Application of a Modified PSO Algorithm to Self-Tuning PID Controller for Ultrasonic Motor

Transcription

1 The Proceedngs of the st Internatonal Conference on Industral Applcaton Engneerng Applcaton of a Modfed PSO Algorthm to Self-Tunng PID Controller for Ultrasonc Motor Djoewahr Alrjadjs a,b,*, Kanya Tanaa b, Shota Naashma b, Shengln Mu b a Electronc Engneerng Polytechnc Insttute of Surabaya Jl. Raya ITS, Keputh, Suollo, Surabaya, Indonesa b Department of Electrcal and Electronc Engneerng, Yamaguch Unversty -- Towada, Ube, Yamaguch -, Japan *Correspondng Author: rwe@yamaguch-u.ac.jp Abstract The ultrasonc motor (USM) has a heavy nonlnearty and tme-varyng characterstcs whch vary wth drvng condtons. Because of no-exact mathematcal model of USM, t s dffcult to control USM. PID controller can be desgned wthout usng the expresson model of plant, but t s hard to compensate the nonlnearty and characterstc changes of USM. Ths paper presents a self-tunng scheme usng a modfed partcle swarm optmzaton (MPSO) for PID controller to overcome the dynamc characterstcs of USM. A modfed PSO employs the strategy that nonlnearty decreases the value of nerta weght from a large value to a small value. Ths strategy s to mprove the performance of the standard PSO n global search and fne-tunng of the soluton. The effectveness of the proposed method s verfed by numercal smulaton and expermental nvestgaton. The results demonstrate that the proposed method can mprove the accuracy of USM. Keywords: PID controller, self-tunng scheme, partcle swarm optmzaton (PSO), ultrasonc motor (USM), nerta weght. Introducton The ultrasonc motor (USM) s a new type motor, whch s drven by the ultrasonc vbraton force of pezoelectrc elements. USM has excellent features, such as compact sze, lght-weght, hgh poston accuracy, slence operaton, hgh torque even at the low speed, hgh retenton torque, quc response, power off self-brae, and unaffected by external magnetc feld. Therefore, USM s capable as an excellent actuator n many applcatons (-). However, t s dffcult to control USM because of no-exact model of USM. Snce the modern controls are hard to be appled on USM, PID controller has been wdely used n USM applcatons (-). However, there are lmtatons of the control performance usng the conventonal fxed-gan type PID controller because USM causes serous characterstc changes durng operaton and contans non-lnearty. In that case, t s dffcult for the conventonal one to compensate such characterstc changes and non-lnearty of USM. To overcome those problems, the research on self tunng of PID controller usng an ntellgent soft computng, such as Neural Networ (NN) and Genetc Algorthm (GA), are proceedng. Nevertheless, there are stll possbltes for PID gans easly to get stuc n a local mnmum and very slow convergence. Meanwhle, partcle swarm optmzaton (PSO) as an alternatve to GA has been actvely researched (-). Moreover, despte the smple algorthm n PSO compare to genetc algorthm, PSO s able to solve the nonlnear optmzaton problem effcently. Although PSO has the characterstcs of fast convergence, good robustness, strong commonalty, and has been successfully appled n many areas, t has the shortcomngs of premature convergence, low searchng accuracy and teratve neffcency, especally the problems nvolvng multple pea values, and t s lely to fall n local optma. In order to overcome the aforementoned lmtatons, many researchers have attempted to mprove the PSO algorthm (-). In ths wor, a modfed PSO whch employs nonlnear decreasng of nerta weght was appled to optmze the PID parameters for a postonng control of USM. show the effectveness of our proposed method, the numercal smulaton and the experment n real system To DOI:./cae. The Insttute of Industral Applcatons Engneers, Japan.

2 were compared wth that of the prevous methods (fxed-gan type PID and PSO wth lnear decreasng nerta weght based PID).. Partcle Swarm Optmzaton PSO s a populaton based stochastc optmzaton method usng the concept of cooperaton nspred by the behavor of organsm, such as brds flocng, n search for food. The outlne for PSO s mared as follows. Let consder the optmzaton problem of maxmzng the evaluaton functon f : M M' R for varable x M Rn. Let there be N partcles (mass pont) on M dmensonal space, where the poston vector and velocty vector of (=,,,...,N) th partcle for searchng number are x and v. The best poston for each partcle n the evaluaton functon f(x) of x,x, x searchng pont s represented as Pb (Pbest), whle the best poston of f(x) n the searchng pont for the whole partcle s represented as gb (gbest). The partcles are manpulated accordng to the followng recurrence equatons: v + x = wv = x.{ Pb x } + c. R. { gb x }. + c. R + v + + where w s the nerta weght; c and c are cogntve and socal constant; R s unform random numbers from the nterval [, ]. pbest ( P ) b x x + v gb x + x c. R.( P b x ) ( ) c R. g b x w.v. Fg.. Search example of PSO. () () The example for optmzed soluton search usng PSO s shown n Fg.. The movement of partcles s governed by three parts: () the nertal part, w.v ; () the cogntve part, (Pb - x ); () the socal part, (g b x ). gbest The velocty vector of v + s formed based on three vectors as shown n Eq. (). The frst one s nerta vector, whch s the vector from weghtng factor w and the velocty vector v. The remanng two are vectors for each (P b x ) and (g b - x ), whch formed from learnng factor c as well as c, and also [, ] of unform random numbers R. From those nteractons, velocty vector v + act so that the partcle moves to new poston, x +.. The Proposed Modfed PSO An nerta weght s mportant parameters to balance the exploraton (global search) and explotaton (local search) ablty of PSO. performances of PSO. Ths balancng s a ey to mprove the However, the adjustng of nerta weght s stll unclear and more need nvestgaton. prevous research, a lnear decreasng nerta weght (PSO-LDW) was ntroduced and was shown to be effectve n mprovng the fne-tunng characterstc of the PSO for determnng the gans of PID controller on USM (). In ths method, the value of w s lnearly decreased from an ntal value (w max ) to a fnal value (w mn ). In ths research, we wll adopt the nonlnear decreasng of nerta weght method to enhance the ablty of PSO and the adjustment of w s gven as: w = w mn + ( w w ). max mn ter ter max max ter where x s the nonlnear modulaton ndex. The value of x wll determne the degree of non-lnearty functon of nerta weght. x () In Ths method s nown as PSO wth nonlnear decreasng nerta weght (PSO-). The effectveness of PSO- s used to determne the gans of PID controller for postonng control of USM n real system experment.. Numercal Smulaton To evaluate the performance of the proposed algorthm and compare to the prevous method (PSO-LDW), we heren tae a -D Sphere functon: ( x ) + ( ) f ( x, y) = y () where the global best soluton for the above problem s zero whch s acheved when x = and y =. We decded to use -D Sphere functon n order to dsplay partcle s flyng process on the computer screen to get a vsual understandng of the PSO performances. For the purpose of comparson, all the smulaton deploy the same parameter settngs n both PSO-LDW and PSO-, such as the maxmum number of teratons, ter max = ; cogntve constant, c =.; socal constant, c

3 =.; and the dynamc range for all elements of a partcle s defned as (, ), that s, the partcles cannot move out of ths range n each dmenson. dmenson s. For the Sphere functon, the The nonlnear modulaton ndex of PSO- was vared as follows:.,.,,, and. Snce PSO s a stochastc algorthm that randomly searches the best soluton, so for testng we have done as much as runs. The nfluence of the dfferent nerta weghts for PSO algorthm wth number of partcles, n = and, s lsted n Table. In Table, the error shows an average error n runs. In order to easy see the nfluence of the nerta weght and number of partcles, the data of Table can be converted nto Fgure. In the event that the number of partcles s, the error becomes smaller when the nerta weght s.. Other event that the number of partcles s, the error becomes smaller when the nerta weght s.. It means that n order to determne the value of the nerta weght should consder the number of partcles. If the number of partcles s a bt, we should use the hgher nerta weght and vce versa. Also by usng the more partcles, the error wll be smaller because the ablty of searchng and the probablty to fnd the soluton s greater. Consequently, the tme calculaton s longer. For further smulaton, we use the number of partcles, n =. The nfluence of the dfferent range of nerta weght for PSO-LDW algorthm wth number of partcles, n =, s lsted n Table. In Table, PSO-LDW wth the range of nerta weght from. to. shows the smaller error among other ranges. Moreover, the error of PSO-LDW wth ths range s smaller than the error of the standard PSO wth nerta weght, w =.. Based on the Table, PSO- algorthm also uses the range of nerta weght from. to.. The nfluence of the dfferent of nonlnear ndex number s lsted n Table. It can be seen that the proposed PSO- wth nonlnear ndex number, x =., has smaller error among other nonlnear ndex number. smaller than the PSO-LDW. Also, the error of PSO- s It means that the soluton accuracy of the proposed PSO- s better than the PSO-LDW. The result can be explaned that the nonlnear modulaton ndex s a new mportant parameter n PSO- and the proper adjustment of x can sgnfcantly mprove the performance of PSO. Error Table. Error over nerta weght. Error Inerta weght n = n =...E-...E-...E-..E-.E-..E-.E-..E-.E-..E-.E-..E-.E E- E- E- n = E- n = E Inerta weght Fg.. The nfluence of nerta weght. Table. Influence of the dfferent range of the nerta. weght PSO-LDW w max - w mn Error. -..E-. -..E-. -..E-. -..E-. -..E-. -..E-. -..E-. -..E- Table. Influence of the nonlnear ndex number. PSO- (. -.) nonlnear ndex, x Error.E-..E-..E-.E-.E-.E-.E-

4 Durng searchng process, all partcles wll try to approach a best soluton and have a dfferent dstance to the best soluton over teraton. The average dstance of all the partcles shows a pattern of spread of partcles to fnd a soluton. We call t as dsperson. Fgure shows the dsperson of each method durng teraton. The fnal dsperson of the PSO- and the prevous methods s lsted n Table. It s clear that the partcles of PSO- have more aggressve to fnd the best soluton. In Table, the success rate (SR) shows the success of partcles n reachng a predetermned soluton value wthn runs. If the partcles can reach ths value or smaller then we can say t as success. In ths case, we used x - as a predetermned soluton value. Ths facts show that the proposed PSO- has hgher success rate than other prevous methods. The convergence speed can be seen from the speed the partcle that has the gbest value as shown n Fgure. We can see that the convergence speed of the proposed PSO- s faster than the prevous methods. Table. Comparson of partcle s characterstcs. Item PSO PSO-LDW PSO- Fnal.E-.E-.E- Dsperson SR (<e-). Applcaton of PSO- n PID Controller In ths wor, the PID controller was used as controller. It s comprsed of three components: a proportonal part, a dervatve part and an ntegral part. uses the followng control equaton: The PID controller K C( s) = K p + + K d. s s () where the K p s proportonal constant, K s ntegral constant and K d s dervatve constant. The man problem n PID controller s tunng process to determne the gans K p, K and K d. system depend on ths process. to unexpected performances. The performances of Improper tunng wll lead The tunng process s an optmzaton problem to obtan the best possble performances. The tunng process wll be more complex for the plant whch has nonlnear propertes such as USM. The ablty of PSO for solvng the optmzaton problem can be appled to the case of determnng the optmal PID parameters for a poston control of USM. r(t) + e(t) PSO- PID [x,x,x ] = [Kp, K, Kd] USM y(t) Dsperson gbest value PSO PSO-LDW PSO- Iteraton Fg.. Dsperson of partcles over teraton. PSO PSO-LDW PSO- Iteraton Fg.. Speed convergence of gbest. Fg.. PSO- based PID controller. Desgn of the PSO- tuned PID controller for USM s shown n Fgure. In ths system, three PID parameters (K p, K, K d ) wll be tuned automatcally by PSO- algorthm. Because there are three parameters that should be adjusted, the PSO- algorthm has three dmensons and each partcle of the algorthm s canddate soluton of the PID parameters. The sgnal e() wll be entered for PSO- algorthm and subsequently evaluated n the ftness functon to gude the partcles durng the optmzaton process. proposed method s gven as: Ftness = + e( ) The ftness functon for the Ftness shows the followng-up of evaluaton functon for the object nput. The purpose s to decrease the steady-state error by maxmzng the functon. The ftness s updated by each mllsecond accordng to the value of e(). ()

5 The USM control for clocwse (CW) rotaton and the counter clocwse (CCW) rotaton use the dfferent PSO n tracng the object nput. Snce the characterstcs of USM s dfferent depends on the rotaton drecton, we evaluate both rotatons separately.. Experment Result The USM servo system constructed n ths study s shown n Fgure. USM, the electromagnetc brae and the encoder are connected on a same axs. The poston nformaton from an encoder s transmtted to the counter board embedded nto a Personal Computer (PC). Meanwhle, accordng to error resulted from the comparson between the output and reference sgnal, the control nput sgnal whch s calculated n PC s transmtted to the drvng crcut through the I/O board and oscllator. In each experment, the load s added or not s dscussed to observe the changes of the USM s characterstcs. Whle the voltage of [V] s mported, the force of. [N.m] could be loaded to the shaft of the USM. The specfcatons of USM servo system s shown n Table.. A Conventonal Hand-Tuned PID Controller Frstly, we used the conventonal method for tunng PID controller on USM servo system. Ths method s ntroduced by Ells [] and called the zone-based tunng. It means that the low and hgh frequency parts of the controller can be tuned separately, startng wth the hgh frequency part. For a PID controller, ths means that frst the P- and D-acton are tuned and then the I-acton. The procedure wth steps to follow to tune a PID controller s gven as follows:. Set K p low, whle K = and K d =. Apply square wave reference at about % of the desred bandwdth. but avod saturaton. Use large ampltude,. Rase K p for approxmately % overshoot.. Rase K d to elmnate most overshoot.. Rase K to elmnate steady-state We found that K p =., K =. and K d =., for the best performance after many experments. Then, we started on USM servo system wth runs of clocwse (CW) drecton (.e., + deg) and runs of counter clocwse (CCW) drecton (.e., - deg) for no-load condton. After that, we repeat agan for wth load condton,.e.,. [N.m]. Table. USM Encoder Load Tm es Fg.. USM servo system. The Specfcaton of USM Servo System. Rated rotatonal speed : [rpm] Rated torque :. [N.m] Holdng torque :. [N.m] Resoluton :. [deg] to. [N.m] <+ deg> Ess [deg] <- deg> Ess [deg] Fg.. Poston accuracy of USM usng PID controller (no-load) <+ deg> Ess [deg] <- deg> Ess [deg] Fg.. Poston accuracy of USM usng PID controller (load. N.m).

6 <+ deg> Ess [deg] <+ deg> Ess [deg] <- deg> Ess [deg] Fg.. Poston accuracy of USM usng PSO-LDW PID controller (no-load) <- deg> Ess [deg] Fg.. Poston accuracy of USM usng PSO- PID controller (no-load) <+ deg> Ess [deg] <+ deg> Ess [deg] <- deg> Ess [deg] Fg.. Poston accuracy of USM usng PSO-LDW PID controller (load. N.m) <- deg> Ess [deg] Fg.. Poston accuracy of USM usng PSO- PID controller (load.). Table. Comparson of the average steady-state error. Methods Frequency of Zero Ess Ave Ess n runs Load Load No-load No-load. Nm. Nm PID.. PSO-LDW.. PSO-.. F tness.... PSO- PSO-LDW Tme [s] Fg.. Convergence of ftness.

7 Fgure and present the poston accuracy of USM n hstogram for no-load and wth load condton. We can say that the poston accuracy of USM usng a hand-tuned PID s good and relable n no-load condton, but becomes poor and naccurate n wth load condton. The gans have been determned prevously only applcable to no-load condton. If the plant s behavor s changed (.e., due to the loadng), t s necessary to re-tune PID and t s drawbac of the fxed-gan PID.. Self-Tunng PID Controller usng PSO- The used parameters n PSO- algorthm are as follows: partcles number, n = ; cogntve constant, c =.; socal constant, c =.; maxmum nerta weght, w max =.; mnmum nerta weght, w mn =.. Usng the same test condton as before, the PSO- algorthm wll tune automatcally to determne the optmal gans of PID controller. For comparson, we also used the common PSO method, called the PSO wth lnear decreasng nerta weght or PSO-LDW wth same parameters. Fgure shows the nfluence of the nonlnear ndex number. From ths fgure, the nonlnear ndex number s. gves the best result. So, based on ths result, we used ths value for the next experment. Fgures - show the poston accuracy of USM n hstogram for no-load and wth load condton. It can be seen clearly that self-tunng PID controller can compensate the characterstc changes of USM due to the loadng effect. The gans PID are automatcally adjusted accordng to the plant s behavor. Thus, the poston accuracy of USM can be mantaned stll good and relable even though there are the characterstc changes of plant. We also found that the CW drecton s characterstc s slghtly dfferent from the CCW drecton s characterstc. By usng self-tunng, we can easly fnd the optmal gans of PID. Compared to a fxed-gan PID and PSO-LDW tuned PID, the PSO- tuned PID have the frequency dstrbuton of steady-state error tends towards zero error. Average Ess [deg] x=. x=. x=. x=. x=. x=. No-Load Wth-Load x=. Fg.. Influence of nonlnear ndex number. x=.. Comparson of the Average Steady-state Error Table shows the comparson of the proposed method and prevous methods n term of average Ess and frequency of zero error n runs. The self-tuned PID controllers (.e., PSO-LDW, PSO-) can outperform a hand-tuned PID or a fxed-gan PID. The average Ess of the PSO- s smallest or.% (no-load condton) and.% (wth load condton) lower than PSO-LDW (common strategy of PSO). Moreover, the frequency of zero Ess of PSO- s more often than the prevous methods. It means that the proposed method can mprove the poston accuracy of USM.. Comparson of the Average Steady-state Error Fgure shows the ftness convergence characterstcs of PSO-LDW and PSO-. It seen clearly that the partcles PSO- acheve faster convergence than the prevous methods. The partcles of PSO- acheved convergence n. seconds, whle the partcles of PSO-LDW acheved convergence n. seconds.. Conclusons In ths paper, the performances of a modfed PSO wth varaton nonlnear decreasng nerta weght based PID controller have been nvestgated and extensvely. The results are compared wth the PSO wth lnear decreasng nerta weght based PID controller and the fxed-gan PID controller by expermentng on postonng control of USM. The nonlnear decreasng nerta weght concept has contrbuted to gettng mnmum ftness functon and to quc convergence ablty wth better accuracy. In general, ths concept has produced a great mprovement n quc convergence ablty and aggressve movement narrowng towards the soluton regon wth dfferent of nonlnear modulaton ndex (x). The experment results show that the nonlnear modulaton ndex (x) plays an mportant role n searchng for optmal soluton n PSO-. Acnowledgment The authors acnowledge the support of Drectorate General of Hgher Educaton, Mnstry of Natonal Educaton, Republc of Indonesa and Electroncs Engneerng Polytechnc Insttute of Surabaya (EEPIS ITS).

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