An Experimental Evaluation of Generalized Predictive Control for Tiltrotor Aeroelastic Stability Augmentation in Airplane Mode of Flight

J L/ / 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 a joint 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 to rapid 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 _i, fli coefficient matrices (OMP) appearing in A, 13, T matrices (formed t?om ai and 13i)in multi- ARX equation and determined by SID step output prediction equation blade kinematic pitch-flap coupling angle control horizon h# error function (difference between target prediction horizon hp and predicted responses) J objective function to be minimized k time index Abbreviations: I number of data time steps used for SID ARX autoregressive with exogenous input MIMO multi-input/multi-output Ell number of feedback outputs to GPC OMP observer Markov parameters order of ARX equation describing system P SID system identification Q,R weighting matrices for inputs and outputs r number of control inputs from GPC u(k) vector of digitized input time histories INTRODUCTION vector of computed control inputs II c V matrix of input and output measurements Tiltrotor aircraft operating at high speeds in the W_., W r weighting factors tbr inputs and outputs 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_vright is asserted in the United States under Title 17, U.S.

Bell with the threefold objective of providing a physical Code. The U.S. Government has a royalty-free license under understanding of the phenomenon, developing an ana- the copyright claimed herein for Government Purposes. All lytical method to predict such instabilities, and identify- other rights are reserved by the copyright owner

ingcorrective design changes. By 1965, bothanexpla-

nation for anda means for eliminating theinstability

werefound(refs.1-2). TheXV-3 aircraft wasthen

modified to incorporate changes indicated by analyses

and retested successfully in the NASA-Ames 40-by80-

footWindTunnel in 1966.Theproprotor/pylon dy-

namic studies initiated byBellin support of theXV-3

investigation werefollowed by several otherstudies

conducted by government andindustry overthenext

decade (see, forexample, re['.,;. 3-6).Taken asawhole,

these studies helped to establish thetechnology base

needed tolatersuccessfully address theissue ofpropro-

tor/pylon/wing aeroelastic stability in thedesign of the

XV-15tiltrotorresearch aircraft in themid-1970s and

theV-22Osprey in the mid-1980s.

Figure1. - Perturbation rotor-induced aerodynamic

forces acting ona tiltrotoraircraft during pitching and

Themechanism of the proprotor/pylon instability

yawing oscillations (rotors interconnected).

experienced by theXV-3wasidentified byBellduring

theinvestigation of thatincident andwasreported by

Proprotor/pylon aeroelastic stability continues to

Hall (ref. 1). The primary destabilizing factorwas

bea primary consideration in thedesign of tiltrotor air-

found tobehubinplane shear forces resulting fromro-

craft. Because wingtorsional stiffness is the major

torprecession when operating athigh forward speeds in

structural design parameter influencing thisphenome-

theairplane mode offlight. The hubshears identified as

non,thestability requirements of current tiltrotorair-

being responsible forthesubject instability areactually

craft(XV-15, V-22,BA-609) havebeen attained by

a subset of a larger setof hubforces andmoments that

using thick, torsionally stiff wings having a 23%thick-

actona tiltrotor aircraft (ref.4). Disturbances occurring

ness-to-chord ratio. Suchwings provide thetorsional

in flight canexciteeitherthe elasticor rigid-body

stiffness required forstability in those aircraft, albeit at

modes of anaircraft in anoscillatory manner. Fora

theexpense of cruise efficiency andmaximum speed.

tiltrotoraircraft,anymotions of thistypeeffectively

Theuseof thinner wingswouldpermithigher cruise

represent oscillatory translational and rotational motions

speeds, increased range, andimproved productivity.

oftheproprotor shaft in space. Thisleads toproprotor-

However, the attendant reduction in wing stiffness

generated aerodynamic forces andmoments thatarea

wouldbringwithit theproblem of proprotor/pylon in-

function of these oscillatory motions.Figure1,from

stability. Thus,aeroelastic instability of the propro-

reference 4,shows theperturbation rotor-induced aero-

tor/pylon/wing system stands asa major barrier to in-

dynamic forces acting atthehubs of a tiltrotor aircraft

creasing themaximum speed capability of tiltrotorair-

executing small pitching andyawing motions when op-

craft. Bothpassive andactive methods for extending

erating in anairplane mode offlight. From theposition

of these forces ontheaircraft, it isclear thattheforces proprotor/pylon stabilityboundaries for conventional

tiltrotoraircraft have been investigated. References 7-

shown caninfluence aircraft longitudinal andlateral-

12,whichdescribe studies into theuseof wing and

directional stability. However, theshear forces H and Y

bladeaeroelastic tailoringtechniques for improving

can, quiteindependently of anyaircraft rigid-body mo-

stability, areillustrative ofpassive approaches.

tions, alsodestabilize theproprotor-pylon-wing system

aeroelastically. These aretheshear forces identified by

Whilethere have been a number ofstudies deal-

Hall(ref.1)asthedrivers forproprotor-pylon instabil-

ingwiththeuseof active controls for improving the

ity. Theyareadirectconsequence of airload moments

aeroelastic behavior of tiltrotoraircraft, mostof these

thataregenerated to precess therotorin space in re-

have addressed theproblem of gust andmaneuver load

sponse toshaftpitching andyawing motions. Inaddi-

alleviation andtherehasbeenonlylimitedattention

tionto a truewhirl instability involving bothpitching

given totheuse of active controls forstability augmen-

and yawing motions ofthepylon, proprotor/pylon insta-

tation(refs.13-16). Reference 13investigated theap-

bilitycanoccur in a single plane (either pitchoryaw),

plication of swashplate feedback for augmentation of

depending on the pylonsupport stiffnesses. Propel-

aeroelastic stability aspartof a broader study of feed-

ler/pylon whirl flutter,ontheotherhand, is driven by

backcontrolfor improving theaeroelastic andrigid-

aerodynamic cross-stiffness moments andcanoccur

body flightcharacteristics oftiltrotor aircraft. Thebase-

onlyif thereis pylon flexibilityin bothpitchand yaw.

lineconfiguration of those studies wasa Boeing soft-

A discussion of these andother important differences in

inplane tiltrotor known astheModel 222. Bode analy-

theacromechanical behavior of propellers andpropro-

ses were used todefine theappropriate gains and phases

torsisgiven inreference 4.

tobeapplied tothewingresponses thatwere tobefed

A 1/5-scale, semispan aeroelastic model of the

back totheswashplate cyclicinputs.References 14-t5

V-22 designed and built by Bell in 1981 is being used in

were analytical studies of feedback control for increas-

the cxperimental evaluation of GPC. That model, on

ingproprotor/pylon stability onanXV-15sizctiltrotor

loan to the AB from the Navy, has been refurbished to

aircraft. A feedback gain matrix was introduced intothe

form a tiltrotor research testbed called the Wing and

formulation by adding a feedback looptotheequations

Rotor Aeroelastic Test System (WRATS) tor use in the ofmotion linearized about a flightcondition ofinterest.

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.

Figure 2. - WRATS tiltrotor testbed installed in TDT.

The Aeroelasticity Branch (AB) and the Army Research Laboratory's Vehicle Technology Directorate Three exploratory experimental studies to investigate at NASA Langley Research Center, in collaboration the use of a multi-input/multi-output (MIMO) GPC- with Bell Helicopter Textron Inc (BHTI), is evaluating 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 hover test with the baseline stiff-inplane gimbaled rotor lk_r actively controlling the swashplate of tiltrotor air- (May 1998), a ground resonance test employing a soft- craft to enhance aeroelastic stability in both hclicopter inplane variant of the gimbaled rotor (October 1999), and airplane modes of flight. The term adaptive as used and a wind-tunnel test with the baseline stiffness rotor here refers to a control system in which the parameters (April 2000).

describing the model are identified under operational 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 sults of the joint NASA/Army/BHTI investigation that new set of control gain matrices and the next set of was conducted in the TDT in April 2000 in the initial commands to be sent to the swashplate actuators, with 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- The essential features of the adaptive control

tigated overarange oftunnel airspeeds forasingle rotor

rotational speed. Therobustness of theactive control process used in the present GPC investigation are de- picted in figure 3. The system has r control inputs u, m

system to rapidvariations in tunnel airspeed androtor

measured outputs y, and is subject to unknown external speed, bothsinglyandin combination, wasstudied.

Theeffectof GPC onblade loads andpitchlinkloads

was notassessed.

Disturbances (d)_

Thepertinent equations underlying themethod

arepresented anddiscussed andthestrategy of com-

Uicl _._+ ,)Inputs (u),.._lv I Plant I Outputs (y)

puterimplementation is described, including system

identification, calculation of control lawmatrices, and

calculation of thecontrol commands sent totheswash-

u C _l System Identification & _. Disturbance Estimation

plate actuators. Considerations related to implementa-

tionfor on-line computations arediscussed. Experi-

,

mental results arethen presented illustrating thestability

augmentation thatwasobtained with the GPC-based

(Feedack/Feedforward) _ _[ Predictive Control active control system.

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- turbances acting on the system. In this way, an estimate dictive control method that uses a linear difference 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 control parameters, there is no need for measurement of next time step is determined by minimizing the devia- the disturbance signal (ref. 28). The parameters of the tion of the predicted controlled plant outputs from the identified model are used to compute the predictive con- desired (or target) outputs, subject to a penalty on con- trol law. A random excitation Uia (sometimes called trol effort.

dither) is applied initially with u,. equal to zero to iden- tify the open-loop system. Dither is added to the A novel version of the GPC procedure was de- closed-loop control input u, if it is necessary to re- veloped at NASA Langley Research Center in 1997 fl)r identify the system while operating in the closed-loop efficient computation and unknown disturbance rejec- mode.

tion by Dr. Jet-Nan Juang and his co-workers. Their 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

byanARXmodel that has the form where

y(k) = a, y(k-1) + _2 y(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 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) ii I v(p.-2) ... v(/-3) u(0) u(l) u(2).., u(p) ... u(l-1) / this equation are referred to as observer Markov pa- *'" ip v(O) v(l- -1) rameters (OMP) or ARX parameters and are the quanti- 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- 1 and 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 (r+m} x I problem has the general form given by 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) steps (/) must be specified by the user. Some guidelines (2) for their selection are given later. The sizes of the vec- +,8;,(k-D +¢_;y(k-2) + ... + ¢_i;,(k-p) tors and arrays are noted.

Equation 2 indicates that the current control input u,(k) In forming the matrices given in equations 4 and may be computed using p sets of the previous input and 5, it has been assumed that the state matrix A is asymp- output measurements. The coefficient matrices c( and totically stable so that for some sufficiently large p, A k = if appearing in this equation are the control gain matri- ces. 0 for all time steps k > p, and that an observer has been added to the system. It is through these expedients that the matrix V is reduced to a size amenable for practical System Identification numerical computation of its pseudo-inverse. The SID process yields OMP rather than system Markov parame- System identification in the presence of the op- ters (SMP) because of the inclusion of an observer. A erational disturbances acting on the system is the first of complete discussion of these aspects of the development the two major computational steps. The external distur- may be found in reference 21.

bances acting on the system are assumed to be unknown (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 t?l x r tH x r 71l x IH ?H x r m x IH m x r 71l x m tH x r nl x t?l V in the equation (7) mx[r +{r +m}p] y= Y V (3) and fig. Equation 9 shows that the output y(k+j) at time

Thesolution fbr Y is obtained by solving equation 3

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 future inputs, u(k), u(k+l) ..... u(k+j). The GPC algo- = (8) -y y Vt= y vr[v vT_ -1 rithm is based on system output predictions over a finite where 4i denotes the pseudo-inverse. If the product V V r 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 about future control inputs. In determining the future dinary inverse can be taken as shown. Otherwise, a control inputs for GPC, it is assumed that control is ap- pseudo-inverse must be used• It should be noted that because the size of V V r is much smaller than V, a plied over a finite horizon h,. known as the control hori- zon. Beyond the control horizon the control input is pseudo-inverse might be appropriate even if the product assumed to be zero. In GPC, the control horizon is al- is well conditioned.

ways equal to or less than the prediction horizon. Let- ting j in equation 9 range over the set of values j = I, 2, Multi-Step Output Prediction Equation .... hi,-1, the resulting equations can be assembled into a multi-step output prediction equation having the form The one-step ahead output prediction equation given in equation I is the starting point for deriving the multi- ' (k)= 7- uj,(k)+ B uv(k-p)+ .,4 yp(k-p) step output prediction equation that is needed for de- )hp h_tlxh "c hl, nlXl., hl,mXl_t I " hpm x [ ' h r x I pr x I pin x I signing a GPC controller• Using equation l, the output (10) at time step k+j may be written in the form The coefficient matrices T, B, and ..4 are formed from .... + t j! , v(k + j)=o'li_v(k-I) + o'_j_v(k-2) + "'" o: )(k-p) combinations of the observer Markov parameters ai and Ijl (9) + fl,,u(k + j)+fl',"u(k + j - I) +..-+ fl,, u(k) fl,. The quantity y;,;,(k) is the vector containing the fu- ture outputs, whereas Uh,(k) is the vector containing the +fl,"u(k-l)+fl_i'u(k - 2) + ... +fll/'u(k-l,) future control inputs yet to be determined. The quanti- ties ul,(k-p) and yp(k-p) are vectors containing the previ- where the coefficient matrices are given by recursive 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

A_ y(k) A:" A, y(k +1) u(k) ] _{/i-I) /_f(jq-2) .. AJ y(k + q - 1) hc< hp u(k.+ 1) /

A(//) j_f I/-I } ., AI"

y(k + q) u(k +/t - l)J : - -.

[/_e -h, ) y(k + h_, - 1) _(,h,-l) t_,{/,,,-2) . .

(11) _2 "" 1_I,I /8/, O' I O_ 2 • ' • O'p_ I 19//, .. RI_ BII_ ]_i( I ) u(k - 1) 0_/I) 0(_ I) ... _il)l _'/)) 2 y(k -l) _p - I rp : : ".. : : u(k - 2) y(k -2) (q-I) .• /_(q-l) _(q-I) • q- ]_l ('l-l) OYl qM) _2/T(q-I ) " " " _pgY(q-[)l _l'(q-II 2 + _p rp (q) (q) j_(q) ... R(q) /_lql t,_l I q ) 2 u(k-p+l) al" a_" "" Gp_, a'p y(k-p+l) t" p-I _p ".. : : : : ".. - : u(k - p) y(k - p) _/_Ihe-I) t_( h r -I )

•.. P2,-" E"-"

I 2 '_(q) -- _(q-I) "l- o(lq-l)j_/,_l ] = Gr + GI'I t_Gv-Il t"p-I -- I p The OMP ai and fli determined in SID form the first ul_ (k) = - ( 7-r RT+ 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 as the control sequence to be applied to the system over of predicted responses. The goal of the GPC control the next h,. time steps. However, only the first r values algorithm is to determine the set of future commands (corresponding to the first future time step) u(k), u(k+l) ..... u(k+h,.-l) that are required to achieve a desired predicted response y(k), y(k+ 1)..... y(k+hl,- I ).

u, (k) = -7 _ vT(k) + fl'u,, (k - p) a v,,(k - p) rxl It should be remarked that the system Markov 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 hp resulting 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,stem states disturbance states (16) p>ceil (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 in the presence of noise. In practice, simply choosing p (13) J = E 'T R _" + u_ Q uj_.

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 hp -> p hc _<hp (17) system; R (symmetric and positive semi-definite) is 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 h l, is set equal to p one obtains SID should be done with the external distur-

a so-called deadbeat controller (ref.25). Byextending

theprediction andcontrol horizons (andhence p, the bances acting on the system so that information about the disturbances is embedded in the OMP. However, order of the ARX model) to very large values, the GPC solution approaches that of the linear quadratic regulator depending on the nature of the external disturbances, it may be possible to perform a SID on a system without (LQR). Thus, GPC approximates an optimal controller the external disturbances and still determine a control [or large p (ref. 17).

law that results in satisfactory closed-loop performance.

Weighting matrices Q and R are used to weight The computation of pseudo-inverses should be the control effort and to weight the relative importance performed using Singular Value Decomposition (SVD) of the differences between the target and predicted re- because of the latter's ability to deal with matrices that sponses, respectively. As mentioned earlier, Q and R are numerically ill conditioned. The use of pseudo- are usually assumed to be diagonal and for Q to have inverses (via SVD) is recommended even in cases the same value w,. along its diagonal and R to have the where the ordinary inverse may seem appropriate (such same value w, along its diagonal. The control weight as in the operation (VV r) n indicated in equation 8).

must be tuned to produce an acceptable solution without going unstable. Reducing w, increases control authority A number of decisions also need to be made.

and performance but eventually drives the system un- For example, should the system be re-identified and the stable.

control law matrices recalculated in every sampling period, every specified number of time steps (block up- Key Features of Present Method dating), or only if some event requiring a re-ID occurs'?

Should all the calculations be done in real time, or can An ARX model is employed to represent the sys- some be done in near real time'? Should updating of the tem and is used for both system identification and con- OMP be done in batch mode or recursively?

troller design. The SID process used makes recourse to an observer to enable numerical computation of the Microprocessor speeds are such that it is now of- pseudo-inverse needed for calculation of the OMP that ten possible to complete the full cycle of GPC computa- comprise the coefficients of the ARX equation. The tions and apply the commands to the actuators within controller is thus inherently observer-based but no ex- one sampling period. However, the need for efficient plicit consideration of the observer needs to be taken computational algorithms and attendant coding is not into account in the implementation. In practice, the expected to diminish.

disturbances acting on the system are unknown or un- measurable. However, as discussed in reference 28, by Implementation Considerations performing the SID in the presence of the external dis- turbances acting on a system, a disturbance model is Several considerations must be taken into ac- implicitly incorporated into the identified observer count when actually implementing GPC algorithms in Markov parameters. However, the identified model hardware for active controls work.

unust be larger than the true system model to accommo- date the unknown disturbances. Thus, the effects of the The measured response time histories must be unknown disturbances acting on the system are embed- passed through a low-pass filter with a cut-off frequency ded in the matrices .,4 and/3, and hence the control law 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 due to a loss of frequency resolution. A sampling rate The solution for Y indicated in equation 8 involves between 2 to 3 timesfN is generally sufficient. Once the forming the matrix products yV T and VV r. 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.

ness is provided by means of a hollow, rectangular cross

Thecomputing tasks canbedistributed among

section, composite spar having chordwise flanges. The

computers ordifferent CPUs onasingle computer. The

choice will influence theextent of userinvolvement. 4.6-foot spar, which lies along the calculated elastic axis of the wing, has segmented, nonstructural, aluminum

Thevalues of theinputandoutput datathatarebeing

aerodynamic fairings that provide the spanwise distribu-

used duringclosed-loop operations mustbecarefully

tion of airloil contour. To provide surface continuity

monitored to ensure thattheyfall withinacceptable

over the lifting surface of the wing, the space between

bounds. Thisis easily done using IF/THEN-type checks

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- serve its overall mass and inertia properties. The pylon The experiment was conducted in the Langley is attached to the wing tip by means of a "racetrack" Transonic Dynamics Tunnel, which is a continuous spring assembly that simulates the combined stiffness of flow, single return, variable pressure tunnel having a the lull-scale conversion actuator and downstop lock test section 16-feet square with cropped corners. The mechanism.

control room and test section walls are provided with large windows for close viewing of the model. The The 3-bladed, 7.6-foot diameter stiff-inplane ro- tunnel is capable of operation at stagnation pressures tor has a gimbaled hub that is connected to the mast by a from near vacuum to slightly above atmospheric and at constant speed joint (2 coincident universal joints). The Mach numbers from near zero up to about 1.2. Either rotor yoke has 2.5 degrees of built-in precone and is air or a heavy gas (R-134a) can be used as the test me- flexible to allow the blade coning angle to adjust under dium. Both the density and test-section Math number centrifugal loading. The blades have a nonlinear distri- are continuously controllable. The present investigation bution of built-in twist with an overall root-to-tip twist was conducted in air near atmospheric conditions and at of 47.5 degrees (leading edge down).

free-stream Math numbers less that 0.30.

A large plywood panel through the vertical plane Model 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 Mode Frequency, Hz Langley Transonic Dynamics Tunnel lor this study is 5.83 Wing Beam Bending shown in figure 2. The model has a length scale factor 8.67 of 1/5 and was designed to maintain full-scale values of Wing Chord Bending 12.0 the Froude, Lock, and Strouhal numbers when operated Wing Torsion 19.4 in air (Ref. 30). The wing and rotor are dynamically Pylon Yaw (2nd wing chord) 7.20 and aeroelastically scaled; the pylon is only dynamically Blade 1st Elastic Flap 12.5 scaled. The fuselage is rigid and only maintains the Blade 1st Elastic Lag 95.6 scaled external aerodynamic shape. The model is at- Blade Rigid Body Torsion llO.

tached to a support structure that is effectively rigid, its Blade 1st Elastic Torsion 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- eluding gimbal flapping angles and downstop spring

frequencies oftherotor system (e.g., at V= 100 knots and

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 Control System Architecture about 1.7 P. The present test was run with the drive system disconnected so the collective lag mode had zero A schematic diagram of the control system for frequency.

active stability augmentation is shown in figure 5. The flapping response of the rotor is displayed Active Swashplate Hardware Three high-frequency electro-hydraulic actuators having a bandwidth of 50 Hertz were used in the control Pilot I _ 1P Conlrol _ flapping system to drive the swashplate (fig. 4). The actuators are positioned azimuthally at the 12, 4, and 8 o'clock oo OP Irim SyNem responses ]_ positions around and slightly aft of the nonrotating swashplate. These actuators were used to input the ,con.

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 _ I Feedback fig. 4). Conlrol ] _ I responses I sy._,. I Servo-valve system Figure 5. - Schematic of active stability control system.

High-frequency 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- latory actuator commands calculated by the GPC system Active swashplate (after those commands are converted from digital to analog signals).

Figure 4. - Major hardware components of the GPC active control system.

Data Acquisition and Processing The data acquisition system (DAS) that is part of Feedback Signals and Instrumentation the WRATS testbed is a 64-channel, PC-based system consisting of a National Instruments data acquisition Feedback signals for the active control system board housed in a 750 MHz Pentium processor com- were developed from the measured responses of the puter with a per-channel sampling rate of 1000 Hz. The pylon/wing system. The suite of candidate responses WRATS DAS is controlled using a LabVIEW front included wing strain gages (beam, chord, and torsion) panel that has been customized to support WRATS and six pylon accelerations (normal, lateral, and axial).

model testing. LabVIEW applications have been devel- The beam and chord strain gages are located 17.0 inches oped for on-line FFT analyses, harmonic analyses, ex- out from the root and the torsion strain gages are located citing the model via stick-stir inputs to the swashplate, 23.0 inches out from the root. The wing bending and damping estimations using moving block and logarith- torsion gages were the sensors used for feedback. A mic decrement analyses, and plotting. The digitized number of other measurements were monitored and data records are written in a binary file format to the recorded either to trim the rotor or for flight safety, in-

hostPC'sharddiskfor access duringthetestandto

Test Conditions and System Configurations

CDsfor permanent storage. A MATLAB-based pro-

gram isused forpost-test data reduction and plotting.

The GPC investigation constituted only a portion of a much broader aeroelastic investigation that was

TheGPC computer consisted ofa500MHZdual

conducted with the WRATS model during the April

processor PCthatincluded a dSPACE DSP1103 add-in

2000 test in the TDT. One of thc other objectives of cardforperforming real-time data acquisition and com- that test was to establish new baseline stability bounda-

putations. A sampling rate of 100 Hz was used for the

ries tbr the model that included the hydraulic compo- nents that had been installed earlier for active controls GPC data channels. The GPC computational tasks were distributed among these three computing elements in the testing. This baseline stability investigation was con- manner indicated in figure 6. The entire process was ducted with the pylon in both the on- and off-downstop managed using ControlDesk, dSPACE's graphical user configurations, over a range of rotor rpm, in both air and interface (GUI), which was installed on CPU #2. The heavy gas.

portion of the GPC algorithm that computes the control gain matrices a" and if is written in MATLAB and runs All GPC testing was conducted with the pylon on CPU #1. The calculation of the control inputs u,. that oriented in the airplane mode of flight and locked to the are sent to the swashplate actuators is made by executa- wing downstop. The rotor was unpowered (wind- ble code that is installed on the DSP card. This code is milling). The stability of the model with the GPC-based algorithm was measured over a range of tunnel air- generated by using Real-Time Workshop to convert a sf_eds up to a maximum of 120 kt, for a single rotor Simulink block diagram of the control law calculation to rotational specd (888 rpm), and two values of kinematic C code. The C code then goes to the dSPACE compiler, pitch-flap coupling (8.s) angles (-15 ° and - 45°). Be- which generates the executable code that is placed on cause this was the first wind-tunnel test of the GPC al- the DSP. The user specifies appropriate values for the gorithm, a conservative approach was adopted wherein parameters 1, p, h;,, h,, w,., and Wr needed by the GPC all closed-loop testing was conducted at conditions that algorithm and initializes the control gain matrices a" and were slightly within (more stable than) the correspond- ff to zero using the ControlDesk front panel GUI that ing open-loop stability boundaries.

runs on CPU #2. The DSP card continuously collects the r-input and m-output data sets used by the GPC al- Test Procedure gorithm. On user command, the DSP sends the set of input/output data needed for system identification to The primary pylon/wing structural frequencies of CPU #1 where the SID computations are performed and interest lie between 4 and 25 Hz, with the critical wing the control gain matrices & and ff are computed. The beam mode frequency varying from 5.8 Hz down to as control gain matrices are automatically sent to the DSP, low as 4.5 Hz with increasing airspeed. Based on the which uses the p latest data sets to (continuously) com- implementation guidelines discussed earlier, the Nyquist pute the control commands to be sent to the swashplate frequency fN was taken as 30 Hz, the sampling fre- actuators. Re-identification of the system, if needed, is quency f_. was set to 100 Hz, and the number of time done closed-loop and on user command.

steps I of input/output data used for SID was taken as 300.

CPU #2 [ Select 1,p, hc, hp, w c, w r

I

t For this test, the control horizon h,. was set equal to the prediction horizon he, hp was set equal to p, the ._ Collect Imput/Oulput data (Continuous calculation)

I

order of the ARX model, and the response weight w_ was set to unity. The assumed order of the ARX model DSP Ca_ (p) and the control weight (w,.) were varied to tune the Compute Uc using the the p latest controller to the nuances of the WRATS model in the UO data se(s (Continuous calculation) on-downstop configuration with _5._ = -15 ° while operat- ing at an airspeed of 80 kt, which is well below the open-loop flutter speed of that configuration. The tun- ing process was conducted at both 770 and 888 rpm and CPU #1 Update ,ac _ I$c fed to the selection of'the fbf[owing best values for these

I-I I (0_ us_" con'cnand)

quantities: p ---30 and w,. = 1.0. Active control was in- troduced into the fixed-system swashplate using three hydraulic servo-actuators mounted aft of the swashplate Figure 6. - Distribution of computing tasks in GPC inside the pylon fairing. Wing bending (vertical and computer for active control studies.

chordwise) and torsion strain gages located near the sufficient for a wide range of rotor speeds and tunnel

wingrootwere thesensors used toprovide theresponse

velocities.

measurements needed by theactive control algorithm.

Thus, there were three control inputs (r = 3) andthree

An indication of the effectiveness of the GPC- feedback outputs (m=3).

based active control system in increasing the stability In addition toitsuse toactively control therotor (damping) of the critical wing beam mode of the model is given in figures 7 and 8. The figures show a com-

for GPCstability augmentation, theactive swashplate

wasalso used to impose "stickstirs"toexcite thewing parison of the measured open-loop and closed-loop modes of interest for damping determination. These wing beam mode damping versus airspeed for 63 values

stick-stir commands weresuperimposed on the GPC

of-45 ° and -15 °, respectively. The mean and standard

andpilotcontrol commands sent to theswashplate ac-

tuators.Control of thestickstiris integrated intothe

GUI thatis partof theWRATS DASsoftware. The

procedure employed for theGPC testing is asfollows: 20

I i i [ r I i

(1)WithGPC off,thetunnel airspeed isbrought uptoa

selected value;(2) GPCis turned on (theassociated

reduction in model response is visiblyquite dramatic);

I:GPO-Onl

GPC-OffJ

(3) Data acquisition is initiated; (4)A stick-stir excita-

Wing beam 12

tionis imposed atthefrequency of thewingmode of

interest; (5)The resulting wing beam (orother) response mode

is observed visually by thetestengineer andthestir

damping, 8

terminated when sufficient amplitude hasbeen attained; % critical

(6)The response isallowed todecay freely toitssteady-

state value ; and (7)Data acquisition is turned off. The

total data acquisition timeistypically 5-8seconds long.

4" I " • - -

Thefree-decay portion of therecorded timehistory is

45 55 65 75 85

thenanalyzed by movingblockandlog decrement

Velocity, kts

methods to obtain anestimate for themodal damping

andthedamped natural frequency. Thisprocess is re-

Figure 7. - Comparison of wing beam mode damping

peated untilfive acceptable estimates of thedamping

versus airspeed with GPC on and off for 8_ = -45 °.

areobtained. GPC is then turned off and the entire pro-

cedure beginning with step (1) is repeated for all of the airspeeds of interest.

I i i i F i q The performance and robustness of the GPC al- [ [ o-GPC-On gorithm were assessed in transient conditions for 8._ = -15 ° by rapidly changing tunnel airspeed and rotor rota- tional speed, both singly and in combination. The rotor Wing 8 beam speed was varied between 770 and 888 rpm and the tunnel airspeed between 60 and 145 kt. mode 6 damping, % critical 4 RESULTS AND DISCUSSION The GPC-based active control system was found highly effective in increasing the stability (damping) of 80 100 120 140 6O the critical wing mode for all of the configurations of Velocity, kts the model tested. In particular, GPC was able to consis- tently yield a closed-loop system in which the critical Figure 8. - Comparison of wing beam mode damping wing mode had a minimum of about four-percent modal versus airspeed with GPC on and off for 8, = -15 °.

damping over the range of test conditions investigated, without visible degradation of the damping in the other modes. The algorithm was robust with respect to its deviation of the five values of damping that were meas- performance in the tracking of rapid changes in both the ured at each airspeed were determined. The mean val- rotor speed and the tunnel airspeed. System identifica- ues of damping are indicated by the symbols, and the tion done at a low-speed flight condition was generally vertical lines indicate the standard deviations about

these mean values. Theplotted curves aretheresult ofa 3000 _ T _ ! [

I i , [_-- GPC Off spline fit applied to themean values. Ascanbeseen,

withGPCturned onthewingbeam mode damping is

considerably higher thanwithGPC off overtheentire

range oftunnel airspeeds tested. It should beremarked

thatwithGPC on,it wasoften difficultto estimate the Wing

beam 1000

damping fromthefree-decay responses aftertermina-

bending

tionof thestick-stir because thelarge damping levels

moment,

resulted in very fewcycles of motion (sometimes asfew

asoneor two)being available forthedamping calcula- in-lbs 0

tions.Thisisthereason forthelarge standard deviation

inthemeasured values oftheclosed-loop damping. In

contrast, because theopen-loop damping of thewing

-1000 ' ' J _

beam mode is small, thestandard deviation ofthemeas- 0 1.5 3.0 4.5

Time, eec

ured damping is correspondingly verysmall.It should

alsobe remarked thatthe effectiveness of GPCwas

Figure 10. - Effect of GPC on damped response of wing

such thatit triedto quelltheimposed stick-stir excita-

beam bending mode at 120 kt for 5_ = -15 °.

tion,thusrequiring considerable amplitude of excita-

tion. Thereason for thedrop-offandthenrisein

closed-loop damping at 100knots in figure8 is not

known. Another example of the effectiveness of GPC is

given in figure I l, which shows a time history of the

Themeasured timehistories of thewingbeam wing beam bending moment while repeatedly cycling

GPC on and off for the 120-knot airspeed case of figure

bending moments duringandaftera typical stick-stir

excitation withGPConandoff areshown in figures 9 8. The response is seen to adapt quickly to the changing conditions, suggesting a certain level of robustness.

and10.Figure 9 corresponds tothe75-knot airspeed of

figure 7 andfigure10corresponds tothe120-knot air-

speed of figure 8. The closed-loop time histories clearly

illustrate theeffectiveness of theGPC-based active con-

1300 [ trolsystem torapidly reduce theresponse.

1- Wing 900 beam bending lO00 moment, 700 in-lb Wing beam

S°°l-

bending 0 3000 1 2 3 4 5 6 7 8 moment, Time, sec in-lbs -500 Figure I I. - Time history of wing beam bending mo- ment while cycling GPC on and off at 120 kt for 83 = -15 °.

.1000 0 I I J I I 1 2 3 4 5 6 7 8 Time, sec An indication of the swashplate pitch angles that are associated with the GPC commanded actuator inputs Figure 9. - Effect of GPC on damped response of wing is given in figure 12. The figure shows the time history beam bending mode at 75 kt for 53 = -45 °.

of the longitudinal and lateral cyclic pitch angles for the 120-knot airspeed condition of figure 8 from a time near theendof a stick-stir excitation toa timewellintothe 3) All system identification and control law computations were done on-line, permitting rapid adap-

steady-state condition. Thesteady-state oscillatory cy-

tation to changing conditions.

clicpitchangles are seen tobemodest.

4) The swashplate angles required were modest,

2.0

generally less than 0.3 degrees.

These results, in combination with equally suc-

1.0

cessful results obtained in an earlier ground resonance test, suggest that a GPC-based active control system is a Swashplate 0 viable candidate ibr stability augmentation in advanced cyclic pitch st_c,,k GPC only -stir tiltrotor systems. Additional wind-tunnel tests of the angles, WRATS model are planned to evaluate the GPC meth- deg -1.0

lLIIlt ,, ,+ra, oyc,,c

odology over a broader range of operating conditions for the baseline stiff-inplane gimbaled rotor, as well as -2.0 an advanced soft-inplane design.

-3.0 0 1 2 3 4 5 6 7 8 REFERENCES Time, sec 1. Hall, W. E.: Prop-Rotor Stability at High Advance Figure 12. - Measured time histories of closed-loop Ratios. Journal of the American Helicopter Society,, swashplate cyclic pitch angles at 120 kt for 53 = -15 +.

June 1966, pp. 11-26.

CONCLUSIONS 2. Edenborough, H. K.: Investigation of Tilt-Rotor VTOL Aircraft Rotor-Pylon Stability. Journal of Air- cmt% Vol. 5, No. 6, March-April 1968, pp. 97-105.

A joint NASA/Army/Bell Helicopter Textron wind-tunnel investigation was conducted in a prelimi- 3. Gaffey, T. M.; Yen, J. G.; and Kvaternik, R. G.: nary assessment of the potential of generalized predic- tive control (GPC) for active stability augmentation in Analysis and Model Tests of the Proprotor Dynamics of a Tilt-Proprotor VTOL Aircraft. Air Force V/STOL tiltrotor aircraft operating in the airplane mode of flight.

The studies were made in the Langley Transonic Dy- Technology and Planning Conference, Las Vegas, NV, September 23-25, 1969.

namics Tunnel using an unpowered I/5-scale semispan aeroelastic model of the V-22 modified to incorporate 4. Kvaternik, R. G.: Studies in Tilt-Rotor VTOL Air- an adaptive, GPC-based, MIMO active control algo- craft AeroelasticiO,. Ph.D. Dissertation, Case Western rithm to individually control the actuators of the fixed- Reserve University, June 1973.

system swashplate. Closed-loop stability of the model resulting from GPC inputs computed using feedback 5. Johnson, W.: Dynamics of Tilting Proprotor Air- from wing-root strain gages was measured over a range craft in Cruise Flight. NASA TN D-7677, May 1974.

of steady tunnel airspeeds for a single rotor rotational speed and two values of kinematic pitch-flap coupling.

6. Alexander, H. R.; Amos, A. K.; Tarzanin, F. J.; Performance of the system was also assessed in tran- and Taylor, R. B.: V/STOL Dynamics and Aeroelastic sient conditions by rapidly changing tunnel velocity and Rotor-Airframe Technology. Volume I1: Description rotor speed about a nominal operating condition. Based and Correlation of New Methodologies. AFFDL-TR- on the results obtained in this investigation, the follow- 72-40 (Volume II), September 1972.

ing conclusions are indicated: 7. Popelka, D.; Lindsay, D.; Parham, T.; Berry, V.; 1) The GPC algorithm employed was highly ef- and Baker, D.: Results of an Aeroelastic Tailoring Study fective in increasing the stability (damping) in the criti- for a Composite Tiltrotor Wing. American Helicopter cal wing mode of the model tested.

Society, 51" Annual Forum, Forth Worth, TX, May 9-11, 1995.

2) The GPC algorithm employed was also robust with respect to rapid changes in both the rotor speed and 8. Corso, L. M.; Popelka, D. A.; and Nixon, M. W.: the airspeed, either singly or in combination.

Design, Analysis, and Test of a Composite Tailored Tiltrotor Wing. American Helicopter Society 53 m An- nual Forum, Virginia Beach, VA, April 29 - May 1, 19. Richalet, J.; Rault, A.; Testud, J. L.; and Papon, J.: 1997.

Model Predictive Heuristic Control: Applications to Industrial Processes. Automatica, Vo[. 14, 1978, pp.

413-428.

9. Nixon, M. W.; Piatak, D. J.; Corso, L. M.; and Popelka, D. A.: Aeroelastic Tailoring for Stability Augmentation and Performance Enhancements of 20. Clark, D. W. (Ed): Advances in Model-Based Pre- Tiltrotor Aircraft. American Helicopter Society 55 _h dictive Control. Oxford University Press, New York, 1994.

Annual National Forum, Montreal, Canada, May 25-27, 1999.

21. Juang, J.-N., Applied System Identification, Pren- 10. Barkai, S. M; and Rand, O.: The Influence of tice Hall, Inc., Englewood Cliffs, New Jersey, 1994.

Composite Induced Couplings on Ti[trotor Whirl Flutter Stability. Journal of the American Helicopter Society, 22. Eure, K.W.; and Juang, J.-N.: Broadband Noise Volume 43, No. 2, April 1998, pp. 133-145. Control Using Predictive Techniques. NASA TM 110320, January 1997.

l I. Srinivas, V.; Chopra, I.; and Nixon, M. W.: Aeroe- lastic Analysis of Advanced Geometry Tiltrotor Air- 23. Juang, J.-N.: State-Space System Realization With craft. Journal of the American Helicopter Society, Vol. Input- and Output-Data Correlation. NASA TP 3622, 43, No. 3, July 1998, pp. 212-221. April 1997.

12. Acree, C. W., Jr.; Peyran, R. J.; and Johnson, W.: 24. Juang, J.-N.; and Phan, M. Q.: Recursive Deadbeat Rotor Design for Whirl Flutter: An Examination of Op- Controller Design. NASA TM 112863, May 1997 tions for Improving Tiltrotor Aeroelastic Stability Mar- (also, Journal of Guidance, Control, and Dynamics, gins. American Helicopter Society 55 tl' Annual Forum, Vol. 21, No. 5, Sept.-Oct. 1998, pp. 747-753).

Montreal, Quebec, Canada, May 25-27, 1999.

25. Juang, J.-N.; and Phan, M. Q.: Deadbeat Predic- 13. Alexander, H. R.; Eason, W.; Gillmore, K.; Mor- tive Controllers. NASA TM 112862, May 1997.

ris, J.; and Spittle, R.: V/STOL Tilt Rotor Aircraft Study.

Volume 7: Tilt Rotor Flight Control Program Feedback 26. Eure, K. W.: Adaptive Predictive Feedback Tech- Studies. NASA CR- 114600, March 1973.

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14. Nasu, K.: Tilt-Rotor Flutter Control in Cruise Flight. NASA TM 88315, December 1986.

27. Phan, M. Q.; and Juang, J.-N.: Predictive Control- 15. Van Aken, J. M.: Alleviation of Whirl-Flutter on lers for Fcedback Stabilization. Journal of Guidance, Tilt-Rotor Aircraft Using Active Control. American Control, and Dynamics, Vol. 21, No. 5, Sept.-Oct. 1998, Helicopter Society 47 rj' Annual Forum, Phoenix, AZ, pp. 747-753.

May 6-8, 199 I, pp. 1321-1344.

28. Juang, J.-N.; and Eure, K. W.: Predictive Feed- 16. Vorwald, J. G.; and Chopra, I.: Stabilizing Pylon back and Feedforward Control for Systems with Un- Whirl Flutter on a Tilt-Rotor Aircraft. 32 ''t known Disturbances. NASA/TM-1998-208744, De- cember 1998.

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Baltimore, MD, 29. Phan, M. Q.; and Juang, J-N: Predictive Feedback April 8-10, 1991 (Paper AIAA-91-1259-CP).

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Vol. 23, No. 2, 1987, pp. 137-160.

30. Settle, T. B.; and Kidd, D. L.: Evolution and Test 18. Kvaternik, R. G.; Juang, J., and Bennett, R. L.: History of the V-22 0.2-Scale Aeroelastic Model. Jour- nal of the American Helicopter Society, Vol. 37, No. 1, Exploratory Studies in Generalized Predictive Control for Active Aeroelastic Control of Tiltrotor Aircraft.

January 1992, pp. 31-45.

American Helicopter Society, Northeast Region Special- ists' Meeting on Active Controls Technology, Bridge- port, CT, October 4-5, 2000 (Paper is also available as NASA/TM-2000-210552).