Adaptive scaling factors algorithm for the fuzzy logic controller


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Adaptive scaling factors algorithm for the fuzzy logic controller
  Adaptive Scaling Factors Algorithm for the Fuzzy Logic Controller José Victor Escola Superior de Tecnologia e GestãoInstituto Politécnico de LeiriaMorro do Lena, Alto Vieiro2400 Leiria, Portugal António Dourado Centro de Informática e Sistemas da Universidade Coimbra Departamento de Engenharia InformáticaPólo II, Pinhal de Marrocos3030 Coimbra, Portugal Abstract  An on line method for adapting the scaling factors of the fuzzy logic controller is presented. Since most of thetimes tuning these controllers is not an easy task and verytime consuming, the solution is to design an adaptive fuzzy controller. The objective of the proposed algorithmis to adapt the scaling factors according to a performance measure in order to fine tune the controller and improve the performance of the control system. An application example is presented for thetemperature control of a heated air stream, processtrainer PT326. 1.Introduction The design of fuzzy logic controllers involves theappropriate definition of a parameter set. This includesthe inputs and outputs of the fuzzy logic controller, thenumber of linguistic terms and the respective membershipfunctions for each linguistic variable, the inferencemechanism, the rules and the fuzzification anddefuzzification methods.Even for someone who is not much familiar with thistechnology the design phase of these controllers is not toohard to learn. The main problem arises when it isnecessary to tune the controller and some choices inherentto the controller parameter structure must be made. Thistask some times is very hard and can take several days,even weeks, in order to achieve a good performance. Itdepends much on the process that is being controlled [ 9 ] .Another important issue is the fact that most processdynamics changes with time or are nonlinear in nature. Inspite of the robustness of the fuzzy logic controllerssometimes this is not sufficient to deal with it. Thecontroller needs to be retuned to achieve goodperformance.The fuzzy logic controllers contain a set of parametersthat can be altered on line in order to modify thecontroller performance [ 1 ] . These include the scalingfactors for each controller variable, the membershipfunctions of the linguistic terms and the rules. Themethod presented in this paper uses the scaling factors tofine tune the fuzzy logic controller.The paper is organised as follows. In section 2 thefuzzy logic controller parameters of our basic controllerare briefly described. Section 3 describes the adaptivescaling factor method. Section 4 presents an applicationexample and section 5 the conclusions. 2.Fuzzy logic controller In the structure definition of a fuzzy logic controller itis necessary to define the inputs and outputs. The fuzzylogic controller implemented has two inputs and oneoutput. The inputs are the error (1) and change of error(2). ery kkk  = − (1) ∆ eee kkk  = − − 1 (2) The output is the change of control (3). ∆ uuu kkk  = − − 1 (3)The universes of discourse of the controller variablesare  E  , ∆  E   and ∆ U   respectively. Their values range is [ -10,+10 ] .The number of linguistic terms for each linguisticvariable is 5, { }  LELELUNBNSZOPSPB = = =∆ ∆ ,,,,  (4)Triangular membership functions were used torepresent the meaning of the linguistic terms (  NB   ≡  Negative Big,  NS    ≡  Negative Small,  ZO   ≡  Zero, PS    ≡ Positive Small and PB   ≡  Positive Big), shown in Figure 1. Figure 1. Membership functions of the linguisticvariable error. Associated to each linguistic variable is a scalingfactor. K  e   ≡  scaling factor for the error K  e ∆  ≡  scaling factor for the change of error K  u ∆  ≡  scaling factor for the change of controlScaling factors enable the use of normalised universesof discourse and play a role similar to that of the gaincoefficients in a conventional controller [ 2 ] .The control policy is established in the rule base andexpresses the knowledge of the designer about theprocess. Figure 2. Rule base. The rules are of the form:If e k   is  LE  i  and ∆ e k   is  LE   j ∆  then ∆ u k   is c ij  (5)where c ij  is a crisp value instead of a fuzzy set. c ij represent the point of minimum fuzziness in theconsequent parts of the rules, i.e., the membershipfunction centres.Figure 2 presents the rule base used in our controller.The rule base is complete, but it does not have necessarilyto be since certain regions of the input domain aresometimes not of interest. In practical applications of fuzzy control almost no rule base is complete.To put all this working together is necessary aninference mechanism that gives the output signal to sendto the process. The authors use the Mamdani ( min-max )inference mechanism [ 3 ] , even though the  product-sum may enable better results. The option by the first derivesfrom the fact that it is more appropriate to test theproposed method.Suppose that in a certain instant k  , the values for theerror and change of error are e k   and ∆ e k   respectively.Using the Mamdani inference, the firing strength of afuzzy control rule will be ( )  fee ijLEkLEk  ij = min(),() µ µ  ∆  ∆ (6)where µ  LE  i and µ  LE   j ∆ represent the membershipfunctions of the linguistic values  LE  i  and  LE   j ∆ .The output of the inference mechanism is a fuzzyvalue, so it is necessary to convert this fuzzy value into areal value, since the physical process cannot deal with afuzzy value. This operation is called defuzzification; theinverse operation is fuzzification and the method usedwas the singleton fuzzifier. There are severaldefuzzification methods to convert a fuzzy value to a realvalue [ 10 ] . The height defuzzification method was usedsince it is simpler and, most of all, computationally veryefficient. u fc f  ijijijijijij =  ∑∑ ,, (7)The main idea of this method is that the larger thefiring strength of a rule ( )  f  ij , the more this rulecontributes to the global fuzzy output. 3.Adaptive scaling factors method The structure described in the previous section is thebasic structure of the fuzzy logic controller. Depending onthe choices made in the design phase we can havedifferent types of fuzzy logic controllers: P, PD, PI or PID  fuzzy logic controller. In [ 6 ]  it is shown that a fuzzy logiccontroller with the same characteristics of the traditionalPID controller can be designed, by using only the error ( ) e k  and the change of error ( ) ∆ e k   as its inputs. One wayto achieve this goal is to serially connect an integrator tothe output of the fuzzy logic controller, as shown inFigure 3. This controller is a PI fuzzy logic controller.   ProcessFLCr k +- e k ∆ e k K e K ∆ e K ∆ u  ∑ Figure 3. PI fuzzy logic controller. To design a PID fuzzy logic controller the PI fuzzylogic controller and the PD fuzzy logic controller must beconnected in parallel, as in Figure 4.   ProcessFLCr k +- e k ∆ e k K e K ∆ e K d K ∆ u ∑ Figure 4. PID fuzzy logic controller. The role of the scaling factors in the fuzzy logiccontroller is very similar to that of the conventional PIDparameters [ 2 ] . The rules for tuning the fuzzy logiccontroller by adjusting the scaling factors are generallyderived by analogy from the rules for tuning of PIDcontrollers.In PID control the integration component has animportant role in the performance of the control system.The system response will be slow if the integrationcomponent is too weak and the system will becameunstable if the integration component is too strong. So theideal objective is to keep the integration component insuch a way that our control system response is fast enoughwithout overshoot. To achieve this the integrationcomponent does have necessarily to change with time. Atan early stage of response the integration component takesa larger value and it is gradually reduced with time inorder to increase the damping of the system and make thesystem more stable. Figure 5 presents the control systemresponse, which can be divided into different phases bythe peak value times. From the start time to the time whenoccurs the first peak value t  1 , the error of the systemcovers the whole universe of discourse. From t  1  on, theerror will be in the interval − 11 , ,where 1  is theabsolute peak value at time t  1 . Figure 5: Response of a control system. The reasoning is analogous when another peak valueoccurs at t  2 . The integral control component can bedecreased at each peak value time according to theabsolute value of each peak.Figure 6 presents the structure of the parameteradaptive controller purposed in [ 6 ] .   ProcessFLCr k +- e k ∆ e k K e K ∆ e K d K ∆ u PeakObserverParameterRegulator   ∑ Figure 6. Parameter adaptive PID fuzzy logiccontroller. The parameter adaptive fuzzy controller is constitutedby a PID fuzzy logic controller, a peak observer and aparameter regulator. The peak observer determines thepeaks at the control system output and measures theabsolute value of the peak. The parameter regulator tunesthe controller parameters, scaling factors K  e ∆  and K  u ∆ ,for each peak according to the peak value at that time.The algorithm for tuning the scaling factors K  e ∆  and K  u ∆ of the PID fuzzy logic controller is: K K  eek  ∆∆ = 0 δ  (8)  KK  uku ∆ ∆ =δ 0  (9)where K  e ∆ 0 is the initial value of K  e ∆  and K  u ∆ 0 is theinitial value of K  u ∆ ; δ k   is the absolute overshoot at time ( ) tk  k   = 12,, K . This algorithm reveals some virtues but also somelimitations. The algorithm substantially improves theperformance of the control system if the value for thescaling factor K  e  is appropriate. A careful analysis of thealgorithm shows that when the peak value becomes verysmall the value for the scaling factor K  e ∆  becomes verylarge and the control system can reach instability.In the present paper the authors propose an algorithmto improve the overall performance of the control system.While in the algorithm presented in [ 6 ]  the scaling factor K  e  remains unchanged, when the first peak value occurs,the value of K  e  is changed in this work according with:If δ ε lk  r  >  then KK  eel = −εδ  (10)where δ l  is the overshoot, r  k   is the reference and ( ) ε ε > 0  is the final percentage allowed for the steadystate error.The tuning of the scaling factors K  e ∆  and K  u ∆  is asfollows:if δ k   > 1  then K K  eek  ∆∆ =δ  (11) K K  A uk uU  ∆∆∆ =δ  (12)else K  A ek  E  ∆∆ = δ  (13) KK  uku ∆ ∆ =δ  (14)where  A U  ∆  and  A  E  ∆  are the universes of discourseamplitude of the linguistic variable change of control andchange of error respectively. 4.Application example This section presents the results of the algorithmsdescribed before when applied to the temperature controlof a heated air stream (process trainer PT326), Figure 7.This type of process is found in many industrial systemssuch as furnaces, air conditioning, etc. and in theliterature can be found several references [ 4, 5 ] . Figure 7. Pilot process PT326. The process is very versatile with variable thermal timeconstants and variable time transport lag. It consists of aheating element controlled by a thyristor circuit that feedsheat into the air stream circulated by a centrifugal fanalong a polypropylene tube. A thermistor sensor, whichmay be placed at one of three points along the tube length,senses the temperature at that point. The volume of airflow is manually controlled by a shutter on the fan inlet.A change in setting represents a supply side disturbanceand the effects are easily demonstrated.The fuzzy logic controller receives from an A/Dconverter the sampled signal of the air-streamtemperature sensor. The control signal is applied with aD/A converter to the thyristor power amplifier. Thesignals range is [ -10,+10V ] .The algorithms described in the previous sections wereimplemented in Wuzzy   [ 7, 8 ] , a real-time fuzzy controltool for Windows 󰂮 95 environment developed by theauthors. It is suited to assist in the project, developmentand tuning of controllers, particularly fuzzy ones.Figure 8 presents the results of applying the fuzzy logiccontroller, described in section 2. The control systemresponse presents almost no steady state error. Theperformance of the controller is quite good because thecontroller parameters were carefully tuned. For instance if the reference changes substantially, in spite of therobustness of the controller, the controller performancedegrades a little bit. In Figure 8 other information besidesthe reference, output and the control signals is presented.On the top right are presented the fired rules in eachiteration and down on the right the state space defined bythe error and change of error.  Figure 8. Fuzzy logic controller Now consider the same controller but not well tuned.Figure 9 presents the results. The control system responseis very poor, oscillating, showing that the adaptive versionof the fuzzy logic controller can achieve goodperformance with that initial set of values. The values forthe scaling factors were the following: K  e  = 02. ; K  e ∆  = 05. ; K  u ∆  = 01.; (15)Figures 10 and Figure 11 shows the results of theapplication of the fuzzy logic controller with the adaptivemechanism. On Figure 10 presents the first 250 iterationsof the control system response and Figure 11 thefollowing 250 iterations. Figure 9. Fuzzy logic controller without adaptivemechanism. With the adaptive mechanism the system oscillates atfirst but immediately the oscillations suffer strongresistance. After the 250 initial iterations the overshoot issmall and after 500 iterations the error is practically null. Figure 10. Fuzzy logic controller with adaptivemechanism.Figure 11. Fuzzy logic controller with adaptivemechanism.Figure 12. Scaling factors evolution. Figure 12 shows the evolution of the scaling factors.The values assumed by K  e  are progressively reduced. K  e ∆  increases when a peak occurs to compensate theovershoot. K  u ∆  assume large values at the beginningsince the initial value is big.
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