Model and Controller Order Reduction for Infinite Dimensional Systems

Transcription

1 IT J. Eng. Sci., Vol. 4, No.,, -6 Model and Contolle Ode Reduction fo Infinite Dimensional Systems Fatmawati,*, R. Saagih,. Riyanto 3 & Y. Soehayadi Industial and Financial Mathematics Goup fatma47@students.itb.ac.id; obed@math.itb.ac.id Analysis and Geomety Goup yudish@math.itb.ac.id 3 School of Electical Engineeing and Infomatics biyanto@lskk.ee.itb.ac.id Institut Teknologi andung, Jalan Ganesa andung, Indonesia. Abstact. This pape pesents a educed ode model poblem using ecipocal tansfomation and balanced tuncation followed by low ode contolle design of infinite dimensional systems. The class of systems consideed is that of an exponentially stable state linea systems ( A,, C ), whee opeato A has a bounded invese, and the opeato and C ae of finite-ank and bounded. We can connect the system ( A,, C ) with its ecipocal system via the solutions of the Lyapunov equations. The ealization of the ecipocal system is educed by balanced tuncation. This esult is futhe tanslated using ecipocal tansfomation as the educed-ode model fo the systems ( A,, C ). Then the low ode contolle is designed based on the educed ode model. The numeical examples ae studied using simulations of Eule-enoulli beam to show the closed-loop pefomance. Keywods: balanced tuncation; infinite-dimensional systems; low ode contolle; model eduction; ecipocal tansfomation. Intoduction The physical system descibed by patial diffeential equation has eceived much attention in ecent yeas, including design contol fo that system. Examples of these systems ae flexible stuctue, fluid dynamics, obotic, wave, and so on. In implementation, designing contol fo that system is difficult since it is of infinite state dimension. Moden contolle designs, such as H and linea quadatic Gaussian (LQG) techniques, lead to a contolle with the same state dimension as the models fom which they ae designed. Hence, it is impotant to have low ode contolle fo infinite-dimensional state space. Thee ae two possible altenatives to achieve a low ode contolle of infinite dimensional system, i.e. model and contolles' eduction. In the fist appoach, * Pemanent addess: Depatment of Mathematics, Faculty of Science and Technology, Univesitas Ailangga, Kampus C Mulyoejo Suabaya, Indonesia. Received Apil 3 th, 9, Revised Septembe 8 th, 9, Accepted fo publication Novembe th, 9

2 Fatmawati, R. Saagih,. Riyanto & Y. Soehayadi an infinite ode of model is educed and then the low ode contolle is designed. In the second appoach, an infinite ode contolle is found and then the ode of the contolle is educed. In this pape, we will focus on the fist appoach. Thee ae many ways to obtain model eduction which have been poposed ove the yeas. Existing methods, such as balanced tuncation [], optimal Hankel nom appoximation, singula petubation appoximation (SPA) [] ae often used to educe model ode of finite dimensional linea time-invaiant (FDLTI) systems. It is known that the balanced tuncation method gives a good model appoximation at high fequency, while SPA method is known to be supeio at low fequency. Using popeties of Hankel opeato, seveal eseaches have extended the balanced tuncation method [3,4,5] to educe the model ode of infinite dimensional case. The ecent pape [6] discussed a computational appoach of the balanced tuncation method fo infinite dimensional system via pope othogonal decomposition (POD). alanced tuncations method consists of tuncating a balanced ealization. The balanced ealization of infinite dimensional systems is the ealization fo which the contollability and obsevability gamians ae both equal to some diagonal opeato [3]. Suppose G( s) = C( si A) D is a tansfe function of infinite dimensional systems with finite ank input and output opeato. The -th ode balanced tuncation of G is the finite dimensional system with the tansfe function G ( s) = C ( si A ) D. If the ealization of G is exponentially stable then D = lims G( s) exists (see [7, Lemma 7.3., p.364]). Since theoy of balanced ealizations and tuncations exists only fo exponentially stable systems, then G( ) = G ( ) = D. It is clea that all educed-ode models obtained by tuncation of infinite dimensional systems have pefect matching at infinite fequency. Unfotunately, this method yields the geatest appoximation eo at the low fequency which is undesiable in many applications. It is vey inteesting to impove model eduction method fo infinite dimensional systems with compaable popeties as SPA eduction method. It is known that SPA method fo FDLTI systems can be analyzed by using popeties of a ecipocal system. In the ecent papes [8,9], ecipocal tansfomation has been applied to study popeties of infinite dimensional system via thei ecipocal systems. Motivated by these woks, we popose a method to designing low ode contolle of infinite dimensional systems via educed-ode model using balanced tuncation and ecipocal tansfomation. The pape is oganized as follows. Section intoduces the abstact fom of infinite dimensional systems and some notation used in the pape. Reduced ode model via balanced tuncation of infinite dimensional systems ae eviewed in Section 3. Reduced ode model using ecipocal tansfomation and balanced tuncation of infinite dimensional systems ae deived in Section 4

3 Infinite Dimensional Systems 3 which constitutes the main esult of this pape. Numeical computation fo obtaining a low ode contolle of infinite dimensional system is poposed in Section 5. The simulation esults ae pesented in Section 6. Finally, concluding emak ae given in Section 7. Peliminaies The infinite dimensional systems dynamics can be pesented in the following abstact fom: x( t) = Ax( t) u( t) y( t) Cx( t) Du( t) whee A is the infinitesimal geneato of C -semigoup T () t on Hilbet space a m Z, and opeatos and C ae finite ank and bounded; L( C, Z), k C L( Z, C ). Hee L( U, Y ) is the space of bounded linea opeato mapping fom U to Y. In this pape, we will assume the system () is exponentially stable means that the opeato A geneates the exponentially stable C - semigoup T () t on Z. The C -semigoup T () t on Z is exponentially stable a a [7] if thee exist positive constants M and such that T () t t. a () t Me, fo all We shall denote the state linea system given by () as ( A,, C, D ) and tansfe function given by G, with ealization G( s) = C( si A) D. We omit the * " D " tem if it is not elevant. The adjoint of opeato A is witten as A and domain of A is denoted by D ( A). A symmetic opeato A is self-adjoint if * D( A ) = D ( A). A self-adjoint opeato A on the Hilbet space Z with its inne poduct, is nonnegative if Az, z fo all zd ( A) and positive if Az, z > fo all nonzeo zd ( A). We will use the notation A fo nonnegativity of the self-adjoint opeato A, and A > fo positivity. The esolvent set of A is the set of all complex numbes fo which ( I A) exists and bounded linea opeato on Z. 3 Reduced Ode Model via alanced Tuncation is In this section, we eview some esult on balanced ealization and tuncation of infinite dimensional system which has been constucted by [3,4]. The constuction of the balanced tuncation of system ( A,, C ) can be obtained by eliminating any states that ae difficult to contol and to obseve based upon the contollability and obsevability gamians. The contollability and obsevability gamians ae defined by

4 4 Fatmawati, R. Saagih,. Riyanto & Y. Soehayadi * * * * = ( ) ( ), = ( ) ( ). a a C a a L T t T t dt L T t C CT t dt Futhemoe, the contollability gamian L and the obsevability gamian L C of an exponentially stable system ( A,, C ) ae the unique solutions of thei espective Lyapunov equations: AL z L A z z z A * * * =, D ( ) () A L z L Az C Cz =, z D ( A). (3) * * C C The pocedue of balanced ealization of infinite dimensional system is based on the singula value and Schmidt vectos of the Hankel opeato of the system. Conside tansfe function G is the Laplace tansfom of (, ; mk h L L C ), with ealization m k opeato : L (, ; C ) L (, ; C ) defined by h ( )( t) = h( t s) u( s)d s, h( t) = CT ( t). h G( s) = C( si A) a. The Hankel Since (, ; mk hl C ) implies that is a compact [7, Lemma 8..4, p. 399] h and so has countable many singula values (squae oots of the eigenvalues h of ) and these ae also the Hankel singula * h h values of G. The singula values of ae also equal to squae oots of the h eigenvalues of LL. The Hankel singula values ae ealization invaiant; C i i.e., they compise a popety of tansfe function only and ae independent of the ealization. The balanced ealization has the contollability and obsevability gamians both equal to =diag (,,...,,...). Futhemoe, the tansfe function G( s) = C( si A) has a balanced ealization ( A,, C ) = ( A,, C ) on the state, the space of squae summable sequences [3]. In [], it is shown that an exponentially stable infinite dimensional system with finite ank input and output opeatos has a nuclea Hankel opeato; i.e., its singula values satisfy <. (4) i i= Using the infinite matix epesentation fo opeatos on, the balanced system ( A,, C ) can be patitioned confomably with L = L = = diag(, ), = diag(,..., ) C, = diag(,,...) > as

5 Infinite Dimensional Systems 5 A A A A A,., C C C The -th ode balanced tuncation of ( A,, C ) ae given by the finite dimensional system ( A,, C ). The coesponding educed ode tansfe function is G. Note that the educed system ( A,, C ) is stable and balanced with gamian. If system ( A,, C ) has output nomal ealization which is constucted in [4], whee L = and L = I (the identity opeato), then the tuncation eo can be epesented as C G( s) G ( s). (5) i i= y (4), the tansfe function of the balanced tuncations convege in the H nom. 4 Reduced Ode Model Using Recipocal Tansfomation In this section, we develop educed ode model of infinite dimensional systems via ecipocal tansfomation. The concept of a ecipocal systems of infinite dimensional systems ( A,, C, D ) was intoduced in papes [8, 9]. If zeo is in the esolvent set of A, i.e. opeato A has a bounded invese, then all the geneating opeato of the ecipocal systems A, A, CA ae bounded. This motivates the following definition. Definition Let the system ( A,, C, D ) is exponentially stable, such that A has a bounded invese. Its ecipocal system is the system ( Aˆ, ˆ whee ˆ ˆ ˆ ˆ A = A, = A, C = CA, D = D CA = G(). (6) Let G be tansfe function of the systems ( A,, C, D ) and suppose Ĝ is tansfe function of the systems ( Aˆ, ˆ. Fom the above definition, we have ˆ ˆ ˆ ˆ ˆ ( ) = = = ( ). G s CA I A A CA D C I A D G s s s (7) Let H ( L( U,Y )) denote the space of L( U, Y) -valued functions of complex vaiable defined in the open ight half-plane which ae bounded and analytic in

6 6 Fatmawati, R. Saagih,. Riyanto & Y. Soehayadi the open ight half-plane, and G H ( L( U,Y )). Fom (7), it is clea that G H ( L( U,Y )) if only if G ˆ H ( L( U,Y)). Lemma If the system ( A,, C, D ) is exponentially stable with its ecipocal system ( Aˆ, ˆ, then thei contollability and obsevability gamians ae equal. Poof. In Lemma 4..4 of [7] it was shown that the contollability and obsevability gamians, L, L, of the exponentially stable ( A,, C, D ) ae the C unique positive definite solutions to the Lyapunov equations * * *,( ) =, ( ) z AL L A zz D A (8) z,( A L L A C C) z =, z D ( A). (9) * * C C Letting * z = ( A ) x elements in Z gives in (8) and z = A x in (9) whee x, x is abitay x,( L ( A ) A L A ( A ) ) x =, * * * x,( L A ( A ) L ( A ) C CA ) x =. * * * C C Theefoe, fom equation (6) we obtain following equations x,( AL ˆ L Aˆ ˆˆ ) x =, () * * x,( Aˆ L L Aˆ Cˆ Cˆ ) x =. () * * C C This shows that the contollability and obsevability Lyapunov equations of the systems ( A,, C, D ) and its ecipocal systems ( Aˆ, ˆ have the same solutions. Fom the above Lemma, the systems ( A,, C, D ) and its ecipocal systems ( Aˆ, ˆ can be connected via the solutions of the Lyapunov equtions. We can apply it to obtain a balanced ealization of ( Aˆ, ˆ, Dˆ) on. It is easy to see that if the ealization ( A,, C, D) = ( A,, C, D) is balanced with gamian, then ( Aˆ, ˆ, Dˆ ) = ( Aˆ, ˆ, Dˆ ) is also balanced with the same gamian. We continue the balanced tuncation method to poduce a low ode model of the balanced ealization ( Aˆ, ˆ. We will use the infinite matix epesentation fo the opeatos with espect to the standad

7 Infinite Dimensional Systems 7 othonomal basis on. Choose a positive intege such that > and patition the systems ( Aˆ, ˆ compatibly with the gamian = diag(, ) whee diag diag become = (,,..., ), = (,,...) Aˆ Aˆ Aˆ Aˆ Aˆ, ˆ ˆ ˆ ˆ ˆ ˆ, C C C In this pape, we assume that the, i =,,..., ae all distinct. Since we i choose >, then we have >. The -th ode balanced tuncation of the systems ( Aˆ, ˆ ae given by the finite dimensional systems ( Aˆ, ˆ. Denote its tansfe function by ˆ G. Theoem The tuncated system ( Aˆ, ˆ is balanced with gamian and asymptotically stable. Poof. The Lyapunov equations have solutions which coespond to the contollability and obsevability gamians of the system ( Aˆ, ˆ via () and (), we have Aˆ Aˆ ˆˆ =, () * * A ˆ A ˆ C ˆ C ˆ =. (3) * * Theefoe, ( Aˆ, ˆ is balanced with gamian. To pove stability, assume that of A ~ˆ v = v, with be an abitay eigenvalues A ~ˆ. Since > and system A ~ˆ, ~ˆ, C ~ˆ, ) is finite dimension, it follows ( Dˆ that ( Aˆ, C ˆ ) is obsevable, then Cv ˆ. Using the obsevation Lyapunov equation (3) we have ( ) = Cˆ v. * / v

8 8 Fatmawati, R. Saagih,. Riyanto & Y. Soehayadi * Hence we obtain ( ) = Re( ). If Re( ) = then Cv ˆ, which is a contadiction. We conclude that Re( ) <, i.e., the tuncated system ( Aˆ, ˆ is asymptotically stable. The following esult shows that a low ode model using a ecipocal tansfomation is a good candidate fo a educed ode model fo systems ( A,, C, D ) and hence fo G. Now define the ecipocal system ( A,, C, D ) of the system ( Aˆ, ˆ by A = Aˆ, = Aˆ ˆ, C = Cˆ Aˆ, D = Dˆ Cˆ Aˆ ˆ, (4) assuming that  is invetible. Note that G ˆ is tansfe function of ( Aˆ, ˆ with ealization ˆ ˆ ˆ G ( ) = C I A ˆ Dˆ. s s Let G be tansfe function of the systems ( A,, C, D ) with ealization G ( s) = C ( si A ) D. Fom (4), the elationship between the tansfe function G ˆ and the tansfe function G, is given by ˆ ˆ ˆ ˆ ˆ ˆ G ( ) = D C A C I Aˆ Aˆ ˆ s s = Dˆ Cˆ Aˆ ˆ Cˆ Aˆ ( si Aˆ ) Aˆ ˆ (5) = G ( s ). The stability and balanced ealization of the systems ( A,, C, D ) ae given in the following theoem. Theoem If the system ( Aˆ, ˆ is asymptotically stable and balanced with gamian so is the ecipocal system ( A,, C, D ) with the same gamian.

9 Infinite Dimensional Systems 9 Poof. Since the system ( Aˆ, ˆ is balanced with gamian, then satisfy the two Lyapunov equations () and (3). Multiplying () fom the left * by and fom the ight by ( A ˆ ), and then multiplying (3) fom the left by  ( ˆ ) * A and fom the ight by Â, yields ( Aˆ ) Aˆ ( Aˆ ˆ )( Aˆ ˆ ) =, (6) * * A ˆ ( A ˆ ) ( C ˆ A ˆ ) ( C ˆ A ˆ ) =. (7) * * Substituting (4) to (6) and (7), we have A A (8) * * =, A * A C * C =, (9) which implies that the ecipocal system ( A,, C, D ) is balanced with gamian. Since  is asymptotically stable, then Re( ) <, with be any eigenvalue of Â. Then the eigenvalue of A is since A = Aˆ. We hence conclude that Re( ) <, i.e., the ecipocal system ( A,, C, D ) is asymptotically stable. The next theoem gives the eo bounds of educed ode model and convegence in the H nom. Theoem 3 Assume that G is the tansfe function of the -th educed ode model of the balanced and output nomal ealizations of G. Then we have G( s) G ( s). () i i= In paticula G( s) G ( s) as. Poof. We have G s G s G s Gˆ Gˆ Gˆ Gˆ G s ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). s s s s Fom equations (7) and (5) we have ( ) = ˆ G s G ( ) and ˆ ( ) = G G ( s ), such that we obtain s s

10 Fatmawati, R. Saagih,. Riyanto & Y. Soehayadi G s G s Gˆ Gˆ ( ) ( ) ( ) ( ). s s Fom Lemma, if ealization of G have output nomal ealization, then ealization of Ĝ also have output nomal ealization. Meanwhile, G ˆ is the tansfe function of the -th ode tuncations of the balanced and output nomal ealizations of Ĝ. Theefoe, fom equation (5), we conclude that y (4), as. ˆ ˆ ( ) ( ) ( ) ( ). s s i i= G s G s G G () < i= i, and so () would imply that ( ) ( ) An impotant popety of this appoach is that the tuncated model to G at zeo fequency, since G () = C A D ˆ ˆ ˆ = ( Cˆ Aˆ ) Aˆ ( Aˆ ) D Cˆ Aˆ G s G s, G is equal = Cˆ Aˆ ˆ G() Cˆ Aˆ ˆ () = G (), with assumption that zeo is in the esolvent set of A. It is clea fom the discussion above that the educed ode model using ecipocal tansfomation of infinite dimensional systems gives cetain accuacy at low fequency. 5 Numeical Algoithm This section addesses the question of computing the balanced ealization and ecipocal tansfomation followed by low ode contolle design of infinite dimensional systems based on the poposed method. In the pevious section it was assumed that the opeato-valued Lyapunov equations can be deived analytically. In geneal, we cannot obtain an exact solution of the infinite dimensional Lyapunov equations. In this case, a convegent numeical scheme must be utilized as in [5, ]. The following algoithm is poposed to find the educed ode model descibed in the pevious section: n n n n. Find appoximating sequence ( A,, C, D ) to ( A,, C, D ), whee n is the ode of the appoximation scheme.

11 Infinite Dimensional Systems. Compute the numeical appoximations n L and n L to the Lyapunov C n n n n equations coesponding to ( A,, C, D ) fo sufficiently lage n. n n n n 3. Obtain the balanced ealization ( A,, C, D ) of the systems n n n n ( A,, C, D ) using a similaity tansfomation. 4. Find ecipocal tansfomation ˆn ˆn (,, ˆn, ˆ n A C D ) of the systems n n n n ( A,, C, D ) as defined in (6). 5. Apply the balanced tuncation method to detemine the -th ode systems ˆ ˆ ˆ (,,, ˆ A C D ) with as small as possible. 6. Fom the educed ode model ˆ ˆ ˆ (,,, ˆ A C D ), find its ecipocal tansfomation ( A,, C, D ) as defined in (4). 7. Then ( A,, C, D ) is the th-ode educed ode model fo n n n n ( A,, C, D ) and so fo ( A,, C, D ). 8. Constuct a contolle fo ( A,, C, D ). 6 Simulation Result In this section, we apply the educed ode model esult to design a low ode contolle fo Eule-enoulli beam clamped at one end ( = ) and fee to vibate at the othe end ( = l). Let w(, t) epesents the deflection of the beam at time t and position. The deflection can be contolled by applying a toque u (t) at the clamped end ( = ). The extenal distubance v (t) induces a unifomly distibuted load dv() t. Using linea viscous damping ( ) and Kelvin-Voigt damping ( ), the patial diffeential equation of the beam [] is given by w ( t, ) w ( t, ) EI w ( t, ) I w ( t, ) = u( t) dv( t), (3) tt t b b t I with is density of the beam, E is the Young modulus, I b is the moment of inetia, and I h is the hub inetia. The bounday conditions ae w( t,) = w ( t,) =, b b t EI w ( t, l) I w ( t, l) =, (4) EI w ( t, l) I w ( t, l) =, b b t fo t > and < < l. h

12 Fatmawati, R. Saagih,. Riyanto & Y. Soehayadi The values of the physical paametes in this example ae taken as follows: Table Physical constants. E. N / m 4 I b.67 m.975 kg / m I h d. Ns / m. Ns / m.9748kgm.4/ kg Let H be the closed linea subspace of the Sobolev space H w H w w = { (,) : () = () = } H (,) and define the state-space Z = H L (,) with state x( t) = ( w( t, ), w ( t, )). If the tip deflection of velocity ae measued, a state-space fomulation of the above patial diffeential equation can be pesented in the following abstact fom whee with domain D x( t) = Ax( t) v( t) u( t) (5) y( t) = Cx( t ), I A = E I 4 4, =, =, b d I b d 4 4 d d d Ih t Cx( t) = w ( t, l ), ( A) = {(, ) Z : H and M = EI I H (,) b b with M ( l) = M ( l) = }. In ode to fomulate a full ode appoximation, we use a Galekin finite element method with cubic spline []. Fo the finite element method, the spatial inteval is divided into 3 elements, with the length of beam to be l = 7. This coesponds to full ode model of size n = 4. The appoximating system (5) is given by x ( t) = A x ( t) v ( t) u ( t) (6) n n n n n n n t

13 Infinite Dimensional Systems 3 n n n y ( t) = C x ( t ). The pupose of the contolle design is to educe the vibation effects of the distubance acting on the beam. Such an objective can be achieved by solving a weighted mixed sensitivity H contol accoding to [3]. The block diagam of the mixed sensitivity poblem with the inclusion of the weighting function is shown in Figue. The pefomance specification ae obtained by designing contolle Ks () which satisfy W s T s W s S s <, whee Ts () is the tansfe function fom the distubance v to contol input u, Ss () is tansfe function fom v to the output y and W and W ae the two weighting functions to be selected. We take a high-pass filte W and a low-pass filte W, espectively, as 94s 4.56e4s.4 W ( s) =, W = w w w, s.334s9.9 whee w = s a with a = 3. v G(s) W (s) z u + X X Y I / s C W (s) + + z A K(s) Figue lock diagam of H contol. Futhemoe, the ode of model is educed accoding to the algoithm in section 5. ased on the educed ode model, the 6th-ode contolles of the lage-scale finite dimensional systems ae designed. The time esponses of the open-loop and closed-loop system with a tempoay step distubance vt ( ) = and a peiodic distubance v( t) = sin( t) ae given in Figue and 3, espectively.

14 Amplitude Amplitude 4 Fatmawati, R. Saagih,. Riyanto & Y. Soehayadi Fom these figue, we obseve that the 6th-ode contolles can attenuate these distubance. 5 output step distubance no contolle 6th-ode contolle time (sec) 5 output step distubance no contolle 6th-ode contolle time (sec) Figue Time esponse of open and closed-loop to v(t) =.

15 Amplitude Amplitude Infinite Dimensional Systems 5 4 output peiodic distubance no contolle 6th-ode contolle time (sec) 4 output peiodic distubance no contolle 6th-ode contolle time (sec) Figue 3 Time esponse of open and closed-loop to v(t)= sin t. 7 Concluding Remak In this pape, we pesented a low ode contolle design based on educed ode model via ecipocal tansfomation of infinite dimensional systems. The ealization of the ecipocal system is educed by balanced tuncation. This esult was futhe tanslated using ecipocal tansfomation as the educed ode model fo the oiginal system. The poposed method pesented in this pape, peseves the popety of singula petubation appoximation (SPA)

16 6 Fatmawati, R. Saagih,. Riyanto & Y. Soehayadi method of finite dimensional system, i.e. its esult povide a good eduction eos at low fequencies and theefoe it extends [] fo infinite dimensional system. The contolle was designed fo the educed ode model with the low ode contolle. The low ode contolle is applied to eject the distubance acting on the beam. The simulations showed the effectiveness of the 6 th-ode contolle to educe the vibation of the beam. Refeences [] Mooe,., Pincipal Component Analysis in Linea Systems: Contollability, Obsevability, and Model Reduction, IEEE Tans. Automat. Contol, AC-6, 7 3, 98. [] Liu, Y. & Andeson, D. O., Singula Petubation Appoximation of alanced Systems, Int. J. Contol, 5, , 989. [3] Cutain, R.F. & Glove, K., alanced Realisation fo Infinite- Dimensional Systems, in Opeato Theoy and Systems: Poc. Wokshop Amstedam, 87 4, 986. [4] Glove, K., Cutain, R.F., & Patington, J.R., Realisation and Appoximation of Linea Infinite-Dimensional Systems with Eo ounds, SIAM J. Contol and Optim. 6, , 989. [5] Cutain, R.F., Model Reduction fo Contol Design fo Distibuted Paamete Systems, in Reseach Diection in Distibuted Paamete Systems, SIAM, Philadelphia, PA, 95, 3. [6] Single, J.R., & atten,.a., alanced Pope Othogonal Decomposition fo Model Reduction of Infinite Dimensional Linea Systems, Poceedings of the Int. Conf. on Computational and Mathematical Methods in Science and Engineeing, CMMSE, 7. [7] Cutain, R.F., & Zwat, H.J., An Intoduction to Infinite-Dimensional Systems, Spinge-Velag, New Yok, 995. [8] Cutain, R.F., Regula Linea Systems and Thei Recipocals: Applications to Riccati Equations, Systems Contol Lett, 49, 8 89, 3. [9] Cutain, R.F., Recipocals Regula Linea Systems: A Suvey, Poceeding MTNS,. [] Cutain, R.F. & Sasane, A.J., Compactness and Nucleaity of the Hankel Opeato and Intenal Stability of Infinite-Dimensional State Linea Systems, Int. Jounal of Contol, 74, 6 7,. [] Ito, K. & Mois, K.A., An Appoximation Theoy of Solutions to Opeato Riccati Equations fo H contol, SIAM J. Contol and Optim. 36, 8 99, 998. [] Reddy, J.N., An Intoduction to The Finite Element Method, McGaw- Hill, Singapoe, 984. [3] Ka, I.N., Miyakua, T. & Seto, K., ending and Tosional Vibation Contol of A Flexible Plate Stuctue using H -ased Robust Contol Law, IEEE Tans. on Contol Systems Technology, 8, ,.