Robust Fuzzy Fractional-Order PID Controller Design using Multi-Objective Optimization

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1 J. Basc. Al. Sc. Res., 3(s)3-36, 3 3, TextRoa Publcaton ISSN Journal of Basc an Ale Scentfc Research Robust Fuzzy Fractonal-Orer PID Controller Desgn usng Mult-Objectve Otmzaton Mohsen Rahman *, Al Barootha, Mahmou Sama, Al taherkhan Deartment of Scence, Takestan Branch, Islamc Aza Unversty, Takestan, Iran ABSTRACT Receve: June 3 Accete: July 4 3 In ths aer, mult-objectve otmzaton s use for Pareto otmum esgn of fuzzy fractonal-orer PID controllers for lants wth arametrc uncertantes. Two conflctng objectve functons have been use n Pareto esgn of the fuzzy fractonal-orer PID controller. The results clearly show that an effectve trae-off can be comromsngly acheve among the fferent fuzzy fractonal-orer PID controllers obtane usng the methoology of ths work an to acheve a robust esgn aganst the lant uncertantes. KEYWORDS: Robust Control, Fuzzy, Fractonal-orer PID Controllers, Pareto. INTRODUCTION The concet of fractonal calculus has been use n many comlex roblems an research actvtes n recent years [-3]. In orer to obtan hgh erformance-controllers, the control engneers are nvolve n usng that concet to evelo controllers whch have outstanng erformances n comarson wth nteger ones []. Inasmuch as the real ynamc systems are generally fractonal [4], fractonal calculus results n the exact moelng of ynamc systems. The comlexty an teousness of the fractonal-orer calculus cause engneers to use the nteger-orer calculus for moelng ynamc systems; however, nowaays, there are many methos to aroxmate fractonal calculus [5]. The alcaton of fractonal-orer calculus has been sgnfcantly ncrease an fractonal-orer controllers are attractvely become a major toc of control engneerng. However, esgnng hgh-erformance an cost effectve controllers s a very comlex subject whch coul be ractcally mossble to otmally accomlsh by some conventonal tral an error methos conserng fferent esgn crteron n tme an frequency oman. Ths can be more sohstcate when such fractonal-orer controllers wth more comlexty an esgn arameters are nvolve. There are many methos to esgn such controllers whch can be regare as otmzaton roblems of certan erformance measures of the controlle systems [5]. In ths aer, a new control strategy s eveloe by combnng fuzzy aroach an mult-objectve otmzaton to regulate the orers of fractonal-orer PID controllers, an other control arameters are obtane by mult-objectve otmzaton. In orer to show the robustness of roose metho, an uncertan rocess lant wth robablstc uncertantes s consere. The mean an varance of the stochastc ste resonses are consere as objectve functons whch shoul be mnmze smultaneously. Whereas n robust esgn there s a trae-off among these objectve functons; therefore, ths roblem can be formulate as a mult objectve otmzaton roblem (MOP) so that trae-offs between objectves can be foun consequently. Fractonal-Orer PID Controllers One of the most wely use controllers n oeraton toay s PID controllers. Ths s because of the fact that PID controllers are easy to esgn an to mlement, an also erforms reasonably robustly n the resence of uncertantes. In fractonal-orer PID controllers (FOPID), I an D oeratons are usually of fractonal orer; therefore, two extra egrees of freeom from the use of a fractonal-orer ntegrator an fferentator wll be ae to three gan esgn varables K, K, K. These two extra esgn varables are the orer of fractonal ntegraton λ an that of fractonal ervatve µ. Such extra egree of freeom of esgn varables makes these FOPID to surass the erformance of nteger-orer PID (IOPID) controllers [6]. The transfer functon of the PI D controller n the frequency oman can be efne as C u s K K s K s () an the control effort of the these controllers n the tme oman may be gven as t K et K D et K D et () Fgure ects the FOPID controller an exlans how the orer of the ntegrator an the orer of the fferentator can vary versus the horzontal an vertcal axs. It can be conclue that λ =, µ = mles the ornary Corresonng author: Mohsen Rahman, Deartment of Scence, Takestan Branch, Islamc Aza Unversty, Takestan, Iran. E-mal: mohsen_rahman@ymal.com 3

2 Rahman et al., 3 PID controllers, for λ =, µ =, the ornary PI controller can be obtane, an λ =, µ = mles the ornary PD controller. Evently, ornary PI, PD, an PID controller are secal cases of the fractonal-orer PID controller when the values of λ an µ are nteger values of or. FOPID Controllers Fg.. FOPID an IOPID controllers Process Moel an Controller Desgn In ths aer, the arameters of FOPID nclung λ an µ can be regulate by a fuzzy suervsory system. In orer to regulate them, two fuzzy logc systems must be efne. Each fuzzy logc system conssts of four arts, the fuzzfer, the fuzzy rule base, the nference engne, an the efuzzfer. The fuzzy rule base of ths can be wrtten as follows R : IF x s A an x s A THEN y s B (3) where x an x are nut lngustc varables, y s the outut lngustc varable. A, A, an B are the values for each nut lngustc varables an outut lngustc varable, resectvely. In ths aer, x s error an x s change n error. For each nut varable, fve trangular membersh functons are consere. In orer to esgn these membersh functons we conser that all unverses of scourses are normalze to le between - an, an the frst an last membersh functons have ther aexes at - an resectvely. Also, the base vertces of membersh functons are concent wth the aex of the ajacent membersh functons. Therefore, membersh functons can be esgne by one arameter whch s calle sacng arameter [7]. Due to ths arameter (Ps) we can obtan the aexes of each trangular membersh functons as bellow s N C, n,,, n (4) n where N s the number of membersh functons whch s an o number. It can be conclue that f Ps s less than one the centers are sace out an f Ps s more than one, the centers are close together n the center. In the multobjectve otmzaton esgn Ps s consere as a esgn varable to be foun otmally. Fgure ects fve membersh functons wth three fferent Ps..5 Ps= Ps= Ps= Fg.. Desgne membersh functon wth fferent Ps 33

3 J. Basc. Al. Sc. Res., 3(s)3-36, 3 In orer to esgn the fuzzy rule base the eas resente by Park an et al. [7] s use. In ths way, two characterstc arameters are use to esgn a fuzzy rule base. One of them s sacng arameters for the nuts an the outut, an another one s the angle for nuts. At frst, t shoul be consere that extreme nut values cause extreme outut value, also, the m-range of the nut values result n the m-range of the outut value. In aton, smlar combnaton of nuts leas to smlar lngustc value of outut. The sace arameters of the nuts are use to create ossble combnatons of lngustc varables whch are calle gr onts, an the angle s use to ve ths regon nto fferent regons. Each regon contans one outut lngust varable. Fgure 3 shows how the fuzzy rule base can be esgne. In Fgure 3, each nut an outut have fve membersh functons whose sace arameters are, an the angle s 45º. Fg. 3. Rule base constructon Table. The obtane fuzzy rule base from Fgure 3 In In The fuzzy system for both λ an µ are efne as follows e e, e. F, (5). F e (6) where, λ an µ are the base of fractonal-orers an F λ an F µ are fuzzy functons. The sace arameters, angle an the base of fractonal-orers coule wth K, K, K are the esgn varables, whch are all obtane usng a multobjectve genetc algorthm. Process Moel an Controller Desgn Many nustral systems can be aequately resente by a frst-orer lus n ntegrator as k (7) G( s) s( Ts) In the case of stochastc robust esgn [8], arameters of the rocess lant vary accorng to a ror known robablstc strbuton functons aroun a nomnal set of arameters. In ths work, beta strbutons wth the coeffcents of an wth the lmts of ±5% of the nomnal values of lant arameters k=t= have been selecte. A goo ste resonse behavor of the system s one of the tme oman erformance metrcs n controller esgn roceure that llustrates how system acts n transent an steay state eros. In orer to obtan a goo tme-resonse, there are many crtera base on the ntegral of the ste resonse error, such as the ntegral square error (ISE) crteron, an ntegral tme weghte square error (ITSE) crteron. In ths aer the ITSE whch s gven as follow s use to efne the otmzaton crteron for a goo ste resonse. Tf (8) ITSE te ( t) t where, e(t) s the ste resonse error of each ranom close-loo system. 34

4 Rahman et al., 3 Whereas, the rocess lant n ths aer has uncertan arameters therefore n the Monte-Carlo Smulaton, the ste resonses are stochastc resonses an ITSE for each resonse s stochastc varable. In the robust otmum esgn, both mean an varance of stochastc ste resonses shoul be mnmze. RESULTS The two objectve functons are now consere smultaneously n a Pareto otmzaton rocess to obtan some mortant trae-offs among the conflctng objectves. In a robust esgn aroach, the vector of objectve functons to be otmze n a Pareto sense s gven as follow J mean( ITSE), var ance( ITSE) (9) whch are comute n a quas-monte Carlo smulaton rocess. The evolutonary rocess of the mult-objectve otmzaton s accomlshe wth a oulaton sze of 5 whch has been chosen wth crossover robablty P c an mutaton robablty P m as.8 an., resectvely. The otmzaton rocess of the robust controller s accomlshe for Monte Carlo evaluatons usng Hammersley Sequence Samlng (HSS) [8] strbuton for each canate control law urng the evolutonary rocess. Pareto front of two objectve functons s shown n Fgure 4. Evently, t can be seen that mrovng one objectve wll cause another objectve eterorates. A J 3. x J Fg. 4. Pareto front The mean of ste resonse for otmum trae-off ont A for HSS s shown n Fgure 5. Also, the varance of the ste resonse for ths ont s shown n Fgure 6. It can be conclue from Fgure 5 an Fgure 6 that the roose metho causes the system to have a goo ste resonse wth hgh robustness.. Mean of ste resonse Tme (sec) Fg. 5. Mean of ste resonse of otmum ont A 35

5 J. Basc. Al. Sc. Res., 3(s)3-36, 3 6 x -3 5 Varance of ste resonse Tme (sec) Fg. 6. Varance of ste resonse of otmum ont A The stochastc ste resonse for samles wth the otmum controller A selecte from the obtane Pareto front s shown n Fgure 7. It can be conclue that the roose metho erforms robustly n the resence of the lant robablstc uncertantes..4. Ste resonses Conclusons Tme (sec) Fg. 7. Stochastc ste resonses of otmum ont A for samles In ths work, mult-objectve otmal robust fuzzy fractonal-orer PID controller has been roose an successfully use for an uncertan rocess lant. The mult-objectve otmzaton of fuzzy fractonal-orer PID controller le to the scoverng the trae-off between those objectve functons. The mult-objectve GAs of ths work for the Pareto otmzaton of fuzzy fractonal-orer PID controller usng some non-commensurable objectve functons s very romsng an can be generally use n the otmum esgn of real-worl comlex control systems wth uncertantes. REFERENCES [] Conceco n Monje, A., Blas, M., Vcente, F., an YangQuan, C., 8. Tunng an auto-tunng of fractonal orer controllers for nustry alcatons, Control Engneerng Practce, 6, [] Felu-Batlle, V., Rvas-Perezb, R., an Castllo-Garca, F.J., 9. Fractonal orer controller robust to tme elay varatons for water strbuton n an rrgaton man canal ool, Comuters an Electroncs n Agrculture, 69, [3] Ramro, S., Barbosa, J.A., Tenrero, M., an Alexanra, M., 7. Performance of Fractonal PID Algorthms Controllng Nonlnear Systems wth Saturaton an Backlash Phenomena, Journal of Vbraton an Control, 3, [4] Chen, Y., Petras, I., an Xue, D., 9. Fractonal Orer Control - A Tutoral, Amercan Control Conference, Hyatt Regency Rverfront, St. Lous, MO, USA, [5] Bouafoura, M.K., an Braek, N.B.,. PI l D µ controller esgn for nteger an fractonal lants usng ecewse orthogonal functons, Communcaton n Nonlnear Scence an Numercal Smulaton, 5(5), [6] Xue, D., an Chen, Y., 9. Solvng Ale Mathematcal Problems wth MATLAB, CRC Press, Taylor & Francs Grou. [7] Park, Y.J., Cho, H.S., an Cha, D.H., 995. Genetc Algorthm-Base Otmzaton of Fuzzy Logc Controller Usng Characterstc Parameters, Proc. of the 995 IEEE Internatonal Conference on Evolutonary Comutaton, [8] Hajloo, A., Narman-zaeh, N., an Moen, A., 8. Pareto Otmum Desgn of Robust Controllers for Systems wth Parametrc Uncertantes, Robotcs, Automaton an Control,. 5-6, I-Tech, Venna, Austra. 36