Robust Fractional Order PID Control of a DC Motor with Parameter Uncertainty Structure

Transcription

1 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. Iue 6, Augut Robut Fractional Order PID Control of a DC Motor with Paraeter Uncertainty Structure Hadi Tajbakhh, Saeed Balochian 2 Deartent of Electrical Engineering, onabad Branch, Ilaic Azad Univerity, onabad, Iran tahaed0@gail.co 2 Deartent of Electrical Engineering, onabad Branch, Ilaic Azad Univerity, onabad, Iran aeed.balochian@gail.co Abtract Thi aer reent a robut DC otor eed controller baed on the concet of fractional-order PID (FOPID or ) controller. The ethod for tuning the ue the Bode enveloe of the control yte with araetric uncertainty. The uncertainty caued by the araeter change of DC otor are forulated a five different deign ecification, which are ued in the objective function in the deign. Robut i deigned fro the olution of fulfill five different deign ecification. Siulation reult are given to illutrate the effectivene of thi ethod. eyword: dc otor, fractional order PID, robut control.. Introduction In the at decade, odern control theorie have ade great advance. Control technique including otial control, H H 2 control, fuzzy control, neural network control, redictive control, and o on, have been develoed ignificantly. Neverthele, the roortional-integralderivative (PID) control technique ha till been widely utilized in any indutrial alication uch a roce control, otor drive, flight control, etc. Nowaday, ore than 90% control loo in indutry are PID control. Thi i ainly due to the fact that PID controller oee robut erforance to eet the global change of indutry roce, ile tructure to be eaily undertood by engineer, and eaine to deign and ileent. Recently, there are increaing interet to enhance the erforance of PID controller by uing the concet of fractional calculu, where the order of derivative and integral are non-integer. Fractional calculu i non-local, which ake it able to ehaize atheatically the long-eory. Fractional order PID controller rooed by Podlubny (999) i a generalization of the PID controller uing fractional calculu. A i characterizing by five araeter, i.e., the roortional gain, the integrating gain, the derivative gain, the integrating order and the derivative order. Over the lat year, find any alication in irrigation canal control (Doingue, Valerio, & da Cota, 2009), teerature tracking (Ahn, Bhabhani, & Chen, 2009), otion control of DC otor (Xue, Zhao, & Chen, 2006), boot converter control (Tehrani et al., 200), hyeronic flight vehicle control (Changao, Naiing, & Zhiguo, 200), ervo re control yte (Fan, Sun, & Zhang, 2007). The above reearch reult how that ha better erforance and robutne than conventional PID controller. Though it i o, the araeter tuning of i an iortant and critical iue. Coare to conventional PID controller, ha two extra araeter. On one hand, it enable eole have ore degree of freedo to deign, and, on the other hand, it ean that it i ore colex in the ynthei of. In the literature, any aroache have been docuented to deign. Thee aroache can be claified into two clae: analytic ethod and heuritic ethod. In the analytic context, the araeter of are tuned by iniizing a nonlinear objective function deending on the ecification ioed by the deigner. Bouafoura and Braiek (200) rooed a new analytic ethod to deign by exanding the control loo ignal and reference odel inut and outut over a iecewie orthogonal function. In uch a anner, the fractional differential calculu i relaced by the generalized oerational atrice of differentiation related to thee bae, and thu, the controller tuning i elaborated ily with a atrix aniulator anner. However, the tuning of order of integrator and derivation are not conidered in Bouafoura and Braiek (200). In Padula and Viioli (20), a et of tuning rule were devied baed on a firt order lu tie odel of the roce by iniizing the integrated abolute error with a contraint on the axiu enitivity. Ziegler Nichol like tuning rule for were given in Valéio and da Cota (2006). Evolutionary algorith including genetic 223

2 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. Iue 6, Augut 204. algorith (A), article war otiization (PSO), electroagneti - like algorith (EM), differential evolution are alo ued to deign. enetic algorith were adoted by Cao and Meng and Xue (2009) to deign by recating the roble to an otiization roble. In Zaani, arii- harteani, Sadati, and Parniani (2009), article war otiization wa ued to deign for an Autoatic Voltage Regulator (AVR) yte via iniizing an objective function coniting of overhoot, riing tie, ettling tie, teady-tate error, the integral of abolute error (IAE), integral of quared inut, gain argin and hae argin. An iroved electroagneti-like algorith with genetic algorith (IEMA) technique wa rooed by Lee and Chang (200) for deign through iniizing the integrated-quare-error (ISE). Biwa, Da, Abraha, and Daguta (2009) rooed to deign baed on the root locu ethod uing iroved differential evolution. In Cervera, Bano, Monje, and Vinagre (2006), tuning of i recated a a quantitative feedback theory (QFT) loo haing roble, where the otiization objective i the high frequency gain of the noinal loo ubjected to the retriction given by the ecification. Monje et al. (2004) rooed to tuning araeter of by taking into account five condition about hae and gain argin ecification and contrain over the enitivity function. A far a the heuritic ethod, rule-baed ethod and evolutionary algorith baed ethod were exlored by everal author. In C.Yeroglu, N. Tan (20), the Bode enveloe of the firt-order and firt-order lu dead tie (FOPDT) yte with the araetric uncertainty tructure are uccefully cobined with five deign criteria, which Monje Vinagre (2004) et al have ued in their aer, to obtain robut. In thi aer, we ay our attention to deign a for a DC otor yte with uncertain araeter uing C. Yeroglu, N. Tan (20) ethod. The ret of the aer i organized a follow. In Section 2 atheatical odel of DC otor a a controlled object i briefly reviewed. Section 3 reent the baic of controller. In Section 4 robut i introduced. Deign of robut for dc otor yte with araeter uncertainty tructure i given in Section 5. Reult obtained fro iulation and obervation are decribed in Section 6. Finally, concluion reark are given in Section Model of DC Motor We will conider the general odel of the DC otor () which i deicted in Fig.. The alied voltage V a control the angular velocity ( t ). The relation for the arature controlled DC otor are hown cheatically in Fig. 2.Tranfer function (with T d () = 0 ) ha the for: θ () () = = V a () ( L + R )( J + f ) + b However, for any the tie contant of the arature i negligible and therefore we can ilify odel (). A ilified continuou atheatical odel ha the following for: θ () () = = V a () R ( J + f ) + b [ R f + b ] = = ( τ+ ) ( τ+ ) Where the tie contant τ = RJ ( R f + b ) and ( R ) = f + b. It i of interet to note that = b. Fig. eneral odel of a DC otor. Fig. 2 Matheatical odel of a DC otor. For our ini DC otor the hyical contant are: () (2) 224

3 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. Iue 6, Augut 204. R = 6 Ω, = b = 0., f = 0.2 N-, J = 0.0 kg2/ 2. For thee otor contant the tranfer function (2) of the DC otor ha the for: 0.08 () = ( ) (3) 4. Monje Vinagre et al. ethod Monje Vinagre et al. (2004) rooed a controller tuning ethod for the firt-order and FOPDT yte. In their aer Bode enveloe of the firt-order and FOPDT yte are cobined with five deign criteria to obtain the. Thi five deign criteria, uch a agnitude at gain croover frequency, hae argin, robutne to lant uncertain tie, high frequency noie attenuation and enitivity function are given a follow. 3. Fractional -order PID ( ) controller The tandard PID controller i defined in erie (interacting) for a i C() = + + d (4) where i the roortional gain, i i the integral tie contant, d i the derivative tie contant. The controller i defined a a generaliation of the tandard PID controller (4), naely, i C() = + + d (5) λ Where λ and are the non-integer order of the integral and derivative ter reectively. It can be eaily noted that, by electing λ = =, a tandard PID controller i obtained. Indeed, in order to ileent the fractionalorder controller, the Outalou continuou integer-order aroxiation (Outalou et al. 996) ha been eloyed. It conit in uing the following aroxiation baed on a recurive ditribution of zero and ole: + N υ z, n k, υ > 0 (6) n = +, n which i valid in a frequency range [, ] and where the gain k i adjuted o that the right ide of (6) ha unity gain at the gain croover frequency of υ. In thi aer the value N = 2 ha been choen, while l = 0.00 and h = Deign of robut l h 4... Phae argin ( ϕ ) and gain croover frequency ( ) ecification: ain and hae argin have alway erved a iortant eaure of robutne. It i known that the hae argin i related to the daing of the yte and therefore can alo erve a a erforance eaure (ee Franklin, Powell, & Naeini, 986). The equation that define the hae argin and the gain croover frequency are: C (j ) (j ) = 0dB (7) db Arg( C(j ) (j )) = π + φ (8) 4..2 Robutne to variation in the gain of the lant: The next contraint can be conidered in thi cae (ee Chen & Moore, 2005): Thi condition force the hae of the oen-loo yte ( arg( (j ) (j )) ) d C d = = 0 Thi condition force the hae of the oen-loo yte to be flat at and hence to be alot contant within an interval around. It ean that the yte i ore robut to gain change and the overhoot of the reone i alot contant within a gain range (io-daing roerty of the tie reone). It ut be rearked that the interval of gain for which the yte i robut i not fixed with thi condition. That i, the uer cannot force the yte to be robut for a articular gain range. Thi range deend on the frequency range around for which the hae of t the oen-loo yte kee flat. Thi frequency range will be longer or horter, deending on the reulting controller and the lant. (9) 4..3 High frequency noie rejection: 225

4 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. Iue 6, Augut A contraint on the coleentary enitivity function T can be etablihed: C(j ) (j ) T(j ) = AdB C(j ) (j ) (0) + db Miniu and axiu lot of the gain are hown by (), () and iniu and axiu lot of R R2 hae are hown by R3(), R4() reectively. () k R τ +, R () k 2 τ + (3) Whit A the deired noie attenuation for frequencie t rad/. R () k 3 τ +, R () k 4 τ + (4) 4..4 To enure a good outut diturbance rejection: A contraint on the enitivity function S can be defined: S(j ) = BdB C(j ) (j ) () + db In order to deign robut, Equation (7), (8), (9), (0), () hould be atified with the tranfer function R 2 ( ), R 4 ( ), R 4 ( ), R ( ), R 2 ( ) reectively, naely ut be taken at the oint a in Fig. 3 and the oint b in Fig. 3 hould be the iniu hae argin of the yte. With B the deired value of the enitivity function for frequencie rad/ (deired frequency range). For fractional order controller uch a a PI λ or a PD, three deign ecification could be et (one for each araeter). Therefore, for the general cae of a the deign roble i baed on olving the yte of five nonlinear equation (given by the correonding deign ecification) and five unknown araeter k, ki, kd, λ., 4.2 C. Yeroglu, N. Tan robut tuning algorith Owing to the tuning ethod, which C. Yeroglu, N. Tan (20) have ued in their aer, we referred to ue C. Yeroglu, N. Tan ethod a a title of thi ection. A robut controller i le enitive to the araeter change of the controlled yte. The uncertainty can be caued by nonrecie identification. The fractional order controller are le enitive to change of controlled yte araeter. C. Yeroglu and N.Tan (20) rooed a robut tuning algorith for firt-order and FOPDT yte with araetric uncertainty tructure. The FOPDT yte with araetric uncertainty can be rereented a [ k, k] [ LL, ] () [,] ττ+ (2) k τ and L are lower liit, k, τ and L are uer Where, liit of the araeter, reectively. Fig. 3 Bode enveloe of a FOPDT lant We leave the detail of thi ubection due to ace liitation. For ore detail leae refer to C.Yeroglu, N. Tan (20). 5. Deign of robut for DC otor with uncertain araeter Thi ection reent a tuning ethod for robut for dc otor yte with araeter uncertainty tructure. In thi aer change in the araeter of the DC otor are taken into account a interval uncertainty, and the robutne of the controller i 226

5 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. Iue 6, Augut 204. analyzed. We aue that the value of the hyical contant of the DC otor ay change in an interval. The rooed ethod benefit fro deign ecification given in Monje Vinagre et al.(2004) tuning ethod and C. Yeroglu and N. Tan(20) ethod that rooed a robut tuning algorith for firt-order and FOPDT yte with araetric uncertainty tructure. Preliinary tudy of thi ection ha been reented in the C. Yeroglu, N. Tan (20).The tranfer function of the DC otor ha the for (3). A dc otor yte with araetric uncertainty can be rereented a [ k, k] 2() = ([ ττ, ] + ) (5) Where k, τ are lower liit and k, τ are uer liit of the araeter, reectively. The fractional-order controller i deigned to obtain the deired erforance for the given interval yte. Now ince a firt order lant yte i required for uing the C. Yeroglu, N. Tan tuning ethod, a MATLAB file called getfod. i ued to aroxiate the econd order yte (5) by firt order lant yte (2). Now we can deign robut to control dc otor with araetric uncertainty tructure. 6. SIMULATION RESULTS τ = Deign ecification for the yte are given a follow: ain croover frequency = 0.rad, hae 0 arginφ = 90, robutne to variation in the gain of the lant ut be fulfilled, deired noie attenuation A = 20dB for the t = 0rad and deired value of enitivity function B = 20dB for the frequency = 0.0rad. The robut to control () i obtained a C () = (8) Ste reone of 0.5 C () () are hown in Fig Perforance of DC otor Syte with robut Controller Conider the dc otor yte (7).We aue that the value of the hyical contant of the DC otor change in an interval. The dc otor yte with araetric uncertainty ha the following for: () k = (6) ( τ + ) Where k [ k, k] = [0.06,0.], τ [ ττ, ] = [0.04,0.06]. Now the getfod. MATLAB file i ued to aroxiate a econd order yte by firt order lant yte. The aroxiated FOPDT yte obtained i given a: () k τ + Where (7) k [ k, k] = [600,000],, L [ LL, ] = [0.04,0.06], Fig. 4 Ste reone of the C () () 6.2 Coarion with other Controller In thi ection, we coare robut with other. For coarion, we conider deigned fractional controller in A.Rajaekhar et al. (20) that i baed on Artificial Bee Colony (ABC) and odified verion of ABC naely L-ABC algorith. The ABC and L-ABC algorith are ileented a in A.Rajaekhar et al. (20). The te reone of the yte controlled by and are hown in Fig. 5. The bet controller araeter and the erforance indice are hown in Table. ABC -, L-ABC- robut 227

6 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. Iue 6, Augut Fig. 5 Ste reone of the ABC -, L-ABC- and robut Table : Bet controller araeter and erforance indice of different Controller k i k d λ E T r T M (%) ROBUST ~ ~ L-ABC ABC With oberve Fig. 5 and Table, we derive that rooed robut ha better erforance than the ABC- and L-ABC-. Thi figure how that the ettling tie and rie tie and overhoot fro the robut at wort i better fro ABC- and L-ABC- while the teady tate error fro the rooed controller i alot the ae a the L-ABC-. It can be een that the te reone of the dc otor yte with araetric uncertainty controlled by robut λ ha very good erforance. PI D 7. Concluion In thi aer we rooed a robut to control DC otor yte with araeter uncertainty tructure. The rooed ethod benefit fro deign ecification given in Monje Vinagre et al. tuning ethod and C. Yeroglu and N. Tan ethod that rooed a robut tuning algorith for firt-order and FOPDT yte with araetric uncertainty tructure. The iulation reult illutrate that the rooed robut for DC otor yte ha bet control erforance than other otial that o far i reented, and alo, for odel uncertaintie, the robut i ore robut. The rooed technique ay aly a an efficient ethod to deign robut otial ractical fractional-order controller for ractical yte to reject external diturbance. Reference [] Ahn, H. S., Bhabhani, V., & Chen, Y. Q. (2009). Fractional-order integral and derivative controller for teerature rofile tracking. Sa dhana,34, [2] Anguluri Rajaekhar, Prata unathi, Ajith Abraha,Millie Pant (20). Fractional Order Seed Control of DC Motor Uing Levy Mutated Artificial Bee Colony Algorith. 20 World Congre on Inforation and Counication Technologie. 228

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