The p, method for flutter analysis is extended to com-pute stability margins of aeroser. voelastic dynamics. Thismethod uses flight data to determine operators whichdescribe errors in a model. The resulting stability mar-gins directly account for these errors to compute a worst-case range of the flight conditions for which the aircraftis guaranteed to be free of aeroservoelastic instabilities.The p, method is used to analyze an F/A-18 aircraft anddemonstrate the stability margins of the nominal dynam-ics are quite large; however, accounting for modeling er-rors can dramatically reduce the stability margins.
Introduction
Aeroservoelastic (ASE) stability analysis considers thecoupling of aerodynamic, inertial, structural, actuation,and control system dynamics. The closed-loop interac-tions of these elements may introduce unexpected insta-bilities if the analytical model used for synthesis andanalysis is not accurate. Stability margins should becomputed to determine safety factors in flight regimesand robustness levels with respect to modeling errors.
Models of aeroservoelastic dynamics can be generated us-ing computational packages such as NASTRAN,ADAM, ICAS,and STARS [1, 8, 16, 19]. These packages combine a finiteelement model with unsteady aerodynamic force theoryto produce a linear representation of the open-loop dy-namics. A closed-loop model is generated by including acontroller and associated sensors and actuators.
Aeroservoelastic stability margins have traditionallybeen generated using classical stability analysis [15].Such analysis utilizes concepts of gain and phase marginsas generalized indicators of closed-loop performance andsensitivity. This type of approach was adopted to ana-lyze the X-29 aircraft and define requirements for notchfiltering of the flexible modes [20]. A similar approach
JNRC Research Fellow, MS 4840 D/RS, Edwards CA 93523-0273, (805) 258-3075, [email protected], Member AIAA
for an F-15 combined the ASE analysis with an optimiza-tion algorithm to achieve desired gain and phase margingoals at modal frequencies [5].
A method has been developed based on system normconcepts that uses singular value tests to indicate sta-bility margins for a multiple-input and multiple-outputmodel of an F/A-18 aircraft [3]. This method introducesuncertainty operators to an analytical model to describeerrors and unmodeled dynamics. Multivariable stabilitymargins based on singular values associated with sensi-tivity and complementary sensitivity are used to indi-cate some measure of stability robustness with respectto these operators. This method used a somewhat adhoc procedure to derive the uncertainty description andthe resulting stability margins may be misleading.
An approach known as the \i method was recently intro-duced for analyzing stability margins of flexible aircraftdynamics [11]. This method is based on a formal math-ematical concept of robustness that guarantees a level ofmodeling errors to which the aircraft is robustly stable.A realistic representation of errors can be formulated bydescribing differences between predicted responses andmeasured flight data [12]. The structured singular value,p,, is used to compute a margin which is robust, or worst-case, to these errors [10].
This paper extends the /j, method to consider aeroservoe-lastic stability margins. Uncertainty operators are asso-ciated with a closed-loop dynamics model and fj, com-putes the range of flight conditions for which the modelis robust to the set of uncertainties. These margins aresuperior to gain and phase margins that cannot be in-terpreted as flight condition information. Another ad-vantage to the \i, method is the ability to simultaneouslycompute ASE and flutter margins. Also, controllers maybe designed and analyzed in the same \i framework.
The // method is used to compute aeroservoelastic sta-bility margins for the F/A-18 High Alpha Research Ve-hicle (HARV). Nominal stability margins indicate theclosed-loop dynamics are safe from any instabilities. Ro-bust stability margins computed with respect to a set ofmodeling errors indicate instabilities may lie significantlycloser to the flight envelope than previously anticipated.
Any aeroservoelastic model is an approximate represen-tation of the aircraft dynamics. Uncertainty operatorsare associated with the model in the framework of thestructured singular value, /^, to account for inaccuraciessuch as errors and unmodeled dynamics through a feed-back relationship. The uncertainty operator, A, is al-lowed to lie within a norm bounded set. Weighting ma-trices are usually included in the ^ framework to nor-malize the uncertainty norm bound to unity.
A = {A : < 1}
Uncertainty and system operators are interconnected us-ing linear fractional transformation relationships [2]. Acommon interconnection is denoted JF) (P, K) which de-fines the closed-loop model of a plant whose lower loopis connected to a controller via feedback.
Define /u with respect to the set A [17].___________1__________min (a(A) : A e A, det(7 - PA) = 0}
with /x(P) = 0 if there exists no A € A such that det(7 —PA) = 0.
Define elements of P as transfer function from distur-bances to errors P22, transfer function relating uncer-tainty feedback PH, and similar Pi2 and P2i.
/j, is an exact measure of robustness for systems withstructured uncertainty. The inverse of p, can be inter-preted as a measure of the smallest destabilizing pertur-bation. The system is guaranteed to be robustly stablefor all uncertainty operators bounded by the smallestdestabilizing value [6].
Theorem 3.1 A stable system P is robustly stable forall A 6 A if and only «//z(Pn) < 1.
Model validation algorithms can verify the uncertaintydescription is sufficient to describe variations betweenthe model and measured data. A /u-analysis method ofmodel validation has been developed that determines ifthere exists some perturbation to the system dynamics,within the allowed set of uncertainty, for which the per-turbed system could produce the measurement data inresponse to the indicated input data [9].
Theorem 3.2 Given measurements y generated by in-put u, then a system P with uncertainty A is not inval-idated if (j, (PII - P12w (P22u - y)~l P2i) > 1
V /
Unfortunately, JJL is difficult to compute. Upper and lowerbounds have been derived which may be solved efficientlyusing convex optimization and a power iteration [2].
The fj, Method for AeroelasticStability Margin Analysis
The generalized equation of motion for the nominalaeroelastic system can be expressed in a form suitable forusing [j, analysis to compute a flutter margin. A fluttermargin is computed by analyzing the robustness of thesystem to changes in flight condition and a set of mod-eling errors. The smallest destabilizing change in flightcondition for which the model is not robustly stable tothe uncertainty is the robust flutter margin.
Consider the generalized equation of motion for thestructural response of the aircraft at a constant Mach [8] .
Mrj qQ(s~)rj = 0
An additive perturbation, <%• € R, on the nominal dy-namic pressure, q0, can be introduced in a linear feedbackmanner to describe variations in flight conditions [11].
Uncertainty operators are associated with the nominalmodel to describe errors and unmodeled dynamics. Astructured set, A, represents the uncertainty such thatevery A g A represents a possible variation between themodel and the aircraft dynamics. A may include com-plex multiplicative operators affecting inputs and out-puts along with real parametric operators affecting ele-ments such as stiffness and damping matrices [12, 13].
H is computed with respect to a single structured opera-tor to compute a robust stability margin. This operatoris a block diagonal matrix with the perturbation to dy-namic pressure, <%, and the uncertainty set, A, as thediagonal blocks as in Figure 1.
Figure 1: Linear Fractional Transformation System for Ro-bust Stability Analysis in the fj. Framework with Perturbationin Dynamic Pressure and Structured Uncertainty
Robust aeroelastic flutter margins are computed by si-multaneously considering the dynamic pressure pertur-bation, 6g, and the uncertainty operators, A, in the sta-bility analysis. The y. method iterates over scalings forthese operators until an appropriate p, condition is sat-isfied and a valid stability margin is computed [13].
The [i Method for AeroservoelasticStability Margin Analysis
The /j, method can be directly applied to aeroservoelasticsystems to generate a robust stability margin in dynamicpressure [14]. This quantity relates the smallest pertur-bation to dynamic pressure which destabilizes at leastone member of the family of plants generated by thenominal closed-loop dynamics and the uncertainty set.
The uncertainty descriptions for the closed-loop modelscan be generated in a manner similar to that used to gen-erate the open-loop uncertainty. These uncertainty de-scriptions for closed-loop models generally contain moreoperators than for open-loop models to account for theadditional elements associated with closed-loop systemssuch as sensor and actuator models. The linear frac-tional transformation framework is especially useful forgenerating the closed-loop aeroservoelastic model withuncertainty. The open-loop aeroelastic model with its as-sociated uncertainty is combined with the feedback andcontrol models, each with their respective uncertainties,into a single closed-loop model with a structured uncer-tainty description using standard LFT operations.
Magnitudes of the uncertainty descriptions can be deter-mined using flight data. Levels of uncertainty operatorsare chosen and the resulting range of model responsesis compared to the flight data responses. These levelsare increased or decreased until reasonable magnitudesof uncertainty are chosen such that the condition of The-orem 3.2 is satisfied and indicates the uncertain modelis not invalidated by any flight data. Thus, the uncer-tainty is chosen to account for errors observed betweenthe model and flight data.
The model used for stability analysis of the aeroservoe-lastic dynamics is formulated as in Figure 1. The singlestructured uncertainty set, A, contains the perturbationto dynamic pressure from the open-loop model, q, andthe structured uncertainty set, A, describing modelingerrors in both the open-loop and feedback models. Theplant model for aeroservoelastic analysis using Figure 1is actually the closed-loop model P = Fi(P0ip,K).
Robust stability margins are computed by analyzing ro-bustness of the closed-loop dynamics to the structureduncertainty set A. Lemma 5.1 demonstrates a p, con-dition to compute an aeroservoelastic stability marginfor this model. Note this Lemma considers the criti-cal perturbation as having a negative sign so the criticalinstability is assumed to be occur at a lower dynamicpressure than the nominal dynamic pressure. Certainlythe Lemma can be altered with a simple modificationto consider positive perturbations; however, the primaryinterest of this paper is to analyze ASE instabilities atlow dynamic pressures.
Lemma 5.1 Given the open-loop plant P0ip derived atnominal dynamic pressure q0 and a feedback controllerK, define the closed-loop plant P = Fi(P0ip,K). Givenan uncertainty description A with ||A||oo < 1 for allA € A along with a perturbation to dynamic pressure S-qassociated with P and arranged in the feedback relation-ship of Figure 1, define the plant P with real diagonalmatrix W-g scaling the feedback signals relating 5-g and P.
Wg 0o i
If and only if ^(P) = 1, then
arob —a — W-'iase — <do vv 1
is the dynamic pressure at the critical aeroservoelasticcondition and
represents the least conservative robust aeroservoelasticstability margin for dynamic pressure.
Proof
•4= (necessary)
The condition /J.(P) = 1 implies the smallest destabiliz-ing perturbation to P is described by some A e A withIIAHoo = 1 so there is no destabilizing A e A and thesmallest positive destabilizing perturbation to dynamicpressure is at least 6-g = — 1 which corresponds to dy-namic pressure q = q0 - WgS-g = q0 - W-g. Thus, P isguaranteed to be robustly stable to the uncertainty setA for any perturbation to dynamic pressure less thanW-g so Wq is a robust ASE margin.
=>• (sufficient)
Assume /u(P) > 1. Define real scalar a < 1 such thatp.(P) = ^ which implies, from Theorem 3.1, that P isrobustly stable to all uncertainties A £ A with HAHoo <a < 1. Thus, P is not guaranteed to be stable over theentire range of modeling uncertainty defined by the unitynorm bounded set A so this is not a valid robust ASEmargin.
Assume /x(P) < 1. Define real scalar a > 1 such thatH(P) = ^ which implies, from Theorem 3.1, that P is ro-bustly stable to all uncertainties A G A with || A^ < a.Thus, P is robustly stable to an uncertainty descriptionlarger than that defined by the unity norm bounded setA so this condition defines a valid ASE margin but it isnot the least conservative robust ASE margin.
Nominal stability margins for dynamic pressure are com-puted using an extension of Lemma 5.1. These marginsrelate the range of flight conditions for which the nominalaeroservoelastic dynamics are stable by determining thesmallest destabilizing perturbation to dynamic pressure.
The condition for computing nominal stability marginsuses a scaled closed-loop plant similar to the plant usedin Lemma 5.1; however, the scaling matrix is different.The identity term of the scaling is replaced with a zeromatrix when computing nominal margins. This zero ma-trix has the effect of eliminating the feedback connectionsbetween the plant and the uncertainty so the /j, analysisconsiders perturbations to dynamic pressure as the onlyoperator affecting the system.
Lemma 5.2 demonstrates a condition to compute a nom-inal aeroservoelastic stability margin based on the LFTof the closed-loop dynamics with associated uncertainty.
Lemma 5.2 Given the open-loop plant P0ip derived atnominal dynamic pressure q0 and a feedback controllerK, define the closed-loop plant P = Fi(P0ip,K). Givenan uncertainty description A with HA^ < 1 for allA € A along with a perturbation to dynamic pressure <%associated with P and arranged in the feedback relation-ship of Figure 1, define the plant P with real diagonalmatrix Wq scaling the feedback signals relating S-q and P.
= P0 0
= 1, then
-norn _ - _ TI^_lase — "/o ''q
is the dynamic pressure at the critical aeroservoelasticcondition and
T- — W-1 q — VVq
represents the nominal aeroservoelastic stability marginfor dynamic pressure.
The n condition in Lemma 5.2, unlike the condition inLemma 5.1, is presented as a sufficient, but not neces-sary, condition to determine a stability margin. The p,value was restricted to be unity to match the size of theuncertainty set when computing robust stability mar-gins; however, the uncertainty set is not considered whencomputing nominal stability margins. Thus, this Lemmacould be rewritten to state that if fj, = ^ for some scalar athen the nominal stability margin is actually F = —
Also, nominal stability margins are computed by analyz-ing p, with respect to the single repeated scalar 6q. Theexact value of p can be computed for this type of oper-ator structure so the nominal margins are not affected
by added conservatism resulting from approximating nas an upper bound.
The /j. method may also be used to compute a robust sta-bility margin in uncertainty using another extension toLemma 5.1. This type of stability margin is commonlyused for closed-loop analysis to indicate the amount ofmodeling uncertainty that may exist for which a stabil-ity analysis of the model indicates the stability of thephysical system.
The condition for computing the robust stability marginsin uncertainty again uses a scaled plant like in Lemma 5.1and Lemma 5.2; however, the scaling matrix is differentthan for either of these Lemmas. The scaling matrix tocompute these margins retains the identity matrix butreplaces the weighting Wq with a zero matrix. The iden-tity matrix ensures p considers the uncertainty set Awhile the zero matrix eliminates the feedback intercon-nection relating the perturbation to dynamic pressurewith the plant model.
Lemma 5.3 provides the condition for a robust stabilitymargin in uncertainty.
Lemma 5.3 Given the open-loop plant P0ip derived atnominal dynamic pressure q0 and a feedback controllerK, define the closed-loop plant P = Fi(P0[p,K). Givena perturbation to dynamic pressure <%• and set of uncer-tainty operators A norm bounded by one associated withP and arranged in the feedback relationship of Figure 1,define the plant P with a scaling to eliminate the pertur-bation to dynamic pressure.
P = P 0 00 /
// p(P) = I/a, then a is the stability margin in uncer-tainty indicating the size of allowable perturbations in Awhich do not destabilize the system.
This measure of stability margin is related to traditionalgain and phase margins but there are several well-knownadvantages to p. Gain and phase margins are not welldefined for multivariable systems and they provide noguarantees of stability. They can be considered as neces-sary, but not sufficient, measures of multivariable stabil-ity margins, p is well-defined for multivariable systemsand it provides a condition which is both necessary andsufficient for robust stability.
This type of stability margin does not consider anychanges in flight condition; rather, it considers the ef-fect of errors which are described by the uncertainty forthe model at a particular flight condition. These marginscan not be used to indicate which flight conditions areunstable, but they are useful to indicate the sensitivityof the model to errors.
The F/A-18 High Alpha Research Vehicle (HARV) wasflown at NASA Dryden Flight Research Center to con-duct flight tests at high angle-of-attack flight conditions.This paper considers the F/A-18 HARV with a thrustvectoring system to direct the flow out of the engine noz-zles. This aircraft, shown in Figure 2, is sometimes de-noted as the HARV-TVCS to emphasize the thrust vec-toring control system, but for convenience the additionalTVCS notation will not be used here.
Figure 2: F/A-18 HARV with Thrust Vectoring in Flight
The F/A-18 HARV underwent several structural modi-fications to enable controlled high angle-of-attack flightwith the thrust vectoring system. These modificationsinclude additions such as three vanes in each engine ex-haust and associated ballast in the forward fuselage tomaintain a desired center of gravity position; a researchflight control system; an inertial navigation system forangle-of-attack and sideslip rate computation at highangle-of-attack flight conditions; wingtip vanes for sen-sors and air data probes; and additional instrumentation.The F/A-18 HARV has significantly different structuraldynamic properties from a standard F/A-18 aircraft asa result of these modifications.
The control system consists of a basic flight control sys-tem and a research flight control system (RFCS). The ad-ditional RFCS controller is used to command the thrustvectoring at high angle-of-attack flight conditions [7].Separate state-space controllers are developed for longi-tudinal and lateral-directional control with distinct gainscheduling laws determining the controller gains at dif-ferent conditions within the flight envelope [18]. Thesecontrollers are discrete-time systems which operate at an80 Hz update rate.
The longitudinal controller, Kiong, is a 15 state systemwhich operates on the feedback measurements given inTable 1. The outputs from this controller result in thecontrol inputs and their respective units given in Table 2.
pitch ratealpha derivativealpha
q deg/seca deg/seca deg
Table 1: Feedback Measurements for the Thrust VectoringLongitudinal Controller Kiong
Table 2: Control Commands for the Thrust Vectoring Lon-gitudinal Controller Kiong
The lateral-directional controller, Kiat, is a 29 state sys-tem which operates on the feedback measurements givenin Table 3. The outputs from this controller result in thecontrol inputs and their respective units given in Table 4.
roll rateslide slip rateyaw ratelateral acceleration
PPrny
deg/secdeg/secdeg/sec9
Table 3: Feedback Measurements for the Thrust VectoringLateral-Directional Controller Kiat
differential stabilatorailerondifferential leading edge flapdifferential trailing edge flaprudderyaw thrust vectoring
$deSaSdlefSdtefSrud
51
degdegdegdegdegdeg
Table 4: Control Commands for the Thrust VectoringLateral-Directional Controller Kiat
The flight envelope for the F/A-18 HARV is a small sub-sonic box within the flight envelope of a standard F/A-18as given in Figure 3.
50
^40
•§30
<20
10
00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 l.f
Mach
Figure 3: Flight Envelopes of a F/A-18 Aircraft : StandardProduction F/A-18 (—) and F/A-18 HARV with RFCS (—)
A finite element model of the F/A-18 HARV is usedto generate a linear time-invariant model to representthe structural dynamics. This model contains 10 elas-tic symmetric and 10 elastic antisymmetric structuralmodes along with 6 rigid body modes. High order mod-els are used to represent the actuator dynamics and add90 states to the model. There are additionally 12 controlsurface modes. Including the state-space representationof the unsteady aerodynamic forces introduces 20 statesfor the symmetric modes and 20 states for the antisym-metric modes.
Table 5 lists some elastic modal natural frequencies inthis state-space model for the F/A-18 HARV at heavy-weight condition with full fuel. There are also severalmodes associated with the thrust vectoring vane dynam-ics that have natural frequencies slightly above 20 Hzbut they are not confidently accepted because of model-ing difficulties and so are not included in Table 5.
Table 5: Natural Frequencies in Hz of the Structural FiniteElement Model of the Heavyweight Full Fuel F/A-18 HARVwith Thrust Vectoring System
Separate models represent the closed-loop aeroservoelas-tic dynamics at the corners of the research flight enve-lope shown in Figure 3. These corners represent the mostextreme flight conditions with respect to Mach and dy-namic pressure so the stability margins are anticipatedto be smallest at these conditions. Additionally, onlydecreasing values of dynamic pressure will be consideredsince any aeroservoelastic instabilities are anticipated tobe associated with low dynamic pressures [3].
Stability margins are computed by considering perturba-tions to dynamic pressure for the models with constantcontroller gains. This is valid because RFCS gain arescheduled within the envelope but are fixed to their val-ues from the closest edge of the envelope when flyingbeyond the design ranges to lower dynamic pressures.Thus, the stability margins indicate the flight conditionsnearest the envelope for which instabilities may be en-countered.
Uncertainty Description
An uncertainty description, A, is formulated that uses 3operators to describe errors in a HARV analytical model.A complex operator, Ain, is a multiplicative uncertaintyin the control inputs to the plant and accounts for actu-ator errors and unmodeled dynamics. Another complexoperator, Aadd, relates the control inputs to the feedbackmeasurements to account for uncertainty in the magni-tude and phase of the computed plant responses. Theremaining uncertainty operator, A,4, is a real parametricuncertainty affecting the modal parameters of the open-loop state matrix to describe errors in natural frequencyand damping parameters.
The block diagram for robust stability analysis of theF/A-18 HARV aeroservoelastic dynamics is shown inFigure 4. This Figure includes an operator, £,, that af-fects the nominal dynamics to describe changes in flightcondition and is used to interpret ^ as a stability mar-gin. Additional operators, Wadd and Win, are shownas weightings to normalize the frequency varying uncer-tainty operators, A.add and A«n. The system model alsocontains an extraneous 5% noise signal corrupting thesensor measurements.
noise
Figure 4: Closed-Loop F/A-18 HARV Model with Uncer-tainty
This uncertainty description is used to describe errors inthe models from each corner of the flight envelope. Twosets of magnitudes are generated with one set describingthe amount of uncertainty in the symmetric dynamicsmodel and the other set describing the amount of un-certainty in the antisymmetric dynamics model. In thissense, A is a worst-case uncertainty description that con-tains the largest errors for the symmetric or antisymmet-ric models at any point in the research flight envelope.
The magnitudes of the uncertainty are chosen from anal-ysis of measured flight data using a model validation pro-cedure based on the condition in Theorem 3.2. This pro-cedure determines if the measured flight data could havebeen generated by some plant comprised of the nomi-nal model and a perturbation described by the uncer-tainty [11]. Values of uncertainty are iteratively testeduntil the family of plants described by Figure 4 has arange of responses which can sufficiently bound the mea-sured flight data transfer functions.
The modal parametric uncertainty for the antisymmetricdynamics model is determined by considering flight datasets generated in response to lateral-directional excita-tion. Consider the transfer functions from yaw thrustvectoring command (8%v) to normal acceleration (ny) forseveral analytical models and data taken at Mach 0.3and 30000 feet shown in Figure 5.
12 14 16Frequency (Hz)
Figure 5: Transfer Functions from 5\v to ny : experimentalflight data (—), nominal model (---), perturbed models (_)
The dashed line in Figure 5 represents the nominal modeltransfer function and clearly does not match the solidline showing the experimental flight data transfer func-tion. The two thick solid lines in this Figure demonstratethe range of transfer functions that can be generated byintroducing modal parametric uncertainty. Each thickline corresponds to a different value of the modal pertur-bation and it can be seen that both perturbation valuesare needed to validate the model. Introducing one per-turbation ensures the model closely matches the datanear 16 Hz while the other perturbation is required near18 Hz.
Separate modal parametric uncertainty levels are chosenfor each mode of the open-loop state matrix to reflectdifferent levels of accuracy. Multiple flight data sets areanalyzed and the largest errors observed between theo-retical and experimental modal parameters are used todetermine the uncertainty levels. The nominal modalparameters and the amount of variation admitted by theparametric uncertainty for the antisymmetric dynamicsmodel are given in Table 6.
Table 6: Modal Parameters and Uncertainty Variations forAntisymmetric Model
The amount of variation needed to describe modal pa-rameter errors is fairly significant for all modes. Thewing fore-aft and fuselage 1st torsion modes have proper-ties which are particularly poorly modeled so there is upto 4% error in natural frequency and 70% error in damp-ing. The remaining modes are relatively better modeledand have only 1% error in natural frequency but stillrequire at least 50% error in damping.
A similar procedure is performed to compute the modalparametric uncertainty levels for the symmetric dynam-ics model. Transfer functions from the pitch thrust vec-toring command ((5^) to normal acceleration (nz] areused to observe errors between the model and the mea-sured flight data. The nominal modal parameters andthe amount of variation admitted by the generated para-metric uncertainty are given in Table 7.
Table 7: Modal Parameters and Uncertainty Variations forSymmetric Model
The amount of variation introduced to the modal param-eters is again quite significant. The natural frequencyvariations are larger for the symmetric model than for theantisymmetric model with the largest error of 8% for thewing fore-aft mode. The damping variations, however,are smaller for the symmetric model than for the anti-symmetric model with errors in damping between 15%and 45% for all modes.
The weighting functions for the input multiplicative andadditive uncertainties are chosen to account for any er-rors between the model and the flight data that can notbe covered by the parametric modal uncertainty.
Stability margins are computed for the F/A-18 HARVsystem in Figure 4 to determine the lowest dynamic pres-sures at which an aeroservoelastic instability occurs. Thestability margin, F-q, computed from the p, method indi-cates the biggest reduction in dynamic pressure that maybe considered before the onset of an ASE instability.
Nominal stability margins are computed for the aeroser-voelastic model by computing the smallest destabilizingperturbation to dynamic pressure for the nominal model.fi is computed for this model to measure robustness withrespect to the S-q operator in Figure 4 but ignoring theA uncertainty operators.
Table 8 lists the nominal stability margins for the lon-gitudinal model describing the symmetric dynamics.These margins are all significantly large to indicate areduction of more than 2000 lb/fl? is required before thenominal dynamics become unstable. For each of thesecases, the nearest instability is actually at a negative dy-namic pressure, which is physically unrealistic, to indi-cate the nominal model with the longitudinal controlleris effectively free from ASE instabilities.
Table 8: Nominal Stability Margins for Symmetric Dynam-ics Model with Longitudinal Controller Kiong
Nominal stability margins for the lateral-directionalmodel are computed in Table 9. These margins are lowerthan the margins for the symmetric dynamics; however,the flight envelope is still clear of ASE instabilities be-cause the nearest unstable flight condition is associatedwith a negative dynamic pressure.
Flight ConditionMach altitude
0.3 30 kft0.3 10 kft
Stability MarginlY ^unstable
-268 lb/fl? 14.8 Hz-757 Ib/ft2 13.7 Hz
Table 9: Nominal Stability Margins for Antisymmetric Dy-namics Model with Lateral-Directional Controller Kiat
Robust stability margins are computed from /j, analy-sis of the model in Figure 4. The smallest destabilizingperturbation to dynamic pressure, iy, is computed suchthat some plant within the family of plants becomes un-stable. Thus, the ASE dynamics are robust to the set ofmodeling errors, A, for dynamic pressures less than iyaway from the nominal flight condition.
Table 10 lists the robust stability margins for the longitu-dinal model describing the symmetric dynamics. Theserobust margins are lower than the nominal margins listedin Table 8 because of the conservatism introduced by theuncertainty; however, they are still significantly large.These margins demonstrate each plant model is robust tothe indicated amount of modeling uncertainty despite re-ductions in dynamic pressure of at least 130 lb/fl? whichgenerally corresponds to a negative dynamic pressure.
Table 10: Robust Stability Margins for Symmetric Dynam-ics Model with Longitudinal Controller Kiong
Table 10 also demonstrates the frequencies of the un-stable modes associated with the robust stability mar-gins are different than the frequencies associated withthe nominal margins in Table 8. The critical instabil-ity is the wing torsion mode for the models at Mach 0.3whereas at Mach 0.7 the fuselage bending mode is thecritical instability.
Robust stability margins are also computed for the an-tisymmetric dynamics models with respect to the uncer-tainty description in Figure 4. These margins, given inTable 11, are quite small and indicate the stability mar-gins of antisymmetric model are very sensitive to model-ing errors. Indeed, the lateral-directional controller mayincur an aeroservoelastic instability if the aircraft reducesthe dynamic pressure by as little as 4 Ib/ft? from the topleft corner of the research flight envelope.
Flight ConditionMach altitude
0.3 30 kft0.3 10 kft
Stability Marginiy ^unstable
-4 lb/fl? 15.4 Hz-17 lb/fl? 8.9 Hz
Table 11: Robust Stability Margins for Antisymmetric Dy-namics Model with Lateral-Directional Controller Kiat
The robust stability margins are considerably less thanthe nominal stability margins and indicate the impor-tance of considering uncertainty. The nominal dynamicsdid not indicate any potential ASE problems but a robustanalysis clearly shows the stability margins are stronglyaffected by errors in the model. This paper utilized aworst-case uncertainty description based on several flightdata sets; however, a wavelet filtering technique has alsobeen developed to reduce the conservatism in the uncer-tainty description and generate more confident stabilitymargins [4].
The p, method is also used to compute robust stabilitymargins for the uncertainty description. These marginsare computed by measuring robustness of the model withrespect to the uncertainty description but ignoring theeffects of a perturbation to the dynamic pressure. Thus,these margins are similar in nature to classical gain andphase margins since each relates a property of the closed-loop dynamics at a particular flight condition.
The stability margin is computed as the scalar value ofH which relates the smallest destabilizing perturbationat the worst-case frequency; however, there is a greatdeal of information that can be extracted by consideringthe upper bound for ^ at every frequency. Peaks in theupper bound function indicate frequencies at which themodel is sensitive and may be associated with subcriticalinstabilities. Also, the frequency variation may indicatestability problems which are caused by bandwidth issuesassociated with the controller.
This Section will only present the results for the dynam-ics models of the closed-loop system at the upper left cor-ner of the flight envelope corresponding to Mach 0.3 and30000 feet. The stability margin for dynamic pressurefor the symmetric model at this condition was similarto the margins for models at other conditions; however,the stability margin for dynamic pressure was particu-lar small for the antisymmetric model at this condition.Thus, the sensitivity of this model to uncertainty is ofparticular interest.
Figure 6 presents the /j, upper bound relating the robuststability of the antisymmetric model with respect to theuncertainty description of Figure 4. The peak in thisFigure agrees with the unstable frequency informationin Table 11. Figure 6 indicates the model is especiallysensitive to modeling errors at 15.4 Hz with a n valueof .77 for the robust stability measure. This informationdemonstrates the stability of the wing fore-aft mode ofthe closed-loop dynamics is sensitive to perturbations.
0.6
lo,0.4
0.3
0.2
10 15Frequency (Hz)
The n value, even considering the peak at u> = 15.4 Hz,is less than 1 in Figure 6. This indicates the system isrobustly stable to the amount of modeling uncertaintydescribed by the set A used to compute /j,. The inverseof the peak value, /j, = .77 can interpreted as a stabilitymargin. This margin indicates the system is robustlystable for any perturbations allowed by the set A witha magnitude of 1.29 or less.
The /j, upper bound relating the robust stability of thesymmetric model with respect to the uncertainty descrip-tion is shown in Figure 7. This plot does not show a clearmaximum peak; rather, there are peaks of similar mag-nitude at several frequencies. This indicates the modelhas similar sensitivities to to modeling errors at thosefrequencies.
One of the maximum peaks occurs at a frequency of12 Hz which corresponds to the critical unstable fre-quency for this model given in Table 10. This frequencyis the natural frequency for the wing torsion mode. Thesensitivity of the model at this frequency indicates thewing torsion mode couples with the controller gains andthe closed-loop system is sensitive to errors in the modalparameters for this mode.
There are also peaks near the other frequencies com-puted as unstable modes associated with both nominaland robust margins in Table 8 and Table 10. In par-ticular, there are noticeable peaks at the wing bendingmode near 8 Hz and at the fuselage bending mode near15 Hz. These peaks, although are not associated withthe worst-case instability dynamics of the wing torsionmode, indicate subcritical sensitivities of the modal dy-namics.
The upper bound for fi is less than 1 at all frequencies toindicate the symmetric model is robustly stable to the setof uncertainty given in Figure 4. The inverse of /z = .56demonstrates the system is stable for all perturbationsstructured as in A and bounded by 1.79 magnitude.
0.8
0.7
0.6
lo,0.4
0.3
0.2
0.1
10 15Frequency (Hz)
Figure 6: p. Relating Robust Stability Margins for Uncer-tainty for Antisymmetric Modes with the Lateral-DirectionalController Ktat at Mach=0.3 and Altitude=30 kft
Figure 7: p, Relating Robust Stability Margins for Uncer-tainty for Symmetric Modes with the Longitudinal ControllerKiong at Mach=0.3 and Altitude=30 kft
A /j, method for evaluating aeroelastic stability is ex-tended in this paper to compute aeroservoelastic stabil-ity margins. An uncertainty description can be intro-duced to account for errors and unmodeled dynamics inthe structural, aerodynamic, sensing, actuation and con-trol system dynamics, p, computes a worst-case stabilitymargin that considers the interactions of all uncertaintiesin the closed-loop system. Two types of stability mar-gins can be generated to relate information about dy-namic pressure or uncertainty. The stability margin indynamic pressure indicates what flight conditions near-est the flight envelope result in instabilities. The stabilitymargin in uncertainty indicates what amount of variationin the modeled dynamics is allowed before an instabilityis possible. Robust stability margins are computed forthe F/A-18 HARV to demonstrate potential instabilitiesnear the flight envelope for high angle-of-attack flight.
Acknowledgments
The authors would like to acknowledge the generous fi-nancial support of the Structural Dynamics Branch ofNASA Dryden Flight Research Center. Rick Lind wassupported by a National Research Council PostdoctoralFellowship.
References[1] W.M. Adams and S. Tiffany-Hoadley, "ICAS - Atool for Aeroservoelastic Modeling and Analysis," AIAAStructures, Structural Dynamics and Materials Confer-ence, La Jolla CA, AIAA-93-1421-CP, April 1993.[2] G. Balas, J. Doyle, K. Glover, A. Packard and R.Smith, ^-Analysis and Synthesis Toolbox - Users Guide,The Math Works, Natick, MA, 1991.[3] M.J. Brenner, Aeroservoelastic Modeling and Val-idation of a Thrust-Vectoring F/A-18 Aircraft, NASATP-3647, September 1996.[4] M.J. Brenner and R. Lind, "Wavelet Filtering toReduce Conservatism in Aeroservoelastic Robust Sta-bility Margins," AIAA Structures, Structural Dynamicsand Materials Conference, Long Beach CA, AIAA-98-1896, April 1998.[5] P.Y. Cheng and T.J. Hirner, "Aircraft Aeroservoe-lastic Compensation Using Constrained Optimization,"AIAA Structures, Structural Dynamics and MaterialsConference, Dallas TX, AIAA-92-2399, April 1992.[6] J. Doyle, "Analysis of Feedback Systems withStructured Uncertainty," IEE Proceedings, Part D,vol. 129, November 1982, pp. 242-250.[7] W. Gilbert, L. Nguyen and J. Gera, "Control LawResearch in the NASA High Alpha Technology Pro-gram," AGARD CP-465, April 1991.
[8] K.K. Gupta, M.J. Brenner and L.S. Voelker, Devel-opment of an Integrated Aeroservoelastic Analysis Pro-gram and Correlation with Test Data, NASA TP-3120,May 1991.[9] A. Kumar and G. Balas, "An Approach to ModelValidation in the n Framework,". Proceedings of the 1994American Controls Conference, Baltimore MD, June1994, pp. 3021-3026.[10] R. Lind and M. Brenner, "Worst Case FlutterMargins from F/A-18 Aircraft Aeroelastic Data," AIAAStructures, Structural Dynamics and Materials Confer-ence, Kissimmee FL, AIAA-97-1266, April 1997.[11] R. Lind and M. Brenner, "Robust Flutter Marginsof an F/A-18 Aircraft from Aeroelastic Flight Data,"AIAA Journal of Guidance, Control and Dynamics,Vol. 20, No. 3, May-June 1997, pp. 597-604.[12] R. Lind and M. Brenner, "Incorporating FlightData into a Robust Aeroelastic Model," accepted toAIAA Journal of Aircraft, expected March-April 1998.[13] R. Lind and M. Brenner, Robust Flutter MarginAnalysis that Incorporates Flight Data, NASA-TP, ex-pected April 1998.[14] R. Lind and M. Brenner, "Robust Stability Mar-gins of Aeroservoelastic Dynamics," submitted to AIAAJournal of Guidance, Control and Dynamics, January1998.[15] E. Livne, "Integrated Aeroservoelastic Optimiza-tion: Status and Direction," AIAA Structures, Struc-tural Dynamics and Materials Conference, KissimmeeFL, AIAA-97-1409, April 1997.[16] T. Noll, M. Blair and J. Cerra, "ADAM - AnAeroservoelastic Analysis Method for Analog or DigitalSystems," AIAA Journal of Aircraft, Vol. 23, No. 11,November 1986, pp. 852-858.[17] A. Packard and J. Doyle, "The Complex Struc-tured Singular Value," Automatica, Vol. 29, No. 1, Jan-uary 1993, pp. 71-109.[18] J. Pahle, B. Powers, V. Regenie, V. Chacon, S. De-groote and S. Murnyack, Research Flight Control SystemDevelopment for the F-18 High Alpha Research Vehicle,NASA TM-104232, 1991.[19] W.P. Rodden, R.L. Harder and E.D. Bellinger,Aeroelastic Addition to NASTRAN, NASA CR-3094,March 1979.[20] A. Zislin, E. Laurie, K. Wilkinson and R. Gold-stein, "X-29 Aeroservoelastic Analysis and Ground TestValidation Procedures," AIAA Aircraft Design Systemsand Operations Meeting, Colorado Springs CO, AIAA-85-3091, October 1985.
