Substructure Synthesis: A Controls Approach

8 Pages·1999·0.24 MB·English

by  Maghami, Peiman G.

AIAA-99-3957
Substructure Synthesis: A Controls Approach
(cid:3) y
Peiman G. Maghami and Kyong B. Lim
NASA Langley Research Center, Hampton, VA 23681
Substructure synthesis from a controls perspective is considered. An eÆcient and
computationallyrobust methodfor synthesis of substructures is developed. The method
considers the interface forces/moments as control inputs and rede(cid:12)nes the synthesis
problemin terms ofobtainingaconstantgain compensatorwhichwould ensure the con-
nectivityrequirementsofthe combinedstructure. Orthogonalsimilaritytransformations
are used to provide simpli(cid:12)ed synthesized dynamics of the combined system. The sim-
plicityofformintransformedcoordinatesareexploitedtoe(cid:11)ectivelydealwithmodeling
parametric and non-parametric uncertainties at substructure level. Uncertainty models
of reasonable size and complexity are derived for the combined structure from those in
the substructure models.
Introduction substructure synthesis methods has not changed, i.e.
Substructure synthesis is a well established area in to accurately predict the combined modal parame-
the modeling of (cid:13)exible structures as is evident in the ters: structuralresonantfrequenciesandmodeshapes.
voluminous literature of which a sample of tutorial Consistent with the primary goal, most attention has
papers are given in [1-6]. It is concerned with the beengiventotheissueofselectingamoste(cid:11)ectivesub-
modeling of the structural dynamics of substructures set of component (or assumed) modes and the related
(or components) and then synthesizing them to pre- issue of modeling the substructure interfaces. Thus,
dict the combined structural response. This synthesis relatively little attention has been given to issues ger-
is accomplished by enforcing de(cid:13)ection compatibility mane to active control.
and force equilibrium at all substructure interfaces. In this paper, we re-examine the substructure syn-
There are several advantages of modeling via sub- thesis method fromacontrolsperspective. Inparticu-
structure synthesis, among these, (1) it allows much lar,ourendgoalistodevelopaphysicallyrealisticand
independence in the design and analysis of substruc- highly eÆcient way of modeling and controlling large
tures, which is especially helpful if substructures are (cid:13)exiblestructuresbasedonsubstructure(subsystems)
designed, fabricated, and even tested by di(cid:11)erent or- models. As in any studies in multivariable control,
ganizations. An example is the damping synthesis for we are concerned with the issues of closed loop sta-
7
the Space Shuttle and more recently the component bility and performance robustness due to inevitable
modulesforthe InternationalSpace Stationwhichare modeling errors, con(cid:12)guration changes, or exogenous
built by di(cid:11)erent companies in several countries. (2) disturbances. Since the stability and robustness is
it increases the power of existing (cid:12)nite-element anal- strongly dependent on the degree of accuracy of the
ysis and design programs by allowing analysis/design mathematicalmodeltothephysicalstructure,wealso
bycomponentsespeciallyinproblemswheretoomany examine the issues of modeling uncertainties for the
(cid:12)nite element degrees of freedom are required to per- substructuresand their in(cid:13)uence on the output of the
form a dynamic analysis/design of the complete sys- connected structure.
tem. (3) It allows a direct synthesis of substructure
test data. This is particularly useful for very large
Substructure Synthesis
structural systems (such as the International Space
Station) that cannot be tested as a whole. This also For simplicity of presentation, consider two compo-
meansthatitcanbeusedasapartofanexperimental nents, shown in Fig. 1, that have to be synthesized.
veri(cid:12)cation tool for substructures before deployment Consider the interface forces and moments (between
as a connected structure. the two components) as control input forces yet to
1
Since its earliest work in, the primary goal of be determined. Moreover, assume that collocated
and compatible with the interface forces/moments
(cid:3)
Senior Research Engineer, Guidance and Control Branch,
are linear/angular velocity outputs from each com-
SeniorMemberAIAA.
ySeniorResearchEngineer,GuidanceandControlBranch. ponent. With these considerations, the linear and
Copyright(cid:13)c 1999bytheAmericanInstituteofAeronauticsandAstro- time-invariant dynamics of the two components are
nautics,Inc.NocopyrightisassertedintheUnitedStatesunderTitle17,U.S.
Code.TheU.S.Governmenthasaroyalty-freelicensetoexerciseallrightsun- represented as follows
derthecopyrightclaimedhereinforGovernmentalPurposes. Allotherrights
arereservedbythecopyrightowner. component 1
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p y p
1 p 2 yp ypi =L~di(cid:8)iqri +L~ri(cid:8)iq_ri +L~ai(cid:8)iq(cid:127)ri (9)
1
2
where qri is the vector of modal amplitudes ob-
Actual Structure tained from transformation zi = (cid:8)iqri; Mri, Dri,
Kri are, respectively, the generalized inertia, damp-
ing and sti(cid:11)ness matrices; and (cid:8)i is a matrix
p1 yp u, y -u, y p2 y whose columns are the r open-loop eigenvectors as-
1 1 2 p2 sociated with the included modes. If the mode
shapes are normalized with respect to the inertia ma-
trix, and modal damping is assumed, then Mri =
Ir(cid:2)r, Dri = Diagf2(cid:16)1i!1i;::: ;2(cid:16)ri!rig, and Kri =
Substructure Configuration 2 2
Diagf!1i;::: ;!rig, where !ji and (cid:16)ji, j = 1;::: ;r
are the open-loop frequencies and damping ratios for
Fig. 1 Synthesized substructures the ith component. De(cid:12)ning the state vector xi =
T
f qri q_ri g ;i = 1;2, the second-order dynamics of
the components can be rewritten in (cid:12)rst-order forms
M1z(cid:127)1+D1z_1+K1z1 =B~1u+H~1p1 (1) as follows
component 1
y1 =C~r1z_1 (2)
x_1 =A1x1+B1u+H1p1 (10)
yp1 =L~d1z1+L~r1z_1+L~a1z(cid:127)1 (3)
y1 =C1x1 (11)
component 2
M2z(cid:127)2+D2z_2+K2z2 =(cid:0)B~2u+H~2p2 (4) yp1 =L1x1+Du1u+Dp1p1 (12)
component 2
y2 =C~r2z_2 (5)
x_2 =A2x2+B2u+H2p2 (13)
yp2 =L~d2z2+L~r2z_2+L~a2z(cid:127)2 (6)
y2 =C2x2 (14)
where, for component i, zi denotes the displacement
vector; Mi is the positive de(cid:12)nite inertia matrix;
Di is the damping matrix; Ki is the non-negative yp2 =L2x2+Du2u+Dp2p2 (15)
de(cid:12)nite sti(cid:11)ness matrix; B~i is the in(cid:13)uence matrix
for the interface forces/moments; u is the interface where Ai, Bi, Hi, Ci, and Li, i = 1;2, respectively
force/moment vector; yi is the interface velocity vec- denote the state matrix, input in(cid:13)uence (interface
tor; C~ri is the corresponding output in(cid:13)uence matrix; forcesormoments)matrix,exogenousdisturbancesin-
ypi is the performance output vector; which could be (cid:13)uence matrix, velocity output in(cid:13)uence matrix, and
a combination of displacement, velocity, and acceler- non-interface output in(cid:13)uence matrix for each com-
ation outputs; and L~di, L~ri, and L~ai are the corre- ponent. The matrices Du1, Dp1, Du2, and Dp2,
spondingin(cid:13)uencematrices. Sincetheorderofalarge represent feedthrough matrices associated with non-
(cid:13)exible space structure (LFSS) can be quite large, for interfaceoutputsofeachcomponent;x1 andx2 denote
design and analysis purposes the order of the system the state vectors of the components; y1 and y2 denote
isreducedtoadesignsizeusingmodelreductiontech- the interface velocity output vectors; yp1 and yp2 de-
niquessuchasmodaltruncationormodalcostanalysis notethenon-interfaceoutputvectorscomponents;and
to obtain for component i, i=1,2 p1 andp2 representtheexogenousdisturbancesacting
on the components. The system matrices in the (cid:12)rst-
Mriq(cid:127)ri +Driq_ri +Kriqri =(cid:6)(cid:8)iTB~iu+(cid:8)iTH~ipi orderforms arerelated tothose in second-orderforms
(7) as follows:
0 I 0
Ai = ; Bi = T (16)
yi =C~ri(cid:8)iq_ri (8) (cid:20) (cid:0)Kri (cid:0)Dri (cid:21) (cid:20) (cid:6)(cid:8)i B~i (cid:21)
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Hi =(cid:20) (cid:8)iT0H~i (cid:21) ; Cri = 0 C~ri(cid:8)i ; (17) Tcohmeedsynamics of the system in new coordinates be-
(cid:2) (cid:3)
(cid:11)_ A^1 A^2 (cid:11) B^1 H^1
Li = L~di(cid:8)i(cid:0)L~ai(cid:8)iKri L~ri(cid:8)i(cid:0)L~ai(cid:8)iDri (cid:26) (cid:12)_ (cid:27)=(cid:20) A^3 A^4 (cid:21)(cid:26) (cid:12) (cid:27)+(cid:20) B^2 (cid:21)u+(cid:20) H^2 (cid:21)p
(cid:2) (cid:3)(18)
(25)
Dui =(cid:6)L~ai(cid:8)i(cid:8)iTB~i ; Dpi =L~ai(cid:8)i(cid:8)iTH~i (19) p1
where p= and
(cid:26) p2 (cid:27)
Now, for the two components to be connected, in a
dynamicalsense,onehasto(cid:12)ndacontrolinputvector A^1 A^2 T A1 0
u,representingtheinternalforcesandmomentsatthe = Nc Rc Nc Rc
(cid:20) A^3 A^4 (cid:21) (cid:20) 0 A2 (cid:21)
interface,suchthatthedisplacementsandrotationsof (cid:2) (cid:3) (cid:2) (cid:3)
(26)
thetwocomponentsattheinterfaceremainidentically
the same for all p1 and p2, and all compatible initial
conditions. To achieve this, append the dynamics of B^1 T B1
the components, to obtain = Nc Rc (27)
(cid:20) B^2 (cid:21) (cid:20) B2 (cid:21)
(cid:2) (cid:3)
x_1 A1 0 x1 B1
= + u+
(cid:26) x_2 (cid:27) (cid:20) 0 A2 (cid:21)(cid:26) x2 (cid:27) (cid:20) B2 (cid:21)
H^1 T H1 0
H1 0 p1 (20) (cid:20) H^2 (cid:21)= Nc Rc (cid:20) 0 H2 (cid:21) (28)
(cid:20) 0 H2 (cid:21)(cid:26) p2 (cid:27) (cid:2) (cid:3)
Note that the linear and angular velocities at the in-
x1 terface are collocated and compatible with the forces
y (cid:17)y1(cid:0)y2 = C1 (cid:0)C2 (cid:26) x2 (cid:27) (21) andmoments,i.e.,C1 =B1T and(cid:0)C2 =B2T,suchthat
(cid:2) (cid:3) along with Eq. (23), one has
yp1 L1 0 x1 Du1 T B1 T T
yp (cid:17)(cid:26) yp2 (cid:27)=(cid:20) 0 L2 (cid:21)(cid:26) x2 (cid:27)+(cid:20) Du2 (cid:21)u+ Nc (cid:20) B2 (cid:21)=Nc C1 (cid:0)C2 =0 (29)
(cid:2) (cid:3)
Dp1 0 p1
(22) resulting in
(cid:20) 0 Dp2 (cid:21)(cid:26) p2 (cid:27)
Thenew output vectory isthe di(cid:11)erence betweenthe B^1 =NcT B1 =0 (30)
velocityofthe twocomponentsattheirinterface. The (cid:20) B2 (cid:21)
problem is to (cid:12)nd u such that y is identically zero for
Now,thecontrolinputvectorumustbechosensuch
all p1 and p2, and compatible x1(0) and x2(0). From
that (cid:12)(t), which represents the coordinates incom-
Eq. (21), for y to remain identically zero for all p1
patible with the connectivity of the two components,
x1
and p2, must remain identically in the right remains identically zero for all t, p, and "compatible"
(cid:26) x2 (cid:27)
initial conditions, i.e., (cid:12)(0) = 0. To accomplish this,
null space of the matrix C1 (cid:0)C2 . This means
theinputvectorumustbechosentorender(cid:12)unreach-
that an appropriate contr(cid:2)ol vector u (cid:3)must be found
able from p, that is
thatwouldrenderthesystemidenticallyunobservable,
i.e., x1 identically remains in the undetectable u=(cid:0)B^2(cid:0)1[A^3(cid:11)+H^2p] (31)
(cid:26) x2 (cid:27)
subspaceofthesystemgiveninEqs. (20)and(21)for Note that the feedback gain associated with (cid:11) and
all p1 and p2. Let Nc denote an orthonormalbasisfor thefeedthroughtermassociatedwithpmustbeinthe
the right null space of C1 (cid:0)C2 , i.e., formgiveninEq. (31)torender(cid:12) unreachablefromp.
(cid:0)1
(cid:2) (cid:3) Although,anyuintheformofu=(cid:0)B^2 [A^3(cid:11)+H^2p]+
C1 (cid:0)C2 Nc =0 (23)
R(cid:12), with R as a non-destabilizing arbitrary matrix,
(cid:2) (cid:3)
and let Rc denote an orthonormalcomplement to Nc. would also be feasible, the condition for synthesizing
Transform the system in Eq. (20) via an orthogonal thedynamicsofthecombinedsystemnecessitatesthat
similarity transformation, such that (cid:12) = 0 for all t, thereby reducing the expression for u
to that given in Eq. (31). The matrix inversion in
x1 (cid:11)
= Nc Rc (24) Eq. (31)isguaranteed,aslongasbothinputin(cid:13)uence
(cid:26) x2 (cid:27) (cid:26) (cid:12) (cid:27) matricesB1 andB2 arefullrank,whichtheyaresince
(cid:2) (cid:3)
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the interface forces are distinct. Therefore, since B1 Using Eq. (37) in Eqs. (20) and (22), results in the
and B2 arefull rank, so is B1 , and so isB^2, since combineddynamicsofthetwocomponentsintheorig-
(cid:20) B2 (cid:21) inal coordinates.
B^2 = RCT (cid:20) BB12 (cid:21), and Rc is an orthonormal basis for x_1 A1(cid:0)B1G1 (cid:0)B1G2 x1
= +
B1 (cid:26) x_2 (cid:27) (cid:20) (cid:0)B2G1 A2(cid:0)B2G2 (cid:21)(cid:26) x2 (cid:27)
the column space of . Using the control input
(cid:20) B2 (cid:21) H1(cid:0)B1D1 (cid:0)B1D2 p1
(38)
u of Eq. (31) in Eq. (25), with the aid of Eq. (30) (cid:20) (cid:0)B2D1 H2(cid:0)B2D2 (cid:21)(cid:26) p2 (cid:27)
simpli(cid:12)es to
(cid:11)_ A^1 A^2 (cid:11) H^1
(cid:26) (cid:12)_ (cid:27)=(cid:20) 0 A^4 (cid:21)(cid:26) (cid:12) (cid:27)+(cid:20) 0 (cid:21)p (32) yp = L1(cid:0)Du1G1 (cid:0)Du1G2 x1 +
(cid:20) (cid:0)Du2G1 L2(cid:0)Du2G2 (cid:21)(cid:26) x2 (cid:27)
It is obvious from this equationthat if the initial con- Dp1 (cid:0)Du1D1 (cid:0)Du1D2 p1
(39)
ditions are compatible, such that (cid:12)(0) = 0, then (cid:12) (cid:20) (cid:0)Du2D1 Dp2 (cid:0)Du2D2 (cid:21)(cid:26) p2 (cid:27)
remains identically zero for all p and t, which means
that, in the physical coordinates, y1 = y2 for all p Model Uncertainty
and t. Therefore, compatibility conditions between
In this section, issues of parametric and nonpara-
the two components would be satis(cid:12)ed for all p and
metricuncertaintiesin thecomponentmodelsandthe
t. The synthesized dynamics of the combined sys-
waytheye(cid:11)ectthesynthesizeddynamicsofthesystem
tem can be written in a reduced-order form in the
will be considered. Parametric uncertainties include
transformedcoordinates,orintheoriginalcomponent-
uncertainties in modal frequencies, damping ratios,
based coordinates. In the transformed coordinates,
and mode shapes of the components. The nonpara-
these dynamics are given as follows
metric uncertainties considered here are the unmod-
(cid:11)_ =A^1(cid:11)+H^1p (33) eleddynamicsofthecomponents. Ifmodalmodelsare
used to represent the component dynamics, nonpara-
with the non-interface output de(cid:12)ned from Eqs. (12), metric uncertainties would be due to the truncated
(15), (24), and (31) modes of the components. The development in this
section uses the representation of the synthesized dy-
yp (cid:17) yp1 =N^1(cid:11)+D^1p (34) namics of the system in the transformed coordinates,
(cid:26) yp2 (cid:27) as given in Eq. (33). This would make the treatment
of parametric and nonparametric uncertainties easier.
where
Here, only the case wherein there are no feedthrough
N^1 = L1 0 Nc(cid:0) Du1 B^2(cid:0)1A^3 (35) terms (acceleration terms) associated with the non-
(cid:20) 0 L2 (cid:21) (cid:20) Du2 (cid:21) interface outputs (see Eq. (34)), is considered. The
treatment for the general case would be similar but
more involved.
D^1 = Dp1 0 (cid:0) Du1 B^2(cid:0)1H^2 (36) Uncertainty in Frequency and Damping
(cid:20) 0 Dp2 (cid:21) (cid:20) Du2 (cid:21)
Asmentionedearlier,theuncertaintiesinmodalfre-
The dynamics of the combined system may also be quency and damping are embedded in the elements of
presentedin theoriginalcoordinates. Rede(cid:12)ne the in- thestatematricesA1 andA2. Onecanrepresentthese
terfacefeedbacklaw(Eq. (31))intermsoftheoriginal uncertaintiesviaadditiveuncertaintyblocks(cid:1)A1 and
coordinates, to obtain (cid:1)A2, such that the true state matrices are given by
u=(cid:0)B^2(cid:0)1[A^3NcT x1 +H^2p] A1 !A1+(cid:1)A1 (40)
(cid:26) x2 (cid:27)
x1 p1
(cid:17)(cid:0) G1 G2 (cid:0) D1 D2 A2 !A2+(cid:1)A2 (41)
(cid:26) x2 (cid:27) (cid:26) p2 (cid:27)
(cid:2) (cid:3) (cid:2) (cid:3)
(37)
Note that no assumption is made on the structure of
with the constant gain matrices G1 and G2, and the the uncertainty blocks, (cid:1)A1 and (cid:1)A2, i.e., they can
feedthrough terms D1 and D2 de(cid:12)ned as be structured or unstructured. Using Eqs. (40)-(41)
in Eq. (33), and assuming that there are no uncer-
G1 G2 =B^2(cid:0)1A^3NcT ; D1 D2 =B^2(cid:0)1H2 tainties in the mode shape data and nonparametric
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uncertaintiespresent,oneobtainstheuncertaintyrep- yp =(N^1+LMc(cid:1))(cid:11) (47)
resentation in the transformed coordinates.
where
(cid:11)_ =(A^1+(cid:1)A^1)(cid:11)+H^1p (42)
(cid:1)A^1 =(cid:1)TMcTAMc(cid:1)+(cid:1)TMcTANc+NcTAMc(cid:1)
where
(48)
(cid:1)A^1 =NcT (cid:1)A1 0 Nc =
(cid:20) 0 (cid:1)A2 (cid:21)
A1 0 H1 0 L1 0
NcT1(cid:1)A1Nc1 +NcT2(cid:1)A2Nc2 (43) A=(cid:20) 0 A2 (cid:21); H =(cid:20) 0 H2 (cid:21); L=(cid:20) 0 L2 (cid:21)
Nc1 and Nc2 are column partition of Nc according to (49)
thecomponents. Thisindicatesthattheadditivepara-
Fromtheseequations,itisobservedthattheinputand
metricuncertaintiesinthecomponentdynamicsinthe
outputparametricuncertaintiesinthedynamicsofthe
originalcoordinatescarriesthroughtothesynthesized
components, in the original coordinates, di(cid:11)use into
systemintransformedcoordinates,inthesensethatit
input, output, and state parametric uncertainties in
stillremainsintheformofanadditiveparametricun-
thetransformedcoordinatesrepresentationofthesyn-
certainty. However, it is observed from Eq. (43) that
thesized system dynamics. Moreover, the state para-
the structure of the uncertainty does not remain the
same,itssizedecreasestomatchA^1,butitsboundre- metric uncertainties involve a quadratic form of the
uncertainty (cid:1). This form of uncertainty can be dealt
mainsinvariant,sincecolumnsofNc areorthonormal.
In other words, (cid:1)A^1 would have the same bounds as with the technique presented in [8-9], wherein poly-
nomial uncertainty representations are represented in
that of (cid:1)A, i.e.,
optimal linear fractional representation (LFT) form
j(cid:1)Aj(cid:20)Æ ! j(cid:1)A^1j(cid:20)Æ (44) approximations. The remaininguncertainties given in
Eqs. (46) and (47) may easily be put in a linear frac-
TheparametricuncertaintyproblemgiveninEq. (43)
tional representation (LFT) form for robust control
may easily be put in a linear fractional representation
design and/or analysis.
(LFT) form for robust controldesign and/oranalysis.
Also, one could observe from Eq. (38) that, in this
Unmodeled Dynamics
case, treating the uncertainty in the transformed co-
The traditional approach in substructure synthesis
ordinates would be cleaner and easier than treating it
has been to use higher bandwidth in the dynamics
in the original coordinates.
of the components to be synthesized, roughly twice
Uncertainty in Mode Shapes the bandwidth of interest for the combined structure.
The parametric uncertainties in the mode shape This approachhas worked successfully in many appli-
data a(cid:11)ect the elements of the input and output in- cations. However, it is somewhat ad hoc, with some
(cid:13)uence matrices, which include matrices C1 and C2. potential shortcomings. First, it does not provide
This means that there would be uncertainty associ- guaranteedlevelsofaccuracyforthepredicted system
atedwiththeelementsofthebasisvectorsNc,asseen parameters, such as frequencies, damping ratios, etc.
formEq. (23). Hereweassumethattheuncertaintyin Second,itmaysu(cid:11)erfrom(cid:12)niteelementmethod’spo-
themodeshapes,whetherstructuredorunstructured, tentiallossofnumericalaccuracyforhigherfrequency
can be translated into an uncertainty representation modes in addition to a loss of correlation with mea-
for Nc. With this assumption, the actual basis vec- suredfrequencies. Asmentionedearlier,controldesign
torsNc may be representedin variousrepresentations requires accurate assessment of model parameters or
of uncertainty, such as additive, multiplicative, etc. potential uncertainties. Traditionally, unmodeled dy-
Here, let Nc be represented as follows namics, which typically include the higher frequency
modes not included in the design model, have been
Nc !Nc+Mc(cid:1) (45)
treated via additive uncertainties in the plant model,
where Mc is a chosen basis for uncertainty in(cid:13)uence, which forces the control system to roll o(cid:11) to avoid
and (cid:1) represent the uncertainty, which can be struc- potentially destabilizing spillover problems. Unfortu-
tured or unstructured. Using Eq. (45) in Eq. (33), nately, unmodeled (truncated) dynamics at the com-
and assuming that there are no uncertainties in the ponent level generally do not correspond to the same
frequency and damping data or nonparametric uncer- in the synthesized (combined) system. Consequently,
tainties present, one obtains itisimperativethatthee(cid:11)ectsofunmodeleddynamics
inthecomponentsbecharacterizedatthesystemlevel
(cid:11)_ =(A^1+(cid:1)A^1)(cid:11)+(H^1+(cid:1)TMcTH)p (46) for proper dynamics and controlsdesign and analysis.
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Theapproachtakenhereissimilartothetreatment where
for parametric uncertainty, which essentially consid-
A1 0 0 0
ered the e(cid:11)ects of uncertainty in the transformed co-
ordinates. Rewrite the dynamics of the components, A^1 =NcT 2 0 A2 0 0 3Nc (56)
0 0 A1 0
including the unmodeled dynamics, in a (cid:12)rst-order 6 7
6 0 0 0 A2 7
form. 4 5
component i= 1 or 2
H1 0
H^1 =NcT 2 0 H2 3 (57)
x_i Ai 0 xi Bi Hi H1 0
= + u+ pi 6 7
(cid:26) x_i (cid:27) (cid:20) 0 Ai (cid:21)(cid:26) xi (cid:27) (cid:20) Bi (cid:21) (cid:20) Hi (cid:21) 6 0 H2 7
4 5
(50)
L1 0 L1 0
N^1 = Nc (58)
(cid:20) 0 L2 0 L2 (cid:21)
xi
yi = Ci Ci (cid:26) xi (cid:27) (51) Itisreasonabletoexpect that both C1 (cid:0)C2 and
(cid:2) (cid:3) C1 (cid:0)C2 C1 (cid:0)C2 are full r(cid:2)ank. De(cid:12)ne(cid:3) the
n(cid:2)ull space of the matrix(cid:3) C1 (cid:0)C2 by the matrix
xi Nc1, whose columns are o(cid:2)rthogonal. T(cid:3)hen, it can eas-
ypi = Li Li (52)
(cid:26) xi (cid:27) Nc1
(cid:2) (cid:3) ily be shown that the columns of matrix is
(cid:20) 0 (cid:21)
Here, the overbars indicate terms associated with the includedinthenullspaceof C1 (cid:0)C2 C1 (cid:0)C2 ,
unmodeleddynamics. It shouldbenotedthatin most represented by the matrix N(cid:2) c, from Eq. (53). Now(cid:3),
applications an accurate knowledge of the parameters choose and partition Nc, such that
associated with the unmodeled dynamics is not avail-
able, thus they are typicallycharacterizedin the form Nc = Nc1 Nc1 (59)
ofuncertainty. Also,notethat, similartotheprevious (cid:20) 0 Nc2 (cid:21)
treatments,thefeedthroughtermsinthenon-interface Here, the (cid:12)rst column partition corresponds to the
outputs have been omitted. Following the synthesis statesof themodeleddynamics, andthe secondparti-
procedureoutlinedpreviously,thecomponentdynam- tion to the states of the unmodeled dynamics. More-
ics,asgiveninEqs. (50)-(52),areappendedtoobtain over, the (cid:12)rst row partition corresponds to the null
a system similar to the one given by Eqs. (20)-(22), spaceof matrix C1 (cid:0)C2 . UsingEq. (59)in Eqs.
exceptthattheorderofthestatesarerearrangedsuch (54) and (55), an(cid:2)d separating(cid:3)the states, gives
that the modeled dynamics’ states appear (cid:12)rst, fol-
lowed by the unmodeled dynamics’ states, i.e., the (cid:11)_m =A~1(cid:11)m+A~2(cid:11)u+H~mp (60)
T T T T T
statevectorisgivenby x1 x2 x1 x2 . Then,
the space of feasible s(cid:2)tates (in the connect(cid:3)ivity and (cid:11)_u =A~3(cid:11)m+A~4(cid:11)u+H~up (61)
compatibility sense) are characterized by an equation
with
similar to Eq. (23), but for the expanded system, as
follows yp =N~m(cid:11)m+N~u(cid:11)u (62)
C1 (cid:0)C2 C1 (cid:0)C2 Nc =0 (53) where
(cid:2) (cid:3) A1 0 0 0
Following the coordinate transformation (similar to A~1 A~2 Nc1 Nc1 T 2 0 A2 0 0 3
Eq. (24)) and deriving the interface input vector u =
(cid:20) A~3 A~4 (cid:21) (cid:20) 0 Nc2 (cid:21) 0 0 A1 0
toguaranteethecompatibilityofthecomponents’dis- 6 7
6 0 0 0 A2 7
placementsandrotationsattheinterface,resultsinthe 4 5
synthesized dynamics of the combined system, which Nc1 Nc1
(63)
in transformed coordinates takes the form (cid:20) 0 Nc2 (cid:21)
(cid:11)_ =A^1(cid:11)+H^1p (54) H1 0
T
H~m Nc1 Nc1 2 0 H2 3
= (64)
(cid:20) H~u (cid:21) (cid:20) 0 Nc2 (cid:21) H1 0
6 7
yp =N^1(cid:11) (55) 6 0 H2 7
4 5
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AIAA-99-3957
p
y
2
p metricpropertiesofthesystemwerechosentoprovide
2
considerable modal content in the low-mid frequency
range to make the synthesis and control design task
Actual Structure more challenging, and are provided in Table 1. The
structure and its two substructures were modeled as
Euler-Bernoullibeams. Eachof the two substructures
p
y
u, y 1 -u, y 2 2 p wasmodeledusingthe(cid:12)rst40(cid:13)exiblemodes,resulting
2
inan80th-orderstatespacemodelforeachofthesub-
structures. The 40modes used weredeemed suÆcient
to provide a basis for describing the (cid:12)rst 20 modes
Substructure Configuration of the combined structure. The combined structure
wasalsomodeled using its (cid:12)rst 20modes, resulting in
40th-order state space model. This model was used
Fig. 2 Synthesized cantilevered beam problem
for validation of the synthesis approach.
N~m N~u = L1 0 L1 0 Nc1 Nc1
(cid:20) 0 L2 0 L2 (cid:21)(cid:20) 0 Nc2 (cid:21)
(cid:2) (cid:3)
(65)
Note from Eq. (60) that A~1 is the same as A^1 of the
reduced dynamics for the baseline combined model
(see Eq. (33)), so that if no unmodeled dynamics
were present, Eq. (60) would represent the assem-
bled structure. Furthermore, the state equations for Table 1. Geometric and structural properties
(cid:11)m and (cid:11)u are coupled. One way of looking at this is
to imagine that there are two systems, one attributed Property Substr. 1 Substr. 2 Whole
to the modeled dynamics ((cid:11)m states) and the other Length 150 50 200
to the unmodeled dynamics ((cid:11)u states), which are in Mass/Length 1000 1000 1000
feedback connection. As mentioned earlier there are Rigidity, EI 1.0e7 1.0e7 1.0e7
uncertainties associated with the unmodeled dynam- No. of modes 40 40 20
ics parameter, namely, A~2, A~3, A~4, H~u, and N~u, i.e.,
they are not known accurately. It can be said that
the uncertainties, due to unmodeled dynamics, ap-
pear in feedback connection to the nominal plant. In
otherwords,theuncertaintiesassociatedwithunmod-
eled dynamics in the components, which are typically
representedby additive uncertainty to the component
models,takestheformofanuncertaintyinafeedback
loop with the nominal combined structure, when the Using the procedure outlined in Eqs. (23)-(39), the
synthesized dynamics are considered. dynamics of the two substructures were synthesized
(combined), resulting in a 160th-order model in the
Numerical Results original coordinates or a 158th-order model in trans-
Theproposedmethodologyforsubstructuresynthe- formed coordinates. Note that the synthesized model
sis and controls is applied to a two-dimensional can- may be further reduced in order by using model re-
tilevered beam problem to assess its e(cid:11)ectiveness and duction techniques, such as modal cost analysis or
feasibility. The current application involves substruc- Hankel norm techniques. Table 2 compares the natu-
ture synthesis and does not include the substructure ralfrequenciesofthesynthesizedmodelvs. thatofthe
uncertainty treatment. The cantilevered beam is as- validation (truth) model. Comparison of the frequen-
sumed to consist of two substructures, a cantilevered cies indicate a good match between the two models,
beam and a free-free beam, as indicated in Figure 2. thusdemonstratingthefeasibilityoftheproposedsyn-
The two interface forces/moments as well as the ver- thesis technique. It should be pointed out that the
tical displacement and slope are indicated, alongwith degreeofmatchingbetweenanytwofrequenciesinthe
anexogenousdisturbanceatthetipofthesecondsub- table depends on whether the modal content used in
structure. A performance output representing the tip the dynamics of each of the components provides a
de(cid:13)ection is also considered. The material and geo- suÆcient basis to express a mode at the system level.
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AIAA-99-3957
104
Table 2. Natural frequencies 102
Mode No. True Synthesized 100
1 0.0088 0.0088
23 00..01555412 00..01555539 Magnitude10−2
4 0.3022 0.3053
10−4
5 0.4996 0.5004
6 0.7464 0.7477
7 1.0425 1.0522 10−6
8 1.3879 1.4006
9 1.7827 1.7850 10−8
10−3 10−2 10−1 100 101
10 2.2268 2.2315 Frequency, rad/s
11 2.7203 2.7469
12 3.2631 3.2939 Fig. 3 Bode magnitude plot from exogenous dis-
13 3.8553 3.8600 turbances to tip displacement; true(solid), synthe-
14 4.4968 4.5075 sized(dashed)
15 5.1877 5.2407
of being more amenable to uncertainty and robust-
16 5.9279 5.9856
ness analysis and design as well as being smaller in
17 6.7175 6.7252
order. Using this realization, uncertainty models of
18 7.5564 7.5769
reasonable size and complexity have been derived for
19 8.4447 8.5348
thecombinedstructurefromparametricandnonpara-
20 9.3823 9.4767
metricuncertaintiesinthesubstructuremodels. These
models can andwill be used forrobust controldesign.
An application of the proposed synthesis technique to
a cantilevered beam problem demonstrated its feasi-
In ordertoobtainabetter assessmentoftheappro- bility.
priateness of the synthesized model, the Bode magni-
tude plotsof thetransferfunction from the exogenous References
disturbance to the tip displacement were computed 1Hurty,W.,\VibrationsofStructuralsystemsbyComponent
andareshowninFigure3forthesynthesizedandtruth Mode Synthesis," Journal of Engineering Mechanics Division,
models. The two transferfunctions agreeveryclosely, Pro2c. of ASCE,1960, pp.51{69.
Craig, R.R., J. and Bampton, M., \Coupling of Substruc-
asobservedfromFigure3. Note that the peaks ofthe
turesforDynamicAnalysis,"AIAAJournal,Vol.6,No.7,1968,
magnitude plots are (cid:12)nite although undamped mod- pp.1313{9.
3
els were used to compute the transfer function. The Hintz, R., \Analytical Methods in Component Mode Syn-
reason for the contradiction is the coarseness of the thesis,"AIAA Journal,Vol.13,No.8,1975,pp.1007{16.
4
frequency points, i.e., no frequency point was chosen Craig,R.R.,J.,\Methods ofComponent ModeSynthesis,"
Shock and Vibration Digest,Vol.9,No.11,1977,pp.3{10.
exactly at a resonance. 5
Craig,R.R.,J.,StructuralDynamics: IntroductiontoCom-
Theseresultsdemonstratethe feasibilityofthepro- puter Methods,Wiley,1981.
6
posed synthesis approach. Meirovitch,L.andHale,A.,\OntheSubstructureSynthesis
Method,"AIAA Journal,Vol.19,No.7,1981,pp.940{47.
7
Kana, D. and Huzar, S., \Synthesis of Shuttle Vehicle
Concluding Remarks
Damping using Substructure Test Results," AIAA Journal of
A novel and robust method for synthesizing the Spacecraft and Rocket,Vol.10,Dec.1973,pp.790{7.
8
dynamics of substructures has been developed. The Belcastro, C., \Parametric Uncertainty Modeling: An
Overview," Proc. of American Control Conference, Philadel-
interface forces and moments were considered as con-
phia,PA,1998,pp.992{6.
trol inputs, and used to design a constant gain com- 9Belcastro,C.,\OntheNumericalFormulationofParametric
pensator which synthesizes the combined dynamics of LinearFractionalTransformation(LFT)UncertaintyModelsfor
the substructures. The synthesized structure can be Multivariate Matrix Polynomial Problems," Tm-1998-206939,
NASA,Nov.1998.
realized in original coordinates, or alternatively, in
a transformed coordinates (obtained via orthogonal
transformations). The realization of the system dy-
namicsintransformedcoordinateshastheadvantages
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