ROBUST ATTITUDE DETERMINATION AND CONTROL SYSTEM OF MICROSATELLITE. Principle of the control system construction

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1 30 UDC: О. V. Zbrutsk, О. М. Мelascheko, L. М. Rhkov ROUS AIUDE DEERMINAION AND CONROL SYSEM OF MICROSAELLIE Itroductio Improvemet of microsatellite cotrol motio is carried out after a few basic directios. he first directio is associated with the use of moder methods of iformatio evaluatio cocerig microsatellite attitude. For the solutio of this problem the Kalma filter is used, as a rule []. he secod directio icludes developmet ad perfectio of methods of cotrol momets formig. Usig of reactio-wheel cotrol which is more effective tha magetic cotrol, is limited to the ecessit of the periodic uloadig of reactio-wheels, that complicates a cotrol algorithm substatiall. Fiall, the third directio cosists i drawig o accomplishmets of moder theor of automatic cotrol, i particular, powerful methods of robust cotrol [2]. Problem statemet he comple problem of qualit icreasig of the satellite spatial orietatio cotrol is eamied b the use of moder methods of iformatio treatmet, algorithms of cotrol momets formig ad usig of robust cotrol methods. Priciple of the cotrol sstem costructio Let us describe satellite motio with the followigs equatios: q 2 q,, I I I O where diag(,, ) is a iertia tesor of satellite; I () is a vector of absolute agular rate of satellite resolved i the bod frame; O is a vector of agular rate of satellite i relatio to the orbital frame, resolved i the bod frame;, q is quaterio of orietatios; is a total mechaical momet, forced o the satellite. he momets are created with magetic coils ad reactio-wheels. For the eceptio of the uloadig mode of reactio-wheels a itegral actio is etered

2 3 i the cotrol law, thus it is give ol o coils. For eplaatio of epediece of such cotrol will write dow the simplified model of satellite motio roud oe ais: k k2 k3 dt, (2) where k, k 2 are coefficiets i epressio for a momet, which is formed b the reactio-wheel; k 3 it is a coefficiet i epressio for a momet, which is formed b the coil. We will write dow equatio of motio of reactio-wheel as follows: h k k, (3) 2 where h is a mometum of reactio-wheel. From the equatios (2), (3) we fid epressio for a stead value of mometum: k 2 h уст. k3 It is see that as a result of itegral itroductio a stead mometum will be limited eve at presece of stead momet which is forced o satellite. It is meat that the mode of reactio-wheels uloadig is ot eeded, that substatiall simplif a cotrol algorithm. At the accessible estimatio of complete state vector ˆ, ˆ O q of satellite it is suggested to choose a cotrol momet as: (4) ctr RW coils, (5) where RW Kˆ O K ˆ, ˆ ˆ ˆ coils L O L LI dt momets which are created with reactio-wheels ad electromagetic coils Β is a vector of iductio of the Earth magetic field; L, L, LI accordigl; are coefficiets of eerg-based cotroller; diagoal matrices K ad K i obediece to the desired descriptios of step respose. It is suggested to fid coefficiets eerg-based cotroller o the basis of the mathematical programmig methods, that to fid them as a solutio of the followig optimiatio problem: mi F, uder restrictio : Gi 0, i,..., m, (6)

3 32 where F is a objective fuctio, G i are limitatios o equatios () ad requiremets to the trasitioal process. I default of agular rate sesor, it will be to decide the problem of satellite full state vector estimatio o the groud of sigals ol positio sesors magetometer ad Su sesor with the use of algorithm of Eteded Kalma filter or algorithm of set-valued estimatio. Set-valued algorithm of satellite state vector estimatio Essece of set-valued satellite state vector estimatio cosists i fidig of its estimatio as to the ceter of the ellipsoid, which the true value of state vector belogs to assuredl. I it the ke differece of this approach cosists classic Kalma filtratio i approach, where search poit estimate of state vector. For the basis of the costructio of procedure of set-valued estimatio of state vector a algorithm, described i [3], is take, where for the estimatio of the state of the oliear sstems higher order terms i the epasio of oliear fuctios i the alor series of it is plugged i a guarateed set which disturbace belogs to. For the receipt of algorithm of set-valued estimatio, we will write dow the model of errors of satellite i state space of cosistig of kid [4]: where t, t Att, is a vector of errors; ˆ ˆ ; A t ˆ 0.5I33 ; Fg Fe L ˆ L ˆ L ˆ dt, O 2 2 c23 c33 c3c23 c23c3 F g c3c23 c33 c3 c33c23 c3c33 c23c33 c3 c ˆ ˆ 0 Fe ˆ 0 ˆ, ˆ ˆ 0 Algorithm of set-valued estimatio of satellite state vector with the multiplicative update of estimatio of quaterio orietatio is preseted o fig.. I O ; (7)

4 33 ˆ, / / State Ellipsoid matri Predictio: ˆ ˆ ˆ / / f / dt ˆ / Q / / / Correctio state: mes orb ˆ / R + RO dˆ / / mes orb + R S OS ˆ ˆ + / qˆ / + +/ +/ + / / 2 qˆ +/ Correctio ellipsoid matri: Rˆ / / / / / mes orb mes orb ˆ + RO / R + RO mes orb + R S OS mes orb S + RO S / / ˆ, / / Fig.. Multiplicative set-valued filter implemetatio Equatios of measuremet for the sstem (3) will be writte dow i a form: O O t ( ) 0 t 33 t 2, ( ) 0 S t 33 where ad S accordigl vector of iductio of the Earth magetic field ad Su-vector, resolved i the orbital coordiate sstem. ˆ ˆ ˆ g ctr I I It is oted o fig. : f ˆ /. 0.5 O q

5 34 he iitial coditios of algorithm, give i fig., are writte dow i a ˆ 0, 0, ˆ v 0, R ˆ. d 0/0 0/0, 0/0 0/0 form: w Q, 0/0 0/0 Numerical desig of satellite motio he umerical aalsis of satellite motio is eecuted for a eerg-based cotroller, epected i obediece to a algorithm (6), at availabilit of full state vector. he problem was solved with methods of umerical optimiatio with the use of epasio of Global Optimiatio oolbo eviromets of Matlab. It was cosidered that satellite with iertia diag (,44,48 0,76) [kg m 2 ] moved o a circle orbit o height of a 650 km with the orbit agle icliatio of As the model of the Earth magetic field the model of WMM2005 is accepted [5]. It was cosidered that satellite i forced permaet revoltig momet τ ( 2 2) 0 7 [N m]. d O fig. 2 the chage of Euler agles are built at the suppositios doe higher ad at the iitial 5 coditios of 0-5 kid (,, ) ( 20, 0, 20 ). he dotted curves o this graphic correspods the desig of cotrol sstem with a eerg-based cotroller, the coefficiets of which are foud b trial-aderror procedure 6 ( L,5 0 ; L 400 ), w s,,, Орбіти Fig. 2. ehavior Euler agles for eerg based cotroller ad cotiuous to the desig with a cotroller, the coefficiets of which are 6 foud as a decisio of optimiatio task ( L,804 0 ; L 960,). he dotted curve o fig. 2 accordig to the estimatio of agles orietatio received based o set-valued algorithm. From fig. 2 we see, that the eerg-based cotroller, obtaied from the solutio of optimiatio problem (6), allows to decrease cosiderabl the time of trasitioal process comparativel with a regulator, obtaied b other method. I additio, such approach allows fleible form performace of cotrol sstem through the proper choice of limitatios G 0 i (6). i w s

6 35 Coclusios Desig of the attitude determiatio ad cotrol satellite sstem with the use of reactio-wheels ad electromagetic coils allows to arrive at accurac of agular orietatio at the level of te of agular miutes, ad applicatio of eerg-based cotroller with a itegral actio allows to avoid the ecessit of procedures implemetatio of reactio-wheels uloadig. hus, estimatio of satellite state vector with multiplicative update of quaterio orietatio estimatio allows set-valued filter to provide high accurac of estimatio at ucertai eviromet of operatig o a satellite. he applied method of fidig of the cotrol sstem regulator coefficiets allowed substatiall to simplif ad accelerate the process of its developmet, providig the high performace of cotrol. he effort i subsequet researches is epediet to poit the set-valued appraiser of the augmeted satellite state vector at developmet, icludig disturbace momet which is forced o satellite. Refereces. Melascheko О. М., Rhkov L. M. Microsatellite gravit-magetic stabiliig sstem research with evaluatio of phase vector b Kalma filter // he Scietific ews of NUU KPI P Melascheko О. М., Rhkov L. M. Mied 2 / optimiatio attitude cotrol sstem of microsatellite // he Scietific ews of NUU KPI , P Scholte E, Campell M. E. A Noliear Set-Membership Filter for O-lie Applicatios // Iteratioal oural of Robust ad Noliear Cotrol, 2003, 3(5): Melascheko О. М., Rhkov L. M. he aalsis of a magetic attitude cotrol sstem of a microsatellite at costructio of a Kalma filter with multiplicative ad additive update of a quaterio estimatio // Матеріали VII науково-технічної конференції Приладобудування 2008: стан і перспективи, Київ, 2008, с