A Robust Control Design Framework for Substructure Models�

Kyong B. Limy

NASA Langley Research Center

Hampton, Virginia

SUMMARY

A framework for designing control systems directly fromsubstructure models and uncertainties is proposed. Thetechnique is based on combining a set of substructurerobust control problems by an interface sti�ness matrixwhich appears as a constant gain feedback. Variationsof uncertainties in the interface sti�ness are treated as aparametric uncertainty. It is shown that multivariablerobust control can be applied to generate centralizedor decentralized controllers that guarantee performancewith respect to uncertainties in the interface sti�ness,reduced component modes and external disturbances.The technique is particularly suited for large, complex,and weakly coupled exible structures.

1 Introduction

1.1 Problem Motivation

The motivation for studying this problem can be ex-plained by considering the docking of the Shuttle witha space station as shown in �gure 1. For the purpose

ISS SST

DURINGBEFORE

ISSSST

AFTER

ISSSST

Figure 1: Docking/berthing of Shuttle and Station

of this study, we de�ne the process of sti�ness couplingbetween the Space Shuttle and station as docking. Itshould be noted that the coupling process is muchmore complicated than is assumed in this study, seefor example reference [1, 2, 3]. During docking orjoining of structures in space, a signi�cant problem to

�Original version appeared as AIAA Paper No: 94-1699yResearch Engineer, Guidance & Control Branch, Flight Dy-

namics & Control Division, k.b.lim@larc.nasa.gov

be considered is the stabilization of the coupled systembefore, during, and after the docking of the two systems.An additional complication faced by the engineer is thedi�culty in accurately predicting the dynamics of thecoupled system because of the complexity in the physicsof the interface or docking mechanism.

A further motivation for this work arises fromthe peculiar problem faced by the engineer in thedevelopment and validation of design models for a large exible structure such as the space station. Due tothe 1-g testing environment, it may not be possibleto test the assembled structure on the ground so thatonly components can be tested. This means that onlymodels of substructures can be re�ned directly usingexperimental data. However, component testing can beused to develop substructure uncertainty models arisingfrom inadequate or incomplete component modes andinconsistencies in the substructure boundary condi-tions.

1.2 Relation to Previous Work

From the viewpoint of large scale systems theory manyrelevant results exist. Early results in [4] derive con-ditions for the existence of robust decentralized con-trollers for a general linear, time-invariant, intercon-nected subsystems. The controllers are assumed decen-tralized in the sense of local output feedback structure.Stability robustness is considered but a quantitativetreatment of robustness is not given. The extensionin [5] of the existence conditions to large exible spacestructures are for colocated sensors and actuators. Theresults are further relaxed in [6] so that collocatedsensors and actuators are not necessary. The existenceconditions are used to guide the choice of actuatorsand sensors needed which is then followed by parameteroptimization to obtain controller gains. Another decen-tralized controller con�guration which is closely relatedto the class of dissipative controllers are developedfor large exible space structures [7, 8]. The resultsin [9] incorporates subsystem disturbance/performancevariables and modeling uncertainties for general inter-connected system. Controller existence conditions arederived and a discussion of the e�ect of structureduncertainty on stability robustness is given. It ishowever not clear how optimal robust controllers canbe obtained.

From the viewpoint of structural modeling, severalrecent methods for decentralized control of large exiblestructures, that are based on �nite-elements and com-ponent modes synthesis exists. A technique wherebysubstructure controllers for each structural components

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are independently designed and then synthesized isgiven in [10]. The independent substructure controllerdesigns are dependent on approximating the interfaceboundary conditions and component modeling of itsadjacent components. To improve the approximation,internal boundary motion is minimized by feedbackcontrol. This method in general do not guaranteenominal stability and the report cites robustness issuesas a current research direction, clearly recognizing in-evitable component modeling errors. A substructurecontroller synthesis technique is proposed in [11, 12]whereby the controller is an assembly of subcontrollerdesigns based on individual uncoupled substructures.However, as in [10], closed-loop stability is not guar-anteed when the substructures are connected. This isnot unexpected because the interface sti�ness couplingis not taken into account in the synthesis. Morerecently, a substructure-based controller design ap-proach has been proposed [13]. The basic idea is tocombine a set of optimal component controllers whichare designed independently. Recognizing the in uenceof neighboring substructures, the substructure plantused in the control design is appended by a simpli�edmodel of neighboring substructure dynamics. Whilethis approach is computationally e�cient, closed-loopstability is not guaranteed in general. In summary,note that since all structural models or for that matter,since all mathematical models are approximations ofreal physical systems, controllers that guarantee robuststability have a clear advantage over controllers that donot.

Although the previous work just described by nomeans represent all pertinent past results in the openliterature, several issues appear to remain unresolved.Among these that are addressed in this paper include:accounting for component modeling errors, variationsand/or undermodeling or modeling errors in the sub-structure interface, and performance robustness. Aswill be evident, the proposed framework amounts to anintegration of well established results in substructuremodeling and recent advances in multivariable robustcontrol. The advantages in this synergism include: (1)nominal model and uncertainties at the substructurelevel can be incorporated directly for control analysisand design, and (2) a class of problems involvingvariable sti�ness coupling between various structuralsystems can be treated conveniently. This approachallows substructure (or component mode synthesis)dynamic models (see for example [14], [15], [16], [17])to be used directly in the analysis and design of controlsystems. Perhaps the major advantage in the aboveapproach is the reduction in the dependence on on-orbitsystem identi�cation testing of the assembled structureby an easier and less costly testing of substructures onthe ground.

1.3 Organization of Paper

Section 2 outlines brie y the dynamic model of sub-structures which is then used in section 3 to de�nethe uncoupled substructure robust control problem (forsimplicity sake the acronym SRCP will be used) inthe framework of modern multivariable robust con-trol. A novel element in sections 2 and 3 is theinclusion of substructure boundary forces/moments asexternal disturbances on the substructure, and thedisplacement and rotations at the substructure bound-aries as substructure output variables. The nominal

substructure model and the associated uncertaintiesare de�ned along with the input and output controlvariables for the substructure. In section 4, a modelfor sti�ness coupling is introduced and substructureinterface sti�ness matrix is de�ned. This interfacesti�ness matrix can be viewed as a transfer functionmatrix which maps the displacements and rotationsat the interface to the corresponding interface forcesand moments. Section 4 shows how displacements androtations at the interface can be realized into a blockdiagram form suitable for connecting subsystems. Forthe case where the substructure interface involves avariable and or unknown sti�ness coupling, the interfaceconditions can be treated as a parametric uncertainty ina multivariable robust control framework. In principle,this interface block could be extended to include certainor uncertain dynamic models. In section 5, the controlproblem for the connected system is de�ned by connect-ing the SRCP using the substructure interface block. Todemonstrate the utility of the ideas introduced in thispaper, a sequence of control design examples involvingcontroller designs for two connected exible beams areoutlined in section 6 and section 7 contains a fewconcluding remarks.

2 Substructure Model

The following formulation is discussed in more detailin [15, 16]. Let M (i) and K(i) denote the mass andsti�ness matrices corresponding to the i-th substructureso that the �nite element model is given by

M (i)��(i)+K(i)�(i) = D(i)u u(i)+D(i)

I g(i)+D(i)r r(i)(1)

where �(i) denote the physical nodal displacement vec-

tor. For substructure i, the matrices D(i)u , D(i)

I , and

D(i)r , denote respectively the force/moment distribution

matrix for control input u(i), the substructure interfaceinput distribution matrix for interface forces/momentsg(i), and the force/moment distribution matrix forexternal command and/or disturbances r(i). It is im-portant to note that in component mode synthesis, theinterface forces and moments are not included or carriedthrough in the substructure equations because they areabsorbed in the synthesis process as internal forces andmoments. However, the explicit consideration of theseinternal variables at the substructure interfaces is a keyingredient in the SRCP formulation.

The substructure eigenvalue problem for substruc-ture i, for an assumed set of boundary conditions, isgiven by

K(i) (i)j = �jM

(i) (i)j (2)

where j denotes the structural mode number. Usingonly a subset of substructure or component modes (i)

where

�(i) = (i)�(i) +(i)tr �

(i) (3)

the Ritz approximation leads to the substructure re-duced modal model,

��(i) + (i)2�(i) = (i)TD(i)u u(i) +(i)TD

(i)I g(i)

+ (i)TD(i)r r(i) (4)

where �(i) is the modal amplitude vector, and (i)2 =

diag(!(i)2

1 ; . . . ; !(i)2

n ). The columns of (i)tr denote trun-

cated modeshapes for substructure i.In general, the synthesized component modes

model do not lead to a model with su�cient �delitywhen only the truncated sets of \normal" componentmodes (represented by (i) in Eq.(3)) are used in thesynthesis. Therefore, the reduced set of \normal"component modes are typically augmented with \con-straint" modes. For a more comprehensive treatmenton adding constraint modes to normal modes, the text-book [15] is recommended. Indeed, the selection of asubset of component modes or assumed shape functionsis an important element in substructure synthesis andthe work in [17, 13] is recommended for a more detaildiscussion.

In the sections to follow, the e�ect of the truncated

substructure modes, (i)tr �

(i) in Eq.(3) (i.e. the Ritzapproximation error), are included as additive uncer-tainties about the nominal model. The closed-looprobustness is partly with respect to this model error. Itis signi�cant to note that if the entire set of componentmodes span the entire substructure con�guration vectorspace, a Ritz approximation plus the additive uncer-tainty will be su�cient to span this vector space. Animplication of this truth is that potential spillover intothe truncated modes can be properly accounted for inthe control design although closed-loop performance islimited by the inaccurate set of reduced normal modes.

The displacement and velocity outputs for sub-structure i are given by

y(i) =

(y(i)d

y(i)v

)=

"D

(i)Td �(i)

D(i)Tv

_�(i)

#(5)

�

"D

(i)Td (i)

D(i)Tv (i)

#��(i)

_�(i)

�(6)

The displacement and rotation at the interface forsubstructure i are

�(i)b = D

(i)Tb �(i) (7)

� D(i)Tb (i)�(i) (8)

Each column ofD(i)b corresponds to the individual inter-

face degree of freedom (DOF) location for substructurei. The outputs of interest are written as

e(i) =hD

(i)Ted D

(i)Tev

i� �(i)

_�(i)

�(9)

�hD

(i)Ted (i) D

(i)Tev (i)

i��(i)

_�(i)

�(10)

In the �eld of structural dynamics, the main pur-pose of substructure modeling via component modesynthesis appears to be the prediction of frequenciesand modeshapes and possibly damping for the as-sembled structure. The modular nature of this ap-proach also allows independent structural analysis andre�nement and component testing even by separateorganizations. Although much attention has been givento the basic problem of component mode selection,issues pertaining to the use of substructure models forcontroller design has not been fully addressed. As a

result, two problems that are addressed in this study aresubstructure model reduction errors and the modelingof variable or uncertain substructure interface both inthe context of performance robustness of closed-loopcoupled substructures.

3 Uncoupled SRCP

De�ne the states for substructure i as

x(i) =��(i)1 _�(i)1 . . . �

(i)n _�(i)n

�T(11)

The substructure state equations can be written as

_x(i) = A(i)x(i) +B(i)1 g(i) +B

(i)2 r(i) +B

(i)3 u(i) (12)

The coe�cient matrices, B(i)1 , B

(i)2 , and B

(i)3 , are state

space forms of D(i)u , D(i)

I , and D(i)r , in Eq.(1). The dis-

placement measurement, pointing error, and interfaceboundary DOF can be written in terms of substructurestate vector,

y(i) = C(i)3 x(i) (13)

e(i) = C(i)2 x(i) (14)

�(i)b = C

(i)1 x(i) � zi (15)

Similarly, the coe�cient matrices, C(i)3 , C

(i)2 , and C

(i)1 ,

are the corresponding state space forms of the relevantoutputs in Eqs.(5) to (10). Denote the transfer functionmatrix of substructure i as

G(i) =

�A(i) B(i)

C(i) 0

�(16)

where

B(i) =hB

(i)1 B

(i)2 B

(i)3

i; C(i) =

264 C

(i)1

C(i)2

C(i)3

375(17)

The transfer function matrix of the substructure, G(i),is then appended with weighting matrices that de�nesubstructure closed-loop performance and uncertaintiespeculiar to that substructure to form augmented sub-structure plant, P(i).

For the special case where the substructures areuncoupled (for example at the onset of docking), Figure2 shows the subsystem plants, P1 and P2, substructurecontrollers, k1 and k2, and uncertainties associatedwith each substructure, �1 and �2. Although thefollowing development is based on a system with onlytwo substructures, it should be clear that the method-ology presented applies also to a system with arbitrarynumber of substructures. The subsystem plants con-sist of substructure models which are augmented withperformance, disturbance and uncertainty weightingmatrices that are used to de�ne the substructure controlproblem. The uncertainty blocks are assumed to benormalized to unity in terms of their maximum singularvalues (see for example [18]). The structure in theuncertainty block is problem dependent and is selectedby the engineer.

P1 P2

K2

∆1

K1

2∆

g1r1

u1

z1e1

y1

g2

r2

u 2

z2e2

y2

subsystem 1 subsystem 2

Figure 2: Uncoupled SRCP for two substructures

The uncoupled SRCP can be described as seekinga controller k(i) which maintains stability and perfor-mance for the set of all closed-loop systems de�nedby the given uncertainty set �i. The performanceconsidered is de�ned in terms of a frequency weightedH1 norm, of the transfer function matrix from thedisturbances, r1, r2 to outputs of interest e1, e2. Theabove form of the performance can physically representregulation, tracking, and/or disturbance rejection prob-lems.

In general, any suitable multivariable design ap-proach can be used to generate a substructure robustcontroller. References [7, 19] describe several controldesign approach including parameter optimization vianonlinear programming. In this paper, we consider therobust performance measure in terms of the structuredsingular value. The problem then reduces to minimizing� (see for example [20, 21, 22, 23]).

4 Substructure Interface

In this section, a model of a substructure interface isdeveloped for the purpose of formulating the coupledSRCP.

4.1 Static Interface

Consider the structural interconnection between twosubstructures. Denote those DOF at the interfaceboundaries for the two substructures by superscripts (1)

and (2), and the sti�ness interfaces denoted by super-

script (I). The variables �(1)b1 ; :::; �(1)bm denote the DOF at

the interface for substructure 1 while �I=(1)b1 ; :::; �

I=(1)bm

denotes the structural interface DOF adjoining sub-

structure 1. Similarly, the variables �(2)b1 ; :::; �(2)bm cor-

respond to the DOF at the interface for substructure 2

while �I=(2)b1 ; :::; �I=(2)bm denotes the DOF at the adjoining

interface structure. The interface sti�ness matrix, SI ,is de�ned as follows

SI��I=(1)

�I=(2)

�=

�fI=(1)

fI=(2)

�(18)

where all the variables without subscripts denote vec-tor representation of corresponding DOF. As in anysti�ness matrtix, the (i; j)th element of SI physically

represents the force (or moment) at DOF i due to a unitdisplacement (or rotation) at DOF j. In the generalcase where there are DOF internal to the interfacesubstructure itself, the interface sti�ness matrix in theboundary input/output form, as in Eq.(18), can beobtained by a static condensation procedure [24].

As an example of a static substructure interface,consider the attachment of a exible aircraft wing toits fuselage as shown in �gure 3. A narrow strip of

Fuselage

Wing

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. ..

.

.

.

. .

. .

Interface Substructure

InterfaceSubstructureElement

ξI/(2)

i+3 ξI/(2)

i+1

ξI/(2)

i+2

ξI/(1)

j+3

ξI/(1)

j+1

ξI/(1)

j+2

Figure 3: Structural interface between a wing and itsfuselage.

structure located between the wing and the fuselagecan be seperately modeled by �nite-elements just as is

done for the main substructures. �I=(k)m denotes the

mth �nite-element DOF at the physical node shared bythe substructure interface and kth substructure. Theinterface sti�ness for this example relates the interfacedisplacements, forces, and rotations, moments, in thesame way as Eq.(18) if we let the superscripts, (1) and(2) denote the fuselage and wing respectively.

From a controls perspective, the interface sti�nesscan be viewed as a collocated, constant gain outputfeedback of the displacements and rotations at thesubstructure interface to the forces and moments onthe same substructure interface. Figure 4 show this

-I SI

forces/momentson substructures

forces/momentson interface

disp/rotat interface

Figure 4: Substructure interface block diagram

constant gain feedback block given in terms of forcesand moments on the substructures. Note that since theforces (and moments) in the right hand side of Eq.(18)are the forces (and moments) acting on the sti�nessinterface, the reaction forces (and moments) on thesubstructures will be in opposite directions.

4.2 Varying/Uncertain Interface

In the event that the substructure interface is uncertainand/or varying, the coupling block can be viewedas a constant linear uncertainty with speci�ed normbounds. One approach is to model the uncertainty asan independent variation in the interface forces abouta nominal sti�ness, so.

For sti�ness interface let the variation in the in-terface sti�ness be modeled by the following equationbetween the substructure boundary displacements (and

rotations) (�(1)b ; �

(2)b ), and the corresponding forces (and

moments) (g1; g2) on the substructures at the bound-aries�

g1g2

�= s

(�(1)b

�(2)b

)(19)

where

s = �( + ��)SI (20)

denotes the e�ective interface sti�ness while �; �; areparameters which are de�ned as follows:

j�ij � 1; i = 1; . . . ;#boundaryDOF (21)

�i = �ui (22)

The parameter is a factor appearing in the nominalinterface sti�ness

so = � SI (23)

Notice that when = 0 is selected, it means thatthe nominal interface sti�ness is zero. However, thisdo not necessarily imply that the substructures areuncoupled. The combined term, �� denotes the factorof interface sti�ness variation about the nominal. �uidenotes the upper limit of change in the ith channel.Note that the ith channel physically corresponds to theith interface DOF. Figure 5 shows in block diagram

Σ

β α

γ -SIg1

g2

z1z2

Figure 5: Interface sti�ness uncertainty/variation

form the e�ective interface sti�ness parameterized as astructured uncertainty.

4.3 Dynamic Interface

So far we assumed a static model in the form of asti�ness coupling and a substructure interface sti�nessmatrix is formally de�ned. In principal, the interfacesti�ness matrix can be extended to a transfer functionmatrix which maps the dynamic displacements androtations at the interface to the corresponding interfaceforces and moments. In other words, an inverse kine-matics model can be developed whereby the kinematicvariables (displacements, velocities and accelerations)at the DOFs adjacent to the substructures are mappedinto the corresponding forces and moments.

Similar to the static sti�ness interface case, thedynamic interface block could be extended to includecertain or uncertain dynamic models. As an example,the interface dynamics associated with a remote manip-ulator arm between the orbiter and the space station(�gure 1) could be modeled as a exible substructurewith its own nominal model and uncertainty. Moreinterestingly, it may be su�cient to treat the complexvariations due to a slow con�guration change in themanipulator arm during docking or when grabing apayload, as a set of plants not di�erent from uncertaintydescriptions.

5 Coupled SRCP

In this section, the substructure interface model iscombined with the uncoupled SRCP to form a coupledSRCP. It is important to note that a single synthesizedmodel is not formed by removing the interface DOFto predict system frequencies and mode shapes as istraditionally done in component modes synthesis (seefor example [15, 16]).

The two substructures connected by a sti�ness in-terface can be represented graphically in block diagramform as shown in �gure 6. The control objective remains

P1 P2

∆

k1

∆

g1r1

u 1

z1e1

y 1

g 2

r2

u 2

z 2e2

y 2

stiffness interface

Interface forces

Interface displacements

substr model unc substr model unc

substr controller 1 substr controller 2

SI

k2

Figure 6: Sti�ness coupled SRCP for two substructures.

the same as the uncoupled SRCP, i.e. to optimizedisturbance rejection performance under substructuremodel uncertainties. However, the objective for coupledSRCP becomes more complicated due to the substruc-ture coupling which may also be slowly varying and/oreven uncertain. The disturbance rejection performanceis in the form of a frequency weighted H1 norm, of thetransfer function matrix from the disturbances, r1, r2to outputs of interest e1, e2. The disturbance rejectionperformance is to be guaranteed under all modeleduncertainties.

Figure 7 shows a general interconnected substruc-tures. The system plant, P , consists of nominalsubstructure models, P1, P2, . . ., while the systemuncertainty, �, consists of individual substructure un-certainties, �1, �2, . . .. The system controller, K,consists of substructure controllers, k1, k2, . . .. A block-diagonal substructure interface however couples thenominal substructures, component uncertainties and

∆1

∆2 ...P1

P2 ...k 1

k 2 ...

Block-diagoalSubstructure

Interface

u yr e

g z

Figure 7: General Coupled SRCP.

substructure controllers. The width dimension of theblock diagonality of interface coupling is dependent onthe degree of physical coupling and topology of theinterconnections. Notice that even if each of the sub-structure and interface uncertainties are unstructured,globally, the uncertainties will be highly structured.This is the basis for applying structured singular valuetechniques (see for example [20, 21, 22, 23]) for robustperformance controller design in this study.

5.1 Decentralized Control

Figure 7 shows the decentralized nature of the plant,uncertainty, and controller. The decentralized con-troller structure (see for example [25, 26, 27]) is enforcedby the classical loop-at-a-time design, namely, designa substructure controller while holding the remainingsubstructure controllers constant. Figure 8 shows the�rst two steps of the design sequence to incorporateboth robustness and decentralization of the overallsystem for a system with two substructures.

The �rst step involves the design of substructurecontroller, k1, while assuming open loop for substruc-ture 2. At this step, the nominal dynamics of the sub-structure 2 is included while the component uncertaintyand performance for the remaining substructure 2 areignored. Ignoring the adjacent component's uncertaintyand performance is crucial to obtain a reasonableperformance controller for substructure 1 because ofthe localized constraint on substructure controller 1.Accounting for nominal substructure 2 should result insubstructure controller 1 which takes into account theprimary coupling e�ects of the neighboring substruc-ture.

In the second step, the controller for substruc-ture 2, k2, is designed by holding the controller forsubstructure 1, k1, constant. The �xed controller k1should strongly complement controller k2 in the control

1rΣ

α

G1 G2

∆11W

1e2z

1z

1y

1u

2g1g

Wr1 We1

-SI

k 1

1rΣΣ

α

G1 G2

∆1 ∆22W1W

2e1e2z

1z

2y1y

2r

2u1u

2g1g

Wr1 Wr2We1 We2

Step 2

k 1

k 2

Step 1

-SI

Figure 8: Design sequence for robust decentralization

of substructure 2. At this step, all uncertainties and dis-turbances with the corresponding errors are included.This essentially guarantees performance robustness ofthe overall system.

Step 3 is similar to step 2 except substructure con-troller 2 is held constant while substructure controller 1is re�ned. Step 4 is the same as step 2 and the sequencefollows. For this design example, iterations up to step 3are carried out for the decentralized control design. Inthe sequential design process described above, the orderof the controller will increase with each iteration if thecontrol design technique chosen results in a controllerof the same order as the current augmented plant (suchas LQG or central H1). Hence, model order reductionis recommended after each step.

5.2 Design via � Synthesis

The disturbance rejection performance from r to ein terms of frequency weighted H1 norm is to beguaranteed (or be robust) with respect to a boundedand structured set of component uncertainties, �1, �2,and interface uncertainty, �s. The controller can becentralized or decentralized as outlined in the previoussection. The worst case (over frequency) � quanti�esthe degree of robust performance. Designing controllersby �-synthesis involves an iterative minimization ofthe upper bound using H1 methods. The underlyingtheory which forms the basis of this method is discussedin detail in [23, 28, 29, 30].

The �-design problem is summarized as follows:

minimize

K;D

kDFl(P;K)D�1k1(24)

where fK : Fl(P;K) 2 H1g; D 2 D. The setof scaling matrices, D, has a similar structure as �(the structured uncertainty matrix) with an appendedidentity matrix. The terms, Fl, P , and K, denotethe lower linear fractional transformation, augmentedplant, and the controller, respectively. To minimizethe weighted H1 norm in Eq. 24, the D-K iteration

technique is used. In this approach,D orK is optimizedindependently and sequentially. Optimizing forD whilekeeping K �xed involves the search for the minimalupper bound on �, whereas, optimizing for K while�xing D involves the minimization of an approximationof � itself. Although this approach is iterative in natureand convergence to a global minimumis not guaranteed,recent numerical studies show excellent convergence(see for example [21, 31, 32]). The Glover-Doylealgorithm [33, 34] is used to solve the H1 problem. TheMATLAB toolbox, �-Tools [22] is used for the analysisand synthesis of the controllers.

6 Example

6.1 Description of Structure

Motivated by earlier work on substructure and com-ponent mode synthesis, a beam which is cantileveredat both ends is used to illustrate the ideas introducedin this paper. The structure is assumed to consist oftwo cantilevered Euler-Bernoulli beams joined at thefree ends by a short sti�ness interface beam element asshown in �gure 9.

Substructure 1(20 elements)

1z 2zInterface stiffness

(1 element)

Substructure 2(10 elements)

g1

e1u1r1 y1

2y e2 u2 r2

g2

Figure 9: Joined cantilever beams

The structural properties and con�gurations ofthe beams are given in table 1. The sti�ness and

Property Substr 1 Substr Inter Substr 2

length 2 .2 1mass density (�) 10 .1 10sti�ness (EI) 10�2 10�4 10�2

No. of elements 20 1 10

Table 1: Structural Properties and Con�guration

mass density properties are selected such that theresonant frequencies are su�ciently spaced, and thelower frequencies to be controlled are su�ciently smallto allow reasonable numerical integration. No deepsigni�cance should be attributed to the choice of thenumerical values in table 1 since this is obviously acontrived example to clarify the proposed approach.The beam substructures modeled by 20 and 10 beamelements resulted in 40 and 20 structural modes. Thefull state space models of the two substructures are then

of orders 80 and 40, respectively. The interface sti�nessis modeled by a single beam element with four DOF.

The connected structure has 60 structural modesfrom the total of 31 beam elements. A truncatedstructural model consisting of the lowest 30 structuralmodes is used as the evaluation model and �gure 10show the maximum and minimum singular values of

10−2

100

102

10−4

10−3

10−2

10−1

100

101

102

103

Freq (rad/sec)

Max a

nd M

in SV

Figure 10: Frequency response of evaluation model

the frequency response matrix from r1, r2, to e1, e2.Notice that there is no clear frequency gap where modelreduction by modal truncation can be done. It is alsoclear from the �gure that the displacement responsesrapidly drop as a function of frequency; this is thebasis for modal truncation of higher frequency modesin addition to the well known inaccuracy of the �niteelement model in predicting the dynamics at higherfrequencies.

6.2 Control Design

6.2.1 Objective

The control objective for this problem is to optimizedisturbance rejection performance. In particular, aminimal desired upper bound of frequency weightedH1 norm, of the transfer function matrix from thedisturbances, r1, r2 to outputs of interest e1, e2, isto be guaranteed under truncated component modesand uncertainties and/or variations in the substructureinterface. To optimize the robust performance, �synthesis is used.

6.2.2 Design Con�guration

The overall system block diagram is shown in �g-ure 11. The two nominal component modes model,(G1,G2), with their corresponding high frequency trun-cated modes which are treated as substructure un-certainties (�1W1, �2W2), are shown. The respec-tive substructure disturbances, (r1,r1), and outputs ofinterests, (e1,e2), are also distinguished. The powerspectrum of the disturbances are represented by Wr1 ,and Wr2 which frequency weights the all-pass inputdisturbances. The output error weighting matrices,We1 , and We2 were chosen as unity but in general maybe chosen to signify its relative importance with respectto other requirements. The unstructured uncertaintyblocks, �1, �2, and diagonally structured interfacesti�ness uncertainty block [�], are assumed to be ampli-tude bounded by unity 2-norm. The interface sti�ness

1rΣΣ

α

G1 G2

∆1 ∆22W1W

2e1e2z

1z

2y1y 2r

2u1u

2g1g

Wr1 Wr2We1 We2

-SI

k k11 12

21 22k k

Figure 11: Block diagram for control design

block couples the substructures and acts like a constantgain feedback. The controller shown in �gure 11 arecentralized and are used for performance comparisons.Substructure decentralization is de�ned by k12 = k21 =0. Figure 12 shows the augmented plant, P , which

unc substr 2

unc substr 1

stiffness interface

substr 2 meas

substr 1 meas

unc substr 2

unc substr 1

stiffnessinterface

substr 1 disturb

substr 1 contr inp

substr 2 contr inp

substr 2 disturb

substr 1output

substr 2output

P

AugmentedPlant

Figure 12: Augmented Plant for � synthesis

includes all performance and uncertainty weights.

6.2.3 Design Weights for SRCP

For all cases, only the lowest 8 modes (16 states)for substructure 1 and lowest 4 modes (8 states) forsubstructure 2 are targeted for control while the restof the component modes (32 and 16 modes for sub-structures 1 and 2) are treated as additive uncertainties.These are \normal"modes from substructure cantileverboundary conditions. No attempt is made to improvethe reduced component modes (nominal) model byadding constraint modes or component shape functionsto approximate the residual exibility. As is typical of exible structures, most higher frequency componentmodes are truncated. Figures 13 show the maximumsingular value of the transfer function matrix of thetruncated higher frequency component modes (solid)and a stable, real-rational, second-order weight functionused as its upper bound (dotted) for both substructures.The weights representing upper bounds for substructureuncertainties have the realization:

W1 =

0BB@

�2:8622 �7:5077 j �0:64587:5077 �2:1794 j 0:4064

0:6458 0:4064 j 0:0735

1CCA (25)

10−2

100

102

10−4

10−3

10−2

10−1

100

101

FREQ (rad/sec)

Y1/

U1

Substr 1

10−2

100

102

10−3

10−2

10−1

100

101

FREQ (rad/sec)

Y2/

U2

Substr 2

We1 We2

Wr 1

W1

W2

Wr 2

Figure 13: Substructure uncertainties and weights

W2 =

0BB@

�5:0127 �7:1225 j �0:77687:1225 �2:8930 j 0:3537

0:7768 0:3537 j 0:0819

1CCA (26)

The output weights, We1 , and We2 , (dash-dot) arechosen as unity. The �gures also show a �rst-order fre-quency weighting function (dashed), Wr1 , Wr2 , where

Wri =s + 1000

s+ 2� :008 i 2 [1; 2] (27)

which is representative of the disturbance spectra at r1and r2. They are chosen to reject external disturbancesat lower frequencies (� 2 rad/sec).

6.3 Case Studies

Five cases are considered to illustrate the utility of theanalysis and design framework based on substructuremodels as listed in Table 2. The cases are listed in

Case # Structure Interface Controller

1 = 0; � = 0 k12 = k21 = 02 = 1; � = 0 centralized3 = 1; � = 0 k12 = k21 = 04 = 1; j�j � 1 centralized5 = 1; j�j � 1 k12 = k21 = 0

Table 2: Design con�gurations (� = 1)

increasing order of design di�culty. In this table, = 0refers to zero nominal sti�ness while � refers to thevariation in the interface sti�ness. The terms kij, i; j 2[1; 2] refer to the controller component that maps themeasured outputs from substructure j to control inputsof substructure i.

6.3.1 Uncoupled SRCP, = � = 0 (Case 1)

This is the limiting case in performance and simplicitysince the subsystems are completely uncoupled. Thehigher frequency truncated modes of the two substruc-tures are the only model uncertainties assumed in thisproblem. The weighted frequency response plots in�gures 14 show that disturbances are rejected very wellat the low frequencies by two orders of magnitude over

10−2

100

102

10−4

10−3

10−2

10−1

100

101

102

103

e1/r1

Freq (rad/sec)10

−210

010

210

−4

10−3

10−2

10−1

100

101

102

103

e2/r2

Freq (rad/sec)

Figure 14: Frequency response for open loop (dotted)and closed-loop for controller 1 (solid), = � = 0

the open loop response while avoiding spillover. Theweighted open and closed loop frequency responses werecalculated using the full model (80 and 40 states) for thesubstructures. Since the substructures are uncoupled( = � = 0) and the controller is decentralized for thiscase, there is no cross coupling response. The �guresalso show a at weighted closed-loop frequency responsewhich is characteristic of optimal H1 controller re-sponse. In the transition bandwidths between the onsetof performance rollo� (� 2 rad/sec) and the onset ofadditive uncertainties (� 7 rad/sec for substructure 1and 2), rapid performance decay occurs.

6.3.2 Coupled SRCP, = 1, � = 0 (Cases 2,3)

This con�guration assumes a known �xed sti�ness in-terface ( = 1, � = 0) between the two substructures.As in case 1, the higher frequency truncated componentmodes are the only model uncertainties assumed in thisproblem. The controller in case 2 is centralized whilethe controller in case 3 is decentralized. Only steps1 and 2 of the sequential design was implemented fordesigning the decentralized robust controller in case 3.Controller order reduction at each step was done viabalanced realization [35] for case 3.

Figure 15 shows the weighted closed-loop frequencyresponses from the combined disturbances to the out-puts of interest when the structure is connected ( =1,� = 0). A full model of the connected structureconsisting of the �rst 30 modes were used in theopen and closed loop frequency response. The openloop response (dotted) is shown for reference. At lowfrequencies, both controllers show excellent disturbanceattenuation. Due to the controller constraint for case3 (solid), a performance degradation is signi�cant,especially in the transition bandwidth.

The closed loop response for the decentralizedcontroller also show strong coupling between distur-bances from one substructure to response of the otheralthough the controller itself is uncoupled. This im-plies signi�cant structural coupling as evidenced bythe magnitude of the o�-diagonal open-loop response.This is surprising since the nominal sti�ness of thecoupling section is two orders of magnitude less sti�than an equivalent length main substructure (see table1). In fact, cantilever beams have large excitability anddetectability at the free ends due to the dominatingfundamental structural mode response. When the twofree ends are connected, even with a \soft" interfacesti�ness, the system is highly coupled as just described.Therefore, although the relative numerical values ofinterface coupling sti�ness may indicate weak coupling,

10−2

100

102

10−4

10−3

10−2

10−1

100

101

102

103

e1/r1

Freq (rad/sec)10

−210

010

210

−4

10−3

10−2

10−1

100

101

102

103

e1/r2

Freq (rad/sec)

10−2

100

102

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10−3

10−2

10−1

100

101

102

103

e2/r1

Freq (rad/sec)10

−210

010

210

−4

10−3

10−2

10−1

100

101

102

103

e2/r2

Freq (rad/sec)

Figure 15: Frequency response for Cases 2 (dashed) and3 (solid), = 1, � = 0; open loop (dotted)

it is clear from the cross-terms that it is not for thisparticular example.

6.3.3 Coupled SRCP, = 1, j�j � 1 (Cases 4,5)

This con�guration is the same as cases 2 and 3 exceptthat the substructure interface sti�ness is assumed to beuncertain and/or variable over a known range (j�j � 1).The uncertainty in the interface sti�ness is assumed tovary about a nominal of = 1 which gives the nominalinterface sti�ness

so =

2664

�0:3 �0:3 0:3 �0:3�0:3 �0:4 0:3 �0:20:3 0:3 �0:3 0:3�0:3 �0:2 0:3 �0:4

3775 (28)

The nominal sti�ness can be viewed as an intermediatesti�ness condition between a fully coupled interface andwhere the two substructures are structurally uncoupled.From a physical standpoint, cases 4 and 5 are especiallyinteresting because it covers the conditions before (� =�1), during (�1 < j�j < 1) and after (j�j = 1)docking using �xed (spatially decentralized for case5) controllers. Steps 1 to 3 of the sequential designwas implemented for designing the decentralized robustcontroller in case 5. Controller order reduction at eachstep was also done via balanced realization.

Figure 16 shows the weighted frequency responseat nominal sti�ness where � = 0. The open loopweighted frequency response (dotted) is also shownfor reference. Both controllers 4 and 5 gives gooddisturbance rejection at low frequencies but at transientand higher frequencies, the decentralized controllersigni�cantly loses performance due to its controllerconstraint. The �gure also show that the cross response(e1=r2 and e2=r1) is similar in magnitude to the localresponse (e1=r1 and e2=r2) even when the controller isdecentralized for case 5.

10−2

100

102

10−4

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10−2

10−1

100

101

102

103

e1/r1

Freq (rad/sec)10

−210

010

210

−4

10−3

10−2

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100

101

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103

e1/r2

Freq (rad/sec)

10−2

100

102

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100

101

102

103

e2/r1

Freq (rad/sec)10

−210

010

210

−4

10−3

10−2

10−1

100

101

102

103

e2/r2

Freq (rad/sec)

Figure 16: Frequency response for Cases 4 (dashed) and5 (solid); open loop (dotted)

6.3.4 Sensitivity to Interface Sti�ness

The variations with � in the largest real component ofthe closed loop eigenvalues for all �ve controllers areshown in �gure 17. A negative largest real componentindicates stability and the �gure shows the approximaterange of interface sti�ness variation over which theclosed loop system remains stable using the evaluationmodel. Only controllers 4 and 5 are designed to ac-comodate speci�ed variations in the interface sti�ness.As expected, controller 1 (x) remains stable in a smallneighborhood about � = �1 when the substructuresare completely uncoupled. Controller 2 (�) remainsstable in a small skewed neighborhood about � = 0.Controller 3 (- -), which is the same as controller 2except with decentralized control, gives a larger andmore even neighborhood of stability than controller 2.Controller 4 (� � �) is the only controller that guaranteesrobust stability over the whole range of �. Withdecentralization, controller 5 (- .) gives a smaller regionof robustness than controller 4 but is still signi�cantlylarger than controllers 2 and 3.

Figure 18 shows the variation in the magnitudeof the unweighted frequency response from input r1to output e1 due to a change in interface sti�ness for = 1, and � = (�:4; 0; :4). The open loop frequencyresponse variation of the evaluation model is shown in�gure 18(a) for reference. The changes in the interfacesti�ness signi�cantly a�ect the frequency response ofthe structure over a wide frequency. Controller 1 wasunstable for all three � values considered and is notshown. Controllers 2 and 3 were closed loop unstableat � = �:4 and :4 and only the stable � = 0 case isshown (dotted). Controllers 4 and 5 were closed loopstable for all three � values. This is not unexpectedsince they were the only controllers that accounted forthe variation in the interface sti�ness. Controller 4 givesthe smallest variation (the three lines almost overlap)in the frequency response and is the most robust asexpected.

Figure 19 show the dependence of the disturbance

−1 −0.5 0 0.5 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

α

open loop

contr 2

contr 3

contr 1

contr 4

contr 5

Figure 17: Variation in largest real component of closedloop eigenvalue with �; open loop (o), controller 1(x),2(�), 3(- -), 4(� � �), 5(- .)

rejection performance (measured in terms of RMS er-ror) on interface sti�ness. The left, center, and rightbar plot for each controller show the RMS value for� = �:4; 0; :4, respectively. Unstable responses aredenoted by 0:1 RMS error in the �gure. The RMSerrors for controllers 4 and 5 show only a small variationover the range of sti�ness. There is a noticable dropin RMS performance for controller 5 due to controllerdecentralization. The nominal performance (at � = 0)for controller 2 and 3 are slightly better than controller4 and 5. This demonstrates the tradeo� in nominal per-formance for robustness over a larger set of structuralcon�gurations in the latter cases.

7 Concluding Remarks

There is currently no established means to quantita-tively account for model errors or uncertainties fora set of reduced component mode models. In re-sponse, a modularized control design framework hasbeen proposed such that substructure data can beutilized directly. This development could prove usefulbecause it takes advantage of the existing signi�cantbody of results in substructure modeling of large ex-ible structures. Although substructure controllers arehighlighted, centralized controllers can also be directlydesigned from substructure data. In addition, synthesisof the substructures, as is usually done in componentmodes synthesis, is not necessary for control design.

Although the numerical examples are based on aone dimensional structural system and is designed onlyto illustrate the proposed concept, it demonstrates adirect way to incorporate nominal substructure modelsand their corresponding uncertainties along with thesubstructure interface dynamics. The examples showthat variations in the interface sti�ness strongly a�ectsstability and disturbance rejection performance. Asmall loss in nominal performance can be traded-o�

10−2

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100

101

100

10−2

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101

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100

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100

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100

open loop

controller 2

controller 3

controller 4

controller 5

Freq (rad/sec)

Figure 18: Variation in frequency response (e1=r1) withinterface sti�ness, = 1, � = �:4 (solid), 0 (dotted),:4 (dashed)

for a signi�cant gain in robustness. The results alsodemonstrate the feasibility of designing decentralizedrobust controllers by a sequential process althoughdecentralization signi�cantly reduced performance. Ofcourse in the limiting case of uncoupled substructures,centralized controllers cannot be any better than decen-tralized controller.

Due to the 1-g testing environment, it may not bepossible to test an assembled large exible structure onthe ground so that some comprehensive form of on-orbitsystem identi�cation is critical. Since substructurescan typically be tested independently in the laboratory,it is in principle possible to develop through testingcomponent uncertainty models arising from inaccurateor reduced component modes and inconsistencies inthe substructure boundary conditions. Perhaps themain advantage then in the proposed technique is inthe reduction of the dependence on on-orbit systemidenti�cation of the assembled structure by an easierand less costly testing of substructures on the ground.In addition, control designs that are based on a singlemodel of the assembled structure cannot be experi-mentally validated on ground. These controllers woulddepend on nominal and uncertainty models that cannotbe experimentally developed or validated.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

openloop

Contr 1 Contr 2 Contr 3 Contr 4 Contr 5

RMS E

rror

Figure 19: RMS error dependence on interface sti�ness� = �:4(left), 0(center), :4(right)

The technique is particularly suited for large, com-plex, exible structures that are weakly coupled. Thedegree of weakness in the coupling should exceed athreshold such that its overall stability cannot be guar-anteed if decentralized controllers are designed indepen-dently or if its adjoining substructures are accountedfor inaccurately. In addition, performance limitationsdue to controller decentralization should be signi�cant.In practice however, logistical constraints may make itimpossible to implement a centralized controller.

Several important aspects of the coupled substruc-ture robust control problem that are not adequatelyaddressed and are open for further research include:modeling substructural interfaces beyond static sti�-ness, optimal substructure model reduction, role oferrors in the low frequency modes of the subsystems,quanti�cation of the degree of suboptimality of thedecentralized design.

Acknowledgements

The author would like to thank the anonymousreviewers for their many valuable comments in therevision of this paper and D. Cox at NASA LaRC orhis help with �gures.

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