NONLINEAR H HELICOPTER HOVERING CONTROL AND IMPLEMENTATION USING CSIA

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1 NONLINEAR H HELICOPER HOVERING CONROL AND IMPLEMENAION USING CSIA Chien-Chng Kng iann-dong Yang National Defene Univerity National Cheng Kng Univerity Chng Cheng Intitte of ehnology Intitte of Aeronati and Atronati ao-yan 335, aiwan ainan 7, aiwan el: el : ext Fax: Fax : kng@it.ed.tw dyang@mail.nk.ed.tw Keyword: Nonlinear H Control; Heliopter Hovering Control; Control Srfae Invere Algorithm (CSIA); Separation Priniple of Nonlinar H Hovering Control. Abtrat hi paper applie the nonlinear H ontrol theory to heliopter hovering manver whoe omplete ix degree-offreedom nonlinear eqation of motion with opled rotor and inflow dynami are onidered diretly withot lineariation. Under thi approah, the exat nonlinear eqation governing both the longitdinal and lateral motion are onidered. he ontrol rfae invere algorithm (CSIA) developed in thi paper to implement the nonlinear H ontrol rfae defletion i baed on Moore-Penroe generalied invere. he implement tehniqe i ontrted baed on minimiing the global error between the nonlinear H ommand and atal ontrol fore and moment. he eparation priniple of nonlinear H hovering ontrol i verified by CSIA nder atator with atration and rate limit. Nomenlatre R rotor radi (bript denote tail rotor) (m) air denity (kg/m 3 ) main rotor peed (bript denote tail rotor) (rad/) β,β, β olletive, longitdinal and lateral flapping angle of rotor blade (bript w denote hb/wind axe) in mlti-blade oordinate (rad) θ, θ longitdinal and lateral yli pith C, γ C w () F () F () F () F () F (bript w denote hb/wind axe) (rad) main rotor and tail rotor thrt oeffiient haft angle(poitive forward, rad) rotor idelip angle (rad) one-per-rev ine omponent of ot-of-plane rotor balde fore two-per-rev ine omponent of ot-of-plane rotor balde fore two-per-rev oine omponent of ot-of-plane rotor balde fore one-per-rev ine omponent of in-plane rotor balde fore one-per-rev oine omponent of in-plane rotor balde fore β w tail rotor yli flapping angle in tail rotor hb /wind axe (rad) a, a main rotor and tail rotor blade lift rve lope (/rad) µ f nodimenional total veloity inident on felage nodimenional felage plane area S p C xf normalied felage x-axi fore oeffiient θ pith angle µ normalied veloity in x-axi in hb-wind axi ytem µ normalied veloity at tail rotor µ total normalied tail rotor inflow veloity θ,θ Τ main and tail rotor olletive pith angle (rad), oliditie of main rotor and tail rotor θ Longitdinal yli pith (bript denote hb/wind axe)(rad) λ, λ main rotor and tail rotor niform inflow veloitie in hb/haft axe (normalied by ΩR) θ tw main rotor blade linear twit (rad) rotor blade hord (m) I β flap moment of inertia (kg-m ) K β Center-pring rotor tiffne (N-m/rad) δ 3, k 3 mehanial hinge et angle (rad), k 3 =tanδ 3 γ n, γ Lok nmber of main rotor and tail rotor λ tail rotor flap freqeny ratio β γ rotor firt harmoni longitdinal inflow veloity in hb/haft axe (bript denote hb/wind axe) (normalied by ΩR) Introdtion Many tdie were etablihed for dynami of heliopter. Padfield [] ha mmaried the modeling of a heliopter, and died abot heliopter flying qalitie. Baed on Padfield relt, we bilt or heliopter model. Several referene an be fond that e robt ontrol tehniqe for heliopter hover-ontrol deign, e.g. Apkarian et al []. he pratial appliation of nonlinear H ontrol theory i limited de to the diffiltie in olving the aoiated Hamilton- Jaobi partial differential ineqality (HJPDI). Reently, nonlinear H ontrol ha been applied to nonlinear paeraft [3], flight ontrol [4]. he paper here derived ontrol rfae invere algorithm (CSIA) to implement nonlinear H

2 heliopter hovering ontrol, whih ha not been onidered in the literatre before. Expliit Aerodynami Model A heliopter an be modeled a the ombination of five interating b-ytem: mainrotor, felage, empennage, tailrotor and engine[]. he ix degree-of-freedom rigid body motion of heliopter an be deribed a following: mu = m( WQ VR) Fx dx (a) mv = m( UR WP) Fy dy (b) ( ) mw = m VP UQ F d () IxxP = Ix R PQ I Q xy PR Iy R Q Iyy I QR L d l (d) I Q yy = I P xy QR I R y PQ Ix P R I Ixx PR M dm (e) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I R = Iy Q PR Ix P QR Iyx Q P Ixx Iyy PQ N dn (f) where U, V, W, and P, Q, R are tandard notation for linear and anglar veloitie, repetively, and all referred to the felage (body-fixed) axe ytem, I, I,, et, are the xx x moment of inertia of the heliopter; m i the heliopter' ma. Fore (F x, F y, F ) and moment (L, M, N) onit of (a) aerodynami fore and moment, (b) gravitational fore, () proplive fore and moment. hey an be deribed a the m of the ontribtion from the five b-ytem F = X X X X X m ginθ x R F tp fn (a) F = Y Y Y Y Y m ginφ oθ y R F tp fn (b) F = Z Z Z Z Z m goφ oθ () R F tp fn L = L L L L L (d) R F tp fn M = M M M M M (e) R F tp fn N = N N N N N (f) R F tp fn where the bript tand for: rotor (R), tail rotor (), felage (F), horiontal tail plane (tp), and vertial fin (fn). he orientation of felage i defined in term of Eler Angle θ and φ with repet to an earth-fixed axe ytem. In the following, eah term in the right-hand-ide of Eq.() will be expreed a an expliit fntion of the ontrol variable ed in the nonlinear H approah. o frther implify the notation, the following definition are ed: Σ ( t) = [ U V W] = [ U V W ] [ v w] =Σ σ ( t) Ω () t = [ P Q R] = [ P Q R ] [ p q r] =Ω ( t) Σ ( t) = [ Fx Fy F] = [ Fx Fy F ] [ f ] ( ) x fy f = Σ σ t ( t) = [ L M N] = [ L M N ] [ l m n] = ( t Ω Ω ) dσ = [ dx dy d], d = [ dl dm dn] where the ymbol with bript ero denote the vale at eqilibrim point (trim ondition), and the lower-ae ymbol denote the deviation from the eqilibrim point. However, it need to be noted here that we do not make the amption of mall deviation, i.e., the nonlinear term h a σ σ and are not negligible when ompared with the linear term σ and. he moment of inertia matrix I and the roprodt matrix S() inded by the vetor =[p q r] M are defined a Ixx Ixy Ix r q IM = Ixy I yy I y, S( ) r p = Ix I y I q p he trim fore Σ and trim moment Ω an be olved from Eq.() by = = = = d = d =. Sbtitting Σ and Ω into Eq.(), yield the nonlinear eqation of motion with repet to the eqilibrim point a σ = S( ) σ S( Ω) σ S( ) Σ m σ m d (3a) σ = IMS( ) IM IMS( ) IMΩ IM S( Ω ) IM IM IMd (3b) Aording to the vale of Σ and Ω, three heliopter ontrol mode an be defined: () Veloity and Body-Rate Control Mode: Σ and Ω ; () Veloity and Attit de Control Mode: Σ and Ω = (3) Hovering Mode: Σ =Ω =. Letting Σ =Ω = in Eq.(3), we have d σ S( ) σ mi d 3 mi σ 3 σ = (4) dt I S( ) I I d I M M M M It an be een that, even for hovering mode, the eqation of motion are inherently nonlinear. he aoiated flight ontrol problem i to deign the ontrol fore σ and the ontrol moment o a to nllify the veloity and body rate of the heliopter and at the ame time to trak the attitde ommand [ φ θ ψ ] in the preene of the external ditrbane. Eq.(4) an all be reat into the following tandard tate-pae form: x = f ( x) g ( x) d g ( x) (5) where = [ ] i the nonlinear H σ ommand of ontrol fore and moment to be determined in the next etion. 3 Nonlinear H Hovering Control We will how that nder nonlinear H ontrol trtre, the hovering attitde ontrol an be eparated from the hovering veloity ontrol. Firt, we onider Hovering Attitde Control Loop. We need inorporating the qaternion parameter ( ε, ε, ε, η) to deribe the attitde dynami of the flight 3 vehile. Applying the repreentation of qaternion variation a a fntion of heliopter anglar rate, and removing the veloity dynami from the ix degree-of- freedom eqation of motion, Eq.(3b) yield ( ηi S( ε) ) ε d η ε d dt = = I IMS( ) IM IM D (, I3 ) h ρ ρ = η Π = ρ ρ f ( x) g ( x) d ( ) g x M IM (6), D( Π, I ) = Co η (7) where Π i the rotation matrix from the body axe (felage oordinate ytem) to the inertial axe. For notational onveniene, let the inertial axe be defined a the deired orientation of the body axe, i.e., let the target vale of Π be 3

3 I 3. he geodei metri D(Π,Ι 3 ) meare the ditane between Π and the identity matrix I 3. he main prpoe of thi ontrol mode i to find h that the ytem' L -gain L / d L <γ and keep D(Π,Ι 3 ) a mall a poible, i.e., to keep the pertrbed body frame loe to the inertial referene frame nder the ation of exogeno ditrbane. It an be hown (ee Ref. [5]) that / <γ i ahieved if there L d L exit a alar C fntion E: R n R with E()=, atifying the following HJPDI E E E f g g gw g h h < (8) x x x E / x = E / x E / x E / x n. W E =diag(w x E γ where [ ] w y w w l w m w n ) and ρ, ρ, ρ η in Eq.(7)~(8) are weighting oeffiient onerning the trade-off between traking performane and ontrol effort. By hooing weighting oeffiient properly, it i poible to obtain an aeptably mall h withot onming a lot of ontrol effort. If h a qalified E an be fond, the nonlinear H flight ontroller i then given a = W g E x Sbtitting the orreponding f(x), g (x), g (x), and h (x) from Eq.(6)~(7) into Eq.(8). Motivated from the linear relt, we earh for a poible qadrati oltion in the form E( x) = C IM Cε IMε Cη( η) > () i determined by Eq.(9) and the relt i = ( C Cεε ) ρ () Where C and C ε atify C ρπ η ργ ργ ε ( 3 Cε ρ /) IM γ ρ > γ ρ () Next, we onider Hovering Veloity Control Loop. From Eq.(3a), the hovering veloity dynami i given by σ = S( ) σ dσ σ = f ( x) g ( x) d g ( x) σ σ m m (3) m h ρσ σ σ = = ρ σ ρσ (9) (4) And the orreponding ontrol problem i to determine σ h that the ytem' L -gain / L L < γ. he andidate oltion of the HJPDI i hoen a ( ) = σ σ (5) E x C σ m At the firt glane, the veloity dynami interat with the attitde dynami via the term S() in Eq.(3); however, the interating term dring the formation of HJPDI with tate x=σ in Eq. (8) diappear by mean of the property: he relt mean that the ix degree-of-freedom hovering ontrol deign an be divided into two independent loop: the veloity ontrol loop wherein only veloity dynami i onidered, and the attitde ontrol loop wherein only attitde dynami i onidered. It i o-alled nonlinear H hovering ontrol eparation priniple. 4 Heliopter Pith Control he deired fore and moment ommand of a heliopter to withtand the exogeno ditrbane have been obtained in Eq.() and Eq.(6) by nonlinear H ontrol theory. For a typial heliopter with for pith ontrol [], it i not poible to exatly implement the nonlinear H ommand, whih ha ix independent omponent. he bet fore and moment b whih an be generated by heliopter pith ontrol to follow the ommand i determined h that the following ommand traking error i minimied: J ( b ) ( b error = Q Q ) (7) where weighting matrix Q i ed to filter the fore and moment ommand in order that the pith ontrol atator an be operated in their natrated region. Ually we an expre b (inlding fore and moment, oming from the vario heliopter omponent) expliitly in term of blade pith angle ( θ, θ, θ, θ ), flapping angle ( β, β, β ), and thrt oeffiient ( C ). In general, only for of thee parameter are independent; one any for of thee parameter are given, the remaining parameter are determined aordingly. Carefl inpetion of the expliit model[] indiate that the fore and moment an be mot natrally expreed in term of the for parameter: β, β, C, and a). where C. We will tart with x-axi total fore F x in Eq.( F = F F C F β F β F C (8) F x x xc x β x β x C 4 F x = F /( ρπω R ), and x = ( β oψ o γ β inψ o γ in γ ) / x w w w w C () () F = a (o γ )( F o ψ F in ψ ) / 8 x w w β () () = γ ψ ψ x w w β 4 F = νβ, ν = ( Ω / Ω) ( R / R) x w C F a (o )( F in F o ) / 8 F = a (o γ )( F β oψ F oψ F β inψ () ( ) () x w w w () 4 F in ψ ) / 4 ( µ / ) S in /( ) w f pc mg θ ρπω R xf he entity with over-bar ymbol i non-dimenionalied by the following rle: fore i divided by ρπω R 4 ; moment i E divided by ρπω R 5, area i divided R ; length i divided by f = Cσ mσ ( S( ) σ) = σ R ; veloity i divided by RΩ ; anglar rate i divided by Whih make the final hovering veloity ontroller σ Ω. he derivation of F, y F, L, M and N are imilar independent of : to F x. We repreent fore ( F x, F, mρσ ργ y F ) and moment σ = C σσ σ (6) ρ ( γ ρ ) ( L, M, N ) in the form of (9)

4 where Fx F y F b = L M N, A b * = δ () Fx F C x Fx Fx β β C Fx F C y Fy Fy β β C Fx F C F F β β C A= LC L β L β L C M M M M β β C N N N N β β C C =,, β δ β C X Y Z * = L M N Applying Moore Penroe invere formla, we an olve the optimal δ minimiing J error a ( ) * A A A ( Q ) δ = () It i notied that entry of A alo depend on the flapping angle and pith angle that annot be known in advane. herefore, an initial ge of δ and an iteration algorithm to obtain the teady-tate δ are neeary. he proedre of implementing heliopter pith ontrol from the nonlinear H ommand omprie (a) Chooe an initial ge of θ, β, β, and θ to form vetor δ. (b) o iterate the main-rotor inflow. () Searh felage aerodynami oeffiient: C y fn tp x f y f f l f m f, and Cn f from aero-data table. (d) o iterate the tail-rotor inflow. (e) Evalate Jaobean matrix A and * in Eq.(). (f) Apply Eq. () to olve δ = C β β C. (g) Ue C to allate θ a C µ p θ a () θ = /3 µ / µ λ µ θ tw 4 (h) Ue θ to allate β a µ 4 θ ( µ ) 4θ µθ tw γ β = (3) 8λβ 4 ( µ λ ) µ ( p λ ) 3 3 (i) Evalate main-rotor pith θ β (4) θ M M w = θ β β M w r θ β w w (j) Evalate the effetive tail-rotor olletive pith * 3 C µ λ θ = (5) 3 µ / a and the new tail-rotor olletive pith * θ = θ k γ 3 ( µ ) /( 8 λ ) k γ 3 ( µ λ ) /( 6λ β β ) (6) (k) Stop the iteration if δ δ ε i ahieved; otherwie, reet δ = δ and repeat the tep (b) ~(k). 5 Flight Simlation Nmerial imlation environment will be ontrted for the validation of the nonlinear H ontrol law deigned earlier. he imlation flow hart i depited in Fig.. he following are the proedre for heliopter () Evalate β, β, β, λ and λ aording to the rrent tate variable and ontrol variable θ, θ, and θ. () Searh for felage aerodynami oeffiient: C, x f C, y C, f C, f l C, f m C, f n C, f C. tp y fn θ, (3) Compte aerodynami fore and moment by the formla. (4) Sbtitte fore and moment into Eq.() and integrate the reltant to obtain U, V, W, P, Q, and R. (5) Ue P, Q and R to ompte qaternion by integration. (6) ranform qaternion into Eler angle. (7) Callate the traking error of veloity,, v, w, and body rate p, q, r. (8) Feed, v, w, p, q, r bak to the nonlinear H ontroller to obtain the ommanded fore and moment by employing Eq.() and Eq.(6). (9) Determine C, β, β and C from Eq.(), and determine pith angleθ, θ, θ from Eq.(4) Fig. Flow hart for nonlinear H heliopter ontrol.

5 5. Reglation of Nonlinear H Hovering Control (m/e) w (m/e) 3 atal deopling ontrol ideal deopling ontrol q (rad/e) o illtrate that the onvergene of the nonlinear H hovering ontrol with atator ontraint i not merely loal, we pertrb the ix DOF nonlinear motion to an initial ondition far from trim ondition, and then verify it onvergene. he pith ontrol atator are modeled a two aade eond order ytem with damping ratio.7 and natral freqenie 4 and 7 for fly-by-wire and primary atator, repetively, a hown in Fig.. he main motor and tail rotor olletive pithe are limited between [ ]. he allowable interval for longitdinal and lateral yli pithe are [ ] and [-7 8 ], repetively. Fig. Atator model Initial pertrbed ondition: Σ=[U V W] =[ ] (m/e), Ω=[P Q R] =[.5.5 ] (rad/e) and deired trim ondition: Σ =[U V W ] =[ ] (m/e), Ω =[P Q R ] =[ ] (rad/e).he pper bond of the L -gain i eleted to γ=; the weighting oeffiient ρ ε i et to.5, and the weighting Q i oniting of Q = ρ W E. 3 ideal deopling ontrol atal deopling ontrol ideal deopling ontrol atal deopling ontrol time(e interval) 4.I3.5 ρ =, WE. I = Sbtitting the above given data into Eq.() and Eq.(6) yield the admiible bond for the feedbak gain C σ =.3978 and C =.898 whih are, in trn, ed in Eq.() and Eq.(6) to give the reqired ontrol fore σ and moment. v (m/e) p (rad/e) r (rad/e) - ideal deopling ontrol atal deopling ontrol ideal deopling ontrol atal deopling ontrol ideal deopling ontrol -5 atal deopling ontrol time(e interval) Figre 3 State repone of hovering ontrol. Solid line denote normal hovering ontrol. Dahed line denote variable θ and θ failing to work he initial ondition i very far from the eqilibrim tate. We plot the tate repone in Fig.3 for ontrol inpt of = a. a i the ontrol rfae defletion ommand ( b ) with both amplitde and rate ontraint. he longitdinal tate, w, q onverge at abot 5 e and downward peed w approahe the teady tate abot m/e, while the lateral tate v, p, r onverge rapidly at abot.5 e withot teady tate error. It reveal that the ontrol ability in the longitdinal diretion i not very well. he drawbak an be overome by adjting the haping and weighting matrix. Dahed line in Fig 3 how the repone when ontrol variable θ and θ fail to work. We ame θ and θ are fatened at. Under thi itation, onvergent rate i very low in forward peed deviation and ae large teady-tate error m/e in downward peed deviation w, while there are little effet in pith rate deviation q. Withot the ontrol variable θ and θ defetion, nonlinear H ontroller till ha ffiient ability to tabilie the lateral tate. 5. Verifiation of Deopling Control in Hovering Mode We have mentioned the property that nder nonlinear H ontrol trtre, the hovering attitde ontrol an be eparated from the hovering veloity ontrol. o illtrate the eparation priniple of the nonlinear H hovering ontrol with/withot atator ontraint, we pertrb the ix DOF nonlinear motion to an initial ondition far from trim ondition, and then verify it onvergene. A ha been hown the hovering attitde ontrol loop and the hovering veloity ontrol loop for vertial take-off and landing vehile an be deigned independently nder the framework of nonlinear H ontrol. o verify thi deopling property nmerially, we intentionally dionnet the hovering attitde ontroller, and let only nonlinear H veloity ontroller obtained from Eq.(6) remain operational. he trim ondition for hovering mode i Σ o =Ω =, and let the initial ondition for the pertrbation be [() v() w()]=[ ] (ft/e) and [p() q() r()]=[...] (rad/e). he RMS time repone of the orreponding ix DOF motion for = ae i plotted (olid line). A expeted, the nontrolled attitde loop i divergent, bt the veloity loop i onvergent nder the nonlinear H ontrol, being not inflened by the divergent attitde dynami. For = a ae, it i diffilt for pith blade defletion to reate h ontrol fore and moment that only ontrol veloitie bt not for body rate. he fat an be oberved in the linear ontrol matrix B that all ontrol variable defletion affet all veloitie, v, w and body rate p, q, r. Hene, we let ontrol variable defletion θ,θ, θ, and θ only trak ontrol fore F x, F y, F, and dionnet the hovering attitde ontroller, hene only nonlinear H veloity ontroller remain operational. On the other hand, for the trim ondition Σ o =Ω =, the ontrol rfae matrix A an be allated from Eq.() a follow A =

6 (m/e) w (m/e) q (rad/e) One an find that fore are mainly proded by C and, while β and β almot have no effet to them. herefore, only C and are hoen to trak reqired C 3 = o = hovering ontrol = o = hovering ontrol = o = hovering ontrol C fore. We ame that β and β are fixed at trim ondition.8 and.4, repetively. he ρ and W E are et a following:.5.5.i3 WE =, ρ = * * where the ymbol * here denote that we don t are the vale. It i notied that the ontrol variable defletion θ,θ, θ are tranformed from β, β, β in Eq.(4). It mean that even β and β are fixed, θ and θ will not be ontant. he deopling ontrol verifiation for ontrol inpt of = a ae how in Fig.4 where initial ondition are Σ=[U V W] =[ ] (m/e), and Ω=[P Q R] =[.5.5.5] (rad/e). hi deopling phenomenon an be oberved from time(e interval) v (m/e) p (rad/e) hovering ontrol = o = hovering ontrol = o = r (rad/e) hovering ontrol = o = time(e interval) Figre 4 State repone with veloity ontroller and withot attitde ontroller for = and = a ae. veloitie are tabilied and body rate are divergent. It i notied that the deopling effet in nonlinear H hovering ontrol ha not the amption of mall pertrbation. he initial ondition i far from the eqilibrim tate. hi deopling property i different to linear flight ontrol deign wherein longitdinal ontrol an be deopled from lateral ontrol when vehile' pertrbation i mall, epeially body rate. While the eparation priniple i valid only dring the 3 e hown in Fig. 5. he inded fore from divergent body rate exeed the ontrol fore reated by ontrol variable defletion. It an be realied that the ontrol variable defletion θ, θ, θ, θ are all atrated for t>3.5 e. It implie that atator retrain deopling ontrol ability. he dereae of Ω, whih i dependent on the inreae of θ. One an notie t>4.35 e that the heliopter will tend to rah with a too low Ω beae of the long time atration of olletive pith. Another limitation of atator to eparation priniple i the onvergent rate. For = ae, veloitie onverge at abot.5 e and body rate remain arond teady tate ondition, bt there i low onvergent rate for = a ae. However, the repone till how the tendeny of eparation property for nonlinear H hovering ontrol. 6 Conlion In thi paper heliopter hovering ontroller were deigned nder the omplete ix degree-of-freedom eqation of motion withot the amption of mall pertrbation. he key featre of thi paper i the pratial diion on how to implement blade pith angle from the nonlinear H ommand. Reglation and eparation priniple of nonlinear H hovering ontrol were validated via ontrol rfae invere algorithm (CSIA). he eparation priniple i valid no more than atator being working. In ideal ae, if atator old otpt arbitrarily, the eparation priniple of nonlinear H hovering ontrol wold be alway valid. Referene [] Padfield, G. D., Heliopter Flight Dynami: he heory and Appliation of Flying Qalitie and Simlation Modeling, AIAA, (996). [] Apkarian, P. et al Deign of a heliopter otpt feedbak ontrol law ing modal and trtred -robtne tehniqe, Jornal of Control, V. 5, (989). [3] Kang, W., Nonlinear H Control and It Appliation to Rigid Spaeraft, IEEE ranation on Atomati Control, Vol.4, pp.8-85, (995). [4] Yang. D. and Kng. C., Nonlinear H Flight Control of General Six Degree-of-Freedom Motion, AIAA Jornal of Gidane Control, and Dynami, Vol.3, No., pp , (). [5] Van der Shaft, A. J., L -gain Analyi of Nonlinear Sytem and Nonlinear H Control, IEEE ranation on Atomati Control, Vol. 37, pp , (99) time(e interval) time(e interval) o time(e interval) time(e interval) Figre 5 Hitorie of ontrol variable defletion with veloity ontroller and withot attitde ontroller for = a ae.