CONTROL LAW DESIGN IN A COMPUTATIONAL AEROELASTICITY ... · INTRODUCTION Aeroelasticity is the mutual interaction between aerodynamics and a flexible body. Aeroelasticity has been

American Institute of Aeronautics and Astronautics1

CONTROL LAW DESIGN IN ACOMPUTATIONAL AEROELASTICITY ENVIRONMENT

Jerry R. Newsom Harry H. Robertshaw Rakesh K. KapaniaNASA LaRC Virginia Tech Virginia TechHampton, VA Blacksburg, VA Blacksburg, VA757-864-6051 540-231-7196 540-231-4881

ABSTRACT

A methodology for designing active control laws in acomputational aeroelasticity environment is given.The methodology involves employing a systemsidentification technique to develop an explicit state-space model for control law design from the output ofa computational aeroelasticity code. The particularcomputational aeroelasticity code employed in thispaper solves the transonic small disturbanceaerodynamic equation using a time-accurate, finite-difference scheme. Linear structural dynamicsequations are integrated simultaneously with thecomputational fluid dynamics equations to determinethe time responses of the structure. These structuralresponses are employed as the input to a modernsystems identification technique that determines theMarkov parameters of an “equivalent linear system”. The Eigensystem Realization Algorithm is thenemployed to develop an explicit state-space model ofthe equivalent linear system. The Linear QuadraticGuassian control law design technique is employed todesign a control law. The computational aeroelasticitycode is modified to accept control laws and performclosed-loop simulations. Flutter control of arectangular wing model is chosen to demonstrate themethodology. Various cases are used to illustrate theusefulness of the methodology as the nonlinearity ofthe aeroelastic system is increased through increasedangle-of-attack changes.

INTRODUCTION

Aeroelasticity is the mutual interaction betweenaerodynamics and a flexible body. Aeroelasticity hasbeen and continues to be an extremely importantconsideration in many aircraft designs1. To addressundesirable aeroelastic effects or phenomena, thestiffness of the wing is often increased, adding weightto the aircraft and decreasing the overall performance.

The control of aeroelastic response through feedback tocontrol surfaces, or more recently through feedback toactive materials, is an alternative to “passive control”

through increased stiffness. Noll2 presents a review ofactive control methods, wind-tunnel experiments, andflight experiences associated with feedback control andaeroelasticity. Currently, most aeroelastic analyses areroutinely conducted using linear aerodynamic and linearstructural models. However, within the last fewdecades, a significant increase in advancing methods toconsider nonlinear aeroelasticity, especially nonlinearaerodynamics in the transonic region, has taken place.Dowell3 presents an excellent overview of nonlinearaeroelasticity and major problems where nonlineareffects should be considered.

With the maturity of CFD codes, their incorporationinto aeroelastic analyses, both static and dynamic, isbeginning to occur, primarily in the researchcommunity. This relatively new field has been termed“computational aeroelasticity” and involves couplingstructural elasticity (static and/or dynamic) andcomputational fluid dynamics together to perform timedomain analyses, where the aerodynamics are anonlinear function of the deformation of the aircraft.Most of the work in computational aeroelasticity hasemployed solutions to either the nonlinear potentialequation or the Euler equation for aerodynamic forces.Bennett and Edwards4 provide a status of recentdevelopments in computational aeroelasticity, with anemphasis on unsteady transonic flow, and results ofsome applications.

Computational aeroservoelasticity involves couplingstructural dynamics, computational fluid dynamics, andactive control systems together. Batina and Yang5

were perhaps the first researchers to examine control ofan aeroelastic system in a computational aeroelasticityenvironment for transonic flow. They conductedstudies with a 2-d airfoil with plunge and pitchdegrees-of-freedom and a 2-d small-disturbancetransonic CFD code. The effect of a simple constantgain control law utilizing displacement, velocity, andacceleration feedback on the time responses wasdetermined. Comparison with linear theory indicatedthat the frequency and damping values weresignificantly different for transonic and linear subsonic

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theory results. References 6-10 are other examples ofresearch in control of aeroelastic systems within acomputational aeroelasticity environment. Similar toBatina and Yang, these studies also only involvevarying the gains of simple feedback control laws tostudy their effect on the response of the aeroelasticsystem. Two-dimensional and three-dimensionalsmall-disturbance and Euler CFD codes are used inthese studies. The studies show that feedback controlcan be effective in suppressing transonic flutter.

Guillot and Friedman11, 12 employed adaptive controltheory to design control laws using a CFD technique.A 2-d airfoil model, with a trailing-edge controlsurface, was used with an Euler CFD code to performthe computational aeroelastic solutions. An adaptivecontrol law was used because of the assumed nonlinearbehavior of the system in the presence of nonlineartransonic flow with large shock motions. Theadaptive control law involved identifying a linear auto-regressive moving average (ARMA) model and thendetermining an optimal full-state control law. At theend of a learning period, the control law was engagedand the ARMA model and control law were thenupdated at each subsequent time step during thecomputational aeroelasticity simulation. An adaptivecontrol law was shown to be quite effective insuppressing transonic flutter with strong shocks.

The objective of this paper is to describe a generalmethodology to design control laws in the context of acomputational aeroelasticity environment. Thetechnical approach involves employing a systemsidentification technique to develop an explicit state-space model for control law design from the output ofa computational aeroelasticity code. Although thereare many control law design techniques available, thestandard Linear Quadratic Guassian technique isemployed in this paper. The computationalaeroelasticity code is modified to accept control lawsand perform closed-loop simulations. Numericalresults for flutter suppression of the Benchmark ActiveControl Technology wind-tunnel model are given toillustrate the approach.

DESCRIPTION OF TECHNICALAPPROACH

The overall methodology begins with performing acomputational aeroelasticity simulation (uncontrolled)with prescribed control surface inputs to obtain a set of

corresponding output time histories. The next step isto employ a system identification technique, using thetime histories of outputs and inputs from the firststep, to determine an “equivalent linear system” for useas a control law design model. Next, design of acontrol law can be performed using any control lawdesign technique. Finally, the control law is evaluatedin the computational aeroelasticity simulation. If thecontrol law performance is not adequate, the controllaw can be redesigned and evaluated again until thedesired performance is obtained. A detailed descriptionof the technical approach is given in Ref. 13. A briefoverview will be given in this paper.

Computational Aeroelasticity Simulation

Computational aeroelasticity simulation involvesintegrating the structural, aerodynamic (CFD), and forthis paper, control equations simultaneously. Thispaper focuses on the transonic case where theaerodynamic fluid flow equations contain nonlinearterms. The particular code that is employed is theCAP-TSD14 code that has been developed at the NASALangley Research Center. The CAP-TSD code is afinite-difference code which solves the transonic small-disturbance aerodynamic equation. The primary outputsof the CAP-TSD code are time histories of thepressures and the generalized coordinate displacements,velocities, and accelerations. It has been used on awide variety of configurations for steady and unsteadypressure distribution calculations and for calculatingtransonic flutter characteristics, including nonlinearlimit-cycle instabilities.

The basic equations of motion implemented in CAP-TSD are;

[ ]{ ˙} [ ]{ } [ ]{ } { }M q C q K q F+ + = (1)

The primary difference between linear aeroelasticitymethods and computational aeroelasticity is in thecomputation of the aerodynamic pressure coefficientCp

used in computing the generalized aerodynamic

force vector { }F .

The CAP-TSD code is a finite difference code thatsolves the general-frequency modified TSD potentialequation. The TSD equation is solved within CAP-TSD by a time-accurate approximate factorization (AF)algorithm developed by Batina15. The algorithmconsists of a Newton linearization procedure coupledwith an internal iteration technique. The CAP-TSDcode is capable of treating configurations with


multiple lifting surfaces and bodies. A relativelysimple Cartesian grid is input along with thecoordinates defining the geometry of the configurationand the corresponding surface slopes. The pressurecoefficient is calculated at each time step and isemployed to calculate the generalized force vector ateach time step.

Equation 1 is rewritten in state-space form as

X A X B u{ } = [ ]{ } + [ ]{ } (2)

The numerical algorithm, employed in CAP-TSD, forsolving equation 2 is

X X B u un n n n+ −= + −1 13 2φ θ ( ) / (3)

where

φ

θ ττ

=

= −∫

e

e d

A T

A TT

∆

∆∆

( )

0

and ∆T is the integration time step.

Control Design Model Development

Most control law design methods require an explicitmathematical model of the system to be controlled.A computational aeroelasticity simulation providestime histories of the variables of an aeroelasticsystem, but does not generate an explicitmathematical model of the system. Computationalaeroelasticity simulations are analogous toperforming experimental investigations where theonly direct outputs are time responses. Therefore, toemploy the various control law design methods thatare available, a control design mathematical model ofa “computational aeroelasticity system” (CAS) mustbe developed. System identification techniques arewidely employed for developing a mathematicalmodel given experimental data. Therefore, since aCAS simulation is analogous to performing anexperiment, system identification is a logical choice.Juang16 provides a description of many of theadvances in system identification for vibration modaltesting and control of space structures. TheObserver/Kalman filter Identification (OKID)technique17 was developed primarily for developmentof a mathematical model for control law design. The

OKID technique has been applied to space structures,such as the Hubble Space Telescope16 and the ShuttleRemote Manipulator System18, and to theidentification of linear rigid aircraft models19. Becausethe OKID technique was developed primarily foridentifying models for control law design, it is thetechnique employed in this paper.

Description of Observer/Kalman FilterIdentification Technique. Given a set of inputand output data, the Observer/Kalman FilterIdentification (OKID) algorithm is shown in Fig. 1.One of the keys to the OKID algorithm is theintroduction of an observer into the identificationprocess. The fist step of the process is thecalculation of the observer Markov parameters. Thenthe system Markov parameters are obtained.

Consider a discrete time state-space model of a systemdescribed by a set of first order difference equations ofthe form

x k Ax k Bu k

y k Cx k Du k

( ) ˆ ( ) ˆ ( )

( ) ( ) ( )

+ = += +1

(4)

Solving for the output y k( ) in terms of the previous

inputs, with the assumption that the system isinitially at rest, i.e. x( )0 0= , yields

y k h u k iii

k

( ) ( )= −=∑

0

(5)

where the parameters

h D h CA B kkk

01 1 2 3= = =−, ˆ ˆ, , , ,....

are the system Markov parameters, which are also thesystem pulse response samples. To reduce thenumber of Markov parameters needed to adequatelymodel a system, an observer is introduced into theOKID technique. Adding and subtracting the term

Ky k( ) to the right hand side of equation (4) yields

x k A KC x k

B KD u k Ky k

y k Cx k Du k

( ) ( ˆ ) ( )

( ˆ ) ( ) ( )

( ) ( ) ( )

+ = + +

+ −= +

1

(6)

The matrix K can be interpreted as an observer gain.The parameters defined as


Y k C A KC B KD Kk

k k

( ) ( ˆ ) [ ˆ , ]

[ , ]

= + + −=

−1

β α (7)

are the Markov parameters of an observer system.

Consider the special case where K is a deadbeat

observer gain such that all eigenvalues of A KC+are zero, the observer Markov parameters will becomeidentically zero after a finite number of terms. Forlightly damped systems, this means that the systemcan be described by a reduced number of observerMarkov parameters. Furthermore, an unstable systemcan be represented using this technique. This isobviously a major advantage for this research.

After the Markov parameters are computed using aleast squares technique, a state-space model of thesystem is developed using the EigensystemRealization Algorithm (ERA). A completedescription for the computation of the Markovparameters and the ERA is given in Refs. 17 and 20.

Control Des ign. There are many control lawdesign methods available. These range from classicalcontrol law design to the LQG method to H∞ robust

control law design methods to nonlinear control lawdesign methods. Because of its ease of use, the LQGdesign method is employed in this paper. Basically,the LQG design process involves minimizing a costfunction of the form

J y Qy u Ru dtT T= +∞

∫ [ ]0

(8)

where y is an output vector (e.g. accelerations, loads)of the system and u is the control input. Theresulting control law is u=-Gx where G is the statefeedback gain matrix that minimizes the cost functionand x is the state vector. Since the states of a systemare generally not all available for feedback, a Kalmanfilter is employed to estimate the states. Theresulting control law is of the form:

˙ ˆ

ˆ

x A x L y

u G x

{ } = [ ]{ } + [ ]{ }{ } = −[ ]{ }

(9)

where is the estimate of the state vector xand L is the Kalman filter gain matrix.

RESULTS AND DISCUSSION

The example employed in this paper to demonstratethe overall control law design methodology is activeflutter suppression for the Benchmark ActiveControls Technology (BACT)21 wind-tunnel modelshown in Fig. 2. The BACT model is a rigid,rectangular wing with a NACA 0012 airfoil section.It is equipped with a trailing-edge control surface andupper and lower surface spoilers that are controlledindependently by hydraulic actuators. Only thetrailing-edge control surface is employed in thispaper. For this study, actuator dynamics are ignored.For the control law designs, an accelerometer locatednear the outboard trailing edge is the assumed sensoremployed for feedback. The wing is mounted to adevice called the Pitch and Plunge Apparatus (PAPA)which is designed to permit motion in principallytwo modes – pitching and vertical translation(plunge). Therefore, the BACT has structuraldynamic behavior very similar to the classical twodegree-of-freedom problem in aeroelasticity. Thevibration frequencies, computed from a NASTRANmodel of the BACT, are 3.4 Hz (plunge) and 5.2 Hz(pitch). Structural damping is assumed to be zero.There are two CAP-TSD aerodynamic representationsof the BACT wind-tunnel model used in this paper.The first one, shown in Fig. 3, is a 3-d model wherethe CFD grid contains 140 points in the chordwisedirection, 40 points in the spanwise direction, and 92points in the vertical direction for a total of 515,200grid points. The second one is an equivalent 2-dmodel where the CFD grid contains 140 points in thechordwise direction, only 2 points in the spanwisedirection, and 92 points in the vertical direction for atotal of 25,760 grid points.

Although some basic aerodynamic and fluttercalculations will be presented for the 3-d model, thecontrol law design and evaluation process results willbe demonstrated with the 2-d model. However, theexact same process would be applied to the 3-d case.

Basic Aerodynamic and UncontrolledFlutter Results

In order to calculate rigid aerodynamic pressures, thefluid flow equation (TSD potential equation) isintegrated in time by CAP-TSD without coupling thestructural dynamics equations (i.e. no aeroelasticeffects). Figure 4 shows the chordwise pressure

x{ }


distribution at the 60% span location at M=0.77 for 2degrees angles of attack. Results from CAP-TSD areshown with the experimental results of Ref. 21.Linear CAP-TSD results are also shown. At 2 degreesangle of attack, a weak shock near the 20% chord hasdeveloped indicative of transonic flow. At 5 degreesangle of attack (not shown), the shock moves aft toapproximately the 35% chord and becomes moderatelystrong. The CAP-TSD results capture the shocklocation and strength reasonably well. Of course, thelinear CAP-TSD results do not indicate a shock andare not in agreement with the experimental data onthe upper surface forward of the shock. However,both the CAP-TSD and linear CAP-TSD resultscompare very well with the experimental data aft ofthe 40% chord on the upper surface and compare wellalong the entire chord for the lower surface. Theapparent dip in the experimental upper surfacepressures near the leading edge has been attributed tothe transition strip at that location21. These resultsprovide some degree of confidence in the aerodynamicpredictions of CAP-TSD for angle-of-attack changes.

Figure 5 shows the chordwise pressure distribution atthe 60% span location at M=.77 for trailing-edgecontrol surface deflections of 2 degrees. Again,results from CAP-TSD are shown with theexperimental results of Ref. 21. CAP-TSD slightlyoverpredicts the supper surface pressure peak near thecontrol surface hinge line. CAP-TSD considerably(approximately 50%) overpredicts the upper surfacepressure peak for δ=5 degrees (not shown).Overprediction of control surface forces is typical forinviscid codes. Therefore, care should be used in realapplications to account for this overprediction whenusing an inviscid code.

For dynamic aeroelastic analyses in a computationalaeroelasticty environment, two steps are employed inperforming the calculations22. In the first step, astatic aeroelastic deformation is calculated to providethe initial flowfield for the dynamic aeroelasticsolution. The dynamic solution is a perturbationabout a converged static aeroelastic solution for eachMach number and dynamic pressure of interest. Theprocedure for calculating static aeroelastic solutions isto allow the structure and aerodynamics to interactwith no initial condition on the modal displacementsand velocities and no external excitation. A verylarge value (0.99) for structural damping is used toprevent divergence of the solution and to accelerateconvergence. Once a static aeroelastic solution iscomputed, the next step is to prescribe either an

initial condition on the modal displacements orvelocities or an external input. For fluttercalculations, initial conditions on the velocities areused to begin the dynamic structural integration.CAP-TSD was run using a value of 1 on both thedimesionless plunge and pitch modal velocity initialconditions and then allowed to decay freely. The ERAsystem ID method of Ref. 20 was employed toestimate modal dampings and frequencies at variousvalues of dynamic pressure. Figure 6 shows thedampings as a function of dynamic pressure (M=0.77)for a Doublet Lattice linear model, the CAP-TSD 3-dlinear model, and CAP-TSD 3-d model.

The CAP-TSD 3-d linear model shows approximatelya 9% greater flutter dynamic pressure than theDoublet Lattice linear model. The Doublet Latticelinear model shows approximately a 3% lower flutterdynamic pressure than the experimental value. TheCAP-TSD linear model shows approximately a 6%greater flutter dynamic pressure than the experimentalvalue. The CAP-TSD model shows approximately a2% greater flutter dynamic pressure than theexperimental value.

CAP-TSD 2-d Results

This section presents uncontrolled flutter results,initial system identification results, and resultsillustrating the sensitivity of the identified systemmodel. All of the results are for a Mach number of0.77 and employ the CAP-TSD 2-d model previouslydescribed. All of the CAP-TSD simulation resultswere calculated with an integration time increment of.000173 seconds and 8000 time steps.

Uncontrolled Flutter Results The ERAsystem ID method was also employed to estimatemodal dampings and frequencies for the 2-d cases. Theoverall character of the damping vs. dynamic pressurecurves (fig. 7) is very similar to the 3-d case. TheCAP-TSD linear 2-d model shows a flutter dynamicpressure which is approximately 19% lower than thecorresponding linear 3-d case. The CAP-TSD 2-dmodel shows a flutter dynamic pressure that isapproximately 21% lower than the correspondingnonlinear 3-d case. This magnitude of reductionbetween 3-d and 2-d is similar to that described inRef. 23.

Initial System ID Results In order to performa system identification (ID), an appropriate controlsurface input signal must be used to excite the system


modes. After exploring several input signals(sinusoid sweep and random), an exponential pulsesignal provided a good input signal for identifying anequivalent linear model of the CAP-TSD outputs. Inorder to test the system ID technique, CAP-TSD wasrun in the linear mode for several of the initial resultsthat will be shown. A dynamic pressure of 5.75 kPawas chosen to identify all of the 2-d system IDmodels. This dynamic pressure is between the flutterdynamic pressure predicted by CAP-TSD (linear) andthe flutter dynamic pressure predicted by CAP-TSD(nonlinear). The linear system ID models wereidentified using the plunge and pitch displacements,velocities, and accelerations as the measurementsignals. Figure 8 shows the results from the systemID procedure in terms of a comparison between theCAP-TSD outputs (linear) and two system ID modeloutputs (using the same exponential pulse input).

A 5-state system ID model was the first oneidentified. This 5-state model matches the CAP-TSDplunge responses well. However, it does not matchthe CAP-TSD pitch response well during theexponential pulse time period. The order of thesystem ID model was increased and an 8-state modelwas identified. As can be seen in Fig. 8, the 8-statesystem ID model provides a very good representationof the CAP-TSD results for both the plunge and pitchresponses over the entire time history. Severaladditional system ID models with higher orders wereidentified, but no overall improvement (over the 8-state model) in the ability to match the CAP-TSDresults were seen. Most of the subsequent resultsemploy an 8-state system ID model.

In order to assess the sensitivity of the system IDmodel to changes in the control input signal, thesystem ID model (derived from the exponential pulseinput) was used in a simulation where the controlsurface input was a sine wave. This result employedthe outputs of CAP-TSD run in the nonlinear mode(including the aerodynamic nonlinear terms). Boththe outputs used as input to system ID procedure andthe simulation for the sine wave input were for thenonlinear CAP-TSD mode. Figure 9 shows resultsof the simulation using a 5 Hz and 10Hz sine waveinput in terms of a comparison of the system IDmodel with the CAP-TSD output. Only the plungeand pitch displacements are shown. The agreementbetween the system ID model and the CAP-TSDoutputs are excellent for the 5 Hz case. There issome slight mismatch of the peaks (especially for thepitch displacement) between the system ID model and

the CAP-TSD outputs for the 10 Hz case. However,the overall ability of the system ID model to matchthe CAP-TSD outputs for these different inputs isquite good.

Control Law Design and Evaluations

Five different cases will be presented to illustrate thedesign methodology. All of the cases are for a Machnumber of 0.77 and a dynamic pressure of 5.75 kPa. The cases begin with the linear case at 0 degrees angleof attack and then proceed with nonlinear cases at 0degrees angle of attack and 0.6 degrees angle of attack.For each angle of attack, a system-ID model isdeveloped and the control law designed with thesystem ID model for that angle of attack is comparedwith the control law for the linear case. Thecomparison is in terms of gain and phase margins,acceleration time history (using the exponentialcontrol input), and control surface deflection. Thecontrol law designed with the linear model is intendedto represent a control law using the state-of-the-artmethodology.

The primary design goal for all of the control lawdesigns is to increase the damping of the systemwhile exhibiting at least 6 dB gain margins and 60degrees phase margins. The gain and phase marginsare to account for uncertainty in the model. Inaddition, the control surface displacements, due to thefeedback command, should be less than 1 degree inorder to stay within a somewhat linear range of thecontrol surface displacement. In each of the LQGdesigns, the weighting on the output and input duringthe regulator design and the intensities of the noisematrices during the Kalman filter design were variedby trial and error until a design that met the goal wasdetermined.

Case 1 Case 1 involves developing a system IDmodel from the CAP-TSD (in a linear mode) outputs,designing an LQG control law, and evaluating theLQG control law using CAP-TSD in the linear mode.The results of the system ID process were previouslydescribed.

Figure 10 shows a comparison of CAP-TSD linearand the system ID linear model outputs for theuncontrolled and controlled (using the LQG controllaw) case. The results using the system ID model andthe CAP-TSD simulation for the controlled case arealmost identical. The controlled case indicates a muchhigher degree of damping than the uncontrolled case.


A gain margin of 8.52 dB and a phase margin of76.28 degrees were determined using a Bode plot (notshown) of the open-loop system (system ID modeland LQG control law). Figure 10 also shows thefeedback control surface command for Case 1. Themaximum control surface displacement isapproximately 0.3 degrees and occurs during theexponential pulse excitation.

Case 2. Case 2 involves evaluating the controllaw designed in Case 1 using the CAP-TSD nonlinearsimulation at an angle of attack of 0 degrees. In orderto calculate stability margins and design a control lawusing the results of the CAP-TSD nonlinearsimulation (Case 3), the CAP-TSD (nonlinear)outputs are used as the input to the system IDprocedure. Figure 11 shows the results from thesystem ID procedure in terms of a comparisonbetween the CAP-TSD (nonlinear) displacementoutputs and the system ID model. Similar to thelinear case, the system ID model provides a very goodrepresentation of the CAP-TSD outputs.

Figure 12 shows a comparison of CAP-TSD outputsfor the uncontrolled and controlled (using Case 1control law) case. The controlled response using theCAP-TSD nonlinear model is not as highly dampedas the linear model results. A gain margin of –4.75dB and a phase margin of 31.34 degrees weredetermined using a Bode plot (not shown) of theopen-loop system Although stable, the control lawdesigned using the linear model does not providesatisfactory stability margins. Figure 12 also showsthe feedback control surface command for Case 2.Similar to the Case 1, the maximum control surfacedisplacement is approximately 0.3 degrees and occursduring the exponential pulse excitation.

Case 3 Case 3 employs the system ID modelderived from the nonlinear CAP-TSD results of Case2 to design a control law. After some trial and errorduring the LGQ design process, a satisfactory controllaw was determined. Figure 13 shows a comparisonof CAP-TSD outputs for the uncontrolled andcontrolled case. A gain margin of –10.84 dB and aphase margin of 64.72 degrees were determined usinga Bode plot (not shown) of the open-loop system.Figure 13 also shows the feedback control surfacecommand for Case 3. Similar to the Cases 1 and 2,the maximum control surface displacement isapproximately 0.3 degrees and occurs during the

exponential pulse excitation. The results using thiscontrol law clearly show much better results, inparticular stability margins, than using the controllaw designed using the system ID model of the linearCAP-TSD outputs.

Case 4 Case 4 involves evaluating the control lawdesigned in Case 1 using the CAP-TSD nonlinearsimulation at an angle of attack of 0.6 degrees. Inorder to evaluate the stability margins and design acontrol law for Case 5, the system ID procedure isapplied to the outputs of CAP-TSD at 0.6 degreesangle of attack. Obtaining a good system ID modelfor this case was a significant challenge because ofthe sensitivity of the system ID procedure to aperceived bias (nonzero value at t=0), sensitivity tothe order of the model, and sensitivity to the value ofthe sampling time. In order to obtain a good systemID model, it was necessary to increase the samplingtime by a factor of forty (decrease the sample rate bya factor of forty). When the perceived bias wasremoved, the sample rate was decreased and the modelorder was 5, a good system ID model was obtained. Figure 14 shows the results from the system ID interms of a comparison between the CAP-TSDoutputs and the system ID model. Although thesystem ID model provides a good representation ofthe CAP-TSD results, it is not as good as the 0degree angle of attack case.

Figure 15 shows a comparison of CAP-TSD outputsfor the uncontrolled and controlled (using Case 1control law) case. The controlled system can be seento be slightly unstable. Figure 15 also shows thefeedback control surface command for Case 4. Theunstable character is clearly seen in the feedbackcontrol surface command.

Case 5 Case 5 employs the system ID modelderived from the nonlinear CAP-TSD outputs at 0.6degrees angle of attack to design a control law. Figure16 shows a comparison of CAP-TSD outputs for theuncontrolled and controlled case. There is asignificant reduction in the acceleration response withthe controlled case. A gain margin of –9.66 dB and aphase margin of 64.17 degrees were determined usinga Bode plot (not shown) of the open-loop system.Figure 16 also shows the feedback control surfacecommand for Case 5. Similar to the previous cases,the maximum control surface displacement isapproximately 0.3 degrees and occurs during theexponential pulse excitation. The results using thiscontrol law clearly show much better results, in


particular stability margins and damping, than usingthe control law designed using the system ID modelof the linear CAP-TSD outputs.

CONCLUSIONS

In summary, the objective of this paper was thedevelopment of a general methodology for designingactive control laws in a computational aeroelasticityenvironment. The methodology involves using amodern system identification technique to develop anequivalent linear model from the nonlinear simulationresults. Standard control law design techniques arethen used to design control laws. The resultingcontrol laws are then incorporated into the nonlinearcomputational aeroelasticity simulation forevaluation. Results of a numerical study applyingthis methodology to flutter control of the BACTwind-tunnel model were presented.

The major conclusions of this research are:

1. Equivalent linear models developed by employinga system identification technique can represent theinput-output relationship of a computationalaeroelasticity simulation very well.

2. For the BACT model used in this study, thesystem ID model represents the input-outputrelationship very well until an angle of attack of .6degrees where the transonic flow conditions cause theshock on the upper surface to move aft of the 40%chord. At this point, extreme care was needed toobtain a good system ID model.

3. A control law designed using a system ID modeldeveloped from a nonlinear simulation can control thenonlinear model better than a control designed using asystem ID model developed from a linearcomputational aeroelasticity simulation.

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[18] Scott, M; Gilbert, M;, and Demeo, M.: “ActiveVibration Damping of the Space Shuttle RemoteManipulator System,” Journal of Guidance, Control,& Dynamics., Volume 16, No. 2, Mar-April 1993,pp. 275-280.

[19] Chen, W. and Valasek, J.: “Observer/KalmanFilter Identification for On-line System Identificationof Aircraft,” AIAA-99-4173, August 1999.

[20] Juang, J.N. and Pappa, R.S.: “An EigensystemRealization Algorithm for Modal ParameterIdentification and Model Reduction,” Journal ofGuidance, Control, and Dynamics, Vol. 8, No. 5,1985, pp. 620-627.

[21] Scott, R. C., Hoadley, S. T., Wieseman, C. D.,and Durham, M. H.: “The Benchmark ActiveControls Technology Model Aerodynamic Data,”AIAA 97-0829-CP, January 1997.

[22] Silva, W. A. and Bennett, R. M.: “Investigationof the Aeroelastic Stability of the AFW Wind-TunnelModel Using CAP-TSD,” NASA TM 104142,September 1991.

[23] Bisplinghoff, R., Ashley, H., and Halfman, R.:Aeroelasticity, Addison Wesley PublishingCompany, Reading, Massachusetts, 1954.

Figure 1. Flow Chart for OKID

Figure 2. Photograph of BACT Wind-Tunnel Model

Input and OutputTime Histories

Observer MarkovParameters

SystemMarkov

Parameters

ObserverGain MarkovParameters

System Matrices A,B,C,D &Observer Gain Matrix G


Figure 3. CAP-TSD Mesh of BACT Model

Figure 4. Aerodynamic results α=2 degrees, M=0.77.60% semi-span

Cpu

Figure 5. Aerodynamic results δ=2 degrees, M=0.77,60% semi-span

Figure 6. Dampings as a function of dynamicpressure, 3-d

Figure 7. Damping as a function of dynamic pressure, 2-d

Figure 8. System ID results for displacements, α=0 degrees

0 1 2 3 4 5 6 7 8-1

0

1

2

3

4

5

6

7

8

9

10

Dynamic Pressure, KPa

Doublet LatticeCAP-TSD (Linear)CAP-TSD (Nonlinear)Experiment

Dam

ping

(%

)

Mode 1

Mode 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.1

-0.05

0

0.05

0.1

0.15

System ID Model (5 states)System ID Model (8 states)CAP-TSD Output (Linear)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.01

-0.005

0

0.005

0.01

0.015

Time (sec)

System ID Model (5 states)System ID Model (8 states)CAP-TSD Output (Linear)

Pitc

h0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1.5

-1

-0.5

0

0.5

1

1.5

x/c

CAP-TSDCAP-TSD (Linear)Experiment

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

2

3

4

x/c

CAP-TSDCAP-TSD (Linear)Experiment

Cpu

Cpl

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

x/c

CAP-TSDExperiment

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

1.5

2

x/c

CAP-TSDExperiment

Cpl

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25

30

Dynamic Pressure, KPa

CAP-TSD (Linear)CAP-TSD (Nonlinear)

Mode 1

Mode 2

Cpu

Cpl

Plun

ge

Dam

ping

(%

)


Figure 9 (b). Comparison of displacement responsesfor a 5 Hz sine wave input

Figure 9 (a). Comparison of displacement responsesfor a 10 Hz sine wave input

Figure 10. Case1 controlled results

Figure 11. Case 2 system ID results fordisplacements, α=0 degrees

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

CAP-TSD OutputSystem ID Model Simulation

0 0.2 0.4 0.6 0.8 1 1.2 1.4-4

-2

0

2

4

6

8x 10

-3

Time (sec)


0 0.2 0.4 0.6 0.8 1 1.2 1.4

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25


0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Time (sec)


0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

System ID Model UncontrolledSystem ID Model Controlled

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

Time (sec)

CAP-TSD (Linear) UncontrolledCAP-TSD (Linear) Controlled

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time (sec)

Fee

dbac

k C

ontr

ol S

urfa

ce C

omm

and

(deg

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.05

0

0.05

0.1

0.15

Time (sec)

System ID Linear ModelCAP-TSD

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time (sec)


Pitc

hPl

unge

Pitc

h

Plun

gePi

tch

Acc

eler

atio

n (g

)

Plun

ge

Acc

eler

atio

n (g

)


Figure 12. Case 2 controlled results

Figure 14. Case 4 system ID results fordisplacements, α=.6 degrees

Figure 13. Case 3 controlled results

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

UncontrolledControlled

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Fee

dbac

k C

ontr

ol S

urfa

ce C

omm

and

(deg

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)


0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time (sec)

Fee

dbac

k C

ontr

ol S

urfa

ce C

omm

and

(deg

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)


0 0.2 0.4 0.6 0.8 1 1.2 1.4-5

0

5

10

15x 10

-3

Time (sec)


Plun

geA

ccel

erat

ion

(g)

Acc

eler

atio

n (g

)

Pitc

h


Figure 15. Case 4 controlled results Figure 16. Case 5 controlled results

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)


0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time (sec)

Fee

dbac

k C

ontr

ol S

urfa

ce C

omm

and

(deg

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)


0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time (sec)

Fee

dbac

k C

ontr

ol S

urfa

ce C

omm

and

Acc

eler

atio

n (g

)

Acc

eler

atio

n (g

)

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CONTROL LAW DESIGN IN A COMPUTATIONAL AEROELASTICITY ... · INTRODUCTION Aeroelasticity is the mutual interaction between aerodynamics and a flexible body. Aeroelasticity has been