(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

AIAA Guidance, Navigation, andControl Conference and Exhibit14-17 August 2000 Denver, CO

AOO-37013

2000-3953

SPILLOVER SUPPRESSION IN A FLEXIBLE STRUCTUREUSING EIGENSTRUCTURE ASSIGNMENT

Jae Weon Choi* and Un Sik Parkf

School of Mechanical Engineering and Research Institute of Mechanical TechnologyPusan National University, Pusan 609-735, KOREA

ABSTRACT

For the active vibration control of large space struc-tures such as a space station, large solar arrays of asolar power station satellite, etc., a controller is nor-mally designed on the basis of a spatially discretizedmodel which is an approximation to the distributedparameter system. This approximated model, inspite of its being a reduced representation of an infi-nite dimensional system, is generally too high orderfor a control designer to cope with due to the limita-tion of computing resources. Hence, in many cases,this model still needs to be reduced further, thenthere occurs a fundamental problem that an infiniteor large finite dimensional system must be controlledwith a much smaller dimensional controller. Thiscauses the spillover phenomenon which degrades thecontrol performances and reduces the stability mar-gin. Furthermore, it may destabilize the entire feed-back control system. In this paper, we propose anovel control design method for spillover suppres-sion in large flexible structures by using eigenstruc-ture assignment scheme. Its effectiveness in spilloversuppression is investigated and verified by numeri-cal experiments using an example of the simply sup-ported flexible beam which is modeled to have fourcontrolled modes and eight uncontrolled (residual)modes.

INTRODUCTION

In general, large space structures(LSS) such as aspace station and large solar arrays of a solar powerstation satellite have the characteristics of a flexi-ble structure by the demands of their light weightand large size. Hence, the control of large space

* Assistant Professor, Member AIAAt Graduate StudentCopyright ©1999 by Jae Weon Choi. Published by the Amer-ican Institute of Aeronautics and Astronautics, Inc. withpermission.

structures which are characterized by their inherentnatures -infinite dimension, distributed parameter,low stiffness, small damping, and densely spaced lowresonant frequencies- and stringent performance re-quirements has recently been a topic of major con-cern [1][2][3]. In large flexible structures, there arcmany vibration modes within the frequency band ofdisturbance and control input. When they are dis-turbed, these modes are likely to remain excited fora long time because of their low resonant frequen-cies and small damping. This might hamper theirmissions in space. Therefore, the concept of an ac-tively controlled large flexible structure with plentyof sensors and actuators located on the structure isneeded to be introduced.

To this end, a controller is normally designed onthe basis of a discretized model which is an approx-imation to the distributed parameter system. How-ever, this approximated model, in spite of its beinga reduced representation of an infinite dimensionalsystem, is generally too large for a control designer tocope with. Thus, the size of model must be reducedfurther to perform the controller design and the gen-eration of an appropriate reduced-order model forrepresenting an infinite dimensional and distributedstructural systems is the elementary challenge incontrol of large flexible structures.

Any reduced-order controller of this kind will in-teract with the residual (uncontrolled) system whenit is in the actual control operation and cause theproblems that the uncontrolled modes are exciteddue to the interactions between the control input andresidual modes and the sensor outputs are contami-nated by measurement of the residual modes. Balasshowed that a model reduction in flexible structuresystems can result in, so-called, spillover which isboth control and measurement coupling between thecontrolled and residual systems [4] [5]. Furthermore,spillover causes the possible instability in flexiblestructure systems, which is known as the spilloverinstability [4]-[6].

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

In control of large flexible structures with dis-cretized sensors and actuators, spillover is unavoid-able and undesirable, hence the effective spillover re-duction or suppression methods are required in thecontrol system design of large flexible structures. Al-though much effort has been made in controller de-sign, pre-filtering, and optimal placement of sensorsand actuators to reduce or suppress the spillover ef-fect, the fundamental problems have not been to-tally solved. Several representative approaches incontroller design are presented in the below.

The direct velocity feedback control(DVFB) wassuggested by Balas [4] and favorable spillover sup-pression is achieved, but there is still a danger ofspillover instability due to the rigid body modes.This method is a special case of the direct out-put feedback control(DOFB), thus sufficiently largenumber of sensors and actuators are needed toachieve the same performance with modal con-trol. The Independent Modal Space Control(IMSC),a method of modal control, was developed byMeirovitch, et al. [7] [8]. However, this approachneeds one actuator for each mode and the spilloverproblem still exists. The other methods such as thepositive position feedback [9], the model error sensi-tivity suppression method [10], and LAC/HAC [11]-[13] method are suggested by several researchers tosolve the spillover problem in control of large flexiblestructures.

In this paper, a novel control design method is pre-sented to overcome the detrimental effect of spilloverthat is a drawback of the reduced-order controllerfor large flexible structures. By using an eigenstruc-ture assignment scheme, we design an observer thathas the spillover suppressibility, which is achieved byletting the left eigenvectors of observer be orthogo-nal to the column space of the observation spilloverterm and reducing the observer eigenvalues' sensi-tivity to the perturbation due to spillover. However,the observer gain enters the observation spilloverterm, so that the observer design for spillover sup-pression may have to be solved by a iterative nu-merical method. To avoid this difficulty, a noveleigenstructure assignment is proposed to solve thespillover suppression problem analytically.

Numerical simulations are performed on a simplysupported flexible beam which has four controlledmodes and eight residual(uncontrolled) modes. Insimulations, we show that the proposed method hasgood performance and is effective in spillover sup-pression of large flexible structures.

SPILLOVER

In this section, we carefully examine the spillover

phenomenon and its detrimental effect on control oflarge flexible structures and also address the spilloverinstability.

Spillover Analysis

Large flexible structures can be described in thestate space representation of a linear, time-invariantsystem as follows:

x(t) = Ax(t)+Bu(t), x(Q)=x0y(t) = Cx(t) (1)

where x £ R", u 6 Rm, and y £ Rl are the state,control input, and output measurement vectors, re-spectively. A, B, and C are constant matrices of sys-tem, input influence, and output measurement withappropriate dimension, respectively. It is assumedthat (B,A) is controllable, (C,A) is observable, andB is an n x m matrix with a full column rank.

By using approximation techniques such as the fi-nite element method, large flexible structures of infi-nite dimension can be approximated into a finite di-mensional model of order n as Eq.(l). However, thenth order model is generally too high order to imple-ment a controller due to the limitation of computingresources. Therefore, the ncth controlled modes areselected by the model reduction method, and theother nrth modes are remained uncontrolled thenbecome the residual modes. Thus, the entire mod-eled (discretized) modes of order n in Eq.(l) can bepartitioned as follows:

= [xc(t) xr(t)}

xc(t)xr(t)

ArtAr

Xc(t)Xr(t)

Bc

0 u(t](2)

where xc(t] 6 R"" is the states of controlled sys-tem and xr(t) € RHr is the states of residual sys-tem, and Ac € Rn^n- and Ar € Rn^n- representthe system matrices of the controlled and residualmodes, respectively. In addition, Acr 6 ^"••Xn= andArc € Rn'*n'- are the correlation matrices that rep-resent the interactions between the controlled andresidual modes. It can be seen that the control inputmatrix Bc £ Rn-=xm appears only in the controlledmodes in Eq.(2).

By using a certain state transformation (x (t) =T v(t)) in Eq.(2), the modal state space model canbe achieved as follows:

= [vc(t) vr(t)]

vr(t)0

Ar Vr(t)Be

Bru(t]

(3)

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

where vc(t) 6 Rn" and vr(t) £ Rn" representthe controlled and residual modes, respectively. InEq.(3), it can be seen that both the controlled andresidual systems are decoupled and the correlationmatrix terms in Eq.(2) are reformulated as externalinput matrices Bc and Br.

Then, system's output can be obtained as follow-ing:

y(t) = Ccvc(t)+Crvr(t) (4)

If a state feedback is applied to Eq.(3), the followingdeterministic estimator(Luenberger observer) mustbe introduced for the states that are not measuredby sensors.

Controller

= Ac vc(t) + Bc u(t)vc(0) = 0

K(y(t] - y(t))(5)

Here, vc(t) £ Rnc is the estimated states and K isthe observer gain matrix. In observer from Eq.(5),the estimated output can be established as follows:

y(t)=Ccvr(t) (6)

Then the state feedback control u(t) can be con-structed by using the estimated states as follow:

u(t) = -Gvc(t) (7)

where G is the control gain matrix.If we define the state estimation errors as ec(t) =

i)c(t)—vc(t), then the following equation can be easilyderived from Eq.(3) and Eq.(5).

ec(t) = (Ac - KCC) ec(t) + KCr vr(t) (8)

From Eq.(8), if the term of residual modes vr(t) isnot present, we can easily obtain the observer gainmatrix K that stabilizes the estimation process inobserver. Generally, if the eigenvalues of observerare chosen to be two or three times faster than thoseof controller, the state estimation process does notinfluence the state feedback process and, thus, a sta-ble observer does not destabilize the controller by theseparation principle [14].

The entire feedback control system is depicted inFig. 1 with representing the spillover phenomenondue to the residual system, thus we can constructa composite system that consists of the controlledmodes vc(t) in controller, the state estimation errorec(t) in observer, and the residual modes vr(t) asfollows:

ec(t)vr(t]

'AC-BCG -BCG _0_0 AC-KCC KCr

Vc(t)'

ec(«)Vr(t)

(9)

Observer

Fig. 1 Spillover mechanism

From Eq.(9), we investigate the spillover, the inter-actions between the controlled and residual systems,intensively.

First, for examining the control spillover, we con-sider the dynamics of the residual modes from Eq.(9)as follows:

vr(t) = Ar vr(t) - BrG vc(t) - BrG ec(t)= Ar vr(t) - B_rG vc(t) (10)= Ar vr(t) -Bru(t)

In Eq.(lO), it can be easily seen that the residualmodes vr(t) are excited by control input u(t) throughthe residual input matrix Br. As a result, throughthe matrix term —BrG, the residual modes vr(t) in-teract with the controlled modes vc(t) and the stateestimation error ec(t) and can be excited by the feed-back control operation, which is called the controlspillover [4]-[6]. However, the control spillover doesnot destabilize the entire feedback control systembut degrades the control performances of vibrationsuppression.

Second, the state estimation process in Eq.(9) isconsidered from Eq.(8). In Eq.(8), the estimationprocess is disturbed by the excitation of_ residualmodes vr(t) through the matrix term KCT. Thisdisturbance is mainly from that sensor outputs arecontaminated by the residual modes in Eq.(4) byCr vr(t), which is called the observation spillover.We notice that the observation spillover may desta-bilize the estimation process in observer, even theentire feedback control system.

Spillover InstabilitySpillover tends to reduce the stability margin by

shifting stable eigenvalues of the entire feedback con-trol system in Eq.(9) toward the right half complexplane and the magnitude of shift depends primar-ily on the spillover terms BrG and KCr. Since the

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stability margin of flexible structure systems is es-pecially small, even a small shift of eigenvalues canmake them unstable.

Bounds on the stability margin reduction due tospillover are given in [4]. Such bounds help to quan-tify the effect of the residual modes on the entireclosed-loop system and indicate tolerable levels ofspillover for stable control operation in large flexiblestructures.

Form Eq.(9), we can rewrite the entire feedbackcontrol system with four block matrices as follows:

ec(t) v r ( t ) ]

w(t) = Hw(t) =#11 : #12

'• H-22 J

w(t)

where

#11 =0

KCr

AC ~ BCG '. —BCG

0 : AC-KCC.r - - . - -i— D f~* • D f~* .[—£frL, . — ±frOrj '

In Eq.(ll), we can always construct the stableclosed-loop system by designing the control gain Gand observer gain K appropriately, if the observa-tion spillover is not present (.K'C'r = #12 =0) . Inthat case, the eigenvalues of composite system areexactly those of Ac — BCG, Ac — KCC, and Ar by theblock triangularity.

When both the control and observation spilloverare present, the composite system matrix H can bewritten as

' = H0 + A#

most observable, controllable, and are closest to thecontrol bandwidth. More importantly, the eigenval-ues' sensitivity of certain modes are closely relatedwith the spillover instability [4]- [6]. Hence, we donot always expect that the control and observationspillover produce an instability, but all modal con-trollers have the potential to generate instabilitiesunless the observation spillover can be eliminated orsuppressed.

Bounds on the stability margin reduction due tospillover are given in [4] and described as follows:

Theorem 1 Let $0 be the right modal matrix ofH0 and the condition number of $0 is K0 =

(11) H^° Ill^p"^!! — 1- If the norm of observation spilloverterm \\KCr\\ is sufficiently small and satisfy

= \\KCr (13)

then H is stable^ and has stability margin 6'0 where5'0 > 6 — K0\\KCr\\. Here, S0 is the stability marginofH0.

a

Note that Theorem 1 places a stringent requirementon the amount of observation spillover that can betolerated for stable closed-loop system. Of course,this sufficient condition may be too conservative butit does give an idea of the relationship of spillover tothe shifting of eigenvalues.

SPILLOVER SUPPRESSION

In this section, the condition that the left eigen-vectors of observer have to satisfy to suppress thespillover and a novel spillover suppression methodusing an eigenstructure assignment scheme are de-veloped for large flexible structures.

#21 #0 1 A#= |° Hl2\ ^ Spillover Suppression Conditions22 0 0

and # = H0 when the observation spillover is ab-sent. The eigenvalues of # depend continuously onthe (real) parameters of A# and hence on those ofKCr. That is, there are continuous functions of theparameters of KCr with starting values equal to theeigenvalues of Ac — BCG, Ac - KCC, and AT.

Note that even small perturbations of the originaleigenvalues due to spillover may lead to instabili-ties, particularly for the eigenvalues of Ar which liecloser to the imaginary axis due to extremely smalldamping and nearly have no stability margin. Thisphenomenon is known as the spillover instability.

However, not all the residual modes are poten-tially critical from spillover, but only those which are

The state estimation error equation in Eq.(8) isconsidered again as follows:

ec(t) = (Ac-KCc)ec(t)+KCrvr(t) (14)

As seen in Eq.(14), the excitation of residual modesvr(t) can be considered as a disturbance to the es-timation process in observer, then the observationspillover problem can be solved by using the generaldisturbance suppression techniques [15] [16].

In Eq.(14), the observer's system matrix can bedescribed by using modal decomposition as follows:

= * A(15)

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

where A £ pncxnc js a diagonai matrix of eigen-values, $ € Rn'xnc is the right modal matrixwhose columns consist of the right eigenvectors ofAc - K Cc, and * £ Rn"xn<= is the left modal ma-trix, similarly. The columns of $ and ^ are de-termined by the solution of the standard eigenvalueproblem. Then, we can obtain the following equa-tion from Eq.(14) by using Eq.(15).

ee(t) = ec(t) + KCr vr(t) (16)

If the initial values are assumed to be zero, then thesolution of Eq.(16) can be obtained as

ec(t) = $ /„*' • KCr) vr(r) drec(0) = 0 (17)

In Eq.(16), the excitation of residual modes vr(t),which is induced by the control spillover in Eq.(lO),can be regarded as a disturbance to the state esti-mation process. Then, from Eq.(17), we can con-clude that the state estimation error ec(t) are freefrom the excitation of residual modes which is_ ap-plied through the observation spillover term KCr,if the spillover suppression condition is satisfied asfollows:

*T • (KCr) = [0] (18)

The condition in Eq.(18) represents that the lefteigenvectors i/Ji must be orthogonal to the vectorspace spanned by the column vectors of KCr. Ifthis condition is satisfied for all left eigenvectors ofobserver, then the observation spillover can be com-pletely eliminated.

However, due to the lack of freedom in matrixcalculation, this condition is difficult to be satisfiedcompletely. This is because, in general case, the di-mension of residual modes (nr) is much larger thanthat of controlled modes (nc). That is, the numberof column vectors of KCr is much larger than thatof left modal matrix <?.

Hence, using the concept of modal disturbancesuppressibility [15] [16], we can suppress the modalspillover which is induced by the specified jthspillover term to the specified ith controlled mode.Then the condition in Eq.(18) can be rewritten asfollows:

• (Kcrj) = 0 for any i, j (19)

where CTJ is the jth column vectors of output ma-trix of the residual system Cr. From here, we cansuppress the ijth modal spillover that contributessignificantly to the total spillover based on the con-cept of modal controllability and observability.

Prom the other approach, we can consider thespillover suppression as a robust stability problemwhich can specify the bound of spillover for guar-anteeing stable feedback control system. Since thespillover phenomenon is originally caused from themodeling errors which include both the unmodeleddynamics of infinite dimensional structure systemsand the residual modes of modeled system, we needhave to consider the stability robustness of observerfor spillover suppression.

Theorem 1 in previous section indicates clearlythat the stability robustness is directly related tothe smallness of condition number of the associ-ated modal matrix. From this, it is evident thatthe maximum stability robustness condition for afixed stability margin (eigenvalues) corresponds tothe least sensitivity of closed-loop eigenvalues, whichis achieved by the minimum condition number ofclosed-loop modal matrix. It is well known that nearorthogonality of the closed-loop eigenvectors is verypreferable to minimize the sensitivity of eigenvalues[17]. Hence, for spillover suppression, any orthog-onal (unitary) matrix is desired for the left modalmatrix of observer.

Now, it is obvious that our goal for suppressingthe observation spillover in_the observer design is tochoose the observer gain K so as to make the lefteigenvectors fa satisfy the condition in Eq.(19) andconcurrently make the left modal matrix orthogo-nal(unitary). However, there is some critical prob-lem in obtaining the observer gain K. Since the ob-server gain K enters the observation spillover termKCr in Eq.(14), the complex numerical methodsneed to be involved to simultaneously obtain boththe observer gain matrix K and the left modal ma-trix $ which satisfy the spillover suppression con-dition in Eq.(19). Hence, to avoid the drawback ofiterative numerical methods, a new eigenstructureassignment which can achieve the observer gain forspillover suppression analytically is proposed in thefollowing subsection.

Eigenstructure Assignment forSpillover Suppression

In this subsection, we present a new eigenstructureassignment method that can make the left eigenvec-tors of observer satisfy the spillover suppression con-ditions presented in the previous subsection.

Let A = { AI , A2, • • •, \nc } be a self-conjugateset of distinct eigenvalues for simplicity. Then, theright and left eigenvalue problems for the observerin Eq.(14) can be defined by [14]

Right : (Ai/ne - Ac + KC_C) fa = 0Left: V? (Ai/ne - Ac + KCC) = 0

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

where Inc is an (nc x nc) identity matrix. In the casea system has repeated eigenvalues, the eigenvalueproblem can be easily generalized [18][19].

Using the dual concept, the left eigenvalue prob-lem in Eq.(20) can be rewritten as follows [14] [20]:

- ATC

-Ac + KCC)}T

CTC KT] fr = 0 (21)

Note that Eq.(21) has the form of the right eigen-value problem in Eq.(20). Hence, we can solve theleft eigenstructure assignment for the observer de-sign by using the right eigenstructure assignment forthe controller form (A£ — C^KT) as seen in Eq.(21).From, here, the problem of right eigenstructure as-signment is to choose the feedback gain matrix KT

such that the required conditions for the eigenvaluesand right eigenvectors of (A£ - C^KT) are satis-fied, and therefore may be considered as an inverseeigenvalue problem.

First, we define

and a compatibly partitioned matrix

(22)

(23)

where the columns of matrix R\t form a basis for thenull space of S\t. If Cj has the full rank, it can beshown that the columns of N\f and M\f are linearlyindependent [21].

Then, the right eigenvalue problem can be rewrit-ten from Eq.(21) as follows:

= 0 (24)

Here, we can introduce the parameter vector hi € Rl

defined by

Then, Eq.(24) can be rewritten as

hi

(25)

= 0 (26)

From Eqs. (22), (23), and (26), we need to find thecoefficient vectors Zi that satisfy the following equa-tion:

MA (27)

where z$ is the linear combination coefficient vectorthat can generate both the achievable eigenvectors^j° from the column space of N\t and the achievableparameter vectors hf from the column space of M\.simultaneously.

To find the coefficient vectors Zi, we must choosethe desired eigenvectors ipid and the desired param-eter vectors hf satisfying the spillover suppressionconditions presented in the previous subsection andsummarized as follows:

Condition 1. The desired parameter vectors hid

satisfyCrj 'hi = 0

where crj is the jth column vectors of Cr =[CTI cr2 ••• crnr] and hi is the ith col-umn vectors of Hd = [hid h2

d ••• hnd] =

Condition 2. The desired modal matrix ^d =[if}f ^ • ' • V'rac ] is orthogonal (unitary).

In here, Condition 1 can be derived from Eq.(19)by transposing both sides and substituting with theparameter vector hi as follows:

= cr

,nr(28)

The desired parameter vectors hf satisfying the Con-dition 1 can be obtained from the following equation.

Ker(cr2r)

i = 1,2,- • ,nc(29)

where a.i is the weighting vector which combinesthe null spaces of each column vectors of the resid-ual output matrix CT and Ker(-) represents the nullspace of (•).

When the desired parameter vector hi satisfyingthe Condition 1 is obtained, then there may be aplenty of eigenvectors which satisfy Eq.(25). Thus,by using the redundant freedom in this set of eigen-vectors, we can select the desired eigenvectors to sat-isfy the Condition 2.

If the desired eigenvectors V'f and the desired pa-rameter vectors hf that satisfy the spillover suppres-sion conditions are selected, then the coefficient vec-tors Zi can be obtained by the following equation:

Zi —

-it

hd(30)

= 1,2, •where the superscript [•]* denotes the pseudo-inverseof a given matrix [•]. Although the desired eigen-

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

vector ifjf does not reside exactly in the achievableeigenspace, the achievable eigenvector i/}f can be ob-tained in the least square sense by using z, fromEq.(30) as follows:

1>i=N\t ^ (31)

In addition, the achievable parameter matrix hf canbe obtained in the least square sense by using zi asfollows:

hai=MXi Zi (32)

Then, the feedback gain KT can be obtained fromEq.(25) and given by the matrix equation as follows:

KT = Ha ,(33)

where K is the observer gain matrix.The approaches taken in the proposed spillover

suppression method are based on the concept ofboth disturbance suppression and sensitivity reduc-tion problem. The proposed eigenstructure assign-ment scheme for designing an observer which hasthe spillover suppressibility is completely analyticaland systematic design method, and simple to imple-ment. The advantage of the proposed design methodfor spillover suppression is that the individual modecan be dealt with easily.

A NUMERICAL EXAMPLE

In numerical experiments, a simply supportedflexible beam model is used to verify the effective-ness of the proposed spillover suppression method.The beam is modeled to have four controlled modeswhich is controlled by two collocated actuators andsensors ,and eight residual modes as well. The statespace representation of the system under considera-tion is described by

0 SO 100 150 200 25O 3OO 35O 4OOTime(sec)

Fig. 2 Response of the open-loop system

0 00 0

43 00 -6.3439

0 00 0

0.5211 0.52110.4179 —0.4179

1 00 1

-0.0047 00 -0.0073

0000

-0.0065

0000

-0.4179-0.5211

0.2319-0.2319

0 -0.0010 00 0 -0.0011

0'000

-0.41790.52110.23190.2319

y = Oc vc(t) +C,.

0 0.52110 0.5211

0.4179"!-0.4179J '

0 0 -0.4179 -0.5211 0.2319 -0.23190 0 -0.4179 0.5211 0.2319 0.23

191-19 J

The eigenvalues of the open-loop system are

-0.0037 ± 2.5187J-0.0023 ±1.9221;-0.0013 ± 1.2737;-0.0008 ± 0.7244;-0.0006 ± 0.3229;-0.0005 ± O.OSOSi J

and the responses of initial value are depicted inFig.2.

Let the desired eigenvalues of controller be

^controller = { -3.8 -4.0 -4.2 -4.9}

Then, the control gain matrix G can be obtainedby a normal eigenstructure assignment method asfollows:

10.9224 19.3525 7.4486 11.4526]11.1229 -14.7373 7.5014 -10.3159J

. . _ _ . , = 34.4102

where || • ||fro represents the Frobenious norm of [•].In observer design, let the desired eigenvalues be

A L — / — i n ^ —11 11 ^ 19\^'•observer — \ J.u.*j J.J. —xx .u — /

In this simulations, for the verification of the pro-posed spillover suppression method, we now considerthe next two cases of the observer design. One isthe case in which the entire closed-loop system isdestabilized by spillover, and the other is the case inwhich this spillover is suppressed and stable feedback

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control system is achieved by the proposed spilloversuppression method. By comparing these two cases,we can verify that the proposed spillover suppres-sion method using a novel eigenstructure assignmentmethod in the observer design has the good sup-pressibility against spillover. Note that these twocases are compared in the same conditions of eigen-values and control gain which are previously pre-sented.

First, the spillover instability case is presented.When a normal eigenstructure assignment is applied,then the following left modal matrix is achieved.

65.0694 69.9818 81.2671 54.4606157.9645 102.8248 200.1102 59.8676184.9395 227.3159 241.9754 169.5290261.4521 194.5013 346.9806 108.5258

fc(*°) = 1412.02.Then, the observation spillover term KCr and its

norm are obtained as follows:

KCr =

|D 0 0 0 38.8870 21.1692 -21.5807 9.42120 0 0 0 -6.0744 12.6486 3.3710 5.62920 0 0 0 -22.0006 -6.8693 12.2094 -3.05710 0 0 0 3.4011 -21.8891 -1.8875 -9.7416

\\KCr\\fIO = 63.5102

In this case, the observation spillover destabilizethe entire closed-loop system, and this can be seenfrom the eigenvalues of composite system in Eq.(9)as follows:

-27.5276 ± 3.3158*-3.1740 ±11.6502*

0.1278 ±9.0056*-0.2096 ±1.2944*

^composite = -0.0941 ± 2.0636*-0.0405 ± 0.6250*-0.0236 ±1.1554*

-0.0184-0.0045

The underlined eigenvalue in ^-composite which isshifted by spillover makes the entire feedback con-trol system unstable.

Next in the second case, we apply a novel eigen-structure assignment method for spillover suppres-sion. Then, the achievable parameter matrix Ha

can be obtained to satisfy the spillover suppressioncondition (Condition 1) as follows:

""[-I2478 0.2516 -0.1891 -0.1963]2142 0.2101 0.2735 -0.2655 J

In addition, the achievable left modal matrix \l/a canbe obtained to satisfy the spillover suppression con-dition (Condition 2) as follows:

-24.6405 25.1415 1.2776 -0.5217'_ 4.9863 3.0496 -9.4753 -9.6486~ -76.7027 74.8595 4.1498 -1.4829

9.0389 5.2878 -17.9232 -15.9697.fc(*a) = 174.0702

Then, the observer gain K can be achieved by using\Pa and Ha from Eq.(33) and given by

-32.5653 -31.1287-22.4155 22.7787

21.8050 21.330526.8253 -27.0364

From here, the observation spillover term KCrand its norm can be obtained as follows:

TO 0 0 0 26.6181 0.7487 -14.7719 0.333210 0 0 0 -0.1518 23.5516 0.0842 10.48150 0 0 0 -18.0266 -0.2473 10.0040 -0.11000 0 0 0 0.0882 -28.0685 -0.0490 -12.4916

KCr =

| | f ro= 54.4149

In this case, spillover is suppressed by the pro-posed method, then the stable composite system isobtained. The eigenvalues of composite system inEq.(9) are as follows:

-28.3653 ± 0.6363*-1.3723 ±11. 8422*-0.8390 ± 8.8641*-0.2051 ± 1.3006*-0.0962 ± 2.0605*-0.0391 ± 0.6250*-0.0244 ± 1.1551*

-0.0191-0.0043

Aco;mposite —

Here, the vibration responses of the composite sys-tem in the second case are depicted in Figs. 3, 4,and 5. In Fig. 5, one of the residual modes is seento remain excited for a long time because this modehas the smallest damping due to the correspondingeigenvalue of —0.0043. Consequently, it can be seenthat the observer which has spillover suppressibilityis working properly. The spillover suppressibility ofobserver could be increased by increasing the num-ber of independent sensors.

CONCLUSIONSThis paper presents a new control design method

that can stabilizes the flexible structural systemswhich includes the neglected dynamics of highermodes of vibration by suppressing the spillover. Theproposed method has been proven to be successful inboth the spillover suppression and vibration controlof large flexible structures by numerical simulations.

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

Fig. 3 Response of the controlled modes vc

VX"

2O 25 3O

Fig. 4 Response of the state estimation error ec

0 50 100 150 200 250 3OO 35O 4OO

Fig. 5 Response of the residual modes vr

ACKNOWLEDGEMENT

This work was supported by the Brain Korea 21Project.

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