Eigenstructure Assignment Approach to the Reconfiguration of Flexible Structure Control Systems

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EIGENSTRUCTURE ASSIGNMENT APPROACH TO THE RECONFIGURATION OF FLEXIBLE STRUCTURE CONTROL SYSTEMS H. Okubo, K. Sunohara and Y. Murotsu IJI'/IIIJ'IIIII' II I

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Abstract, An eigenstructure assignment approach is proposed for r ecove rin g a structural control system subjected to component failures . The obje ct iv e of t he cont r o ller redesign is to make the failed system fun c tion as much like the nominal syste m as possible in the sense that the dynamics of the recove red closed-loop syste m is as similar as possible to that of the undamaged system . Us e of the eigenstructure assig nm ent algorithm for a state fe edback controller allows th e full specification of a set of c losed loop eigenvalues and the allowable e le me nt s of the cor responding closed - loop e i ge nvectors so that they are as close as possible to th ose of the original e i ge nsys t em, This approa ch is suited to the control system r econ figuration for the l arge flexible s pace structures where many sensors and actuators are to be placed on th e flexible body for active vibration co ntrol, Verifi ca tion of th e proposed eigenstructure assig nm e nt a lgorithm is given with a numerical example. i ,e . . ac tuator failure acco mm oda ti on in the vibration control for a flexible beam . Keywords, Fle x ible s tructure control; fault-torelant control system; eigenst ru ctu r e assignment; restru c turable control; la rge space structures

INTRODUCTION Fault-torelanc e of control systems has receiv e d much attention most r ece ntly in relation to the requirement of high r e li~bility for the automatic control systems in aerospace. For example. mu c h work has been done on the fault-torelant or restructurable flight co ntrol systems for control configured vehicles (e , g. Eterno and others. 1985), The present paper dea ls with this topic in association with th e active co ntrol technology for large space st ru ct ur es (LSS), Dimensions of th e future LSS. such as large space antennae and solar powe r satellites. will be o f the order of hundred meters to several kil omet ers. Since such a huge stru c ture in space wi II hav e a high level of me c hani ca l fl ex ibility. undesired vibrations and de formations of the flexible body can easi Iy induc e d by external disturban ces or control inputs for attitude stabilization, Therefore. needs for the vibration suppression and shape determination by the use of active co ntrol technology thus arise, While a considerable amount of work has been done on the co ntrol of LSS in the last decad e (e.g, Atluri and hos. 1988). the problem of unr e l iabi I i ty of the contro l components considering possible failures has received very little attention (Vander Verde and Car ignan. 1984; Okubo and ot he rs . 19 87), Neve rth eless. th e fault-torelance is one of th e ke y aspects of the active control technology for th e LSS . Reliability of a syste m against component failures can be improved by in corporat ing a hardwar e redundancy. a bank of ba c kup hardwares. wh ich however increases the system weight and volume, This will cause serious di sadvantag es in space app li catio ns, In contrast. use o f analytic redund a ncy can provide the same improv e ment in the system rei iabi I i ty wi thout in c r eas ing th e number of hard wa r e backups , Ther e f ore . this type of r edund anc i es wi I I be of primar y imp ortance in space systems, Thi s approach utilizes the analytical relationships between the dif ferent co ntr ol compo nent s ,

For example. in the a na l yt i c a l redundancy appro ac h. redund ant contro l actuators a~ dis tributed on a fl ex ibl e body at different Idt ations rather than they are p l aced at th e same position as the multiple ha r dware back up s, The accommodation to a failur e i s ma de with an appropriate failur e dete c ti o n algol'i thm f o llowed by th e con trol system reconfi gur at ion by using the remaining act ua tors. Th e r e are seve r al me th ods proposed for th e reconfiguration o f feedba c k contro l syste ms (Vander Velde. 1984 ; Gavit o a nd Collins. 1987), In this paper . an e i genstructure assign ment approach is developed for the purpose of reconfigurin g th e failed contro l syste m so that the e igen va lu es a nd eigenvectors of t he r ecovered ~ osed loop system are as close as possible to , ~h ose of the nom i na I system , .1

FLEXIBLE STRUCTURE CONTROL Consider a finite order discrete mod e l of a lin ear sys tem of the structural dynamics of the form

z

[M]

+

Z +

[ C )

[K]

z =

[0]

u

Cl)

and output equation y

=

[ S ,]

z

+ [ S 2] z

(2)

where [M]. [ C). [K] respectively are mass. damping and stiffness matrices. [M] >O. [C];;;O. [K];;;O. and z(t ) u( t ) ye t ) [0] [Si]

nx I conf igurat ion vector mx I contro l vector I' x I output vec t or nxm co ntr o l influ ence matrix rXIl outp ut infl ue nce ma t I' i x

Th e structural eigenvalue prob l em associated wit h the z=.pexp(A t ) solutions for the undamped free vi bra t ion case of Eq,(I ) is given by eigenvalues: det [

~

[ M]

,2= 1,2

- (J

+

,2 [ K]

o

(3)

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346 eigenvectors:

~,

[[M]A,2+[K]]~,=0

The eigenvectors are orthogonal with respect to [M] and [K]; the orthogonality equations are written in matrix forms (5) (6)

where

1\ = diag [w ,2.W 22.-_-.W

n

2]

(8)

We introduce a transformation of Eq. (I) into the modal space associated with the undamped eigensystem:

z

=

(9)

7J

The modal space equation of motion is derived in the form (10)

where TJ is the nX I vector of modal coordinates. The n second order system of equations. Eq. (10). can be written as a system of 2n first order equations in the standard form and

x = A x + B u

(Il)

y = C x

(12)

where the 2nx I state vector x is defined as x=col( TJ. TJ}. and A= [ _1\0

C =

[

] •B _ TC

0 TO

1.

[ S , • S 2 ]

In this paper we are primary concerned with design of the constant mXr gain matrix. G. for a linear feedback control law of the form u(tl=Gy(I)=GCX(I)

(13)

Substituting the feedback law. Eq. (13). Eq.(lI). the closed loop system becomes X= [A+BGC]

into

1\ = d i ag [ A ,. A 2. - - - • A n]

del[[A+BGC]-AI]=O

right eigenvectors: [A+BGC]

X

=

V,=

(15)

[v ,.v 2.---.V n] A,V,

(16)

The eigensolutions (A.V) occur in complex conjugate pairs. and the 2nX I eigenvectors vi have the structure such as vi=col(vi. A iVi} (Vi is an n X I comp I ex vector). In terms of the distinct closed loop eigenvalues A i. and eigenvectors vi. the output vector is given by y(t)=CX(t) [ Cv , ( 1

FAILURE ACCOMMODATION VIA EIGENSTRUCTURE ASSIGNMENT In this paper. we consider the fai lure accommodation for flexible structure control systems where the fai led closed loop system is reconfigured in order that the eigenstructure (i.e .• the set of eigenvalues and eigenvectors) of the recovered system is as close as possible to that of the original system. The proposed problem can be stated as follows : "Determine the gain matrix G such that the eigenvalues and eigenvectors of the reconfigured closed loop system are as close as possible to those of the or iginal (unfai led) system." On the assignment of the eigenstructure using a feedback control. the following theorem is given by (Srinathkumar. 1982): Theorem. Given the controllable and observable system by Eq.(II) and Eq.(12) and the assumptions that the matr ices Band C are of full rank (rank B=m;o!oO. and rank C=n=O). then max(m.r) closed loop eigenvalues can be assigned and max(m.r) eigenvectors can be partially assigned with min(m.r) entries in each vector arbitrarily chosen by using the gain output feedback. Eq. (13) . Thus the assignabi I ity of the eigenvectors depends on the number of available control devices (i.e., sensors and actuators) . Therefore. this approach is useful for the design of MIMO controllers having a sufficiently large number of sensors and actuators. This characteristics is suited for the active control of large space structures where many control sensors and actuators are to be distributed on the flexible body for the purpose of vibration control and/or shape determination.

(14)

x

The closed loop eigenvalue problem. associated with x=v'exp( A t) solutions of Eq. (14). is eigenvalues:

Equation (17) is the quantitative description of the modal expansion theorem, and it is noted that every solution representing a free response of the closed loop system. Eq. (14). depends on the three terms. i.e. closed loop eigenvalues. eigenvectors. and initial values. Thus it is apparent that if feedback is to be used to make the transient response of the fai led s)'stem as much simi lar to that of the original system. eigenvector selection must be considered as wel I as pole (eigenvalue) placement (Moore. 1976).

In the case of state feedback, i.e .• when u=Gx (C=I). and if (A. B) is controllable. all of the system eigenvalues are arbitrarily assigned and eigenvectors are partially assigned with m entr ies. There are several methods for the eigenstructure assignment, in which a set of allowable eigenvectors are chosen for the pole placement and the feedback gains are determined (Kautzky and others, 1985; Junkins and Rew, 1988). The right eigenvalue problem for the state feedback is given by [ A •I

exp (A • t)

(17)

v,=BGv,

( 18)

or in a matrix form

B G = X 1\ X - ,

- A

( 19)

Construct the singular value decomposition (SV D) of the matrix B: B

,T X 0)

A]

= U 2: V

T=[ U

0

U ,]

Il 0Z

'J

(Z

= 2:

V T) (20)

with U=[UO, UI] orthogonal and Z nonsingular.

~-17

Recollfigllratioll of Flexihle Structure COlltrol SYstems

Then the range and nu II space of B can be spanned by Uo. UI' respectively. Premultiplication of Eq.(19) by UT gives the following two equations. U oT(XI\X

Z G

o

-

A)

-

A)

(21) (22)

Equation (22) shows that each eige nvect or vi must be a member of the following right null space V

(23)

£

1

illustrated in Fig. I. A set of tendon actuator consists of a couple o f for c e actuators (or an equivalent torque generat or ) at t he beam root and moment arms located on the be am with I inkages of tensile wires. Tendon actuat or is a kind of torquer which appl ies control torques to the beam at a distance from the beam-root. The authors proposed the use of such type of actuator for the applications in space. i.e .. active vibration control of flexible appendag e structures deployed from a large spacecraft. A hardware model of the tendon control s;'stem has been constructed for th e labor a tory experiment of

The gain matrix G is given by Eq.(21) as G=V o L - 'U OT(P

- A)

(24)

f<---

where P=XAX- 1 is computed by solving the following I inear algebraic eq uatio ns for the co I umns of pT in order to avoid the matrix in version.

Step I

m ment ar m

. "'\ sen sor

\J )

::::....::: force actuat or

:::.

The algorithm for the reconfiguration marized as fol lows:

te ndon

/

~

:::::

(25) We now consider the origina l (undamaged) system (AO. BO) and state feedback. uO=GQx. with the closed loop eigenvalues a nd eigenvectors (AOi' vOi)' Assume that a fai lur e changes the system matr ices to (Af. Bf) wh ich causes a def ic iency to the system due to the varied eige nstructure . (Afi. vfi)' The design objective of a selfreco nfi gur ing contro ll er is therefore to regain t he undamaged modal structure (AOi. vOi) subsequent to such system deficiencies via a reconfigured set of feedback gains. Therefore. we use the eige nst r ucture ass i gnment a l gorithm so that the eigenval ues of the damaged system are exactly assigned to those of the nominal system. and simultaneously. th e norm of difference between the reconfig ured c lo sed l oop eigenvectors and the nominal ones are minimized.

la

-:::

M

~

tens; on

Rs

R Fl g. 1 Te ndon control system for beam vi brat ion suppression. Table

Beam characteristics. Stainless steel

Material

2.4 [m)

Length

150 [mml

Width

1 [mm)

Thickness

2.39 [Nm') \

F1exural rigidity El

is sum-

Mass density

1.14 [Kgm- '

Weight

2.74 [Kg)

I

Obtain a matrix Ui which spans the space. Si . of admiss i b le eigenvectors for pla c ing the closed loop eigenvalues exactl y to those of the undamaged system: S

.=

N {U

HT

(A, -

)., o . I )}

(26 )

Step 2 Determine the reconfigured system eigenvector vri=Uiwi £ Si which minimizes the residual I vri-vOi I (27)

( 28 ) where Ui U is th e pseudo-in verse of Ui. Step 3 Compute the reconfiguring gain matri x Gr by us i ng the determ i ned e i ge nvecto r s Xr [v r l·v r 2.---. vr2 n J: G r= Z - . U

0 T

(P

-

A ,)

BEAM

(29) Fig. 2

Experimental tendon control system.

(30)

Table 2

Natural frequencies for beam .

NUMERICAL EXAMPLE To verify the f easibi li ty of the proposed algorithm in reconfiguring a flexible structure control system. a numerical example is given in this chapter. The examp I e is taken from the pre v ious studies of the authors. i .e .. a lab oratory experiment of tendon control system for a flexible beam (Okubo and others. 1987. 1988; Murotsu and others. 1988. 1989). Gener ic configuration of the tendon cont rol system for a canti levered flexible beam is

modes

st nd rd th th th th th

Eule r- Bernoulli Theory (wi thout 9 1 ( .... ith 9 1 0 . 14 0 . 88 2 .4 2

4 74 7 83 11 . 70 16 3 4 21 . 76

O. 4 2 1 .28 2 90 5 26 8 39 12 29 16 93 22. 35

Timoshen k o

Ex per iment

( w i th gl

0. 4 2 1. 2 8 2 . 90 5 . 26 8 . 39 12 . 28 16 . 93 22 . 36

0 _42 0.2 8 2. 9 2 5. 35 8. 55 12 . 45 1 7 . 33 22. 8 5

348

H. Okllho. K. SlIllohara alld Y. MlIrolSlI

a beam (Fig.2). A cramped-free thin stainlesssteel beam hanged in the vertical direction is used for the test structure. Dimension and properties of the flexible beu is given in Table I. Natural frequencies of the beam are computed with an FEM analysis taking account of the effect of the gravi tational force and I isted in Table 2 up to the 8th mode together with the experimentally measured values. It is assumed in the numerical simulation that three sets of tendon actuators are used for the vibration control and that each tendon actuator has a sufficiently wide bandwidth as compared to the frequency range of the beam vibrations, i.e., the dynamics of the tendon actuators are not included. This assumption is introduced for si mp I i c i t y a I t h0 ughit is 0 bs e r ve d fro m the experiment that the dynamics of the tendon actuator is of importance in the controller design (Okubo and others. 1987) . The three sets of moment arms are placed at the 30% and 70% of the beam length from the root and the beam-t i p (i. e.. I alI =0.3. 0.7.1.0). The moment arm locations are shown in Fig. 3 together with shapes of the vibration modes. Time history of the transverse displacement at the beam-tip is illustrated in Fig. 4. which shows the free vibration subsequent to an impulsive input torque appl ied at the same point. Many lightly damped vibration modes are observed in the free impulse response. A state feedback controller for the vibration suppression is

designed through the LQ (linear quadratic) regulator theory . Closed loop response of the beal to the same impulse input is shown in Fig. 5 as well as the tile histories of the control torques applied to the three actuators. The assumption is made that the actuator placed at the beam-tip is damaged and the effect of this actuator is removed from the system. Figure 6 shows the free impulse response of the damaged system. It is noted that. as can be seen in the figure. the 3rd vibration mode is nearly uncontrollable after the failure of the beam-tip actuator. Now we redesign the system to function without the failed actuator so that th~ reconfigured system may compensate for the decreased damping in the 3rd mode. The reconfiguring gains are obtained by using either of the LQ regulator algorithm or the eigenstructure assignment approach proposed in this paper. The results of the re--configuration using these two different approaches are respectively shown in Figs . 7 and 8. The recovery of the closed loop system is very satisfying for the eigenstructure assignment approach. Fig . 8. where the transient responses are very close to those of the undamaged system. On the other hand. the redesigned LQ regulator is insufficient in the damping of the 3rd mode (Fig. 7). The weights for the 3rd mode output in the LQ cos t function should have been increased for improving the redesigned closed loop dynamics. Thus the redesigning

1 st

o T

,; I

beam-tip displacement

i~~r_'__~______________________

2nd

o.

00

2.00

.,00

TIME

01 i

~or

4th

O,ou

l. n

Fig. 3 Mode shapes and actuator locations.

control input ( l Il 0.3) ____________a___

~~I--------------------~----­

'1 0.7

10. 00

e

~~I

0.3

6.00

o

3rd

'1

15.00

(SEC)

2.00

<11. 00

e. oo

8.00

10.00

o

e

~: I ,..0

Za ll

control input (

0.7)

~~~

o

:;:

~~I 0,00

__________~____ 2. 00

ci

oil.

00

e,

00

8.00

control input (

4:0

....JO

Za.ll

10. 00

1.0)

0.0 ({)

o N

o

f 0 ~--+-----------~------~--------------~ . 00 2. 00 <4 , 00 e.oo B.OO 10.00 TIME

(SEC)

Fig. 4 Tile histor¥ of beal-tip displacelent (open-loop).

2.00

oil . 00

TIME

e,

00

8. 00

10.00

(SEC)

Fig. 5 Closed loop response of undalaged s¥stel.

349

Reconfiguration of Flexible Structure Control S\'stems

o

o T

r~

beam-tip displacement

:5 ;1\\\j

~

c"

o

c

o

~0 ~-----------------------------------.00 2 . QC -4 ,00 6. 00 8.00 10,OC

~~--------------------~----------~ 0.00 2.00 04,00 8,00 8,00 10.00 T I ME

TIME

(SECI

(SECI

c o

";1 1

control input ( Z Iz

f-

a

zo WO ~

o

control input ( Za Iz

0.3)

0.3)

o·

:;;: o

m

~~----------------------------------e,oo 0.00

2.00

4.00

8.00

2.00

10. 00

".00

8.00

8.00

10.00

o G

o

control input ( Za lZ

fZo

0.7)

control input ( ZalZ

fZ c

0.7)

WO ~o

WO ~o

o

o

:;;:

:2

o o

o

m

,

~~----------------------------------00 e,oo 0 . 00

2. 00

~.

TIME

8.00

0 , 00

10.00

2.00

Fig. 6 Closed loop response of da.aged syste •.

"'. 00

TIME

(SEC)

6, 00

8.00

10.00

(SECI

Fig. 8 Closed loop response of reconf\Bured syste. (eigenstructure assignlent) .

~ 0

beam-tip displacement c..i

beam-tip displacement

<{o

C'

0.. 0

0..0

<0 -,0

-,0

Cl)

Cl)

o

0

~

~~----------------------------------0.00 2.00 04,00 6.00 8 .00 10.00 T I ME

~

0 , 00

2.00

(SECI

4 , 00

TIME

8 ,00

8.00 '

10.00

(S ECI

0 0

control input ( Z Iz a

f-

zo

0.3)

z

WO ~ 0

:;;:

:;;:

o

0.3)

0

0 0

o

m

,

~~--------------------~------------0.00 2.00 04. 00 8,00 8 . 00 10.00

0,00

o o

2.00

.... 00

8,00

8 ,OC

10.00

0 0

.:

.:

control input ( Za/ Z

f-

Z

control input ( ZaIz

0

:2 0

WO

1

f- .:

0.7)

control input ( Za lZ

f-

z

0

WO ~o

:;;:

:;;:

o

0.7)

0

WO ~ d

0

o o

0 0

,

,

0,00

2.00

04, 00

T I ME

8 . 00

8.00

10.00

(SEC)

Fig. 7 Closed loop response of reconfigured syste. (LQ regulator).

0.00

2.00

.tI.oo TIME

0,00

8.00

10 . 00

(SECI

Fig. 9 Closed loop response of reconfigured syste. (eigenstructure assign.ent with input saturation).

H. OklliJo. K. SlIllobara alld Y. \illrotslI

through the LO regulator approach sometimes needs tuning of the cost function. which is not preferable for the purpose of autonomous selfreconfiguration. Closed loop poles for the nom inal. damaged. and reconf igured systems are respectively shown in Fig. 10. It also should be noted that. although the eigenstructure assignment approach shows the satisfying result. the magnitude of the reconfiguring control inputs is I ikely to be increased especially at the initial time. Therefore. it might cause the deficiencies due to the input saturations. Figure 9 shows the result of simulation where the input torque saturations are considered. The impulse response of the reconfigured system is sti II satisfying for this case. CONCLUSIONS An eigenstructure assignment algorithm is deve loped for the fai lure accommodat i on of structura I contro I systems. Feas i b i I i ty of th i s approach in reconfiguring state feedback control is demonstrated with a numerical example. It is shown in the numerical example that the proposed algorithm Yields a satisfying closed loop dynamics which are very similar to those of the nominal system. whereas the LO regulator redesigning approach does not assure such a good recovery and is likely to give insufficient dampings for the nearly uncontrollable vibration modes . Since the theory developed in this paper is limited within the framework of the state feedback control. further studies are needed on the fai lures in the output feedback control systems and more sophisti cated gener~1 dynamic compensators. Investigations on the robust stab i I i ty of the reconf igured SyStem. taking into account the high frequency residual modes and parameter uncertainties. are also important.

References Atluri. S.N. and hos. A.K. (Eds.) (1988). Large Space Stru c tures : Dynamics and Control. Springer-Verlag. Eterno. J.S. et al. (1985). Design issues for tolerant restructurable aircraft control. NASA TP-225. Gavito. V.F. and Collins. D.J. (1987). Appl ication of eigenstructure assignment to self reconfiguring aircraft MlMO controllers . AlAA Paper 87-2235. Junkins. J.L. and Rew. 0.'11 . (1988). Unified optimization of structures and controllers. Large Spa ce Stru c tures : Dynamics and Control. Atluri. S.N. and Amos A.K. (Eds . ). Springer-Verlag . 323-353. Kautzky. J .• Nichols. N. K. Van Dooren. P. (1985). Robust pole assignment in I inear state feedback. Int. J. Control. 4..1. 1129-1155. Moore. B.C. (1976) On the flexibi I it y off e red by state feedback in multivariable s ystem s beyond closed loop eigenvalue assignment. IEEE Trans. on Automatic Control. AC- 2I. 10 . 689- 692 . Murotsu. Y. et al. ( 1988) . Dynamics and c ontrol o f e xperimental tendon control system for fl ex ible space stru c ture. AlAA Paper 88-4154. Murotsu. Y. Okubo . H. Terui. F. (1989). Low-authority control of large space structures by using a tendon control system. J.Q.w:..:. nal of Guidance. Control a nd DYnamics. If.. 2. 264-272. Okubo. H. • Murotsu. Y. • and Terui . F. (1987). Failure detection and identification i n the control of larg e s pac e s tru c tur es . 10th IFAC World Congr ess on Autom a t ic Control. Munich . Q. 172-175 . Okubo . H. et al. (1988 ) . Structural control by using active tendon control systems. 16th Int e rnational Symposium on Space Technology and Sci ence . Sapporo. Japan.

-10 . 00

REAL

-'0.00

-:0.00

0 ,00

REAL

PART

-'0. 0 0

0,00

PA RT

DAMAGED

UNDAMAGED SYSTEM

Srinathkumar. S. (1978). Eigenvalue / eigenv ec tor ass ignment using output feedback . IEEE Tr a ns . Autom a t ic Control. ~ 23 . I. 79-81. Vander Velde. W. E. (1 98 4) . Contro I syst e m reconf igur a t ion. Amer ica n Con trol Confer ence . Sa n Di ego . 1741 - 1745. Vander Velde. W. E. a nd Ca ri gna n. C.R. ( 1984) . Number and plac e ment o f control system c omponents considering poss ibl e fai lures. Journal of Guidance Control a nd Dynami cs. 1. 6. 703709 .

-1 0 . 00

REAL

- '0.00

0 . (:(1

- 10 . 0 0

PAR T

R~Al

RECONFIGURED (LQ R DESIGN)

Fig.IO

P ." RT

RECONFIGU RED (EIGENSTRUCTURE ASSI GNMENT)

Closed loop pole locations.