JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: VOL. 59, No. 2, NOVEMBER 1988

Dynamically Similar Control Systems and a Globally Minimum Gain Control Technique: IMSC I H. Oz 2

Communicated by L. Meirovitch Abstract. The concept of dynamically similar control systems is introduced. The necessary and sufficient conditions to minimize a quadratic modal gain measure are given for dynamically similar closed-loop control systems. The globally minimum modal gain is obtained when the independent modal space control (IMSC) is used. Corollaries of the results for the control of infinite-dimensional structural distributed parameter systems (DPS) are given. Based on the results, a modal interaction parameter (MIP) is defined for all control systems. The minimum value of MIP is zero and uniquely corresponds to the IMSC. A nonzero value of MIP corresponds to all other coupled control (CC) designs and implies suboptimality relative to the IMSC design. The relative optimality of the real-space gain matrices of the IMSC and the CC designs depends on the actuator locations for the IMSC. Based on this, a real-space interaction parameter (RIP) is defined. A positive value of RIP renders IMSC optimal in its real-space gain matrix. The MIP and RIP are indications of suboptimality of a particular control technique and can be used to tune-up the control design via actuator locations. Actuator distribution criteria are suggested for both CC and IMSC designs, based on the values of MIP and RIP, respectively. Key Words. Distributed parameter system control, modal control, control gain optimization. 1. Introduction M e t h o d s o f active c o n t r o l o f d y n a m i c systems can be classified as c o u p l e d c o n t r o l s ( C C ) a n d i n d e p e n d e n t m o d a l s p a c e control ( I M S C ) ; see Ref. 1-10. C o u p l e d c o n t r o l s are t h o s e w h i c h i n t r o d u c e external c o u p l i n g o f the m o d e s via f e e d b a c k c o n t r o l inputs. In the I M S C t e c h n i q u e , m o d a l r e s p o n s e s o f the c o n t r o l l e d system are i n d e p e n d e n t o f e a c h other, i.e., the t This work was supported by the National Science Foundation, Grant No. MEA-82-04920. 2 Associate Professor, Department of Aeronautical and Astronautical Engineering, Ohio State University, Columbus, Ohio. 183 0022-3239/88/1100-0183506.00/0 © 1988 Plenum Publishing Corporation

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modal control gain matrix is block diagonal so that no external modal coupling occurs due to feedback control inputs. A comparison of the two approaches to the control problem was done recently in Ref. 2. Reference 2 compares the techniques from a design computational point of view as well as through the work done and the quadratic performance measure for particular choices of control weighting matrices. Even though it is not explicitly stated in the paper, Ref. 11 can also be regarded as a comparison of the IMSC and CC techniques. In this paper, the domain of comparison, or, rather, the optimality of control techniques is expanded to a new horizon via the concept of dynamically similar control systems. Dynamically similar control systems are defined to be the set of all different control techniques which result in the same closed-loop eigenvalue locations for the same dynamic system. Ultimately, the comparison of control techniques reduces to a comparison of coupled controls and independent modal space control. For this reason, the norms of the control gain matrices of the CC and IMSC techniques are compared in this paper. The norm of a control gain matrix is measured as the sum of the Euclidean norms squared of its rows (columns). It is proven in this paper that, in this sense, among all admissible control design techniques, the one that satisfies a number of orthogonality conditions meets the requirements for minimization of the norm of modal control gain matrix. It is also proven that the IMSC technique is the only design technique that satisfies all of the orthogonality conditions, and hence its modal gain matrix is the global minimum in the sense mentioned above. Determination of optimality among the set of dynamically similar control designs represents a broader perspective which encompasses and accommodates other definitions of optimum performance. The measures of optimality discussed do not stand in the way of using other appropriate (problem-dependent) definitions of performance criteria. The reason is that, regardless of what physical significance a particular optimal performance measure and its optimal control law may have, it is always possible to find its dynamically similar IMSC design and reassess the optimality of that particular performance measure in terms of the norm of the control gain it requires, in comparison to the globally minimum norm gain of IMSC. The amount of control gains required by the two control approaches to locate the eigenvalues of the closed-loop system at the same positions has practical significance from the control power, response, and model reduction points of view (Refs. 12-15). More recent work (Refs. 12 and 13) shows that the globally optimal modal gain control technique is also the one with a globally optimal quadratic performance measure. This globally optimal performance measure, which uses globally minimum norm gain, controls an infinite-dimensional distributed parameter system (DPS), whatever the control objectives are, by using a continuously distributed input

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profile. All other dynamically similar control designs which use spatially discrete inputs on a reduced-order model of the DPS are suboptimal with respect to the perfbrmance of the globally minimum norm gain control (Ref. 15). The globally optimal performance of the globally minimum gain control technique is coordinate-frame indifferent; that is, regardless of whether it is expressed in terms of modal coordinates or physical coordinates, it has the same scalar value for the complete DPS dynamics (Refs. 12, 13, 15). This feature demonstrates that modal performance measures can be meaningful physically and provide even further insight to accomplish the control task, if they are defined properly for a DPS system (Ref. t5). This paper presents results from the perspective of required control gains to control a DPS and does not concentrate on quadratic performance integrals. However, with the provision of the above background as to how the measures of control gains and performances are mutually dependent, it is hoped that the reader will view the results of this paper as more significant and far-reaching in controlling a distributed parameter system than might be expected initially (Ref. 15). Because the globally optimal IMSC requires the use of a spatially continuous input profile, it is difficult to implement. In practice, an approximate version of IMSC implemented by point inputs is preferred. The most widely known such version uses as many point inputs as the number of controlled modes. In this paper "real-space gain matrices of the IMSC" and "actuator locations for the IMSC" will refer only to this most commonly known implemented IMSC; no other version of IMSC is considered herein. Based on the proofs given in the paper, we define a modal interaction parameter (MIP) for a controlled system due to external coupling because of feedback inputs. The minimum value of the parameter is zero, which implies no modal interaction and hence that the control technique is the IMSC. Another significance of the results is that no actuator locations and number exist for coupled control techniques that can yield a lower measure of the modal control gains for CC than that of the IMSC. This fact is alluded to in the global minimum characterization of the IMSC. It is proposed that an actuator location criterion, as is often sought by CC designs (Ref. 11), can be taken to be the modal interaction parameter (MIP) defined in this paper. The MIP can never be made to vanish by coupled controls, however, its value can be lowered by changing the actuator locations. On the other hand, the relative optimality of the norms of real-space gain matrices of the IMSC and CC designs depends on the actuator locations for the IMSC. To this effect, a real-space interaction parameter (RIP) is defined. A positive value for RIP implies an optimal IMSC design and a negative value for RIP implies a suboptimal IMSC design relative to the CC design. Accordingly, an actuator location criterion is given for the IMSC

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in that actuator configurations which render RIP positive are favored. The values of MIP and RIP can be considered to be the degree of inefficiency of a control technique. Numerical examples are given to demonstrate the minimality of the IMSC modal gains relative to those of CC for dynamically similar systems, and MIPs are computed. It is demonstrated that modal gains of a CC technique, based on optimization of a quadratic performance measure in the classical sense, are suboptimal relative to the modal gains of IMSC, based on the concepts advanced in this paper. Similarly, the values of the RIPs are computed, and the effects of IMSC actuator locations are shown. No numerical example for actuator locations for CC is provided, as it is not the mainstay of this work.

2. Two Fundamentally Different Approaches to the Control Problem The spatially discretized modal state-space equations of a linear DPS can be written in the form ~=Aw+

W,

(la)

in which w = ( ~ , n l , . • . , ~,n°) T

is the modal state vector and W ( t ) is the modal input vector; 2n denotes the number of controlled modes in the state space. For an undamped system, the open-loop dynamic matrix is given by A=blockdiagAr,

At=

[0] -wr

O)r

0

'

r = l , 2 . . . . ,n,

(lb)

in which tot are the natural frequencies of the system. Equations (1) may follow from either an exact solution or an approximate solution to any desired accuracy of the eigenvalue problem of the DPS. Denoting a set of exact or highly accurate eigenfimctions by ~br(P) and the spatially discretized real input vector by F ( t ) , one has W = BF, (2a) in which B is the 2n x m input distribution matrix whose elements are the eigenfunctions evaluated at corresponding locations of m spatially discrete real inputs. In this paper, we restrict our results to nongyroscopic systems. However, the results, with some modifications, can be extended to gyroscopic systems as well. For nongyroscopic systems, the odd components of W and the odd rows of B must be zero identically. Thus, the general form of the B matrix is B = [ 0 , b,,0, b 2 , . . . , 0 , b,,] "r,

br

=

OJr'['Yrl,

"Yr2, • • " , ~brn] T

~brj= 4'r(P/),

(2b)

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in which ~ is the position of point actuator j, 1, 2 , . . . , m. There are two distinct approaches to the state feedback control of systems: in one of the methods the real space input F is designed first as a linear state feedback of the form of Eq. (3a); in the other method the modal space input W is designed first as a linear state feedback of the form of Eq. (3b): Fc =GRcw, (3a) w , = G ,Mw. (3b) The superscripts R and M on the gain matrices G denote that the transformations are from the modal state to real-space input and from the modal state to modal-space input, respectively. Thus, the former method determines G~ first, and the latter determines G , first. For the first method, the modal-state to modal-input gain matrix can be obtained subsequently as in Eq. (4a); for the second method, the modal state to real input gain matrix can be obtained as in Eq. (4b), in which ? implies a generalized inversion of B, and Eq. (4b) essentially represents a modal synthesis: G~ BG~, (4a) R ~" M G. = B,G. . (4b) The first approach, characterized by Eqs. (3a) and (4a), yields a fully populated modal gain matrix GcM, thereby coupling the modal responses of the feedback controlled system. Therefore, this technique is referred to as coupled control (CC) technique (Ref. 2), which should explain the subscript C in Eqs. (3) and (4). The second approach, characterized by Eqs. (3b) and (4b), is known as independent modal space control (IMSC, Refs. 1-10), in which the matrix G~ is chosen in the form =

G~=blockdiag G,M,

Mr0 01

G,~=

&r,2r-1 g2.... J '

r = 1 , 2 , . . . , n, (5) thereby insuring the independence of the modal responses from each other. Quantities pertaining to the IMSC technique carry an asterisk as a subscript. The particular form of the G ~ matrix yields superior computational advantages for IMSC (Ref. 2). In case of linear state feedback, the closed-loop dynamics becomes (6) rOc = A w c + BcG~wc = (A+ Gr~)Wc = AcL,cWc ~ , = Aw,+G,Mw, =(A+Gr~)w,=AcL.w,. (7) for CC and IMSC, respectively. Here, CL refers to the closed loop dynamic matrix. The corresponding real-space input and modal-space input are Fc = G~wc, Wc = BcO~wc = O~wc, (8a) Fg~ t M R if,,= G ,Mw,. = B,G, w,- G,w,, (8b)

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Because of the nature of the control technique, ACL,C is fully populated; therefore, in essence, CC unnaturalizes the modal coordinates w which are naturally uncoupled pairs of conjugate modal coordinates ~:r, ~Tr in the uncontrolled form. In contrast, ACL.* is a 2 × 2 block diagonal matrix, which preserves the naturality of each pair ~:r, */r by retaining them uncoupled from each other, even in the controlled form. From the above discussion, it is clear that the IMSC and CC techniques essentially differ in how their respective modal gains G M are generated. In the sequel, absence of subscripts C o r , on any quantity should be interpreted as that the quantity is discussed with regard to both IMSC and CC techniques. For a desired set of closed-loop eigenvalues, G M is such that its elements satisfy the 2n-invariants of the closed-loop matrix AcL = A + GM; with ICLk=~lil,i2

....

k= 1,2,...,2n,

,ikl=Vk,

il < i2 < . . • < i2n,

(9)

Here, ICLk is the kth-invariant operation of the closed-loop matrix ACL; [il, i2,. • •, ik[ is the determinant of the matrix formed from rows and columns i l , i 2 , . • • , ik ; and Vk denotes the value of the invariant. For a specific set of closed-loop eigenvalues, the VkS can be computed readily in terms of the desired eigenvalues. In particular, we note that the 1st and 2nth invariants are 2n

ICL1 = -TrAcL =

-TrG

M = - ~

p~ = v l ,

i=1 2n

ICL2n = IA-~- GM[ = l~ pi = 1J2n,

/Jl ~> 0 ,

b'2n]>0 ,

i=1

in which p~ is the ith eigenvalue of the closed-loop system and Tr denotes the trace of the matrix. The signs of v~ and v=, follow from the assumption that the closed-loop system is at least critically stable with no eigenvalues at the origin and from the fact that, for physical systems, complex eigenvalues must occur in conjugate pairs. Theoretically speaking, the control problem via pole assignment is to determine the elements of the G M matrix either directly or indirectly such that all invariants of the closed-loop system satisfy the 2n Eqs. (9). For the 2nth order formulation in coordinates w, G M will have 4n 2 elements to be determined from the 2n Eqs. (9). For nongyroscopic systems, the formulation dictates that all odd rows of G ~ be zero, Gu=0 , i-- 1 , 3 , 5 , . . . , 2 n - 1 , j = 1,2, 3 , 4 , . . . , 2 n . (10) Thus, one has only 2n 2 unknown elements, even rows of GM; to determine.

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Therefore, the general control design is undetermined; there is no unique solution for a desired closed-loop eigenvalue spectrum. The argument given above assumes a direct determination of the G M matrix. However, the determination of G M can also be indirect in terms of another gain matrix G, such that G M= CG, in which C plays the role of a 2n x m constraint matrix and the m x 2n elements of the G matrix are the unknown gains. If m = 1 (e.g., if the modal feedback Gw represents a single input), then there are only 2n elements of G and one is able to determine the elements of G from Eqs. (9) uniquely for any given constraint matrix C. Changing the matrix C will yield a different solution for G. Clearly, if m > 1, regardless of the choice of C, the equations will be underdetermined, and the system of Eq. (9) must be augmented to determine G. Depending on whether a direct or an indirect approach is used, the modal gain matrix G M will have distinct characteristics. All indirect approaches in the form

GM=c2,×,,G,~×2~,

m
will be rank-deficient designs with regard to the rank of G M. We define the full-rank design to be the one that generates G M with rank n. On the other hand, inspection of Eq. (5) reveals that the IMSC will generate a full-rank gain matrix G M. In view of this, with the proper understanding of what is meant, one may safely refer to the IMSC and CC techniques as the full-rank [i.e., R(GM)=n] and the rank-deficient [i.e., R ( G M ) = m
3. Dynamically Similar Closed-Loop Control Systems and Control Gain Measures Let us assume that we have an uncontrolled dynamic system in its modal form such as in Eq. (la), with W = 0 and an initial disturbance

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w(O) = wo. Let us also assume that a number o f control designs are available to control the same dynamic system. One of the control designs is IMSC and all others are CC, which may have been designed by any algorithm such as the Simon and Mitter pole-allocation technique or by a solution of a Riccati equation via linear optimal control theory. If all the CC techniques and the IMSC technique are to control the same dynamic system, all inheriting the same initial state Wo, the trajectory of the controlled system will evolve according to WC=~CL,C(t)Wo W,=~CL*(t)Wo

~CL,c=exp(AcL,Ct)

(lla)

qbCL*= exp(AcL*t),

(llb)

in which qbCL is the closed-loop transition matrix and the initial time is at zero. Since the transition matrices qbCL.C,qbCL* will be different for both systems, the trajectories will be different in the modal state space in spite of the same initial conditions. This difference in responses will lead to different performances and is attributed to the differences in the control gain matrices G M in Eqs. (6) and (7). Hence, it is clear that the size of the control gains will have profound effects on performance measures, however they are defined. As examples, see Refs. 12, 13, and 15 for global performance measures for flexible systems. To express the size of the control gain matrices, we use the Frobenius norm II" II2 of a matrix, which is the summation of the squared Euclidean norms of the row (column) vectors of the matrix. From the definition of Frobenius norm, we can write s = 11~tt2= Tr ~gT~ = Tr GTRG = IIGllZR,

~3= R~/2G,

(12)

and we note that 11NI is a weighted norm of the control gain matrix G. We refer to s as the quadratic gain measure (QGM). Either the modal-space gain matrices or the real-space gain matrices and corresponding weighting matrices can be used in Eq. (12). The norm IIGll~ is important from an implementation point of view, as it is an indication of total gain required by the controller. The QGM of Eq. (12) is also important with regard to any well-defined performance measure, since performance is a function of control inputs, which depend on control gains. Our analysis will determine which control technique establishes the optimal use of control gains in both the modal space and the real space. To realize our objective, we first establish a common basis for comparison of the gain norms defined by Eq. (12) of the CC and IMSC techniques. We require that the control systems be similar in some sense and introduce the concept of dynamically similar closed-loop systems. We define the dynamically similar control designs to be the set of all control techniques which result in the same closed-loop eigenvalue locations for

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the same dynamic system. This would be another meaningful way of comparing different control techniques fairly. The requirement of dynamic similarity of the control designs is compatible with the design techniques in other engineering disciplines, based on the concept of similarity. For dynamically similar IMSC and all other CC designs, we can write p{AcL,c}=p{AcL*}, IAce,c[ =IAcL,*I, TrIAcL,cI =TrlAcL,*I,

(13)

in which f I is the determinant operation and p{- } is the set of eigenvalues of the matrix {. }. We compare the gain norms of the dynamically similar IMSC and CC techniques according to Eq. (12), subject to the conditions of Eqs. (13). One can compare either the modal-space gain norms [[GM[I~M or the real-space gain norms l[GR[[~R, depending on the interest. The R R and R M matrices do not have to be the same. If modal-space gains are compared, the same weighting matrix R M must be used; or if real-space gains are compared, the same weighting matrix R R must be used for both the IMSC and CC gains. However, there atso remains the question of what to choose for the weighting matrices. As long as they are symmetric and positivedefinite, the comparisons will be valid. However, some good, physically meaningful choices exist (Refs. 12 and 13). In any case, in the absence of guidelines, the simplest choice would be to take the weighting matrices as unit matrices. In this paper, we take them as unit matrices. 4. Globally Minimum Modal Gain Matrix In this section, the necessary and sufficient conditions are given for the minimization of a quadratic modal-gain measure (QMGM). The extension to a quadratic real-space gain measure (QRGM) follows from the Q M G M after a few manipulations. We write the Q M G M as

s

IIOMII -- Tr GMTc M.

(14)

Note that we make no reference at this point as to how G M is designed. The basic question is the following: For a desired closed loop eigenvalue spectrum, among all dynamically similar control designs, what are the conditions that the matrix G M must satisfy so that s M is minimized? Theoretically, one may augment Eq. (14) with 2n invariants as constraints so that the eigenvalues will be as desired and solve for the optimality conditions. However, we choose an alternative way. Because we are interested in only dynamically similar designs, all candidate G M must at least satisfy and have the same 1st and 2nd invariants //'1 = ICLI > 0,

/J2n : ICL2n ~" 0

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of the characteristic polynomial. We are not concerned with the satisfaction of the other 2 n - 2 invariants; instead, we attempt to minimize s M subject to the 1st and 2nth invariants as constraints. All nondestabilizing gains G M which minimize s M, subject to these two constraints, will be referred to as admissible gains for the optimal gain matrix in the class of dynamically similar systems with a specified set p{AcL}. Note that satisfaction of the 1st and 2nth invariants is necessary but not sufficient for similarity; therefore, a candidate for the optimal modal gain matrix in a class of dynamically similar systems must first of all be admissible; otherwise, it does not belong to the set of dynamically similar systems under consideration. If the admissible gain matrix also happens to satisfy the remaining 2 n - 2 invariants, then it will be the optimum gain matrix that is sought (see Ref. 16, Chap. 7). Let us denote the ith row vector of G M by gJ and the ith column vector of G M by g¢. Similarly, ALL and hCL k denotes the kth row vector and the kth column vector of ACL, respectively. Also, an overbar denotes the version of these vectors with some of their elements deleted.

Proposition 4.1. The necessary and sufficient conditions that the admissible gains G M must satisfy are giT)t~L =

O,

g T h c L k = 0,

go=O,

i, k = 2, 4, 6 . . . . ,2n,

k ~ i,

(15)

i, k = 1, 3, 5, . . . , 2n - 1,

k ~ i,

(16)

i , j = 2 , 4 , 6 . . . . ,2n,

j#i.

(17)

In Eq. (15), an overbar implies that all even elements of g~ and ,~kL must be deleted; similarly, in Eq. (16), all odd elements of gi and ACLk must be deleted; these can be ascertained from the nature of associated indices i, k. The proof is given in Appendix A.

Proposition 4.2. The form of the admissible gain G M which can minimize s M, subject to the constraints /CL1 = 1"1

ICL2n ---=P2n,

is uniquely given by 0 G M= block diag [Igi(i-1)

O i] ,

i=2,4,6,...,2n;

(18)

therefore, the admissible G M has no modal interaction. Note that the proposition gives the form of G M, rather than specify specific values for gi(i-i) and g,. The proof is given in Appendix B.

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Proposition4.3. The global optimum gain matrix G Mwhich minimizes the Q M G M s M, while placing the eigenvalues p{AcL} at desired positions, is that of the IMSC given by • M G ,M = block dlag[ G,~]

= block diag

[0

g2r,2r- !

0 3,

r = 1 , 2 , . . . , n,

(19)

= 20~r,

(20)

g2r,2r

in which -lr 2 / 2-g2r,2r..1 = O~r 1_09r--tCgr-r-l~2r)],

g2r,2r

with

pr=eer+i~Sr,

fir=eer-ijSr,

p{AcL} = {pr,/5~}.

(21)

Alternatively put, among the set of dynamically similar control designs, the globally minimum-norm modal-gain matrix is the one that has no modal interaction among the controlled modal coordinates. Furthermore, the optimal solution G M has full rank. The associated Q M G M is given by s,M =

Wr v~g ~'rMT[~rM ~ , ~--

{4OZrr +e t0 --2[0)r(OL2_[_/~2r)]}2.2 __

(22)

i=l

Here, an asterisk denotes the optimal gain matrix. Proof. The IMSC G~4 is admissible, because it has the form of Eq. (t8). Furthermore, the gain elements g2r~2r-l) and g2r2r, given by Eqs. (20), are unique in placing the eigenvalues at desired positions c~± i/3r, r = 1 , 2 , . . . , n; therefore, all other 2 n - 2 invariants of the characteristic equation is satisfied. The minimum is global, because the necessary conditions for minimization of s M are the same as the minimization of z = , / s M = IIGMII, and z constitutes a norm. Any norm is a convex functional (Ref. 17); hence, z , , and therefore s ,M, are globally minimum. []

Proposition 4.4.

All CC techniques are suboptimal in their QMGM.

Proof. There are two cases of concern. If m < n, the G cM is rankdeficient; therefore, it cannot be optimal according to Proposition 4.3. If CC is full-rank, i.e., m = n, then, by definition of a CC technique, it does not have the form given by Eq. (18); therefore, it cannot be optimal. []

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JOTA: VOL. 59, NO. 2, NOVEMBER 1988 In view of Propositions 4.3 and 4.4, we state the following corollary.

Corollary 4.1. In the presence of uncontrolled or unmodeled modes, the optimum control gain effort is that given by Eqs. (19)-(21), with r = 1, 2, 3 . . . . , n + 1, n + 2 , . . . , oo, in which r > n are the residual modes with eigenvalues pr, fir = ±ioJr, ar = 0,/3r = tot ; that is, the gain norm for an infinite-dimensional DPS is optimized when there is no control spillover into the residual modes.

Proof. It is straightforward. All one has to do is expand the set p{AcL} to include those of the residual modes and view the control problem as that of the new expanded system; the optimum G ~ will still be given by Eq. (19), with the range of index r extended to include the new modes, as many of them as desired. [] Corollary 4.2. Any control spillover into the residual modes necessarily implies a suboptimal Q M G M for the complete DPS system with controlled and uncontrolled modes combined. Proof. Denote the sets of controlled and uncontrolled modes with the subscripts N and R, respectively. The closed-loop equations will have the form

[wq

w=

WR =

olrw l+ro ,

0

ARJLWR_I [_GR M OJLwRJ

in which the matrices A and G M should be evident. Existence of GRM # 0 implies control spillover, which is another form of modal interaction between controlled and uncontrolled (residual) modes. Furthermore, it is known that control spillover will not move the eigenvalues of the residual system. Again expanding the system as in Corollary 4.1 so as to include the residual eigenvalues, one can view G M as the modal gain matrix of the aggregate infinite-dimensional system. For optimality of the gains G M for the complete system, G M must have the form of Eqs. (19)-(21), with r expanded to include the residual modes. But since G M# 0, G M can never be of the form of Eq. (19). []

Proposition 4.5. For a given number of actuators rn < n, the pursuit of finding optimal actuator locations, regardless of what the objective function is, will always produce a suboptimal QMGM, thus ultimately resulting in a suboptimal control effort.

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Proof. Our main result is obtained regardless of how the numbers in the modal gain matrix G M are generated. Since when rn < n,

G M = Grff = B c G ~ ,

the matrix G M will always be rank-deficient, regardless of how Bc and G R are changed around. Any rank deficiency in G M automatically implies suboptimality according to Proposition 4.4. [] Corollary 4.3. The ultimate improvement in any objective function expected to result from the distribution of actuators in controlling a DPS will come only through an increase in the number of actuators. Proof. It hinges upon the rank deficiency of the G M matrix, which can only be effectively increased by an increase in the number of actuators. []

In essence, the implication of corollaries 4.1, 4.2, 4.3 is that, to relocate the eigenvalues of the 2n modes of an infinite-dimensional nongyroscopic DPS, the globally optimal modal gain matrix is given by Eq. (19), regardless of residual modes, modeled or not; therefore, Eq. (22) can be used as an absolute global measure of modal gain. Whether G ~ can be realized exactly in practice is not the question posed here; If " G ,M cannot be realized exactly, then the control design is necessarily suboptimal, and its degree of suboptimality can always be measured by computing the associated s M and by comparing it to the s ,M of Eq. (22).

5. Modal Interaction Parameter

It is clear from the results of Section 4 that the globally optimal G ~ requires no coupling of the modal responses; it is an IMSC in form, and any modal coupling due to the form of G M produces suboptimality. Thus, we define a modal interaction parameter /~(MIP) as an indication of suboptimality of the Q M G M of a control design: M tr G M T G M trG,

G,

tr G R T B T B G P" 1 ---

try,

tr G M T G M >-- tr ("~rMTI'~M

v,

1,

(23a) (23b)

in which G ~ is given by Eqs. (19), (20) and G M is the modal gain matrix of any control design. Since the traces will be equal if and only if G M= G ~ (that is, if and only if the control design is an 1MSC), it follows that the

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absolute minimum value of the MIP /x M is zero, signifying no modal interaction. Any value /xM> 0 automatically implies G M ¢ G~, a nonoptimal CC design with modal interaction. Thus, the MIP tz M should serve as a convenient indicator of the quality of the control design. By using Eqs. (22) and (23), the MIP can be written in the alternative forms M tr GMTG M tr ~ M T ~ M # - Er=l{4 c~2+ -2 2 1 = o~r [O~r--(O~r+t3r)]2 2 2} Er=," Ibg*II=- 1 , (24) in which we defined the n-dimensional n vectors g* in the configuration space in the form g* = (g2r,2r_, + g2,,2~)'/2er,

(25)

and er is an n-dimensional null vector, except for its rth component which is unity; g2r,2~-I and g2~,2r are given by Eqs. (20). The MIP can be computed readily for any given control technique with a specified set p{Ace}. The form of/~M on the far right-hand side of (24) is physically very instructive. The n vectors g* are orthogonal, and they form a basis for the gain space spanned by the column vectors of ~M, where ~M is obtained from G M by deleting its odd (zero) rows. After expressing the columns of ~M as linear combinations of g*, we can define a modal interaction quotient (MIQ) as M

/zq

tr (~rMT(~rM

rt 9 :g E r = l C;lIg, [[2

MT M tr G , G ,

,, E,=I llg*II 2 '

M

M

/* =/*q --1.

(26)

The coefficients G represent coupling of modal responses of the controlled system. This coupling necessitates larger control gains and leads to inefficiency or waste in the amount of control energy expenditures. We refer to G's as control coupling coefficients. The lowest value of the MIQ is 1, and this can happen according to Proposition 4.3 if and only if

c~=c2 . . . .

=cn=l

simultaneously, which will necessarily imply an IMSC design. If a mode r can be designed independently from all other modes, its corresponding c~ is necessarily 1; however, the converse is not true. The coefficients G can be used to tune up a control design to change its #M. Ideally speaking, a zero value f o r / z M is desirable, but this can only be accomplished through the use of an increasing number of inputs. However, it is possible that the control designer will settle for a low, yet nonzero, value o f / x M, reflecting a degree of compromise in the design. If a CC design is employed to design the G M = B G ~, the tuning of the coefficients c~ can be brought about by relocating the actuators a n d / o r

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changing the number of physical inputs. Therefore, we propose a criterion for locating the actuators in CC: that a feasible technique is to distribute them such that the MIQ (or MIP) is the lowest that can be attained within other constraints of the design. It must be noted that, with fewer actuators than the number of controlled modes (i.e., if the design is rank-deficient), the absolute minimum values of 1 (for MIQ) and 0 (for MIP) can never be achieved.

6. Optimization of Real-Space Gain Matrix and Actuator Location Criterion for IMSC We now extend the previous results to state their counterparts in terms of real-space gain matrices G R. To this end, define a real-space interaction parameter (RIP) /z R in the form

/z

R tr ,-'c "-,c tr ,-,c --,c ~rRT/,~R 1 --MT -T -1 - M try, ~, trG, B, B, G,

--

1,

(27)

in which B , is the n x n actuator location matrix for the IMSC obtained from B [Eq. (2b)] by disregarding its zero (odd) rows and is needed to find the real-space gain matrix G R of the IMSC according to the modal synthesis implied by Eq. (4b). G-M , is the n x 2n modal gain matrix obtained from G ~ by deleting its zero (odd) rows. Assume that G~ has been computed by any CC design corresponding to n x m actuator location matrix B C=[bCl

, bc2,...,bcm].

For a given set p{. }, the matrix Bc must be specified; therefore, the computed G~ is unique for the given CC technique and given Bc. On the other hand, for the IMSC corresponding to the same set p{. } (hence the control designs are similar), there is an infinite set of G ,R depending on the choice of B , . By a poor choice of B , , the matrix G ,R can be made arbitrarily large. In the IMSC, the choice of B , does not alter the modal gains and the designed closed-loop eigenvalues. This being the case, in contrast to a given G R, one can generate many G R matrices for arbitrary choices of B , , possibly rendering the IMSC suboptimal in its real-space gain measure relative to the CC design, in which case/z R becomes negative. This last statement needs no proof. An example is given in Section 7 demonstrating its validity. However, the same statement, with an apparent negative tone for IMSC, also gives an optimistic message to the contrary, the clue being the location of actuators. Here, through the definition of RIP, we have a motivation for making the actuator locations

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of concern for the IMSC design and its is relevant to propose an actuator location criterion for the IMSC. Proposition 6.1. For a given set p{. } in an IMSC design, actuator locations can be chosen to r e n d e r / z R > 0, i.e., in a way that will make the real-space gain measure o f the IMSC optimal relative to dynamically similar CC designs. Again, the proof of the statement is in the numerical examples of Section 7. The optimality of the IMSC relative to the CC design depends not only on the IMSC actuator locations but also on the specified set p{. }. In other words, if a given actuator configuration yields an optimum IMSC (/~ R> 0) design over a CC for a given set p{. }, for a new set p{, } the same actuator locations may render the IMSC design suboptimal relative to the CC design (/z R< 0). The conclusion is that, insofar as the norms of real-space gain matrices are concerned, no claim as to the global optimality o f the implemented real-space IMSC gain G , can be made. The search for proper actuator locations in the IMSC to optimize itsreal-space gain matrix can be facilitated to some extent by making its B , matrix similar in configuration to the Bc matrix of the corresponding CC design. This can be accomplished by embedding the set Bc in the set B . . Then, one needs only to search for locations for the actuators in excess of m actuators of the CC design. Hence, the matrix B . may be chosen such that B , = [Bc i bem+l, ' ' ' ,

b , , ] = [Bc ] B].

(28)

By augmenting the G R matrix by n - m rows of zeroes, we can write

Next, noting that trL 0 J L 0 j = t r O c

,<

Oc=trOc

_.r

-,-,,,,

B, B, Gc,

(30)

and substituting the result into Eq. (27), we obtain --MT

g /z

-

trGc -MT tr G,

T --1 - M (B,B,) Gc x -1 -M-(B,B,) G,

1.

(31)

A positive value for/.t R in Eq. (31) implies that the IMSC is optimal relative to the CC design. As soon as a positive/x R is attained, the search for actuator locations for the IMSC can be stopped, or one may wish to

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continue the search to yield larger positive values f o r / . ~ , which implies better optimization of the IMSC gains. In essence, the process can be interpreted as tuning the IMSC design. At this point, the reader is urged to notice the duality of the results that we obtained for the IMSC and CC designs: A CC design can be tuned via actuator locations to yield lower values of IzM; and an IMSC design can be tuned via actuator locations to yield larger values o f / z R.

7. N u m e r i c a l E x a m p l e s

A pinned-pinned uniform beam in flexural vibration was considered with unit length, mass/length, and flexural rigidity. The closed form expressions for the natural frequencies and normalized eigenfunctions are ~Or= r27r2,

4~r = ~/2 sin(r~rx).

As a first example, we consider a sixth order model (n = 3, r = 1, 2, 3) with three actuators, each located at x~ = 0.2, x2 = 0.4, x3 = 0.6. For CC designs corresponding to m = 1, 2, 3, the actuators were at Xl; xl, Xz and xl, x2, x3, respectively. For the IMSC designs, all three actuators were considered. Hence, the Bc matrices of the CC designs were all embedded in B . of the IMSC. All three CC designs (m = 1, 2, 3) were based on the linear-optimal control theory, with the state and control weighting matrices being compatible unit matrices in each case. Corresponding to each optimal CC design, the closed-loop eigenvalues p{. } were found, and an IMSC design was designed with the same p{- }. The corresponding modal-space gain matrices and real-space gain matrices were computed to be used in Eqs. (23) and (27) to evaluate/z M a n d / z R. the results are shown in Table 1. The second case corresponding to m = 2 in Table 1 is a CC design, based on the pole-allocation algorithm. For the same beam, another case was run with n = 2 (r = 1, 2) with a single actuator, located at x~ = 0.25. For the CC design, a single-input (at x~) pole-positioning algorithm was used to locate the eigenvalues p { - 1 :~ j l 0 , - 1 ±j40}. The IMSC design with the same eigenvalues requires two inputs. The first actuator at xl was embedded in the matrix B , of the IMSC design. The location of the second actuator required for the IMSC was varied to optimize its real-space gain matrix. Table 2 shows the results. For the case in which x~ = 0.25, x2 = 0.75, it is rather instructive to look at the modal gain matrices and compute the control coupling coefficients cr in Eqs. (26), a:g

-

blockdiag

[-0.3638 0 0 0 02t - 2 [ -1.0754 [

-

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(~=

[-0.3475 1-0.1229

-2.063 -0.7294

-3.3404 -1.t81

-5.4785]. -1.9369.1

The following quantities were computed:

IIg*t12= 4.1324,

llg*ll 2= 5.1565,

c 2= 11.022,

c~

= 1.1031. Equation (26) is /z

M_ ll'02211g*l] 2+ l'1031llg*ll 2 -

IIg*l12+ IIg*ll 2 Comparing the coefficients of Ilg*ll 2 in the nominator and denominator, a control coupling coefficient of 11.022 for the first mode implies that this mode is coupled strongly to the second mode by the controls. On the other hand, a control coupling coefficient of 1.1031 for the second mode implies that this mode is weakly coupled to the first mode. A simple observation of the elements of G~ verifies the descriptive quality of the control coupling coefficients. T a b l e 1.

Illustrative comparison o f g a i n n o r m s o f dynamically similar IMSC and C C designs.

Closed-loop eigenvalues Number of inputs

p{. }

/x M (MIP)

~R (RIP)

m= 1

-0.0595 ±j9.8696 -0.024t ±j39.4784 -0.0107 ±j88.8264

2.005

0.036

m=2

-0.1133 ±j9.8696 -0.0283 ±j39.478 -0.0126 ±j88.8264

0.804 11.65 x 106

-0,350 4.002 × 106

m=3

-0.1487 ±j9.8696 -0.0320 ± 39.478 -0.0142 ±j88.8264

0.196

-0.527

T a b l e 2.

Actuator allocation f o r I M S C via RIP. /z R (RIP)

Number of inputs

p{ .}

#M (MtP)

x2 = 0.53

xz =0.55

xz = 0.75

m = 1 at x I = 0.25

-1 ± j l 0 - 1 ±j40

4.516

-0.089

0.050

1.008

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In all of the cases studied, a positive value for/x implied that the IMSC was optimal. F o r / z M calculations, the IMSC is a global optimal design in modal-space gains. For R calculations, the optimality of the IMSC realspace gains depends on p{-} and its actuator configurations. In Table 1, no actuator location search for optimizing the IMSC design was undertaken. All three cases m = 1, 2, 3 were compared to the IMSC design without changing the actuator positions. When the closed-loop eigenvalue locations were changed, the IMSC became suboptimal in going from m = 1 to m = 2, 3. Table 2 shows an example of actuator location criterion for the IMSC. With the second actuator located at x2=0.53, a negative value of/~R shows a suboptimal IMSC. Placing the actuator at x2= 0.55 improves the design and a positive/x R implies that the IMSC is now optimal. Putting the actuator at x2 = 0.75 further improves the design by making/x R still a larger positive value. Thus, the IMSC design has been tuned to be optimal relative to the CC design via an actuator location criterion. In other words, in addition to having smaller modal gain parameters than the CC, the IMSC design, after tuning, has smaller real-space gain parameters also.

8. Summary The optimality of the gain matrices of the IMSC and CC techniques has been investigated. It was found that, among the set of all dynamically similar control designs, the globally optimal modal gain matrix is obtained when the IMSC is used. Implications of the results for the control of infinite-dimensional structural DPS were given. It was also established that relative optimality of the real-space gain matrices of dynamically similar IMSC and CC designs depends on the actuator locations for the IMSC. In view of these results, modal and real-space interaction parameters (MIP) and (RIP) were defined. Actuator location criterion for CC and IMSC designs were suggested, based on MIP and RIP, respectively.

9. Appendix A: Proof of Proposition 4.1 Consider Eq. (14), 2n

sM= IIGMI]2=zr G MTGM = Y j=l

2n

Z

g~.

(32)

i-2,4,..,

Because s M is a positive-definite, quadratic functional, the necessary conditions for its optimality are sufficient for a minimum. We minimize s M subject

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to the discussions of Sections 3 and 4. The optimal solution belongs to the set of admissible gains which must satisfy the constraints 2n

2n

us = - Y Pr = - T r AcL = r= S

Y

gi~ > 0,

(33a)

i = 2, 4 , . . .

2n

~2. = 17 p, = IAcd = IA+ GMI > 0.

(33b)

i=1

Satisfaction of these constraints based on the 2nth.and 1st invariants of the characteristic equation of the closed-loop system is necessary (but not sufficient) for dynamic similarity. The constraint Eq. (33b) is needed, because we note from the form of AcL (see the end of Appendix A ) t h a t ICL2. is independent of g~, i = 2, 4, 6 , . . . . An attempt to minimize s M subject to only Eq. (33b) will yield trivially g~ = 07 which is not permissible, because the first invariant cannot vanish for a stable closed-loop system. Next, introducing the notation w

=

[~s, n , , ~ , n2, . . . , ~o, ~ . ] ~

for the modal state vector formulation Eq. (la), we can rearrange the closed-loop modal dynamics in the form

[ ~] = [--[/--.0]0'-'~Gs [~o] = diag[oJ1,

O)2,...

G2J

'

, f.On] ,

,7 = [n~,- --, n.]7~ where G~ and G2 are n x n modal gain matrices formed from G M. Disregarding the zero rows, G1 is formed by taking the odd columns and G2 is formed by taking the even columns of G M. By using the equivalent closed-loop form above, we note that ICL~.=IAcLI=(--1)"

1] 'orlo,--[o~]l="=o. r=l

Is = - T r ACL = - T r G: = Ul. Thus, the problem of minimizing s M subject to constraints (33) is the same as minimizing s M subject to the constraints IcL2, - IAcL, I = IG, - [oJ]I = G .

(34a)

v~, = (-1)"v2, fi w7 l,

(34b)

r== |

- T r G2 = t-h.

(34c)

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We now note that the constraints (34) are independent of the off-diagonal terms of G2 (that is, all elements of even columns of G M, except for the diagonal elements). Therefore, any nonzero off-diagonal elements of G2 will only increase the value of s M. The conclusion is that the admissible G M must have go =0,

i,j=2, 4 , 6 , . . . 2n, i # j ,

(35)

which are conditions (17). Hence, the problem now reduces to minimizing s M subject to constraint (34a), involving only gu, i = 2, 4, 6 , . . . , 2n, j = 1, 3, 5 , . . . , 2n - 1, as variables, regardless of g,. Introducing the augmented functional sY, s .M = s ' M +

~.{IAcLII- ~;.},

s'M=TrG~GI =

2n

2n--1

2

~.

i=2,4,,., j-

(36a) g~,

(36b)

1,3 ....

the conditions for minimization are

Os~/Og o = os'M/ogo + tz, (OI'cc2,/Ogo), i=2,4,6,...,2n,

j= 1,3,...,2n--1,

Os'~/Otx, = {AcL~ -- P~} = 0.

(37a) (37b)

Next, we note that os'M/ogo - =

2g o,

(38a)

a I'cc2,/ ago = (a/ ago )[Accd =

]Accli,j

= Co(AcE1),

(38b)

where ]AccllO is the determinant of the matrix after annihilating the ith row and replacing the jth column element in that row by unity. The result is the C o. (Ace0 element of the cofactor matrix C of Acc~, with the property C0=0,

j=2,4,6,...,2n#i.

In view of Eqs. (38), Eqs. (37) yield 2g 0 + iz,C 0 = 0, i=2,4,6,...,

2n,

j = 1, 3, 5 , . . . , 2 n - 1 .

(39)

From Eqs. (39), solve once for go and once for C o to obtain g~ = - ( ~ . / 2 ) c ~ ,

co = - ( 2 / ~ ° ) g 0 ,

i = 2 , 4 , 6 , . . . , 2n,

j= 1,3,...,2n-1.

(40)

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Forming the vectors

gi=[gi,,g~3 . . . . . g,2,_1] T,

C~=[Ci,, Ci3,... ,

Ci2n_l]T~

(41)

and by using Eqs. (40) take the inner product 2n--I

g'TC'= E [-(2/~.)g,j]go---(z/~)[Ig'll =,

(42a)

j=t,3 2n--1

g ~rCi=

2

[-(tz,/Z)C~]Cu=-(tx./2)llCill 2,

(42b)

j = 1,3,...

in which II" lJ is the Euclidean norm of a vector. Multiply Eqs. (42) with each other and then take the square root to obtain

g 'TC'=- Ilg'll" IIC'tl. Thus, C ~ and g~ must be collinear. Note that C~¢ 0, otherwise I~Len = IAcL,I = 0 would result, and such a system would not be admissible in the class of dynamically similar closed-loop designs. Next, denote the proportionality constants between g~ and C ~by d~, i = 2, 4, . . . , and take the inner product o f g i with any k-row vector )'~L~ of ACE1, k = 2, 4, 6 , . . . # i, to obtain A kT CLlg

i

.l ~ r r x - - u i ~-"

k 'tCLl

=

kT diAcLlC

i

,

i,k=2,4,...,i~k.

(43)

We conclude that the inner product must vanish, because the right-hand side of Eq. (43) is the expansion by row i of the determinant of a modified ACE ~ in which the ith row is replaced by the kth r o w AkL~. In this form, Eq. (43) must vanish because it is literally the determinant of ACE1, which has its ith row identical to its kth row. Hence, gir)tkL1 = 0,

i, k = 2, 4 , . . . ,

i ~ k,

(44)

that is, the partition G~ of the modal gain matrix G M must be such that the ith row of G~ must be orthogonal to every kth row of ACL~, k # i. Equations (43) correspond to the conditions stated by Equation (15). It remains to discover the necessary conditions that the odd columns of G M must satisfy to be an admissible gain matrix. To this end, note that s M = T r GroG, = T r GIG[,

[AcL,] = ]ArL,t,

and the previous derivation can be repeated for the rows of G~r and AcrL~. In this case, the counterparts of Eqs. (41) are

gj = [g2j, g , j , . . . , g2,~]r,

Cs = [C2:, C4j, • • •, C2.j] T,

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and one obtains gfAcLlk =0,

j,k=l,3,5,...,2n-1,

k#j;

(45)

that is, the jth column of GI must be orthogonal to every kth column of AcL1, k #j. Equations (45) correspond to the conditions stated by Eq. (16). [] In Appendix B, we obtain the explicit form of the globally minimum modal gain matrix which satisfies the conditions of Eqs. (15)-(17). Before this is done, it will be prudent to consider whether there are any other conditions that we might have missed for optimality of G M. Equations (15)-(t7) each constitute n 2 - n conditions for optimality. They add up to 3(n 2- n) equations. But in Section 2, we showed that, in order to be able to locate the eigenvalues of the closed-loop system at desired locations, at most 2(nZ-n) additional equations could be used to augment the 2n invariants equations to find the 2n 2 elements of G M. But conditions (15)-(17) constitute 3(n 2- n) additional equations, i.e., n 2 - n equations in excess of what is allowable at the most. Apparently, if anything, the conditions that we have found may have overspecified the problem within the context of dynamically similar systems. However, it turns out that, as we show in Appendix B, one half of the conditions that are given by Eqs. (15) and (16) are redundant, so that the optimality conditions indeed constitute only a total of2(n 2 - n) equations, which is just the maximum number of additional equations that can be allowed in conjunction with the 2n invariants equations for the closed-loop characteristics equations. This then guarantees that any modal gain which satisfies the optimality conditions (15)-(17) and the 2n invariants is unique, which necessarily makes it globally optimal without any resort to convexity arguments. The form of a typical closed-loop matrix ACL, for n = 2, is given below:

AcL = A +

G M =

[,?

21 ("01 g22

0 L g41

Gl=[g21 g23], Lg41

g43.J

g42

Gz=[g22 k g42

0

g24

0

w2]'

g23 g43 - -

(1)2

01

g44

g24]. g44 .J

Note that g0=0,

i=1,3,...,

C~ = 0,

i, j even.

j=1,2,3,4,...,

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10, Appendix B: Proof of Proposition 4.2 The necessary and sufficient conditions [Eqs. (15), (16)] can be written explicitly in terms of the elements of G M as follows: g

iT, k

A'CLI ~-

2n-I 2 j= 1,3,...

i,k=2,4,6,...,

gijgkj--('Ok/2gi(k-l)=0, k~i,

(46)

2n

gThcLlk--= ~

gjigjk--O)k+l/2g(k+l)i =0,

j=2,4

i, k = 1,3, 5 , . . . ,

k ¢ i.

(47)

Introducing the inner product notations or (i, k) = g,rgk = (k, i), (i, k) = gT g k = (k, i)

(48)

Eqs. (46), (47) become ( i, k ) - o)k/2gi(k-l~ = O,

(i, k) - Wk+l/2g(k+l)i= O,

i,k=2,4,6,...,iCk, i,k=l,3,5,...,i#k,

(49a) (49b)

respectively. If one interchanges the indices i and k, we have

(k, i) - o)i/2gk(i_l) = O, (k,i)-~o,+~/2g(~+~)k=O,

i, k = 2, 4, 6 , . . . , k ~ i, i,k=l,3,5,...,k#i.

(50a) (50b)

Using (48), subtracting Eq. (50a) from Eq. (49a), and subtracting Eq. (50b) from Eq. (49b), we obtain -- (.Ok/2gi(k_l)-k-

Wi/2gk(i_

~ ) = 0,

(51a)

- - (.Ok + l / Z g ( k + l )i "I- O , ) i + l / 2 g ( i + l ) k = O. (5tb) If the indices i and k are interchanged in Eqs. (51), one ends up with the same equations; therefore, one half of the information contained in Eqs. (51) is redundant. It follows that Eqs. (51) will yield only (1/2)(n 2 - n) useful equations, respectively. Hence, Eqs. (15)-(17) constitute a total of 2 n 2 - n independent conditions, as was already stated in Appendix A. By assigning values to i and k in Eqs. (51) as required and assuming that the natural frequencies are distinct, the unique solutions of Eqs. (51) for the elements of the admissible G M are

g0=0,

i=2,4,6,...,

j=l,3,5,...,j~i-1.

[]

References 1. MEIROVITCH, L., and Oz, H., Modal-Space Control of Distributed Gyroscopic Systems, Journal of Guidance, Control and Dynamics, Vol. 3, No. 2, pp. 140-150, 1980.

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2. MEIROVITCH, L., BARUH, H., and (3Z, H., A Comparison of Control Techniques for Large Flexible Systems, Journal of Guidance, Control, and Dynamics, Vol. 6, No. 4, pp. 302-310, 1983. 3. ME1ROVITCH, L., and BARUH, H., Control of SeIf-Adjoint Distributed Parameter Systems, Journal of Guidance, Control, and Dynamics, Vol. 5, No. 1, pp. 60-66, 1982. 4. 0z, H., and MEIROVITCH, L., Stochastic Independent Modal-Space Control of Distributed Parameter Systems, Journal of Optimization Theory and Applications, Vol. 40, No. t, pp. 121-154, 1983. 5. MEIROVITCH, L., and BARUH, H., On the Problem of Observation Spillover in Distributed Parameter Systems, Journal of Optimization Theory and Applications, Vol. 39, No. 2, pp. 611-620, 1981. 6. BALAS, M. J., Active Control of Flexible Systems, Journal of Optimization Theory and Applications, Vol. 25, No. 3, pp. 415-536, 1978. 7. MEIROVITCH, L., and BENNINGHOF, J. K., Control of Traveling Waves in Flexible Structures, Proceedings of 4th VPI/AIAA Symposium on Dynamics and Control of Large Structures, Blacksburg, Virginia, 1980. 8. MEIROVITCH, L., BARUH, H., MONTGOMERY, R. C., and WILLIAMS,J. P., Nonlinear Natural Control of an Experimental Beam, Journal of Guidance, Control, and Dynamics, Vol. 7, No. 4, pp. 437-442, 1984. 9. HALLAUER, W. L., SKIDMORE, G. R., and MESQUITA, L. L., ExperimentalTheoretical Study of Active Vibration Control, Proceedings of the International Modal Analysis Conference, Orlando, Florida, pp. 39-45, 1982. 10. BARUH, H., and SILVERBERG, L., Robust Natural Control of Distributed Systems, Journal of Guidance, Control, and Dynamics, Vol. 8, No. 6, pp. 717-724, 1985. 11. LINDI3ERG, R. E., and LONGMAN, R. W., On the Number and Placement of Actuators for Independent Modal-Space Control, Paper No. 82-1436, AIAA/AAS Astrodynamics Conference, Gatlinburg, Tennessee, 1982. 12. Oz, H., A New Concept of Optimality for Control of Flexible Structures, Ohio State University, Aeronautical and Astronautical Engineering Research Report in Dynamics Control, No. AAE-RR-DC-101-1988. 13. MEIROVITCH, L., and SILVERBERG, L., Globally Optimal Control of Self-Adjoint Distributed Systems, Optimal Control Applications and Methods, Vot. 4, pp. 365-386, 1983. 14. CHEN, C. T., Linear System Theory and Design, Holt, Rinehart, and Winston, New York, New York, 1984. 15. Oz, H., FARAG, K., and VENKAYYA, V. B., Efficiency of Structure-Control Systems~ Journal of Guidance, Control, and Dynamics (to appear). 16. DORNY, N., A Vector Space Approach to Models and Optimization, John Wiley and Sons, New York, New York, 1975. 17. LUENBERGER, D., Optimization by Vector Space Methods, John Wiley and Sons, New York, New York, 1969.