October 2000

NASA/TM-2000-210552

Exploratory Studies in Generalized

Predictive Control for Active Aeroelastic

Control of Tiltrotor Aircraft

Raymond G. Kvaternik and Jer-Nan Juang

Langley Research Center, Hampton, Virginia

Richard L. Bennett

Bell Helicopter Textron, Inc., Fort Worth, Texas
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October 2000

NASA/TM-2000-210552

Exploratory Studies in Generalized

Predictive Control for Active Aeroelastic

Control of Tiltrotor Aircraft

Raymond G. Kvaternik and Jer-Nan Juang

Langley Research Center, Hampton, Virginia

Richard L. Bennett

Bell Helicopter Textron, Inc., Fort Worth, Texas
Available from:

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)

7121 Standard Drive 5285 Port Royal Road

Hanover, MD 21076-1320 Springfield, VA 22161-2171

(301) 621-0390 (703) 605-6000

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and Space Administration.

This paper was presented at the American Helicopter Society Northeast Region Specialists' Meeting on

Active Controls Technology, Bridgeport, CT, October 4-5, 2000. The full paper, which consisted of a set

of charts and a text explanation of each chart, is reproduced in this report.
iii

Exploratory Studies in Generalized Predictive Control

for Active Aeroelastic Control of Tiltrotor Aircraft∗

Raymond G. Kvaternik and Jer-Nan Juang

NASA Langley Research Center

Hampton, VA

and

Richard L. Bennett

Bell Helicopter Textron, Inc

Fort Worth, TX

Abstract

The Aeroelasticity Branch (AB) at NASA Langley Research Center has a

long and substantive history of tiltrotor aeroelastic research. That research has

included a broad range of experimental investigations in the Langley Transonic

Dynamics Tunnel (TDT) using a variety of scale models and the development of

essential analyses. Since 1994, the tiltrotor research program has been using a

1/5-scale, semispan aeroelastic model of the V-22 designed and built by Bell

Helicopter Textron Inc (BHTI) in 1983. That model has been refurbished to form

a tiltrotor research testbed called the Wing and Rotor Aeroelastic Test System

(WRATS) for use in the TDT. In collaboration with BHTI, studies under the

current tiltrotor research program are focused on aeroelastic technology areas

having the potential for enhancing the commercial and military viability of

tiltrotor aircraft. Among the areas being addressed, considerable emphasis is

being directed to the evaluation of modern adaptive multi-input multi-output

(MIMO) control techniques for active stability augmentation and vibration

control of tiltrotor aircraft. As part of this investigation, a predictive control

technique known as Generalized Predictive Control (GPC) is being studied to

assess its potential for actively controlling the swashplate of tiltrotor aircraft to

enhance aeroelastic stability in both helicopter and airplane modes of flight. This

paper summarizes the exploratory numerical and experimental studies that were

conducted as part of that investigation.

 

∗

 Presented at the American Helicopter Society Northeast Region Active Controls Technology Conference,

Bridgeport, CT, October 4-5, 2000.
1

Exploratory Studies in Generalized Predictive Control

for Active Aeroelastic Control of Tiltrotor Aircraft

Raymond G. Kvaternik and Jer-Nan Juang

NASA Langley Research Center

and

Richard L. Bennett

Bell Helicopter Textron

Presented at the

American Helicopter Society Northeast Region

Active Controls Technology Conference

Bridgeport, CT

October 4-5, 2000

Outline

● Introductory remarks

● Theoretical development

● Features characterizing method

● Computational considerations

● Implementation considerations

● Illustrative numerical and experimental results

● Concluding remarks

● Status and plans
2

Some Relevant Background

● AB/TDT has a long history of tiltrotor aeroelastic research

● Research has included a broad range of experimental

investigations and the development of essential analyses

● Since 1994, tiltrotor research program has been using a

1/5-scale, semispan aeroelastic model of the V-22

● AB/BHTI cooperative study of promising tiltrotor

aeroelastic technology areas underway since 1994

● Evaluation of modern adaptive MIMO control techniques

for active stability augmentation a major activity

● GPC-based method for actively controlling swashplate to

enhance stability under investigation since 1997

The Aeroelasticity Branch (AB) at NASA Langley Research Center has

a long and substantive history of tiltrotor aeroelastic research (ref. 1). That

research has included a broad range of experimental investigations in the

Langley Transonic Dynamics Tunnel (TDT) using a variety of scale models

and the development of essential analyses. Since 1994, the tiltrotor research

program has been using a 1/5-scale, semispan aeroelastic model of the V-22

designed and built by Bell Helicopter Textron Inc (BHTI) in 1983. That model

has been refurbished to form a tiltrotor research testbed called the Wing and

Rotor Aeroelastic Test System (WRATS) for use in the TDT. In collaboration

with BHTI, studies under the current tiltrotor research program are focused on

aeroelastic technology areas having the potential for enhancing the commercial

and military viability of tiltrotor aircraft. Among the areas being addressed,

considerable emphasis is being directed to the evaluation of modern adaptive

multi-input multi-output (MIMO) control techniques for active stability

augmentation and vibration control of tiltrotor aircraft. As part of this

investigation, a predictive control technique known as Generalized Predictive

Control (GPC) is being studied to assess its potential for actively controlling

the swashplate of tiltrotor aircraft to enhance aeroelastic stability in both

helicopter and airplane modes of flight. This paper summarizes the exploratory

numerical and experimental studies that were conducted as part of that

investigation.
3

Approaches to Control

● Conventional control approach

– System modeling and verification (off-line)

– Controller design and verification (off-line)

● Adaptive control approach

– On-line/real-time system identification

– On-line/real-time controller design

– User specification of model order, horizons, and control weights

● Autonomous control approach

– System identification/controller design combined into one step

– Automated estimation of system order and disturbances

– Automated adjustment of control weights and control gains

– All done on-line and in real-time

The approaches to active control may be grouped into three general

categories: conventional control, adaptive control, and autonomous control.

Conventional approach: Uses an explicit plant model represented by a

transfer function or state-space model. Determination and verification of the

system’s transfer function and the design of the controller are done off-line.

LQR, H∞, H2DQG V\QWKHVLVDUHH[DPSOHVRIVXFKDSSURDFKHV

Adaptive approach: The term adaptive as used here refers to a real-time

digital system that updates the system identification (SID) parameters based on

input/output data, uses the updated system parameters to compute a new set of

control parameters, and then computes the next set of commands to be sent to

the actuators, with all computations being done on-line. The user specifies the

order of the system model, the prediction and control horizons, and the control

weights.

Autonomous approach: SID and controller design are combined into one

step. Estimation of system order and disturbances, and adjustment of control

weights and control gains (e.g., via fuzzy logic, neural nets, and genetic

algorithms) are all automated. All computations are done on-line and in realtime. Autonomous control is a very active area of research at this time.

The work addressed in this paper falls into the second category.
4

Generalized Predictive Control

● Belongs to class of model-based predictive controllers used

in process control industries since late 1970s

● Method introduced in 1987 by Clarke of Oxford University

● Input-output map of system used to form multi-step output

prediction equation over a finite prediction horizon while

subject to controls imposed over a finite control horizon

● Optimal control sequence determined by minimizing

deviation of future system outputs from desired (target)

response, subject to penalty on control power

● A novel version of GPC algorithm developed at NASALangley in 1997 for efficient computation and unknown

disturbance rejection

Predictive controllers introduced in the chemical industries for controlling

chemical processes have found applications in a wide variety of industrial processes

(e.g., ref. 2). Predictive control refers to a strategy wherein the decision for the

current control action is based on minimization of a quadratic objective function that

involves a prediction of the system response some number of time steps into the

future. A variety of predictive controllers have been proposed (e.g., ref. 3). Among

these, Generalized Predictive Control (GPC), which was introduced in 1987 (refs. 4-

5), has received notable attention by researchers. GPC is a time-domain multiinput-multi-output (MIMO) predictive control method that uses a finite-difference

representation for the input-output map of the system. The input-output equation is

used to form a multi-step output prediction equation over a finite prediction horizon

while subject to controls imposed over a finite control horizon. The control to be

imposed at the next time step is determined by minimizing the deviation of the

predicted controlled plant outputs from the desired (or target) outputs, subject to a

penalty on control effort.

A novel version of the GPC procedure was developed at NASA Langley

Research Center in 1997 for efficient computation and unknown disturbance

rejection by Dr. Jer-Nan Juang and his associates. Their work has resulted in a suite

of MATLAB m-files that have been collected into a Predictive toolbox that can be

used by researchers for GPC studies. A summary of the SID and control theory

underlying their development is found in references 6-13, among others. These

references were the primary sources of information for the work reported in this

paper. 
5

Adaptive Control Process

System Identification

&Disturbance Estimation

Predictive Control

(Feedback/Feedforward)

Input (u)

Disturbance (d)

Output (y) Plant

uc

uid

The essential features of the adaptive control process used in the present

GPC investigation are depicted in the diagram above. The system (plant) has r

control inputs u, m measured outputs y, and is subject to unknown external

disturbances d. Measurement noise is also present. There are two fundamental

steps involved: (1) identification of the system; and (2) use of the identified

model to design a controller. A finite-difference model in the form of an AutoRegressive moving average with eXogenous input (ARX) model is used here.

This model is used for both system ID and controller design. System

identification is done on-line in the presence of any disturbances acting on the

system, as indicated in the center box of the diagram. In this way, an estimate of

the disturbance model is reflected in the identified system model and does not

have to be modeled separately. This approach represents a case of feedback with

embedded feedforward. Because the disturbance information is embedded in the

feedforward control parameters, there is no need for measurement of the

disturbance signal (ref. 13). The parameters of the identified model are used to

compute the predictive control law. A random excitation uid (sometimes called

dither) is applied initially with uc equal to zero to identify the open-loop system.

Dither is added to the closed-loop control input uc if it is necessary to re-identify

the system while operating in the closed-loop mode.
6

Form of Model and Control Law Equations

● ARX model used for system identification

● Control law equation

yk yk yk yk p

uk uk uk p

p

p

( ) = aa a

bb b

1 2

0 1

1 2

1

( - ) + ( - ) + + ( - ) 

 + ( ) + ( - ) + + ( - )

L

L

u k yk yk yk p

uk yk uk p

c c

p

c

c c

p

c

c ( - ) + ( - ) + + ( - ) 

 + ( - ) + ( - ) + + ( - )

( ) = aa a

bb b

1 2

1 2

1 2

1 2

L

L

The relationship between the input and output time histories of a MIMO

system are described by the time-domain AutoRegressive eXogenous (ARX)

finite-difference model shown in the first equation. This equation states that the

current output y(k) at time step k may be estimated by using p sets of the

previous output and input measurements, y(k-1),..., y(k-p) and u(k-1),...,u(k-p);

and the current input measurement u(k). The integer p is called the order of the

ARX model. The coefficient matrices i and i appearing in this equation are

referred to as observer Markov parameters (OMP) or ARX parameters and are

the quantities to be determined by the identification algorithm. Closed-loop

robustness is enhanced by performing the system identification in the presence

of the external disturbances acting on the system, thereby ensuring that

disturbance information will be incorporated into the system model. The goal of

SID is to determine the OMP based on input and output data. The OMP may be

determined by any SID techniques that returns an ARX model of the system.

The ARX model is used to design the controller and leads to a control

law that in the case of a regulator problem has the general form given by the

second equation. This equation indicates that the current control input uc(k) may

be computed using p sets of the previous input and output measurements. The

coefficient matrices and appearing in this equation are the control gain

matrices. The derivation of the parameters appearing in the SID and control law

equations is described below.

a i

c

b i

c
7

System Identification

● The digitized input and output time histories at l time

 points are used to form the data matrices y and V in

 the equation

 according to

y YV =

[ ( )]

(0) (1) (2) ( ) ( 1)

u(0) (1) (2) ( ) ( 1)

(0) (1) ( 1) ( 2)

V= (0) ( 2) ( 3)

(0) ( 1)

m l

r r mp l

y y y y y p yl

u u u p ul

v v v p vl

v v p vl

v vl p

   

×

 

 

 

++ ×

= −

−

− −

− −

− −

L L

L L

L L

L L

OML M

L

System identification in the presence of the operational disturbances

acting on the system is the first of the two major computational steps. The

external disturbances 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 order of the ARX model (p) and the number of time steps (l)

must be specified. Some guidelines for their selection are given later.

The system is excited with band-limited white noise for SID. These

random excitations are input to all r control inputs simultaneously and the m

responses measured. The digitized input and output time histories (u and y) at l

time points are then used to form the data matrices y and V as indicated in the

slide. The sizes of the resulting arrays are noted. The equation shown follows

from writing the discrete-time state-space equations for a LTI system at a

sequence of time steps k = 0, 1, ..., l-1 and grouping them into the above matrix

IRUP7KHYHFWRU LVGHILQHGRQWKHQH[WVOLGH

In forming the matrices above, it has been assumed that the state matrix A

is asymptotically stable so that for some sufficiently large p, Ak ≈ 0 for all time

steps k ≥ p, and that a observer has been added to the system. It is through these

expedients that the matrix V is reduced to a size amenable for practical

numerical computation of its pseudo-inverse. The SID process yields OMP

rather than system Markov parameters (SMP) because of the inclusion of an

observer. A complete discussion of these aspects of the development may be

found in reference 6.
8

The vector N that appears in the data matrix V is formed from the

vectors u(k) and y(k) as indicated above. Because of the introduction of an

observer into the discrete state equations for the system (ref. 6), the quantities i

and i in are referred to as observer Markov parameters (OMP). The

solution for , the vector of observer Markov parameters, is obtained by solving

the equation y = * V for as shown in the bottom equation in the slide

above. If the product VVT

 is a well-conditioned matrix, the ordinary inverse can

be taken as shown. Otherwise, a pseudo-inverse must be used. It should be

noted that because the size of VVT

 is much smaller than V, a pseudo-inverse may

be appropriate even if the product is well conditioned.

Y

Y

Y

Y

System Identification (Concluded)

● The vector in the matrix is formed according to

● is the matrix of observer Markov parameters (OMP)

which are to be identified and has the form

● The solution for is given by

v k( ) V

( )1

( ) ( ) ( )

r m

u k v k y k

     

   

+ ×

=

Y

0112 2 3 3 p

[ ( )]

 p

m r m r m mm r m m m r m m m r m m

m r r mp

Y   ββαβα βα β α  

× ×× ×× ×× × ×

×++

= L

Y

-1 † y y = T T Y V V VV =    

where † denotes the pseudo-inverse.
9

Derivation of Multi-Step Output Prediction Equation

 The input-output map is given by the ARX equation

yk yk yk yk p

uk uk uk p

p

p

( ) = aa a

bb b

1 2

0 1

1 2

1

( - ) + ( - ) + + ( - ) 

 + ( ) + ( - ) + + ( - )

L

L

Using this equation, the output at time step k+j can be written as

where the coefficient matrices are combinations of αi and βi

() () ( )

1 2

(1) ( )

0 o

( ) (j) ( )

1 2

( ) ( -1) + ( -2) + + ( - ) 

 + ( )+ ( 1) ( )

 + ( -1)+ ( 2) + ( - )

j j j

p

j

o

j j

p

yk j yk yk ykp

uk j uk j uk

uk uk ukp

αα α

ββ β

ββ β

+ =

+ +− + +

− +

L

L

L

The one-step ahead ARX equation describing the input-output map of the

system is the starting point for deriving the multi-step output prediction equation

that is needed for designing the MIMO controller. Using the one-step ahead

equation shown at the top of the slide, the output at time step k+j may be written

in the form shown in the second equation, where the coefficient matrices are

given by recursive expressions involving the matrices i and i appearing in the

ARX equation (ref. 10). The matrices i and i are determined by the system

identification process described earlier.

The second equation states that the output y(k+j) at time step k+j may be

estimated by using p sets of the previous output and input measurements,

y(k-1),..., y(k-p) and u(k-1),...,u(k-p), and the (unknown) current and future

inputs, u(k), u(k+1),...,u(k+j). Letting j range over the set of values j = 1, 2,

..., hp-1, the resulting equations can be assembled into a multi-step ahead output

prediction equation having the form indicated in the next slide.
10

Derivation of Prediction Equation (Concluded)

Letting j range over the set of values j = 1, 2, ..., q, ..., hp-1, the resulting

equations can be assembled into the multi-step output prediction equation:

Current &

Future Output Past Input

Past Output

Current &

Future Input

1 1 1 1

() () ( ) ( )

p c pc p p

p c

h hp p h m x h r h m x pr h m x pm h mx hrx pr x pm x

y k u k uk p yk p = + −+ − WE D

where the coefficient matrices are formed from combinations

of the observer Markov parameters αi

 and βi appearing in the ARX

equation describing the input-output map of the system, and where hp is

the prediction horizon, hc is the control horizon, and hc ö hp.

W ED , ,

GPC is based on system output predictions over a finite horizon hp known

as the prediction horizon. To predict future plant outputs, some assumption

needs to be made about future control inputs. In determining the future control

inputs for GPC, it is assumed that the control is applied over a finite horizon hc

known as the control horizon. Beyond the control horizon, the control input is

assumed to be zero. In GPC, the control horizon is always equal to or less than

the prediction horizon.

The general form of the multi-step ahead prediction equation indicating the

sizes of the constituent matrices is shown in this slide. The coefficient matrices

 , , and are formed from combinations of the observer Markov parameters

i and i appearing in the ARX equation describing the input-output map of the

system. The objective is to predict the output for hp time steps ahead, given the

input for hc steps ahead, where hc  hp. The quantity yhp(k) is the vector

containing the predicted future outputs, whereas uhc(k) is the vector containing

the future control inputs yet to be determined. The quantities up(k-p) and yp(k-p)

are vectors containing the previous p sets of control inputs and outputs,

respectively.

W E D
11

The expanded form of the multi-step output prediction equation for the

case in which the control horizon is less than the prediction horizon is shown in

this slide. The OMP i and i determined in system identification form the first

block rows of the coefficient matrices , , and in the equation. The terms

in the remaining rows are computed using the recursive relations indicated in the

boxes (ref. 10). All terms in the above equation are known, except for the hc

sets of future commands, and the hp sets of predicted responses. The goal of the

GPC control algorithm is to determine the set of future commands u(k), u(k+1),

..., u(k+hc-1) that are required to achieve a desired predicted response y(k),

y(k+1), ..., y(k+hp-1).

It should be remarked that the system Markov parameters (SMP), which

are commonly used as the basis for identifying discrete-time models for linear

dynamical systems, form the first block column in the matrix ; the remaining

block columns are formed from subsets of the SMP. The Markov parameters

are the pulse response of a system and are unique for a given system. The

discrete-time state-space matrices A, B, C, and D are embedded in the SMP.

The multi-step output prediction equation is used to define an objective

function whose minimization with respect to uhc(k) leads to the control law from

which a vector of future control inputs can be computed using the p sets of

previous control inputs and measured outputs.

W D E

W

(1)

( 1)

( )

( 1)

( 2)

( 1) (1)

( 2

1

) (

2

)

( )

( 1) ( )

( 1) ( 1)

( ) ( 1)

( 1) p p p c

o

q

o

q

o

h

o

o

q

o o

q

o o

c

h hh

p o

o

p

o

y k

y k u k

u k yk q

yk q uk h

yk h

β

β β

β β

β

β

αα α

β

β

β

β β

−

−

−

−

− −

−

   

    +            +    + − =

   +

  + −  

  + −    

+

M M MO

L M L

M M MO M

L

L

(1) (1) (1) (1)

12 1

( 1) ( 1) ( 1) ( 1)

12 1

() () () ()

12 1

( 1) ( 1) ( 1) ( 1)

1 2

1 2

1

1

( 1)

( 2)

( 1)

( )

pp pp

p p

qq qq

p p

qq qq

p p

hh hh

p p

p

y k

y k

yk p

yk p

αα αα

αα αα

αα αα

αα α

α

α

β β

−

−− −−

−

−

−− −−

−

 

    −

  −  

  +

  − +  

    −

 

L

L

M MOM M

L M

L

M MOM M

K

(1) (1) (1) (1)

12 1

( 1) ( 1) ( 1) ( 1)

1 2

() () () ()

12 1

( 1) ( 1) ( 1) ( 1)

1 2

1

1

( 1)

( 2)

( 1)

( )

pp pp

p p

qq qq

p p

qq qq

p p

h h

p

p p

p

h h

u k

u k

uk p

uk p

ββ ββ

ββ ββ

ββ ββ

ββ ββ

β β

−

−− −−

−

−− −−

−

  −

    −

  −  

 

  − +  

    −

 

L

M MOM M

L M

L

M MOM M

K

Expanded Form of Multi-Step Output Prediction Equation

1 ( 1)

1

( ) ( 1)

1 − − −− = + p

q q

p

q

p α α α α

0 ( 1)

1

( 1)

1

( )

0 β β α β − − = + q q q

1 ( 1)

1

( ) ( 1)

1 − − −

− = + p

q q

p

q

p β β α β

h h c p <
12

Derivation of Control Law

c c

T T

h h JR Q = + ε ε u u

Define an error function :

() () () () ( ) ( ) TT p p c h h p ε = − = − − −− − yy y u u y k k k k kp kp WE D

Define an objective function J:

Minimize J with respect to :

( ) †

1

() ( ) () ( ) ( )

C

c

T T

h Tp p

h r

u k R Q R yk uk p yk p

×

=− + − + − + − WW W E D

Retain the first component (first r rows) of :

1

() () ( ) ( ) Tp p

cc c

c

r

u k yk uk p yk p γβ α

×

=− + − + −

( ) hc u k

( ) hc u k

The predictive control law is obtained by minimizing the deviation of the

predicted controlled response (as computed from the multi-step output

prediction equation) from a specified target response over a prediction horizon

hp. To this end, one first defines an error function that is the difference between

the desired (target) response yT (k) and the predicted response yhp(k) and forms

an objective function J quadratic in the error and the unknown future controls.

Two weighting matrices are included in the objective function: Q (symmetric

and positive-definite ) is used to limit the control effort and stabilize the closedloop system; R (symmetric and positive-semidefinite) is used to weight the

relative importance of the differences between the target and predicted

responses. Typically, Q and R are assumed to be diagonal and for Q to have the

same value wc along its diagonal and R to have the same value wr along its

diagonal. Minimizing J with respect to uhc(k) and solving for uhc(k) gives the

control sequence to be applied to the system over the next hc time steps. The

first r values (corresponding to the first future time step) are applied to the r

control inputs, the remainder are discarded, and a new control sequence is

calculated at the next time step.

The target response is zero for a regulator problem and non-zero for a

tracking problem. Q must be tuned to ensure a stable closed-loop system.

Typically, hc is chosen equal to hp. However, hc may be chosen less than hp

resulting in a more stable and sluggish regulator.
13

Features Characterizing Method

● Predictive controllers employ a strategy wherein current

control actions are based on prediction of system response

at a number of time steps into the future

● Approach allows one to work directly with an input-output

(ARX) model of the system

● SID process employs an observer to enable numerical

computation of pseudo-inverse for calculation of OMP

● Controller is thus inherently observer-based but no explicit

consideration of its presence needs to be taken into account

● System identification is performed in the presence of any

disturbances acting on system

Predictive controllers such as GPC employ a strategy wherein the values of

the current control inputs are based on the predicted responses at a number of

time steps into the future. Such an approach lends itself to the use of an ARX

model to represent the system for both system identification and controller

design. The SID process used makes recourse to an observer to enable

numerical computation of the pseudo-inverse needed for calculation of the OMP

that comprise the coefficients of the ARX equation. The controller is thus

inherently observer-based but no explicit consideration of the observer needs to

be taken into account in the implementation.

In practice, the disturbances acting on the system are unknown or

unmeasurable. However, as discussed in reference 13, by performing the SID in

the presence of the external disturbances acting on a system, a disturbance

model is implicitly incorporated into the identified observer Markov parameters.

However, the identified model must be larger than the true system model to

accommodate the unknown disturbances.
14

Features Characterizing Method (Continued)

● Disturbance model is implicitly incorporated into the

identified coefficients (OMP) of the ARX model

● Effects of unknown disturbances embedded in coefficient

matrices and (and hence and ) because SID

done with disturbances acting on system

● Estimated order of ARX model used for system ID given

by

● Prediction and control horizons are set according to

c α c β

p ceil

n n

m

system states disturbance states   +



 





D E

h p hh p cp  

The disturbance model is implicitly incorporated into the identified

coefficients (OMP) of the ARX model. Thus, the effects of the unknown

disturbances acting on the system are embedded in the matrices and , and

hence the control law matrices c and c.

An expression for estimating the order of the ARX model that is to be used

for SID is given in the slide. The number of system states is typically chosen to

be twice the number of significant structural modes; the number of disturbance

states is set to twice the number of frequencies in the disturbance; m is the

number of output measurements. If measurement noise is of concern, the order

of p so computed should be increased to allow for computational poles and zeros

to improve system identification 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 horizons are set according to the relations

indicated at the bottom of the slide. Although hp can be set equal to p, hp is

typically set greater than p to help limit control effort so as not to saturate the

control actuators. If the control horizon is greater than the system order a

minimum energy (minimum norm) solution is obtained wherein the commanded

output is shared so that the control actuators don’t fight each other. If hp is set

equal to p one obtains a so-called deadbeat controller (ref. 10).

D E
15

Features Characterizing Method (Concluded)

● Extending hc and hp (and hence p) to very large values,

GPC solution approaches LQR solution; hence GPC

approximates an optimal controller for large p

● Matrices Q and R are control effort and tracking error

weighting matrices

● Matrix (and hence ) is invariant if system does not

change because it is formed from SMP (which are unique)

● Solution for obtained by an efficient computational

procedure

W c γ

Y

By extending the prediction and control horizons (and hence p, the order of

the ARX model) to very large values, the GPC solution approaches that of the

linear quadratic regulator (LQR). Thus, GPC approximates an optimal

controller for large p (ref. 5).

Weighting matrices Q and R are used to limit the control effort and to

weight the relative importance of the differences between the target and

predicted responses, respectively. As mentioned earlier, Q and R are usually

assumed to be diagonal and for Q to have the same value wc along its diagonal

and R to have the same value wr along its diagonal. The GPC solution offers no

guarantee of stability and the control weight must be tuned to produce an

acceptable solution without going unstable. Reducing wc increases control

authority and performance but eventually drives the system unstable.

If only the disturbances acting on the system change, there is no need to

recalculate (and hence c ) because it is formed solely from the SMP, which

are unique for a given system (ref. 13).

The solution for described earlier involves forming the matrix products

yVT and VVT. Here, these products are obtained using the computationally

efficient procedure described in reference 8.

W

Y
16

GPC Computational Considerations

● System ID in presence of external disturbances for implicit

inclusion of disturbance model

● Pseudo-inverse based on SVD techniques

● Basis for recalculation of matrices

● On-line/real-time versus on-line/near-real-time

● Batch versus recursive calculation of OMP

● Efficient computational algorithms and coding

Several considerations dealing with computations should be kept in mind

when developing algorithms for GPC applications.

 SID should be done with the external disturbances acting on the system

so that information about the disturbances is embedded in the OMP. However,

depending on the nature of the external disturbances, it may be possible to

perform a SID on an undisturbed system and still determine a control law that

results in satisfactory closed-loop performance.

 The computation of pseudo-inverses should be performed using Singular

Value Decomposition (SVD) because of the latter’s ability to deal with matrices

that are numerically ill-conditioned. The use of pseudo-inverses (via SVD) is

recommended even in cases where the ordinary inverse may seem appropriate

(such as in the operation (VVT)

-1 indicated in the expression for the OMP).

The basis for recalculation of matrices needs to be selected. For example,

should the system be re-identified and the control law matrices recalculated in

every sampling period, every specified number of time steps (block updating),

or only if some event requiring a re-ID occurs? Should all calculations be done

in real time, or can some be done in near real time? Should updating of the

OMP be done in batch mode or recursively?

Microprocessor speeds are such that it is now often possible to complete

the full cycle of GPC computations and apply the commands to the actuators

within one sampling period. However, the need for efficient computational

algorithms and attendant coding is not expected to diminish.
17

GPC Implementation Considerations

● Low-pass filtering of measured time histories

● Cut-off frequency of filter set to Nyquist frequency fN

● Nyquist frequency chosen so that maximum frequency of

interest is about 75% of fN

● Sampling frequency selected to be between 2-3 times fN

● Length of data stream for SID determined by need for 5-10

cycles of lowest frequency mode in measured responses

● Scaling (normalization) of digitized input and output data

● Distribution of computing tasks among computers/CPUs

● Extent of user interaction/involvement

● Constraints on acceptable values of input and output

Several considerations must be taken into account when actually

implementing GPC algorithms in hardware for active controls work.

The measured response time histories must be passed through a low-pass

filter with a cut-off frequency fc set equal to the Nyquist frequency fN. The latter

is chosen so that the maximum frequency of interest is about 75% of fN. The

sampling frequency fS should be at least twice fN to prevent aliasing. However, if

fS is made too large the low frequency modes will be poorly identified due to a

loss of frequency resolution. A sampling rate between 2 to 3 times fN is

generally sufficient. Once the sampling frequency has been selected, the

minimum number of data points that should be used for SID follows from the

requirement of having 5-10 cycles of the lowest frequency mode in the measured

response time histories.

Normalization of the input and output data that is used for SID on the

maximum actual or expected values of the data is often helpful numerically. The

procedure will depend on whether the computations are being done in batch mode

or recursively.

The computing tasks can be distributed among computers or different CPUs

on a single computer. The choice will influence the extent of user involvement.

The values of the input and output data that are being used during closedloop operations must be carefully monitored to ensure that they fall within

acceptable bounds. This is easily done using IF/THEN-type checks in the code.
18

A variety of numerical and experimental studies comprised the

exploratory evaluation phase of the GPC investigation during the period 1997-

1999. These studies included: (1) Numerical simulations using 3- and 9-DOF

mass-spring-dashpot systems varying the number and location of external

disturbances, the number and location of control inputs, and the number,

location, and type of response measurements (displacement, acceleration); (2)

Extensive experimental simulations using the BHTI Cobra stick model in which

the number and location of the shakers used for control and disturbance were

varied; (3) Preliminary assessment of GPC for simulated tracking response using

an XV-15 math model determined using BHTI’s COPTER (COmprehensive

Program for Theoretical Evaluation of Rotorcraft) flight simulation program

(ref. 14); (4) A hover test (April 1998) of a stiff-inplane rotor on the WRATS

testbed to explore the vibration reduction capability of GPC (excitation caused

by rotor downwash and rpm near resonance); and (5) Ground resonance test

(Oct. 1999) of a soft-inplane rotor on the WRATS testbed.

Illustrative Numerical and Experimental Results

● Simulations using lumped-mass-spring-dashpot systems

● Experimental dynamic studies using Cobra stick model

● Simulated tracking control of XV-15 in roll maneuver

● Hover test of stiff-inplane rotor on WRATS

● Ground resonance test of soft-inplane rotor on WRATS
19

The 3-DOF mass-spring-dashpot system used in the 3-DOF simulations is

depicted at the top of the slide.

The masses were equal to 1.0 and the spring rates were all equal to 1000

lb/in. The damping coefficients of dashpots 1 and 2 were equal to 0.1 lb/in/sec,

but the damping of dashpot 3 was set to – 0.1 lb/in/sec to make the system

unstable (one pair of complex conjugate eigenvalues had a positive real part).

Control was imposed at masses 1 and 3 (r = 2). Accelerations were

measured at masses 1 and 2 (m = 2). Sinusoidal disturbances were applied at

masses 1 and 2 at a frequency f = 6.276 Hz, which coincided with a natural

frequency of the system. The expressions used to represent the disturbances in

the simulation are ud1 = 2*cos(2*pi*f*t) and ud2 = 1*sin(2*pi*f*t) and are

shown in the plot at the lower left. Also, Ðt = 0.05 sec, l = 300, and p = 5.

The oscillatory diverging behavior of the open-loop system is shown in the

plot at the upper right, which shows the time history of the acceleration

responses of masses 1 and 2. The loop was closed after 100 time steps. The

time histories of the closed-loop responses and control commands are shown in

the plots in the lower right and upper left, respectively.
20

In parallel with the extensive numerical simulations that were being

conducted using the 3- and 9-DOF mass-spring-dashpot systems, bench tests

were conducted by Bell Helicopter using the “Cobra stick model” shown in the

slide. This model is a 36-inch long, 50-lb, multiple degree-of-freedom lumpedmass dynamic model that approximates the dynamics of a Cobra helicopter.

The dominant vertical bending modes of the model have natural frequencies of

about 8 and 23 Hz. Electromagnetic shakers were used in various combinations

for imposing external periodic and random disturbances, random excitations

needed for system identification, and the control inputs called for by the GPC

algorithm.

Cobra Stick Model

External disturbance shaker

Control shakers
21

An illustrative experimental result obtained using the Cobra stick model is

shown in the slide above. The case shown is for sinusoidal excitation of 23.5 Hz

(near a model natural frequency) at the nose of the model; random excitation at

the main rotor hub for SID; vertical response at the tail rotor location; control

input at the hub; fs = 512 Hz; l = 300; m = 1; r = 1; p = 8; wc = wr = 1.0. For the

case shown here, 300 time points were used to ID the system and imposition of

closed-loop control was then (arbitrarily) delayed for 50 time steps.

The top plot shows the time history of the tail boom vertical acceleration.

The bottom plot shows the time history of the control input during SID, the

delay, and after closing the loop. The response is reduced dramatically in about

0.2 sec.
22

A cursory examination of using the GPC algorithm as a means for

actuating the controls necessary to execute a prescribed maneuver was made.

The subject aircraft was the XV-15 cruising at 150 kts in the airplane mode of

flight. The controls considered were collective pitch, longitudinal and lateral

cyclic pitch, and pedal. Responses of interest were aircraft pitch, roll, sideslip,

and forward velocity. Bell Helicopter’s COPTER flight simulation program

(ref. 14) was used to compute the linearized mass, damping, and stiffness

matrices for a trimmed fight condition as well as components for the B matrix.

MATLAB was used to form the A, B, C, and D state matrices and generate the

simulated input-output time histories needed for SID and GPC. A one-cycle

saw-tooth-type roll maneuver was prescribed to be the target response, yT.

Simulated Tracking Control of XV-15 Tiltrotor

● Aircraft in airplane mode of flight;

150 kts airspeed; sea level

● Control inputs: Collective, pedal,

longitudinal & lateral cyclic pitch,

● Responses: Pitch, roll, sideslip,

and forward velocity

● COPTER used to generate linearized mass, damping, and stiffness

matrices for trimmed flight condition and components for B matrix

● MATLAB used to form A, B, C, and D state matrices and generate

simulated input-output time history needed for SID/GPC

● Roll maneuver prescribed
23

One cycle of a saw-tooth-type roll maneuver was prescribed to be the

target response, yT. The time histories of this desired response and the actual

roll response produced by GPC are shown for comparison at the bottom of the

slide. The remaining three responses (pitch, sideslip, and forward velocity) had

negligible magnitudes and are not plotted.

The time histories of the rudder and lateral cyclic control inputs called for

by the GPC algorithm to perform the prescribed roll maneuver are shown at the

top of the slide. The remaining control inputs (collective pitch and longitudinal

cyclic pitch) had negligible magnitudes and are not plotted.

The primary structural natural frequencies of the aircraft lie between 0.8

and 10 Hz. The values of pertinent GPC parameters are: wc = 1.0, p = 8, hp = hc

= 2p, l = 400, Ðt = .025 sec, fs = 40 Hz.
24

WRATS Model in Aeroelasticity Branch

Rotorcraft Hover Test Facility

The initial evaluation of GPC on the WRATS testbed was conducted in

April 1998 during a one-week hover dynamics test conducted in the AB

rotorcraft hover test facility located in a building adjacent to the TDT. A stiffinplane gimballed rotor was employed in this investigation. The model is shown

above with its rotor blades removed and replaced by equivalent lumped weights

for a ground vibration test that was conducted prior to the test. Emphasis in this

initial evaluation of GPC was on active control of vibration using only the

collective control. To provide a rigorous test of the GPC algorithm, the openloop response of the model was exaggerated by running the rotor at an rpm that

nearly coincided with the natural frequency of the wing vertical bending mode.

Additional excitation of the wing was provided by the downwash associated

with running the rotor at a high thrust level.
25

Implementation of GPC Algorithm for

WRATS Hover Dynamics Test

Identify system parameters

Compute αc and βc

Dual-computer approach with user interaction:

Computer #1

Computer #2

User sends αc and βc

to first computer

User sends data to

second computer

Compute uc using the p latest

I/O data sets

(Continuous calculation)

Collect necessary I/O data 

 (on user command)

User selects l, p, hc, hp, wc, wr

For the hover dynamics investigation, the GPC computations were

distributed between two computers and user interaction was required to transfer

the data from one computer to the other. On user command, data required for

system identification was collected on computer #1 and sent to computer #2

where SID was performed and the control gain matrices c and c were

computed. On user command the control parameters were sent to computer #1

which used the p latest data sets to (continuously) compute the control

commands that were sent to the swashplate actuators. If re-identification of the

system was required, the process was repeated on user command.

All computations were done using MATLAB on PCs with 500 MHz

CPUs.
26

Some experimental results obtained during the WRATS hover dynamics

test that illustrate the effectiveness of GPC in reducing rotor-induced vibrations

are shown in this slide, which shows the measured time histories of the open- and

closed-loop responses of the vertical bending moment rear the root of the wing. It

is seen that the response is dramatically reduced within a second after the control

is turned on.
27

Implementation of GPC Algorithm for

WRATS Ground Resonance Test

Initialize αc and βc

Collect necessary I/O data

(on user command)

Compute uc using the p latest

I/O data sets

(Continuous calculation)

Identify system parameters

Update αc and βc

Single-computer/dual-processor approach:

CPU #1

CPU #2

User selects l, p, hc, hp, wc, wr

For the ground resonance investigation, the GPC computations were

distributed between two CPUs on a single computer with no user interaction

required for data transfer as in the earlier hover test. On user command, data

required for system identification was collected on CPU #1 and sent to CPU #2

where SID was performed and the control gain matrices c

 and c were

computed. The control parameters were automatically sent to CPU #1 which

used the p latest data sets to (continuously) compute the control commands to be

sent to the swashplate actuators. If re-identification was required, the process

was repeated on user command.

All the algorithms were implemented in dSPACE software on a PC with

500 MHz CPUs.
28

A brief investigation of the use of GPC to actively control the ground

resonance behavior of a soft inplane tiltrotor was conducted in the AB hover test

facility in October 1999. For this test, the model blades were modified by

replacing the stiff inplane flexure at the root of each blade with a spindle

incorporating a lag hinge and an adjustable viscous damper and spring. The

open-loop behavior (frequency and damping versus rotor speed) was compared

with its closed-loop behavior for several values of collective pitch. A GPCbased algorithm was used to actively control the cyclic inputs into the (fixedsystem) swashplate in a manner which produced a whirl of the rotor tip-pathplane in the direction and frequency needed to stabilize the critical body mode.

For the open-loop configurations of the model in which a definitive ground

resonance instability was observed, use of GPC was found to be strongly

stabilizing. In particular, damping levels of about 2% critical were noted in the

rpm range where the open-loop system was unstable. The results for the

configuration with eight-degrees of collective pitch are shown at the right in the

slide above. A photo of the model obtained from the video recording made

during the test is at the left. A pneumatically-actuated snubber system,

consisting of four horizontal cables attached to the upper end of the pylon in an

X-pattern, was used to arrest instability and is partially visible in the photo.

WRATS model in hover test cage.

(8 degrees collective at 75% r/R)

Ground Resonance Test of Soft-Inplane Rotor

on WRATS Testbed
29

Concluding Remarks

● Exploratory studies of GPC for active aeroelastic control

of tiltrotor aircraft completed

● Studies included numerical simulations and experimental

investigations

● GPC-based MIMO active control system was highly

effective in increasing stability and reducing response

● GPC appears to be a viable approach for active stability

augmentation and vibration control of tiltrotor aircraft

● Wind-tunnel tests of WRATS model with GPC needed to

more fully evaluate method

Exploratory numerical and experimental studies into the use of GPC for

active aeroelastic control of tiltrotor aircraft have been completed. A GPCbased MIMO active control system was demonstrated to be be highly effective

in increasing ground resonance stability and reducing vibratory response. While

these results are quite encouraging with respect to establishing the viability of

the method, it is recognized that a broader evaluation of the methodology is

needed to validate GPC-based algorithms for active stability augmentation and

vibration control of tiltrotor aircraft.
30

Status and Plans

● Initial wind-tunnel evaluation of GPC on WRATS for

airplane-mode stability augmentation completed

● Method highly effective in increasing stability (damping)

of critical wing mode

● Continue development and evaluation of GPC-based

method for active aeroelastic control

● Improve suite of computational algorithms

● Establish robustness to system nonlinearities

● Conduct additional ground and wind-tunnel tests

The initial wind-tunnel evaluation of GPC on the WRATS testbed for

airplane mode stability augmentation has recently (April 2000) been completed.

The method was highly effective in increasing stability (damping) of the critical

wing mode for all of the model conditions tested.

Plans are to continue development and evaluation of GPC for active

aeroelastic control. Emphasis will be on improving the suite of computational

algorithms comprising the current GPC software system developed for WRATS.

In this regard, work is underway on providing for the calculation of the closedloop eigenvalues using closed-loop input-output data and for the recursive

calculation of the OMP.

It is expected that work will also be initiated to establish the robustness of

GPC-based methods to system nonlinearities. Both numerical simulations and

experimental studies are anticipated.

Additional ground and wind tunnel tests are to be conducted as necessary

to evaluate the GPC-based methodology over a broader range of simulated flight

and operating conditions.
31

References

 1. Yeager, W. T., Jr.; and Kvaternik, R. G.: Contributions of the Langley Transonic

Dynamics Tunnel to Rotorcraft Technology and Development. Presented at the AIAA

Dynamics Specialists Conference, April 5-6, 2000, Atlanta, GA (AIAA Paper 2000-

1771).

 2. Richalet, J.; Rault, A.; Testud, J. L.; and Papon, J.: Model Predictive Heuristic Control:

Applications to Industrial Processes. Automatica, Vol. 14, 1978, pp. 413-428.

 3. Clark, D. W. (Ed): Advances in Model-Based Predictive Control. Oxford University

Press, New York, 1994.

 4. Clark, D. W.; Mohtadi, C.; and Tuffs, P. S.: Generalized Predictive Control, Part 1.

The Basic Algorithm, Automatica, Vol. 23, No. 2, 1987, pp. 137-148.

 5. Clark, D. W.; Mohtadi, C.; and Tuffs, P. S.: Generalized Predictive Control, Part 2.

Extensions and Interpretations, Automatica, Vol. 23, No. 2, 1987, pp. 149-160.

 6. Juang, J.-N., Applied System Identification, Prentice Hall, Inc., Englewood Cliffs,

 New Jersey 07632, 1994, ISBN 0-13-079211-X.

 7. Eure, K.W.; and Juang, J.-N.: Broadband Noise Control Using Predictive Techniques.

NASA Technical Memorandum 110320, January 1997.

References (Concluded)

 

 8. Juang, J.-N.: State-Space System Realization With Input- and Output-Data Correlation.

 NASA TP 3622, April 1997.

 9 .Juang, J.-N.; and Phan, M. Q.: Recursive Deadbeat Controller Design.

 NASA TM 112863, May 1997 (also, Journal of Guidance, Control, and Dynamics,

 Vol. 21, No. 5, Sept.-Oct. 1998, pp. 747-753).

10. Juang, J.-N.; and Phan, M. Q.: Deadbeat Predictive Controllers. NASA TM 112862,

 May 1997.

11. Eure, K. W.: Adaptive Predictive Feedback Techniques for Vibration Control.

 Ph.D. Dissertation, Virginia Polytechnic Institute and State University, May 1998.

12. Phan, M. Q.; and Juang, J.-N.: Predictive Controllers for Feedback Stabilization.

 Journal of Guidance, Control, and Dynamics, Vol. 21, No. 5, Sept.-Oct. 1998,

 pp. 747-753.

13. Juang, J.-N.; and Eure, K. W.: Predictive Feedback and Feedforward Control for

 Systems with Unknown Disturbances. NASA/TM-1998-208744, December 1998.

14. Yen, J.; Corrigan, J.; Schillings, J.; and Hsieh, P.: Comprehensive Analysis Methodology

 at Bell Helicopter: COPTER, presented at the American Helicopter Society

 Aeromechanics Specialists Conference, San Francisco, CA, Jan. 1994.
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3. REPORT TYPE AND DATES COVERED

 Technical Memorandum

4. TITLE AND SUBTITLE

Exploratory Studies in Generalized Predictive Control for Active

Aeroelastic Control of Tiltrotor Aircraft

5. FUNDING NUMBERS

WU 581-10-11-03

6. AUTHOR(S)

Raymond G. Kvaternik, Jer-Nan Juang, and Richard L. Bennett

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

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National Aeronautics and Space Administration

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AGENCY REPORT NUMBER

NASA/TM-2000-210552

11. SUPPLEMENTARY NOTES

 Kvaternik and Juang: Langley Research Center; Bennett: Bell Helicopter Textron, Inc.

 Presented at the American Helicopter Society (Northeast Region) Active Controls Technology Conference,

 Bridgeport, CT, October 4-5, 2000.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

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13. ABSTRACT (Maximum 200 words)

The Aeroelasticity Branch at NASA Langley Research Center has a long and substantive history of tiltrotor

aeroelastic research. That research has included a broad range of experimental investigations in the Langley

Transonic Dynamics Tunnel (TDT) using a variety of scale models and the development of essential analyses.

Since 1994, the tiltrotor research program has been using a 1/5-scale, semispan aeroelastic model of the V-22

designed and built by Bell Helicopter Textron Inc. (BHTI) in 1981. That model has been refurbished to form a

tiltrotor research testbed called the Wing and Rotor Aeroelastic Test System (WRATS) for use in the TDT. In

collaboration with BHTI, studies under the current tiltrotor research program are focused on aeroelastic technology

areas having the potential for enhancing the commercial and military viability of tiltrotor aircraft. Among the areas

being addressed, considerable emphasis is being directed to the evaluation of modern adaptive multi-input multioutput (MIMO) control techniques for active stability augmentation and vibration control of tiltrotor aircraft. As

part of this investigation, a predictive control technique known as Generalized Predictive Control (GPC) is being

studied to assess its potential for actively controlling the swashplate of tiltrotor aircraft to enhance aeroelastic

stability in both helicopter and airplane modes of flight. This paper summarizes the exploratory numerical and

experimental studies that were conducted as part of that investigation.

14. SUBJECT TERMS

Tiltrotor aeroelasticity; active control; active stability augmentation; active vibration

reduction; generalized predictive control

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