HYBRID FUZZY LOGIC PID CONTROLLER. Abstract

Transcription

1 HYBRID FUZZY LOGIC PID CONTROLLER Thomas Brehm and Kudip S. Rattan Department of Eectrica Engineering Wright State University Dayton, OH Abstract This paper investigates two fuzzy ogic PID controers that use simpified design schemes. Fuzzy ogic PD and PI controers are effective for many contro probems but ack the advantages of the fuzzy PID controer. Design methodoogies are in their infancy and sti somewhat intuitive. Fuzzy controers use a rue base to describe reationships between the input variabes. Impementation of a detaied rue base increases in compexity as the number of input variabes grow and the ranges of operation for the variabes become more defined. We propose a hybrid fuzzy PID controer which takes advantage of the properties of the fuzzy PI and PD controers and a second method which adds the fuzzy PD contro action to the integra contro action. The effectiveness of the two PID fuzzy controer impementations are iustrated with exampes. 1 Introduction Fuzzy controers demonstrate exceent performance in numerous appications such as industria processes [8] and fexibe arm contro [5]. Mamdami's [3] work introduced this contro technoogy that Zadeh pioneered with his work in fuzzy sets [9]. Unike "two vaued" ogic, fuzzy set theory aows the degree of truth for a variabe to exist somewhere in the range [0,1]. For exampe, if pressure is a inguistic variabe that describes an input, then the terms ow, medium, high and dangerous describe the fuzzy set for the pressure variabe. If the universe of discourse for pressure is [0, 100], then ow coud be defined as "cose to 10", "medium" is "around 40", and so on. For contro appications, inguistic variabes describe the contro inputs for dynamic pants and rues define the reationships between the inputs. Thus, precise knowedge of a pant's transfer function is not necessary for design and impementation of the controer. The thrust of earier efforts invoved repacing humans in the contro oop by describing the operators' actions in terms of inguistic rues. There are two steps invoved in the impementation of a fuzzy ogic controer; fuzzification of inputs and determination of a "crisp output." Fuzzification invoves dividing each input variabe's universe of discourse into ranges caed fuzzy subsets. A function appied

2 across each range determines the membership of the variabe's current vaue to the fuzzy subset. Linguistic rues express the reationship between input variabes. Tabe I is an exampe of a matrix of rues to cover a possibe combinations of fuzzy subsets for two input variabes. In this case, each variabe has seven subsets that gives a tota of 49 rues. Defuzzification to determine the "crisp output", resoves the appicabe rues into a singe output vaue. Tabe I: PD Contro Rue Matrix Change In Error Error NB NM NS ZO PS PM PB PB ZO PS PM PB PB PB PB PM NS ZO PS PM PB PB PB PS NM NS ZO PS PM PB PB ZO NB NM NS ZO PS PB PB NS NB NB NM NS ZO PS PM NM NB NB NB NM NS ZO PS NB NB NB NB NB NM NS ZO Recent research into fuzzy contro has appied cassica techniques to stabiity anaysis [1] and design [6,10]. The operation of a fuzzy controer behaves simiar to a cassica PD or PI controer [1,6]. For a cassica PD controer, the position and derivative gains remain constant for a vaues of input. However, for a fuzzy controer, the gains depend on the range where the contro variabes exist at any instant. The piecewise inearity of the fuzzy controer provides better system response than a cassica controer [2,6,7]. Aso, since the operating point of the fuzzy controer is not fixed, it provides improved robustness to changes in the system parameters as compared to a cassica controer. Design concepts for fuzzy controers invove manipuating the fuzzy subsets and rues to obtain the desired effective controer gain. Section 2 discusses design techniques for PI and PD ike fuzzy controers as we as the drawbacks of each. As the number of contro variabes increase, impementation of the design becomes more compicated. Two hybrid PID fuzzy controers that invove a simpified design and impementation are investigated in Section 3. Simuations to demonstrate the effectiveness of the PID fuzzy controers are given in Section 4 and some concusions are drawn in Section 5. 2 Fuzzy Logic Controers Expression of the fuzzy contro action in a cosed form [1,6] provides insight into how the fuzzy subsets and output vaues affect the controer response. The fuzzification of a contro variabe consists of appying a membership function to each of the fuzzy subsets for that

3 contro variabe. For our design process, we choose the trianguar function because of its simpicity and effectiveness. The peak vaue of a membership function is the point where the output is unity and the width is the distance to zero membership as shown in figure 1. If the width extends to the adacent subset's peak vaue and vice versa, the sum of the appied memberships over that interva wi be one. Therefore, if a subsets are impemented with the widths extended to adacent peaks, the tota membership over the universe of discourse for that contro variabe wi aways be equa to one. Figure 1 shows that if the vaue of E is between E and E +1, then these two fuzzy subsets A and A +1 are active and the membership for A is E + 1 e E + E and A 1 +1 is e E + 1 E + E. These memberships sum to one as expected. Figure 1 aso shows 1 the simiar case for the second contro variabe. Figure 1. Membership Functions Two contro variabes of a PD ike controer have four "active" subsets that give four appicabe rues expressed as: R : if e(t) is A and de(t) is B then u(t) is C i R +1: if e(t) is A +1 and de(t) is B then u(t) is C +1 R +1 : if e(t) is A and de(t) is B +1 then u(t) is C +1 R +1+1: if e(t) is A +1 and de(t) is B +1 then u(t) is C +1+1 where e(t), de(t), u(t) are the error, change of error and output respectivey. The crisp output contro action is determined by appying the contro rues to a variation of the centroid of area (COA) defuzzification scheme given by

4 i u ( t ) = µ i iu i µ (1) where µ i is the product rue; [membership of e in A ]x[membership of de in B ] and U i is the output vaue C. The product rue is necessary for obtaining a cosed form of the expression for u(t) [1,4,6] and equation (1) expands to i u( t) = e J + 1 [ U + 1 [ DE, ( E [ U + ( e * de) E DE + de + E U + 1, ( E [ U U + 1, + 1, E, ( E U ( E ] + DE )( DE, E U + 1 ] + E + 1 )( DE, DE U E [ U ] + [ U, E )( DE [ U DE ] + E )( DE ) + 1, + 1 ) + 1, DE U DE [ DE U U, ) ) ] +, U DE + 1, ] ] + 1, DE U + 1, ] (2) The four terms of equation (2) demonstrates how the variation of the contro gains depend on the design of the fuzzification process and rue base. The first two terms are the inear portion of the controer. The error and change of error are mutipied by gains that are set according to the distance between adacent peak vaues of the fuzzy subsets (i.e. E and E +1 ), and the difference between the output vaues (i.e. U, and U +1, ) for the rues of the appied fuzzy subsets. Therefore, the narrower the widths of the fuzzy subsets and greater the difference between output vaues, the higher the effective gain. Aso, if the peak vaues and/or fuzzy outputs between adacent rues do not increase at exact intervas, the contro action wi vary in gain over the universe of discourse of the contro variabes. The third term is a product of the input contro variabes and a factor that is aso based on the fuzzy subsets and output vaues. Finay, the fourth term is constant but depends on the fuzzy subset pacement. If the third and fourth terms are ignored, the resutant expression indicates that a fuzzy controer behaves simiar to a cassica PD controer. 2.1 Design Concepts The fuzzy controer expressed in equation (1) is stabe [1]. However, effects on the stabiity of the system must be considered in the design process. The design of the controer invoves specifying the scae factors and the peak vaues [10]. Scae factor is the maximum peak vaue which determines the universe of discourse for the variabe. Modification of the scae factor shifts a peak vaues by the new scae factor. For exampe, if the maximum peak vaues are one and negative one and the scae factor is five, then the effective maximum peak vaue

5 is five and the universe of discourse wi be between negative five and positive five. Obviousy, a change in the scae factor affects a fuzzy rues. Modification of a singe peak vaue ony affects the rues that invove that particuar fuzzy subset. Since the scae factors affect the whoe rue base, they shoud be determined first. Adustments to individua peak vaues are done to customize the controer response. The first step for PD controer design is to assume that the universe of discourse for each contro variabe is [-1,1] and that the controer output is modeed as e k p + e k d = k p (e+ e k d /k p ). The vaues for k p and k d are seected by using any typica cassica controer design technique. The term k d /k p is the scae factor for the change in error term. The scae factor modification can be accompished by mutipying the change in error input by k d /k p or mutipying each peak vaue of the change in error fuzzy subsets by k p /k d. The error term is input directy into the fuzzy ogic controer. The output of the controer is then mutipied by the gain k p which is the scae factor for the contro action output. The design steps for a fuzzy PI ike controer are simiar, but with k i e+e k p = k p (e+k i /k p e). The second step of the design process is to adust the peak vaues of individua fuzzy subsets. The desired action of a PD type fuzzy controer initiay has ow gain of about one and an increase in the gain as the output reaches desired point. This is demonstrated in equation (2) which shows that the widths (or distance between peak vaues) of the subsets shoud decrease and the around zero subset (ZO) shoud be the narrowest [6]. For a fuzzy controer, the narrower the subset, the quicker the contro action. In the previous step ist was assumed that the gain was constant at one. However, the maximum gain of the fuzzy controer has to ensure that the system is sti stabe. Figure 2. Contro Surface

6 The response of a PD controer is basicay a contro surface as iustrated in figure 2. Modification of the scae factors from the first step either expands or contracts the contro surface aong the dimension of the modified variabe. The iustrated contro surface has narrower subsets as the contro variabes approach ZO and increased difference in output vaues at the maximum subsets. A change in any peak vaue changes the effective gain in the ocaized area of the subset. Figure 2 aso shows that the contro surface is reativey smooth as the gain continuousy increases toward the ZO region. The smoother the contro surface, the smoother the response, Therefore, modification of the subsets to change the regiona gain shoud keep the contro surface smooth. 2.2 PD and PI Controer Limitations Some of the imitations of cassica PD and PI controers carry over to their fuzzy counterparts. Like the cassica PD controer, the fuzzy counterpart can not eiminate steady-state error. The fuzzy contro action drives the pant to the zero subset for both the error and change in error (the center rue of the rue matrix). However, since the zero subset is defined as a range, it is possibe to have a sma error when the controer action is zero. In order to reduce the error, ZO for the error term can be made smaer so that there is contribution from the contro action of adacent rues of the matrix. Narrowing the ZO range corresponds to increasing the gain of the contro surface at the center. Figure 3 shows that if the change in error is zero, and the ZO for the error term decreases to a width of zero, the contro action becomes discontinuous. A discontinuous contro action causes the pant response to osciate. Integra Error Figure 3. Narrow Error ZO Suset

7 The PI fuzzy controer has a sower response due to the integra error contro variabe. Intuitivey, the controer response is a path through the matrix of contro rues. Movement from the currenty appied rue to the next rue is dependent on the current and next subset of each contro variabe. Movement from subset to subset based on the integra error contro variabe is sow because the integra error variabe changes sowy, thus sowing down the fuzzy controer response. 3 Reduced Rue PID Controer The number of rues to cover a possibe input variations for a Fuzzy PD controer is simpy the number of subsets for the error mutipied by the number of subsets for the change in error. The rue matrix expressed in Tabe I has seven subsets for error and seven subsets for change in error which gives 49 rues. Simiary, the PI contro rue matrix has 49 rues. The addition of another contro variabe significanty increases the number of rues. For exampe, if the integra error contro variabe with seven subsets is added to the PD controer, there woud be 343 contro rues. Design of such a arge rue base woud be a tedious task. Logicay, it shoud be possibe to divide the action of the PID controer into two separate contro actions: PD controer for fastest response and PI controer for the eimination of the steady-state error. Obviousy, the pant must be capabe of being compensated by a PI controer. Figure 4 is the impementation of the proposed hybrid fuzzy PID contro scheme. Simiar to separate contro rue tabes for "coarse" and "fine" contro [2], a PD controer provides the "coarse" contro and the PI controer gives the "fine" contro. The PI portion activates ony when the PD portion reduces the error and change in error to where both are in the ZO fuzzy subset range. Therefore, at any instant, cacuation of the contro action invoves ony four contro rues where as a three contro variabe controer (i.e. a typica PID) requires eight. If the three contro variabes of the hybrid controer contain seven subsets each, ony a maximum of sixteen subsets woud be checked to determine the appicabe rues. The rue search first checks the two ZO subsets for the PD portion and then checks at most a fourteen of the PI portion subsets. For the hybrid fuzzy PID controer, the PD and PI portions are designed separatey and ogic contros when to switch between the two controers. The ogic switches to the PI portion when both change of error and error are in the ZO range. The PD portion must not be re-enabed unti the error variabe moves out of the ZO range, regardess of the change in error variabe. The PI portion in the process of reaching steady-state obviousy creates a change in error that might be out of the PD's ZO range and thus reactivate the PD portion.

8 Figure 4. Hybrid PID Fuzzy Logic Controer As pointed out, the pant must be capabe of being compensated by a PI controer for the hybrid controer method to work. In order to ift this restriction, a second approach as iustrated in figure 5 was impemented. The contro action is the sum of a PD controer output and the integra of the error term scaed by a gain. As before, the PD controer provides the "coarse" contro actions but is unabe to eiminate the steady-state error. The integrator provides the additiona signa to correct for error. The first step in design is to buid the PD controer with the desired amount of rise time and the east amount of error. The integra gain determines how rapidy the steady-state error reduces. Figure 5. PD Pus I Fuzzy Logic Controer 4 Simuation Resuts The contro scheme incuding the fuzzy controer was impemented on the System_Buid modue of Matrix x. For the fuzzy controer, each contro variabe has been divided into seven ranges which gives a vector of ength seven as the fuzzification output. An error bock produces the

9 error membership vaues over the seven subsets and a change of error bock outputs the change in error membership vaues. For the PD controer, defuzzification is accompished by matrix mutipying the error membership vector by the rue gain tabe. The inner product between the resutant vector and the change of error bock give the crisp output. The PI controer is impemented in a simiar manner. The hybrid fuzzy PID controer was used to contro the servo motor with the transfer function G( s) = s( s ). The PD controer was designed to have a ramp error of 2 percent and ξ = 1 which gives k p =72 and k d = The peak and output vaues are E1 = -4/3 DE1 = -1 NB = -7/6 E2 = -2/3 DE2 = -2/3 NM = -1 E3 = 1/6 DE3 = -1/3 NS = -2/3 E4 = 0 DE4 = 0 ZO = 0 E5 = 1/6 DE5 = 1/3 PS = 2/3 E6 = 2/3 DE6 = 2/3 PM = 1 E7 = 4/3 DE7 = 1 PB = 7/6 These vaues give a gain of about 1 at the arger peak vaues and a gain of about 8 around the ZO subset. The PI portion was designed with k i /k p to be.1 and k p to be 100; and the peak vaues and output scaars were the same as the PD portion. Figure 6 shows that after a sight overshoot, the ramp response settes to the desired input. The fuzzy PD pus integra controer was used to contro singe ink fexibe arm with the transfer function G( s) = 2 s +. The type of the system is zero which has a steady-state error for a step input. The PD portion design gives a step error of 2 percent and ξ = 1; k p =49 and k d =2.13. The peak and output vaues were the same as the PD controer from the hybrid method. The integra vaue that gave the best response was 30. The step response is shown in figure 7 The PD portion provides a criticay damped response at first and the integra portions provides minima action. At steady-state, the integra term adds the necessary action to eiminate the error and the PD portion provides minima action. 5 Concusions PD and PI fuzzy controers have the same design disadvantages as their cassica counterparts. Therefore, in some cases a fuzzy PID controer maybe required. The fuzzy PID controer entais a arge rue base which presents design and impementation probems. First, a reduced rue fuzzy PID scheme was impemented to take advantage of both PD and PI contro actions. Some further research is required for the process

10 of switching between the contro actions. The second fuzzy PID contro scheme used ony the PD portion with an integra term added to eiminate steady-state error. Resuts from simuations of both contro schemes demonstrate the effectiveness of the PID controers. References 1. Ghoamresa Langari, A Framework for Anaysis and Synthesis of Fuzzy Linguistic Contro Systems, Ph.D. Thesis, University of Caifornia at Berkeey, December Y. Li and C Lau, "Deveopment of Fuzzy Agorithms for Servo Systems," IEEE Contro Systems Magizine, pp 65-71, Apri, E.H. Mamdani, "Appication of Fuzzy Agorithms for Contro of Simpe Dynamic Pant," Proc. IEE 121 Vo. 12 pp , M. Mizumoto, "Reaization of PID Contros By Fuzzy Contro Methods," IEEE Kudip S. Rattan, B.Chiu, V Feiu and H.B. Brown Jr., "Rue Based Fuzzy Contro of a Singe-Link Fexibe Manipuator in the Presence of Joint Friction and Load Changes," American Contro Conference, Pittsburgh, PA, June D. Sabharwa and K. Rattan, Design of a Rue Based Fuzzy Controer for the Pitch Axis of an Unmanned Research Vehice," NAECON Proceedings, Dayton, D. Sabharwa and K. Rattan, Design of a Rue Based Fuzzy Controer for the Pitch Axis of an Unmanned Research Vehice," NAECON Proceedings, Dayton, M. Sugeno, Industria Appications of Fuzzy Contro, North-Hoand, L. A. Zadeh, "Outine of a New Approach to the Anaysis of Compex Systems and Decision Processes," IEEE Trans. Systems, Man Cybernetics, Vo 3, pp , L. Zheng, "A Practica Guide to Tune of Proportiona and Integra (PI) Like Fuzzy Controers," IEEE 1992.

11 Figure 6. Ramp response of the system compensated by a hybrid PID fuzzy controer. Figure 7. Step response of the system compensated by a PD-I Fuzzy Controer.

12