Tilt-rotor multivariable attitude control with rotor state feedback

20th IFAC Symposium on Automatic Control in Aerospace 20th IFAC Symposium on Automatic Control in Aerospace August 21-25, 2016. Sherbrooke, Quebec, Canada 20th IFAC IFAC Symposium on Automatic Automatic Control in Aerospace Aerospace 20th Symposium on Control in August 21-25, 2016. Sherbrooke, Quebec, Canada Available online at www.sciencedirect.com August 21-25, 2016. Sherbrooke, Quebec, Canada August 21-25, 2016. Sherbrooke, Quebec, Canada

ScienceDirect IFAC-PapersOnLine 49-17 (2016) 100–105

Tilt-rotor multivariable attitude control Tilt-rotor Tilt-rotor multivariable multivariable attitude attitude control control with rotor state feedback with rotor state feedback with rotor state feedback S. Panza, ∗∗ L. Guastalla, B. Roda, M. Lovera ∗∗ S. Panza, ∗∗ L. Guastalla, B. Roda, M. Lovera ∗∗ S. S. Panza, Panza, L. L. Guastalla, Guastalla, B. B. Roda, Roda, M. M. Lovera Lovera ∗ ∗ Department of Aerospace Science and Technology, of Aerospace Science and Technology, ∗ ∗ Department Department of Science Technology, Politecnico di Milano, via La Masa 34, and 20156, Milan, Italy Department of Aerospace Aerospace Science and Technology, Politecnico di Milano, via La Masa 34, 20156, Milan, Italy Politecnico di Milano, via La Masa 34, 20156, e-mail: (simone.panza, marco.lovera)@polimi.it Politecnico di(simone.panza, Milano, via La marco.lovera)@polimi.it Masa 34, 20156, Milan, Milan, Italy Italy e-mail: e-mail: (simone.panza, (simone.panza, marco.lovera)@polimi.it marco.lovera)@polimi.it e-mail:

Abstract: In this paper, a multi-objective optimization-based methodology for rotorcraft Abstract: In paper, a optimization-based methodology for rotorcraft Abstract: In this this paper, aismulti-objective multi-objective optimization-based methodology for to rotorcraft attitude control lawpaper, design a applied to a tilt-rotor case; the framework allows enforce Abstract: In this multi-objective optimization-based methodology for rotorcraft attitude control law design is applied to aa tilt-rotor case; the framework allows to enforce attitude control law design is applied to tilt-rotor case; the framework allows to enforce requirements of stability, performance, control action moderation, and safety. The structured attitude control law design is applied to a tilt-rotor case; the framework allows to enforce requirements of stability, performance, control action moderation, and safety. The structured requirements of stability, performance, control action moderation, and safety. The structured H approach is taken into account, and the optimization problem is stated as a mixed-sensitivity ∞ requirements of stability, performance, control action moderation, and safety. The structured H approach is into and the optimization problem is stated as H∞ approach is taken taken into account, account, andto thelimit optimization problem stated as aaa mixed-sensitivity mixed-sensitivity problem. Rotor state feedback is used flap motion, and is stated introduced as an additional ∞ H is taken into account, and the optimization problem as mixed-sensitivity ∞ approach problem. Rotor state feedback is used to limit flap motion, and is introduced problem. Rotor state feedback is used used to limit limit flap flap motion, motion, and and is is introduced introduced as as an an additional additional loop to a Rotor classical attitude control law.to problem. state feedback is as an additional loop to a classical attitude control law. loop to aa classical attitude control law. loop to classical attitude control law. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Attitude control, H-infinity optimization, Aircraft control, Sensitivity functions Keywords: Keywords: Attitude Attitude control, control, H-infinity H-infinity optimization, optimization, Aircraft Aircraft control, control, Sensitivity Sensitivity functions functions Keywords: Attitude control, H-infinity optimization, Aircraft control, Sensitivity functions 1. INTRODUCTION Tilt-rotor prototypes possess a wider instrumentation than 1. Tilt-rotor prototypes possess a wider instrumentation than 1. INTRODUCTION INTRODUCTION Tilt-rotor prototypes possess wider instrumentation than productionprototypes ones; in possess additionaa wider to a instrumentation classical IMU (iner1. INTRODUCTION Tilt-rotor than production ones; in addition to aa classical IMU (inerproduction ones; in addition to classical IMU (inertial measurement unit) which to provides measurements of production ones; in addition a classical IMU (inertial measurement unit) which provides measurements of tial measurement unit) which provides measurementsand of attitude angles and rates, andprovides linear accelerations tial measurement unit) which measurements of angles and rates, and accelerations and In the near future of civil aviation, tilt-rotors are expected attitude attitude angles and rates, and linear linear accelerations and velocities,angles often and a blade flapping measurement system is attitude rates, and linear accelerations and In the near future of civil aviation, tilt-rotors are expected velocities, often a blade flapping measurement system In the near future of civil aviation, tilt-rotors are expected to the playnear a relevant role in the class of VTOL (vertical velocities, velocities, often aprototype, blade flapping flapping measurement system is is mounted on the and is used for monitoring In future of civil aviation, tilt-rotors are expected often a blade measurement system is to play aa relevant role in class of VTOL on the and is monitoring to play and relevant role in the the classand of Padfield VTOL (vertical (vertical take-off landing)role aircraft (Meyer (2005)), mounted mounted the prototype, prototype, and is used usedof for for monitoring reasons; inon order to limit the amount flapping, such to play a relevant in the class of VTOL (vertical mounted onorder the prototype, and is used for monitoring take-off and landing) aircraft (Meyer and Padfield (2005)), reasons; in to limit the amount of flapping, such take-off and landing) aircraft (Meyer and Padfield (2005)), since they can overcome the shortcomings of conventional reasons; in order to limit the amount of flapping, such measurement system could potentially be used to feed flap take-off and landing) aircraft (Meyer and Padfield (2005)), reasons; in order to limit the amount of flapping, such since they overcome the of system could potentially be used to feed flap since they can can overcome the shortcomings shortcomings of conventional conventional helicopters in terms of speed and range; however, the de- measurement measurement system could potentially be used to feed flap measurements back to the attitude control law in addition since they can overcome the shortcomings of conventional measurement system couldattitude potentially be used to feed flap helicopters in terms of and range; the back control law addition helicopters inthis terms ofofspeed speed andrequires range; however, however, the dede- measurements velopment of kind aircraft special attention measurements back to to the the attitude control load law in in addition to IMU measurements, or to the structural alleviation helicopters in terms of speed and range; however, the demeasurements back to the attitude control law in addition velopment of this kind of aircraft requires special attention to IMU measurements, or to the structural load alleviation velopment of this thisqualities kind of of aircraft aircraft requires special especially attention to as for handling and flight dynamics, to IMUcontrol measurements, oristo toevidence the structural structural load alleviation active law: there in theload literature that velopment of kind requires special attention IMU measurements, or the alleviation as for handling qualities and flight flight dynamics, especially active control law: there is evidence in the literature that as for handling qualities and dynamics, especially in the conversion maneuver, see Manimala et al. (2004). active control law: there is evidence in the literature that the V-22 tilt-rotor already used such flap feedback many as for handling qualities and flight dynamics, especially active control law: there is evidence in the literature that in the conversion maneuver, see Manimala et al. (2004). the V-22 tilt-rotor already used such flap feedback many in the conversion maneuver, see Manimala et al. al. (2004). (2004). In the airplane configuration andsee during conversion, it may the the V-22 tilt-rotor already used suchRotor flap feedback feedback many yearsV-22 ago,tilt-rotor see Miller et al. used (1991). state feedback in conversion maneuver, Manimala et already such flap many In airplane configuration and during conversion, it may ago, et (1991). Rotor state feedback In airplane configuration and show during it happen thatconfiguration large rotor loads up conversion, in demanding ma- years years ago, see seebyMiller Miller et al. al. (1991).see Rotor state feedback was studied different authors, Takahashi (1994), In airplane and during conversion, it may may years ago, see Miller et al. (1991). Rotor state feedback happen that large rotor loads show up in demanding mawas studied by different authors, see Takahashi (1994), happen that large rotor loads show up in demanding maneuvers in the form of in-plane bending moment, which was studied by different authors, see Takahashi (1994), Horn et al. (2012), Ivler (2014), Panza and Lovera (2014), happen that large rotor loads show up in demanding mawas studied by different authors, see Takahashi (1994), neuvers in the the form form of of in-plane in-plane bending bending moment, moment, which which Horn et al. (2012), Ivler (2014), Panza and Lovera (2014), neuvers in in turn are related toofout-of plane moments and lead, as Horn Horn etand al. Lovera (2012), (2015), Ivler (2014), (2014), Panza and Lovera (2014), Panza Panza et al. (2015), as a mean neuvers in the form in-plane bending moment, which et al. (2012), Ivler Panza and Lovera (2014), in turn to plane and lead, and Lovera (2015), Panza et al. (2015), as aa mean in turn are are related related to out-of out-of plane moments moments and lead, as as Panza a consequence, to large flap angles; these large oscillatory Panza and Lovera (2015), Panza et al. (2015), as mean to improve performance of the attitude loop without losing in turn are related to out-of plane moments and lead, as Panza and performance Lovera (2015), the Panza et al.loop (2015), as a losing mean aa consequence, to large flap these large oscillatory improve consequence, toto large flap angles; angles; these large oscillatory loads could leadto increased fatigue usage of oscillatory structural to to improve performance of of the the attitude attitude loop loop without without losing losing stability margin. a consequence, large flap angles; these large to improve performance of attitude without loads could lead to increased fatigue usage of structural stability margin. loads could lead lead to increased increased fatigueMoreover, usage of of structural structural components, see Miller et al. (1991). large flap stability stability margin. margin. loads could to fatigue usage components, see Miller et al. (1991). Moreover, large flap In this paper, a rotorcraft attitude control law design components, see Miller et al. (1991). Moreover, large flap motion can result, in extreme cases, in contact between this rotorcraft attitude control law design components, see Millerextreme et al. (1991). Moreover, large flap In motion can result, cases, contact between In this paper, paper,is aaaapplied rotorcraft attitude control lawThe design methodology to the tilt-rotor case. full this paper, rotorcraft attitude control law design motion canwing. result,It in inis extreme extreme cases, in in contact between blade and thus essential, forcontact safety between reasons, In methodology is applied to the tilt-rotor case. The full motion can result, in cases, in blade and wing. It is thus essential, for safety reasons, methodology is applied to the tilt-rotor case. The full coupled four axes control problem in the hover condition is methodology is applied to the tilt-rotor case. The full blade and wing. It is thus essential, for safety reasons, to limitand thewing. bladeItflap motion amplitude: in this respect, coupled four axes control problem in the hover condition is blade is motion thus essential, for in safety to limit the blade flap amplitude: this reasons, respect, coupled coupled fourand axes control problem in isthe the hover condition condition is considered, rotor stateproblem feedback employed as a mean four axes control in hover is to limit the blade flap motion amplitude: in this respect, structural load alleviation is envisaged, through both pasand state is as aa mean to limit theload blade flap motion amplitude: in thisboth respect, considered, structural is considered, and rotor rotor state feedback feedback is employed employed astrade-off mean to achieve safety requirements: it turns out that aas and rotor state feedback is employed a mean structural load alleviation alleviation is envisaged, envisaged, through through both both paspas- considered, sive and active techniques.is to achieve safety requirements: it turns out that a trade-off structural load alleviation envisaged, through passive and active techniques. to achieve safety requirements: it turns out that a trade-off exists between performance and safety. Aspects of control to achieve safety requirements: it turns out that a trade-off sive and and active active techniques. techniques. between performance and safety. Aspects of sive A standard document defining requirements and han- exists exists between performance and safety. Aspects of control control oriented modeling, definitionand of safety. controlAspects architecture and exists between performance of control A standard document defining requirements and hanoriented modeling, definition of control architecture and A standard document defining requirements andunlike han- oriented dling qualities for tilt-rotors does not exist yet, oriented modeling, definition of control architecture and control law synthesis are taken into account. The control A standard document defining requirements and hanmodeling, definition of control architecture and dling qualities for tilt-rotors does not exist yet, unlike law are taken account. control dling qualities for tilt-rotors tilt-rotors does not exist exist yet, (2000) unlike control the widely accepted standard does document ADS-33 control law synthesis synthesis are takenasinto into account. The Theproblem control law synthesis problem are is stated an optimization dling qualities for not yet, unlike control law synthesis taken account. The control the widely accepted standard document ADS-33 (2000) law synthesis problem is stated stated asinto an optimization optimization problem the widely accepted standard document ADS-33 (2000) for helicopters. Due to the hybrid nature of tilt-rotors, law synthesis problem is as an problem and is tackled by means of the structured H approach. the widely accepted standard document ADS-33 (2000) ∞ law synthesis problem is stated as an optimization problem for helicopters. Due to the hybrid of is by of H approach. for helicopters. Due tofrom the both hybridthenature nature of tilt-rotors, tilt-rotors, inspiration was Due takento helicopter ADS-33 and and is tackled tackled by means means of the the structured structured H∞ In this framework, requirements of performance, control ∞ approach. for helicopters. the hybrid nature of tilt-rotors, and is tackled by means of the structured H approach. ∞ inspiration was taken from both the helicopter ADS-33 In this framework, requirements of performance, control inspiration was taken from from both the the helicopter helicopter ADS-33 moderation, (2000) and was fixed-wing MIL-HDBK-1797 (1997) standard In this framework, requirements of performance, control robustness can be taken into account, and inspiration taken both ADS-33 In this framework, requirements of performance, control (2000) and fixed-wing MIL-HDBK-1797 (1997) standard moderation, robustness can be taken into account, and (2000) and fixed-wing MIL-HDBK-1797 (1997) standard documents, and efforts MIL-HDBK-1797 were made towards the definition moderation, robustness can be taken takenweights. into account, account, and are encoded in the form of frequency This paper (2000) and fixed-wing (1997) standard moderation, robustness can be into and documents, and efforts were made towards the definition are encoded in the form of frequency weights. This paper documents, and efforts were made towards the definition of tilt-rotor handling qualities: studies like Meyer and are encoded in the form of frequency weights. This paper extends the work of Panza and Lovera (2014), Panza and documents, and efforts were made towards the definition are encoded in the form of frequency weights. This paper of tilt-rotor handling qualities: studies like and the work of Panza and Lovera (2014), Panza and of tilt-rotor handling qualities: studies Padfield like Meyer Meyer and extends Padfield (2005), Padfield et al. (2006), (2008), extends the work of Panza and Lovera (2014), Panza and Lovera (2015), Panza et al. (2015) where the framework is of tilt-rotor handling qualities: studies like Meyer and extends the work of Panza and Lovera (2014), Panza and Padfield (2005), Padfield et al. (2006), Padfield (2008), Lovera (2015), Panza et al. (2015) where the framework is Padfield (2005), Padfield et al. al. (2006), Padfield (2008), Cameron (2005), and Padfield (2010) show the results obtained so applied Lovera (2015), Panza et al. (2015) where the framework is to the case of helicopter roll attitude control. Padfield Padfield et (2006), Padfield (2008), Lovera (2015), Panza et al. (2015)roll where the framework is Cameron and Padfield (2010) show the results obtained so applied to the case of helicopter attitude control. Cameron and Padfield (2010) show the results obtained so far in this and research. One(2010) conclusion of these studies is that, applied to to the the case case of of helicopter helicopter roll roll attitude attitude control. control. Cameron Padfield show the results obtained so applied far in this research. One conclusion of these studies is that, The paper is organized as follows: Section 2 briefly introfar ininthis this One conclusion of studies likein theresearch. helicopter case, flight control augmentation is paper is organized as follows: Section 22 briefly introfar research. One conclusion of these theseaugmentation studies is is that, that, like in the helicopter case, flight control control is The The paper follows: Section introduces how is theorganized tilt-rotoras controlled; Section The paper is organized asattitude follows: is Section 2 briefly briefly intro-3 like in the helicopter case, flight augmentation is necessary, if an adequate level of handling qualities is to duces how the tilt-rotor attitude is controlled; Section 3 like in the helicopter case, flight control augmentation is necessary, if an adequate level of handling qualities is to duces how the tilt-rotor attitude is controlled; Section 3 addresses the problem of obtaining a reduced order model duces how the tilt-rotor attitude is controlled; Section necessary, if an an adequate levelcontrol of handling handling qualities is as to addresses the problem of obtaining a reduced order model3 be obtained. Active tilt-rotor is thusqualities envisaged necessary, if adequate level of is to be obtained. Active tilt-rotor control is thus envisaged as addresses the problem of obtaining a reduced order model suitable to control law synthesis; in Section 4 the control addresses the problem of obtaining a reduced order model be obtained. Active tilt-rotor control is thus envisaged as a mean to achieve both high handling and safety to control law synthesis; in Section 44 the control be obtained. Activeboth tilt-rotor control isqualities thus envisaged as suitable aa mean high handling qualities and safety safety suitable to law synthesis; in law architecture defined and control requirements are suitable to control controlis law synthesis; in Section Section 4 the the control control mean to to achieve achieve both high handling qualities and requirements. law architecture is defined and control requirements are a mean to achieve both high handling qualities and safety requirements. law architecture is defined and control requirements law architecture is defined and control requirements are are requirements. requirements. Copyright©©2016, 2016 IFAC (International Federation of Automatic Control) 100Hosting by Elsevier Ltd. All rights reserved. 2405-8963 Copyright © 2016IFAC IFAC 100 Copyright 2016 IFAC 100 Copyright ©under 2016responsibility IFAC 100Control. Peer review © of International Federation of Automatic 10.1016/j.ifacol.2016.09.018

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stated; Section 5 introduces the H∞ approach and the choice of weights; finally, in Section 6 simulation results are shown. 2. ROTOR AND PILOT COMMANDS The aircraft attitude is controlled by properly modulating the magnitude and direction of the thrust vector of the rotors, which in turn are controlled by acting on blade pitch; each of the two rotors is provided with a swashplate mechanism, so that it can be controlled by means of a collective θ0 and lateral θ1c and longitudinal θ1s cyclic commands, resulting in six control inputs [θ0,R , θ1c,R , θ1s,R , θ0,L , θ1c,L , θ1s,L ] which from now on will be referred to as rotor commands. However, from the pilot point of view, this is not an intuitive way to control the aircraft, unlike the classical single main rotor helicopter configuration. Rather, when the tilt-rotor is in the hover configuration the pilot can control it by means of four inputs (from now on pilot commands): • heave velocity w is controlled by means of the common collective command θ0,C which acts increasing or decreasing simultaneously the collective command on the two rotors, thus acting on the resultant thrust magnitude; • roll angle ϕ is controlled by means of the differential collective command θ0,D , which acts on the thrust magnitude of the two rotors in a differential way, thus producing a rolling moment; • pitch angle θ is controlled by the common longitudinal cyclic command θ1s,C , which tilts the thrust vector of the two rotors longitudinally in the same direction, thus producing a pitching moment; • yaw rate r is controlled by the differential longitudinal cyclic command θ1s,D , which tilts the thrust vectors in the longitudinal direction but in opposite ways, so as to produce a yawing moment. It should be remarked that the control strategy above described does not involve the usage of the lateral cyclic rotor commands. A transformation matrix T can be used to transform pilot commands into rotor commands: 

    θ0,L θ0,C 1  θ0,R   θ0,D   1 θ =T θ  = 0 1s,L 1s,C 0 θ1s,R θ1s,D

1 −1 0 0

0 0 1 1

  0 θ0,C 0   θ0,D  . (1) −1   θ1s,C  1 θ1s,D

3. CONTROL-ORIENTED MODELING The starting point of the modeling process for this study is a state space, linearized model of the XV-15 tilt-rotor in the hover condition, provided by MASST (Modern Aeroservoelastic State-Space Tools), see Masarati et al. (2010). MASST is a collection of tools developed by Politecnico di Milano for the linearized aeroservoelastic analysis of fixed and rotary-wing aircraft, based on the state-space approach; it follows a modular approach and combines models of subsystems so as to build aircraft models of arbitrary architecture, which can be used, for instance, for the design of control systems for flutter 101

101

Table 1. Separation of axes: dominant modes. Roll-Yaw

Pitch-Heave

RS (−1.05) LD (−0.0635) DR (+0.0187 ± 0.153)

PS (−0.609) VD (−0.470) LP (+0.18 ± 0.434)

suppression, vibration reductions and load alleviation. The MASST tilt-rotor model is a 32 states linear model, which encompasses: • fuselage dynamics: (inertial) rigid body, fully coupled 4 axes fuselage dynamics are described by the model (heave, roll, pitch, yaw); • rotor dynamics: each of the two gimballed 3 bladed rotors are described by the states of flap and blade pitch, expressed in MBC (multi-blade coordinates). Rigid blade flapping is considered. A model order reduction approach has been adopted: starting from an accurate model of the vehicle, the model is then reduced so as to obtain a simplified model suitable to control law synthesis, which retains the most relevant dynamics in the frequency range of interest. The approach undertaken was based both on physical assumptions and on the inspection of the dynamics of the system on the different channels: it was noticed that the eigenvalues of the model can be divided into low frequency modes (< 10 [rad/s]) related to flight mechanics, medium frequency modes (10 ∼ 20 [rad/s]) related to slow rotor dynamics (i.e., regressive flap modes) and high frequency modes (> 20 [rad/s]) related to fast rotor dynamics (i.e., progressive flap and blade pitch modes). While the low and medium frequency modes are regarded as relevant to the attitude dynamics of the fuselage, the high frequency modes are not; hence, a residualization technique was employed in order to reduce the order of the model from 32 to 20 states, supposing that the states related to blade pitch were much faster (hence at steady state) with respect to the other states. For each of the four axes, the on-axis SISO transfer function can be computed, based on the reduced order model, from the inputs to the outputs as stated in Section 2. As a further step in model order reduction, such SISO transfer functions can be inspected by means of modal decomposition and eigenvector analysis, following an approach similar to that used in Panza and Lovera (2015), so as to highlight the most relevant dynamics into that channel. It turns out (see Table 1) that the roll and yaw axes are dominated by the same set of modes (namely, roll subsidence RS, lateral displacement LD and dutch roll DR), while the pitch and heave axes are dominated by a different, independent set of modes (pitch subsidence PS, vertical displacement VD and longitudinal phugoid LP). This suggests to separate the four-axes attitude control problem into two subsets of two-axes problems, which are easier to manipulate. Based on these considerations, two reduced order submodels of order 4 were obtained to be used in control law design: • a roll-yaw model, which takes [θ0,D , θ1s,D ] as inputs and [ϕ, r] as outputs;

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• a pitch-heave model, which takes [θ1s,C , θ0,C ] as inputs and [θ, w] as outputs. Moreover, with a similar approach, a reduced order fouraxes coupled model was obtained, which includes regressive flap dynamics as well in addition to the modes listed in Table 1. Finally, as the MASST model does not encompass actuator and sensor dynamics, transfer functions which account for these dynamics were cascaded upstream and downstream the model, for each of the input/outputs of the system; a pure time delay was considered as well, in order to consider digitalization and signal processing. 4. ATTITUDE CONTROL LOOP DESIGN The automatic flight control system (AFCS) of a rotarywing aircraft has in general the objective to assist the pilot and reduce his workload, so that he can concentrate on higher level tasks than the one of stabilizing the aircraft. In particular, the attitude control loop of the tilt-rotor must satisfy several requirements: • stabilization: the attitude control law is in charge of stabilizing the unstable modes related to attitude dynamics; • performance: the aircraft shall satisfy performance requirements, often stated in terms of handling qualities (see ADS-33 (2000)); in this class can also be included requirements of disturbance rejection (e.g., as a response to a wind gust); • control action moderation: since the SCAS (stability and control augmentation system) actuation authority is limited, the control action shall not exceed these limits, in order to avoid saturations; • safety: flap angles should be limited for reasons stated in the introduction.

of negative feedback on flap rotor measurements limits the amplitude of the out-of-plane blade motion, which on one hand helps meeting safety requirements, but on the other hand limits the achievable performance. 4.2 Control law architecture Based on the considerations carried out in Section 3, the control problem can be naturally decomposed into two subproblems: • roll attitude ϕ - yaw attitude rate r; • pitch attitude θ - heave velocity w;

In the present work, an ACAH (attitude command, attitude hold ) response was chosen for the roll and pitch axes, and a RCAH (rate command, attitude hold ) one for yaw and heave (see ADS-33 (2000) for details). In the following, the roll-yaw control problem will be addressed; the pitchheave problem can be undertaken in the same way. Two different control law structures were defined: • a baseline control law, which uses only measurements from the fuselage (i.e., roll angle ϕ and rate p and yaw rate r); • a RSF control law which, in addition to fuselage measurements, uses rotor flap measurements as well; in particular, it was chosen to use longitudinal flap measurements β1c,L , β1c,R from the two rotors, which affect the yaw dynamics. Figure 1 shows the RSF control law architecture as the composition of a fuselage and a rotor measurements loop. y0 −

ϕ, p, r e

Kf uselage

u

T

−

G(s)

β1c

Krotor 4.1 Measurements Rotorcraft attitude control laws usually exploit measurements related to fuselage motion provided by an IMU, such as attitude angles and rates, and linear accelerations and velocities. Studies have been carried out (Takahashi (1994); Horn et al. (2012); Ivler (2014); Panza and Lovera (2014, 2015); Panza et al. (2015)) about the possibility to introduce feedback measurements from the rotor states in the attitude control law of a helicopter (rotor state feedback, RSF). Indeed, the helicopter attitude dynamics is intrinsically coupled to rotor dynamics at “medium frequency” (namely with modes such as regressive flap), and rotor states contain information about this coupled dynamics, introducing phase lead in the attitude loop. This in turn could potentially improve performance of the attitude control loop by increasing the bandwidth without deteriorating the stability margins. In particular, feedback from the main rotor MBC flap measurements was studied. In the tilt-rotor case, results from Section 3 show that the regressive flap rotor modes lie at higher frequency with respect to low frequency attitude modes, thus the dynamic coupling is looser with respect to the helicopter case. Still, it should be remarked that a trade-off inherent to RSF exists between safety and performance: the introduction 102

Fig. 1. RSF control architecture, roll-yaw axes. The baseline control law is made up of static gains on roll angle ϕ and roll rate p, and a gain on the feedback measurement of the yaw rate r. The error between the i-th reference yi0 and measured variable yi is defined as ei = yi0 − yi . In matrix form, this results in 

θ0,D θ1s,D



= Kf uselage



eϕ ep er



=



Kϕ Kp 0 0 0 Kr

  eϕ  ep . (2) er

It should be remarked that this control law does not introduce any decoupling terms between the control actions of the two axes; moreover, the control law computes control action in terms of pilot commands rather than rotor commands, which is the most natural choice for attitude feedback. The RSF control law, in addition to the loop on fuselage measurements, introduces an additional negative feedback loop which can be engaged or disengaged independent of the fuselage loop; the rotor feedback control action is

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computed in terms of rotor commands rather than pilot commands: this is the most intuitive choice, since β1c,R should be fed back only to θ1s,R , and the same holds true for the left rotor, in order to avoid cross-couplings between the two rotors induced by cross-feedback closure. The control law contribution from the rotor feedback is thus 

θ1s,L θ1s,R



= Krotor



−β1c,L −β1c,R



=



Kβ 0 0 Kβ



−β1c,L −β1c,R





  θ0,L Kϕ Kp 0  θ0,R   −Kϕ −Kp 0 θ = 0 0 −Kr 1s,L 0 0 Kr θ1s,R

0 0 Kβ 0

Table 2. Sensitivity weights parameters. Bandwidth [rad/s] DC gain HF gain

   eϕ 0  ep  0   er  (4)  0    −β 1c,L Kβ −β1c,R

5. STRUCTURED H∞ CONTROL DESIGN

In Sections 3 and 4 the control requirements and control law architecture have been defined; in particular, the controller is a matrix of static gains on the measurements. Such gains need to be tuned, so that the closed-loop system satisfies the requirements. The proposed methodology is based on the structured H∞ approach; such approach allows the user to define a priori an arbitrary control law architecture, and to select the tunable parameters; control requirements are defined in the form of frequency weights; like in the classical H∞ approach, performance inputs and outputs are defined on the closed-loop system, and the weights are imposed on the proper input/output signals in order to set up a mixed sensitivity problem; tunable parameters are optimized so as to minimize the infinity norm of the closed-loop system from performance inputs to performance outputs. This allows to overcome the limitations intrinsic to the classical H∞ approach (controller is a full matrix, dynamic and of high order) making this approach suitable to rotorcraft control design, especially in the case the architecture of the existing control law cannot be modified and the parameters of the control law can only be re-tuned, or an additional feedback channel is to be implemented on top of the existing FCS. The interested reader can refer to Apkarian and Noll (2006) for more theoretical and implementation details about the structured H∞ technique and algorithms, and to Apkarian (2013), Panza and Lovera (2014), Panza and Lovera (2015), Panza et al. (2015) for applications of such approach to practical cases. In this paper, all frequency weights are first order and were enforced in the range [0.1, 100] [rad/s]. 5.1 Performance weights Let ny , nu respectively be the number of controlled outputs and control inputs of the plant; the MIMO [ny × 103

Roll

Yaw

Pitch

Heave

1 10 0.9

1 10 0.9

1.5 10 0.7

1.5 10 0.7

Table 3. Control sensitivity weights parameters.

(3)

where the gain Kβ is chosen to be the same on both the left and right rotors, for reasons of symmetry. The overall RSF control law, given by the composition of fuselage and rotor measurements loops, can be written in matrix form by summing the two contributions, after performing the proper transformations:

103

HF gain Pole [rad/s] HF/DC gain

Roll

Yaw

Pitch

Heave

2.5 26 2.6 × 105

2 26 2.6 × 105

0.5 26 2.6 × 104

50 26 2.6 × 104

ny ] sensitivity function can be interpreted as the transfer matrix from the disturbances on the ny outputs, to the outputs themselves; the relevant terms of such transfer matrix are the diagonal terms corresponding to the controlled outputs, i.e., in this case, ϕ and r; performance is characterized by the so-called disturbance rejection bandwidth (DRB, see Blanken et al. (2008)), i.e., the bandwidth of the sensitivity function, rather than the classical bandwidth definition from ADS-33 (2000). Weights are chosen as low-pass filters, where the inverse of the weight can be interpreted as an upper bound on the magnitude of the sensitivity function; in this sense, the desired bandwidth is the frequency at which the inverse of the weight cuts the −3 [dB] axis. A particular importance is given to the choice of the high frequency gain of the weight too: the higher the HF gain, the lower the inverse of the weight will be, and as a consequence the lower the magnitude of the sensitivity function will be at high frequency, avoiding peaks related to low-damped modes. Parameter values of the sensitivity weights are shown in Table 2; they were chosen on the basis of past experience with tuning of helicopter attitude control laws, due to the lack of a proper standard document which specifies similar requirements for tilt-rotors. 5.2 Control moderation weights A square diagonal weight matrix is chosen, of dimension [nu × nu ], such that each of the weights is associated to one of the pilot commands; this weight matrix will be imposed on the output of the control sensitivity transfer matrix. The frequency weight is chosen as a high-pass filter, according to the actuators’ bandwidth; the high frequency gain of the filter is tuned on the basis that, the higher the HF gain, the lower the gains of the control law will be. Values of control sensitivity weight parameters are shown in Table 3. 5.3 Safety weights In order to limit the out-of-plane motion of the rotor blades, which can be due to transient flapping in demanding maneuvers, weights were imposed upon the closed-loop transfer functions from references to flap angles; in particular, since longitudinal flap is associated to the generation of force components along the longitudinal axis, thus to fuselage pitch and yaw dynamics, it was chosen to impose weights Wβ,i (s) on the elements of the complementary sensitivity function from the reference on variable i (where

S. Panza et al. / IFAC-PapersOnLine 49-17 (2016) 100–105

i can indicate θ, r) to outputs β1c,L , β1c,R . Weights are symmetrical between the two rotors, and were chosen to be constant over frequency, following the rationale that the higher Wβ,i (s), the lower the amplitude of the flap response will be: indeed, the inverse of the weight can be interpreted as an upper bound to the magnitude of the closed-loop transfer function from reference i to longitudinal flap angle(s). A weight Wβ,θ (s) = 1 was imposed on the pitch channel, since it is the most demanding one in terms of control action required and, in turn, of longitudinal flap angle amplitude, while for the yaw channel a weight Wβ,r (s) = 0 was chosen. It should be remarked that a trade-off exists between performance and safety: indeed, lower flapping is associated to lower control power, which in turn leads to reduced performance. Safety requirements were introduced only in control laws with RSF.

Table 4. Disturbance rejection bandwidth (DRB, [rad/s]) on the four axes. Axis

baseline

RSF

RSF2S

Roll Yaw Pitch Heave

1.0235 1.6298 1.5557 4.9770

1.5557 0.8111 1.2328 5.4623

1.0235 0.8498 1.3530 4.9770

Step response r 10 8 r [deg/s]

IFAC ACA 2016 104 August 21-25, 2016. Quebec, Canada

6 4 2 0

0

0.5

1

1.5

2

2.5

3

3.5

4

6. RESULTS

Gains of the baseline control law were tuned separately in two subsets Kϕ , Kp , Kr and Kθ , Kq , Kw , based respectively on the reduced order models of the roll-yaw and pitch-heave attitude dynamics obtained in Section 3. Gains of the RSF control law were tuned based on the full coupled four axes reduced order model, also introduced in Section 3; indeed, flap dynamics need to be included in the model in order to tune the Kβ gain.

Such two-step tuning approach may turn out to be useful in case a rotor feedback loop is to be closed on top of an existing baseline attitude control law, but without modifying the values of the fuselage gains. Table 4 shows DRB for each of the four axes, computed for the different control laws. In order to verify the performance achievable with the synthesized control laws, simulations were carried out on the closed-loop full order model, choosing reference values of small enough amplitude in order for the analysis on the linear system to be meaningful; namely, values of 5 [deg] were chosen for the roll and pitch angle references, 10 [deg/s] for yaw rate and −1 [m/s] for vertical velocity.

Figure 2 shows the on-axis response of yaw rate along with the differential longitudinal cyclic required; it can be noticed that, even though the performance requirement were the same for all the control laws, a degradation of settling time occurs when the rotor feedback loop is 104

2

0

0.5

1

1.5

2 [s]

2.5

3

3.5

4

Fig. 2. Response to r0 : r (top) and θ1s,D (bottom). Step response θ 6

Moreover, a third RSF2S control law was synthesized according to the following two-steps procedure:

4 2 0

0

0.5

1

1.5

2

2.5

3

3.5

4

15 θ1s,C [deg]

• first, the loop is closed on the full order model using the fuselage measurements, by using gains obtained for the baseline law and assuming no feedback from rotor states, obtaining a “partially closed-loop” system; • gains of the rotor feedback loop are tuned based on this “partially closed-loop” system.

baseline RSF RSF 2 step

4

0

θ [deg]

• a set of gains for the baseline control law, with no feedback from rotor states (Kβ = 0); • a set of gains for the RSF control law.

θ1s,D [deg]

6

The weights described in Section 5 were imposed in the optimization routine; two sets of control law gains were obtained:

baseline RSF RSF 2 step

10 5 0 −5

0

0.5

1

1.5

2 [s]

2.5

3

3.5

4

Fig. 3. Response to θ0 : θ (top) and θ1s,C (bottom). closed, with similar performance for the RSF and RSF2S laws. As the contribution from RSF comes into play, control action is sensibly reduced. Figure 3 shows the corresponding quantities for the pitch axis; in this case, a slight degradation in damping of the response of the RSF2S law can be observed with respect to the RSF law. Moreover, Figure 4 shows the response of β1c,R to a step reference on the pitch angle: rotor feedback contributes to limit the amount of transient flapping, reducing its peak value; this is confirmed by the values of infinity norm of the transfer function from θ0 to β1c,R listed in Table 5 along with the reduction with respect to the baseline case: the peak value of magnitude of such frequency response (i.e., the infinity norm) is reduced by means of RSF.

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Response of β1c,R to a step on θ

6

baseline RSF RSF 2 step

4

2

[deg]

0

−2

−4

−6

−8

0

0.5

1

1.5

2 [s]

2.5

3

3.5

4

Fig. 4. Flap response (β1c,R ) to a step on θ0 . Table 5. Values of infinity norm of the transfer function from θ0 , r0 to β1c . Pitch Yaw

baseline

RSF

RSF2S

3.7603 0.5620

1.2591(−67%) 0.2598(−54%)

1.2959(−66%) 0.2699(−52%)

As a final comment, it can be noticed that the RSF2S control law can still achieve adequate performance, even though it is sub-optimal with respect to RSF since in the former case the only parameter upon which to act to satisfy all requirements is Kβ , while in the latter case all control law gains can be tuned simultaneously. Indeed, RSF2S achieves very similar reduction to RSF in both the infinity norm and the peak flap angle. In both the RSF laws, the weight Wβ,θ (s) can be seen as a “tuning knob” to trade performance for safety, which is accomplished through the optimization of the value of Kβ gain. 7. CONCLUDING REMARKS An optimization-based methodology for rotorcraft attitude control law design has been applied to a linearized tilt-rotor model in hover; key features are the possibility for the user to choose arbitrarily the structure of the control law, the possibility to enforce a wide set of requirements inspired by standard documents and by literature, and the possibility to synthesize innovative control laws which make use of rotor state feedback. Robustness requirements were not included but can be easily incorporated in the framework if an uncertainty description were available, in the form of multiplicative uncertainty. ACKNOWLEDGEMENTS The Authors wish to thank Prof. Giuseppe Quaranta and Dr. Vincenzo Muscarello of Department of Aerospace Science and Technology, Politecnico di Milano for providing the MASST model which was used herein. REFERENCES (1997). MIL-HDBK-1797 - Flying qualities of piloted aircraft. 105

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(2000). ADS-33E-PRF, Aeronautical Design Standard, Performance Specification. Handling Qualities Requirements for Military Rotorcraft. Apkarian, P. (2013). Tuning controllers against multiple design requirements. In American Control Conference, Washington DC, USA. Apkarian, P. and Noll, D. (2006). Nonsmooth H∞ synthesis. IEEE Transactions on Automatic Control, 51(1), 71–86. Blanken, C.L., Hoh, R.H., Mitchell, D.G., and Key, D.L. (2008). Test guide for ADS-33E-PRF. Technical Report AMR-AF-08-07, AMRDEC. Cameron, N. and Padfield, G.D. (2010). Tilt rotor pitch/flight-path handling qualities. Journal of the American Helicopter Society, 55(5), 42008. Horn, J.F., Guo, W., and Ozdemir, G.T. (2012). Use of rotor state feedback to improve closed-loop stability and handling qualities. Journal of the American Helicopter Society, 57(2), 1–10. Ivler, C.M. (2014). Development and comparison of explicit and implicit rotor-state feedback control systems for a fly-by-wire UH-60. In AHS Rotorcraft Handling Qualities Specialists Meeting, Huntsville, USA. Manimala, B., Padfield, G.D., Walker, D., Naddei, M., Verde, L., Ciniglio, U., Rollet, P., and Sandri, F. (2004). Load alleviation in tilt rotor aircraft through active control; modelling and control concepts. The Aeronautical Journal, 108, 169–184. Masarati, P., Muscarello, V., and Quaranta, G. (2010). Linearized aeroservoelastic analysis of rotary-wing aircraft. In 36th European Rotorcraft Forum, Paris, France. Meyer, M.A. and Padfield, G.D. (2005). First steps in the development of handling qualities criteria for a civil tilt rotor. Journal of the American Helicopter Society, 50(1), 33–45. Miller, D.G., Black, T.M., and Joglekar, M. (1991). Tiltrotor control law design for rotor loads alleviation using modern control techniques. In 1991 American Control Conference, Evanston, USA. Padfield, G.D. (2008). Capturing requirements for tiltrotor handling qualities - case studies in virtual engineering. The Aeronautical Journal, 112, 433–448. Padfield, G.D., Brookes, V., and Meyer, M.A. (2006). Progress in civil tilt-rotor handling qualities. Journal of the American Helicopter Society, 51(1), 80–91. Panza, S., Bergamasco, M., Vigan`o, L., and Lovera, M. (2015). Rotor state feedback in rotorcraft attitude control. In 41st European Rotorcraft Forum ERF 2015, Munich, Germany. Panza, S. and Lovera, M. (2014). Rotor state feedback in helicopter flight control: robustness and fault tolerance. In IEEE Multi-Conference on Systems and Control, Antibes-Nice, France. Panza, S. and Lovera, M. (2015). Rotor state feedback in the design of rotorcraft attitude control laws. In 3rd CEAS Specialist Conference on Guidance, Navigation and Control, Toulouse, France. Takahashi, M.D. (1994). Rotor-state feedback in the design of flight control laws for a hovering helicopter. Journal of the American Helicopter Society, 39(1), 50– 62.