A NEW INTELLIGENT, ROBUST AND SELF TUNED CONTROLLER DESIGN (II).

Transcription

1 IJISET - Iteratioal Joural of Iovative Sciece, Egieerig & Techology, Vol. Iue 9, November. ISSN 3 79 A NEW INTELLIGENT, ROBUST AND SELF TUNED CONTROLLER DESIGN (II). Farha A. Salem Mechatroic egieerig program, Departmet of Mechaical Egieerig, College of Egieerig, Taif Uiverity,, Taif, Saudi Arabia. Alpha ceter for Egieerig Studie ad Techology Reearche, Amma, Jorda. Abtract Thi paper propoe a ew, imple, itelliget, robut ad elftued cotroller deig approach for gettig a proce uder cotrol, while achievig a importat deig compromie; acceptable tability, ad medium fate of repoe. The propoed approach i baed o aalyzig plat tep repoe to calculate it' ad baed o calculated, calculate cotroller', feed it to cotroller, ad repeat proce util achievig a acceptable tability, ad medium fate of repoe. The propoed approach wa tet uig MATLAB m.file ad Simulik model for differet ytem Keyword: Cotroller deig, Modelig/Simulatio.. Itroductio The term cotrol ytem deig refer to the proce of electig feedback gai (pole ad zero) that meet deig pecificatio i a cloed-loop cotrol ytem. Mot deig method are iterative, combiig parameter electio with aalyi, imulatio, ad iight ito the dyamic of the plat (Katuhiko Ogata,997),(Ahmad A. Mahfouz, et al,3), A importat compromie for cotrol ytem deig i to reult i acceptable tability, ad medium fate of repoe, oe defiitio of acceptable tability i whe the uderhoot that follow the firt overhoot of the repoe i mall, or barely obervable(farha A. Salem,3), Beide world wide kow ad applied cotroller deig method icludig Ziegler ad Nichol kow a the proce reactio curve method (J. G. Ziegler, et al, 93). ad that of Cohe ad Coo (G. H. Cohe, 953) Chiei-Hroe- Rewick (CHR), Wag Juag Cha, may cotroller deig method have bee propoed ad ca be foud i differet text icludig (Atrom K,J,et al,99)(r. Matouek, )(Sumita Da, et al, )(Saeed Tavakoli, et al, 3)(Atrom K,J, et al, 99)(Norma S. Nie,)(Gee F. Frakli, et ak, )(Dale E. Seborg, et al, ), each method ha it advatage, ad limitatio. Thi paper exted previou work (Farha A. Salem,3)(Farha A. Salem,*)(J. G. Ziegler,et al,93)( G. H. Cohe, et al,953)(atrom K,J, et al,99), ad propoe a ew imple ad itelliget cotroller deig approach, that i baed o relatig cotroller()' ad plat' to reult i meetig a importat deig compromie; acceptable tability, ad medium fate of repoe i term of miimum PO%, 5T, T S, ad E SS.. Propoed approach (Farha A. Salem,3)(Farha A. Salem,*) propoed (P-, PI-, PD- ad PID-) cotroller deig method with correpodig expreio for calculatig cotroller' (K P, K I, ad K D ), a log with oft tuig (α, β ad ε). The propoed method are baed o relatig ad calculatig cotroller' from plat' (ζ, ω, T), to reult i overall repoe with acceptable tability, medium fate of repoe ad miimum overhoot. For PID cotroller deig, the propoed expreio derived by (Farha A. Salem,3)(Farha A. Salem,*) are give i Table, where : parameter α i repoible for peedig up repoe, ad reducig Error, meawhile, ε i repoible for tuig overhoot, where icreaig ε will icreae overhoot, ad vie vera, fially parameter α i repoible for reducig both overhoot ad error. A how o block diagram repreetatio (Figure ),the propoed method i accomplihed a follow; firt by ubjectig the cloed loop ytem with oly proportioal cotroller with gai et to K P =, where K I ad K D are et to zero, to tep iput of R()=A/, the feedig reulted tep repoe to module with program, that will aalyze repoe curve to calculate plat' (e.g. ζ ω,t), the ue thee to aig to cotroller tuig (α, β ad ε) ad calculate correpodig cotroller gai (K P, K I ad K D ) accordig to Table, ad the feed calculated cotroller to cotroller, to refie the reulted repoe, the program will ubject the overall cloed loop ytem with calculated gai, to the ame tep iput, ad feed reulted repoe to module with program ad fid correpodig cloed loop ytem ad cotroller' gai, repeatig thi proce to fially calculate cotroller', that will reult i acceptable tability, medium fate of repoe with miimum overhoot, ad maitai thi repoe by cotiuouly aalyzig reulted repoe ad maitaiig it. I cae of diturbace iput, the propoed method will repeat the metio proce, to retur the cloed loop repoe to previou tate. For example for ecod order ytem, the program will ue ytem tep repoe to calculate ytem dampig ratio ζ, udamped atural frequecy ω, ad time cotat T. for higher order ytem, the program to ue ytem tep repoe to aalyze repoe ad apply domiat pole approximatio, ad fid correpodig plat approximated, ad proceed a metioed. 3

2 IJISET - Iteratioal Joural of Iovative Sciece, Egieerig & Techology, Vol. Iue 9, November. ISSN 3 79 Table : Propoed expreio for PID calculatio Plat PID K P K I K D T D T I N ζ ω β Tuig limit :if ω ξ ξω ω ε, ε =. α, α =.5.5 ξ ξω ξω α ξω ξ ω ξ εω R() Feedig gai (K P, K I,K D ) P-Term I-Term D-Term + D() Plat Calculatig gai (K P, K I,K D ) Aalyzig For Plat :ζ ω,t C() Output Module & Program H(). Tetig propoed approach To clarify the operatio of the propoed method, MATLAB/Simulik will be ued to imulate the propoed method, write program m.file for repoe aalyi, calculatig plat', calculatig cotroller, feedig calculated gai to cotroller, ubjectig overall cloed loop ytem with calculated gai to tep iput, ad repeatig the proce util reachig, acceptable tability, medium fate of repoe ad miimum overhoot. To tet the propoed approach, Simulik model how i Figure i developed, thi model i with four plat of differet order, icludig PMDC motor ytem a prime mover to be ued for both, robot peed cotrol, ad robot arm poitio cotrol.. Tetig for robot liear peed cotrol The DC motor ope loop trafer fuctio without load attached relatig the iput voltage, V i (), to the motor haft output agular peed i give by Eq.(). To model, Simulate ad aalyze the ope loop plat ytem, the total equivalet iertia, J equiv ad total equivalet dampig, b equiv at the armature of the motor are give by Eq.(). Sice, the geometry of the mechaical part determie the momet of iertia, to compute the total iertia, J equiv platform, ytem ca be coidered to be of cylidrical hape, with the iertia calculated a give by Eq.(3). To calculate the load torque of platform, the followig force ca be icluded; the hill-climbig reitace force F climb, aerodyamic Figure Block diagram repreetatio of the propoed approach Drag force, F aerod ad the liear acceleratio force F acc, a give by Eq.(), with their correpodig torque. The robot ha the followig omial ; M=5 Kg, height, h= M, width, a=. M, the wheel radiu i to be.75. Figure (c) how Simulik model of Mobile robot torque (Farha A. Salem,3*)(Farha A. Salem,*). The tachometer cotat i elected a give by Eq.(5), to reult i liear velocity of.5 m/ for iput voltage. The followig omial for the variou of eclectic DC motor ued: V i = Volt; J m =.7 kg m²; b m =.7;K t =. N-m/A; K b =.5 V-/rad; R a =.557 Ohm; L a =.Hery. Baed o Eq.(,5), propoed approach, ad refereeig to(farha A. Salem,3*)(Farha A. Salem,*)(Ahmad A. Mahfouz, et al, 3)(*-Farha A. Salem(3). the Simulik model coitig of ubmodel how i Figure are developed to be ued to tet propoed approach. ω( ) Kt / Gpeed ( ) = = () V ( ) LJ + ( RJ + b L) + ( Rb + KK ) i a m a m m a a m t b N N equiv = m + Load, equiv = m + Load N N b b b J J J () 3 bh N Jload = Jequiv = Jmotor + J gear + ( Jwheel + mr ) N (3) dυ F =.5ρAC υ + Mg i( α) + m dt () K tach max D = Vimax Vimax.7 ω = υ /r =.5/. 75 = (5) max 3

3 IJISET - Iteratioal Joural of Iovative Sciece, Egieerig & Techology, Vol. Iue 9, November. ISSN 3 79 Cotrol Curret Torque Agular peed Load torque Diplay.5*rou*frot_area*Cd Derivative Cotrol igal Robot_arm.mat To File La.+Ra Trafer /(L+R)3 Robot_arm.mat curret Kt To File Torque cotat Torque Robot_arm.mat de() To File Trafer /(J + b)3 Robot_arm3.mat To File3 agular peed Agle Agular diplacemet Itegrator. gear ratio. i(u) r*m*g co(u) SiCo Termiator Icliatio agle du/dt r Agular peed 3 To Seor Kb EMF cotat gear ratio, Agular peed Figure (a) DC machie ubytem model for robot ad robot arm Pul e Geerator Kp Ki Kd., du/dt PD() PID Cotroller Cotrol igal y.mat Load Torque Figure (c) Mobile robot torque Icliatio agle Icliatio Arm agle- Radia Mobile robot Load torque Icliatio agle Robot_arm.mat Load Torque Agular peed Arm agle- degree ARM Load torque -Kraddegree T_arm Arm agle du/dt Agular acceleratio Step,V=, Kp /Ti Filter (Td/N)+ Td du/dt Agular diplacemet Load torque Cotrol igal Agular peed DC Subytem To Seor du/dt Robot_arm5.mat.mat To File3 r wheel r Liear peed Td. (Td/N)+ Filter Td du/dt Arm agle feedback, Kpot Agular peed feedback, Ktach Figure (d) Simulik model ued to tet propoed approach. -K- -K- Clock Tetig_patet.mat patet_tet To Workpace patet_tet To Workpace 3

4 IJISET - Iteratioal Joural of Iovative Sciece, Egieerig & Techology, Vol. Iue 9, November. For third program elf-ru, with ecod calculated gai,the reulted repoe curve i how i Figure 3(c), ad overall ytem ISSN 3 79 Switchig Simulik model to robot peed cotrol, Ruig program (m.file), for cotrollig the output liear peed of robot to be.5 m/ for iput of V. Firt, the iput to the program, i the reulted cloed loop repoe curve of overall cloed loop ytem with oly Figure 3(a) firt program elf- ru: plot of both origial ad calculated tep repoe Step repoe of rpbot with PID Cotroller..5 proportioal gai et to K P =, ad K I ad K D are et to zero, program will aalyze the repoe curve ad. calculate plat', repoe meaure (lited i Table (a)) approximated ecod order trafer fuctio (give by Eq.()), ad plot both origial ad calculated.3. repoe (how Figure 3(a)), ad feed calculated gai to PID cotroller, the the program will ubject the cloed. loop ytem with PID cotroller with calculated gai, to the ame tep iput, ad the, program will take a iput reulted repoe curve, aalyze it, to calculate overall cloed loop approximated ecod order ytem, calculate Time () cotroller gai ad calculate tuig (lited i Figure 3(b) Secod program elf- ru: Step Table ) ad feed it to cotroller, ad the, oce agai ru repoe of robot with PID cotroller the imulatio ad repeat the aalyi, calculatio ad feedig proce, if acceptable repoe i reulted, maitai it, if ot, calculate the ew cotroller gai baed o ew cloed loop repoe curve aalyi, repeat the Step repoe proce utill acceptable repoe i achieved For ecod program elf-ru, with the calculated gai, the reulted repoe curve i how i Figure 3(b), ad overall ytem are lited i Table (b),.5. approximated cloed loop trafer fuctio i give by Eq.(7) Step repoe : ad Liear peed Liear peed.. Liear peed Time () Figure 3(b) Third program elf- ru: Step repoe of robot with PID cotroller Time (ecod) Table (a): calculated ytem,, repoe meaure, firt ru robot ytem ε.53 ω.39 T R.9 M P T P Peak β ε α K P K I K D gai PO% DC gai.75 33

5 IJISET - Iteratioal Joural of Iovative Sciece, Egieerig & Techology, Vol. Iue 9, November. ISSN 3 79 Table (b): ytem, repoe meaure, ecod elf-ru with calculated gai robot ε ω T R M P T P Peak PO% ytem e e- β ε α K P K I K D DC gai.9753 Table (c): ytem, repoe meaure, Third elf-ru with calculated gai ε ω T R M P T P Peak PO% DC gai robot e e ytem 3 β ε α K P K I K D G ( ) = G ( ) = G ( ) = Eq.() Eq.(7) Eq.(). Tetig propoed approach for differet ytem Cae (): Tetig the propoed approach for third order ytem with trafer fuctio give by Eq.(9), for deired output of, for tep iput of R()=/, will reult i approximated ecod order trafer fuctio give by Eq.(), ad calculated plat ad PID ad repoe meaure, how i Table 5(a), both repoe plot of origial ad compeated ytem lited i Table (b), with approximated cloed loop trafer fuctio give by Eq.(), ad fial cloed loop repoe how i Figure (b), thi i a acceptable repoe, that the program will maitai, for each ext ru, ad retur the repoe to it i cae of diturbace iput. + G ( ) = Eq.(9) cotrolled ytem are how i Figure (a). The calculated plat are ued by program to calculate cotroller', feed it to cotroller ad ubject ytem to tep iput, ad repeat proce. For third program elf-ru, with calculated gai, will reult i repoe curve how i Figure (b), ad overall G() = Eq.() Table 5(a): calculated ytem,, repoe meaure, firt ru robot ytem ε.9 ω.95 T R 3.33 M P.5 T P.3 Peak.3 PO%.33 3 β ε α K P K I K D DC gai.3 3

6 IJISET - Iteratioal Joural of Iovative Sciece, Egieerig & Techology, Vol. Iue 9, November. Table 5(b): calculated ytem,, repoe meaure, firt ru robot ytem ε.93 ω.79 5 T R.5 M P.59 9 T P 3.53 Peak.3 PO%.753 β ε α K P K I K D ISSN 3 79 DC gai.3 5 cloed loop Step repoe Amplitude Magitude Time (ecod) Figure (a) repoe plot of origial ad approximated ytem.5.5 Time() Figure (b) cloed loop repoe with PID Cae (): Tetig the propoed approach for fourth order ytem with trafer fuctio give by Eq.( ), for deired output of, for tep iput of R()=/, will reult i approximated ecod order trafer fuctio give by Eq.(3), ad calculated plat, PID ad repoe meaure, how i Table 5(a), both repoe plot of origial ad compeated ytem are how i Figure 5(a). The calculated plat are ued by program to calculate cotroller', feed it to cotroller ad ubject ytem to tep iput, ad repeat proce, util a acceptable repoe i achieve (how i Figure 5(b)) G ( ) = G ( ) = Eq.() Eq.(3) Table (a): calculated ytem,, repoe meaure, firt ru ε ω T R M P T P Peak PO% DC gai robot ytem β ε α K P K I K D

7 IJISET - Iteratioal Joural of Iovative Sciece, Egieerig & Techology, Vol. Iue 9, November. Overall loed loop repoe ISSN 3 79 Amplitude Magitude 5 5 Time (ecod) Figure 5(a) repoe plot of origial ad approximated ytem 3 5 Time() Figure 5(b) cloed loop repoe with PID.3 Program code Kp=;Ki=;Kd=; while() clc,cloe all format bak dip('====================================') dip( ' Tetig Repoe Data ') dip( ' ') dip('====================================') plot(patet_tet,patet_tet,'r'); figure plot(patet_tet,patet_tet,'r'); repoe_data=[patet_tet,patet_tet ]'; if ay(repoe_data(,[:ed])<);% udamped repoe Kp=;Kd=5 ;Ki=5;% ele [a,b]=max(repoe_data(,:)); fial_teady_atae_=repoe_data(,ed); fpritf(' fial teady atae = %g \',fial_teady_atae_) ww=repoe_data(,:); for z= :legth(repoe_data(,:)) if repoe_data(,z)>= fial_teady_atae_*.95 rie_time=repoe_data(,z); break, ed, ed fpritf(' Rie time = %g \',rie_time) overhoot= a-fial_teady_atae_; fpritf(' Overhoot = %g \',overhoot) if (overhoot <= )== ZETA= ; omega_=(.7*zeta-.)/rie_time; ettlig_time=(9.3*zeta-3.)/omega_ Percet_overhoot=; fpritf(' Udamped atural frequecy, omega_ = %g \',omega_); fpritf(' Dampig ratio, ZETA = %g \',ZETA) %================= alpha=; %================= ele Peak_= a ; fpritf(' Peak Value = %g \',a) Peak_time=repoe_data(,b); fpritf(' Peak Time = %g \',Peak_time) Percet_overhoot=((Peak_fial_teady_atae_)/fial_teady_atae_);% *; fpritf(' Overhoot = %g \',overhoot) fpritf(' Percet overhoot = %g \',Percet_overhoot) dip(' ========================================== =========================') ZETA=-log(Percet_overhoot) /(qrt((log(percet_overhoot))^+ (pi)^)); fpritf(' Dampig ratio, ZETA = %g \',ZETA) omega_=pi/(peak_time*qrt(-zeta^)) ;%calculatig from peak time fpritf(' Udamped atural frequecy, omega_ = %g \',omega_);ed trafer_fuctio=tf(fial_teady_atae_*omega_^, [, *ZETA*omega_, omega_^]) hold, tep(trafer_fuctio), grid leged('', ''), ed if (Percet_overhoot >.7) beta=;epilo=;alpha=; eleif (Percet_overhoot >.)&&(Percet_overhoot <=.7) beta=;epilo=95;%, 5, alpha=; eleif(percet_overhoot <.)&&(Percet_overhoot >=.5) beta=;epilo=7;alpha=; eleif(percet_overhoot <.5) %<.5 beta=%, epilo=;alpha=;ed Kp=beta ;% Speed up repoe Kd=(/(*ZETA*omega_))* epilo;% Reduce overhoot Ki=(omega_/*ZETA)*alpha;% dip('====================================') fpritf(' alpha= %g \',alpha) fpritf(' epilo= %g \',epilo) fpritf(' Kp = %g \',Kp) fpritf(' Ki = %g \',Ki) fpritf(' Kd = %g \',Kd) paue(3) %============================ im( 'patet_tet ')% ruig iulik model etitled patet_tet %============================ paue() ed 3

8 IJISET - Iteratioal Joural of Iovative Sciece, Egieerig & Techology, Vol. Iue 9, November. Cocluio A ew, imple, itelliget, robut ad elf-tued cotroller deig approach for gettig a proce uder cotrol, while achievig a importat deig compromie i propoed ad teted. the propoed approach i baed o relatig cotroller()' ad plat' to reult i meetig a importat deig compromie; acceptable tability, ad medium fate of repoe i term of miimum PO%, 5T, T S, ad E SS.. The propoed approach wa tet uig MATLAB m.file ad Simulik model for differet ytem, the theoretical reult how the implicity ad applicability of the propoed approach. A future work, to modify program, uch that it ca be applied for mot type of ytem, ad to ugget ad built a circuit deig to read ad aalyze plat repoe. Referece Katuhiko Ogata (997), moder cotrol egieerig, third editio, Pretice hall,. Ahmad A. Mahfouz (3), Mohammed M. K., Farha A. Salem, "Modelig, Simulatio ad Dyamic Aalyi Iue of Electric Motor, for Mechatroic Applicatio, Uig Differet Approache ad Verificatio by MATLAB/Simulik", IJISA, vol.5, o.5, pp Farha A. Salem(3),, New cotroller efficiet modelbaed deig method, Europea Scietific Joural May ed. vol.9, No.5. J. G. Ziegler (93), N. B. Nichol. Proce Lag i Automatic Cotrol Circuit, Tra. ASME, 5, pp G. H. Cohe (953), G. A. Coo. Theoretical Coideratio of Related Cotrol, Tra. ASME, 75, pp Atrom K,J (99), T. Hagllud, PID cotroller Theory, Deig ad Tuig, d editio, Itrumet Society of America, R. Matouek (), HC: Efficiet PID Cotroller Deig, Egieerig Letter, pp -, :, Sumita Da(), Aya Chakraborty,, Jayata Kumar Ray, Soumyedu Bhattacharjee. Biwarup Neogi, Study o Differet Tuig Approach with Icorporatio of Simulatio Apect for Z-N (Ziegler-Nichol) Rule, Iteratioal Joural of Scietific ad Reearch Publicatio, Volume, Iue, Augut Saeed Tavakoli (3), Mahdi Tavakoli, optimal tuig of PID cotroller for firt order plu time delay model uig dimeioal aalyi, The Fourth Iteratioal Coferece o Cotrol ad Automatio (ICCA 3), - Jue, Motreal, Caada. Atrom K,J (99), T. Hagllud, PID cotroller Theory, Deig ad Tuig, d editio, Itrumet Society of America, ISSN 3 79 Norma S. Nie(), Cotrol ytem egieerig, Sixth Editio Joh Wiley & So, Ic, Gee F. Frakli (), J. David Powell, ad Abba Emami-Naeii, Feedback Cotrol of Dyamic Sytem, th Ed., Pretice Hall,. Dale E. Seborg(), Thoma F. Edgar, Duca A. Mellichamp,Proce dyamic ad cotrol, Secod editio, Wiley *Farha A. Salem () New cotroller' deig method for firt order ytem ad ytem that ca be approximated a firt order. *Farha A. Salem() New efficiet cotroller' deig method for ecod order ytem ad ytem that ca be approximated a SECOND order. *Farha A. Salem () New Cotroller deig method for FOPDT/SOPDT procee Iteratioal Joural of Egieerig Reearch Ad Maagemet (IJERM) ISSN : 39-5, Volume-, Iue-7, October *Farha A. Salem(3), Dyamic ad Kiematic Model ad Cotrol for Differetial Drive Mobile Robot, Iteratioal Joural of Curret Egieerig ad Techology, Vol.3, No., PP53-3,Jue **Farha A. Salem (3), Refied model ad cotrol olutio for Mechatroic deig of robotic platform, Etoia Joural of Egieerig, 3, 9, 3, 3 **Farha A. Salem,(3) Dyamic Modelig, Simulatio ad Cotrol of Electric Machie for Mechatroic Applicatio, it. joural of cotrol, automatio ad ytem, vol. No., pp33-, April 3 Ahmad A. Mahfouz(3), Mohammed M. K., Farha A. Salem, Modelig, Simulatio ad Dyamic Aalyi Iue of Electric Motor, for Mechatroic Applicatio, Uig Differet Approache ad Verificatio by MATLAB/Simulik (I). I.J. Itelliget Sytem ad Applicatio, 3, 5, *-Farha A. Salem(3), Mechatroic Deig of Motio Sytem; Modelig, Cotrol ad Verificatio. Iteratioal Joural of Mechaical & Mechatroic Egieerig IJMME-IJENS Vol:3 No:, pp-7. Farha A. Salem: Now with Taif Uiverity, college of egieerig, Dept. of mechaical Egieerig, Mechatroic egieerig. 37