Multidisciplinary Techniques and Novel Aircraft Control Systems

AIAA 2000-4848

Multidisciplinary Techniques and Novel

Aircraft Control Systems

Sharon L. Padula, James L. Rogers, and David L. Raney

NASA Langley Research Center

Hampton, VA 23681

8 th AIAA/NASA/USAF/ISSMO Symposium on

Multidisciplinary Analysis and Optimization

September 6-8, 2000 / Long Beach, CA

AIAA 2000-4848 MULTIDISCIPLINARY TECHNIQUES AND NOVEL AIRCRAFT CONTROL SYSTEMS Sharon L. Padula." James I_. Rogers, _ and l)avid I_ Raney _ NASA l_angley Research (?enter Hampton, VA 23681 aircraft conceptual design (see for example Ref. 1) assumes a small number of conventional control The Aircraft Morphing Program at NASA Langley devices: ailerons (roll effectorsl, elevator (pitch Research Center explores opportunities to improve effector), and rudder (yaw effector). The size and location of these devices can be estimated b} using airframe designs with smart technologies. Two elements of this basic research program are historical databases for component weight and control effectiveness information. The detailed control law multidisciplinary design optimization (MDO) and advanced flow control. This paper describes examples design is postponed until the aircraft configuration is where MDO techniques such as sensitivity analysis, frozen and until precise control moments are measured automatic differentiation, and genetic algorithms or predicted.

contribute to the design of novel control systems. In the test case, the design and use of distributed shape- The Aircraft Morphing Program envisions ne_ types of control devices, such as inflatable bladders or change devices to provide low-rate maneuvering capability for a tailless aircraft is considered. The oscillatory .jets distributed over the wing surface, _ hich ability of MDO to add value to control system achieve control by changing the real or virtual shape of development is illustraled using results from several the wing. 2 These novel control s_stems require an equally revolutionary control design process. This years of research funded by the Aircraft Morphing Program. paper suggests that MDO techniques, such as automatic differentiation for calculating sensitivities Introduction and genetic algorithm (GA) optimization procedures, are essential components of that new process.

Researchers who specialize in multidisciplinary design optimization (MI)O) and those who specialize in Control System Design Process optimal control have a natural affinity, since all use mathematical optimization techniques. Yet, few The proposed control system design process is examples of MI)() research including flight control as illustrated b3 application to a tailless fighter aircraft one of the disciplines exist. This apparent concept. The numerous control devices (effector contradiction stems from the traditional process of arrays) were m_xieled as generic shape-change devices designing aircraft flight control systems. Traditional that respond immediately to commands from the * Senior research scientist, Multidisciplinary Optimization Branch, MS 159, Senior Member AIAA.

: Senior computer scientist, Multidisciplinar3 Optimization Branch, MS 159.

Senior research scientist, D3 namics and Control Branch. MS 406, Member AIAA.

Copyright © 20(X) b 3 the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17. U.S. Code. The tl.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental Purposes. All other rights are reserved by the copyright o_vner.

I American Institute of Aeronautics and Astronautics height h along that normal are described by Park in

controller.Thiskindof simplification wasjustified

because the aim of the paper is to demonstrate MDO Ref. 7. For the present ICE configuration model, these calculations required the derivatives of 3 outpu!

techniques.

quantities with respect to 1394 input quantities.

Figure I illustrates the proposed control system design Calculating such a large number of derivatives was practical because of the adjoint option in version 3.0 of proc'css. In Fig. 1, the arrows indicate the flow of data the AI)IFOR software. _ Park estimates that he spent a and the boxes represent steps in the design process.

total of about one week modifying the PMARC codc For example, the first step is to develop the vehicle concept and the final step is to develop and test the and applying ADIFOR-3.0. Calculating the control law. The arrow which connects the final box to sensitivities required about one hour of CP[I time on a the initial box indicates that unsatisfactory results from high speed engineering workstation.

the control law design may necessitate changes in the Given a 3xl394 matrix of partial derivatives for grid vehicle concept. In Fig. 1, the boxes with a dark border represent steps where MDO techniques can add value. locations on the right wing, the partial derivatives for The boxes with a light border represent steps where corresponding locations on the left wing were engineering judgement is especially important. Each constructed by' assuming the),' had same magnitudes but step in the process is described in this section. the roll and yaw derivatives had opposite sign. Given this matrix of all partial derivatives, the moments m Development of Vehicle Concept resulting from activation of any set of effectors was estimated with a matrix multiplication: The first step in this multidisciplinary control system design process was to create a computational fluid ,"It -- [ B]U m u (I) dynamics (CFD) model of the vehicle and the control devices. In the present research, generic shape-change where U,,,,d is an n-vector of device heights h and the devices controlled a representative aircraft matrix B is constructed by' selecting the n columns configuration called ICE (Innovative Control associated with those devices. Note that this methcxl of Effectors), created by Lockheed Martin _. The ICE estimating moments implies linear superposition, design, described in Refs. 3 and 4 and shown in Fig. 2, which neglects control effector interactions. On the was used under a cooperative agreement with other hand, if m,m d represents a vector of desired roll, Lockheed Martin. The effectors were shape-change pitch, and yaw moments, then U,,,d can be calculated by devices modeled as bumps on the surface of the wing.

using the pseudo-inverse allocation method discussed For the current proof-of-concept studies, existing grid in Ref. 9: points were deformed in the direction of the surface normal to represent potential shape-change devices.

= m_,,,,d (2) The CFD model of the ICE configuration is evaluated U md BT[BBr] -l by an aerodynamic panel code called PMARC. 5 where T denotes the matrix transpose. As in Ref. 9, Prediction of Cotttrol Moments heights were restricted to positive values less than some maximum achievable device height. Therefore, The second step was to predict the sensitivities of the if any element of U,,,d was negative that element could control moments to a shape change at each grid point be set to zero, and the corresponding clement on the of the CFD model. The sensitivities were obtained by opposite wing could be increased by, the same value.

automatic differentiation of PMARC with the ADIFOR Then the resulting vector was normalized so that thc code generation tool." The sensitivities required were maximum element equaled the maximum achievable device height. Finally, Eq. (1) was evaluated and the g,C_ OC,,, 8C,,/, the partial derivatives of roll (CO,

7" ' ,,h j

target moments m,,,,_ were compared with the achievable moments m.

pitch (C,,,), and yaw (C,,) moments with respect to a displacement h. The details of calculating a surface Definition of Effector Array Candidates normal at evcry grid point and apply'ing a change in Equations (i) and (2) were used to estimate the control The use of trademarks or names of manufacturers in moments produced by an array of effectors on the ICE this report is for accurate reporting and does not vehicle. An interactive design tool was created to let a constitute an official endorsement, either expressed or researcher quickly' build up and analyze potential implied, of such products or manufacturers by the locations (i. e., effector arrays) by' selecting grid points National Aeronautics and Space Administration.

American Institute of Aeronautics and Astronautics and assigning device heights. This interactive tool, would create a very similar plot. The candidate which uses MATLAB software developed by The effector arrays were located in 7 regions: on the upper MathWorks, Inc., helps the control law designer to and lower leading edge (I,E), on upper and lower determine good potential locations so that each device trailing edge (TE), on upper and lower wing tips, and can produce the forces and moments required to on the upper surface near the middle of the wing. The maneuver the vehicle. Figure 3 illustrates the graphical arrays were not necessarily' disjoint. In fact, some user interface for the MATLAB-based effector array devices were members of several effector arrays.

design tool. The wing planform on the left side of the display presents the CFI) grid. The three contour plots The ellipses in Fig. 5 indicate a suite of four effector on the right side of the display present the sensitivity arrays that were studied in Ref. 9. This suite was used data produced by step 2. The top contour plot indicates in a six-degree-of-freedom dynamic simulation to roll moments on the top and bottom wing surfaces, the investigate the unaugmented and augmented aircraft middle plot indicates pitching moments, and thc dynamics. Results of that simulation are reported in bottom plot indicates yaw moments. Ref. 9 and indicate that a 10-degree-per-second roll rate is the maximum achievable with this suite of Once an array of effectors were defined by, specifying effectors. Those results suggest that this particular locations and heights, the designer obtained a suitc could be valuable for mild maneuvering or could preliminary prediction of the arrays effcctiveness based bc used in an autopiiot to keep the wings level but on Eqs. ( I ) and (2) and AI)IFOR sensitivities. Next, he could not take the place of conventional control surfaces.

generated a perturbed geometry grid that included the deployed effector array. This geometry, file could then be reevaluated with the PMARC aerodynamic analysis Selection of Optimal Effcctor Arrays program. New sensitivity derivatives were calculated based on the perturbed PMARC results. These The manually selected effector suite studied in Ref. 9 PMARC-based sensitivities could be used with Eqs. (1) required 82 individual devices and did not completely and (2) to further assess the effectiveness of the array'. meet the goals set by the designer. Given enough time For the present study', perturbed device heights were and good intuition, the designer might have selected other effector suites with better characteristics.

less than or equal to 0.2 feet.

Alternately., the initial exploration for candidate effcctor suites can be accomplished _ith discrete Figure 4 displays a 17x41 grid of candidate device optimization techniques. For example, Ref. 10 locations on the right upper wing. Circles indicate one selected effector array'. This array: contains 28 devices. contains a literature survey of actuator placement research that indicates gocud results for a wide range of A right and left pair of these wing tip arrays can provide the estimated roll, pitch, and yaw moments applications.

shown in the first column of Table 1. Reanalysis of With the MATLAB-based tool, the designer can define these same devices with a deformed grid in PMAR(, a large number of arra3s and then can predict their gives slightly' modified estimates shown in the second column of Table I. In this way, 34 different effector control moments by using the PMARC analysis code.

However, selecting the best set of arrays from this arrays were selected and evaluated. The two methods of estimating moments were quite consistent: in potential pool is a combinatorial problem that can be solved by using MDO techniques. The current study general the A1)IFOR estimates were smaller than the PMARC estimates. Therefore, reliance on ADIFOR selected a GA-based optimization approach. The goal of the optimization was to reduce the number of estimates for optimization should produce a conservative design. devices required and to satisfy control effectiveness criteria. The GA is not suggested as a replacement for Table I. Control Moment Estimates Compared with the designer, but as a tool for screening potential PMARC Reanalvsis Results. effcctor arrays and thus allowing the designer to Moment ADIFOR PMARC consider only the most promising.

Roll - 1.44E-04 - 1.56E-0-1- A multilevel GA, used to select control device Pitch -2.02E-04 -2.20E-04 locations, is described in Ref. I 1. The goal of the GA Yaw -0.22E-04 -0.28E-04 is to select the minimum set of devices that can provide the required uncoupled control moments (e. g., provide The PMARC estimates for the effector arrays are sufficient pitching moment without adverse roll or plotted in Fig. 5. Estimates based on Eqs. (1) and (2), yaw). That GA technique was tuned and validated with sensitivity information generated by ADIFOR, with a simplified wing model, t H-_ American Institute of Aeronautics and Astronautics this problem, the GA was allowed to pick at most one In the current study, the GA described in Ref. 11 was array from each of the 7 regions shown in Fig. 5. "Ihe adapted to select effector arrays on the ICE model. GA implementation would be simplified if each region had the same number of arrays. So, because the upper Again, the objective was to find the minimum number of devices required to provide uncoupled roll, pitch, TE region contained 8 arrays, arrays in the other and yaw moments. Each member in the GA population regions were duplicated until each region contained 8 represented one possible effector suite. Each arrays. In this way, the string length for the phase !

individual was evaluated three times to determine roll, GA was set to 7 and each digit in the string could have a value between 0 and 8. Thus, the number of possible pitch, and yaw moments and to compare these achievable values with the target values. Failure to combinations was 9 7 (approximately 4,800,000), meet any of these targets caused a penalty to be added although not all of these combinations represented to the fitness (i. e., objective) function. unique effector suites. The duplication of effector arrays should not have had a significant impact on the convergence of the GA. Although. duplication does In Ref. 11, the penalty has a fixed size. After some experimentation, a step-linear penalty was determined produce many members of the population with the to work better. Thus the fitness function, J, can be same fitness value, a GA is especially well suited for written as follows: optimization problems with this characteristic.

The GA population size for phase I was 200. An initial J=n+ w, +w,,,( cm +w,, population of members was produced randomly and their fitness evaluated. Successive generations were produced by the GA operations of tournament selection, uniform crossover, and mutation, with a where n is the number of devices, w_ are the minimum mutation rate of 5%. The maximum number of penalties, C_ are the moments and C,* are the targets.

generations was set to 300. A single execution For cases reported in this paper, w_ = 150 if the consisting of 300 generations of the GA procedure moments are less than the targets (e. g., if C_ < C_* ) requires about one hour on a engineering workstation.

and w, = 0 otherwise. The number 150 is an For complete descriptions of GA techniques and appropriate size for the minimum penalty because it definitions of GA parameters, see Ref. 13.

has the same order of magnitude as n judging from the number of devices in the manually-selected suite.

Typical results of the GA are shown in Fig. 6. Notice Thus, the GA will be encouraged to drive all the that the scales on each figure are different in order to penalty terms to zero by choosing a sufficient number emphasize several points about the convergence of devices.

history. Figure 6a shows the value of the fitness function averaged over all population members in each For the current study, the GA was developed and tested generation. The maximum and minimum fitness is also in two phases. During phase 1, the 34 effector arrays plotted. Figure 6a shows clearly that the population selected by Raney were defined as the set of possible maintained adequate diversity for 300 generations.

arrays. The GA was allowed to select up to 7 arrays Figure 6b shows the average and minimum fitness for from the original 34. The effectiveness estimate for the first 20 generations. Notice that the average fitness each suite of effector arrays was based on the PMARC reduced dramatically over this interval and approached reanalysis data used in Ref. 9. The target values (see the value of 150. This trend suggests that all members Eq. 3) were set to C/* = 6.0E-04, C,,,* = 5.0E-04, and of the population were converging towards designs C,,* = 3.0E-04. During phase II, a set of 349 whose achievable moments were close to or better than individual devices could be selected independently.

the target values. Figure 6c shows the best fitness ever The effectiveness estimate for each stfite of devices calculated as a function of generation number. Figure was based on the ADIFOR data. The target values for 6,: indicates the original population contained at least phase I1 are set to Ct* = 5.0E-04, C,,,* = 5.0E-04, and one member with a fitness less than 150, suggesting C,* = 1.0E-04; the targets were reduced because the that some members met all the performance targets.

estimates based on ADIFOR data consistently Notice also that the best individual had 96 devices and underestimated roll and yaw.

was foufid before generation 20.

Obviously, phase 1 was a much smaller combinatorial Even though 300 generations and about 60,000 problem, but it had its own complications. For different effector suites (i. e., about 1.25% of all example, some of the arrays overlapped and so not all possible combinations) were evaluated by the GA, possible combinations were allowable. To circumvent apparently no combination with fewer than 96 devices American Institute of Aeronautics and Astronautics cotdd be found. This global best design is pictured in proportional fashion: each device in an effector arra 5 Fig. 7. The design had three arrays on the upper wing was set to the same height. As greater moments were surface and two on the lower surface for a total of 96 required by the control system, the height of a given devices in 5 arrays as compared to 82 devices in 4 effector array was increased until the limiting height arrays selected by the manual method. However, this was reached. By using these arrays, the control system was able to stabilize and maneuver the vehicle without 96-device design met all the targets while those selected by the manual method did not.

conventional moving surfaces such as ailerons or a rudder. The predicted authority of these devices was Based on encouraging results from phase I, the GA was still rather low when compared with that of a rudder or used to select individual devices from a set of 349.

aileron, so the control system generated relatively love- This set contained all of the devices that made up the rate maneuvers (roll rates of 5 to 10 degrees per 34 arrays in phase 1, plus some other devices that second). Future research will focus on experimental seemed promising. validation of the predicted authority of various flow control devices and on better estimates of their effectiveness.

The phase II GA was tested with a variety of crossover strategies and mutation rates. A typical execution of the phase II GA requires about 8 hours of CP[I time.

Figure 10 contains time history plots comparing the The choice of strategy does not seem to affect the phase I GA suite (thick solid), the original manually- results or the efficiency very much. The results in Fig. selected suite (thin dashed), and the ideal moment 8 were produced with a single-point crossover and a targets (thin solid). The roll and yaw moments coming mutation rate of 1 _.

from the GA suite Icx)k very good-slightly better than those from the original suite. Both suites cause an The members of the phase II GA population were undesired pitch perturbation (see Fig. 10b), but the GA initially assigned random numbers from 0 to 349. If" results in smaller pitch transients during the maneuver.

the same random number was generated more than The crosswind gust capabilit) of the GA effector suite once in any member, then all the duplicates were set to was about the same as the original: the GA suite could zero. Duplicates were similarly set to zero following tolerate 29.5 ft/s crosswind gust, while the original crossover and mutation operations. The string length suite could withstand a 28 ft/s gust. From this analysis, was set to IO0, so the number of possible combinations the phase I GA appears to have found a good solution.

Further testing is required to evaluate the phase II (iA solution.

(,0O) ,0_, of 100 devices chosen from 349 was 349 ,-4x Because the number of combinations was much larger Concluding Remarks than in phase I, the population size was increased to 300 and the maximum number of generations was set This paper summarizes several years of research to 500. Thus, about 150,000 individual members were supported by the Aircraft Morphing Program at NASA evaluated during each repetition of the GA. This l.angley Research ('enter. The paper emphasizes the number represented a tiny percentage of all possible use of MDO techniques applied to the control system combinations.

design process for novel aircraft configurations. One such novel aircraft concept is the Lockheed-Martin ICE Figure 8 shows typical convergence performance for configuration with distributed shape-change devices to the GA. and Fig. 9 shows the 45 devices selected by generate control moments. Starting with a CFI) model this execution of the GA. Notice in Fig. 8, that even of the ICE configuration, this paper shows hmv the the "best ever" fitness value was initially above 500; control moments were estimated, promising effector this high initial value means that all individuals in the locations were proposed, optimal subsets of these population were heavib penalized. After about 100 proposed locations were chosen and the control s3stem generations, the best fitness dropped sharpb, indicating was simulated and tested.

that a design meeting the targets had been found.

Obviously, the advanced flow control research is not Simulation of the Control System complete. The concepts must be tested in the wind tunnel, and more effective shape-change devices must Several effector suites defined with the MATLAB- be sought. Moreover, both CFD predictions and flight based tool have been applied to the ICE vehicle in a control simulations need to be improved and validated.

simulation and used in a stability augmentation and These improved prediction capabilities may influence control system design (Ref. 9). For the present study, the fitness function used for optimization as well as the the control system deployed the effeclors in a American Institute of Aeronautics and Astronautics implementation details used for automatic University, Department of Computational and differentiation. Applied Mathematics, Jan. 2000.

9. Raney, David L., Montgomery, Raymond C., Park, While acknowledging that flow control research is in Michael A., and Green, l,awrence L., "Flight its infancy, this paper demonstrates useful MDO Control Using Distributed Shape-Change Effeclor techniques that will be available to the control Arrays," AIAA Paper 2000-1560, Apr. 2000.

designers of the future. The automatic differentiation 10. Padula, Sharon L., and Kincaid, Rex K.: techniques demonstrated with the ICE model and PMARC code are easily adaptable to other CFD "Optimization Strategies for Sensor and Actuator models and codes. Likewise, the genetic algorithms Placement," NASA TM- 1999-209126, Apr. !999.

developed herein can be used with improved control- 1. Rogers, James L., "Optimum Actuator Selection effectiveness measures to find the best locations for a with a Genetic Algorithm for Aircraft Control," wide variety of shape-change effectors. Thus, Intelligent Engineering Systems Through Artificial simulation techniques such as CFD, as well as MDO Neural Networks, Vol. 9, edited by Dagli. Buczak, techniques such as GA and automatic differentiation, Ghosh, Embrechts, and Ersoy, ASME Press, New empower engineers to explore revolutionary control York, 1999, pp. 355-360.

concepts for aircraft of the future.

12. Cook, A., and Crossley, W., "Genetic Algorithm References Approaches to Smart Actuator Placement for Aircraft Flight Control," AIAA Paper 2000-1582, 1. Raymer, Daniel P., Aircraft Design: A Apr. 2000.

Conceptual Approach. AIAA, Washington, I)C, 13. Goldberg, 1)., Genetic Algorithms in Search, 1992.

Optimization and Machine Learning, Addison- 2. Wlezien, R. W., Horner, G. C., McGowan, A. R., Wesley Publishing Co., New York, 1989.

Padula, S. L., Scott, M. A., Silcox, R. J., and Simpson, J. O., "The Aircraft Morphing Program," AIAA Paper 99-1927. Apr. 1998.

3. Scott, M., Montgomery, R., and Weston, R., "Subsonic Maneuvering Effectiveness of High Performance Aircraft Which Employ Qtmsi-Static Shape Change Devices," SPIE 1998 International Symposium on Smart Structures and Materials, Paper 3326-24, pp. 223-233.

4. Dorsett, K. M., and Mehl, D. R., "Innovative Control Effectors (ICE)," Wright Laborator3: Report, WL-TR-96-3043, Jan. 1996.

5. Ashby, D., Dudley, M., lguchi, S., Browne, L., and Katz, J., "Potential Flow Theory and Operation Guide for the Panel Code PMARC_12," NASA Ames Research Center, Moffett Field, CA, Dec.

1992.

6. Carle, A., Fagan, M., and Green, L., "Preliminary Results from the Application of Automated Adjoint Code Generation to CFL3D," AIAA Paper 98-48078, Sept. 1998.

1. Park, M., Green, L., Montgomery, R., Raney, D., "Determination of Stability and Control Derivatives using Computational Fluid Dynamics and Automatic Differentiation," AIAA Paper 99-3136, June 1999.

8. Carle, A., and Fagan, M., "Overview of ADIFOR-3.0," CAAM-TR 00-02, Rice American Institute of Aeronautics and Astronautics Fi__..._.l re s Develop concept vehicle _1 _ Predict control moment sensitivities .......................................................... _....................................................... IIFrom CFD model i, Define i !i effector array candidates Use GA to select best arrays ........... _ 1_ , Simulate and test controls Fig. 1 Control system design pr_xzess.

American Institute of Aeronautics and Astronautics Area - 75.12.m 2 (808.6ft 2) Span - 11.43m (37.5ft) Aspect Ratio - 1.74 LE sweep - 1.134 rad (65 deg) Fig. 2 Lockheed-Martin Innovative Control Effector (ICE) configuration.

Fig. 3 Effector array design tool graphical user interface.

American Institute of Aeronautics and Astronautics 20 i i i f i i i i i O88081_ O00EZ33D Span, ft 10 0 -- i __i 1 .--J. 1 L J --J- 0 5 10 15 20 25 30 35 40 45 5o Chord, ft Fig. 4 One of the 34 candidate effector arrays on the right _ving upper surface.

8.E-04 I ! I• Roll 6.E-04 I Pitch i i wya w 4.E-04 c" d) •-- 2.E-04 O d) 0 0.E+00 O t- -2.E-04 d_ E O -4.E-04

'1

I

4;-.

-6.E-04 Lower I ' Upper tip Upper mid Upper TE V i i Lower TE tip : Upper LE i -8.E-04 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Design number Fig. 5 Control authority plots for candidate effector arra3 s: 7 regions.

American Institute of Aeronautics and Astronautics

l,tl ,ik ,lJl,,,Atll tit;,ili

120C 80O Fitness 12 oo

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oo 40C oo oo Span, fl 10 oo oo 0 eo .... , .... . ....... L ........

oo 0 100 200 300 oo _o Generation number I---.Mmimurn -iAvera_le -- MaxirnumJ (a) Population extremes | 5 10 15 20 25 30 35 40 45 50 1600.

Chord. It 1400- 1200.

(a) Upper surface arrays 1000- 2( 800.

Fitness 600- 400- 200- 0. 14 0 5 10 15 20 oo Generation number @o I'-,',- Minimum _Avera_le I Span, _ 10 oo eo oo oo (b) Minimum and average 5 10 15 20 25 30 35 40 45 50 Fitness 8o Chord ft 4O (b) Lower surface arrays 2O Fig. 7 0 Best effector suite found by phase I GA.

5 10 15 Generation number ;;i!i_i,;_ii!iii_!i _i _ i_ III_ _ ili _i i_i!_i'!i (c) Global best Fitness Fig. 6 Genetic algorithm convergence history.

2OO 100 2oo 30o 4oo Generation number I_Maximum IAVe_age ........ Best ever I Fig. 8 Genetic algorithm convergence history.

American Institute of Aeronautics and Astronautics X 104 o. ..........

o ..............

o e o _ ooo .... o oo .... o

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-1 e -2 -3 J 0 -4 0 5 10 15 5 10 15 20 25 30 35 40 45 50 Chord, It Time, sec (b) l.ower surface devices (c) Ya_v moment time histories Fig. 9 Best set of devices found by phase II GA.

Fig. 10 Simulation time histories generated in reslxmse to a +/- 20-deg bank angle doublet command.

(_ ideal _ Phase 1 GA ....... Ref. 9) 10 `3 i 0.6 0.4 Roll o -02 -04 -06 !

-08 i i -1 5 10 15 Time, sec (a) Roll moment time histories I1 American Institute of Aeronautics and Astronautics