HYBRID GA-PSO BASED PITCH CONTROLLER DESIGN FOR AIRCRAFT CONTROL SYSTEM

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1 Jr. of Industrial Pollution Control 33()(017) pp Rsarch Articl HYBRID GA-PSO BASED PITCH CONTROLLER DESIGN FOR AIRCRAFT CONTROL SYSTEM VAIBHAV SINGH RAJPUT 1*, RAVI KUMAR JATOTH, NAGU BHOOKYA 3 AND BHASKER BODA 4 1 Dpartmnt of Mtallurg and Matrials Enginring, National Institut of Tchnolog, Andhra Pradsh, India Assistant Profssor, Dpartmnt of Elctronics and Communication National Institut of Tchnolog, Warangal, India 3 Assistant Profssor, Dpartmnt of Elctrical Enginring National Institut of Tchnolog, Warangal, India 4 Assistant Profssor, Dpartmnt of Elctronics and Communication Enginring K.G Rdd Collg of Enginring, Hdrabad, India (Rcivd 07 August, 017; accptd 0 Sptmbr, 017) K words: Pitch rat, Elvation angl, PID controllr, GA, PSO, Phugoid ABSTRACT In this papr proportional, intgral, drivativ (PID) controllr is usd to control th pitch angl of th aircraft whn th lvation angl is changd or modifid. Th pitch angl is dpndnt on lvation angl; a chang in on corrsponds to a chang in th othr. Th PID controllr hlps in rstrictd chang of pitch rat in rspons to th lvation angl. Th PID controllr is dpndnt on diffrnt paramtrs lik Kp, Ki, Kd which chang th pitch rat as th chang. Various mthodologis ar usd for changing thos paramtrs for gtting a prfct tim rspons pitch angl, as dsird or wishd b a concrnd prson. Whil rckoning th valus of thos paramtrs, trial and gussing ma prov to b futil in ordr to provid comfort to passngrs. So, using som mta huristic tchniqus can b usful in handling ths rrors. Hbrid GA-PSO is on such powrful algorithm which can improv transint and stad stat rspons and can giv us mor rliabl rsults for PID gain schduling problm. INTRODUCTION Ths das aircraft, jts, missils mainl rl on automatic mods for its convnincprovidd to th passngrs and to th pilot himslf. Howvr, thsautomaticcontrol sstms nd to b accurat and fast nough to rspond. Th control of aircraft pitch is dpndnt on dflction of lvator hingd at th tail of aircraft. Th dflction causs th air to gt rdirctd. This causs a forc and as a rsult aircraft rvolvs about th pitch axis. Th lvator is displacd b a control stick. Th lvator dflcts in proportion to th stick (dg) and rsulting rotation about th pitch axis is calld pitch rat masurd in dgr of rotation pr scond (dg/s) (Cadwll, 010). Th dnamics of aircraft control sstm compriss of longitudinal and latral dirctions. Th pitch control is a longitudinal control. (Vishal and Joti, 014) Hnc, our focus will b onl in th longitudinal modl and longitudinal quations of motion. In past man rsarchrs hav don work in ordr to control th pitch, aw and roll angl of th aircraft (Amit and Sharma, 009; Vishal and Joti, 014; Saad, t al., 01; Wahid and Rahmat, 010). This paprfocuss on PID controllr tuning with GA,PSO and hbrid GA-PSO to control th pitch rat of th aircraft. Using PSO algorithm can b a smart choic as it is fast and convrgs vr quickl. (Karaboga and Okdm, 004) Howvr, on th othr hand, thr is high possibilit of PSO to gt stuck on th local valus rathr than sarching for a global valu. This problm is liminatd in GA as it dosn t convrgs on a local valu and hunt for global valus. *Corrsponding authors mail: vaibhav.sr@gmail.com

2 1596 RAJPUT ET AL. Th vr problm GA offrs is, slow convrgnc rat for computationall xpnsiv functions. (Ebrhart and Shi, 000; Musrrat, t al., 009) In ordr to ovrcom ths complications hbrid of PSO and GA provd to b a bttr altrnativ. Th spd of PSO mrgd with th global hunting charactristics of GA can bring out th bst solution within a limitd tim and itrations. Hbrid GA-PSO whn tund for PID paramtrs for pitch rat dtction gav much bttr rsults than GA and PSO alon. Th simulation is don in MATLAB and SIMULINK for th analsis of PID paramtrs. Th ris tim, sttling tim, stad stat rror, maximum ovrshoot ar calculatd for ths algorithms and ar compard. PROBLEM FORMULATION Th fdback liminats rror but givs poor transint and stad stat rspons (Fig. 1). This is improvd b PID controllr. Th comfort for th passngrs includs: 1. Ovrshoot lss than 10%.. Ris tim lss than sc. 3. Sttling tim lss than 10 sc. 4. Stad stat rror lss than %. (t) = r (t)- (t)..(i) t d() t u() t = Kp.() t + Ki. (). t + Kd Λ (ii) Whr, r (t) is th rquir valu, (t) is th output valu and (t) is th rror, u (t) is th nw valu PID convrts rror. Thn rror ( (t)) of th plant isoptimid using PID controllr which is thn optimid using mtahuristics tchniqus as mntiond arlir. Ths rquirmnts for plant ar not fulfilld without using PID controllr. Th PID controllr is placd in a control loop fdback mchanism usd to valuat rror continuousl btwn dsird valu and masurd variabl. Th PID controllr tris to rduc rror b using its thr paramtrs Kp, Ki, Kd, Kp trm accounts for th prsnt valu of rror. Ki trm accounts for th past valus of rror. Kd trm accounts for th futur valus of rror. Stting valus for ths paramtrs is calld tuning of th PID paramtrs. Trial and rror mthod can giv rsults but can bcom cumbrsom and dos not guarant prfct valus. Hnc, tuning is rquird to st th PID paramtrs. MATHEMATICAL MODEL Th quations of motion of aircraft can b dividd into latral and longitudinalquations. As statd arlir our major concrn will b mainl on longitudinal dirction and its quation (Torabi, t al., 013; USAF, 1988). Dscriptionof pitch conduct Th pitch control of aircraft can b thought as a combination of two scond ordr transfr quation, dscribing short and long priod stabilit charactristics calld mods (Cadwll, 010). Th combination of two, rsults in 4th ordr transfr function. Th long priod mod calld th phugoid is a nondivrgnt oscillation of aircraft (tim priod gratr than 10 sconds) about th pitch axis (Fig. ). This trm is onl prsnt whn th aircraft is positivl stabl and simplifis into divrgnc if it is ngativl stabl (USAF, 1988; Cadwll, 010). This divrgnc can b ignord in comparison to th short priod mod as its ffct is gratr than long priod mod. So, w will ngotiat long priod oscillation for ths rasons. Eliminating long priod oscillation rstricts our work and quation to scond ordr transfr function (Liu and Lampinn, 00). Equations of motion Th quations of motion of aircraft ar vr complx and complicatd to undrstand. Hnc, in ordr to appl ths quations som modifications could b bnficial. Th dvlopmnts of quations ar shown blow (Fig. 3): Translational motion quation From Nwton s scond law, appling from ground fram (Fig. 4): dv F = m XYZ Fig 1. PID block diagram in fdback loop. Fig. Phugoid mod. (1)

3 HYBRID GA-PSO BASED PITCH CONTROLLER DESIGN FOR AIRCRAFT CONTROL SYSTEM 1597 Fig 3. Viwing axis of aircraft x,, from ground fram with X, Y, Z axis of ground. Fig 4. Pitch axis( axis), Roll axis(x axis), Yaw axis( axis). Whr w= Nt Angular vlocit of Aircraft; V t =Total vlocit (Tru Vlocit) of Aircraft; m=mass of th aircraft. dv F = m( + w v t ) XYZ Taking componnts of w and in roll, pitch,aw axis: Hnc, w = piˆ+ qiˆ+ rkˆ (3) () whr, p, q, r ar angular vlocitis in x,, dirction V = uiˆ+ vj ˆ= ωkˆ (4) t Whr, u, v, ω ar vlocitis in x,, dirction B using quations, (1), (), (3), (4) Fx = m(u rv+ qw) (5) F = m(v + ur p ω) (6) F = m( ω + pv uq) (7) Rotational motion quation dr Vt = ( ) x + w r (8) whr, L t =Total angular momntum of aircraft r = Position vctor from th cntr of gravit of aircraft. dr Vt = ( ) x + w r As, dr ( ) x = 0 (9) Lt = m(r (w r)) (10) As m is variabl on th whol aircraft, intgrating angular momntum for a dm mass (Fig. 4), ovr th whol aircraft will bring out th nt angular momntum, Lx = pi x qi x ri (11) Whr, σ is th mass dnsit of aircraft, which is sam vrwhr V is th Volum of aircraft On solving quation (11) Lx = pi x qi x ri (1) L = ri qi pi (13) L = ri qi pi (14) Whr, I x, I, I ar th momnt of inrtia about thir rspctiv axis Using Nwton s rotational quation, dl T = + w L x As most aircraft hav plan of smmtr about x- plan, so I x = 0, I =0; Thn quations (1),(13),(14) changs to (15) Lx = pi x ri (16) L = qi (17) L = ri pi (18) On putting ths valus to q. (15) rsults into M = qi + rp + r (Ix I ) I (p ) M = qi + rp + r (Ix I ) I (p ) N = ri + pq(i I ) + I (qr p ) x Whr, L, M, N ar Torqus/Momnts about roll, pitch, aw axis. Rlation btwn angls and angular vlocit Ψ = angl of aw Θ = angl of pitch Φ = angl of roll ϒ = angl of lvation α = angl of attack Calculating angls (also calld Eulr s angls) rlation with rspctiv angular vlocit can b don b taking on of th angls as ro for ach axis

4 1598 RAJPUT ET AL. and solving ach axis mchanics. Thn adding all rlations rsult in quations: p = Φ ψ sin Θ (19) q = ψ sin Φ cos Θ+ Θ cos Φ (0) r = ψ cos Θcos Φ Θ sin Φ (1) Th aircraft is mostl dpndnt on its arodnamic trms for its motion. Howvr, th abov quations ar that rsults from summing forcs and momnts, which ar non-linar, and xact solutions ar impossibl. In ordr to linari thm a linarid modl is suggstd which is basd on small disturbancs and small prturbation thor. This modl givs a boost to nginring problms bcaus arodnamic ffcts ar linar functions of variabls of intrst. To cap it all, th small disturbanc thor is applid in thr stps, 1. Writ quilibrium conditions.. Assuming it has small prturbations. 3. Us first ordr Talor sris xpansion to dtrmin small prturbations ffct. Whn oprating undr small prturbation, longitudinal motion can b xprssd in trms of variabls shown: Longitudinal Motion (D, L, M) = f (u, α, ἀ, q,ϒ ) Whr, D, L, M ar Lift, Drag and momnt in longitudinal dirction Appling small disturbanc thor on abov quations b substituting: u=u o +δu; v=v o +δv; ω=ω o + δω ; P=p o +δp; q=q o +δq; r=r o + δr γ= γ o +δγ; For convninc w hav assumd smmtric flight conditions and no propulsiv forcs ar acting. This implis v o =ω o =p o =q o =r o = γ o =0; Using thrust, gravit and groscopic ffct on D,L,M and appling Talor sris on thos and quating thm with th abov quations of forc and momnt rsults in longitudinal dirction quations. Longitudinal motion quation da Drift : (1 Lq) Lu (1 L a ) Lαα = Lγ γ α () da Lift : (1 Lq) Lu (1 L a ) Lαα = Lγ γ (3) α d θ dθ da Pitch : M q Mu u M α Maα = Mγ γ α (4) As w statd arlir, longitudinal motion is what w car for, now writing quations for longitudinal dirction for short priod tim (mod) in spac stat modl: ẋ= Ax + Bu = Cx + Du δa Z / 1 Z /u a uo δa γ o = + δ q M ZM /u M M δ q + + M + MZ / u a a a o q a γ a a o δα = [ 0 1] δ q whr is th output matrix Now, hading towards our transfr function: L( output) Transfr function = L( input) [ γ ] (5) (6) Now, using Cramr s rul, transfr function can b valuatd for pitch rat and lvation angl using (5), (6): Mα Zγ ( Mγ + M / ) ( / ) αzγ u o s MαZα uo δ q(s) uo = (7) δγ (s) Zα MαZα s (Mq + M α+ )s + ( M α) u u Also, δq o = δθ (8) δq=sδθ (9) Hnc, transfr function for pitch angl and lvation angl is, Mα Zγ ( Mγ + M / ) ( / ) αzγ u o s MαZα uo δθ (s) u = o δγ (s) Z α MαZ α s( s Mq + M α+ s+ M α ) uo uo Now simplifing th transfr function b substituting valus of variabls from Tabl 1, (valus takn from commrcial Boing aircraft). δθ (s) 1.151s = δγ s s s 3 (s) GENETIC ALGORITHM o (30) Optimiation tchniqu rquirs complicatd tchniqus in ordr to accomplish th task. On of th tchniqu in ordr to accomplish optimiation is gntic algorithm (GA). GA is xpctionall good at giving optimid rsults. GA involvs a

5 HYBRID GA-PSO BASED PITCH CONTROLLER DESIGN FOR AIRCRAFT CONTROL SYSTEM 1599 Tabl 1. Valus takn from commrcial Boing aircraft Z-Forc, (F -1 ) Pitching Momnt (FT -1 ) Pitching Momnt (FT -1 ) Rolling Vlocitis X ϒ = Z ϒ = M ϒ = Yawing Vlocitis X W =0.036 Z W =-.0 M W =-0.05 Angl of attack X α =0 Z = α M =-8.8 α X ἀ =0 Z =0 ἀ M = ἀ Pitching Rat X a =0 Z =0 a M =-.05 a Elvator dflction δ X =0 δ Z =-8.15 δ M = vr famous concpt of humanit Survival of th Fitttst. GA uss this concpt to find th global valus of th functions using chromosoms as its population, whos fitnss is to b xamind for survival. It involvs thr major procss slction, mutation and crossovr. Th algorithm involvs th following stps: (i) Gnration of Chromosoms Randoml gnratd chromosoms in binar or ral form ar takn as th functions positions. Ths positions ar usd for tsting th valus and ar furthr valuatd accordingl. (ii) Mutation and Crossovr Th gnratd chromosoms ar thn mutatd and crossovrd with ach othr to includ vr possibilit to find th fittst chromosoms. All th gnratd, mutatd and crossovrd chromosoms ar thn takn for final slction to gt th most fit population. (iii) Slction Th pool of chromosoms gnratd using randomnss, mutation and crossovr ar thn slctd to gt th bst gnration of chromosoms. Th bst fit chromosoms giv th bst cost i.. th global bst. PARTICLE SWARM ALGORITHM Th criticism GA and DE suffrd of bing slow was ovrcom b PSO as it has a vr fast convrgnc rat. Howvr, with this ovrwhlming spd on problm couldn t b ovrcom i.. sticking to a local valu. This mthod is basd on swarm bhavior in natur. All swarms sprad in all dirction in ordr to find food (in this cas global valu). On that finds food calls othr swarms to com to that plac. This is xactl how particl algorithm works. Ths mthods ar also calld natur inspird algorithms, as th ar inspird from natur and its bhavior. Th positions and vlocitis ar gnratd randoml. Thn th ar updatd according to th quations: v t+1 = t *w*rand+c1*rand*(pbst x t )+c*rand*(gbst x t ) (31) x t+1 = x t +v t (30) Whr, w is inrtial mass Є[0,1], pbst is th prsonal bst position, gbst is th global bst position for th swarm, rand is an random numbr. c1, c ar slf-confidnc and swarm-confidnc rspctivl. Th psudo cod for PSO is givn blow: start: Initialiation position, vlocit, prsonal bst, global bst whil { Chck th cost of function Updat prsonal and global bst, if cost < prvious cost Updat positions Updat vlocitis Choos th bst cost swarm positions} End In aircraft dnamic plant Kp, Ki, Kd ar takn as 3-D positions. In (Fig. 5), PSO is chcking th cost of function b passing positions (Kp, Ki, Kd) to plant and chcking th bst solution b optimiing rror. This algorithm can b applid to various nginring filds lik control sstms, intllignt sstms, path planning, tuning of LQR and PID controllrs. HYBRID GA-PSO ALGORITHM Aftr analing GA and PSO alon, on fact that cannot b put asid is that both hav som or th othr problm. This mad GA, PSO algorithms to go futil in various filds. Howvr, whn combind ffcts of both ar usd, it gav much bttr rsults in trms of tim and valus. First th bst population of GA is sortd on th basis of its fitnss. Thn th bst valu for ach vctor is matchd, with its fitnss. Thn thos vctorpositions with thir bst valus ar passd to PSO. PSO

6 1600 RAJPUT ET AL. sarchs for global valus on th positions of thos vctors and not randoml as it usd to do arlir. Th flow chart for hbrid GA-PSO and how it is applid to th aircraft dnamics plant is givn in (Fig. 5). SIMULATION PID paramtrs ar calculatd using th abov mthodologis. (Fig. 6) Th rror which is to b optimid is ISTE (Intgral squar tim rror), givn b: t ISTE = [(t)] (31) o Th rror and tuning of PID paramtrs is don b using th abov algorithms. Th rror form ISTE is optimid for gtting th ris tim, sttling tim, stad stat rror and maximum ovrshoot. Th pitch angl (for opn loop) graph on appling stp rspons is shown (Fig. 7): Th abov graph dosn t mt with rquird dsign (givn in problm formulation) so, using PID ma altr th rsults and can bring closr to dsird modl (Fig. 8). Th (Fig. 9) shows that both th pols ar on th lft sid of imaginar axis. Hnc, th function /plant is stabl. RESULTS AND DISCUSSION Th PID paramtrs wr obtaind using PSO, GA, hbrid GA-PSO algorithms. Bst rsults wr notd for hbrid GA-PSO. Th input (rquird valu) rspons takn was 0. radians (11. ). Th comparison of all th algorithms is givn in Tabl and (Fig. 6 and 10). PSO and GA alon didn t provid good sttling tim for pitch control of aircraft. Passngrs comfort nds rduction in ris Fig 5. Implmntation of GA-PSO in aircraft dnamic plant.

7 HYBRID GA-PSO BASED PITCH CONTROLLER DESIGN FOR AIRCRAFT CONTROL SYSTEM 1601 Fig 6. Simulink modl. Fig 7. Opn loop rspons for pitch angl vs. tim. Tabl. valus of Kp, Ki, Kd Fig 8. Closd loop rspons for pitch angl vs. tim. Rspons PSO GA GA-PSO Ris Tim Sttling Tim Maximum Ovrshoot (%) Stad Stat Error (%)

8 160 RAJPUT ET AL. Fig 10. Comparison of GA, PSO, GA-PSO. tim, sttling tim, stad stat rror and maximum ovrshoot of closd loop rspons. Ths dsign rquirmnts wr mt using natur inspird algorithms rathr than arbitrar choosing valus of Kp, Ki, Kd, which is clarl sn in Tabl : CONCLUSION Aircraft pitch control plant has bn dsignd and anald carfull. Hbrid GA-PSO basd controllr has givn th bst rsults of all algorithms. Th dsird input givn is 0. rad (11 dg). Th ris tim is 0.18s, th sttling tim is 0.3 s, maximum ovrshoot is 1.794%, and stad stat rror is Th rsults in Tabl, justifis thathbrid GA-PSO is th bst mthod b far in ordr to optimi th pitch control of aircraft. REFERENCES Amir, T., Amin, A.A., Ali, K. and Sd, H.K. (013). Intllignt Pitch Controllr Idntification and Dsign. Journal of Mathmatics and Computr Scinc. Amit, M. and Sharma, A. (009). Thr Axis Aircraft Autopilot Control Using Gntic Algorithms: An Exprimntal Stud. IEEE Intrnational Advanc Computing Confrnc. Cadwll, J.A. (010). Control of Longitudinal Pitch Rat as Aircraft Cntr of Gravit changs. A thsis prsntd to th Facult of California Poltchnic Stat Univrsit. Ebrhart, R.C. and Shi, Y. (000). Comparing Inrtia Wights and Constriction Factors in Particl Swarm Optimiation, In Procdings of IEEE Intrnational Congrss on Evolutionar Computation. 1 : Karaboga, D. and Okdm, S. (004). A Simpl and Global Optimiation Algorithm for Enginring Problms: Diffrntial Evolution Algorithm. Turk J Elc Engin. 1 : 1. Liu, J. and Lampinn, J. (00). A fu adaptiv diffrntial volution algorithm. Lappnranta Univrsit of Tchnolog, TENCON '0 Procdings 00 IEEE Rgion 10 Confrnc on Computrs, Communications, Control and Powr Enginring. 1 : Musrrat, A., Milli, P. and Ajith, A. (009). Simplx Diffrntial Evolution. Acta Poltchnic Hungarica. 6 : 5. Saad, M.S., Jamaluddin, H. and Darus, I.Z.M. (01). PID Controllr Tuning Using Evolutionar Algorithms. Wsas Transactions On Sstms and Control USAF (Unitd Stats Air Forc). (1988). Stabilit and Control (Volum II). Vishal. and Joti, O. (014). GA tund LQR and PID controllr for Aircraft Pitch control. IEEE papr. Wahid, N. and Rahmat, M.F. (010). Pitch Control Sstm Using LQR and Fu Logic Controllr. Industrial Elctronics & Applications (ISIEA), IEEE Smposium on Industrial Elctronics Malasia