High Angle of Attack Velocity Vector Rolls

Copyright IQ IFAC Automatic Control in Aerospace. Palo Alto. California. USA . 1994

HIGH ANGLE OF ATIACK VELOCITY VECTOR ROLLS CAPT. K.E. BOYUM, M. PACHTER, CH. HOUPIS Air Force Institute o/Technology/ENC. Department Wright PattersolZ AFB OH 45433- 7655, USA

0/ Electrical and Computer Engineering , 2950 P Street,

Abstract. This paper presents the design of a flight control system for performing the velocity vector roll maneuver at high angles of attack . A six degree of freedom model is developed, and the multiple-input multiple-output Quantitative Feedback Theory (QFT) robust control design method is used to treat both the inherent system nonlinearities and the variability in the system parameters due to changes in flight condition. as structured uncertainty . The development of a weighting matrix aids in this design process, yielding a three-axes rate-commanded inner-loop control system. Nonlinear six degree-of. freedom closed-loop simulations demonstrate the successful initiation and arrest of the maneuver. Key Words . Flight Control. Robust Control

1. INTRODUCTION

longitudinal and lateral/directional channels and on the small perturbation hypothesis cannot be used.

In the quest for supermaneuverability and agility , aircraft designers are exploring the high angle of attack (AOA) regime of the flight envelope. The design of flight control systems (FCS) for maneuvering in this regime are complicated by the decreased effectiveness of aerodynamic control surfaces at high AOA and an increase in the nonlinear behavior of the aircraft in high AOA maneuvers . Thrust vecloring has been shown to provide increased control authority at high AOA , but robust FCSs for high AOA maneuvers have yet to be developed.

The McDonnell Aircraft Company and the NASA High Angle of Attack Research Vehicle (HARV) program office (Pahle et al., 1992) have demonstrated a gain-scheduled flight control system for high AOA flight and Enns, et aI., (1992) have demonstrated a nonlinear control law based on the dynamic inversion concept for the F-18 HA RV aircraft in this regime . However, the former exhibits undesirable cross-coupling effects in the execution of a high AOA velocity vector roll, and the latter requires that the aircraft's complete aerodynamic data base be maintained and accessible on-board the aircraft.

This paper focuses on one specific high AOA largeamplitude maneuver. the velocity vector (or stability axis) roll. This maneuver is difficult 10 control because normal acceleration, tends to draw the heavy nose and tail portions of the aircraft farther from the axis of rotation at high roll rates, resulting in a positive pitch rate and an increasing AOA. Perhaps the most significant difficulty in designing a controller for this maneuver, however, is accounting for the nonlinearities introduced into the system by the maneuver itself. These nonIinearities must be present in the aircraft model used in control system design . Thus, standard linear aircraft models and flight control design techniques that hinge on the separation of the

This paper presents an innovative design approach which reduces the complexity and size of the flight control system while at the same time exhibiting superior performance in the velocity vector roll. The specific objectives of this effort are to: 1) Explore the nonlinearities of the high AOA velocity vector roll, 2) Develop an aircraft model which accurately represents these nonlinearities, and 3) Design a flight control system to control the thrust vectored F-18 HARV aircraft through the velocity vector roll maneuver over a range of flight conditions.

53

This paper is organized as follows . Section 2 discusses the approach taken to achieve the objectives. In section 3 the aircraft model is developed from the nine fundamental equations of motion. The Quantitative Feedback Theory (QFf) robust control design technique is briefly outlined in section 4, and the concept of structured plant uncertainty is used in section 5 to generate a set of linear time-invariant (L TI) plants to be used in the inner-loop FCS design, section 6. Section 7 demonstrates the validity of the design through closed-loop non linear simulations of the velocity vector roll, and section 8 presents the conclusions drawn and the plans for future efforts.

design of the inner-loop control system for the transition regions of the velocity vector roll.

Fig. 2. FCS Block Diagram

3. MODELING To properly account for the nonlinearities present in the system during a velocity vector roll at high AOA, standard linear model approximations cannot be made, and the system model must be rigorously developed from the fundamental nine state equations of motion . The linearization of the nine state equations of motion about trimmed flight conditions is typically based on the small perturbation theory, which assumes that only small excursions in the state variables occur about a given trim condition (Enns, 1972). However, the velocity vector roll consists of a high stability axes roll rate and large excursions in the bank angle, so the linearization technique must be modified .

2. APPROACH The basis of the design is a time-scale separation in the velocity vector roll maneuver. Two primary regions of the roll are identified, viz., the transition and free stream regions shown in Fig. I . The transition regions represent the initiation and arrest of the velocity vector roll, which occur in a relatively short amount of time (on the order of one second). In these transition regions the fast states of the aircraft, the angular rates, are dominant. If the stability axes are chosen for the control system design, the ideal velocity vector roll consists only of roll rate P with no pitch or yaw rates. In the free stream region, the desired roll rate is simply maintained through the desired stability axes bank angle, while the perturbations in AOA and sideslip angle are regulated to zero .

All states in the system other than the roll rate and the Euler angles can still be represented as perturbation quantities (denoted by lower-case variables), and the squares and products of these states can be neglected as is typically done in small perturbation theory linearization. All terms containing the roll rate P and bank angle 4> quantities must remain, however, since there these quantities are not small perturbations.

This time-scale separation leads to the control structure shown in Fig. 2. The inner-loop of the FCS consists of a three-axes rate-commanded control system which controls the fast states of the system. This control is essential in the transition region of the velocity vector roll maneuver. The outer-loop serves to regulate the AOA and sideslip perturbations which occur throughout the entire velocity vector roll. This outer-loop is therefore applicable to both the transition and free stream regions of the roll. This paper focuses on the

I'

The state equations are defined in the stability axes. However, the aerodynamic and control derivatives in the force and moment summation equations, and the moments and products of inertia in the state equations, are generally defined in the body axes. Therefore, the derivatives appearing in the state equations are replaced with equivalent body axes derivatives. Performing the modified linearization, nondimensionalizing, performing the axes transformation, and substituting six of the F-18 HA RV's actual control inputs (differential aileron, differential elevator, pitch elevator, rudder, pitch thrust vectoring, and yaw thrust vectoring) in for generalized control inputs yields a rather complex set of equations. Therefore, the terms premUltiplying each state in the resulting equations are redefined as a "consolidated" stability derivative yielding the following partially linearized six

Fre-e Stream Reg10n

lime

Fig. 1. Velocity Vector Roll Time-Scale Separation

54

degree of freedom (OOF) model suitable for velocity vector roll analysis:

p=

C, P + C, Pq + C, r + C, f3 + C, p

pq

r

fJ

6.

into their Multiple-Input Single-Output (MISO) counterparts, where the coupling between the channels is treated in the same manner as a disturbance (Houpis, 1987). Practically any system can be converted into a set of LTI MISO plants that can be controlled using a QFr designed controller. A MIMO QFr CAO package has recently been developed at AFIT which greatly facilitates this QFr design process (Sating, 1993).

D UT

(I)

tP = P

(2) (3)

r

= Cn". Pq + Cnp P + Cn, r + Cn~ f3 + Cn6 '" Do,

(4)

+Cn6~, D,,+Cn 6.... Dn+Cn "TV, DTVI

a=q-Pf3+(

5. ATTACK ON NONLINEARITIES AND UNCERTAINTY

~,)(I-costP)+c.-,'0 q+c. a

2U -

The six OOF model developed in section 3 contains several nonlinear terms involving the roll rate and bank angle. Additionally. the aircraft's stability and control derivatives (and therefore the "consolidated" stability derivatives) change rather significantly with flight condition (Fe.) in the high AOA regime of the flight envelope. The use of the QFr design technique allows these nonlinearities and changes in the plant parameters to both be treated as structured parameter variations (bounded uncertainties).

(5)

+C':a a+C: iJ,p Drp +C=iJTVP]VP D

f3= pa-r-(

~,)SintP+C

2U -

P+C r+C , f3

>pl,

p

(6)

+C, 6Q1 Da, + C, (J,. . 8" + C". "0 8 n- + C>.6TV," D rv ,

Note that the roll rate is represented by a capital P, indicating that during a velocity vector roll the roll rate is not necessarily small, viz., not a perturbation. An in-depth discussion of the linearization and transformation process, including the definition of each of the "consolidated" stability derivatives, appears in (80yum, 1993). It should be noted that the resulting six OOF model still contains the nonlinearities from the velocity vector roll. QFr and the concept of structured plant parameter uncertainty are used to accommodate these nonlinearities.

The first step is using the concept of structured uncertainty to attack the system nonlinearities. Assuming that the fast states (P. q. and r) are dominant and the AOA and sideslip angles are maintained to near zero values in the transition region of the velocity vector roll (a valid assumption given the short time period of the transition regions), then the six OOF model can be reduced to the nonlinear three OOF model :

(7)

4. QFT OVERVIEW

(8)

Quantitative Feedback Theory (QFr) is a robust multi variable control system design technique developed by Or. Isaac Horowitz in the early seventies, with modifications and improvements continuing to the present (O'Azzo and Houpis. 1988). This design technique uses feedback of measurable states to illicit a desired response from a system even in the face of quantified, structured plant parameter uncertainty and disturbance signals. The structured plant parameter uncertainty is modeled with a set of LTI plants which adequately represent the plant variation over the entire range of uncertainty, and the QFr design technique then yields a single fixed compensator and prefilter guaranteed to meet the closed-loop specifications for all plants over the entire range of structured uncertainty .

(9) whose nonlinearities are all a function of the roll rate P . Based on available data for the HARV (Pahle. 1992), a maximum bound of approximately 30 deg/sec can be placed on the magnitude of P for a 30 degree AOA velocity vector roll. Therefore. introducing a roll rate parameter Pparam which varies over the range of 0 to 30 deglsec (the nondimensional value is actually used) in place of the state variable P in the nonlinear terms allows the roll rate to be treated as structured uncertainty in the plant. Note that the roll rate parameter only replaces one of the state variables in the p2 term, viz., p2 ==p·pparam ' With the introduction of the roll rate parameter, a set of three OOF L TI models which model the nonlinear plant are identified:

Using QFr, Multiple-Input Multiple-Output (MIMO) plants are mathematically decomposed

55

x(t)= Ax(t)+8u(t)

However. since two control derivatives are available in each of the three primary channels (roll . pitch. and yaw). the weighting matrix is modified slightly :

(10)

yet) = Cx(t)+Du(t)

where x(t) represents the three fast states (P , q, r) , u(t) represents the six aircraft control inputs, C is the identity matrix, D is a zero matrix,

[

A = -C'"

C,,, P1'0"'"

C,

P1'0'"'" "

Cn,

8=

[ C" •.

C,

' ..

0

0

Cn,,'

cn,,,

C,

Cm,

C"'" Ppo,a",

Cn" P1'0"'"

Cn,

0

0

C,. " ,

c",,, ... 0

Cm_".,.

0

0

Cn 'no,

1

(11 )

1

c,0"

W=

(12)

Cn

'"

Four values of the roll rate parameter (0, 8. 16. and 24 deg/sec. in dimensional quantities) are used to represent the variation in roll rate. Thus. structured uncertainty is used to account for the nonlinearities introduced by the velocity vector roll maneuver.

(13)

Note that this weighting matrix is based on control derivatives which change with F.c. Therefore. it appears that the weighting matrix must be scheduled. However. substituting the actual derivatives into the weighting matrix shows that the change in elements due to the changes in flight condition are minimal. and the scheduled weighting matrix can be replaced by a constant weighting matrix .

The concept of structured uncertainty is also used to model the variation in plant parameters due to changing F.C.s . This variation is determined rather easily by extracting the nondimensional (body axes) stability and control derivatives from the F-18 HARV Batch Simulation (Evans. 1991) for a desired set of F.c.s. and then forming the "consolidated" stability derivatives for each F.c. For this study. the variations in F.e. represent the HARV trimmed at 30 degrees AOA. at the altitudes of 10.000. 15.000. and 20.000 feet. The combination of four roll rate parameter values and three sets of "consolidated" stability derivatives from the three F.c.s yields twelve LTI plant cases which are used in the QFT design .

The next design step is identifying the actuator dynamics. For this study. a reduced-order set of actuator dynamics which closely match the HARV actuator dynamics was taken from Adams. et al. (1992) . The aileron. rudder. and thrust vectoring actuators are modeled with second-order transfer functions. and the elevator actuators are modeled with fourth -order transfer functions .

6. CONTROLLER DESIGN

The final step is defining the stability and performance specifications. In flight control system design. a phase margin angle of 45 degrees is typically required for stability. The performance specifications are not yet well documented for the high angle of attack regime. however. and are therefore determined from other research efforts and physical insight into the problem , The tracking response of the system to a roll command is overdamped. with a settling time of approximately one-second (hence the one-second period for the transition region). The tracking response for the pitch and yaw channels are to be regulated to zero for a velocity vector roll. so for simplicity the roll channel specifications are placed on the pitch and yaw channels. The maximum cross-coupling effects are chosen to be five percent of the commanded value. Given the short time period of

In most control system design techniques. the number of inputs into a MIMO system must equal the number of outputs of the system. that is, the plant must be square . QFT is no exception to this. and a weighting matrix must therefore be introduced to reduce the six control inputs being used for the HARV down to the number of outputs. three. The most successful weighting matrix design attempted in this study borrows the general flight control concept of an aileron-rudder interconnect. using it to account for the cross-coupling effects found at high AOAs . For instance. a roll command would not only command roll. but also yaw to remove the adverse yaw produced by the aileron deflections , The amount of added yaw is based on the ratio of the yaw and roll control derivatives .

56

the transition region, this means that a maximum of I degree of sideslip angle could be achieved, and this is considered acceptable . To prevent undesired interaction with the bending modes of the aircraft, the frequency at which the open-loop Bode plot of the control system crosses the 0 dB axis (called .the phase margin frequency) is required to be less than 30 rad/sec .

commands appear in Fig. 4 . The commanded roll rate is easily achieved in the desired one second period of the transition region. developing approximately 15 degrees of rolllbank angle. Additionally. the cross-coupling responses in the pitch and yaw rate channels is well below the design specification of 1 deglsec. Most importantly. however. the AOA and sideslip angles remain negligibly small throughout the velocity vector roll initiation. This not only validates the time-scale separation assumption made in the model development. but also indicates that a true velocity vector roll has been initiated. If the sideslip angle f3 and the AOA perturbation a are not regulated to zero. the pertinent maneuver cannot be considered a velocity vector roll. for the aircraft's pointing objectives will not be attained .

With the weighting matrix , actuator dynamics, and stability and performance specifications identified, the design of the diagonal cascade compensator G and diagonal prefilter Fusing QFT is performed . For each channel, the interactive design of the compensator element, gj' is performed graphically using the available CAO package by plotting a nominal open-loop transmission Ljo = g,-qjjo ' where qijo is a nominal plant case , on a Nichols chart, along with the stability, performance, and crosscoupling bounds which represent the aforementioned specifications. These bounds are formed and plotted on the Nichols chart such that a satisfactory design for the nominal plant case guarantees satisfactory performance for all plant cases. The compensator element is "built up'· from a unity gain, by adding poles and zeros and adjusting gain , until all bounds are satisfied . For the roll rate channel, the result is a third-order transfer function which contains a zero (pole at the origin) to ensure tracking .

The arrest of a velocity vector roll is more difficult to simulate than the roll initiation since all of the initial conditions (I.c.s) must properly reflect the states of the system at the desired roll rate. In other words. the aircraft must essentially be "trimmed" in a velocity vector roll. The most straightforward method of obtaining the "trimmed" values for each state is to use the final conditions from the velocity vector roll initiation simulation. The response of the closed-loop system with these I.c.s (but using a 170 degree bank angle rather than a 15 degree bank angle) to a velocity vector roll arrest command appears in Fig. 5. and the corresponding control effector commands appear in Fig. 6. Again, this simulation demonstrates that the roll can be arrested in the desired time period, with negligibly small perturbations in the pitch and yaw rates . As in the roll initiation. the sideslip angle also remains sufficiently small throughout the roll arrest. However. both the AOA and sideslip angle appear to steadily increase. even as the other states come to a steady-state value . Looking back at these state equations in the six OOF equations of motion. the cause of this increase is evident. The nonlinear trigonometric term involving the bank angle becomes dominant at high bank angles. This demonstrates the limitation of the rate-commanded control system-- large perturbations in the angular (slow) states cannot be controlled for large time periods . It is for this reason that this ratecommanded control system serves as the inner-loop of the FCS. and that AOA and sideslip angle control are performed by an outer-loop (see e .g. Fig. 2).

Before moving on to the next channel, the prefilter must be designed . The prefilter is designed interactively using a Bode plot by building the transfer functions up from unity gain until the closed-loop system ·s control ratio falls within the performance specifications. The resulting prefilter element is a second-order transfer function . The design of the pitch and yaw rate channel compensator and prefilter elements are performed in a similar manner, yielding fourth- and third-order compensators and second-order prefilters . These designs complete the diagonal compensator G and diagonal prefilter F definitions for the three-axis rate-commanded control system for the transition regions of the velocity vector roll.

7. RESULTS The compensator and prefilter are validated through six OOF simulations of the closed-loop system which include the nonlinearities introduced by the velocity vector roll. the constant weighting matrix. and the actuator dynamics of the aircraft. The closed-loop response to a 20 deglsec velocity vector roll command (lateral stick) appears in Fig. 3, and the corresponding control effector

These simulations demonstrate rather impressive results for the control of the transition regions of the velocity vector roll. However, one very important nonlinearity has not been accounted for in the closed-loop system; that nonlinearity is

57

control surface saturation limits. The saturation limits for the control effector group commands cannot be modeled in the QFf linear design technique, and are therefore not designed for. However, these limits are quite real in the actual aircraft, so the simulation of the closed-loop system is not complete without this nonlinearity.

9. REFERENCES Adams, Richard J., James M. Buffington, Andrew G. Sparks and Silva S. Banda (1992). An Introduction to Multivariable Flight Control System Design. Technical Report, Flight Dynamics Laboratory, Wright-Patterson AFB, OH. WL-TR-92-311O.

Introducing the aileron command limit of 42 degrees, elevator command limit of 24 degrees, rudder command limit of 30 degrees, and yaw thrust vectoring command limit of 10 degrees yields the closed-loop responses to the velocity vector roll initiation and arrest commands shown in Figs .7-IO. The impact of the saturation limits do not significantly affect the performance of the system in the roll channel. The velocity vector roll is still successfully initiated and arrested within the one-second period of the transition region . However, the yaw rate response exceeds the original design specification of 1 deg/sec perturbations. This specification was rather arbitrarily set at the onset of the design stage, however, and the primary states of interest are instead the AOA and sideslip angle states. In the velocity vector roll simulation, both of these states remain negligibly small throughout the one-second period. As in the previous velocity vector roll arrest simulations, however, the bank angle dominates the AOA state response, indicating the need for the angle controlling outer-loop.

Boyum, Kevin E. (1993). Evaluation of Moderate Angle of Attack Roll of a Dual Engine, Thrust Vectoring Aircraft Using Quantitative Feedback Theory. MS thesis. Air Force Institute of Technology, Wright-Patterson AFB, OH. D'AzZQ, John J. and Constantine H. Houpis (1988) . Linear Control System Analysis and Design -Conventional and Modern (3rd Edition). McGraw-Hill, NY. Enns, Dale F. and Daniel 1. Bugajski (1992) . "First Steps Toward a Robust Nonlinear HARV Flight Control Solution," High-Angle-of-Attack Technology, NASA Conference Publication 3149, pp. 1241-1256. Etkin, Bernard (1972). Dynamics of Atmospheric Flight. John Wiley & Sons, Inc., NY. Evans, Martha (1991). F-18 Simulation Guide . NASA Ames Research Center, Dryden Flight Research Facility, Edwards AFB, CA. Houpis, Constantine H. (1987) Quantitative Feedback Theory (QFT) -- Techniquefor Designing Multivariable Control Systems. Technical Report, Flight Dynamics Laboratory, Wright-Patterson AFB, OH. AFW AL-TR-863107.

8. CONCLUSIONS This paper has demonstrated the effective application of QFf to a unique controls problem, the initiation and arrest of the velocity vector roll maneuver at high angles of attack . The robust QFf design technique has been used to jointly address system non linearity and structured plant parameter uncertainty in the design of the inner-loop control system. with excellent results. Through simulation, several key characteristics of the velocity vector roll have been identified, as well as the limitations of the inner-loop control system. The role that the outer-loop control system must play has also been identified, paving the way for future work in the design of an appropriate compensator and simulation of the entire velocity vector roll maneuver. Future work will also include the extension of the design to additional flight conditions, including higher angles of attack.

Pahle, Joseph W., Bruce Powers, Victoria Regenie and Vince Chacon (1992). "Research FlightControl System Development for the F-18 High Alpha Research Vehicle," High-Angle-of-Attack Technology, NASA Conference Publication 3149,pp. 1219-1239. Sating, Richard R. (1993). MIMOIQFT CAD Program -- User's Manual (2nd Edition). Air Force Institute of Technology, Wright-Patterson AFB, OH.

58

Fig. 3. Velocity Vector Roll

Fig . 7. Velocity Vector Roll, Limited System

~~fZ o ~:rz o

02

O'

0 ,6

0 ,8

02

0"

0 .6

08

~~t~ o O,S: 0

0,2

0

Q .~

~J f;

;

0' 0

08

~ .:g~ o

9.L::::::::::,§

~:[/ 'I- 1~

08

~JS

--

Control Eft.aw Group ConYnInOs 101d1

Control EneclOt Group ConvNnds No UnIt$. 1C1d t

0

~

~ o. O.

0"

0.6

0 .8

0

02

O.

T~tsec)

M,st 20 OegHC Roll 20

R~I.

No Lm«s. 10kt'1

~ 175[O~O' 0 170

a~~~

AnestrlO 20

~

06

~

P

Os

O.

0

02

O.

0

~

0

-8 S

io· ~ t ji~!

O.

02

Tnw

o~

0 ,8

& 17t;;;:=~

a~'~=s:

0 .8

0,6

08

0 ,6

08

0 .6

0 .8

06

08

o

Q.2

Q'

Q.6

21

O.

0 .6

08

~.

Q6

00

O.

0 .6

Q.8

!J! ~ ~oit 0

to~f 0 1S

q! 0

~~f 0

02

~

0

7 /

sa

06

0 .8

0.6

0 .8

2

~

o~ 02

a 1

n

1

1

RoI Rate ' 10kt1

0.4

1 0 .6

08

1

j Q8

j

~.8

1

Q:;;;; Q'

Fig . 9. Roll Arrest. Limited System

Control Enldof GrOlC) Corm\IIndS No Unls, 1C1d 1

0

Q.§

(sec)

Fig. 5. Velocity Vector Roll Arrest

~ :~t ~

~MC

==-----:;;

~ ~f

O '

1

j

Fig . 8. Roll Initiation Command, Limited System

~

10~

J

r~(s.ec.)

Fig. 4 . Velocity Vector Roll Initiation Command

D.

11

Q8

€.l~ 0 .2

1

QI

Q: 0 .2

l

2·1

O.

~ :~[ ~ 0

~ :j~r

g·1

---l

1

~

1

0

~O'

2·6

21

0

~.

Q§

!U!

O.§

0 .8

2!

~

"u~ ~o~

toit ~

~

----~

0

g.§

Q.8

19t 15 . 10 0

0 .6

0.11

0

~.lL

o .~

o.

O·Z

0 .'

02

O'

0 .6

1 1

l 1

~

0 .8

1

Tme(MC)

T..... (MC)

Fig. 10. Roll Arrest Commands , Limited System

Fig . 6 . Velocity Vector Roll Arrest Commands 59