Hybrid Differential Evolution based Concurrent Relay-PID Control for Motor Position Servo Systems

Transcription

1 Hybrd Dfferental Evoluton based Concurrent Relay-PID Control for Motor Poston Servo Systems B.Sartha 1, Dr. L. Rav Srnvas P.G. Student, Department of EEE, Gudlavalleru Engneerng College, Gudlavalleru, Andhra Pradesh, Inda 1 Professor, Department of EEE, Gudlavalleru Engneerng College, Gudlavalleru, Andhra Pradesh, Inda ABSTRACT: Ths paper presents the mplementaton of HDE algorthm for Concurrent Relay-PID (CRPID) controller for motor poston servo systems. CRPID s composed of a deadband relay subcontroller whch mproves the transent performance and the parallel PID subcontroller whch mproves the steady state performance of the system. Controller s mplemented usng modern heurstc tunng technques whch are Dfferental Evoluton (DE) and Hybrd Dfferental Evoluton (HDE). The performance of these optmzaton technques s evaluated by settng ther objectve functon as Integral Square Error (ISE). Usng both DE and HDE optmzaton technques comparson of performance has been made between PID and CRPID. KEYWORDS: PID, CRPID, DE, HDE, deadband relay, servo systems. I. INTRODUCTION Electrcal servo systems are wdely used n robotcs, automated factores, electrcal vehcles. PID controllers are smple n structure relable n operaton and robust n performance however, they may produce hgher overshoots and over oscllatory responses. Hybrd scheme CRPID mproves the system performance n terms of transent as well as steady state. CRPID s composed of two controllers one s conventonal PID subcontroller whch mproves the steady state performance and another one s deadband relay subconroller whch mproves the transent performance of the system. The goal of PID controller tunng s to attan the optmal control parameters so as to meet the closed loop system performance specfcatons over a wde range of operatng condtons. Improper PID control parameter tunng could lead to cyclc and slow recovery and poor robustness wll be the collapse of system operaton [1]. To obtan optmal control parameters of PID may strateges has been proposed. Among conventonal PID tunng methods Zegler-Nchols s most popular method however, ths may produce large overshoot []. Soft computng technques have been proposed to attan the optmal control parameters. Basc Dfferental Evoluton optmzaton technque proposed by Storn and Prnce[3-4]. DE s a powerful yet smple evolutonary algorthm for optmzaton of real valued, mult modal functons. DE s generally consdered as relable, accurate and robust optmzaton technque. However, the algorthm suffers from premature convergence, and large computatonal tme for optmzng the computatonally expensve objectve functons. Emprcal analyss of DE has shown that t may stop proceedng towards a global optmum even though the populaton has not converged to a local optmum [5]. The stuaton when the algorthm does not show any mprovement though t accepts new ndvduals n the populaton s known as stagnaton. Besdes ths, DE also suffers from the problem of premature convergence due to loss of dversty n the populaton. The entre populaton converges to a pont whch may not even be a local optmal soluton. Hybrd Dfferental Evoluton has proposed n ths paper to overcome the lmtatons of DE. Copyrght to IJIRSET DOI: /IJIRSET

2 Ths paper s organsed as fallows. Secton II descrbes the proposed control strategy and secton III descrbes the mathematcal model of the plant. Secton IV descrbes desgn pattern of two subcontrollers. Secton V llustrates smulaton result. Concluson has been made n secton VI. II. CONTROL STRATEGY The block dagram of proposed controller CRPID has been shown n Fg. 3. Control strategy conssts of two subcontrollers. The PID subcontroller s a conventonal PID controller. The deadband relay, lead compensator and a auxlary lmted ntegrator makes up a deadband relay subcontroller. When the system has large error deadband relay automatcally turns on and produces ether postve or negatve maxmum controller output untl ts error becomes small. Whenever the error reaches deadband lmt deadband relay automatcally turns off and the correspondng output becomes zero and PID subcontroller alone handles the system regulaton around the steady state. The lmted ntegrator s to ad the PID subcontroller wth near steady state regulaton. The ntegrator n the PID subcontroller may not accumulate suffcent strength to mantan the system output due to fast response of the deadband relay subcontroller n the transent state after the deadband relay subcontroller swtches off. Lmted ntegrator provdes an ntal value for system regulaton near the steady state where only the PID subcontroller s n effect. Incluson of auxlary lmted ntegrator reduces the settlng tme. Whle desgnng the plant unavodable modelng errors occurred ths may lead the system unstable. Lead compensator mproves the relatve stablty of the resultant closed loop system. Fg. 1 Block dagram of CRPID Here controllers are tuned by Dfferental and Hybrd Dfferental Evoluton technques and the comparson has been between these technques. Tunng of controllers are brefly descrbed n the desgn pattern whch s n the followng sectons. III. MATHEMATICAL MODEL OF THE PLANT Fg. Block dagram of dc servo system Fg. 3 Tme doman block dagram representaton of dc servo system Block dagram of dc servo system s shown n Fg. and the tme doman block dagram s represented n Fg. 3. From fgures (s) s the poston and t can be controlled usng concurrent relay PID controller by tunng the controller wth dfferent tunng technques. The open loop transfer functon of armature controlled dc servo system from Fg. 3 s obtaned as follows [6] Copyrght to IJIRSET DOI: /IJIRSET

3 Where m Ea s Kt R J a m 1 Kt K Dm J m Ra J m = motor equvalent nerta constant=30*10-6 kgm K b =back-emf constant=60*10-3 vsrad -1 K t =motor torque constant=17*10-3 NmA -1 R a =armature resstance=3.ω D m =motor equvalent vscous densty =small (neglected) L a =armature nductance=small (neglected). The open loop transfer functon of the dc servo system s m G E S S a b s (1) () IV. DESIGN PATTERN 1. Desgn of PID subcontroller: Controller PID and CRPID are mplemented usng modern heurstc tunng technques Dfferental Evoluton (DE) and Hybrd Dfferental Evoluton (HDE). The transfer functon of the controller s K G p Kp Kd * s (3) s Where Kp = proportonal gan; K = Integral gan; Kd = Dervatve gan. 1.a. Dfferental Evoluton: DE s a stochastc populaton based search technque. The algorthm has been used n many practcal causes because of ts good convergence and global optmzaton capablty. DE can search for the optmal condton very fast wth mnmal control parameters such as ntalzaton, mutaton, crossover and selecton. All these operatons are brefly descrbed n HDE technque. Mutaton, crossover, selecton contnued untl stopng crteron s reached as shown n Fg. 4. The man dsadvantage of DE s premature convergence. HDE overcomes ths lmtaton by performng mgraton operaton. Fg. 4 Flowchart of Dfferental Evoluton 1.b. Hybrd Dfferental Evoluton: Mgraton strategy s manly added to orgnal DE to perform Hybrd Dfferental Evoluton by Chou and Wang. HDE s an effectve relable optmzaton technque to obtan the optmum control parameters of the controller where the ftness value ISE s mnmzed [7]. Ftness Functon: T ISE { e( t)} 0 The performance ndex ISE reduces the large error quckly and makes the response fast. The proposed method s descrbed brefly below dt (4) Copyrght to IJIRSET DOI: /IJIRSET

4 Step 1. Intalzaton: Intalze upper and lower bounds of each control varables and sze of the populaton. The ntal populatons are chosen randomly n the nterval [X mn, X max ] by unform probablty dstrbuton. Ftness value has been calculated for each set of control varables. Generaton of control varables has been made usng below formula. X 0 X mn X X, 1,... N (5) max Where [0,1] s a random number. The ntal process produces N p ndvduals of X 0 randomly. mn Step. Mutaton Operaton: Mutaton expands the search space. A mutant vector s generated based on the present ndvdual X G as fallows G1 G G G G G Y X F(( X r1 X r ) ( X r3 X r 4)) (6) The mutaton factor was selected as F [0,1.], and the upper lmt of 1. for F was determned emprcally; r 1, r, r 3 and r 4 are randomly selected and dstnct. Step 4. Crossover Operaton: Mutant vector Y G+1 and a present ndvdual X G are chosen by a bnomal dstrbuton to progress the crossover operaton to generate an offsprng. The dversty of populaton has been ncreased. Each gene of the j th ndvdual s G+1 reproduced from the mutant vector Y = (Y G+1 1, Y G+1,...,Y G+1 k ) and the present ndvdual X G = (X G 1, X G,..., X G k ). That s G, 1 X f a random number C G h r Yh (7) G1 Yh, otherwse Where =1,,..., N p ; h=1,,..., n c ; n c s the dmenson of decson parameters; K s the no of genes; and the crossover factor s set to be C [0,1]. r Step 4. Estmaton and Selecton: The parent s replaced by ts offsprng f the ftness of the offsprng s better than that of the parent. Contrarly, f the ftness of the offsprng s worse than that of the parent then parent s retaned for the next generaton. Two forms are presented as follows G1 G G1 G1 G1 X arg mn{ F( X ), F( Y )} & X arg mn { F( X ) (8) h Where arg mn means the argument of the mnmum and X b G+1 s the best ndvdual. Step 5. Mgraton If Necessary: Mgraton strategy s to dversfy a populaton that faled n certan tolerance besdes escapng from local optmal and prevents premature convergence. The new populatons are based on the best ndvduals X G+1 b. The h th gene of the th ndvduals as fallows G1 X X G1 G1 h h mn X 1( mn ), 1 hb X h X hb f G G1 X X max X h h hb (9) G1 G1 X hb 1( X h max X hb ), otherwse 1, Where are randomly generated numbers unformly dstrbuted n the range of [0, 1]; h= 1,..., n c. The mgraton n HDE s executed only f a measure fals to match the desred steadness of populaton dversty. Ths measure s defned as fallows b p Copyrght to IJIRSET DOI: /IJIRSET

5 G1 G1 X X N j jb p nc 1, f G1 X j / nc ( N p 1) 1 ; X j X jb (10) 1 j1 b 0, otherwse 1 [0,1] and [0,1] respectvely express the desred steadness for the populaton dversty and the gene dversty wth respect to the best ndvdual. Step 6. Repeat step to step 5 untl desred ISE s reached as shown n Fg. 5. Fg. 5 Flow char of Hybrd Dfferental Evoluton Fg. 6 Input-Output characterstcs of the deadband relay. Desgn of Deadband Relay: The nput-output characterstcs of deadband relay have been shown n Fg. 6. From fg output s zero f the nput of the deadband relay s wthn deadband lmt and s postve or negatve maxmum controller output f deadband relay exceeds the threshold lmt. Where d s the relay output ampltude and h represents the threshold value. Nonlnear systems can be analysed usng descrbng functon analyss. Descrbng functon of the deadband relay can be derved as [8, 9] 4d h N( X ) 1 ( ) (11) X X Where N(X): descrbng functon of the deadband relay, d : relay output ampltude, h: half of the deadband wdth threshold, x represents nput. Inverse descrbng functon of the deadband relay s 1 X N( X ) 4d h 1 ( X It has a mnmum value at some pont n the admssble set of :{ X X h} descrbng functon can be found at ) X h Where e : maxmum absolute error. max The mnmum value of the descrbng functon s 1 h N( X ) d mn (1) X. The mnmum value of the nverse. The threshold value must be selected n the range (13) 0 h e max Copyrght to IJIRSET DOI: /IJIRSET

6 3. Desgn of Lmted Integrator: In order to reduce the settlng tme of the transent response of the system lmted ntegrator s used after the deadband relay. Thepurpose of lmted ntegrator s to complement the ntegrator already present n the PID subcontroller for fast settlng-down of the system. Hence, the gan of the lmted ntegrator should be some way related to the ntegral gan of the PID subcontroller. Emprcal formula for calculatng the gan of the lmted ntegrator s 6K Ka (14) d Where K a : gan of the lmted ntegrator, K : ntegral gan of the PID subcontroller. Large gan could adversely affect the settlng tme of the entre system so dlgence s always requred whle determnng the gan of the lmted ntegrator. 4. Desgn of Lead Compensator: The lead compensator s to mprove the relatve stablty of the resultant closed loop system. The block dagram of the closed-loop system n terms of transfer functons /descrbng functon s shown n fg. 7. Fg. 7 Block dagram of closed loop system Fg. 8 Equvalent system for the desgn of lead compensator Where G c (s) : transfer functon of the PID subcontroller, G l (s) : transfer functon of the lead compensator, G p (s) : transfer functon of the plant, G p (s) : transfer functon of pseudo-pi controller, N(X) : descrbng functon of deadband relay. The transfer functon of the pseudo-pi controller followng the deadband relay, whch s gven by G p s K s a Fgures 7 and 8 descrbes that the desgn of lead compensator for the deadband relay subcontroller s equvalent to the desgn of lead compensator for a plant wth the transfer functon of G 1 N( X ) p mn G p (1 G G A lead compensator wth unt statc gan s of the form TS 1 G l (0 1) TS 1 Where, T represents tme constant of the numerator, α represents rato of tme constants. V. SIMULATION RESULT Let us consder postve /Negatve maxmum controller output s -./.(volts). The output ampltuse of the deadband relay s set to. volts due to the maxmum controller output(d) s. volts and the threshold(h) of the deadband relay s consdered as the mnmum value of the descrbng functon for the deadband relay[10] s Tunng Ranges: Populaton=10, kp mn =0, kp max =4, k mn= 0, k max =, kd mn =0, kd max =0.5, Rate of crossover (c r )=0.8, Mutaton Factor( F)=0.5, =0., lead =55deg. Case A: The transfer functon of dc servo motor s represented by c p ) (15) (16) Copyrght to IJIRSET DOI: /IJIRSET

7 m 18.3 G( s) (17) Ea S(0.1S 1) Mathematcal model s developed n [9] and the optmal control parameters of the controller are obtaned by Hybrd Dfferental Evoluton optmzaton technque. Control parameters obtaned usng ZN method for both PID and CRPID are kp=0.85; k=.83; kd= Optmal control parameters for PID wth HDE technque are kp=0.855; k=; kd= and CRPID wth HDE technque are kp=3.3075; k=1.84; kd=0.133 and lead =55deg. Fg. 9 Step responses of PID wth ZN and HDE Fg. 9 shows that the performance has been mproved by tunng PID controller wth HDE optmzaton technque. Overshoot decreases compared wth ZN method. Fg. 10 Step responses of CRPID wth ZN and HDE Fg. 10 shows that the performance has been mproved by tunng CRPID wth HDE optmzaton technque an all control aspects lke settlng tme, rse tme, peak overshoot and steady state error. Hence CRPID s more effectve compared wth PID. Case B: The transfer functon of dc servo motor s represented by m G (18) Ea S S Optmal control parameters are obtaned by tunng the controller wth DE and HDE optmzaton technques. Copyrght to IJIRSET DOI: /IJIRSET

8 Fg. 11 Convergence graphs of PID, CRPID wth DE and HDE tunng technques. Fg 11 shows that mnmzaton of ISE wth no of teratons. ISE converges fast by tunng controllers wth HDE technque and especally wth CRPID. Fg. 1 Frequency response of lead compensator wth DE Fg. 13 Frequency response of lead compensator wth HDE Fg 1 and 13 shows the frequency response of lead compensator wth DE and HDE. Relatve stablty has been greatly mproved whle tunng CRPID wth HDE optmzaton technque. Fg. 14 Step responses of PID and CRPID wth DE Fg. 15 Step responses of PID and CRPID wth HDE Fg 14 shows the step responses of PID and CRPID controllers by tunng wth DE technque. The performance characterstcs of CRPID has been greatly mproved n all control aspects. Fg 15 shows the step responses of PID and CRPID controllers by tunng wth HDE optmzaton technque. From both plots the performance s mproved for CRPID n all control aspects compared wth PID. Hence system performance has been mproved wth CRPID by tunng wth HDE Optmzaton technque. Copyrght to IJIRSET DOI: /IJIRSET

9 Tab. 1: Characterstcs of step response wth ISE objectve functon Parameter Specfcatons PID-DE PID-HDE CRPID-DE CRPID-HDE KP KI KD Rse Tme(sec) Settlng Tme(sec) Peak Overshoot(%) Steady State Error e e e e-06 ISE.464e e e e-10 Above table shows the optmal control parameters values and step response characterstcs by tunng PID, CRPID wth DE and HDE optmzaton technques. The above values represents the performance of the system has been effectvely mproved wth CRPID controller by tunng wth HDE optmzaton technques. Tab.: Specfcatons of lead compensator Parameters Phase Margn (db) Gan Margn (db) Ω gc (rad/sec) Ω pc (rad/sec) DE Un Compensaton 33.6 nf 44.5 nf DE Compensaton 7.5 nf 69. nf HDE-Un Compensaton 47.5 nf 39.7 nf HDE Compensaton 81.3 nf 66.5 nf Table shows the specfcatons of lead compensator by frequency doman technque bode plot. From the above values phase margn greatly mproved wth compensaton over wthout compensaton for both DE and HDE controller tunng technques. Hence the desgn of lead compensator for CRPID wth HDE optmzaton technque mproves the relatve stablty of the system. VI. CONCLUSION It has been demonstrated that the tunng of PID and CRPID controllers usng Hybrd Dfferental Evoluton technque for motor poston servo systems s hghly effectve over Dfferental Evoluton. From smulaton results CRPID s more effectve than PID controller n all performance aspects and the relatve stablty of the resultant closed loop system has been greatly mproved. REFERENCES [1] Guomn L and Ka Mng Tsang, Concurrent Relay-PID control Motor Poston Servo Systems : Internatonal Journal of Control, Automaton and Systems, Vol. 5, no 3, pp. 34-4, june 007. [] Bhawna Tandon, Genetc Algorthm Based Parameter Tunng of PID Controller for Composton Control System :Internatonal Journal of Engneerng Scence and Technology, Vol. 3, No.8,Aug,011. [3] R.Storn and K.prce, Dfferental Evoluton - A Smple and Effcent Adaptve Scheme for Global Optmzaton over Contnuous Spaces : Techncal report TR-95-01, March 1995,ftp.ICSI.Berkeley.edu/pub/techreports/1995/tr ps.Z. [4] R.Storn and K. Prce, Mnmzng the Real Functons of the ICEC 96 contest by Dfferental Evoluton, proc. Of IEEE Int. Conf. On Evolutonary Computaton, Nagoya, Japan [5] Lampnen and I. Zelnka, On stagnaton of the dfferental evoluton algorthm, n: Pavel Ošmera, (ed.) Proc. of MENDEL 000, 6th Internatonal Mendel Conference on Soft Computng, pp , June , Brno, Czech Republc. [6] M.S. Raml, M.F. Rahmat and M.S. Najb Desgn and Modelng of Integral Control state-feedback controller for mplementaton on servo motor control, 6 th WSEAS Conference on CIRCUITS, SYSTEMS, ELECTRONICS, CONTROL AND SIGNALPROCESSING, Caro, Egypt Dec 9-31, 007. Copyrght to IJIRSET DOI: /IJIRSET

10 [7] K.Lakshm Sowjanya and Dr.L.Rav Srnvas Tunng of PID Controller usng Hybrd Dfferental Evoluton, IJAR n Electrcal Electroncs and nstrumentaton Engneerng, Vol.3, Issue 1, December 014. [8] G. L, Robust Control Strateges for Motor Servo Systems, Ph.D. Thess, The Hong Kong Polytechnc Unversty, Hong Kong, [9] K. M. Tsang and G. L, Robust nonlnear nomnal model followng control to overcome deadzone nonlneartes, IEEE Trans. Ind. Electron, vol. 48, pp , 001. [10] K.Ogata, Modern Control Engneerng, nd ed., Prentce Hall, Englewoodclffs, NewJersey,1990. Copyrght to IJIRSET DOI: /IJIRSET