Ashish TewariIndian Institute of Technology, Kanpur 208016
India
AbstractAn active flutter suppression system is designed for atypical section in incompressible flow using optimalcontrol methodology. An optimized rational functionapproximation is chosen for the unsteady aerodynamictransfer function matrix. The output and controlweighting factors in the regulator performance indexare selected to yield the best possible closed-loop initialresponse, as well as the maximum controllerrobustness with respect to flight velocity, whilerequiring the least control effort. The failuremechanism of the closed-loop system changes fromflutter to divergence as the output weighting factor isincreased. Stability robustness with respect toatmospheric density and sensor location is alsoexamined. It is observed that an ideal sensor locationexists for the best closed-loop performance androbustness combination. Flutter margin is seen toincrease with an increasing altitude, thereby yieldingthe most critical design point at the sea-level.
Aeroelastic Model
Aeroservoelasticity (ASE) is a multidisciplinary studyof interactions among structural dynamics, unsteadyaerodynamics and control system, and is fast emergingas a primary tool in achieving desirable flightcharacteristics of modern aircraft. It has now becomepossible to organize the ASE mathematical modelingand control system design into a systematic process,with the emergence of new analytical techniques'"8.ASE design is a challenging problem in that the plantdynamics vary with flight conditions, and instabilitycan occur due to control system dynamics interactingwith the plant dynamics at any point in the flightenvelope. One of the favorable ASE applications isactive flutter-suppression of aircraft wings, and isstudied here with the simple example of a typical
section in incompressible flow. Several more designparameters may become important when consideringcompressible subsonic, supersonic or hypersonicspeeds. However, the design methodology presentedhere can be extended to these regimes.
The equations of motion of an aeroelastic system canbe expressed as:
[M]{^t)}+[C]{?(t)}H-[K]{^t)}={QJ(t)} (1)
where [M], [K], [C] are the generalized mass, stiffnessand damping matrices, respectively, (£(t)} is thegeneralized displacements vector, and {Q,(t)> thegeneralized aerodynamic force vector due to structuralmotion, gusts and control inputs, {u(t)>. For equation(1) to be expressed in a linear state-space form givenby
{X}=[A]{X}+[B]{u}+[F]{d} (2)
where {d} is the disturbance (process-noise) appearingat the plant output, the generalized unsteadyaerodynamic matrix transfer function must berepresented by a rational function approximation(RFA) in the Laplace domain, such as:
[Q(s)]=fAo]+[A1]s+[A2Js2+E[A(B+2)]s/(s-H)n)n=l
(3)
The conventional RFA1 of equation (3) uses the samelag-parameter (pole), bn , values for all elements of[Q(s)] while the matrix Fade RFA2 allows bn to varyfrom element to element. The coefficients [Ao],[Ai],...,[A<n+2)l are determined by a least-squares curvefit with the frequency domain data [H(ro)] obtainedfrom an oscillatory aerodynamic theory3'4 at a set offrequencies oo. The normalized squared error of thiscurve fit can be written as:
where RIJ(a)m)=max(lJH1J(cDm)|2). The RFA is verysensitive to the pole values, which are determined by anonlinear optimization process minimizing the totalfit-error:
NM N N
(5)
where wy are weighting factors. For each lag term inthe series of equation (3), several aerodynamic statesare introduced in equation (2). Other RFA methods arealso possible, such as the minmum-state method byKarpel5 . Refs. 7 and 8 showed that when the searchtechniques yield repeated optimum b0 values, thefollowing multiple-pole RFA is required to avoid theill-conditioned eigenvalue problem in equation (2):
Nl[Q(s)]=[Ao]+[A1]s+[A2]s2+S[A(n+2)]s/(s+bn)
Nln=)
n=Nl+1[A^2)Is2/(s+bn)2 (6)
where Nl is the total number of poles, (N2-N1) thenumber of poles repeated twice or more times, and soon. Tewari and Brink-Spalink9 gave an analyticalexplanation for the need for multiple-pole RFA. Amultiple-pole RFA is also more efficientlyoptimized, since each multiple-pole represents onlyone optimization variable as opposed to several for thesimple pole RFA which it replaces for a givenaccuracy.
A typical section with a trailing-edge control surface ischosen as a simple example to demonstrate the ASEdesign process. The mass coupling of the controlsurface without gap or overhang is included in thegeneralized mass matrix, and its rotational stiffness inthe generalized stiffness matrix. The unsteadyaerodynamic forces due to pitch (angle-of-attack), a,plunge (vertical displacement at the elastic axis), h,and control surface rotation, p, are modeled after theincompressible aerodynamics of Theodorsen andGarrick10, and the aerodynamic force vector is given by
, My(t), H(t)}T (7)
where L(t), My(t) and H(t) are the lift, pitching-moment, and control surface hinge-moment,respectively. In frequency domain, [Q(s>] depends uponthe Theodorsen function, C(k), which is approximated
here by a two pole optimum RFA representation"given by
where k=a>b/U is the reduced frequency, U thefreestream velocity and b the semi-chord of the section.The complete modeling of the unsteady aerodynamichinge-moment H(t). due to a, h, and p is used, whichallows the selection of the torque applied by theactuator on the control surface as the single controlinput u. This results in a flexibility in the choice of anactuator, which can be added to the model. The statevector is chosen as
(9)
where {£}={a. h/b, p}T. (zi) and {z2} are theaerodynamic state vectors resulting from the two poles,and obey the following equation:
{zn}={?}-(U/b)bn{zn} (10)The typical section span is taken to be I m withb=0.2m. The elastic-axis, center of mass and controlsurface hinge-line are located 30% 40% and 80% ofchord aft of the leading-edge, respectively. Theresulting plant is of order 12 with two aerodynamicpoles. The in-vacuo natural frequencies of pitch,plunge and control surface modes are 9.8713, 1.9743and 91.97 rad/s, respectively.
Fig. 1 shows a transient response flutter analysis of theopen-loop system. The pitch (a) mode is seen to flutterat velocity, Uf=3.815m/s and frequency, cof=8.2027rad/s.
For flutter suppression, it is decided to measure andfeedback normal acceleration at a sensor(accelerometer) location as the single output variable,y, given by
y = h + Ua (11)
where 1, is the sensor location aft of the elastic-axis.The accelerometer location is chosen ab-initio to be80% of semi-chord aft of the elastic-axis (1,=0.16m).
Compensator Design
Aeroservoelastic feedback control-law derivation isaimed at providing adequate stability and robustness,while minimi/ing control effort. The most commontechnique employed for this purpose is the linear
quadratic optimal control (LQG), carried out iterativelyfor loop transfer recovery (LTR)12. The design isperformed by the separation principle by firstdetermining an optimal observer (Kalman filter) gainmatrix, Kf , for the unmeasurable states, and thendesigning an optimal regulator minimizing the costfunction
J=J(yTQy+uTRu)dto
subject to equation (2). Here
y=CX+Du+m
(12)
(13)
is the plant output, m is the measurement noise and thefeedback control-law is of the form
u=-KX (14)
K being the feedback gain matrix and, X the estimatedstate vector. The output and control weighting factors,Q and R, respectively, must be selected by a nonlinearoptimization process to meet the demands on stabilityand performance robustness, usually expressed as thesingular values of return difference matrices at plantinput and output, given by aOKXsI-Ay'B) anda(C(sI-A)"'K£), respectively. However, for fluttersuppression, one is primarily interested in achievingstable closed-loop response at flight velocities largerthan the flutter speed, Uf. Therefore, the mainconsideration is given to a design which will be themost robust with respect to the flight velocity, whileexhibiting a desirable closed-loop behavior right uptothe critical velocity at which the closed-loop failureoccurs. The compensator design process begins withdetermining the Kalman gain matrix. Kf, according to
Kf=(SAT+FVFT)CT(CFVFTCT)-] (15)
where S is the optimal covariance matrix obtained bysolving the following matrix Riccati equation:
S=AS+SAT-SCTM-' CS+FVF1 (16)
V and M are the spectral densities of process noise, d,and measurement noise, m, respectively. The optimalfeedback gain matrix is then determined by
K = R-1 BTN (17)
where N is obtained as the solution of the followingmatrix Riccati equation:
-N = NA+ATN-NBR-' BTN+Q (18)
From the observer state equation, the estimated statevector is given by
X(s)=(sI-A+BK+KfC)-1 Kfy(s) (19)
thereby yielding the compensator transfer function as
G(s) = K^I-A+BK+KfC)'1 Kf (20)
The plant transfer function is
P(s) = C(sI-A)'1 B+D (21)
Fig.2 is a block diagram of a general ASE system.Alternative compensator design approaches for ASEapplications are Schy13 (constrained minimization ofan objective function including the minimum singularvalue of output sensitivity matrix, amm([I+P(s)G(s)]"1)for the determination of numerator and denominatorcoefficients of the compensator transfer functionmatrix, G(s)), Adams and Tiffany14 (simplexnongradient minimization for the same problem), H*,method15 and structured singular value synthesis16.While the latter two techniques are mathematicallyrefined in that they more directly ensure stabilityrobustness, they appear to be no less computationallyintensive than the LQG/LTR method. Baldelli, et al.17
have recently applied an H* technique for active fluttersuppression of a test wing.
Fig. 3 shows the sensor acceleration at the open-loopflutter velocity at sea level, for an initial pitch-angle.a=0.2 radian.
Since all state-variables are not being measured anoptimum observer (Kalman filter) is necessary toestimate the state of the system. Fig. 4 shows theoptimum observer root-locus as the sensor noisecovariance is increased from 100 to 10,000 whileprocess noise covariance is fixed at 100. The locusindicates that a faster observer dynamics is possible asthe observer gain matrix, K f , is increased. The rightmost pole locations are taken as the final design,because any increase in the observer dynamics is at thecost of robustness.
After the observer design is finalized, the optimalregulator with output weighted performance index(equation (12)) is designed. Fig.5 shows the variationof closed-loop critical velocity as the output weight, Q,is increased from 0.01 to 1, while control costparameter, R, is held constant at 1. For Q=0.01, 0.025and 0.03 the critical closed-loop mode is flutter, while
for values of Q ^0.035 the critical mode is divergence.The maximum critical velocity of 5.72 m/s at sea-levelis observed for Q=O.035. However, for this value of Q,the pitch mode comes very close to flutter at U=4.5m/s, as seen in Fig.6. The extremely low damping inpitch at U=4.5 m/s for Q=0.035 is confirmed by largedecay times in Fig.7 which shows the initial conditionresponse and control torque of the closed-loop systemfor oc=0.2 radian. This rules out Q=0.035 as anacceptable value, although it gives the largest increasein the critical velocity from the open-loop flutter point.From this point of view, Q=0.05 is chosen as it givesthe critical velocity of 5.542 m/s while retaining anadequate minimum damping ratio in pitch at U=4.075m/s seen in Fig.8. Fig.9 shows the initial conditionresponse of a and h to a 0.2 radian initial pitch angleat the open-loop flutter velocity at sea-level. Both thetransients are seen to subside in about 40 seconds. Thecontrol activity at the open-loop flutter-velocity for thisinitial condition is seen in Fig. 10 in the form ofactuator torque, u, and control-surface angle, |3. Themaximum torque magnitude is only 0.5 N-m, while themaximum control-surface deflection p= ±0.13 radianfor stabilising the flutter mode, which appears to be anacceptable design. The control activity and decay timeat the lowest damping in pitch which occurs atU=4.075 m/s do not change appreciably from theiropen-loop flutter point values, as seen in Fig. 11. Fig. 12shows the closed-loop initial response at the closed-loop critical velocity, U=5.542 m/s, indicating thatdivergence is the failure mode. However, Figs. 13 and14, which are initial response plots of a,h and u,|3,respectively, at U=5.5 m/s show that the closed-loopsystem has acceptable behavior right up to the criticalvelocity. Fig. 15 shows the singular value plots of theplant with and without the compensator in the open-loop configuration. The sharp spike in the plot for theplant alone at 8 rad/s indicates the flutter mode, and isseen to diminish in magnitude for the compensatedplant signifying an increase in damping at the samefrequency and flight velocity. A comparison of initialresponses at open-loop flutter point for closed-loopsystems with Q=0.05 and Q=l in Fig. 16 shows that theincreased stability margin for Q=l translates into amanifold increase in the overshoot value, which isunacceptable. Also, as seen earlier, the closed-loopcritical velocity decreases as Q increases beyond 0.035.
It is necessary to examine the effect of atmosphericdensity,p, upon the closed-loop design. Fig. 17 showsthat both open-loop flutter velocity and flutterfrequency decrease as density increases from p=0.2Kg/m3 to p=1.225 Kg/m3 (sea-level). The most critical
design point is thus at sea-level. Fig. 18 shows theclosed-loop critical velocity and frequency variationwith atmospheric density. Below p=0.8 Kg/m3 theclosed-loop failure mode is flutter, while above p=0.8Kg/m3 the critical mode is divergence. The closed-loopsystem is seen to be more stable with decreasing p.
Effect of sensor location on the optimal closed-loopcritical velocity is shown in Fig. 19. It can be observedthat there is an optimal sensor location (65% semi-chord aft of the elastic axis) for which the closed-loopcritical velocity is a maximum. However, this sensorlocation is seen to have an unacceptably small dampingin pitch at U=4.55 m/s (Fig.20) for which the closed-loop initial condition response is shown in Fig.21,exhibiting the long decay time of sensor accelerationand control torque. The best sensor location from thepoint of view of both critical velocity and pitchdamping is 80%, which was also the ab-initio choice.
The design example for the flutter suppression of atypical section highlights some aspects of themultidisciplinary optimization, where trade-offbetween stability and robustness must be made for animplementable design, as well as the variations in thedesign resulting from parameters such as sensorplacement and atmospheric density.
References
'Abel, I., "An Analytical Design Technique forPredicting the Characteristics of a flexible WingEquipped with an Active Flutter-Suppression Systemand Comparison with Wind-Tunnel Data", NASA TP-1367, Feb. 1979.
2Dunn, H.J., "An Analytical Technique forApproximating Unsteady Aerodynamics in the TimeDomain", NASA TP-1738, 1980.
3Albano, E., and Hodden, W.P., "A Doublet-LatticeMethod for Calculating the Lift Distribution onOscillating Surfaces in Subsonic Flow", AIAA J.,Vol.7, 1969, pp.279-285.
''Tewari, A., "Doublet-Point Method for SupersonicUnsteady Aerodynamics of Nonplanar Lifting-Surfaces", J. of Aircraft, Vol.31, No.4, July-Aug. 1994.5Karpel, M., "Design for Active and Passive FlutterSuppression and Gust Alleviation", NASA CR-3482,1981.
'Tiffany, S.H., and Adams, W.M., Jr., "NonlinearProgramming Extensions to Rational FunctionApproximation Methods for Unsteady AerodynamicForces", NASA IP-2776. 1988.
7Eversman, W., and Tewari, A., "Consistent RationalFunction Approximation for Unsteady Aerodynamics",J. of Aircraft, Vol.287 No.9, Sept. 1991.8Tewari, A., "New Acceleration Potential Method forSupersonic Unsteady Aerodynamics of LiftingSurfaces, Further Extension of the Nonplanar DoubletPoint Method, and Nonlinear, Nongradient OptimizedRational Function Approximations for Supersonic.Transient Response Unsteady Aerodynamics", Ph.D.Dissertation, University of Missouri-Rolla, 1992.
"Tewari, A., and Brink-Spalink, J., "Multiple PoleRational Function Approximations for UnsteadyAerodynamics", J. of Aircraft, Vol.30, No.3, May-June1993.
10Theodorsen, T., and Garrick, I.E., "NonstationaryFlow About a Wing-Aileron-Tab Combination ,Including Aerodynamic Balance", N.A.C.A. Rept. 736,1942.
"Eversman, W., and Tewari, A., "ModifiedExponential Series Approximation for the TheodorsenFunction", J. of Aircraft, Vol.28, No.9, Sept. 1991.
12Stein, G., and Athans, M., "The LQG/LTR Procedurefor Multivariable Feedback Control Design", EEEETrans. on Automatic Control, Vol.32, 1987.I3Schy, A.A., "Nonlinear Programming in the Designof Control Systems with Specified Handling Qualities",Proc. 1972 EEEE Conference on Decision and Control,Dec. 1972.
14Adams, W.M., Jr., and Tiffany, S.H., "Control LawDesign to Meet Constraints Using SYNPAC-SynthesisPackage for Active Controls", NASA TM-83264, Jan.1982.
15Glover, K., and Doyle, J.C., "State Space Formulaefor All Stabilizing Controllers that Satisfy an H*, NormBound and Relations to Risk Sensitivity", Systems andControl Letters, Vol.11, 1988, pp. 167-172.
16Doyle, J.C., "Structured Uncertainty in ControlSystem Design", Proc. 24th IEEE Conf. on Decision
and Control, Ft. Lauderdale, FL, Dec. 1985, pp.260-265.
17Baldelli, D.H., et al., "Flutter Margin AugmentationSynthesis Using Normalized Coprime FactorsApproach", J. Guidance, Control, and Dynamics,Vol.18, No.4, July-Aug. 1995, pp.802-811.
18Maciejowski, J.M., "Multivariable Feedback Design",Addison-Wesley, Reading, MA, 1989.19Grace, A., Laub, A.J., Little, J., and Thompson,C.M., "Control System Toolbox-User's Guide", TheMath Works Inc., Natick, MA, 1994.
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