A Gradient Descent Based Online Tuning Mechanism for PI Type Single Input Interval Type-2 Fuzzy Logic Controllers Tufan Kumbasar and Hani Hagras
Abstract—In this paper, we will present design methods for Single input IT2-FLCs (SIT2-FLCs) and we will introduce an online tuning mechanism to enhance their control system performance. The most important feature of the SIT2-FLC is the closed form output presentation which is defined in a two dimensional domain. Based on this structural information, we will present design methods for SIT2-FLCs composed of 3 rules to produce a Smooth SIT2-FLC (S-SIT2-FLC) and an Aggressive SIT2-FLC (A-SIT2-FLC) by only tuning a single parameter. It will be shown that the S-SIT2-FLC will result in a potentially more robust control performance in comparison A-SIT2-FLC. However, the transient state and disturbance rejection performance of the S-SIT2-FLC might degrade in comparison to the A-SIT2-FLC. This drawback will be solved by tuning the FOU size of the SIT2-FLCs to provide a trade-off between the robust control performance of the S-SIT2-FLC and the acceptable transient and disturbance rejection performance of the A-SIT2-FLC structure. Thus, we will present a GradientDescent (GD) based online tuning mechanism to enhance both the transient state and disturbance rejection performances of the SIT2-FLCs while preserving a certain degree of the robustness against nonlinearities and disturbances. We will present simulation results where the GD based SIT2-FLC (GD-SIT2- FLC) is compared with the S-SIT2-FLC and the A-SIT2-FLC structures. Moreover, we will compare the performance GD-SIT2-FLC with a robust self-tuning Type-1 (T1) FLC which has a fuzzy based tuning mechanism. The results will show that the GD-SIT2-FLC enhances both the transient state and disturbance rejection performances when compared to the IT2 and robust self-tuning T1 counterparts. Keywords—Interval type-2 fuzzy logic controllers; Design Methods, Self-tuning controllers
I. INTRODUCTION Type-1 (T1) Fuzzy Logic Controllers (FLCs) have been widely used as alternatives to conventional controllers which can be designed using single; two or three inputs [1-10]. Although the majority of the research work on T1-FLCs focuses on the two-input structures [1-9], it has been shown that single input T1-FLC provide greater flexibility and better functional properties [7-9]. Recently, there has been growing research interest on Interval Type-2 (IT2) FLCs since IT2-FLCs have demonstrated control performance improvements due to the additional degree of freedom provided by the Footprint of Uncertainty (FOU) present in T. Kumbasar is with the Control and Automation Engineering Department, Istanbul Technical University, Istanbul, Turkey (e-mail:
[email protected]) H. Hagras is with the Computational Intelligence Centre, University of Essex, Colchester, United Kingdom (e-mail:
[email protected]) This research is supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under the project (113E206). All of these supports are appreciated.
their IT2 Fuzzy Sets (FSs) [10-17]. The internal structure of the IT2-FLC is similar to its T1 counterpart [17-20]. Nevertheless, the systematic design of IT2-FLCs is still a challenging problem due to the main difficulty in determining the parameters of the FSs and the rulebase [21-25]. In the IT2 fuzzy control literature, the main focus is on double input IT2-FLCs since the design of the IT2-FLC is usually solved by extending/ blurring its T1 counterpart (since IT2-FS is a generalization of T1-FS) [24], [25] or by employing evolutionary optimization methods [10-15]. The main drawback of these approaches is the lack of understanding of how the FOU parameters affect the performance of the IT2-FLCs [22]. However, several studies have been presented to analyze and examine the effect of the FOU parameters on the performance of the IT2-FLC [21-25]. It has been showed in [23-25] that the IT2-FLCs are generally more robust than their T1 counterparts since they result in smoother control surface around the steady state. Although a smooth control surface is probably a common objective in practice, the problem is that the resulting disturbance response might be unacceptable (too slow) since disturbances occurring around the steady state will cause a smaller control output change [25]. Recently, a self-tuning zSlices based general Type-2 FLC has been proposed to achieve a satisfactory robust control and disturbance rejection performance [25]. In this paper, we will present design methods for PI type Single input IT2-FLCs (SIT2-FLCs) and then propose an online tuning mechanism to enhance the performance of its control system. The most important feature of the SIT2-FLC is the closed form presentation of its Fuzzy Mapping (FM) which is defined in a two dimensional domain. Thus, the design problem of the IT2-FLC will be transformed from control surface generation to Control Curve (CC) generation. Then, based on the analytical information of the IT2-FM, design methods will be presented for SIT2-FLCs composed of 3 rules to generate commonly employed CCs such as smooth and aggressive. We will show that it is possible to design a Smooth SIT2-FLC (S-SIT2-FLC) and an Aggressive SIT2-FLC (A-SIT2-FLC) by only tuning the FOU parameters based on the presented derivations. It will be shown that the S-SIT2-FLC will result in a potentially more robust control performance in comparison A-SIT2-FLC. However, its transient state and disturbance rejection performance might degrade in comparison to the A-SIT2-FLC structure. Therefore, to provide an acceptable trade-off between the robust control performance of the S-SIT2-FLC and the acceptable transient and disturbance rejection performance of the A-SIT2-FLC, we will propose a Gradient Descent (GD) based online tuning mechanism which will tune the FOU parameters in an online manner. We will present comparative
simulation results to show that the self-tuning GD based PI type SIT2-FLC (GD-SIT2-FLC) improved both the transient state and disturbance rejection in comparison to the S-SIT2-FLC and A-SIT2-FLC counterparts. Moreover, we will also compare the GD-SIT2-FLC with a robust self-tuning PI type T1-FLC [5] which has relatively more design parameters and an extra fuzzy inference. It will be shown that tuning the FOU size of the SIT2-FLC in an online manner will give the opportunity to enhance the overall control performance of the SIT2-FLC in terms of transient state and disturbance rejection while also preserving a certain degree of robustness against nonlinear dynamics and disturbances when compared to its IT2 and robust self-tuning T1 counterparts. Section II will present the structure and design strategy of the PI type SIT2-FLC, Section III will present the GD based online tuning mechanism. Section IV will present the comparative simulation results, and Section V will present conclusions and future work.
IT2-FSs can be described in terms of upper MFs and which create the FOU in IT2-FSs (the lower MFs extra degree of freedom of the SIT2-FLC) [16-20]. As shown in Fig. 1b, mn’s represent the height of the lower MFs and will be the main design parameters of the SIT2-FLC to be tuned. We will prefer to employ symmetrical MFs, thus the value of will be set to the value of . Thus, the membership grades of the MFs for a crisp input ′ will be as: | |, | ′| ′ ′ ′ ′ (5) |1
′
′|,
where
and ∑
are defined as follows: ∑ ′ ∙ ∑
Hence, the FLC will reduce to a conventional PI controller. is defined such that the Here, the input Scaling Factor (SF) input is normalized to the universe of discourse where the antecedent Membership Functions (MFs) of the SIT2-FLC where are defined. Therefore, is defined as 1⁄ is the maximum allowable error value. Consequently, is converted after normalization into which is the input of the SIT2-FLC while the its output ( ) is converted into the control signal as follows: (2) where (3) Here, and are the baseline PI controller gains, and . It can be seen that, the is the output SF defined as output of the PI type SIT2-FLC is analogous to a conventional PI structure [23]. Note that, the SFs and the baseline PI gains of the SIT2-FLC can directly affect the performance and robustness of the control system [26], [27]. A. The Internal Structure of the SIT2-FLC In this study, a SIT2-FLC which is composed of N=3 rules is employed and preferred for simplicity. The rule structure of the SIT2-FLC is as follows: (4) : IF is THEN is , 1,2,3 are the crisp consequents which are set as where 1, 0 and 1 . The antecedent MFs are defined as shown in Fig. 1b The with triangular IT2-FSs
|1
′|
(6)
In [1], it has been demonstrated that the defuzzified output of the SIT2-FLC is as follows: (7) ⁄2
II. THE PI TYPE SIT2-FLC In this section, the structure and design methods of the PI type SIT2-FLC is presented. The SIT2-FLC is constructed by choosing the input to be the error signal and the output as the control signal as shown in Fig. 1a [23]. In this structure, the SIT2-FLC is cascaded to a baseline PI is a Unit Mapping controller. Thus, if the IT2-FM (UM) as follows: (1)
′
∑
′ ′ ∙
∑
′
′ ∙
∑
′
∑
′ ∙
∑
′
(8)
(9)
Here, R and L are the switching points which minimize/ maximize Equations (8) and (9), respectively [18]. Since it is always guaranteed that a crisp value of always belongs to two successive IT2-FSs, the switching points (R, L) are always equal to “1” (for any crisp input only two rules (N=2) are always activated) [23]. Thus, the FM of the SIT2-FLC can be derived for the input ∈ 0, 1 as follows: 1 2
(10)
Replacing the membership grades given in Equations (5) and (6) into (10), we can obtain (11) ∙ where is the nonlinear gain generated from the IT2-FM and is defined as: 1 1 (12) 2 1 1 Similarly, for the input ∈ 1,0 , the IT2-FM of the SIT2-FLC can be derived as: 1 2
(13)
Now, by replacing Equations (5) and (6) into Equation (13), we can obtain: (14) ∙ It can be observed from Equations (11) and (14), that IT2 FM satisfies the following properties: (i) The IT2-FM increase proportionally with respect to its ⁄ input since 0 is always satisfied. (ii) The IT2-FM is always a symmetrical FM with respect to its input since for ∀ 0 is always satisfied. (iii) If 0, then the output of IT2-FM is always 0 which is necessary to have zero steady state error.
Fig. 1. Illustration off the (a) PI type SIT2-FLC structurre (b) antecedent IIT2-FSs of the IT T2-FLC
B B. Design Strrategy of the SSIT2-FLC In this secttion, we will present desiggn methods ffor the SIT2-FLCs too construct IT T2 fuzzy CCs (CCIT2s). As it has bbeen derived in the Equatioons (11) and (14), the SIT22-FLC ooutput can bee explicitly deerived in the input domainn. This ssimplifies the SIT2-FLC dessign method innto a CC generration, iinstead of a control c surfacee design [26]. Thus, we wiill first eexamine the eeffect of the FOU parametters on the IT T2-FM ggeneration. F For the sake of simplicityy, we will employ and 1 inn the rest of the paper. Thuus, the nnonlinear gainn generated fr from the IT2-F FM given Eqquation ((12) can be redefined as: 1 1 1 (15) 2 1 T The design off the IT2-FM w will be accomp mplished with rrespect tto the UM. In this context, llet us first defi fine: (16) w where iis defined as thhe difference bbetween the U UM and IIT2-FM. The ssign variation of mation will provide inform aabout the agggressiveness aand smoothneess of the CC CIT2 in ccomparison too the CC of thhe UM (U-CC)). In this conteext, let uus firstly redefine Equation (16) as follow ws: (17) 1 T Then, it can be concluded thhat: If 0 , then the IT2-FM is m more aggressiive in comparisoon with the UM M for ∀ ∈ .
∈ 0,1 aand Consequently, an √ √5 1 /2 . C Aggressive C CCIT2 (A-CCITT2) will be geneerated. If , then 0 for ∀ ∈ and 0 for ∀ ∈ where ∈ 0, andd ∈ ,1 . Thus, an inveerse S-shaped CCIT2 will be generated. Notte that, since tthe IT2-FM is symmetrical FM with resppect to itts input, similaar observationns can be madee for the IT2-F FM wheen ∈ 1,0 . In Fig. 2, tthe S-CCIT2 aand A-CCIT2 are skettched for the values v 0.2 and 0.8, resspectively. It ccan be clearly seen that the S-CC CIT2 has relattively low inpput mparison to tthe senssitivity when is close tto “0” in com A-C CCIT2. Thus, w we can concludde that: The S-SIT2-FLC is poteentially moree robust agaiinst parameter vvariations andd disturbancess. However, its resulting dissturbance rejeection responsse might be ttoo slow since diisturbances occurring aroundd the steady sttate will cause a smaller s controol output changge. The A-SIT2--FLC will ressult with a fasst transient sttate response andd disturbancee rejection buut might not be robust againsst nonlinearitiees and uncertaainties. Itt can be conccluded that tuuning the design parameterr (w which createss the FOU) in an onlinee manner migght pprovide an acceptable trade-off between thhe robust conttrol pperformance off a S-SIT2-FL LC and the accceptable transient aand disturbance rejection perrformance of an a A-SIT2-FL LC.
If 0,, then the IT22-FM is smooother in compparison with the U UM for ∀ ∈ . T The neighborrhoods 0 and 0 can be ddetermined byy simply investigating the zzero crossing points oof . Thuus, we will exaamine the signn variation off w with respect too the tuning paarameter forr ∈ 0,1 . Thhe zero ccrossing pointts of a found as: aree derived and are 1 (18) 1 2 2 H Here, is onne of the bounndary points oof the intervall 0,1 . T Therefore, thee IT2-FM willl always reducce to the UM at the ppoint 1 sincee 0 wherreas the poinnt will provide iinformation about sign vvariation of . Thuus, by eexamining thee sign variationn of , wee can observe:: where If 0 , thenn 0 foor ∀ ∈ ∈ 0,1 1 and 3 √5 /2 . Thus, a Smooth CCIT2 (S-CCIT2) will bee generated. where If 1 , theen 0 foor ∀ ∈
Fig. 2. Illuustration of the S--CCIT2, A-CCIT2 and a U-CC
IIII. GRADIENT DESCENT BASSED ONLINE TU UNING METHOD A As it has been asserted in the preceeding section,, a selff-tuning SIT2--FLC might bbe able to ennhance both tthe trannsient state and disturbance rejection perfformances whhile provviding a certaiin degree of roobustness to coontrol system.. In this context, we w will present thhe novel appliccation of the G GD metthod [28] to tuune the FOU parameter off the SIT2-FLC C.
Fig. 3. Illustrration of the (a) G GD-SIT2-FLC struucture (b) OSF-T1-FLC structure
In the GD D method bassed tuning mechanism m thee cost ffunction to bee minimized is defined as: (19) 1⁄2 H Here, is the output of ddesired referennce model whhich is ddefined as: (20) 1 1 1 is thhe time consttant of the desired w where 0 d rreference moddel and is thee current samppling step. Thee FOU pparameter oof SIT2-FLC w will be updatedd as follows: (21)
1
H Here, is the learning rate of GD methood and the. W We will ddefine the derivative in Equuation (21) as follows: f (22) w where
∆ ∆
1 1
(23)
aand 1 2
1
T Thus, an onlline tuning of the FOU pparameter aaccomplished as shown in F Fig. 3a.
(24) c be can
IV. EXPERIM MENTS AND RE ESULTS t control ssystem In this secction, we wiill examine the pperformance of the GD-S SIT2-FLC in comparison to the S-SIT2-FLC aand A-SIT2-FLC structures. Moreover, w we will ccompare the SIT2-FLCs w with a robust self-tuning T1-FLC [5]. The T1-FL LC is construccted with two iinputs, which are a the eerror signal s ∆ , aand the and changee of the error signal ooutput is the control signaal as show wn in Fig. 3bb. The T T1-FLC is com mposed of 9 ruules and has tw wo input SFs annd one ooutput SF. In this structure, the output SF is adjusted online bby the fuzzy parameter p reguulator shown inn Fig. 3b. Thee fuzzy pparameter regulator is a Maamdani type FL LC that consissts of a 77x7 rule basee where its anntecedent andd consequent ppart is ddefined with triangular T1--FSs. Detailedd information about O Output SF tuuned T1-FLC (OSF-T1-FLC C) structure ccan be ffound in [5]. IIt is worth to m mention that thhe OSF-T1- FL LC has
relaatively more design param meters and an extra fuzzzy infeerence in compparison to the SIT2-FLC strructures. W We will examinne and compaare the transiennt state responnse, distu turbance rejecttion performannces and the roobustness agaiinst nonnlinear dynam mics of the PI type FLCs on o the followiing nonnlinear benchm mark system: 0.25
(225)
wheere is the tim me delay whichh has been sett to 0.5 [5]. Inn the simulaation studies, the PI tyype FLCs were impplemented as the discrete ttime versionss with the fixxed sam mpling time off 0.1s and the controllers w were designed for unitt step reference. The A-SIT22-FLC has beeen designed suuch thatt to have a faast transient sttate and distuurbance rejectiion perfformance by setting s 0.3. The S-SIT T2-FLC has beeen tuneed by settinng 0.95 to have a robust conttrol perfformance for tthis operation ppoint. In the ddesign of the G GDSIT T2-FLC, the innitial conditionn of the FOU parameter has h beenn set to the same of vallue of the S--SIT2-FLC onnes 1 0.95 while w the paraameters of G GD based tuniing 0.01 1 and 0.04 4. To make a ffair mecchanism are seet as com mparison, thee SFs and baseline PI parameters of SIT T2-FLCs have been set andd fixed to the same values as folloows 1, 0.3 and 0 0.12. The desiign paraameters of thee OSF-T1-FL LC have been set to the sam me valuues as given inn [5]. We willl compare the performancess of the control system ms with respeect to their Setttling Time (T Ts), Oveershoot (OS%)) and Integrall Absolute Errror (IAE) valuues. Thee simulations w were performeed on a personaal computer w with an IIntel Pentium Dual Core T22370 1.73 GHzz processor, 2.99 GB RAM, and sooftware packagge MATLAB//Simulink 7.4.00. W We will first examine e the trransient state performances of the FLCs for a reeference trajecctory with the values of 1, 00.5 andd 0.9 in successsive order sincce the nonlineaarity is relatedd to the system outpput. Moreoveer, we will investigate tthe turbance rejecction perform mances of thee FLCs. In tthis distu context, we will ffirstly employy a unit step reference and thhen empploy input andd output distuurbances withh the magnituddes of0..2” in 20th annd 40th secondds, respectivelly. The transient statee and disturbaance rejection performancess of the FLCs are illusstrated in Fig.. 4 and Fig. 55, respectivelyy. The calculatted perfformance meaasures of the FLCs F are tabulated in Tablee 1.
OSF-T1-FLC C A-SIT2-FLC C S-SIT2-FLC C GD-SIT2-FLC
TABLE 1. THE CALLCULATED PERFO ORMANCE MEASUR RES OF THE FUZZY Y CONTROL SYST TEMS Transsient State Perforrmance Disturbaance Rejection P Performance 0-1 1--0.5 0.5-0.9 Total IAE Input Disturb bance Outtput Disturbancee Ts O OS Ts OS T Ts OS IAEdu IAEdy 18.5s 19.3% 8.7s 20.5% 92.1860 9.8702 10.9434 13.7s 16.1% 10.8s 24.8% 14.3s 17.0% 9.9s 21.6% 69.1983 6.1037 10.3370 8.8s 8.55% 14.9s 0.0% 188.4s 0.0% 82.4200 16.39988 29.9330 8.6s 9.66% 7.7s 4.38% 4.3s 8.4% 63.9619 8.5402 14.4023
Fig. 4. IIllustration of thee transient state peerformances of FL LC structures for the varying refereence signal
F For instance, if we examiine the resultts for the refference vvariation from m 1 to 0.5, it caan be firstly obbserved that alll three SIT2-FLCs reesulted with beetter Ts and OSS% values whiile also pproviding low wer Total IA AE values in comparison to the O OSF-T1-FLC.. Moreover, as expectedd the S-SIT22-FLC sstructure resuulted with a more robusst system ressponse ((without any ooscillations) inn comparison to the A-SIT22-FLC. H However, in comparison c too the S-SIT2-F FLC, the GD--SIT2F FLC structuree decreased thhe Ts value abbout 47% (inccreased tthe convergennce speed to reeference) while it also reducced the ttotal IAE vaalue about 223%. On thee other handd, the G GD-SIT2-FLC C structure ressulted with ann OS% value of o 4.38 w while the S--IT2-FLC proovided with a system ressponse w without any ovvershoot. Simiilar commentss can be made for the oother two refference variatiions. Note thaat, for all refference vvariations, thee OS% value of the GD-S SIT2-FLC is aalways rrelatively biggger than the S--IT2-FLC oness but still acceeptable ssince it resulted with lowerr Ts value whhich is an acceeptable ttradeoff from a control enggineering pointt of view. How wever, w when we exam mine the distuurbance rejectiion performannces, it ccan be seen that when coompared to tthe S-IT2-FLC C, the G GD-IT2-FLC has better ooutput (IAEdy) and input (IAE ( du) ddisturbance reejection perforrmance valuess. Note that, thhe best ddisturbance rrejection perfformances aree obtained by b the A A-SIT2-FLC and OST-T1--FLC which pperformed relaatively ppoor in their trransient state pperformances iin comparisonn to the G GD-IT2-FLC.. Moreover, the GD-IT2-FLC resultedd with ddisturbance rrejection respponse withouut any oscilllations aalthough it resulted with reelatively biggeer IAE valuess when ccompared to thhe A-SIT2-FL LC and OST-T T1-FLC structuures.
W We have also investigated tthe convergennce of the tuniing paraameter α of the GD-SIT2-FL LC. As it can bbe seen from F Fig. 4 annd Fig. 5, the value of thhe FOU param meters relatively decrreases to accellerate the systeem response ssufficiently whhile reduucing possiblee overshoot aand oscillationns as the system respponse approaches the referennce value. Itt can be clearrly observed tthat the GD based b self-tuniing metthod gives thee opportunity to enhance tthe speed of tthe trannsient state andd disturbance rejection perfformances of tthe SIT T2-FLC while aalso providingg a certain degrree of robustness agaiinst the presennt nonlinearityy and the distuurbances. V. CONCLUSIONS O A AND FUTURE WORK Inn this paper, we w proposed a novel self-tuuning GD-SIT T2FLC C where the FOU size oof the anteceddent IT2-FSs is adjuusted in an on-line mannner. We first presented tthe struucture of the P PI type SIT2--FLC and derived its IT2-F FM withh respect to thhe FOU param meters. We provvided theoretiical explanations show wing how thee size of the F FOU affects tthe controller perforrmance. Thenn, we have ppresented desiign metthods for SIT22-FLCs composed of 3 rulees to construcct a S-SIT2-FLC andd an A-SIT2-F FLC by only tuning the FO OU S w will paraameter α. We have concludded that the S-SIT2-FLC resuult with a poteentially more robust r control performance bbut its transient statte and disturrbance rejection performannce migght degrade inn comparison to the A-SIT22-FLC structuure. Thuus, we presenteed a novel impplementation of o GD methodd to provvide a trade-offf between thee robust controol performancee of the S-SIT2-FLC and a the accepttable transientt and disturbannce rejeection perform mance of the A-SIT2-FLC C. We presentted
ssimulation reesults conduccted on a noonlinear systeem to vvalidate the prroposed approoach where thee GD-SIT2-FL LC was ccompared withh the A-SIT2--FLC, S-SIT2--FLC and the robust sself-tuning O OSF-T1-FLC sstructures. Thhe presented rresults ssupported the effectiveness of the proposeed SIT2-FLC design aapproaches and a the propposed GD baased online tuning m mechanism. It can be concluded that t the prooposed G GD-SIT2-FLC C gives the oopportunity too enhance booth the ttransient statee and disturbannce rejection performances while ppreserving the robustness against nonliinear dynamiccs and ddisturbances in different operating pooints which iis not ppossible with its IT2 and robbust self-tuninng T1 counterpparts. a to focus onn more sophistticated For our futuure work, we aim ttuning mechannisms which m might improvee the performaance of tthe IT2-FLCs and employ thhem in real-tim me applicationns. REFFERENCES [1]
R.-E. Precupp and H. Hellenddoorn, “A survey oon industrial appllications of fuzzy conntrol,” Comp. Indd., vol. 62, pp. 2133–226, 2011. [2] S. Galichet and L. Foullloy, “Fuzzy conntrollers: synthesis and equivalencees,” IEEE Trans. Fuzzy Syst., vool. 3, no. 2, pp.1440–148, 1995. [3] M. Mizumooto, “Realization oof PID controls byy fuzzy control meethods,” Fuzzy Sets aand Systems, vol. 70, pp. 171–1822, 1995. [4] H. Ying, "T Theory and applicaation of a novel fuuzzy PID controlller using a simplifiedd Takagi-Sugeno rule scheme", Inf. Sci., vol. 123, no. 3-4, pp.281-293,, 2000. [5] R. K. Mudii and N. R. Pal, ““A robust self-tuuning scheme for PI- and PD-type fuzzzy controllers,” IIEEE Trans. Fuzzzy Syst., vol. 7, nno.1, pp. 2–16, 1999.. [6] Z. W. Woo, H. Y. Chung annd J. J. Lin, “A PID-type P fuzzy coontroller with self-tuuning scaling facttors,” Fuzzy Setss Systems, vol. 1115, pp. 321–326, 20000. [7] G. K. I. Maann, B. G. Hu, andd R. G. Gosine, ““Analysis of direcct action fuzzy PID ccontroller structurres,” IEEE Trans.. Syst., Man, Cybbern., B, Cybern., vool. 29, no. 3, pp. 371–388, 1999. [8] B. J. Choi, S. S W. Kwak, and B. K. Kim, “Dessign and stability analysis of single-innput fuzzy logicc controller,” IE EEE Trans. Systt., Man, Cybern., vool. 30, no. 2, pp. 303–309, 2000. [9] S. Yordanovva, “Robust stabillity of single inputt fuzzy system forr control of industriaal plants with timee delay,” Journal of Intelligent andd Fuzzy Systems, vool. 20, no. 1, pp. 229-43, 2009. [10] T. Kumbasar and H. Hagraas, “Big Bang-B Big Crunch Optim mization based Intervval Type-2 Fuzzy PID Cascade Conntroller Design Sttrategy“, Inf. Sci., vool. 282, pp. 277–295, 2014.
[11] E. Yesil, “Intervval type-2 fuzzy PID load frequeency controller ussing Big Bang–Big C Crunch optimizatioon”, Applied Softt Computing, vol. 15, pp. 100–112, 20014. P. Melin, “A reviiew on the designn and optimizationn of [12] O. Castillo and P interval type-2 fu fuzzy controllers,”” Applied Soft Coomputing, vol. 12, pp. 1267–1278, 20112. [13] S-K. Oh, H.-J. Jang, and W. Peedrycz, "A comparative experimenntal study of type-11/type-2 fuzzy ccascade controlleer based on gennetic algorithms and pparticle swarm opptimization," Exppert. Syst. Appl., vvol. 38, no. 9, pp. 11217-11229, 20111. [14] R. Martínez, O. Castillo, and L. T. Aguilar, "Opttimization of interrval type-2 fuzzy loogic controllers ffor a perturbed aautonomous wheeeled mobile robot using genetic algoriithms," Inf. Sci., vol. 179, no: 13, pp. 2158-2174, 20099. W. Tan, “Geneticc learning and perrformance evaluattion [15] D. Wu and W. W of type-2 fuzzy logic controllers,,” Int. J. Eng. Apppl. Artif. Intell., vvol. 19, no. 8, pp. 8229–841, 2006. Hierarchical Typee-2 Fuzzy Logic C Control Architectture [16] H. Hagras, “A H for Autonomouss Mobile Robots,”” IEEE Trans. Fuzzzy Syst., vol. 12, no. 4, pp. 524-539, 22004. C. Wagner, “Tow wards the Wide Sppread Use of Typpe-2 [17] H. Hagras and C Fuzzy Logic Systems in R Real World Applications,” A IE EEE Computational IIntelligence Magaazine, vol.7, no. 3, 3 pp.14-24, 2012.. [18] J. Mendel, Unceertain Rule-Basedd Fuzzy Logic Syystems: Introducttion and New Directiions. Upper Sadddle River, NJ: Prenntice-Hall, 2001. M. Mendel, “Interrval type-2 fuzzy logic l systems: theeory [19] Q. Liang and J.M and design,” IEE EE Trans. Fuzzy S Syst., vol. 8, no.55, pp. 535-550, 20000. M. Mendel, “On the continuity off type-1 and interrval [20] D. Wu and J. M type-2 fuzzy loggic systems,” IEE EE Trans. Fuzzy Syst., vol. 19, noo. 1, pp. 179–192, 20011. a analysis of thee analytical structuures [21] X. Du and H. Yiing, "Derivation and of the interval tyype-2 fuzzy-PI annd PD controllers." IEEE Trans. Fuuzzy Syst., vol. 18, noo. 4, pp. 802-814,, 2010. Ying, " A Methodd for Deriving thee Analytical Structture [22] H. Zhou and H. Y of a Broad C Class of Typicall Interval Type--2 Mamdani Fuuzzy Controllers" IEE EE Trans. Fuzzy S Syst., vol. 21, no. 3, 3 pp. 447-458, 20013. [23] T. Kumbasar, ““A Simple Designn Method for Intterval Type-2 Fuuzzy PID Controllers,,” Soft Computingg, vol. 18, no. 7, ppp. 1293-1304, 20014. [24] D. Wu, “On the Fundamental Diffferences betweenn Type-1 and Interrval Type-2 Fuzzy L Logic Controllers,” IEEE Trans. Fuzzy F Syst., vol., 10, no.5, pp. 832- 8448, 2012. A Self-Tuning zS Slices based General [25] T. Kumbasar annd H. Hagras, “A Type-2 Fuzzy PI P Controller,” IE EEE Trans. Fuzzzy Syst., (Articlee in Press). A One to Three Input Mapping IT T2-FLC PID Dessign [26] T. Kumbasar, “A Strategy”, in Prooc. IEEE Int. Conf nf. Fuzzy Syst, Hydderabad, India, 20013. D Type Single Innput [27] T. Kumbasar, ““Robust Stabilityy Analysis of PD Interval Type-22 Fuzzy Control Systems”, in Prroc. IEEE Int. Coonf. Fuzzy Syst, Beijiing, China, 2014.. C. C. Chen, "A hyybrid clustering and a gradient desccent [28] C. Wong and C approach for fuzzzy modeling", IE EEE Trans. Syst., M Man, Cybern. B, vvol. 29, pp.686 -6933, 1999.
Fig. 5. Illustration of the input and output o disturbance rejection responnses of FLC structures
