult its analytic study. Fo this eason, a PI is usually employed instead of a PID in most applications of fuzzy contol. Figue shows the basic diagam of an incemental Fuzzy-PID contolle, whee eo e -the diffeence between the pocess output y, and the efeence signal -, the eo change ce, and the second eo deivative ac, ae the input FLC vaiables, and the incement of the contol action u is the FLC output. The paametes chosen to tune the FLC ae the scale factos GE, GR, GA and GU, gains which weight the input and output vaiables espectively. + - y e d/dt d/dt 2 ce ac GE GR GA Fuzzy Infeence Figue. Fuzzy-PID contolle and tuning paametes u GU du The set of ules which descibes the FLC-PID behavio have thee antecedents and one consequent: Ri: if GE.e is Ei and GR.ce is CEj and GA.ac is Ak then u is Ui whee GE.e is the linguistic vaiable eo (e) weighted by its gain GE, GR.ce is the scaled change of the eo and GA.ac is the weighted second deivative of the eo. The conclusion is equal to the incement contol action u. In ode to simplify this analysis, two pimay fuzzy sets ae assigned to each one of the thee input vaiables, coesponding to the labels P (positive) and N (negative); theefoe, thee ae up to 8 contol ules. The contol output has thee labels adding the linguistic tem Z (zeo). The subscipt could epesent whicheve one of the labels associated to each vaiable (Ep, En, Uz, etc.). The ules ae shown in Table. GE.e(t) Ep En GA.ac(t) Ap An Ap An GR.ce(t) CEp Up Up Uz Un CEn Up Uz Un Un Table. Contol ules of the FLC-PID contolle The membeship functions ae defined in tapezoidal fom but ae symmetical fom thei cente, L. Theefoe, the contol action can be appoximated by linea piece functions [8. The final contol action can be calculated by diffeent defuzzification algoithms; two of them ae used hee: linea (L) and non-linea (Cente of Aea COA), which ae given by the following expessions, whee Uk is the conclusion value coesponding to each contol label (P, Z, N), Nu the numbe of contol tems, pk the cente of the membeship associated to the contol label k (± Lu o 0), and Ei, CEj, Ak ae the membeship value of the input vaiables. ###u L = Nu k= Nu Uk.pk k= ###u COA = = Nu k= Uk.pk Uk = (Up - Un). Lu () (Up Un).Lu (Up + Uz + Un) (2) Up = o(min(ep,cep,ap),min(ep,cep,an),min(ep,cen,ap)) Un = o(min(en,cep,an),min(en,cen,ap),min(en,cen,an)) Uz = o (min (Ep, CEn, An), min (En, CEp, Ap)) Afte each one of the 8 ules has been evaluated applying the connective and as the minimum, and the Lukasiewicz o, six diffeent conclusions can be obtained fo each ule. Theefoe, thee will be 48 zones of linea contol. These zones ae given by the diffeent elations between the sign and the absolute value of each input. The contolle output fo these zones is given in tems of the input vaiables, accoding to the defuzzification algoithm employed, eplacing the expession fo the membeship functions in each equation. The contol output u(t) by any defuzzification method, with Ke, Kc and Ka being some equivalent coefficients obtained fo each zone (see Appendix), is: u(t) = [Ke.GE.e(t)+Kc.GR.ce(t)+Ka.GA.ac(t) (3) An in-depth study of these functions has allowed us to establish an equivalence in each linea contol zone between the paametes of a conventional PID contolle (Ki, Kp, Kd), which also weight the vaiables e, ce and ac, and the FLC-PID output coefficients [6.
This analytic development depends on the stating configuation of the ule table since thee ae othe sets of ules which also coectly descibe the behavio of the FLC, poducing a smooth action o a stonge contol 2. Equivalence between FLC-PID and PID paametes Although the equivalent paametes of the FLC ae diffeent fo each one of the zones whee the contol is linea, thei vaiation ange is bounded, and it is possible to establish the limits of this vaiation both in tems of the PID paametes o the FLC-PID paametes as shown in the following expessions [3, [7,: Linea Defuzzification Method: The ange of vaiation of the gains of the FLC is: 2Ki/3 ### GE.GU ### 2Ki 0 ### GR.GU ### 2Kp 0 ### GA.GU ### 2Kd Based on the analysis of the output behavio in each diffeent contol zone, the initial FLC gains can be appoximated by setting them to the following values: GR = 2Kp/GU GE = Ki/GU GA = 2Kd/GU (4) Non-Linea Defuzzification Method: The ange of vaiation of the gains of the FLC is: 8Ki/3 ### GE.GU ### 8Ki 0 ### GR.GU ### 8Kp 0 ### GA.GU ### 8Kd Theefoe, a good appoach to the initial paametes of the FLC-PID contolle is: GR = 4Kp/GU GE = 5.3Ki/GU GA = 8Kd/GU (5) Even though this appoach has been developed fo this specific FLC-PID contolle, in the following section, we will pove that they ae also valid fo othe diffeent contolles and pocesses. 2.2 Initial paametes evaluation One of the main poblems that aises in this type of egulato is the lack of systematic pocedues fo tuning. The selection of the initial paametes of the FLC is usually caied out by tial and eo, as we can ead in the liteatue about the tuning of Fuzzy-PI o Fuzzy-PD contolles [2. This is a tedious and time-consuming task, which makes it difficult to establish geneal esults and notably inceases the design time. Theefoe, if we have some stating values calculated by any systematic pocedue, it makes it easie to analyze the FLC behavio, although those paametes may not be the best ones. Pefomance index: The validity of these FLC-PID tuning paametes will be detemined by inspecting the system esponse in the tempoal domain. Both, a qualitative analysis of the output, and the evaluation of following pefomance indexes [5 will be consideed: t 2 - I: quadatic eo I = 0 et (). dt - I2: nomalized oveshoot I 2 = y max - I3: ise time I 3 = min t / y(t) = 90% - I4: settling time I 4 =min t/y(t) [95%, 05% These initial paametes have been tested fo seveal systems. Fist of all, a typical 4º ode system with monotonous esponse has been used [6. It eflects the behavio of most industial systems. Afte its pocess model has been estimated, the PID paametes (Kp, Ti and Td) ae calculated by any classical tuning technique in ode to contol the pocess with cetain specifications; in this case, with the Ziegle-Nichols method, they ae: Kp =.654; Ti = 3.7; Td = 0.925 With these paametes, the FLC-PID tuning paametes ae calculated by equations (4) o (5), accoding to the defuzzification method (linea o non linea). These values and the esponse chaacteistics ae shown in Table 2, with thei index values. Conto lle PID FLC nonlinea defuzzification FLC linea defuzzification Gains Kp =.654 Ti = 3.7 Td = 0.925 GE = 0.237 GR = 0.66 GA = 0.62 GE = 0.0447 GR = 0.3308 GA = 0.3060 I 22.0533 20.934 23.875 I2 0.227 (tp = 5.7) 0.4434 (tp = 4.4) 0.96 (tp = 8) I3 (sec) 4 2.9 4.8 I4 (sec) 7.4-0.6 Table 2. Response chaacteistics with a conventional PID and with a FLC-PID (* tp: pick time). In ode to veify the validity of the initial paamete fomulas, the index values have been obtained fo diffeent systems. Table 3 shows simulated esults fo thee plants.
System e 04. s ( s + ) 3 ( s+ )( s+ 2) s + Gains GE = 0.252 GR = 0.0875 GA = 0.42 GE = 0.3723 GR = 0.6 GA = 0.3 GE = 0.3720 GR = 0.5454 GA = 0.0 I 4.656 5.9647 2.255 I2 0.2466 (tp = 4.9) 0.3840 (tp = 2).5484 (tp = 0.9) I3 (sec.) 3.2 0.6 I4 (sec.) 9.7 5 4.8 Table 3. Evaluation of initial tuning paametes fo diffeent systems Although these gains depend on the estimated model, we ae not looking fo the most accuate paametes but some stating values fom which one can obtain the specifications in few steps by qualitative tuning. 3. Qualitative study of the FLC-PID The FLC stuctue descibed in Figue is the stating point fo analyzing the qualitative behavio of the FLC-PID contolle. The influence of each gain (input and output scale factos) is consideed fo seveal systems. This study allows us to poduce some esults that may be used as a guide in the adjustment of fuzzy contolle paametes. The fuzzy system consists of a plant whose tansfe function is known o fo which a good model exists, and a fuzzy incemental PID contolle. The models epesent the moe usual dynamics of industial pocesses. In ode to study the vaiations of the esponse with the FLC scale factos, some initial values must be assigned to these paametes in a pevious phase. They ae then vaied and the behavio of the system is obseved. y(t) = F[GE, GR, GA, GU, e(t), u(t) The contolle has become moe geneal. The numbe of pimay tems of the vaiables has been inceased to 3 labels fo input and 7 labels fo output: PG, PM, PP, Z, NP, NM, NG whee G, M and P ae the modifies of Big, Medium and Small. The chaacteization of the membeship functions is defined by tapezoidal functions, that have 0.5 degee of completeness ove thei coesponding univeses. Although this makes the analytic study quite complicated, the qualitative behavio can be analyzed in ode to impove the pefomance when the application equies a moe complex contolle. The validity of the initial paametes is also poved fo new cases, peviously not contemplated in the mathematical study. The ule base consists of 27 ules of thee antecedents and one consequent that descibes the behavio of the contolle. As the numbe of ules has been inceased, the diffeence between the esults obtained depending on the diffeent intepetation fo the connectives becomes moe evident, since they ae seveal ules which geneate the same output. This diffeence is accentuated with the non-linea defuzzification method. The next simulations ae computed with the Lukasiewicz o function fo geate simplicity. 3. Influence of the scale facto vaiation The geneal effects on the index esponse of vaying the scale factos o gains, which weigh both the input and output vaiables, can be summaized as follows (k = constant, ###: incement): GE GR GA GU Effects on the esponse ### k k k Faste, moe oscillatoy. Impoves the tansient educing the stationay eo and ise time; inceases the isk of instability k ### k k Faste; may educe the oveshoot in a naow ange of values; inceases the quadatic eo. Makes the pefomance moe sensitive aound the set-point k k ### k Low dependence of the oveshoot and quadatic eo. Inceases the ise time,educes the settling time k k k ### Faste ise time, shote integal squaed eo; inceases the oveshoot and settling time. The most destabilizing, significantly influences convegence. 3.2 Influence of the deivative gain GA We ae going to show the specific influence of the gain of the deivative tem on the system esponse fo a given plant. This also makes it possible to pove the fomulas of the initial paametes. The initial value of GA is 0.0306 (by (4)). As we can see (Figue 2), it is vey close to the optimum value, accoding to the next gaphs which show the behavio of that system when this gain changes within an inteval between 0 and 3.
Response (GA = 0-0.3) GA=0.05 GA=0. GA=0 GA=0.3 GA=0.2 Fig. 2.. Quadatic eo Fig. 2.2. Oveshoot x0 - x0 - Reponse (GA: 0.3-3) GA=0.7 GA=3 GA= GA=0.3 Fig. 2.3. Rise time Fig. 2.4. Settling time The indexes of the system esponse show a complex pefomance. In geneal, they all ae inceasing functions of GA fom a value nea to the optimum point 0.2. The low dependence on the oveshoot and the quadatic eo with this facto is emakable, as opposed to the geat vaiation in the settling time. The esponse is hadly oscillatoy except fo vey low values of this gain, o fo values highe than 2, so the stability of the system is impoved. Theefoe, low values of GA (but not so low that this action would be canceled) give a esponse within the most usual specifications, howeve as this gain inceases, the esponse becomes much slowe. Figue 3 shows the system esponse and the contol action fo diffeent acceleation gain values. The est of the vaiables gains ae set to the initial values (4) and only this scale facto is changed. The gaph of the system esponse has been divided in two intevals to show its vaiation with moe detail since its behavio is diffeent fo those intevals. The contol gaph has become unified. The gain values ae: GA = 0, 0.05, 0., 0.2, 0.3, 0.4, 0.7,, 2, 3. Figue 3.. Response vaiation with the scale facto of the eo acceleation Contol GA=0 GA=0.4 GA=3 GA= Figue 3.2. Contol vaiation with the scale facto of the eo acceleation Theefoe, this qualitative tuning could not only be applied in a fist step to get some adequate paametes, but it also allows us to establish a fine adjustment of the initial paametes obtained by othe analytic methods. 4. FLC-PID vs. FLC-PI
The deivative tem is seldom employed even in classical contol, mainly due to the fact that it inceases sensibility to noise and that many times, a PI is good enough. Although most of the egulato stuctues incopoate this action, it is quite usual fo the plant opeatos to inhibit this function. Howeve, this thid eo vaiable gives a new contol action to the fuzzy contolle: the deivative action. This deivative tem makes it possible: - to complete the fuzzy contolles analysis in a simila way to the classical contolles and to establish some elationships between thei paametes. - to impove stability, since the deivative tem can be a geat help fo stabilizing the contol system, which emains one of the main poblems of FLC. - to give moe flexibility to the FLC, since it allows us to expand the vaiation ange of othe contolle scale factos within cetain magins, making it easie to each the esponse system specifications and impoving its behavio. It is also possible to pove that, in contol laws, canceling the deivative action in contol equations (by setting thei coefficient GA = 0) does not poduce a esult like a PI, since the elationship between the inputs and the output is stongly non linea in spite of the fact that the ule table of the FLC-PID may be pactically educed to that of FLC-PI by emoving the acceleation vaiable. Response Contol FLC-PID FLC-PID FLC-PI FLC-PI Figue 4.- Response of Fuzzy-PI and Fuzzy-PID (with GA = 0) contolles This esult can be checked in Figue 4, whee a compaison is made between the system esponse with a Fuzzy-PI, and that with a Fuzzy-PID contolle with quite simila ules except in the acceleation tem, and whee the deivative action gain has been canceled. 5. Conclusions The complete study of the fuzzy contolles should involve all the tems that chaacteize the conventional ones. The addition of the deivative tem makes it possible to show the non-linea chaacteistic of the fuzzy contolle, as well as to enlage the vaiation ange of the othe input vaiables by means of thei gains so as to impove the contolle behavio. Analytic tuning fomulas fo Fuzzy-PID contolles have been obtained. Theefoe, initial paametes fo these contolles have been poposed fo the diffeent defuzzification methods, and thei validity has been evaluated by simulation examples with satisfactoy esults. On the othe hand, the qualitative study of these FLC-PID has helped to poduce some ules fo a fine adjustment by means of the effects of these paametes on the system esponse. Howeve, the subject of the design and tuning of geneal fuzzy contolles is a poblem that emains open. Acknowledgments This wok has been patially suppoted by the Spanish CICYT unde poject TAP94-0832-C02-0. Refeences [ K.J. Astöm, C.C. Hang, P. Peson, W. K. Ho: Towads intelligent PID contol. Automatica 28,, -9, 992. [2 M. Baae, D.A. Ruthefod: Selection of paametes fo a fuzzy logic contolle. Fuzzy Sets Syst. 2, 3, 85-99, 979. [3 S. Domido, M. Santos, A. P. de Madid, F. Moilla: Autosintonía de contoladoes boosos utilizando técnicas clásicas basadas en eguladoes PID. Poc. of III FLAT, España, 993, pp. 27-225. [4 C.C. Lee, Fuzzy Logic in contol systems: Fuzzy Logic Contolle - Pat I, IEEE Tans. Syst. Man. Cyben., vol. 20, n.2, pp. 404-48, 990. [5 W. Pedycz: Fuzzy Contol and Fuzzy Systems. Reseach Studies Pess, England, 993. [6 M. Santos: Contibución a las técnicas de sintonía de los contoladoes basados en la Lógica Boosa. Ph.D. Dissetation, Depatment of Compute Science, Univesidad Complutense de Madid, España, 994. [7 M. Santos, S. Domido, J.M. de la Cuz, J.A. López Oozco: Tuning of fuzzy contolles: application of the Relay Method. Poc. of EUROSIM 95, pp. 3-36, Ed. Elsevie, N-H, 995. [8 H. Ying, W. Sile, J. J. Buckley: Fuzzy contol theoy: a nonlinea case. Automatica 26, 3, 53-520, 990. [9 D.E. Thomas, B. Amstong-Hélouvy: Fuzzy Logic Contol- A Taxonomy of Demonstated Benefits, Poc. of IEEE, Engineeing Applications of Fuzzy Logic, vol. 83, no. 3, pp. 407-422, 994.
Appendix FLC-PID Contol laws in the linea contol zones (Non-Linea Defuzzification Method) 05Lu. [ 2GE. e + GR. ce + GA. 4L GR. ce( t) 2GA. ac( t) ###### sg(ce) ### sg(ac) ###u(t) 05Lu[. GE. e( t) + GR. ce( t) 4L 2GR. ce( t) GA. ac( t) 05Lu[. GE. e( t) + GR. ce( t) 4L GE. e( t) 2GR. ce( t) Zones ###### sg(ce) = sg(ac) ###### sg(e) = sg(ce) 05Lu. [ 2GE. e + GR. ce + GA. 4L GE. e( t) 2GA. ac( t) 05Lu. [ 3GEe + GRce + 2GA 4L 2GE. e( t) GA. ac( t) ###### sg(e) ### sg(ac) ###### sg(e) ### sg(ac) [ GE e t GR ce t 05Lu. 3. ( ). ( ) 4L 2GE. e( t) GR. ce( t) ###### sg(e) = sg(ce) [ GE e t GA ac t 05Lu. 3. ( ). ( ) 4L 2GE. e( t) GA. ac( t) ###### sg(e) = sg(ac) 05Lu. [ 3GEe + 2GRce + GA 4L 2GE. e GR. ce ###### sg(e) ### sg(ce) 05Lu. [ 2GE. e + GR. ce + GA. 4L GE. e 2GR. ce ###### sg(e) ### sg(ce) 05Lu[. GE. e( t) + GA. ac( t) 4L GE. e( t) 2GA. ac( t) ###### sg(e) = sg(ac) 05Lu. [ 2GE. e+ GE. ce+ GA. 4L 2GE. e( t) 2GA. ac( t) ###### sg(ce) ### sg(ac) 05Lu[. GE. e( t) + GA. ac( t) 4L GR. ce( t) 2GA. ac( t) ###### sg(ce) = sg(ac)