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NASA Technical Reports Server (NTRS) 20010059025: An Experimental Evaluation of Generalized Predictive Control for Tiltrotor Aeroelastic Stability Augmentation in Airplane Mode of Flight PDF

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Preview NASA Technical Reports Server (NTRS) 20010059025: An Experimental Evaluation of Generalized Predictive Control for Tiltrotor Aeroelastic Stability Augmentation in Airplane Mode of Flight

L/ J / An Experimental Evaluation of Generalized Predictive Control for Tiltrotor Aeroelastic Stability Augmentation in Airplane Mode of Flight Raymond G. Kvaternik and David J. Piatak NASA Langley Research Center Hampton, Virginia Mark W. Nixon, Chester W. Langston, and Jeffrey D. Singleton Army Research Laboratory Vehicle Technology Directorate NASA Langley Research Center Hampton, Virginia Richard L. Bennett and Ross K. Brown Bell Helicopter Textron Fort Worth, Texas The results of ajoint NASA/Army/Bell Helicopter Textron wind-tunnel test to assess the potential of Gener- alized Predictive Control (GPC) for actively controlling the swashplate of tiltrotor aircraft to enhance aeroelastic stability in the airplane mode of flight are presented. GPC is an adaptive time-domain predictive control method that uses a linear difference equation to describe the input-output relationship of the system and to design the con- troller. The test was conducted in the Langley Transonic Dynamics Tunnel using an unpowered I/5-scale semispan aeroelastic model of the V-22 that was modified to incorporate a GPC-based multi-input multi-output control algo- rithm to individually control each of the three swashplate actuators. Wing responses were used for feedback. The GPC-based control system was highly effective in increasing the stability of the critical wing mode for all of the conditions tested, without measurable degradation of the damping in the other modes. The algorithm was also ro- bust with respect to its performance in adjusting torapid changes in both the rotor speed and the tunnel airspeed. Notation y(k) vector of digitized response time histories o__'if control law gain matrices A, 13,T matrices (formed t?om aiand 13i)in multi- _i, fli coefficient matrices (OMP) appearing in ARX equation and determined by SID step output prediction equation control horizon blade kinematic pitch-flap coupling angle h# error function (difference between target hp prediction horizon J objective function to be minimized and predicted responses) k time index Abbreviations: I number of data time steps used for SID ARX autoregressive with exogenous input Ell number of feedback outputs to GPC MIMO multi-input/multi-output P order of ARX equation describing system OMP observer Markov parameters Q,R weighting matrices for inputs and outputs SID system identification r number of control inputs from GPC u(k) vector of digitized input time histories INTRODUCTION vector of computed control inputs IIc V matrix of input and output measurements weighting factors tbr inputs and outputs Tiltrotor aircraft operating at high speeds in the W_., W r airplane mode of flight are susceptible to a propro- matrix of OMP determined by SID tor/pylon instability akin to propeller whirl flutter. Such an instability was first encountered during full-scale testing of the Bell XV-3 tiltrotor in the NASA-Ames Presented at the American Helicopter Society 57th Annual 40- by 80-foot Wind Tunnel in 1962. Following this Forum, Washington, DC, May 9-11, 2001. Copyright ©2001 incident, extensive analytical and experimental studies by the American Helicopter Society International, Inc. No of small dynamically-scaled models were initiated by COl_vrightisasserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license under Bell with the threefold objective of providing a physical the copyright claimed herein for Government Purposes. All understanding of the phenomenon, developing an ana- other rights are reserved by the copyright owner lytical method to predict such instabilities, and identify- ingcorrectivdeesignchangesB.y1965b,othanexpla- nationforandameanfsoreliminatintgheinstability werefound(refs.1-2). TheXV-3aircraftwasthen modifiedtoincorporactehangeinsdicatebdyanalyses andretestesduccessfuilnlytheNASA-Ame4s0-by80- footWindTunneiln 1966.Theproprotor/pyldoyn- namicstudieisnitiatedbyBellinsupporotftheXV-3 investigatiownerefollowedby severaoltherstudies conductebdygovernmeanntdindustroyverthenext decad(eseef,orexamplere,['.,3;.-6).Takenasawhole, thesestudieshelpedtoestablishthetechnologbyase needetdolatersuccessfualldydresthseissueofpropro- tor/pylon/winageroelastsictabilityinthedesignofthe XV-15tiltrotorresearcahircraftinthemid-1970asnd theV-22Ospreiynthemid-1980s. Figure1.- Perturbatiornotor-induceaderodynamic Themechanisomftheproprotor/pylionnstability forcesactingonatiltrotoraircrafdturingpitchingand experiencbeydtheXV-3wasidentifiedbyBellduring yawingoscillation(rsotorsinterconnected). theinvestigatioonfthatincidenatndwasreportebdy Hall(ref. 1). Theprimarydestabilizinfagctorwas Proprotor/pylaoenroelastsictabilitycontinuetos foundtobehubinplanesheafrorcesresultinfgromro- beaprimaryconsideratiionnthedesignoftiltrotorair- torprecessiownhenoperatinagthighforwardspeedins craft. Becausweingtorsionasltiffnessis themajor theairplanmeodeofflight.Thehubshearidsentifieads structuradlesignparameteinrfluencintghisphenome- beingresponsibfloerthesubjecint stabilitayreactually non,thestabilityrequiremenotfscurrenttiltrotorair- asubseotfalargersetofhubforcesandmomentthsat craft(XV-15,V-22,BA-609h)avebeenattainedby actonatiltrotoraircraf(tref.4).Disturbancoecscurringusingthick,torsionallsytiffwingshavinga23%thick- in flightcanexciteeithertheelasticor rigid-body ness-to-chorardtio. Suchwingsprovidethetorsional modesofanaircraftinanoscillatormyanner.Fora stiffnesrsequirefdorstabilityinthoseaircrafta,lbeiatt tiltrotoraircraft,anymotionsof thistypeeffectively theexpensoefcruiseefficiencyandmaximumspeed. represeonstcillatortyranslationaanldrotationamlotions Theuseofthinnerwingswouldpermithighercruise oftheproprotosrhaftinspaceT.hisleadstoproprotor-speedsin, creaserdange,andimprovedproductivity. generateaderodynamfoicrcesandmomenttshatarea Howevert,heattendanrteductionin wingstiffness functionoftheseoscillatormyotions.Figure1,from wouldbringwithittheproblemofproprotor/pylionn- referenc4e,showtsheperturbatiorontor-induceadero- stability.Thus,aeroelastiincstabilityof thepropro- dynamifcorcesactingatthehubsofatiltrotoraircraft tor/pylon/winsgystemstandassamajorbarriertoin- executinsgmalpl itchingandyawingmotionwshenop- creasintghemaximumspeecdapabilitoyftiltrotorair- eratinginanairplanmeodeofflight.Fromtheposition craft. Bothpassivaendactivemethodfsorextending oftheseforcesontheaircrafti,tisclearthattheforces proprotor/pylosntabilityboundariefosr conventional showncaninfluenceaircraftlongitudinaalndlateral- tiltrotoraircrafhtavebeeninvestigatedR.eference7s- directionastlabilityH. owevetrh,esheafrorcesHandY 12,whichdescribsetudiesintotheuseofwingand can,quiteindependenotlfyanyaircrafrtigid-bodmyo- bladeaeroelastticailoringtechniquefsor improving tions,alsodestabilizteheproprotor-pylon-wsinysgtem stabilitya,reillustrativoefpassivaepproaches. aeroelasticalTlyh.esearethesheafrorcesidentifiebdy Whiletherehavebeenanumbeorfstudiedseal- Hall(ref.1)asthedriversforproprotor-pylionnstabil- ity. Theyareadirectconsequenocfeairloadmomentsingwiththeuseofactivecontrolsforimprovingthe thataregeneratetodprecestsherotorinspaceinre- aeroelastbicehavioorftiltrotoraircraftm, ostofthese sponsteoshaftpitchingandyawingmotionsI.naddi- haveaddressethdeproblemofgustandmaneuvelorad tiontoatruewhirlinstabilityinvolvingbothpitching alleviationandtherehasbeenonlylimitedattention andyawingmotionosfthepylonp,roprotor/pylionnsta- giventotheuseofactivecontrolfsorstabilityaugmen- bilitycanoccurinasingleplane(eithepritchoryaw), tation(refs.13-16).Referenc1e3investigatethdeap- dependinognthepylonsupporsttiffnessesP.ropel- plicationofswashplafteeedbacfkoraugmentatioofn ler/pylonwhirlflutter,ontheotherhandi,sdrivenby aeroelastsictabilityaspartofabroadesrtudyoffeed- aerodynamcicross-stiffnemssomentasndcanoccur backcontrolforimprovingtheaeroelastaicndrigid- onlyifthereispylonflexibilityinbothpitchandyaw. bodyflightcharacteristoicfstiltrotoraircraft.Thebase- Adiscussioonftheseandotherimportandtifferenceins lineconfiguratioonfthosestudiewsasaBoeingsoft- theacromechanibceahlavioorfpropellerasndpropro- inplanteiltrotorknownastheMode2l22.Bodeanaly- torsisgiveninreferenc4e. seswereusedtodefinetheappropriagteainsandphases tobeappliedtothewingresponstehsatweretobefed A 1/5-scale, semispan aeroelastic model of the backtotheswashplactyeclicinputs.Reference1s4-t5 V-22 designed and built by Bell in 1981 is being used in wereanalyticasltudieosffeedbacckontroflorincreas- the cxperimental evaluation of GPC. That model, on ingproprotor/pylsotnabilityonanXV-15sizctiltrotor loan to the AB from the Navy, has been refurbished to aircraft.Afeedbacgkainmatrixwasintroduceindtothe form a tiltrotor research testbed called the Wing and formulatiobnyaddingafeedbaclokoptotheequations Rotor Aeroelastic Test System (WRATS) tor use in the ofmotionlinearizeadbouatflightconditioonfinterest. Langley Transonic Dynamics Tunnel (TDT) (fig. 2). Wing tip vertical velocities and accelerations were used tor feedback. The gains needed to stabilize the system were determined by simply varying the terms in the gain matrix until an eigenvalue analysis of the closed-loop system indicated a stable system. Reference 16 was an analytical study into the use of linear quadratic regulator (LQR) techniques for determining the wing feedback gains needed to stabilize the whirl modes of a tiltrotor aircraft using an active swashplate. The method was studied using a mathematical model that had been de- veloped earlier for a full-size semi-span configuration of the XV-15. Control design was done in modal space. The discrete state-space equations were transformed to modal form to allow for separation of stable and unsta- ble modes during the design of the controller. A Kal- man-Bucy filter was employed as the state estimator to account for disturbances and noise. The Aeroelasticity Branch (AB) and the Army Figure 2.- WRATS tiltrotor testbed installed inTDT. Research Laboratory's Vehicle Technology Directorate at NASA Langley Research Center, in collaboration Three exploratory experimental studies to investigate with Bell Helicopter Textron Inc (BHTI), is evaluating the use of a multi-input/multi-output (MIMO) GPC- based method tbr active control have been conducted on an adaptive control technique known as Generalized the WRATS model to-date (ref. 18). These include a Predictive Control (GPC) (ref. 17) to assess its potential lk_ractively controlling the swashplate of tiltrotor air- hover test with the baseline stiff-inplane gimbaled rotor craft to enhance aeroelastic stability in both hclicopter (May 1998), a ground resonance test employing a soft- and airplane modes of flight. The term adaptive as used inplane variant of the gimbaled rotor (October 1999), and a wind-tunnel test with the baseline stiffness rotor here refers to a control system in which the parameters describing the model are identified under operational (April 2000). conditions using measured input/output data, and the The purpose of this paper is to summarize the re- updated system parameters are then used to compute a new set of control gain matrices and the next set of sults of the joint NASA/Army/BHTI investigation that commands to be sent to the swashplate actuators, with was conducted in the TDT in April 2000 in the initial assessment of the potential of GPC for augmenting sta- all computations being done on-line. GPC is a time- bility in tittrotor aircraft operating in the airplane mode domain method that uses a linear difference equation to of flight. The WRATS model was modified to incorpo- describe the input-output relationship of the system and rate a GPC system for independently controlling the to design the controller. An ARX-type difference equa- actuators of the fixed-system swashplate. Active con- tion is being employed in the present study. The ARX trol was introduced into the fixed-system swashplate model is used for both system identification and control using three high-frequency servo-controlled hydraulic design. The coefficient matrices of the ARX equation actuators mounted aft of the swashplate inside the pylon are the quantities determined by the identification algo- fairing. The actuator commands included the steady rithm. Closed-loop control is enhanced by performing pilot commands as well as the actuator motions called the system identification in the presence of the external for by the active control algorithm. Wing bending and disturbances acting on the system. The coefficients of torsion strain gages located near the wing root were the the ARX model are assembled into a multi-step (finite- sensors used to provide the response measurements horizon) output prediction equation, the desired (target) needed by the active control algorithm. The GPC- response is specified, and the resulting expression is computed commands were summed with the pilot trim used to form an objective function. Minimization of the commands to produce the desired swashplate actuation objective function leads to an expression for the control about the trim state. The effectiveness of the rotor to be applied to the system. swashplate in increasing stability (damping) was inves- tigatedoverarangeoftunnealirspeedfosrasinglerotor The essential features of the adaptive control rotationaslpeed.Therobustnesosftheactivecontrol process used in the present GPC investigation are de- systemtorapidvariationisntunnealirspeeadndrotor picted in figure 3. The system has r control inputs u, m speedb,othsinglyandincombinatiown,asstudied. measured outputs y, and is subject to unknown external TheeffectofGPConbladeloadsandpitchlinkloads wasnotassessed. Thepertinenetquationusnderlyintghemethod Disturbances (d)_ arepresenteadnddiscusseadndthestrategoyfcom- puterimplementatioisndescribedin,cludingsystem Uicl_._+ ,)Inputs (u),.._lIv Plant IOutputs (y) identificatiocna,lculatioonfcontrollawmatricesa,nd calculatioonfthecontroclommandssenttotheswash- plateactuatorsC.onsideratiorneslatetdoimplementa- uC _l SDyissttuermbanceIdentifEicsatitmioantion & _. tionforon-linecomputationasrediscussedE.xperi- , mentarlesultasrethenpresenteilldustratintghestability augmentatitohnatwasobtainedwiththeGPC-based activecontroslystem. _[ (FeePdraecdki/cFtieveedforwCaorndt)rol _ Figure 3. - Block diagram of identification and control GENERALIZED PREDICTIVE CONTROL procedure. Introductory Remarks disturbances d. Measurement noise is also present. There are two fundamental steps involved: (1) identifi- Predictive controllers were initially introduced in cation of the system, and (2) use of the identified model the chemical industries for controlling chemical proc- to design a controller. As mentioned earlier, a linear esses and have found applications in a wide variety of difference equation or model is used to describe the industrial processes (e.g., ref. 19). Predictive control relationship between the input and the output of the sys- refers to a strategy wherein the decision for the current tem. A linear input-output model gives the current out- control action is based on minimization of a quadratic puts as a linear combination of past input and output objective function that involves a prediction of the sys- measurements. The input-output model used in the pre- tem response at some number of time steps into the fu- sent work has the form of what is referred to as an ARX ture. A variety of predictive controllers have been pro- (autoregressive with exogenous input) model. The posed (e.g., ref. 20). Among these, Generalized Predic- ARX model is used for both system identification and tive Control (GPC), which was introduced in 1987 (rel\ controller design. System identification is done on-line, 17), has received notable attention by researchers. GPC but not at every time point, in the presence of any dis- is a time-domain multi-input-multi-output (MIMO) pre- dictive control method that uses a linear difference turbances acting on the system. In this way, an estimate of the disturbance model is reflected in the identified equation to describe the input-output relationship of the system model and the disturbance does not have to be system. The input-output equation is used to form a modeled separately. This approach represents a case of multi-step output prediction equation over a finite pre- feedback with embedded feedforward. Because the diction horizon while subject to controls imposed over a disturbance information is embedded in the feedforward finite control horizon. The control to be imposed at the next time step is determined by minimizing the devia- control parameters, there is no need for measurement of tion of the predicted controlled plant outputs from the the disturbance signal (ref. 28). The parameters of the desired (or target) outputs, subject to a penalty on con- identified model are used to compute the predictive con- trol effort. trol law. A random excitation Uia(sometimes called dither) is applied initially with u,.equal to zero to iden- A novel version of the GPC procedure was de- tify the open-loop system. Dither is added to the veloped at NASA Langley Research Center in 1997 fl)r closed-loop control input u, if it is necessary to re- efficient computation and unknown disturbance rejec- identify the system while operating in the closed-loop tion by Dr. Jet-Nan Juang and his co-workers. Their mode. work has resulted in a suite of MATLAB m-files that have been collected into a Predictive Toolbox that can Form of Model and Control Law Equations be used by researchers for GPC studies. A summary of the SID and control theory underlying their develop- The relationship between the input and output ment is found in references 21-29, among others. digitized time histories of a MIMO system are described byanARXmodetlhat has the form where y(k) = a, y(k-1) + _2y(k-2) + ... +o_, y(k-p) (1) y=[y(O) y(I) y(2)...y(p)...y(l-1)] (4) + fl0u(k) +fl_u(k-1) + ... +fl,,u(k-p) m x l This equation states that the current output y(k) at time and step k may he estimated by using p sets of the previous output and input measurements, y(k-l) ..... y(k-p) and 1 u(k-l) ..... u(k-p); and the current input measurement v(0) v(1) .-. v(p-l) .-- v(/-2) u(k). The integer p is called the order of the ARX model. The coefficient matrices ai and fit appearing in (5) V= v(0) iiI v(p.-2) ... v(/-3) this equation are referred to as observer Markov pa- u(0) u(l) u(2).., u(p) ... u(l-1) / rameters (OMP) or ARX parameters and are the quanti- v(O) *'" v(l- ip -1) ties to be determined by the identification algorithm. [r +(r +m)p] x l Closed-loop robustness is enhanced by performing the system identification in the presence of the external Equations 4 and 5 follow from writing the discrete-time disturbances acting on the system, thereby ensuring that state-space equations for a linear time-invariant system disturbance information will be incorporated into the at a sequence of time steps k = 0, I..... l-1and grouping system model. The goal of SID is to determine the them into matrix form. The vector v(k) appearing in the OMP based on input and output data. The OMP may be data matrix V is formed from the vectors u(k) and y(k) determined by any SID technique that returns an ARX according to model of the system. The ARX model is used to design the controller (6) and leads to a control law that in the case of a regulator problem has the general form given by (r+m} xI The order of the ARX model (p) and the number of time u_(k)=6r[y(k-I) + a';y(k-2) + --- +@ y(k-p) (2) steps (/) must be specified by the user. Some guidelines +,8;,(k-D +¢_;y(k-2) + ... +¢_i;,(k-p) for their selection are given later. The sizes of the vec- tors and arrays are noted. Equation 2 indicates that the current control input u,(k) may be computed using p sets of the previous input and In forming the matrices given in equations 4 and output measurements. The coefficient matrices c( and 5, it has been assumed that the state matrix A is asymp- if appearing in this equation are the control gain matri- totically stable so that for some sufficiently large p, Ak= ces. 0 for all time steps k > p, and that an observer has been added to the system. It is through these expedients that System Identification the matrix V is reduced to a size amenable for practical numerical computation of its pseudo-inverse. The SID System identification in the presence of the op- process yields OMP rather than system Markov parame- erational disturbances acting on the system is the first of ters (SMP) because of the inclusion of an observer. A the two major computational steps. The external distur- complete discussion of these aspects of the development bances acting on the system are assumed to be unknown may be found inreference 21. (unmeasurable). The number of control inputs is r and the number of measured outputs is m. The system is Y is the matrix of observer Markov parameters excited with band-limited white noise for SID, These that are to be identified and has the form random excitations are input to all r control inputs si- multaneously and the m responses are measured. The digitized input and output time histories (u and 3') at l time points are then used to form the data matrices 3'and V in the equation t?lxr tHxr 71lxIH?Hxr m xIH mxr 71lxm tHx r nl xt?l (7) mx[r +{r +m}p] y= Y V (3) ThesolutionfbrY is obtained by solving equation 3 and fig. Equation 9 shows that the output y(k+j) at time step k+j may be estimated by using p sets of the previ- for Y according to ous output and input measurements, y(k-l) ..... y(k-p) and u(k-l) ..... u(k-p), and the (unknown) current and -y = y Vt= y vr[v (8) future inputs, u(k), u(k+l) ..... u(k+j). The GPC algo- vT_ -1 rithm is based on system output predictions over a finite where 4idenotes the pseudo-inverse. If the product V Vr horizon hi, known as the prediction horizon. To predict is a well-conditioned matrix of reasonable size, the or- future plant outputs, some assumption needs to be made dinary inverse can be taken as shown. Otherwise, a about future control inputs. In determining the future pseudo-inverse must be used• It should be noted that control inputs for GPC, it is assumed that control is ap- because the size of V Vr is much smaller than V, a plied over a finite horizon h,.known as the control hori- pseudo-inverse might be appropriate even if the product zon. Beyond the control horizon the control input is is well conditioned. assumed to be zero. In GPC, the control horizon is al- ways equal to or less than the prediction horizon. Let- Multi-Step Output Prediction Equation ting j in equation 9 range over the set of values j = I, 2, .... hi,-1, the resulting equations can be assembled into a The one-step ahead output prediction equation given multi-step output prediction equation having the form in equation I is the starting point for deriving the multi- step output prediction equation that is needed for de- ' (k)= 7- uj,(k)+ B uv(k-p)+ .,4 yp(k-p) )hp h_tlxh "c hl,nlXl., hl,mXl_t I" signing a GPC controller• Using equation l, the output hpm x[ ' hrxI pr xI pinxI at time step k+j may be written in the form (10) The coefficient matrices T, B, and ..4are formed from .v...(k +j)=o'li_v(k-I) + o'_j_v(k-2) + "'" +o:tj!)(,k-p) combinations of the observer Markov parameters ai and +fl,,u(k +j)+fl',"u(k +j - I) +..-+ fl,,Ijl u(k) (9) fl,. The quantity y;,;,(k) is the vector containing the fu- +fl,"u(k-l)+fl_i'u(k - 2) + ... +fll/'u(k-l,) ture outputs, whereas Uh,(k) is the vector containing the future control inputs yet to be determined. The quanti- where the coefficient matrices are given by recursive ties ul,(k-p) and yp(k-p) are vectors containing the previ- ous p sets of control inputs and outputs, respectively. expressions involving the matrices ai and fli appearing The expanded form of this multi-step output prediction in the ARX equation (ref. 25). The system identifica- equation is shown in equation 11. tion process described earlier determines the matrices as = "+a("'%1 y(k) A_ A:" A, y(k +1) u(k) ] y(k + q- 1) _{/i-I) /_f(jq-2) .. AJ hc< hp AI" u(k.+ 1) / y(k +q) A(//) j_f I/-I } ., u(k +/t - l)J : - -. y(k + h_,- 1) _(,h,-l) t_,{/,,,-2) .. [/_e-h, ) (11) O'I O_2 •' • O'p_ I 19//, _2 "" 1_I,I /8/, 0_/I) 0(_ I) ... _il)l _'/)) y(k -l) ]_i( I) 2 .. R_pI_-I BrpII_ u(k - 1) : : ".. : : y(k -2) u(k - 2) + OYlqM) _2/T(q-I ) """ _pgY(q-[)l _l'(q-II •q- ]_l('l-l) 2(q-I) .• _/p_(q-l) _rp(q-I) al" a_" "" Gp(q)_, a'p(q) y(k-p+l) t,_lIq) j_(q2) ... Rt"(q)p-I /__plql u(k-p+l) : : ".. - : y(k - p) ".. : : u(k - p) •.. P2,-"E"-" t_( hr-I ) _/_Ihe-I) I 2 =Gr + GI'I t_Gv-Il t"p'_-I(q) ---- _I(q-Ip) "l-o(lq-l)j_/,_l ] The OMP ai and fli determined in SID form the first ul_(k)=- (7-rRT+ Q )+7-rR (_3y(k)+/3t_,(k- p)+.Ayp(k- p)) block rows of the coefficient matrices T,, ,4, and /3 in : _/ (-3y(k)+13t_,(k-p)+A)),(k-p)) equation 11. The terms in the remaining rows are com- puted using the recursive relations indicated in the I_.rxl boxes (ref. 25). All terms in equation 11 are known, (14) except for the h, sets of future commands and the he,sets of predicted responses. The goal of the GPC control as the control sequence to be applied to the system over the next h,. time steps. However, only the first r values algorithm is to determine the set of future commands u(k), u(k+l) ..... u(k+h,.-l) that are required to achieve (corresponding to the first future time step) adesired predicted response y(k), y(k+ 1)..... y(k+hl,- I). u, (k) = -7_ vT(k) + fl'u,, (k - p) a v,,(k - p) It should be remarked that the system Markov rxl parameters (SMP), which are commonly used as the (15) basis for identifying discrete-time models lbr linear dy- arc applied to the r control inputs, the remainder are namical systems, form the first block column in the ma- discarded, and a new control sequence is calculated at trix U, the remaining block columns are formed from the next time step. subsets of the SMP. The Markov parameters are the pulse response of a system and are unique for a given The target response is zero for a regulator prob- system. The discrete-time state-space matrices A, B, C, lem and non-zero for a tracking problem. The matrix Q and D are embedded in the SMP. must be tuned to ensure a stable closed-loop system. Typically, h,.is chosen equal to hi,. However, h,.may be Derivation of Control Law chosen less than hpresulting in a more stable, but slug- gish, regulator. The predictive control law is obtained by mini- mizing the deviation of the predicted controlled re- An expression for estimating the order of the sponse (as computed from the multi-step output predic- ARX model that is to be used for SID is given by tion equation) from a specified target response over a prediction horizon hi,. To this end, one first defines an number qf + number of error function that is the difference between the desired ._3,stemstates disturbance states p>ceil (16) (target) response YT (k) and the predicted response in yhj,(k): where ceil denotes rounding up the value of the quantity e = YT(k) - Yhp (k) in parentheses to the next higher integer. The number of system states is typically chosen to be twice the num- = Yv(k)-'Tut_ (k)-]3tlp(k - p)-Ayl,(k- p) ber of significant structural modes; the number of dis- (12) turbance states is set to twice the number of frequencies in the disturbance; m is the number of output measure- An objective function J quadratic in the error and the ments. If measurement noise is of concern, the order of unknown future controls is then formed: p so computed should be increased to allow for compu- tational poles and zeros to improve system identification J = E'TR _"+ u_ Q uj_. (13) in the presence of noise. In practice, simply choosing p to be 5-6 times the number of significant modes in the system is often adequate. The prediction and control Two weighting matrices are included in the objective horizons are set according to the relations function: Q (symmetric and positive definite) is used to weight the control effort and stabilize the closed-loop system; R (symmetric and positive semi-definite) is hp ->p hc _<hp (17) used to weight the relative importance of the differences Although hi, can be set equal to p, h_, is typically set between the target and predicted responses. Typically, greater than p to weight the control effort so as not to Q and R are assumed to be diagonal and for Q to have saturate the control actuators. If the control horizon is the same value w,. along its diagonal and R to have the greater than the system order a minimum energy (mini- same value w, along its diagonal. Minimizing J with mum norm) solution is obtained wherein the com- respect to uh,(k) and then solving for uh,.(k) gives manded output is shared so that the control actuators don't fight each other. If hl,is set equal to p one obtains aso-calleddeadbecaotntrolle(rref.25).Byextending SID should be done with the external distur- thepredictionandcontrohl orizon(sandhence p, the bances acting on the system so that information about order of the ARX model) to very large values, the GPC the disturbances is embedded in the OMP. However, solution approaches that of the linear quadratic regulator depending on the nature of the external disturbances, it (LQR). Thus, GPC approximates an optimal controller may be possible to perform a SID on a system without [or large p (ref. 17). the external disturbances and still determine a control law that results in satisfactory closed-loop performance. Weighting matrices Q and R are used to weight the control effort and to weight the relative importance The computation of pseudo-inverses should be of the differences between the target and predicted re- performed using Singular Value Decomposition (SVD) sponses, respectively. As mentioned earlier, Q and R because of the latter's ability to deal with matrices that are usually assumed to be diagonal and for Q to have are numerically ill conditioned. The use of pseudo- the same value w,. along its diagonal and R to have the inverses (via SVD) is recommended even in cases same value w, along its diagonal. The control weight where the ordinary inverse may seem appropriate (such must be tuned to produce an acceptable solution without as in the operation (VVr) nindicated in equation 8). going unstable. Reducing w, increases control authority A number of decisions also need to be made. and performance but eventually drives the system un- stable. For example, should the system be re-identified and the control law matrices recalculated in every sampling Key Features of Present Method period, every specified number of time steps (block up- dating), or only if some event requiring a re-ID occurs'? An ARX model is employed to represent the sys- Should all the calculations be done in real time, or can tem and is used for both system identification and con- some be done in near real time'? Should updating of the troller design. The SID process used makes recourse to OMP be done in batch mode or recursively? an observer to enable numerical computation of the pseudo-inverse needed for calculation of the OMP that Microprocessor speeds are such that it is now of- comprise the coefficients of the ARX equation. The ten possible to complete the full cycle of GPC computa- controller is thus inherently observer-based but no ex- tions and apply the commands to the actuators within plicit consideration of the observer needs to be taken one sampling period. However, the need for efficient into account in the implementation. In practice, the computational algorithms and attendant coding is not disturbances acting on the system are unknown or un- expected to diminish. measurable. However, as discussed in reference 28, by performing the SID in the presence of the external dis- Implementation Considerations turbances acting on a system, a disturbance model is Several considerations must be taken into ac- implicitly incorporated into the identified observer Markov parameters. However, the identified model count when actually implementing GPC algorithms in unust be larger than the true system model to accommo- hardware for active controls work. date the unknown disturbances. Thus, the effects of the unknown disturbances acting on the system are embed- The measured response time histories must be ded in the matrices .,4and/3, and hence the control law passed through a low-pass filter with a cut-off frequency set equal to the Nyquist frequency)'u. The latter is cho- matrices a" and if. If only the disturbances acting on sen so that the maximum frequency of interest is about the system change, there is no need to recalculate 75% offN. The sampling frequency f_should be at least 7" (and hence y') because 7" is formed solely from the twice fN to prevent aliasing. However, iffs is made too SMP, which are unique for a given system (ref. 28). large the low frequency modes will be poorly identified The solution for Y indicated in equation 8 involves due to a loss of frequency resolution. A sampling rate between 2to 3timesfN is generally sufficient. Once the forming the matrix products yVTand VVr. Here, these sampling frequency has been selected, the minimum products are obtained using the computationally effi- number of data points that should be used for SID fol- cient procedure described in reference 23. lows from the requirement of having 5-10 cycles of the lowest frequency mode in the measured response time Computational Considerations histories. Normalization of the input and output data that is used for SID on the maximum actual or expected Several considerations dealing with computa- values of the data is often helpful numerically. The tions should be kept in mind when developing algo- procedure will depend on whether the computations are rithms for GPC applications. being done in batch mode or recursively. Thecomputintgaskscanbedistributeadmong ness is provided by means of a hollow, rectangular cross computeorsrdifferenCtPUsonasinglecomputeTr.he section, composite spar having chordwise flanges. The choicewill influencetheextentofuserinvolvement. 4.6-foot spar, which lies along the calculated elastic axis Thevaluesoftheinputandoutpudtatathatarebeing of the wing, has segmented, nonstructural, aluminum usedduringclosed-looopperationmsustbecarefully aerodynamic fairings that provide the spanwise distribu- monitoretdo ensurethattheyfall withinacceptabletion of airloil contour. To provide surface continuity boundsT.hisiseasilydoneusingIF/THEN-typcehecks over the lifting surface of the wing, the space between inthecode. the segments is filled with strips of foam rubber. The wing-tip-mounted pylon contains the transmission and gearbox components for the rotor drive system, the EXPERIMENTALSETUP lower part of the mast, and the swashplate control sys- tem. Because these internal components can be treated Wind Tunnel as rigid, the pylon is scaled dynamically so as to pre- The experiment was conducted in the Langley serve its overall mass and inertia properties. The pylon Transonic Dynamics Tunnel, which is a continuous is attached to the wing tip by means of a "racetrack" flow, single return, variable pressure tunnel having a spring assembly that simulates the combined stiffness of test section 16-feet square with cropped corners. The the lull-scale conversion actuator and downstop lock control room and test section walls are provided with mechanism. large windows for close viewing of the model. The tunnel is capable of operation at stagnation pressures The 3-bladed, 7.6-foot diameter stiff-inplane ro- from near vacuum to slightly above atmospheric and at tor has a gimbaled hub that is connected to the mast by a Mach numbers from near zero up to about 1.2. Either constant speed joint (2 coincident universal joints). The air or a heavy gas (R-134a) can be used as the test me- rotor yoke has 2.5 degrees of built-in precone and is dium. Both the density and test-section Math number flexible to allow the blade coning angle to adjust under are continuously controllable. The present investigation centrifugal loading. The blades have a nonlinear distri- was conducted in air near atmospheric conditions and at bution of built-in twist with an overall root-to-tip twist free-stream Math numbers less that 0.30. of 47.5 degrees (leading edge down). Model A large plywood panel through the vertical plane of symmetry of the model and attached to a support The model used in the investigation is a modified structure consisting of a spider-like arrangement of steel version of a Froude-scaled semispan aeroelastic model tubes was employed to seal the open back side of the of the V-22 tiltrotor that was used by Bell/Boeing to semi-fuselage and to serve as a reflection plane. The support the preliminary and full-scale design phases of structure also provided the offset needed to position the the aircraft (Ref. 30). Upon completion of that series of rotor axis near the centerline of the wind tunnel. tests, the Navy transferred the model to the Aeroelastic- ity Branch (AB) at NASA Langley under a loan agree- The fundamental natural frequencies of the ment to be used as the experimental testbed of a tiltrotor model (as measured on the test stand) with the pylon aeroelastic research program that was initiated by AB in locked to the downstop in an airplane mode configura- 1994. The tiltrotor testbed has been designated the tion are summarized in Table I. Typical wind-on natural Wing and Rotor Aeroelastic Testing System (WRATS). General Characteristics Table I- System Natural Frequencies* The WRATS tiltrotor testbed as installed in the Langley Transonic Dynamics Tunnel lor this study is Mode Frequency, Hz shown in figure 2. The model has alength scale factor Wing Beam Bending 5.83 of 1/5 and was designed to maintain full-scale values of Wing Chord Bending 8.67 the Froude, Lock, and Strouhal numbers when operated Wing Torsion 12.0 in air (Ref. 30). The wing and rotor are dynamically Pylon Yaw (2nd wing chord) 19.4 and aeroelastically scaled; the pylon is only dynamically Blade 1st Elastic Flap 7.20 scaled. The fuselage is rigid and only maintains the Blade 1st Elastic Lag 12.5 scaled external aerodynamic shape. The model is at- Blade Rigid Body Torsion 95.6 tached to a support structure that is effectively rigid, its Blade 1stElastic Torsion llO. lowest frequency being well above any important elastic Rotor on, flapping locked out, zero rpm. mode frequency of the model. Simulation of the dis- tributed wing beamwise, chordwise, and torsional stiff- frequencioefstherotorsystem(e.g.a,tV=100 knots and eluding gimbal flapping angles and downstop spring load. Pitch-link and blade loads were not measured. f_ = 742 rpm) are: gimbal flapping at 0.85P, first cyclic lag at 1.2P, and collective flapping (coning mode) at about 1.7 P. The present test was run with the drive Control System Architecture system disconnected so the collective lag mode had zero frequency. A schematic diagram of the control system for active stability augmentation is shown in figure 5. The Active Swashplate Hardware flapping response of the rotor is displayed Three high-frequency electro-hydraulic actuators having a bandwidth of 50 Hertz were used in the control Pilot I _ 1P system to drive the swashplate (fig. 4). The actuators ,Conclrol _ ofnlapping . are positioned azimuthally at the 12, 4, and 8 o'clock positions around and slightly aft of the nonrotating oo OPIrim SyNem responses ]_ swashplate. These actuators were used to input the steady pilot commands as well as the swashplate inputs called for by the GPC active control algorithm. Three J/AcI u_o r - 3000-psi servo-valves were used to meter hydraulic fluid to the actuators. These servo-valves were located inside the pylon near their corresponding actuators (see GPC _ IFeedback fig. 4). Conlrol ] _ I responses I sy._,. I Servo-valve system High-frequency Figure 5. -Schematic of active stability control system. to the pilot as 1P lateral and longitudinal flapping of the rotor tip-path-plane. The pilot trims the windmilling rotor by setting both the collective pitch (to maintain the desired rotor speed) and the cyclic pitch (to maintain zero flapping with respect to the shaft). A mixer in the pilot control console calculates the commands to be sent to the individual actuators to trim the rotor based on the pilot stick commands. The GPC control system calcu- lates the actuator commands needed for stability aug- mentation. The steady 0P (DC) swashplate actuator commands required for trim are summed with the oscil- Active swashplate latory actuator commands calculated by the GPC system (after those commands are converted from digital to Figure 4. - Major hardware components of the GPC analog signals). active control system. Data Acquisition and Processing Feedback Signals and Instrumentation The data acquisition system (DAS) that is part of the WRATS testbed is a 64-channel, PC-based system Feedback signals for the active control system consisting of a National Instruments data acquisition were developed from the measured responses of the board housed in a 750 MHz Pentium processor com- pylon/wing system. The suite of candidate responses puter with a per-channel sampling rate of 1000 Hz. The included wing strain gages (beam, chord, and torsion) WRATS DAS is controlled using a LabVIEW front and six pylon accelerations (normal, lateral, and axial). panel that has been customized to support WRATS The beam and chord strain gages are located 17.0 inches model testing. LabVIEW applications have been devel- out from the root and the torsion strain gages are located oped for on-line FFT analyses, harmonic analyses, ex- 23.0 inches out from the root. The wing bending and citing the model via stick-stir inputs to the swashplate, torsion gages were the sensors used for feedback. A damping estimations using moving block and logarith- number of other measurements were monitored and mic decrement analyses, and plotting. The digitized recorded either to trim the rotor or for flight safety, in- data records are written in a binary file format to the 10

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