Modal control of an oblique wing aircraft

APPENDIX A COMPLEX CONJUGATE EQUATION PAIRS We wish to examine the relationship between a pair of equations, p and p + 1 , extracted from the complete diagonalized set of equations, equations (3). This pair of equations represents a second- order mode for which we know that the eigenvalues are complex conjugates, Xp+l = X ; (see eqs.

(8)). We wish to show that the second equation is the complex conjugate of the first.

From equations (3) and (4) we can write directly € = X - ' x [ = A € + M u Let us define

x = [ V I 02 * * * v , ]

x-' = [w1 w2 * * . w , ] T

(A21 G = [g1 92 * * * g m I where the vj and gj are column vectors and the wi are row vectors. From equations (3) and knowing that the eigenvalues are complex conjugates, equations p and p + 1 can be written as €P = X P €P + VP €p+l = A ; €p+l + Vp+l where, because M = X - l G , m u p = C w p g j uj j= 1 (A41 m Vp+l = C W p + l g j uj j= 1 First we show that the wi are the left eigenvectors of F . From the definition of eigenvectors and the matrix X we can write Fvj = X j vj (A51 The V j are called the right eigenvectors because they multiply F on the right. We can write succes- sively F X = X A F = X A X - ' (A61 X-lF = A X - ' From equation A6, we can write for each row The wi are called the left eigenvectors because they multiply F on the left. In particular for row p we can write w,F = X p wp (A81 I Taking the complex conjugate of equation A8 we get successively But because 16 = and for each eigenvalue there is a unique eigenvector, then I Wp+l = w; W 0 ) Now from the definition of 6, equations A1 and A2, we can write

I

€ P = WP x € p l = W p l x But uptl = up*, so we can write €p+l = w; x = € ; Using similar logic, we can write successively j = 1 = v; Therefore, the p + 1 equation can be written as

APPENDIX B OWRA SYSTEM MATRICES This appendix presents in numerical form the system matrices associated with the OWRA at the study condition. Each matrix has a row of symbols across the top and a column of symbols along the side. Each symbol across the top indicates the component of the vector the matrix multiplies. Each symbol down the side indicates the equation corresponding to that row. All numbers are rounded to three decimal places to adequately show the structure of the matrices without inundating the reader in numerical trivia.

The natural-system matrices, F and G are presented first.

r P P 4 --0.996 -7.054 0 0 0.145 0.086 0 0 1 M

1 -0.804 -0.083 0 0 0 0 0 z

0 0.026 -0.015 -0.039 0 0 0 0 F = 1 0 0 0 0 0 0 0 de 1.860 36.427 0 0 -3.117 2.087 -67.201 0 L 0 -1.852 0 0 0.040 -0.710 10.833 0 N 0 0 0 0 -1 -0.293 0.039 0 - 0 0 0 0 1 0 0 0 --0.152 -0.175 0.010 - M

-0.002 -0.002 0 z

0 0 -0.000 x

G = 0 0 0 de 0.206 -0.355 0.266 L 0.028 -0.031 -0.108 N -0.000 0.000 0.001 Y

- 0 0 0 - 0

The matrix of eigenvectors, X , and its inverse, X -', are presented next.

Dutch Roll Short Period (-0.037,O .351) (-0.037, -0.351)

( 0.004 , 0 .loo) ( 0.004 , -0.100)

(0.032,O .002) (0.032, -0.002) (0.134,O ,000) (0.134, -0 .OW) ( 0 .OOo, 0 .OW) (0.000, -0 .OOO) (0.001,O .OOO) (0.001, -0 .OOO) X = (0.031, -0.006) (0.031,O .006) (0.119, -0.035) (0.119,O .035) (-0.631,O .670)(-0.631, -0.670 (0.024, -0.227) (0.024,O .227) (0.074,O .OOO) (0.074,O .OOO) ( 0.238 , 0 .164) ( 0.238 , -0 .164) Roll Mode Spiral Mode (0.039, -0 .00l) (0.039,O .00l) - -0.025 -0 .Ooo M (-0.006, -0 .OOO) (-0.006,O .OOO) 2 0.013 -0 .000 (0 .526,0 .OW) (0 S26, -0 .OOO) X 0 .Ooo 0.001

(-0.106, -0.719) (-0 .lo6 , O .719) 0 .ooo

dB 0.009 (-0.019, -0.014) (-0.019,O ,014) -0.030 L 0.939 (-0.008,O . 0 1 5 ) (-0.008, -0.015) N -0.011 0.038

(-0.002,o .OOl) (-0.002, -0 .ool) Y 0.001 0.002

(-0.206,O .387) (-0.206, -0.387) -0.342 0.999 d d

- (-0.672 , O S71) (-2.137, -1 390) (0.040, -0.062) (0.001

, O .OOO) (-0.672,-0.571) (-2.137,1.890)(0.040,0.062) (0.001,-O.OO0) (0.021, -1.706) (4.575,O .293) (0.043,O. 1 2 7 ) (-0.001,O .OOl) X-' = (0.021,1.706)(4.575,-0.293) (0.043,-0.127) (-0.001,-0.001) 3.012 -7.568 -0.228 -0.003 1.397 0.077 0.414 0.537 (-0.002,O .084) (-0.005, -0.619) (0.950,O .126) (-0.001,O .695) (-0.002,-0.084) (-0.005,0.619) (0.950,-0.126) (-0.001,-0.695) Note that for second-order modes, the adjacent columns of X are complex conjugates whereas adjacent rows of X-' are complex conjugates.

The diagonal eigenvalue matrix, A , is presented next. Note that the second-order modes have complex eigenvalues.

0 0 ( -0.487,3.146) 0 0 (-0.487, -3.146) 0 0

I 0 0 (-1.084,2.618) 0

0 0 0 ( - 1.084,-2.618)

A = 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 dr 0 0 0 0 dr 0 0 0 0 S P 0 0 0 0 S P -2.750 0 0 0 r m 0 -0.030 0 0 S P l 0 0 ( -0.007,O .053) 0 P h 0 0 0 (-0.007, -0.053) P h The complex modal control matrix, M , is presented next. Note that the rows corresponding to second-order modes are complex conjugates of each other as shown in appendix A.

-(0.094,-0.007) (0.134,-0.186) (0.037, -0.253) (0.094,O .007) (0.134,O.186) (0.037,O .253) (-0.026,O .225) (0.012,O.341) (-0.005,O .064) M = (-0.026,O .225) (0.012,O.341) (-0.005,O .064) -0.062 - 1.094 -0.147 -0.063 -0.460 -0.147 (0.000,-0.010) (0.001,-0.016) (0.000,-0.001) - (0.000 , O . O l O ) (0.001,0.016) (O.OO0 , O .OOl) Using the rule established in equation (10) for each second-order mode, we can develop the second-order transformation matrix, P -'.

I

1 (0.487; 3.146) ( 0.487, -3.146) 0 0

1 0 0 0 ( 1.084,2.618) (-1.084, -2.618) 0 0 1 1 p-1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 0 0 0 dr 0 0 0 dr 0 0 0 S P 0 0 0 S P 0 0 0 r m 1 0 0 S P l 0 (0.007,O .053) (0.007,-0.053) Ph 0 1 1 Ph P is just the inverse of P-' . Note that P has the same structure as P-' , that is, 1's opposite real eigenvalues and block 2 x 2 matrices opposite complex conjugate eigenvalues. In practice, it is never actually necessary to calculate P .

( 0 , -0.159) ( 0 . 5 , O .077) 0 0 ( 0 , O .159) (0.5, -0.077) 0 0 0 ( 0 , -0.191) (0.5 , O .207) 0 0 ( 0 , O .191) ( 0 . 5 , -0.207) 0 0 0 0 0 0 0 0 0 0 0 0 0

I : 0 0

0 0 dr 0 0 0 0 dr 0 0 0 0 S P 0 0 0 0 S P 1 0 0 0 r m 0 1 0 0 S P l 0 0 ( 0 , -9.461) ( 0 . 5 , O .066) Ph 0 0 (0,9.461) (0.5, -0.066) Ph ture Proceeding with the analysis, we calculate r as P -' A P . Note that r is real with the stru of equation (11) opposite complex conjugate eigenvalues.

0 -10.135 0 0 0 0 0 0 1 -0.973 0 0 0 0 0 0 0 0 0 -8.030 0 0 0 0 0 0 1 -2.167 0 0 0 0

r =

0 0 0 0 -2.750 0 0 0 0 0 0 0 0 -0.030 0 0 0 0 0 0 0 0 0 -0.003 0 0 0 0 0 0 1 -0.014 The second-order modal control matrix S is just P -'M . Note that S is a real matrix.

0.139 1.299 1.626 ' 0.188 0.268 0.074 - 1.234 - 1.762 -0.347 -0.052 0.024 -0.010 S = -0.062 - 1.094 -0.147 -0.063 -0.460 -0.146 0.001 0.002 O . O o 0 0.000 0.001 0.000 To transform directly from the open-loop system, equations (3), to the transformed modal sys- tem, equation (6), we need the real transformation matrix, T -' . This is just the product of the two previous transformation matrices, that is, T -' = P -' X -' .

4 Q! P 4

UlU1 e P r

0.430 -0.002 -0.227 -15.923 3.882 0.590 -4.248 9.813 15.628 -0.015 -1.344 -4.273 0.081 0.002 -0.004 -0.649 -3.935 -0.148 8.981 8.381 -0.572 -0.009 0.126 4.293 -4.882 0.019 0.043 9.150 0.085 -0.003 -0.116 -0.240 12.580 -0.177 3.012 -7.568 -0.228 -0.003 1.231 4.780 -0.711 0.921 1.397 0.077 0.414 0.537 0.396 2.438 0.000 0.000 -0.009 0.065 -0.000 -0.073 -0.000 -0.003 0.004 -0.002 -0.004 -0.010 1.890 -0.001 -0.001 -0.005 t

T is just the inverse of T -'

dr dr rm S P S P S P l Ph P h 0.032 -0.011 0.134 -0.183 -0.025 -0 .Ooo -0.011 0.039

0.001 0.031 0 .Ooo 0.134 0.013 -0 .ooo -0.006 -0.006

0 .Ooo 0 .Ooo 0 .Ooo 0.001 0 .000 0.001 0.000 0.526 T = -0.002 0.032 -0.013 0.134 0.009 0 .000 - 13.609 -0.01 1 0.213 -0.735 -0.322 0.503 0.939 -0.030 -0.257 -0.017 -0.072 0.059 -0.028 0.118 -0.011 0.038 0.280 -0.009 0 .Ooo 0.074 0.010 0.026 0.001 0.002 0.017 -0.002 0.052 0.213 0.024 -0.322 -0.342 0.999 7.325 -0.257 REFERENCES 1. Alag, Gurbux S . ; Kempel, Robert W.; and Pahle, Joseph W.: Decoupling Control Synthesis for an Oblique-Wing Aircraft. NASA TM 86801, 1986.

2. Alag, Gurbux S . ; Kempel, Robert W.; Pahle, Joseph W.; Bresina, John J.; and Bartoli, Febo: Model-Following Control for an Oblique-Wing Aircraft. NASA TM 88269, 1986.

3. Andry, Jr., Albert. N.; Shapiro, E. Y,; and Chung, Jonathon C.: Eigenstructure Assignment for Linear Systems. IEEE Transactions on Aerospace and Electronic Systems, vol. AES- 19, no. 5, 1983, pp. 711-729.

4. Larson, Gregory L.; and Williston, Kenneth W.: Eigenstructure Synthesis of an Oblique Wing Flight Control System. Proceedings of the 24th IEEE Conference on Decision and Control, Fort Lauderdale, Florida, vol. 1, 1985, pp. 660-662.

5. Enns, Dale F.; Bugajski, Dan; and Klepl, Martin: Flight Control for the F-8 Oblique Wing Research Aircraft. 1987 American Control Conference, Minneapolis, Minnesota, vol. 2, 1987, pp. 1112-7.

6. Porter, Brian; and Crossley, Roger: Modal Control Theory and Applications. Taylor & Francis Ltd., 1972.

7. Military Specification, Flying Qualities of Piloted Airplanes, MIL-F-8785C, 1980.

8. Phelan, Richard M.: Automatic Control Systems. Cornel1 University Press, 1977 9. Kempel, Robert W.; McNeill, Walter E.; Gilyard, Glenn B.; and Maine, Trindle A.: A Piloted Evaluation of an Oblique-Wing Research Aircraft Motion Simulation With Decoupling Control Laws. NASA TP 2874,1988.

2. Government Accession No. 3. Recipient's Catalog No.

1. Report No.

NASA TP-2898 5. Report Date 4. Title and Subtitle

I Modal Control of an Oblique Wing Aircraft I January 1989

6. Performing Organization Code 8. Performing Organization Report No.

7. Author(s)

I James D. Phillips I A-88250

10. Work Unit No.

I

505-60 9. Performing Organization Name and Address 11. Contract or Grant No.

Ames Research Center Moffett Field, CA 94035 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address Technical Paper National Aeronautics and Space Administration 14. Sponsoring Agency Code Washington, DC 20546-0001 15. Supplementary Notes Point of Contact: James D. Phillips, Ames Research Center, MS 237-1 1, Moffett Field, CA 94035 (415) 694-4126 or FTS 464-4126 16. Abstract A linear modal control algorithm is applied to the NASA Oblique Wing Research Aircrafi (OWRA). The control law is evaluated using a detailed nonlinear flight simulation. It is shown thai the modal control law attenuates the coupling and nonlinear aerodynamics of the oblique wing and remains stable during control saturation caused by large command inputs or large external disturbances. The technique controls each natural mode independently allowing single-input/single- output techniques to be applied to multiple-inputlmultiple-output systems.

18. Distribution Statement 17. Key Words (Suggested by Author(s1) Oblique wing, Modal control Unclassified-Unlimited Linear control Flight dynamics Subject Category - 08 120. Security Classif. (of this page) 121. NO. of pages I Z. Price 19. Security Classif. (of this report) Unclassified Unclassified 52 A04 NASA-Langley, 1988