TOWARD THE DEVELOPMENT OF A METHODOLOGY FOR DESIGNING HELICOPTER FLIGHT CONTROL LAWS BY INTEGRATING HANDLING QUALITIES REQUIREMENTS

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1 TOWARD THE DEVELOPMENT OF A METHODOLOGY FOR DESIGNING HELICOPTER FLIGHT CONTROL LAWS BY INTEGRATING HANDLING QUALITIES REQUIREMENTS Jean-Charles Antonioli ONERA Air Base Salon de Provene Frane Armin Taghizad ONERA Air Base Salon de Provene Frane Thomas Rakotomamonjy ONERA Air Base Salon de Provene Frane Mustaha Ouladsine LSIS Av. Esadrille Normandie Niemen 3397 Marseille Frane Abstrat Piloting a helioter is a demanding task for human oerators: autoilots have been introdued by manufaturers to assist ilots in their iloting task. Designing the gains of the assoiated ontrol laws taking into aount Handling Qualities requirements from standards suh as the ADS-33 is a diffiult industrial roblem. NASA has already led many studies on this area. These works have led to the develoment of CONDUIT (Control Designer s Unified Interfae), a omuter aided design tool for rotary and fixed wing airrafts ontrol laws using interative otimization tehniques. The tool has demonstrated its benefits in terms of time of develoment redution. ONERA has been working for several years to the establishment of another roess. One of the first ideas was to lead loal sensitivity studies, and use the results as design guidane. Then these results have been onfronted with some analytial develoments led on simlified models. This aer shows how these analytial studies an be used to initialize the gain tuning roess effiiently, taking into aount the struture of the system and the requirements from handling qualities standards (ADS-33). A tool has been develoed: it generates harts of Flying Qualities. Thanks to the harts generated for eah ase of study, the gains of the ontrol laws an be effiiently initialized for the simlified models, with Flying Qualities objetives. The exeted results are omared with those obtained with the full linear model. Sensitivities an then be used to hel in designing more reisely the gains. Finally, full nonlinear simuions are led in order to omare the results with the exeted Flying Qualities. As a onlusion, all these studies seem to lead to a full and effiient roess of gain tuning taking into aount the onstraints from the ontrol law strutures and the requirements from the handling qualities standard. Furthermore, the roedure an be alied from the very first stage of tuning: the initialization of the gains.. INTRODUCTION. Piloting assistanes Helioters are naturally unstable systems. Pilots must make onsiderable efforts to stabilize the airraft. Piloting assistane systems are now available on board (Autoilots (AP), Automati Flight Control Systems (AFCS), Fly By Wire / Fly By Light ontrol systems (FBW/FBL)). They are designed to imrove the system ilotability and safety. In this goal, the develoment of the assoiated flight ontrol laws must be addressed arefully. More reisely, tuning the gains of the ontrol laws is ruial and remains exensive in terms of time of develoment.. Handling Qualities (HQ) and Flying Qualities (FQ) The ilotability an be evaluated using riteria of Handling Qualities (HQ) []: those qualities or harateristis of an airraft that govern the ease and reision with whih a ilot is able to erform the tasks required in suort of an airraft role. Thanks to years of studies of numerial simuions and analyzes of flight tests around the world, the HQ an now be evaluated using qualitative and assoiated quantitative riteria. Today, these riteria are defined, exlained and lotted in the Aeronautial Design Standard (ADS) []. A helioter showing good results with reset to these riteria is said to have good Flying Qualities (FQ): this means the helioter shows good erformanes not only statially but dynamially as well. Designers usually tune the gains of the ontrol laws in order to handle at best the requirements from this standard..3 Tehniques for tuning the gains of the ontrol laws with FQ objetives No lassial synthesis method (suh as LQ, LQG, LQG/LTR, H, LMI, µ synthesis, et. [3]) seems to solve this roblem. Indeed, these tehniques do not manage the seifiity of the ontrol laws and the seifiity of the requirements. NASA has already led many studies for years to establish methods of tuning taking into aount riteria from standards whih have led to the develoment of CONDUIT [4]: a omuter aided design tool for rotary and fixed wing airrafts ontrol laws using interative otimization tehniques.

2 It is well known that the usual tuning roess used by manufaturers (Fig. ) an be time onsuming. ontrol of the anti-torque (or tail) rotor, and the motion of the helioter around its yaw axis. Tuning hat swith bad HQ test good Imlementation Fig. Usual method of tuning ONERA has been working for several years to the establishment of roesses ontributing to the redution of the ontrol laws develoment yles by integrating HQ requirements from the early hase of gains design (Fig. ) [5] [6] [0]. ol lon ed HQ seifiations Design and tuning roess Imlementation Fig. 3 Main ontrol Inuts used by the ilot Fig. Diret method of tuning foused by ONERA. MATERIALS AND PROBLEM STATEMENT. Notations On Fig. 3 the main ontrol inuts used by helioter ilots are resented. The Fig. 4 shows the main dynamial variables used to reresent the flight dynamis of a helioter. Tab. lists the arameters used for the establishment of the state sae reresentation of the natural system. Tab. List of main arameters ol Colletive inut / Lateral yli inut / lon Longitudinal yli inut / ed Pedals inut / u Longitudinal seed m/s v Lateral seed m/s w Vertial seed m/s Roll rate rad/s q Pith rate rad/s r Yaw rate rad/s ɸ Bank angle rad θ Pith angle rad Ψ Heading angle rad. Basis of iloting a helioter The olletive stik is onneted to the swash-e and ermits the ontrol of the rotor lift fore magnitude by modifying the average omonent of the blades ith angle. The yli stik allows hanging the -er-rev harmonis of the blades ith angle, and thus ontrolling the roll and ith moments of the rotor. This results in the orientation of the rotor around its roll axis when ating on the eral yli inut, and on the ith axis when ating on the longitudinal yli inut. The edals allow the Fig. 4 Main helioter flight dynami arameters.3 State Sae reresentation of a helioter The establishment of this kind of reresentation is well known [7] [8]. Let U, X, and Y resetively be the inut, state and outut vetors of the state sae reresentation of a helioter. Then, the inuts and the states of the helioter are: () v U X t [,,, ] ol q θ lon ed t [ u, v, w,, q, r,, θ, ψ ] Moreover, A, B, C, D are resetively the state, ommand, observation and inut/outut matries. Using usual linear assumtions around equilibrium, they an be alued from full reresentative nonlinear flight dynami models or identified from dediated flight tests. In this study, authors have used HOST (Helioter Overall Simuion Tool [9]), a modelling and simuion tool develoed by Airbus Helioters, and jointly imroved by ONERA. A M B M C I D () 9 x 9 9 x 4 9 x 9 04 x 9 r ψ w u

3 Then, we an write the state sae reresentation of the helioter (3). (3) X & A. X + BU. Y X This 9-by-9 linear model has been used for the develoment of a roess aiming to integrate HQ requirements from the early hase of gains design. The ontrol law studied during this work was an Attitude Command Attitude Hold (ACAH) ontrol strategy.4 Attitude Command Attitude Hold (ACAH) ontrol law The Attitude Command Attitude Hold (ACAH), as shown on Fig. 5 for the roll axis as an examle ermits to ontrol the attitudes of the helioter (ɸ for the roll axis), governed by the ilot yli stik. Fig. 5 ACAH roll axis only Then, we an write the feedbak equation for this axis (4). (4) K + K.( ) + K. ( ) Helioter K K ɸ K iɸ. i The aim of this aer is to set u a method to hel designers in tuning the gains of the ACAH ontrol law, taking into aount HQ requirements as desribed in the ADS-33 standards. Only some of these riteria have been seleted for this study and are resented in the next subsetion..5 Seleted riteria from ADS-33 (all others MTE hover and low seed) The standard seifies HQ riteria for a wide range of helioters. All riteria deend on the ase study, and they are detailed in a seifi lassifiation. For this aer, we will fous on a seifi ase study, exlained hereafter. ɸ ɸ.5. The ase study The helioters are indexed into 4 rotorraft ategories: attak, sout, utility and argo. This aer will fous on the study of a 0-ton lass helioter: argo-tye. A table in the standard seifies whih Mission Task Elements (MTE) the studied helioter should be aable of, and the required agility needed for eah one. For this aer, we will fous on the all others MTE requirements only. We will fous on the hover and low seed ase. A rating of Usable Cue Environment (UCE:, or 3) is rovided. We will study the ase of UCE, whih means that the ilot an make limited orretions with onfidene and reision is only fair [] on attitude and / or transional iloting. This an be evaluated using a Visual Cue rating sale (rovided in the standard as well). Deending on the ase study, a table seifies the required resonse tye. Here, the ACAH is needed to be able to ahieve Level resonse tye rating. The blo diagram reviously resented orresonds to this resonse-tye requirement. The next setions will exlain the requirements for this ase study. The roll axis will be foused. The three seleted riteria are eigenvalues laement (Setion.5.), attitude quikness laement (Setion.5.3) and bandwidth / hase delay laement (Setion.5.4). They resetively evaluate the helioter s stability, agility and ability to follow high frequeny inuts with auray. They onstitute the basi quantitative riteria to evaluate the exeted overall Handling Qualities (HQ) of a helioter. The standard exlains how to alue and lot the requirements for eah riterion. The aim is to lae the resulting oint(s) alued for eah riterion in the LEVEL area on its assoiated lot. LEVEL area is an indiator of medium quality. LEVEL 3 area should be avoided. The rotorraft shall meet the Level standards for all riteria. Vioion of any one requirement is exeted to degrade Handling Qualities []. The bold blak lines on Fig. 6, 8 & 0 reresents a limit between two areas. The green oints on eah of these lots are examles of good tunings, for its assoiated riterion..5. Limits on osilions: stability riterion This riterion is evaluated by aluing the eigenvalues of the dynami matrix of the linearized losed loo system. The requirements for their ositioning, as shown on Fig. 6, determine the quality of the helioter stability. The Level/Level limit an be suffiiently aroximated by ζ0.35 (red dotted line on Fig. 6).

4 q k ( k) [r k] Fig. 7 Definition of the attitude quikness arameters during an attitude hange Q Fig. 6 Stability riterion: ole laement.5.3 Requirements for moderate amlitude attitude hanges: quikness riterion This riterion is evaluated using attitude ature flight tests. For the gain tuning studies, attitude hange simuions are erformed. The riterion is omuted using 3 arameters (Fig. 7): eak roll rate ( k ), eak roll hange ( ɸ k ) and minimum roll hange ( ɸ min ). Then, of these 3 arameters are used to alue the quikness riterion (Q). The requirement for its ositioning deending on the minimum attitude hange is shown on Fig. 8. The Level / Level limit an be aroximated by (red dotted urve on Fig. 8): k k + a (5) Q + b k θ min Fig. 8 Quikness riterion: attitude quikness with k 3deg/se, a 7deg and b 0./se.5.4 Requirements for small-amlitude roll attitude hanges: auray riterion Fig. 9 Definition of small amlitude riterion arameters This riterion is evaluated using frequeny swees ature flight tests. In this aer, Bode diagrams are generated using either equivalent linear models of the losed loo system, if available, or using diretly frequeny swee simuions. For the study of the ACAH, this riterion is erformed using 3 arameters, as detailed on Fig. 9: the bandwidth ω 80 defined by a ut-off of the hase at -80, the bandwidth hase ω BWhase using a +45 hase margin from ω 80 utoff, and the variation of hase ɸ ω80 from -80 at the frequeny *ω 80. Then, these 3 arameters are used to alue the hase delay arameter (τ ), as defined on Fig. 9. The requirement for ositioning τ deending on ω BWhase is shown on the next figure: Fig. 0.

5 Fig. 0 Auray riterion: bandwitdth / hase delay The Level / Level limit an be aroximated by a vertial line at ω BWhase rad/se (red dotted line on Fig. 0). Indeed, the values of hase delay (τ ) seem to be mainly influened by the natural system delays. The gains of the ontrol laws seem to have negligible influene on its value [0], for stabilized systems at least..6 Desrition of the roedure develoed at ONERA The first idea followed in order to establish a diret method of tuning was to use the sensitivities of gains to the riteria as design guidane for full linearized systems. A tool (CAST-HEL-AP: Comuter Aided Setting and Tuning tool for Helioters Autoilots) was develoed at ONERA Salon de Provene for this urose [0]. The advantage of this roedure is: the designer is not influened by an otimization tool that ould tra his design in a loal otimum. The helioter is omlex (hard ouled system), the onstraints from the ontrol laws are hard (nonfull state feedbak) and the requirements from the standard are very seifi (non-ommon). Therefore, it was deided to establish simlified models of the oen loo and the losed loo system. As it is exlained thereafter, these models an be used to hel in establishing the initialization roess. The methodology is exosed through the examle of the tuning of the ACAH ontrol law for the roll axis. The riteria hosen as design objetives are those exosed in setion.5 for the study of a 0T lass helioter, argo-tye, with UCE-tye and all other MTE-tye requirements. As for an initialization roess with suffiient FQ requirements the linear models will ignore atuator models and their saturations. HQ targets However, due to the non-linearity of the reionshi between the gains and the HQ riteria, the sensitivity method an t be effiiently alied if the initial set of gains doesn t already insure the stability of the helioter with a signifiant imrovement of the overall FQ This means that to be fully effiient with the sensitivity analysis toolbox (CAST-HEL-AP), there is a need to set u a roess of gains initialization roviding already fair handling qualities. This aer fouses on the establishment of the methodology to initialize the gains of a ontrol law with FQ objetives. Analyzes and harts Gains initialized Imlementation of gains on full linear model with CAST-HEL-AP Exeted HQ Evaluation of the linear initialization roess The final full roedure foused by ONERA is summarized on Fig.. The setion 3 will fous on the exlanation of the new linear initialization roess (yellow art on the Fig. ). The setion 4 will give an examle of a omlete design and evaluation, using the full roedure. 3. PROCESS FOR INITIALIZING GAINS OF CONTROL LAWS WITH FQ OBJECTIVES As shown on Fig., the idea is to reate a roess that will ermit to initialize the gains of the ontrol law, diretly from a set of FQ riteria, seified at the beginning of the design studies. If the requirements hosen to be ahieved are too restritive, there is a risk of saturation from atuators. Then, the aim is to ahieve suffiient levels of quality for the whole set of requirements, not to imrove them more than needed. Adjustments of gains on full linear model with CAST-HEL-AP using linear sensitivities results Gains and exeted HQ imroved Imlementation of gains on full nonlinear model Obtained HQ Adjustments of gains on full nonlinear model using loal linear sensitivities Final gains and HQ Evaluation of the linear sensitivities roess Evaluation of the initialization of the final system Evaluation of the full roedure Fig. Full roedure of tuning foused by ONERA

6 3. Analytial studies led to hel establishing reionshi between gains and FQ riteria Eah axis-ontrol law is reated for seifi uroses, say for seifi ontrol of the axis. The main aim of a ontrol law for one axis is not to ontrol the other axes, whih are stabilized thanks to their own axis-ontrol laws (or by the ilot at least). Then, the work made for one axis an be re-used for the other axes. Thanks to these onditions, and due to the omlexity of the system, we will onsider the establishment of a simlified one-axis oen loo (and losed loo) system. 3.. Simlified one-axis oen-loo model The axis hosen to be studied here is the roll axis. The assoiated states are (,ɸ). The main inut used to ilot this axis is. Then, from (), let X, X, U and U be: (6) U X t t [ ] U [,, ] ol lon ed [, ] t X [ u, v, w, q, r, θ, ψ ] t Then (3) an be written on another way using ermuted matries of A and B, suh that: X& X& A A A A X. X B + B B B U. U X & A. X + A. X + B. U + B. U X & A. X + A. X + B. U + B U (7) (8). Then, some hyotheses an be used for the establishment of our simlified models. - Hyothesis : X 0. During a roll hange, the ilot or the other ontrol laws take are of the stabilization of the other axes (vertial, ith, yaw). We onsider the assoiated states will only slightly vary from equilibrium, say zero. - Hyothesis : U 0. To ilot the roll axis, the mainly used inut is the roll assoiated inut. For an initialization of gains roedure, even if slight variations of the other inuts are needed to omensate oulings, we onsider that the other inuts do not vary from initial equilibrium, say zero. Alying these two hyotheses on (7) imly: X &. + U A X B. (9) - Hyothesis 3: &. The derivative of the roll angle is aroximated to the roll rate (true while in ure eral flight, and this fits with the onditions of the assoiated evaluations of ADS33 requirements). Alying this new hyothesis, using the definitions from (6) and using sensitivities L (ṗ)/ (), L ɸ (ṗ)/ (ɸ), L (ṗ)/ (), (9) beomes: & L. + L. + (0) L. - Hyothesis 4: L ɸ 0. The dynami between the roll angle and the derivative of the roll rate is negleted (negligible value omared with the other sensitivities: usually less than 0-9 %). Alying this hyothesis on (0) imlies: & L. + () L. 3.. Simlified one-axis losed-loo model Inluding the equation of the ontrol law (4) into (), we obtain: ( K. + K.( ) + K ( ) & L. + L i ) Using hyothesis 3 again, and alying Lalae transformations (s is the Lalae variable here), this last equation an be written: l + L l. s Lˆ L + L. K Ll L. K L l L. K i () s Lˆ.. s + L.( ) (3) with:. Then, the inut/outut transfer funtion is: (4) s 3 Lˆ. s ( L. s + L ) l L l l. s L l We obtain an order three model. By analogy to the usual natural modes of the rotorraft, we hoose to imose one real mode and two omlex modes to the behaviour of the losed loo model. Then, (4) an be written with an equivalent transfer funtion: (5) (6) + τ. s ωn with: + τ. s s +. ζ. ωn. s + ωn τ τ +. ζ and ω n > 0, τ > 0, τ > 0, ζ > 0 ω n

7 A reion between the arameters of the equivalent transfer funtion and the assoiated gains of the one-axis iloting assistane an be established (7). (7) Ki K K ωn L. τ. ζ. ωn + τ. ωn L. τ L +. ζ. ω n. τ + L τ. L Thanks to these develoments, (7) is seifially very interesting. Indeed, if values of ω n, τ and ζ an be hosen, the gains of the simlified model an be diretly initialized with (7), using the sensitivity arameters L and L from the model. Then, the idea is to hoose the values of the arameters of the equivalent transfer funtion in order to satisfy suffiient FQ requirements. A tool an hel in that urose. 3. Develoment of a tool generating harts of FQ for simlified models of helioters This tool has the aim to hel the designer in hoosing values of the arameters of the simlified models suh that the assoiated set of exeted FQ are suffiiently good. The idea for this tool is to generate the riteria for a swee of models suh that the arameters used in (5) (ω n, τ and ζ) are hysially suitable. Then, the data an be used to generate harts of FQ. The designer an then use these harts to find assoiated values of ω n, τ and ζ suh that the exeted FQ are those reommended by the standard. Then the equations from (7) ermit the initialization of the gains. The first ste is to establish a range of values of ω n, τ and ζ hysially suitable to the system. A quik overview on (5) an hel in this. On the one hand, if ω n, τ or ζ are almost equal to zero, the model (5) might hange radially (risk of reation of a ure unstable integrator for examle). Furthermore, the imat on the values of the gains (7) an be unaetable (value of K iɸ an tend to infinity for examle). On the other hand, beause we are studying a hysial and ilotable system, the maximum values of ω n, τ and ζ an be emirially limited. Finally, one an hoose to generate a set of models suh as (5) with: (8) < < ω n τ < 3 < 3 < ζ < Then, all riteria an be alued for the whole set of models using fast linear simuions. Here is an examle of an algorithm for generating harts of FQ: - Choose a value for ζ and a value for the amlitude of the attitude hange ɸ (neessary for attitude quikness aluions). - For ω n from 0. to 3 and for τ from 0. to 3 do: - Create the simlified transfer funtion (5) (using (6)). - Calue the attitude quikness arameters using a linear attitude hange simuion. - Calue the bandwidth / hase delay arameters using a hase Bode diagram. - Calue isoleths of attitude quikness and bandwidth / hase delay arameters. - Calue isoleth of the aroximated Level / Level limit for the attitude quikness riterion (Fig. 8 and equation (5)). - Calue isoleth of the aroximated Level / Level limit for the bandwidth / hase delay arameter (Fig. 0). - Plot the whole set of isoleths in a hart with ω n deending on τ. 3.3 Exlanation of how to read a hart through an examle The Fig. shows an examle of a hart of FQ using the revious algorithm, hoosing ζ0.35 and ɸ 0deg, for the roll axis. A model of natural delays is inluded in the aluions. A oint on this figure is defined by seifi values for ω n and τ. As ɸ and ζ are fixed, then a oint defines a model suh as (5). For eah oint, all riteria and interesting arameters are generated as exlained in revious setions. Then, for eah oint on this figure, we have information on the behaviour of the simlified one-axis model of the helioter, for the roll axis in this examle. The oloured lines lotted are isoleths of riteria and arameters of interest for this aer. For examle, the red bold line reresents all models ossible to hoose if we want to be sure that their erformane in terms of quikness is at the LEVEL / LEVEL limit (as defined by the red dotted line on Fig. 8). This means that if we hose a oint on this line, the assoiated arameters ζ, ω n and τ (and ɸ ) generate a model (5) whih insures to rovide a quikness level verifying equation (5), for this ase study (ɸ ). The examle for the red bold line is the same for the blue bold line: if we hoose a model on this line, the assoiated arameters insure that the simlified model resents a suffiient LEVEL quality with reset to the bandwidth riterion (as defined by the red dotted line on Fig. 0).

8 E E E 3 E 4 Q 3 W 3 W W Q Q Fig. Examle of a hart generated for the roll axis ontrol law of a argo-tye helioter All the oloured lines follow the same rule. Red lines give information on quikness (isoleths of Q arameter). Blue lines give information on bandwidth (isoleths of ω BW arameter). Yellow lines are isoleths of the integral gain K iɸ (used er in the ommuniation). Blak lines give information about the need on atuator energy during the attitude hange simuion made to alue the attitude quikness, using an energeti riterion (exlained in next subsetion). Then, for eah oint on this hart, a model and its assoiated FQ are generated and lotted (via isoleths). One an hoose a oint on this kind of hart, deending on its FQ objetives. Then thanks to (7), one an alue the assoiated gains that will ermit to obtain suffiient FQ levels for the simlified model at least. These gains an be used as initialization for the omlete model. The differenes between the results of the hart and those obtained with full linear models are evaluated in the next setions so that the designer has information on the results he an trust in. The oints on Fig. will mainly be used for these evaluations. be used. We will hek the energy used until the system will be onsidered as stabilized, so that the energy usage will be stabilized as well. We deided to onsider that a system modelled suh as (5) is stabilized one the values of the attitudes do not extend any more over a band of 0% of the final value. The instant from when the system is stabilized is alled settling time to 0%: Ts 0%. To evaluate this settling time, the model (5) an be aroximated by (9). s (9) ω n +. ζ. ωn. s + ωn The Fig. 3 shows a usual attitude hange simuion generated with (5) in yan, and its assoiated aroximated attitude hange simuion generated with (9) in blue. The time exression of the enveloe of the time resonse of the model (9) to a ste inut ɸ an be aroximated by (0) (red urve on Fig. 3). 3.4 Additional riterion for energy evaluation During an attitude hange simuion, an amount of energy is used from the atuators. In order to evaluate this amount of energy sent, a riterion an (0) Env ( t). e ζ. ω. t n ζ

9 The value of _MAX orresonds to the value of stati saturation of atuators. The riterion E an be visualized with the grey area on Fig. 4. Its value will be omared to the maximum of energy the atuator ould furnish during the same time. This an be evaluated thanks to the Ue riterion: Fig. 3 Evaluation of the aroximated settling time to 0%. Finally, we an alue the settling time to 0% of the final value: () Env ( Tr 0% ζ. ωn. Tr e ) ζ log( 0.05 ζ ) () Tr 0% ζ. ω Thanks to all data generated, it is now ossible to evaluate the amount of energy used during an attitude hange simuion (Fig. 4) using the E riterion (alied to the roll axis): Tr0% (3) E min(, ( t ) 0 Ts 0% n _ MAX ) _MAX Ts 0% Fig. 4 Evaluation of the energy used during an attitude hange simuion (model (5)) dt 0% Tr 0% (4) Ue ( _ MAX ) 0 dt Finally, the energy riterion hosen to be onsidered for the tuning will be exressed as a erentage of the maximum amount of energy the atuator ould have generated during the attitude hange simuion until the settling time to 0%. This riterion will be alled Energy Usage Eu (%): (5) Eu 00 E Ue Isoleths of Eu have been lotted on Fig.. If Eu tends to 00%, this means the atuator saturates during the whole attitude hange. Then the designer might look for minimizing its value. Later, a (realisti) filter will be used so that the first eak of the atuator usage an be highly redued, with no signifiant loss on overall exeted FQ. By the way, for more seurity about saturations due to windus, an antiwindu arhiteture ould be used []. 3.5 Evaluation of the hart Many oints of tuning have been hosen on Fig. in order to hek the reliabilities of the exeted FQ. Full linear simuions (inluding natural system delays and atuator models) have been led in that urose (using CAST-HEL-AP [0]). The results are summarized on Fig. 5, 6 & 7: - red dots for the exeted results with hart. - blue stars for the obtained results with full linear simuions, using same gains. - blak, green and grey squares for full linear and nonlinear design led in setion 4. The oints Q, Q and Q3 have been hosen suh that the exeted bandwidth is at the Level / Level limit. The oints W, W and W3 have been hosen suh that the exeted quikness arameter is at Q0.5. The oints E, E, E3 and E4 have been hosen suh that all exeted riteria are at least at the Level / Level limits. Results are summarized in Tab.. The quikness arameter is exeted (simlified model) to be imroved by 40% between Q and Q and by 9% between Q and Q3. The obtained results (full linear model) show that they are imroved by 3% and 9%.

10 W Q E E WQ E3 E4 E4 E4 E4 Q3 Ei W3 Q3 Wi & Q Q E4 E3 E E4 E E3 E4 E4 E4 E E Q3 Fig. 5 Attitude quikness results (i {:4}) Q3 Wi & Q W WQ W3 Q Q W W E4 E E4 E4 Q Qi E W Ei W3 E3 Q3 QW E4 W3 Fig. 7 Poles results (i {:4}) Fig. 6 Bandwidth / hase delay results (i {:4}) Tab. Summary of the results and omarison (Ga row) between simlified and full linear models Points on Fig. Q Q Q 3 W W W 3 E E E 3 E 4 Simlified model Full linear model Ga Gains τ (s) ω n (rad/s) Q (/s) ω BW (rad/s) Q (/s) ω BW (rad/s) ζ Q% -40% -3% -3% -3% -3% -9% -6% -8% -4% -6% ω BW % +8% -5% +% +% -5% -35% +% +% +8% -4% ζ% +9% +4% +9% +43% +4% +0% 43% 43% 9% 4% K K ɸ K iɸ The bandwidth arameter is exeted to be imroved by % between W and W and by 34% between W and W3. The full linear simuions show resetively an imrovement of the bandwidth by 33% and 55%. Then, in eah ase, the sensitivity is verified. It omes out from Tab. that in eah ase, the overall FQ obtained with the full linear model are in an area near the exeted FQ given by the hart. This result demonstrates that the gains generated by the harts are able to ahieve the targeted FQ riteria. As a onsequene, the harts seem to be enough aurate to ermit a fast and effiient initialization of the gains with multile FQ objetives. 3.6 Advies for using the hart in order to initialize the gain tuning roess with HQ objetives In order to meet LEVEL HQ, a designer might want to use a hart for his design. The first ste is to hoose the ase study. Deending on this ase study, the set oint might be fixed, and, in order to meet suffiient LEVEL stability, ζ might be fixed to Then, thanks to the algorithm, a hart an be generated. The Fig. has been generated using the reommendations from the standard with ɸ 0deg and ζ0.35. In order to meet suffiient LEVEL quality for bandwidth hase delay riterion, a oint might be hosen on the blue bold line. However, if a oint is hosen on this line, the assoiated quality for

11 attitude quikness is never at LEVEL for this ase study. None of these designs seem interesting to meet good overall FQ. Then, the designer might hoose a oint on the red bold line. Indeed, all oints on this line generate gains with whih the exeted quality for attitude quikness is suffiiently at LEVEL. Furthermore, the exeted quality for bandwidth hase delay is at LEVEL with a large exeted margin. Then, if a designer hooses any oint on this line, the overall exeted FQ might be at LEVEL. A question might aear then: if any oint an be hosen on the red bold line (for examle E, E, E3 and E4), whih an be onsidered as the best? Indeed, the exeted FQ objetives seem suffiiently good in all ases. Here, we suggest onsidering two additional riteria for the initialization roedure: an energy riterion (the energy usage riterion exlained before) and a new unouling riterion. The blak isoleths on Fig. already give indiations on the exeted amount of energy used for eah design during the attitude hange simuion done to generate the attitude quikness, as exlained in setion 3.4. We suggest minimizing this riterion (in order to minimize the risks assoiated with otential saturations). Then, in this ase study, the left area of this hart might be avoided. The last objetive might be to minimize the ouling of the full system. Eventually, even if the model we established has been simlified for a one-axis study, the ouling of the system an be seen by this model as erturbations. Indeed, the equations (7) and (8) an be seen shematially on Fig. 8, using a framework similar to the individual hannel design []. In setion 3., U and X have been negleted suh as our model has been redued to the red art of Fig. 8. However, these hyotheses are wrong. Then, we an see these ouling signals as erturbation ert from the oint of view of the one-axis ontrol law. Then, if we wish to onsider these oulings during the tuning roess, (9) beomes: & with (6) X A. X + B. U + ert (7) ert & & Then, using hyothesis (3), using Lalae transformations and onsidering ɸ 0, (6) and (7) beome: (8). s L. + L. K ( s). + & and (9). s + with (30) K & ( s ) K. s + K + K i s X E U K (s) B ert X & A X B A B A X E K (s) B U X & X A Fig. 8 Helioter arhiteture with the usual ACAH ontrol law

12 Finally, the new transfer funtions between eah one of the erturbations and ɸ are: (3) (3) & & s s 3 3 Lˆ. s s L s. Lˆ. s L l. s L ( s L ) l. s L. K. K Then, thanks to the ontrol law, the steady state error to ste inut like erturbations is zero. Indeed: ε (33) lim ( t) lim s. ( s) 0 s t ϕ s 0 But the steady-state errors to ram-like erturbations are resetively (using normalized amlitudes): (34) (35) ε ε s _ & i i limϕ( t) lim s. ( s) t s 0 L ϕ lim ( t) lim s. ( s) s _ & t s 0 L. L. K The results from (34) and (35) are ruial for deouling objetives. Indeed, as required by the standard, the tuning must be led by keeing in mind to unoule the system as muh as ossible. Here, we an see that we tend to hel in that unouling by inreasing the value of the integral gain. That s why the isoleths of K iɸ have been lotted on Fig. (yellow lines). Here is the full roedure advised for an effiient initialization of the gain tuning roess: - Choose a ase study in order to fix the FQ objetives as well as ζ and ɸ. - Generate the hart for this seifi ase study of the one axis ontrol law. - Find the area in whih the exeted attitude quikness and bandwidth hase delay riteria are both at LEVEL (this should be near at least one of the two bold lines, red or blue on Fig. ). - Choose a oint in this area, near at least one of these limits, by making a omromise between maximizing as muh as ossible the integral gain (for unouling) and minimizing as muh as ossible the energeti riterion during the oneaxis attitude hange simuion led. K i i 4. APPLICATION AND EVALUATION OF THE FULL PROCEDURE The initialization of gains roedure is alied to the full helioter, for eah axis. Then, an analysis is led for the roll axis only (without modifiations of the gains on ith and yaw). All results obtained during the analyses led for the roll axis in this setion are summarized on Tab. 3. All FQ obtained in this setion are reresented on Fig. 5, 6 & 7: red dot Ei (with i4) for exeted FQ from the hart, blue star E4 for obtained FQ with same gains used on full linear model, blak square E4 for the FQ obtained after modifying the gains on full linear model using loal linear sensitivities, green square E4 for the FQ obtained with the same gains used on the full non linear model, E4 for the final FQ obtained on full non linear model after modifying the gains using loal linear sensitivities. All these stes math with the full roedure exlained by Fig.. All FQ objetives are those exlained in setion.5, for the roll axis. 4. Initialization of the gains of the full linear model using harts Here, we diretly aly the initialization of gains roedure exlained in setion 3.6. We hoose to tune the ACAH for the roll axis with LEVEL FQ objetives. We fix: ζ035 and ɸ 0deg. The hart on Fig. has been generated with these onditions. A oint on the red bold line generates gains with whih all exeted FQ are at least suffiiently in the LEVEL area: we will hoose a tuning on this line. A good omromise between reduing energy onsumtion (blak isoleths of Eu riterion) and inreasing unouling (yellow isoleths of K iɸ riterion) an lead to hoose the oint E4: red dot Ei for i4 on Fig. 5, 6 & 7. The exeted FQ and the assoiated gains are summarized on the first row of Tab. 3. The Fig. 5, 6 & 7 show that the overall exeted FQ with the simlified model are quite well verified with the full linear model (blue star E4). Indeed, the gains obtained with the hart have been imlemented in CAST-HEL-AP [0] to evaluate the FQ (using exatly same onditions) with the full linear model: this inludes model of atuators and natural system delay. The results are summarized in the seond row of Tab. 3. Tab. 3 Summary of the results of the full tuning roedure: last three olumns show the evolution of gains Full tuning roedure Q (/s) ω BW (rad/s) ζ K K ɸ K iɸ Exeted FQ from hart and assoiated gains Obtained FQ with full linear model (same gains) Modifiations of the gains using sensitivities (full linear model) Modified gains Imlemented on the full non-linear model Modifiations of the gains on the full non-linear model

13 The last olumn of Tab. gives reisions about the gas of results between the exeted FQ with simlified model and the obtained FQ with full linear model. There is a ga of 6% between the results of quikness, 4% between the results of bandwidth and 4% between the results of stability. This onfirms the overall obtained FQ with full linear model are in the area of the overall exeted FQ by the hart. 4. Modifiation of the gains of the full linear model using linear sensitivities The blue star E4 on Fig. 7 indiates a lak of stability for the full linear model with these gains. Using loal linear sensitivity results [0], the derivative gain is inreased in order to obtain LEVEL stability for the assoiated slow mode. The results are summarized with the third row of Tab. 3 and with the blak square E4 on Fig. 5, 6 & 7. Thanks to this modifiation, the overall exeted FQ with full linear model are at LEVEL for all riteria. In order to obtain these results, only the derivative gain has been inreased by 0% only. Doing this, the quikness arameter has been dereased by 5%, the bandwidth has been inreased by 4% and the stability has been inreased by 4%. All of this means that we have initialized the gains effiiently in one ste thanks to the hart. The sensitivities results are used only to make slight adjustments at this ste. Now, all exeted FQ are all in LEVEL area, with a good margin for the bandwidth (as it was already exeted by the hart). 4.3 Evaluation of the differenes of results between full linear models and full nonlinear models Then these gains are used with full nonlinear models (inluding atuator model and natural system delay) to evaluate the overall FQ, using exatly same onditions: results are summarized with the fourth row of Tab. 3 and with green square E4 on Fig. 5, 6 & 7. As we an see, the quikness falls by 8% and the bandwidth falls by 47% (the stability is evaluated with full linear model, that s why there is no modifiation on this riterion). The figures show that the bandwidth is still in the LEVEL area. Only the quikness is in the LEVEL area. However, the overall obtained FQ are still in an interesting area, as exeted by linear results. 4.4 Modifiations of the gains on the full nonlinear model using linear sensitivities results The green square E4 on Fig. 7 indiates a lak of quikness for the full nonlinear model. Using loal linear sensitivities results, the roortional gain is inreased in order to obtain LEVEL quikness. All results are summarized with the fifth row of Tab 3. and with grey square E4 on Fig. 5, 6 & 7. Then, by inreasing the roortional gain by 33%, the quikness has been inreased by 7%, the bandwidth has been inreased by % and the stability (obtained with full linear model) by %. The overall exeted FQ is at LEVEL. 4.5 Evaluation of the full diret roess: omarison of exeted results from harts with obtained results from full nonlinear model The aim was to tune the gains of the ontrol law of the helioter in order to obtain LEVEL exeted HQ for the roll axis. More reisely, there were onditions on quikness, bandwidth and stability arameters. Some gains have been hosen for initialization. Then, in order to meet LEVEL limits, the derivative gain has been inreased by 0% and the roortional gain by 33%. During the tuning roess, the quikness arameter has been inreased by %, the bandwidth arameter has been dereased by 5% and the stability arameter has been inreased by %. Finally, with this diret roedure, the overall exeted HQ has been set to LEVEL, in few stes, using the FQ requirements from ADS-33. Only slight adjustments have been neessary. 5. CONCLUSION ONERA has been working for several years to the establishment of an effiient methodology of gain tuning for helioter ontrol laws with Handling Qualities objetives, from the early stes of gains design. In this aer, an examle of the full roedure has been resented. Thanks to the methodology develoed, the use of Flying Qualities harts generated from an analytial study widely ontributes to an effiient initialization of ontrol law gains onstrained by Flying Qualities requirements. Moreover, the hysial basis of these harts insures not only the initialization of the gains, but also a orret sensitivity of the gains to the Flying Qualities riteria, as well as a good estimation of tuning oints near the limits. A tool imlementing all this methodology is urrently in develoment at

14 ONERA Salon de Provene. This tool, CAST-HEL- AP Comuter Aided Setting and Tuning tool for Helioter AutoPilots [0] might effiiently hel a designer in his task of gain tuning of seifi ontrol laws with Handling Qualities objetives. Simlified models, full linear models inluding models of atuators and its natural system delays, and full non linear models an be easily used to generate all neessary Flying Qualities at eah ste of the tuning roess detailed on Fig.. The next stes ould be to use PysHel (Fig. 9), a rototying and simuion form for helioters, urrently in develoment at ONERA Salon de Provene. Indeed, thanks to this form, tehnologies suh as ontrol laws and assoiated oerational logis an be imlemented and tested easily. The gains of these ontrol laws an be tuned using the methodology exosed in this aer. One this is done, the gains an be imlemented in the models (last ste of Fig. ) and they ould be tested by ilots to evaluate the overall Handling Qualities of the airraft with the seifi gains designed. Fig. 9 PysHel: a rototying and simuion form for helioters urrently in develoment at ONERA Salon de Provene 6. COPYRIGHT STATEMENT The author(s) onfirm that they, and/or their omany or organisation hold oyright on all of the original material inluded in this aer. The authors also onfirm that they have obtained ermission, from the oyright holder of any third arty material inluded in this aer, to ublish it as art of their aer. The author(s) onfirm that they give ermission, or have obtained ermission from the oyright holder of this aer, for the ubliation and distribution of this aer as art of the ERF04 roeedings or as individual offrints from the roeedings and for inlusion in a freely aessible web-based reository. This work has been done thanks to funds from ONERA and Conseil Régional Provene Ales Côte d Azur (Pôle Pégase). 7. REFERENCES [] G. E. Cooer and P. Harer and G. O. Young. The Use of Pilot Rating in the Evaluation of Airraft Handling Qualities. Aendix A. NASA TN D-553. Aril 969. [] B.J. Baskett and Dr. L.O. Daniel. Aeronautial Design Standard erformane seifiation Handling Qualities requirements for military rotorraft. United States Army Aviation and Missile Command [3] D. Alazard and C. Cumer and P. Akarian and M. Gauvrit and G. Ferreres. Robustesse et Commande Otimale. CÉPADUÈS ÉDITIONS, 000. [4] M.B. Tishler and J.D. Colbourne and M.R. Morel and D.J. Biezad and W.S. Levine and V. Moldoveanu. CONDUIT A New Multidisilinary Integration Environment for Flight Control Develoment. NASA [5] A. Luzi. Méthodologie de ommande orientée séifiations our héliotères. ONERA, MR /766 DCSD. 00. [6] A. Badatheff. Définition de lois de ommande our héliotères orientées séifiation. ONERA, PFE /8895 DCSD. 0. [7] G.D. Padfield. Helioter Flight Dynamis: The Theory and Aliation of Flying Qualities and Simuion Modelling, Seond Edition. Oxford:Blakwell [8] A.R.S. Bramwell. Helioter dynamis. AIAA. 00. [9] B. Benoit and A.-M. Dequin and K. Kama and W. Gunhagen and P.-M. Basset and B. Gimonet. HOST, a general helioter simuion tool for Germany and Frane. 56 th Annual Forum of the Amerian Helioter Soiety [0] J.C. Antonioli and A. Taghizad and T. Rakotomamonjy and M. Ouladsine. Helioter flight ontrol design tool integrating Handling Qualities requirements. Euroean Conferene for Aerosae Sienes EUCASS. 03. Best student aer of the ategory Flight Dynamis & GNC. [] J-M. Bianni. Limit-Cyles Prevention Via Multile Hinfinity Constraints with an Aliation to Anti-Windu Design. In the roeedings of the 9 th IFAC Symosium on Nonlinear Control Systems. Toulouse. Frane. 03. [] C. E. Ugalde-Loo and E. Liéaga-Castro and J. Liéaga-Castro. x Individual Channel Design MATLAB Toolbox, 44 th IEEE Conferene on Deision and Control, and the Euroean Control Conferene 005.