Document
NASA Technical Memorandum 88269
Model-Following Control for an
Oblique-Wing Aircraft
Gurbux S. Alag, Robert W. Kempel, Joseph W. Pahle,
John J. Bresina, and Febo Bartoli
r
(NASfl-TM-88268) HODEX-FOLLGHJNG CONTBOL FOE N86-2S867 AW OBLIQUE-WOG .AIECBlfT (NASfl) , 13 p CSCL 01C
Unclas
G3/08 43518
August 1986
NASA
National Aeronautics and Space Administration NASA Technical Memorandum 88269
Model-Following Control for an
Oblique-Wing Aircraft
Gurbux S. Alag, Robert W. Kempel, Joseph W. Pahle, John J. Bresina, and Febo Bartoli Ames Research Center, Dryden Flight Research Facility, Edwards, California
NASA
National Aeronautics and Space Administration Ames Research Center Dryden Flight Research Facility Edwards, California 93523-5000 MODEL-FOLLOWING CONTROL FOR AN OBLIQUE-WING AIRCRAFT Gurbux S. Alag* Western Michigan University Kal amazoo, Michigan and Robert W. Kempel ,** Joseph W. Pahle, ' John J. Bresina,** and Febo Bartoli** NASA Ames Research Center Dryden Flight Research Facility Edwards, Cal ifornia Abstract u input vector A variable-skew oblique wing offers a sub- v velocity, ft/sec stantial aerodynamic performance advantage for aircraft missions that require both high effi- x state vector ciency in subsonic flight and supersonic dash or cruise. The most obvious characteristic of the a angle of attack, deg oblique-wing concept is the asymmetry associated with wing-skew angle which results in significant 6 sidesl ip angle, deg aerodynamic and inertial cross-coupling between the aircraft longitudinal and lateral-directional 6 control surface deflection axes. This paper presents a technique for synthe- sizing a decoupling controller while providing the 6 pitch angle, deg desired stability augmentation.
A wing skew angle, deg The proposed synthesis procedure uses the con- cept of explicit model following. Linear quad- <)> bank angle, deg ratic optimization techniques are used to design the linear feedback system. The effectiveness of Subscripts the control laws developed in achieving the desir»d decoupling is illustrated for a given flight con- aL left aileron dition by application to linearized equations of motion, and also to the nonlinear equations of aR right aileron six degrees of freedom of motion with nonlinear aerodynamic data.
hL left horizontal tail Nomenclature hR right horizontal tail system matrices, m model altitude p aircraft cost functional Superscripts gain matrix m number of inputs Mach number n number of states oblique-wing research aircraft T matrix transpose roll rate, deg/sec Introduction state weighting matrix The advantages of an oblique wing were first noted in the 1940's. However, not until recently pitch rate, deg/sec have the interest, technology, and mission of an oblique-wing design evolved into a full-scale control weighting matrix flight research program. Dryden Flight Research Facility of NASA Ames Research Center and the yaw rate, deg/sec U.S. Navy are developing an oblique-wing research aircraft (OWRA). Gregory,1 and Wiler and White2 complex frequency have outlined potential advantages and disadvan- tages of this type of airplane. Theoretical and time wind tunnel studies have shown that a variable- skew oblique wing offers a substantial aerodynamic *Associate Professor, Electrical Engineering performance advantage for aircraft missions that Dept., Member AIAA.
require both high efficiency in subsonic flight **Aerospace Engineer.
and supersonic dash or cruise.
^Aerospace Engineer, Member AIAA.
The most obvious disadvantage of the oblique- as the model would respond to a given pilot com- winy concept is the asymmetry associated with mand. More precisely, the model -fol lowing problem wing-skew angle. This asymmetry results in sig- can be stated as follows.
nificant aerodynamic and inertial cross-coupling between the aircraft longitudinal and lateral - Given the linearized aircraft dynamics, directional axes.
( 1 ] The test bed for OWRA will be the NASA F-8 digital fly-by-wire aircraft. This aircraft will n 171 be modified by the removal of the current high where x eR , UpeR , A , and B are matrices of p p p wing.and installation of a composite wing-and- appropriate dimensions, find the control u '• p pivot assembly. A major part of the OWRA program such that the aircraft states x approximate p will be the synthesis of a flight control system "reasonably well" model state vector x . The m which will provide acceptable stabilization and decoupling across the Mach, angle-of-attack, and model state vector xm is defined by the equation: wing-skew envelope. These aircraft stability and decoupling requirements are idaally suited for the x 8 u ( 2 ) m m m application of modern control theory techniques to the solution of problems associated with OWRA.
where x eRn , UmeR ", Ap,, and are matrices of m The potential advantages of the oblique-wing appropriate dimensions.
concept can only be realized by development of related technologies. Foremost among these is the For OWRA, the state and .input vectors are control system architecture to compensate for given by cross-coupled aircraft responses, while presenting the crew with the feel of a conventional airplane.
velocity, ft/sec Current typical design procedures synthesize angle-of-attack, deg aircraft controllers based on solutions of two, sideslip angle, deg or at most, three degrees of freedom. However, bank angle, deg the added OWRA stabilisation and decoupling prob- pitch angle, deg lems require at least "ive degrees of freedom roll rate, deg/sec simultaneously.
pitch rate, deg/sec yaw rate, deg/sec Model following ha:; been a popular method for the design of multi var able control systems, and left horizontal tail deflection, deg '«HL shown to be amenable tn the solution of many air- right horizontal tail deflection, deg craft control problems.. Commencing with the work hR left aileron deflection, deg «aL of Kalman and Tyler,4 extensive use has been made of the linear optimal control theory in design of right aileron deflection, deg «aR model-following controllers. Yore^ used this rudder deflection, deg «R method for simultaneous stability augmentation and mode decoupling. While optional control the- The desired model of the aircraft .defined by ory provides an extremely flexible synthesis tech- matrices Am and Bm, as well as the aircraft mat- nique, various structural approaches have been rices Ap and Bp, are given in Table 1. The desired adopted in synthesizing model-foil owing control- 6 model used in this study is a modification of the lers. A controller based on perfect model fol- zero-wing-skew configuration at the same flight lowing (such as perfect matching of the dynamics condition. The aircraft matrices correspond to a of the compensated plant to thoss? of the model) flight condition of Mach 0.8 and'an altitude of was presented by Alag, Kempel, and Pahle for 20,000 ft at 45° wing skew. The elements of An, control of OWRA. Thoucjh the desired degree of were modified to increase the short period and decoupling was achieved, the control surface dutch-roll mode damping, and to provide improved activity was excessive and alternate controller roll and spiral mode characteristics.
development was required.
Explicit Model-Foil owing Systems This paper presents, the use of models in design of linear feedback systems by means of There are two configurations of model fol- linear-quadratic optimization. The linear control lowing, known as implicit model following and law developed provides the decoupling as well as explicit model following. As Fig. 1 shows, in the desired stability Augmentation. The effec- implicit model following, the control inputs to tiveness of the control law is illustrated by time the plant are formed from the aircraft states responses from Iineari2:ed equations of motion, and and pilot input. No dynamic coupling exists nonlinear equations of six degrees of freedom of between the model states and the closed-loop aircraft motion for a c.iven flight condition.
plant; the model state % appears only in the performance index.
Model-Fol lowing Systems Figure 2 illustrates the explicit model Problem Definition following in which the model states must be generated for use in forming the control input.
The concept of model following is useful when Explicit model following requires the simulation an ideal set of plant equations of motion can be of the model as a part of feedforward controller.
specified. The objective of model-following Alignment of plant and model in the presence of flight control is to force the aircraft to respond response of pitch and yaw angular rate and bank uncertainties, such as unknown parameters and ran- angle to an.elevator command input. Significant dom disturbances, requires this type of control.
yaw rate and bank angle are generated as a result This enables a continuous correction of the errors of pitch command. Of particular interest is the between model and plant states even in the pres- large change in bank angle, indicating a high ence of unknown disturbances.
degree of cross-coupling.
The dynamics of the plant and model are Table 2 gives the gain matrices for the governed by the following linear state equations: explicit model-foil owing configuration. Figure 4 indicates the linear closed-loop system response B up p .(1) for the command input with the explicit model - following gains. The aircraft response is indi- cated by the solid line, and the model response x = (2) m by the dashed line in Figs 3 and 4. The model- following response is considered satisfactory with Defining an augmented state vector x, the dynamics considerable attenuation in the degree of coupling, of the system are expressed as as indicated by the bank angle response of the air- craft. The model response in bank angle is zero.
Figure 5 indicates the closed-loop response for x = Ax + Bu (3) r the same case but'with the aircraft now repre- sented by a nonlinear simulation of 6 degrees of where freedom for the same flight condition. The non- linearities include the constraints on the con- rx -| l-Ap 0 0] l-Bp p trol surface rate and position.
x = x A = 0 A, B B = 0 (4) m n m Figure 6 indicates the response of the open- Urn LO 0 0 J LO loop system to 1° aileron command input. Bank angle, yaw angular rate, and pitch angular rate The pilot input u is modeled as a constant, m are shown. The relative pitch coupling is not an assumption that does not overly distort as severe for this case as the roll coupling is the reality of the situation and allows a com- for the elevator command; however, coupling is plete analysis of the problem from a theoret- still evident.
ical viewpoint.
Figure 7 indicates the linear closed-loop In explicit model following, tne control func- system response to the same aileron command. A tion up is required to minimize a performance dashed line denotes the model response, and a index given by solid line, the aircraft response. The relatively small coupling in the pitch axis was not con- sidered severe, as indicated by the aircraft if" T T J X , / ^ P - xm) Q(x - xm) + upRup] dt (5) pitch-angular rate response. The model pitch- p f. JQ angular rate response was zero. The objective in this case was to provide adequate lateral- where Q _>. O R > 0 are weighting matrices. Using t directional dynamics. Figure 8 shows the closed- the augmented state vector x, the index J can be loop response for the same case with aircraft non- rewritten as linear simulation.
T Conclusions dt J « i / (x Qx (6) The method presented describes a decoupled control for a highly coupled unsymmetric aircraft.
where The method develops an'explicit model-following f Q -Q 0] control law by means of linear quadratic optimi- -Q Q 0 (7) zation techniques. The results indicate that L 0 0 Oj the method does obtain the decoupling incorpo- rated in the ideal model for the flight condi- If pair (A,B) is stabilizable and pair (A,D with tion considered.
T D D = Q) detectable, the optimal control up which minimizes J is given by Evaluation of the control system on nonlinear equations of six degrees of freedom of motion for Up = Kx = K pXp (8) X the flight condition considered shows promising results for implementation as a candidate control The model equations must be simulated as a part of system for the aircraft. Work is in progress to the feedforward controller because the controller investigate gain scheduling requirements and to requires model states. obtain piloted simulation results.
Results References The degree of coupling in the open-loop air- ^Gregory, T., "Oblique Wing Ready for Research craft is illustrated by response to application Aircraft," Aerospace America, June 1985, pp. 78-81.
of a 1° command input at 1 sec and returned to zero at 3 sec (either by elevator or aileron 2wiler, C.D. and White, S.N., "Projected Advantage input). Figure 3 illustrates the open-loop system of an Oblique Wing Design on a Fleet Air Defense Mission," J. A i r c r a f t , Vol. 22, No. 10, Oct. 1985, Yore, E., "Optimal Decoupling Control," 9th Joint pp. 896-900.
Automatic Control Conference of the American Auto- matic Control Council, New York, 1968, pp. 327-336.
Kalman, R.E., Englar, T.S., and Bucy, R.S., "Fundamental Study of Adaptive Control Systems," ^Morrow, L.D. and Balasubramanian, R., "Real Model ASD-TR-61-27, Vol. II, 1962, Martin Co., Rias Following Control," J. Aircraft, Vol. 12, No. 12, Division, Baltimore, Md.
Dec. 1975, pp. W6-9W.
Tyler, J.S., "The Characteristics of Model - Alag, G.S., Kempel, R.W., and Pahle, J.W., Following Systems as Synthesized by Optimal Con- "Decoupling Control Synthesis for an Oblique-Wing trol ," IEEE Transactions on Automatic Control , Aircraft," NASA TM-86801, 1986.
Vol. AC-8, Oct. 1964.
TABLE 1. - AIRCRAFT AND MODEL MATRICES* (a) Aircraft matrices ' 0.009'. 22.0707 10.5479 -0.1341 -32.1127 0.0057 -0.0265" -0.0005 0.0958 -0.000). -0.7826 0.0000 0.0000 0.0030 0.9926 -0.0003 -0.2908 0.0387 0.000(1 -0.0592 -0.0002 0.0259 0.0001 -0.9920 0.000(1 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0247 a = A = p 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 33.1432 -53.6933 0.0000 0.0000 -3.1250 2.0552 1.7210
-o.ooo;: -8.6816 0.7975 0.0000 0.0000 0.1679 -1.0352 0.1810
. 0.000(1 10.7521 0.0000 0.0000 -0.0213 -1.0092 0.0080 -0.7129.
" 0.0841 -0.0309 -0.2210 -1.2572 3.6598" -0.0974 -0.0974 -0.0198 -0.0302 0.0000
-o.oiei; 0.0166 0.0008 -0.0005 0.0647
0.0000 0.0000 0.0000 0.0000 0.0000 B = p 0.000(1 0.0000 O.OOOC 0.0000 0.0000 12.9804 -22.2664 15.8467 -11.8422 13.2774 -9.407'.t 0.8797 0.5694 -10.8555 -1.2311 1.9854 -2.2579 0.5262 -0.3276 -6.2493.
(b) Model matrices '-0.007' 23.5966 0.0000 0.0000 -32.1129 0.0000 0.0000" 0.0000
-o.ooo:. 0.0000 0.0000 0.0000 0.0000 0.9909 0.0000
-1.1CS2 0.0000 0.0000 -0.2852 0.9387 0.0000 -0.0148 0.0000 -0.9919 0.0001) .0.0000 0.0000 0.0000 1.0000 0.0000 -0.0120 0.0000 A = m 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 -44.3777 0.0000 0.0000 0:0000 0.0000 -4.5000 0.0000 14.6000 0.0001 -12.1514 0.0000 0.0000 0.0000 0.0000 -4.0000 0.0000 0.0000 . 0.0000 0.0000 12.1943 0.0000 0.0000 0.1890 -4.0000.
-0.8354 -0.8354 "-2.203L' -2.2032 0.0000" -0.084U -0.0848 -0.0494 -0.0494 0.0000 -0.0161) 0.0156 0.0000 0.0000 0.0647 - 0.0000 O.COOO 0.0000 0.0000 0.0000 B m 0.0001) 0.0000 0.0000 0.0000 0.0000 11.237!) -11.2379 29.0513 -29.0513 9.6847 0.0000 -7.822') -7.8229 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -6.6502 *M = 0.8; h - 20,000 ft; a = 1.6°; A = 45° TABLE 2. - GAIN MATRICES* a 6 V B r * P
q
r -0.0519 -1.6309 -0.0001 -0.0432 2.1180 -0.0214 0.2558 -0.0239" -0.0614 -1.0436 -0.6015 0.0630 1.9146 0.0475 0.3378 0.1859 K = xp -0.0081 -0.4667 0.1671 -0.0406 0.4541 -0.0268 0.0250 -0.0426 0.6063 -0.0928 0.0333 -0.6926 0.0213 0.0232 -0.0348 0.0273 .-0.0524 -0.3968 -0.1964 -0.0009 0.7165 -0.0090 0.0442 0.1586.
a V 6 e r
P q
* ' 0.0520 1.4288 -0.9317 0.9512 02.1145 0.0205 -0.2196 0.0521" 0.0615 1.2344 0.1713 -0.0608 -1.9159 -0.0348 -0.2036 -0.1015 = KXITI 0.0082 0.3162 -0.0769 0.0431 -0.4550 0.0225 -0.0428 0.0515 -0.0232 -0.4681 0.0552 -0.0362 0.6935 -0.0177 0.0487 -0.0424 . 0.0524 0.4089 -0.0353 -0.0013 -0.7161 0.0092 -0.0130 -0.0256.
<5hL «hR «aL «aR <5R 0.2711 0.3173 -0.5317 " 0.6121 -0.2012' 0.0015 0.2122 -0.3388 0.1843 0.0414 i/ _ 'nun - 0. !607 0.0542 0.2295 -0.2846 -0.0907 -0.0384 -0.1941 -0. !221 0.2630 0.0875 .-0. !622 -0.1349 -0.1708 0.1445 0.0993.
*M = 0.8; h = 20,000 ft; A = 45° Fig. 1 Implicit model-follouing control lau structure.
Fig. 2 Explicit model-following control lean structure.
9 10 (a) Pitch rate.
1.0 .5 -.5 Angle of -1.0 attack, deg -1.5 -2.0 -2.5 -3.0 -3.5 1 2 3 4 5 6 8 9 10 Time, sec (b) Angle of attaak.
-10 -20 Bank angle, -30 deg -40 _ \ \_^- —
1 1 1 I I I I I I I
3 1 2 3 4 5 6 7 8 9 10 Time, sec (c) Bank angle.
Fig. 3 Open-loop airoraft reeponee to elevator ccamand input.
Model 1.5 A 1.0
'\\
— .5 Pitch rate, -.5 deg/sec — ^^ -1.0 1 * -1.5 - \ l -2.0 —
V
1 1 1 1 1 1 ' 1
—2.5 0 1 2 3 4 5 6 7 8 9 1 0 Time, sec (a) Pitch rate.
Aircraft •3 i— -.3 Angle of attack, -.6 dog -1.2 I I I I I -1.5 1 2 3 4 5 6 8 9 10 Time, sec (b) Angle of attack.
Model Aircraft .05 — Bunk dag 1 1 1 1 1 1 I I I I > 1 2 3 4 5 6 7 8 9 10 Time, sec !c) Bank angle.
Fig. 4 Model-follouing response to elevator oontnana input.
Pitch
T
rate, ~ deg/sec
I I I I I I I I I
10 I ' Angle of i •s^ - ^-- attack, «F- deg I I I I I I I I I
J '
Bank angle,
"r
deg I I I I I I I I I
J i
Pitch angle,
IF-
deg I I I I I I I I I
J i
0 1 2 3 4 5 6 7 8 9 1 0 Time, sec Fig. S Nonlinear model-following response to elevator command input.
Bank angle, 9 deg I I I I I I I I I 2 3 4 5 6 8 9 10 Time, sec (a) Bank angle.
.8 .3 A .7 .6 .1
\
.5
A
Dl*r»h riicn Yaw rate, rate, .4 deg/sec deg/sec -.1 v ;
-3. —
.2
-.3 —
.1
u
J I
I I I I I I I
_ A 2 3 4 5 6 8 9 10 0 1 2 3 4 5 6 7 8 9 1 0 Time, sec Time, sec (b) You rate.
Pitch rate.
Fig. 6 Open-loop aircraft response to aileron command input.
Model Aircraft Bank angle, 12 deg I I I J I 2 3 4 5 6 8 9 10 Time, sec (a) Bank angle.
Model Aircraft (b) yau> fate.
Model Aircraft •6 Pitch - rate, deg/sec -.2
I
0 1 2 3 4 5 6 7 8 9 1 0 Time, sec (a) Pitch rate.
Fig. 7 Model-following response to aileron command input.
f Bank angle, 10 - deg
Xl I I I I I I I I
Angle of sideslip, deg
I I I I I I I I I
•
-
Yaw rate, deg/sec
I I I I I I I I I I
Jl
10,
-
Pitch rate, deg/sec
l I I I I I I I I I I
'
-10
-
Angle of attack, deg I
I I I I I I I I I
2 3 4 5 6 7 8 9 1 0 0 1 Time, sec Fig. 8 Nonlinear model-following response to aileron eoiaaand.
1. Report No. $|^ 2. Government Accession No. 3. Recipient's Catalog No.
NASA TM-88269 *§?
4. Title and Subtitle 5. Report Date MODEL-FOLLOWING CONTROL FOR AN OBLIQUE-WING AIRCRAFT August 1986 Performing Organization Code 6.
7. Author(s) 8. Performing Organization Report No.
Gurbux S. Alag, Robert W. Kempel , Joseph W. Pahle, H-1362 John J. Bresina, and Febo Bartoli 10. Work Unit No.
9. Performing Organization Name and Address RTOP 533-06-01 NASA Ames Research Center 11. Contract or Grant No.
Dryden Flight Research Facility P.O. Box 273 Edwards, CA 93523-5000 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address Technical Memorandum National Aeronautics and Space Administration 14. Sponsoring Agency Code Washington, O.C. 20546 15. Supplementary Notes Prepared as AIAA Paper 86-2244CP for presentation at AIAA Conference on Guidance and Control, Williamsburg, Virginia, August 18-20, 1986.
16. Abstract A variable-skew oblique wing offers a substantial aerodynamic performance advantage for aircraft missions that require both high efficiency in subsonic flight and supersonic dash or cruise. The most obvious characteristic of the oblique-wing concept is the asymmetry associated with wing-skew angle which results in significant aerodynamic and inertial cross-coupling between the aircraft longitudinal and lateral-directional axes. This paper presents a technique for synthesizing a decoupling controller while providing the desired stability augmentation.
The proposed synthesis procedure uses the concept of explicit model fol- lowing. Linear quadratic optimization techniques are used to design the linear feedback system. The effectiveness of the .control laws developed in achieving the desired decoupling is illustrated for a given flight condition by application to linearized equations of motion, and also to the nonlinear equations of six degrees of freedom of motion with nonlinear aerodynamic data.
17. Key Words (Suggested by Author(sl) 18. Distribution Statement Asymmetric aircraft control Unclassified - Unlimited Flight control systems Model -foil owing control system Multi-variable control system STAR category 08 19. Security Qassif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price' Unclassified Unclassified 10 A02 *For aale by the national Technical Information Service, Springfield, Virginia 22161.