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CONTROLLABILITY OF INHERENTLY DAMPED LARGE FLEXIBLESPACE STRUCTURES j TECH LIBRARY KAFB, NM

by j

A.S.S.R. teddyGraduate -Research Assistant

Department of Mechanical EngineeringHoward University

Washington, D.C. 20059, U.S.A.

and

Peter M. BainumProfessor of Aerospace EngineeringDepartment- of-Mechanical Engineering

Howard UniversityWashington, D.C. 20059, U.S.A.

Paper No. IAF 82-319

33rd Congress of the InternationalAstronautical FederationParis, Sept. 26- Oct. 2, 1982

s&sz

OK'.CI;>:;.L PASS i"OF POOR QUALITY

COTTPotiABn.rT? or :v3rssrr_LARGE FLEXIBLE SPAC2 STSUCTTXES1

A.S.S.a. Roddy' and Peter HDapirtocne of Mechanical Engineering

Howard Chivorsity. Washington . D.C. 20059, USA

Abstract I. Istraductlon

Craph theoretic techniques are uaed Co studycontrollability of linear systeaa which could re-present largo flexible orbttlag ipaco systems with'Inherent dasplng. The controllability of cho pairof tatrlcea representing Che oystoa ctata and con-trol laflaaica nstrlces U aasared when all statesin cho nodal aro reachable In a digraph scnee fromat least ona Input and also when cha tana rank ofa Boolean oacrlx whooa ooa trivial conponsits arabaaed on too ocate and control Influeaea satricashas a tara rank of the order of cha state vector.It la oecn char the damping aatrlx does oot la-Quanca the required nuober of actoatora but givesflazlblllcy to tha poaolbla locations of cho actu-ators for which tha syntaa la controllable, andChat tho fftiffoaoa oatrlx tars rank deficiencydictates cha ouafaer as well as tha location oftha required actuators. Specific azaoples In-cludo a sodel of a ehallow apharlcal orbitingihall uharo both orlencatian and ahapo control areroquized. and alco a nmller diaansionol nuaarlealaxacple (unrelated to tha shall) which readilydaconacratGa the affect of daoplng.

&I

~

BDO f

KHZR'S0

SoBenelatura

CUB oystoa oacriz2n»7n aystaa oacrlznm control influeaco eitrlcea2nra control influenca eatrtxnn daoplng caerlxno cndlfled daaplng aatrtxBoo loan equivalmts of A.B and D'astriceaOJB »tlfrao«B cncrlzam ease natrtrraachabllity aatrlzUB cabcatrljt of oatrtz Bdiagonal aacrlx (rxr)

-asl-fnput vac tornxn unitary ortha{ sal catrlxam anltary orthogonal aatrlznil vectorOXB catrlx•Bill paraeataritjh singular value of aatrix A

rcb supported by HASA Grant SSC-1414.Suppl. 4

^Senior Grafts Baaaarch Aaalatane**?rofeaaor of A*>u«peca Eoglaaertng

Any linear, elaa invariant dynamical systeocaa to. In general represented by:

X • AX+fiO (1)

whore

X is an ncl state vector of tha aysteaA Is an "*n syotaa state cacrlzB Ls aa OXB control Influence oacrtrC Is an, ml input of tha systea.

TV« aystea described by equation (1) Is saidto be concrollablo if. with finlto 0 and la finiteCiae, che systea (1) can be tranaferred froa any

-Rate to-any other state. Tins-concept van-firstintroduced by Richard E. Calosn." The verifica-tion of controllability is essential for controlsystea design as no control law should be designedfor a systea which is not controllable. The con-trollability concept Is ovos are important farLarge apace structural systess whose dimension-ality la very large. If cho design of cha con-

-trolrsy«tea~is-undertaiiaa~vicQDaff irsr -verifyingcontrollability, a considerable aaounc- of- effortnay ba waned, through eta tall ore to arrive, atany satisfactory control lav for an uncontrollableoyscea.

In-tha-foUonlns sections, tha concept cfcontzollablllty. ao a proparty of the A and 3catrlcea, la reviewed.

, .The systeo, (1) la controllable If and onlyIT'*:

rank (B.AB,A2B A0"1*) - n (2)or tank (B.A-ljI) • a, 1- 1.2,..n (3)

whore X. are tha llgeovalucs of che aatrlx, A.

—Tha detarainatlcn of the rank of che racrlxIn equation (2) poses the problea of salecclng aindependent coluims out of sa columa and thlacould ba dona by such oucarical techniques as thesingular valua decoBpoaleijn of a oatrlz. 3

The singular value ieeorpoeltion Is a nuoar-ical algorltha uaed to (lad tha nuaertcal rank afa rectangular mcrti. A. (am. say n>a) throughCha evaluation of r<o orthosDnal (unitary) n» -ericas. 7, and V, such chat

71 £ 72T

where

C*)

(S)

and S • 4 lag

OFP.*?? 3

OL'AUTV

wi:h J, >0ri . . .>o >0, being singular value* ifA. The o.~.->, ?, are the non ;an> eigen-values ofthe uatrtx AA' (or A«<C as: : '-» therack of the satrlx A. The techal?i« if findingthe singular values of A by evml-j»claj the eigen-values of AA1 or \TA la. In general, lass accuratethai! using singular value decooaosir loa tech-niques and xa*r result ID erroneous conclusions,as L* woocnxcraeed by the folljviza, example. J

Lee l ilj °° -J

3 la the control Influence oacrlx

0 Is the axl vector of Inputs.

Th« dyumical system of aquatlaaa (8) can !>• ta-wrlreen aa a sec of f irst ordardifferentialtlona (in standard state space fora as):

-IT (9Vj

1°(6)

Then [tf l-Hi2*!. buc 1+t^l. for cb9 coopateraccuracy involved ]

A'A • (7)

Equation (9) con b« coualdared aa

X - A T-f BD

vfacra _¥ - [x,i]T

Tha algetvaluoa of this approximation W A'A ara/T and 0, roapoctlvely. vhoreas t&a /reclsa

of ArA are found to ba / Z+iT4 and•j.

Tha UM of cha singular voloa dccoepoatcloatechnique. Itself. ray resale in msasrical pro-bleas vbaa the magnitude of the Istjtjvalue of tho satrlx A, la as orior at(or inre) different froo chat of :ba caallascsingular valuo. [A conpiate dlecaoatja of ehasingular value decaspoalciea~eec!sl^E2~i3-giras —ic Raf . 3. It should be noted chat vtcii the ueaaf this algs rl£ ha, it la unoeceaeuy to ilrectly«w»luBt» too olnsular valuoa of tea A ootrti.The rank detarBlaocloa La accoaalLabcd baaed oncha unitary, orthogonal properties of tSo cs-trlcea-7]^ aad Vj vl^bl^ tba algorlria,]

The application of eha contzolIobllltT con-dition la equation (3) requires tba darenunatiujof tba elgravaluoa of eha oatrlx & aad the eval-uation of cba rank of a catrti of orter fnx(a*a) ]for o«ch of cba a eigenvalues. This cebesa lacore aceractlv* than that of coadlcioa C) aaeha croblaA of rank ovaluatloa of on -""' catrlzla reduced to tbe naL evaluacloaa of a n triceseach of dltienslan [ax(n+o)]. The olgcaTnluasare. la general, needed for ebe strscfjral dyoa-olc anAlyol»-of_tho-57stea-and-say.-:btt», be-already available for CBia phaaa of tba controlsyetea deetjn.

II. Controllability of Large Saete Stfoctures

The dynaalcal equations of a Larga ipaca•truecure ^yitea are. In general, described bya ^<t of linear (econd order coupled iUfarcn-tt.\l equations as:

(10)

(11)

and i •

The controllability condition (2) for this systecan be written as:

rank fi'.AB'..... A2"*^] » (12)

If v« aeeuno I>0, which'la tno for assny IdealUedfrea v^bratlag structuros, the.controllability

C-

bccoeofl

(11)

(14)

It can ba very eatlly s»«n chat C haa a rank 2atf and only U

rank [8.AB.... ,An"lB] • a

which leada to eha following thaoreoi*:

~rieeraa;The pair I H^ I Kl f con-

ot •»• rr • 3 (8)

whereU the oxl vector of Che feceralLiedcoordinaceaU the eass eaerlz (ora*U the imp In j xatrU (ao> (c*a lacludeTtacoua doe^lnf as well as (ynacoplcaffects)

(15)

erollablo If and only if cba pair (A.B) is con-tollable.

Thla cheorea reduces cha detenloaelon of thecontrollability of a Znt-'i order zrotea to the do-Ceralaaeloa of Che controllability of aa equiva-lent acn. order ayaton. In general for Large soecestructure appllcacioaa. n Itself my still be suf-ficcoely Urr.o and. thus, aucarlcjl techniquesvauld be required ia order to determine concrol-L»bllity. Thla cheorea Is baled oa the inherentaisuotlaa that >*0 and no insight con be drawnwhen 3 ts not equal Co rero. The effect of thecatrlz. 0. oo controllability la studied la thispaper 'asIng the graph theoretic definition ofcontrollability.

. ..OF POOR QUALIFY

Crash 3e?ir.iti3tiof

Given the general, linear, tl-e invariantdyaaolcal systea described by equation (1), re-peated bora aa:

X • AX * 3U

the pair [A.B] la controllable if and only

(16)

(1) the tera rank of [r v] -where Ag.Bj are the Boolean equivalents of r.haaatricos A and 3, respectively; and(2) -11 states In tba aystea ara teachable (SOBae leaat one Input In tba digraph senna.

Thaciw-terns, com rank and reachability, are- axpX&iaed- fr^r *•.

Tera rank In tba my!"*"* rank a ratrlx am achievedue to tba Locations of tba 000 xaro, COT fixedelements of tba catrlx rather than daa to thaauaorlcal values of tba elescnts. I cooplete dis-cussion of this concept la provided In tba Appen-dix.

.Reachability. If one-draws-a-digraph for. tha ex-tended square catrlx and finds the Input-

state reachability catrlx aa explained la thaAppendix* thara cuat be at. least one-oca xeroentry for every roo-of-tha subzncriz (I1) in- thareachability catrlx, 8, forced froo tba row andcolum indlcaa 1 to n. and nt-1 to n+o, respectively,aa-abowB la equation (17) where n ia tba auaberof states la the syotea and a la tha sucber ofactuators,

1 alo-H.X ' • • Hs ... x

o on+i

a*o

. Cant roll ability of System»lth ^i

r« dyaaal£a af large (pace syetesu vlrn In-herent deeding can be written aa ( repeat ug e<pu-tloa (?) with tba ootatton defined In eqoatloa(10) )II- (13)

The digraph far the jystta =atrlces A. 3 can bedrawn in general as ihown la Fig. 1.

The eleaenta at A(l,l«o). i*1.2....,n -I.e.those elesaenta of A appearing m the Identityaacrix - are represented in the digraph by the(solid) lines joining :he nodes (i+n) ca i ucere>l,2,...,a. Tha aleoents of .\(i-Ha,j). 1-1.-...3,j«L-,2 a I.e. thoso-aieaents-of j appearingIn A are represented In the dlgrepb by the (dashed)lines joining cho jth oode to tha (i+a)th oode.The el'neDts of A(i*a,J^n), i'i.Z.... ,a. j>l.Z...nI.e. those ale&anta of A appeariag In D' are repre-sented la tha digraph by tha (dotted) line* con-necting tha (J-m)th node to tha (i+o)th node.

Tha eleccoea of 3 ftna lover half of 3 •i"(t<«.J)J. B?(t.J). 1-1.2....n, J-1.2....0. arerepresented In the digraph by tha (double solid)lines joining the Jch actuator [the (2a+j)chnode, hare] to tha_(Mn)th node.

Fran tha reachability condition. It to ob-served that tha lines In tha d__raph due to the 0aatrlx can sapplamcot thosa lines due to thn-3,nacrlx. cor exanale suppose thoc due to tha struc-ture of D. ther .- a directed (dotted) line froo•_oda (n-^3) to t de (n+2) ia Fig. 1. Suppose thatthe at'J actuator represented by node (2n+a) candirectly influence node (o-3). Then it is cleartha actuators nuaber 2 and ) (represented by node2s+2 and 2a+3, rtapeetiveiy) need-not ba-presentIn order to Influence nodes n+2 and n+3. Thustha daoplng oatrlz can allow « greater-flexi-bility in chieselection of tha actuator locations.

-Although-boaad on thla argo&sst one- could construe -that the 3 earrlx influences tha nuaber of actua-tors (that could be reaoved). it should be reasa-bered that-tha alfllaun auabar of acrutorG--roqulred-if dictated froo tho tem rank condition which willnow be dlceuaaed.

For the aystea represented by equation (18)to be controllable, tha tera rank (as explainedla the Appendix) of tba Boolean aatrlx, a

0

-V0

•

I .

Dt8

0

0

'50

be 2n. Sate that the dlseaslooaltty at thestate vector la edition (IS) is 2a. If A haaton-rack l«aa thaa a, then the term r_nk of the(2a<TB)ch order Boolean matrix in aquation (19) canonly be augXEated due to t=a presence of &«. sincethe 0«t (Xol • - Oet [Agj. Thua. 0; can not beused to augnoic tha tars rsajc of the Boolean equi-valent of tba state zatrix I in equation (IS).

tn tuacary, the dosplng satrix. D, haa anaffect oa tha location of the activator*, while thesatrix A haa «n Lapact on the location aa well aa,tba nccfcer of actuatora.

QU.VLii'i

V. Suneneal Examples

The use of graph theoretic techniques la thedeterainatlon of controllability and the aoouatof Jaforsatloa about the location of cbc actuatorsAnd tha nuzbar of actuators needed is deaonstratodusing tha aodel of an orbiting shallow sphericalshell la crhle vl£h and without tha stabilising

_.duoJ)b«ll. (Pig 2).6

The Soolean Equivalent of the ays ten aatrir(A) fat a shallow spheric*! ihall is given laFig. 1. Tho digraph 1» given IB Fig. 4. Froo thadigraph le can ba oeea that tha nodes say beBubgroajxjd aa:

10

1112

123

134

14

S "1 l6 | 7

189

To reach all che IB states froa ae lease oneInptit, control actuators trust directly influencetha following nodes: (a) (10 or 11) ; (b) atleast ana of tho nodes. (12-IS); (c) (16); (d) (IT):

_and-(a) (18). The syocea catrlx, A, haa a tensrank deficiency of 1 (note tho presence of onlyzeros la tie first colus. Fig. 3) and, thus, onaactuator la required foe controllability. Thisactuator sect be placed such that the above asn-Claned states era directly influenced.

Tha BDdel of a shallow spherical shall with• scabi) *-lng duabbell'' Ls considered as anotheraxaeple tor controllability considerations. TheSoolaan equivalent of tha 22°d ordered systcnratrlx A, is givsn la Fig. S cad the digraph Isshown la Fig. 6. Froo the digraph le coa be seenChat tha total states can be subdivided Intocwo groups

€f^

12

1

16

5

17

6

13 19

7 8

20

9

a n10 U

13

2

(2)

14

3

.

15

4

The control actuators oust directly Influence oneor are states from group (1) and ona or morestates froa group (2). Tha systea sacrl* harehas full ten rsnk-and.chu*, ona-accoator-is-suff i-clcat to establish eontrollaAllity.

Tba no practical exaaales considered in thissection to thia point do not specifically Illus-trate tha Independence botveon tha nuofaer ofactuators required and the daatplng aatrtx. Toill tat rat* -this affect en ezoaple of sixth orderLs created and analysed.

It 1-. aocaasad that tha systea catriz isgiwn by

(20)0

0

0

113

^

0

0

0

0

0

0A

0

0

0

0

0

0

1.0

1

0

0

-10.07.0

2

J

8

010

.0

.0

.00

001

;.o6.0

9.0•*

Tha dlgrape is dram aa sbo«n la Fig. 7

"roa the digraph the reachability caniiiian for:or.tT3li*bUi:y :j satisfied if aay one or acreof :he states: ' . ,5,6, are directly iaflumc-o bythe control actuitors. The tera rank of A, hasa deficiency of tvo and thus cvo actu«tors'aturequiritd Cor concrollabillty. Even if the daoplngaatriz X), tha saaa nucber (2) of actuators laneeded for controllability and. :hus. it is shownthet dasping has no effect on the required nuaharof actuators. But, it tho damping satrix 1>0,thao the location of the actuators aust be chanjeisuch that che states «,S&& can be directly in-fliwaicad by the control actuators. la ordor toeopbaaizo tha point, when D"0 the dotted Linesshould ba zeaovcd froa the digraph iihown la Fig. 7.

VI. Conclusions

The definition of controllability asapplied to general linear tino invariant dynamicstystena and largo space systems is revievcd.Tho special nature of tha coupled catrix secondorder differential equations chat ore uced todeserlba large apace systeaa is used to arrive atspecific controllability conditions. The graphtheory approach la eaployad to define controllabi-lity la terns of the torn rank and Inpot-statareachability concepts. This approach is used taflad tho effect of Inherent daoplng present inlarge space sysceaa on tha nuober of tha actuator*and choir locations. It is observed that the

-'doRplag dooo pot 'effect the ulnlnm nuabor ofaccuatora required, but does provide greater flex-ibility la tha possible locations of .the.actuator*.Tha naofaar of actuators reqlrod dapendo on thatern rank of the generalised r/oteo (stlf fnees)catrlz. The stiffAoae catrls aleo Jaflusncea thalocation of tto-actuators.

References

1. Kalman, R.E., "On tha General Theory of ControlSysteaa," la Proc. 1st IFAC Congress, vol. 1,londont Botcereorth, I960, pp. *81-491.

2. Paige, C.C., "Properties of Sntmrlfiil Algo-rithao Related te. Cocpurmg Csntrollabillty."IEEZ_Trano.-on-Autonstic-Control, vol. AC-26,Ho. 1, Feb. 1981, pp. 150-138.

3. Kleoa. 7.C., and Laub, A.J.. "The SingularValue Decomposition: Its deputation andSoaa Applitatlono," I££Z Traao on Aucosatic .Control, vol. AC-25. No. 2. April 1980. pp.164-176.

4. Solas, M.J.. "Feedback Control of Flexible Sys-tem," IEZE Treaa. oa Automatic Control, vol.AC-23. So. 4. Aug. 1973. ?p. 673-679.

5. Schiiss. C., and Evans, F.J.. "Kaak InvariantTransfonatlons and Controliability of LargeScale Systeaa," Electronic Letters, vol. 16,Ho. 1. 3rd Jan.. 1930., pp. 17-20.

6. Kucar, 7.C. and Balaua. P.M.. "Go the Motionof a Flexible Shallow Spherical Shell laOrbit," AIAA 19th Aeroopace Sciences Moating,Jon. 12-15, 1981, St. Louis, itlseorrl. paperno. AIAA-41-0170; also to appear la AIAA Journal.

POOR QUALITY

P.M.. iDi «iJy. A.S.J.S. . "OB :heS.-joe .icd .'rleoeat^jn Control ai in Jrtttin*Shallow aohsrljtl shell structure." to bePresented it cha Joint IFAC'CSA iycpoilbat ooAutooulc Coatrol In Space. .•bor=wl*'cernouc.Cho Netherlands. July 3-?. Hi:.

Appendix

A. — Tera-Sank.of a-J

slaarhe tars r*nk of a scuare aatrix of dlr.za i< le«a chut a U ood only It cite aatrlx

b«a • MID fubsatrlx "0" of dioeaolra rrr' irltbrrz'>n. Tba Cera rank Is dLIf«rcot £rca cba ao-Dorteal rank la etta followiag *eno*. U a tquaroOBtrLx of ardor a ha a tw colucaa or rows that aredependent on each othor, coca lea cam nak la notreduced while Its auoarlcal rank U reduced by onefor each pair of cclums of rowo choc are depend-ent. For. largo opace tjatassa, the toiemlcotionof the auoarlcml volueo of Cho elescait* for tbatjtfea natrlcea ore not exact, and ttma tba prob-ability chac tva coltuae or rov« vauld bi ld«n-ClcjiUy eqnal. or thae ona row la a constant cicaaanother la vary icall. If such a dopeodaicy «xUeathat suot be detected before suojoctlng 1C to thaCera rank costs for aacabllJhlng tha coacrallabl-llcy of large tpace aysteoa.

J. Ia»u:-3tate taachablilty

The auzseBted idjaceacv satrix for the sysiaasatrlx pair IA.3J can be writ cm as

*8

where Aj and 3, are the adjacency aatrlcea ofA and 8. respectively.

Tba acates can be reachoa froa any at thai la-paca and can one bo. of length tare then o. Sacon be ralced to tha power n <nH cbsia cneoyieca reachability eatrU U glvea by

whenFor al

Che lapuc-euta reachability tatrUcho stacoa to bo reached froa at lease ana

la put, every row of K.- out. have ac leaac one aor.xero carry.

2aH

a*l

fig. 1. 31(npb of ;A.B) setrlx pair.

aIMUa

a

J3 '

I

Ij•tf

w

.or peer: wU^Li

1C1112

IJ

14

U

16

1718

13 « 5 6 7 3 9 10 11 i: 13 li 15 16 IT 13

' Hi '• ', i !

I I I

! x l

I I! ' - x

X! X

Xi

j I

i xl

Fig 1. location of notftaro olcacnto In the tyicea a»crLx of cte•ballov apb*rlc*l atoll vteboue tba diobboll In orbit

15 / 16 / IT / 18

1 2 1 4 5 6 7 8 9

Fig. 4. Otjroph at the ahallov &ball systca aatrlx.

ORIGINAL PAGS '^OF POOR QUAL!T*

1 2 3 4 S 4 7 8 9 10 11 12 13 14 15 16 17 13 19 20 21 22

12

34S

6

7

8

910

a121314IS16

17

181920

2122

i s :

; i

! 1*1 I

I ! I £1 JX J I X| X ! I x i x x x

I I

I ll it •' t ' y I t! TJTu; I . x)

Irl

it!

tit S. *ocaeio& of aoo-xaro e'.eaancs of the syotea oascrtr of.shall wtm tha dusab«ll in i-rbtt.

ORIGIN^-OF POOR

Fig 6. Digraph of cha ahailow spharicai shall syscoa tucrlz vlth dusbbo.'.l.

'•if

'• Slgrtph of system aoertx {Ivoa In equacloa (20)

fr? :-.-v.'.'.."-:= •-:"""";i -<--•:*.".',. , \._Y*'~ ^.=, -'*'•"•:-,*.'• ;. , s " % * '/' •' " . • • • . ."'-;-.;;•• '".--^r*' • • - • - . - . . -^ ;. •- -o "•;- " .'.-^.;''. • " • . • - ' . . - • - . - . , ; • • , •• .: •'- " ^'-' -,•^ _ - - - . ' A> O - ' i . , - - ~ • ' - . * - T " l - „-•- . . , , - , J

;: 'x:|i^ft-S''V;iV;fr5^Kv^3 vIt:lwi^A3mfej;K;S^t -:SW; i ftlfl®SS^^?^^^yi--\ ; ;,":-:,v" :ir^^;-:sji^V'-'-^:V.--;-vi:-^V^';"^ -"--"'i":^-: ^-^

;"• ^ '.?,•;_.'.>-• *v^3 -? ° - ,-zJ/;~ >• ' " • ' - • . j- r

5; ^"'V^^^'.cr;'--' •-' :--s'

>\ • * -• , .- * -j. • " , . • - - • • • = ' , - - • - - • •">--.' --r '--^.:^;?--"V">;"-:'»^vy--_\f«fe.:*^ >* - / - 7;*;/-'•?,.;•'. •--'"-...T^^V-'v^^-;,-^;' >-*' { \- -'; "^

. .". '•""''." '-t '.' " t I- "-->: J,-" «'.-'';• '• *-" - • "*"'" .? " ' ~.J>' - . 3