QFT/H Control of a Small Solar Thermal Transfer Vehicle (SOTV) with Inflatable Appendages

Transcription

1 QFT/H Control of a Small Solar Thermal Transfer Vehile (SOTV) with Inflatable Appendages Rémi Drai - Maarten Steinbuh, Douwe Hoekstra - Sohrab Salehi Eole des Mines de Paris - Tehnial University of Eindhoven - European Spae Ageny remi.drai@ma.inria.fr m.steinbuh@tue.nl sohrab.salehi@esa.int Abstrat: Solar-thermal propulsion is reognized as a promising high performane onept for orbit transfer, station-keeping and interplanetary flight. The present paper is a study of the GNC related issues for a mirosatellite spaeraft equipped with two large inflatable Sun onentrators and a solar-thermal engine. Several robust ontrol methodologies have been developed and tested for the design of an AOCS sub-system involving a ontinuous thrusting strategy of the solar-thermal engine in presene of signifiant ontrol/struture interation. Copyright 24 IFAC Keywords : Robust ontrol, Satellites, Attitude ontrol. 1. INTRODUCTION The use of solar-thermal propulsion for either large transfer vehiles or small satellites is a highly attrative tehnology for future spae missions. Operating a solar-thermal engine requires very large Sun onentrators that ollet power from the Sun and fous it into an high Isp thermal engine. Inflatable materials offer unique advantage for the storage and mass of very large appendages: they are urrently onsidered as solutions for several ambitious projets involving so-alled Gossamer spaeraft. The study of the impat on AOCS design of ontrol/struture interation for appliations of inflatable tehnology to solar-thermal vehile has been proposed in a reent ESA Call for Tender (ESA, 22). The present paper is organized as follows: a general desription of the problem onsidered is followed by one setion on modelling and one on the formulation of an AOCS robust ontrol problem. The main results obtained by the two design teams from the Eole des Mines of Paris and the Tehnial University of Eindhoven using LMI/H /QFT ontrol tehniques are then summarized. seleted for the referene mission. The reader is referred to (Wassom, 21), (Kennedy and Palmer, 22) for further disussion on the appliations of solar-thermal propulsion to miro-satellites. Two main engine thrusting strategies an be envisioned for a solar-thermal demonstration spaeraft: i) a ontinuous strategy, in whih the onentrators ontinuously trak the Sun while the engine axis remains aligned with the orbit tangent. ii) an intermittent strategy, where solar energy is first stored before the S/C re-orients to point its nozzle into the desired diretion. The problem onsidered in the present artile is the ontinuous strategy whih is learly the most hallenging from a ontrol point of view. In this ase, the ontrol design problem has some time-varying features: the S/C inertias and flexible harateristis hange around the orbit,.f. Figure 1. Z Sun Diretion 2. PROBLEM DESCRIPTION In order to evaluate, benhmark and demonstrate the benefits of robust ontrol tehniques for the design of robust AOCS a small version of a Solar Thermal Transfer Vehile similar to the one desribed in (VTeh, 21) and (EADS, 22) was Figure 1 β α X Y perigee SOTV (Continuous Traking Strategy)

2 The main ontrol hallenge is to design ontrollers that provide high performane attitude ontrol despite unertainties affeting the inflatable appendages and a strong ontrol/struture interation aused by eah engine firing. In a ontinuous thrusting strategy, the onentrators trak the Sun permanently and the entire S/C rolls along its veloity axis. The orresponding profiles an be analytially omputed for an elliptial orbit. These analytial expressions are given here for onveniene: tanθ tanθ = roll sin β osαγ os β = sin β 2 + (osα sinα γ γ os β os β ) The two profiles depend on the Sun elevation with respet to the orbit β, and of the S/C position on orbit given by the true anomaly, as one has set: e sinθtrue α γ = α γ = θtrue + θsun and tanγ = 1 + e osθtrue where γ is the S/C flight path angle. 2 model adapted and refined from (VTeh, 21) and (EADS, 22). Important quantities are the heating time, i.e. the time required to bring the engine from T min to T max, and the burning time during whih the engine is atually operated i.e. the time for the temperature to fall from T max to T min. These two quantities an be omputed using some simplified temperature equation, like dt mc p = α AW sun εsσt = ( TE T ) / 4τ dt where T Ε = T Ε (α,ε) is the equilibrium temperature and (α,ε) are the body emissivity and absorptane respetively (τ is a harateristi body onstant). Knowing then the temperature evolution, the engine thrust fore an be dedued, as it is given by: max F ( t) = m& g Isp = m& g Isp T ( t) / T A typial thrust profile, orresponding to a single engine firing, will typially look like: max A plotting of the profiles is given, in Figure 2 below, for a typial GTO orbit: Figure 3 Engine Temperature/Thrust Evolution (F=5N) Thus even in a ontinuous strategy, thrusting is ahieved by a suession of on-off sequenes whih an have a destabilizing effet on attitude ontrol. Figure 2 S/C and Conentrators Steering Profiles 3.2 Conentrators Modeling 3. MODELING In order to design and test robust AOCS ontrol laws in the presene of ontrol/struture interation, both disturbane and spaeraft models need to be available. These models are now briefly desribed. The CAD modeling and a preise FEM omputation for the two off-axis paraboloid onentrators was performed by the Toulouse-based ompany IDEAMECH. The CAD modeling of the inflatable (mylar-based and pressurized) appendage is illustrated by Figure 4 below. 3.1 Solar Thermal Engine Modeling The firing of the solar-thermal engine is the main ause of disturbane. The propulsion sub-system of a solar thermal spaeraft is designed for some engine operating onditions inside [T min, T max ]. The temperature evolution of the solarthermal engine has been studied using a simplified engine Figure 4 Conentrator CAD Modeling

3 The oupling partiipation fators and the antilevered flexible modes were omputed for the present saled version of the SOTV vehile. The frequeny of the lowest modes is provided below: 3.3 Spaeraft Modeling Table 1 Appendage Flexible Modes The modeling tasks presented above are then inorporated into the linearized vehile equations of motion, whih are of the form: M( θ ) q&& + Lq& + Kq = B ( θ ) u + B ( θ ) f q = ( θ s Frequeny Mode # (H z) 1,79 2 1,14 3 2,25 4 8,3 5 16, 6 17,7 ', q flex u ')', B = ( I 3 f )', y = θ These equations are not onstant along the orbit as some matries depend of the onentrator rotation angle θ and of the position of the S/C global CoM. They an be written in various referene frames: in a S/C referene frame, the time-varying effet is due to the rotation θ and results in a signifiant migration of the global flexible modes. In the onentrators referene frame, the effet is muh smaller sine aused by the (small) entral body motion. It was thus hosen to work in the onentrators referene frame in order to remain in an LTI robust ontrol framework rather than in an LTV one. s In Table 3 the robustness requirements are given. Spaeraft M.C.I. unertainties Conentrators modes unertainties Engine unertainties Parameters Involved m SC S/C CoM S/C inertias Cross-oupling Terms Frequeny Damping Misalignment Coupling Coeffiients Thrust diretion Table 3 Robustness Speifiations Unertainty Range 6 kg from BoL to EoL 1 m 1% ± 15 kg.m^2 15% deg 2% δθ engine = ±.5 An additional goal is to design a single ontroller for the entire orbit. 5. ROBUST CONTROL DESIGNS For benhmarking purposes, three different robust ontrol methodologies were developed and tested on the SOTV appliation. 5.1 LMI Multi-objetive Control Design The first used was the robust multi-objetive design method adapted from (El Ghaoui - and Folher, 1997), (Sherer et al., 1998). In this state-spae approah the unertain plant is desribed through a Linear Frational Transformation (LFT) as shown in Figure 5 below. (s ) 4. CONTROL PROBLEM FORMULATION w G(s) z The attitude ontrollers to be designed will have to provide high performing ontrol despite model unertainties and ontrol/struture interation. Speifiations w.r.t. performane and robustness of the ontrol laws are presented in this setion. The AOCS ontrollers designed will have to ensure robust losed-loop performane w.r.t. variations of the onentrators flexible harateristis and the usual S/C parameters (MCI, thrust alignment,..). The performane speifiations are traditional performane indiators as ilndiated in Table 2. Attitude pointing auray Frequeny domain stability riteria Time domain stability riteria end of slew manoeuvre Time domain stability riteria disturbane rejetion.3 deg (1σ) bias 6 db gain margin 1 deg max defletion Table 2 Performane Speifiations.3 deg (1σ) random 3 deg phase margin 15 se settling time (to remain within +/-.3 deg) 15 se stabilisation time (to remain within +/-.3 deg) K(s) Figure 5 LFT Unertain Plant A list of ontrol speifiations an then be hosen, for the various performane hannels (w,z). This fairly extensive list of speifiations inludes: Pole-plaement Constraints The losed-loop modes are imposed to be loated in the lefthand omplex plane or in more general LMI regions (like half-planes, irles, setors). H 2 Constraint An H 2 onstraint on the hannel (w 2,z 2 ) onsists to impose an H 2 norm onstraint on the transfer funtion T z2w2.

4 L Constraint / Settling Time Constraint Likewise, an L onstraint on the hannel (w 2,z 2 ) onsists to impose an L norm onstraint on the transfer funtion T z3w3. All the previous ontrol speifiations an be translated into matrix inequalities and were atually imposed to the ontrolled unertain spaeraft. 5.2 H Control Design The lassial 4-blok H design methodology was used for the y-z axes. In this approah, whih is an extension of the mixed-sensitivity design, robustness is ahieved by the shaping of the 4 transfer funtions: w = ( r, d) z = ( z 1, z2) of Figure 7. S-Proedure and Heuristis The above speifiations an be translated into LMIs only for a nominal plant and not for an LFT. The solution of the robust ontrol problem is obtained in two steps: the unertainty blok is eliminated using the so-alled S-proedure. Then a linearizing hange of variables is introdued to obtain a set of onvex LMIs onstraints desribing the nominal design objetives. These onstraints are oupled with some nononvex onstraints of the form: S.T = I whih are addressed using an heuristi taken from the artile (El Ghaoui and Folher, 22). d r W 2 (s) W 1 (s) W 3 (s) z 1 z 2 K(s) P(s) e u Figure 7 4-blok H Control Problem Results A good onvergene of the algorithm was obtained and, as a result, all temporal and stability margins requirements were met despite signifiant parametri unertainties. A typial example of the ontrollers obtained using this methodology is illustrated by the following Nihols harts for the x-axis. The design objetive is to find a stabilizing ontroller K(s) ahieving good referene traking and disturbane rejetion with moderate atuator usage. These speifiations an be insured by imposing the single losed-loop H onstraint: T zw < γ. Results A typial example of the results obtained using this methodology is illustrated by the following Nihols plots Nihols Charts, theta on = 9 deg., Nominal Margins: 14. db, 55.6 deg. Nom inal Inertia +2% Inertia -2% Modes +15% Modes -15% 1 5 Open-Loop Gain (db) Open-Loop Phase (deg) Figure 6 Nihols hart for LMI ontroller Figure 8 Nihols hart for H ontrol The Nihols harts indiate that the LMI approah sueeded to provide here some effiient and robust high-gain phase ontrol in whih the lowest flexible modes are plaed right to the ritial point. The figure above shows that the 4-blok H approah, given the seleted weighting filters, a typial low gain stabilization ontroller, the ontrolled modes being below the ritial point in the Nihols hart, i.e. the db rossing of the open-loop Bode plot is at a lower frequeny then the flexibilities. This type of ontrol is popular among the flexible strutures ontrol ommunity for its high robustness properties. The required

5 bandwidth for the y-z ontrollers is signifiantly lower as the engine disturbane affets mostly the x-axis. 5.3 QFT Control Design A QFT design,.f. (Houpis and Rasmussen, 1999) or (Horowitz, 1992) was onduted along the SC three axes. The main steps of suh a design an be summarized as follows: first plant unertainties are translated into plant templates whih are frequeny dependent. For eah template, the speifiation, using a nominal model, is heked, and this results in so-alled bounds. These frequeny dependent bounds are borders, whih the nominal open loop frequeny response may not ross, as illustrated in Figure 9. similar to the LMI one. In addition, QFT tehniques were used during the study in order to improve the performane of the LMI/H ontrollers. For all three designs, similar performane and robustness levels were obtained for various onentrator angles demonstrating the possibility to use a single ontroller along the entire GTO orbit. 5. PERFORMANCE VALIDATION The above results were further validated through extensive simulation ampaigns in order to hek the robust performane.f. Figure 11 below, and identify the worst-ase onfigurations. Figure 9 QFT Control Bounds Taking into aount these bounds, the loop shaping proess is done by adding filter parts to the ontroller and moving around gain and phase (Borghesani et al., 22). So QFT boils down to onverting the family of plants into one nominal plant, and a family of bounds for whih the nominal plant must hold. Loop shaping tehniques.f. (Steinbuh and Norg, 1998) are then be used in a straightforward manner. Figure 11 Sensitivity Funtions (QFT design) These simulations were omplemented by a mu-analysis performed on the three designs for the flexible modes frequenies,.f. Figure 12. This step was made in order to get information on the atual parametri domains for whih stability is guaranteed in addition of merely heking that the robustness speifiations are met. Figure 1 Nihols hart for QFT Control, inl. bounds A typial Nihols hart of the QFT design, respeting all the bounds, is shown above, indiating a high gain, phase strategy Figure 12 Mu-Upper Bounds (H /QFT design)

6 Moreover, nonlinear Matlab/Simulink simulations, inluding the sensors (noise: 1 arse., delay: 2 ms) and atuators (thrusters pulse-width modulation) main harateristis, were also performed, as illustrated in Figure 13. y x deg y y deg y z deg.4.2 Conentrators Angle: 9 deg, Control: QFT, Configuration: worst u x Nm u y Nm u z Nm F (N) Time (se.) Figure 13 SOTV 3-axis Time-domain Simulation These simulations onfirm that the disturbane rejetion speifiations are met. 6. CONCLUSION This paper presents some of the work done by CMA-TUE in the frame of an ESA study devoted to the design of robust AOCS ontrol laws for a small size Solar Thermal Transfer Vehile. While the main study goal is the benhmarking of various robust ontrol methodologies, speial attention was also given to the modeling of the ontrol/struture interation aused by the solar-thermal engine firing. The main results for eah of the tested ontrol tehniques used were presented. These results largely onfirm the prevailing reeived wisdom of the ontrol ommunity onerning the three retained tehniques, namely that for realisti appliations: 4-blok H design allows ogent robust designs where performane is often traded for robustness. Quantitative Feedbak Theory is a versatile ontrol tehnique well suited for single input/single output design. In any ase, eah of the tested design methodologies onsidered during the study have demonstrated their ability to effiiently address the hallenging AOCS problem raised by the SOTV appliation. REFERENCES Borghesani, C., Y. Chait and Y. Yaniv (22) Users guide for the QFT Frequeny Domain Control Toolbox, 22 EADS (22) Solar Thermal Upper Stage Tehnologies for Future Launher Generation Program (STOTS), EADS- LV Exeutive Summary, Ref. YX/PL El Ghaoui L. and J.P. Folher (1997) Multi-objetive Robust Control of LTI Systems Subjet to Unstrutured Perturbations Systems Control Letter, 1997 ESA (22) AOCS for Large Flimsy Appendages: Statement of Work, ESA Doument Ref. TOS-E/FLAP/1.22 Horowitz I.M. (1992) Quantitative Feedbak Theory, Houpis, C.H. and S.J. Rasmussen, (1999) Quantitative Feedbak Theory, Fundamentals and appliations, 1999 Kennedy, F.G. and P.L Palmer (22 ) Design and Proto- Flight Test Strategy for a Mirosale Solar Thermal Engine, IAC-2-S.6.5 Powell, J.D. and A. Emami-Naeini (1994) Feedbak ontrol of dynami systems, third edition, Sherer C., P. Gahinet and M. Chilali (1998) Multi-objetive Output-Feedbak Control via LMI Optimization, IEEE Trans. Automati Control, 1998 Steinbuh M. and M. Norg (1998) Advaned Motion Control: an industrial perspetive, European J. of Control, pp , 1998 VTeh (21) Solar Orbit Transfer Vehile, Virginia Teh Report, Department of Aerospae Engineering, June 21 Wassom, S.R. (21) Revolutionary Propulsion Conepts for Small Satellites,, 15 th Annual AIAA/USU Conferene, August 21 Robust Multi-objetive LMI design is a powerful ontrol tehnique whose implementation an be intriate.