# Full text of "Approximation theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures"

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i^ASA Contractor Report 181705 ICASE REPORT NO. 88-48 ICASE APPROXIMATION THEORY FOR LQG OPTIMAL CONTROL OF FLEXIBLE STRUCTURES CiJ/08 0105575 J. S. Gibson A. Ad ami an N d 9 - 1 1 7 5 Contract Nos. NAS 1-1 7070 and NAS 1-18 107 August 1988 INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING NASA Langley Research Center, Hampton, Virginia 23665 Operated by the Universities Space Research Association rvi/>sA National Aeronautics and Space Administration unQwy RaMMch Oantaf Hannpton, Virginia 23665 APPROXIMATION THEORY FOR LQG OPTIMAL CONTROL OF FLEXIBLE STRIFCTURES J. S. Gibson and A. Adamian Mechanical. Aerospace and ^'^-^^^^/"^^"""'"^^A University of California, Lo.; Angeles, CA 9002^ ABSTRACT XMs paper presents approximation the - ^^ r;;7irtrrbut:d Gaussian optimal control P-^^- /^^^ 'operators "he main purpose of the n,odels have bounded input ^nd output oP^^atjrs ^^^^^^^ ^^^, ^^^,,^,. theory is to guide the design of finite dt.ensio P ^^^^^^ ^^^^^^_ n>ate closely the optimal compensator ^^^^^^^^\^\^ ^,,,J^^ dimensional quadratic control problem lies in ^^^ ^'^^"^^^^ J^ ^^ ^he paper approximates Riccatl operator equation The ^'°^:^l^^ If flniL dimensional LQG the infinite dimensional LQG P^^^^^^^^^^V dimensional , usually finite element problems defined for a ^^^^^^ J^^'^^.'Alfted'^L^^ 'of the structure. Two MccTt"Vatr^^^eta^tn\"neferm\" t^e solution to each approximating problem. Tbe finite dimensional equations for numeic la ro^ veloped, including formulas °^. ^^/f,; t^'^f.^'pLJLn of gains based on dif- to their functional ^^P^^^^f ^^^°^. ^° ^^^ ^e of the approximating control and ferent orders of approximation. C^";;//;;^,;,,;' dimensional compensators is estimator gains and of the co'rrespondi g fimte ^ ^^^^^^ ^..^uced studied. Also, convergence and stability of the c ^^ convergence with the finite dimensional ^^^^^^^.^l?"' J^ZJ^^f the finite dimensional theory is based on the convergence of the solutions ^^^^^ ^^^^^^^ ^^^^_ f.-:r rn-t^\\rexrpi:ti;t°rflLite^htim, . rotatmg rigid body, and a lumped mass is given. , , v^ the National Aeronautics and Space Administra- This research was supported \^^^_^%f'^^^ NASl-18107 while the first author tion under NASA Contract Nos. NASI 17070 and N^b ^^ g^i^^ee and was in residence at the Institute for ^°''^^'-^' ^^^'^ ^ vA 23665. Also. Engineering (ICASE), NASA ^angleyResearhCener Hampton. ^^ ^^^ ^^^^^ ^^^^^ rPOSrG^ra^nrBro?0.^rd°"yThV.Trpuls^:n Laboratory. Pasadena, CA under Grant 957114. CONTENTS Abstract 1 . Introduction - 2. The Control System ^^ 2.1 The Energy Spaces and the First-order Form of the System The Elastic-strain-energy Space V and the Total-energy Space E j^ The Damping Functional and Operator 15 The Semigroup Generator ^ 2.2 The Adjoint of ^ JJ 2.3 Exponential Stability 3. The Optimal Control Problem ^^ 3.1 The Generic Optimal Regulator Problem 22 3.2 Application to Optimal Control of Flexible Structures 32 4 . The Approximation Scheme 4.1 Approximation of the Open- loop System 32 4.2 The Approximating Optimal Control Problems 3^ 5. Convergence 5 .1 Generic Approximation Results 5.2 Convergence of the Approximating Optimal Control Problems of Section 4.2 Exam pi e 6.1 The Control System 6.2 The Optimal Control Problem 6 .3 Approximation 6 .4 Numerical Resul ts 53 53 58 59 61 PRECEDING PAGE BLAMK NOT FILMED lii 7. The Optimal Infinite Dimensional Estimator, Compensator and Closed-loop System 71 7 .1 The Generic Problem .'.'.*.'.'.*.'.'.'.'!,'.'.'.' .' 71 7.2 Application to Structures '.'.'.'.'. 76 8. Approximation of the Infinite Dimensional Estimator .. 79 8.1 The Approximating Finite Dimensional Estimators . 79 8.2 Stochastic Interpretation of the Approximating Estimators oo 8.3 The Approximating Functional Estimator ' Gai ns ".'.'! .* 85 8 .4 Convergence " g- 9. The Finite Dimensional Compensators and Realizable Closed-loop Systems 9q 9.1 Closing the Loop .*!."!!!.'!.'.'!.'."*" 90 9.2 Convergence of the Closed-loop Systems".'!.'.*!.'.'!.*.' 92 9.3 Convergence of the Compensator Transfer Functions pg 10. Closing the Loop in the Example 99 10.1 The Estimator Problan !!!!!!!!!!! 99 10.2 Approximation of the Optimal Compensator"!!!!!!!! 107 10.3 Comments on the Structure and Dimension of the Implementable Compensators I09 11. Conclusions ,,, References , - - Appendix: Errata for [Gl] Hg iv Introduction „..e„t ye... .ave seen inereaslng researoh In active oontrcl of nexi.le .tnuotures. The pri.a,, .oUvatlon Tor U,!., research 1. eontroX of large ne=..Xe ae.o.pace structure. wMc. are .co.ins larger ana .ore ne,i.le at ^e sa^e ti.e that their .rfor.a„oe r.uir«.ent3 are .coding .ore stringent, .or e^Ple. m tracking ana other applications, satellites wit. large enten- te, solar collectors ana other flexible components .ust perfor. fast slew ^.neuvers while maintaining ti*t control over the vihrations of their fle,i- .ie el^ents. Both of these oonflicti,* ot.ectives can he achievea only with . sophisticatea ocntroUer. .ere are appaications also to control of rototic manipulators with flexible lin.s, ana Poss.hl. to stahilization of large civil engineering structures such as long hriage, a„a tall huiiaings. ^e first question that must he answerea when aesigning a controller for , .,.ther a finite almensional moael is sufficient as a a flexible structure is whether a unite ux ^sis for a controller that will proauoe the rcuirea perfor-anoe. or is a aistributea moael necessary, While some structures can be moaelea well by a f ixea n«ber of aomi.nt moaes, there are structures whose flexible character oan . capturea sufficiently for precise control only by a aistributea moael. Still others - perhaps most of the aeros.ce structures of the future - can . moaelea sufficiently for control pur^ses by some finite almensional approximation, but an aae,uate approximation may . impossible to aete^ine .fore aesign of the »„troller. or .mP^nsator. This pap.r aeals with struc- ..es that are flexible enou* to r.uirc a aistributea moael in the aesign of an optimal LQG compensator. The Unea^,uad.au.-gau=3la„ optical «.„t.ol probl- for distributed, o. inflate dX^enalcnal, a.at»a Is a generalization to „llt«.t space of the now olassioax WO p.oM» for finite dimensional s,st«s. ,.e solution to the infinite dimensional prohl. nelds an Infinite dimensional stat^astlmaton- based oom^nsator, which Is optimal m the context of this pape.. By a =epa.atlon principle tBl, «,, the pnohl» .educes to a deterministic Unea,- quadratic optimal contnol p.o„- and an optimal estimation, o. filtering pnohlem with gausslan white noise. l„ i„f.,,e dimensions, the control syst. dynamics are represented h, a semigroup of hounded linear operators Instead of the matrix exponential operators In flMte dimensions, and the plant noise process may he an Infinite dimensional random process, ^e solutions to ^th the control and fnterl^ prohlema Involve Hlccatl operator equations, which a.e generalizations of the «lccatl matrix equations In the finite dimensional oase. c».re„t results on the Infinite dimensional UiO prohl« are most com- plete for problems where the Input and measur.ent curators are bounded, as this paper requires throuchouf T^,^o k^ ^ ^ cnroughout. This boundedness also permits the strongest approximation results here Pn». «^i»«. ^ here. For related control problans with unbounded input and measurement, see [C3. C5 , LI, L2] , our prlman- objective In this paper Is to approximate the optimal infl^ ite dimensional UO com^nsator for a districted model of a flexible struc- ture with finite dlmenslo,.! com^nsators based on approxlmaUons to the atruoture. and to have these finite dimensional «,mpensators produce near optimal performance of the closed-loop syst.. We discuss how the gains that determine the finite dimensional «.mP.nsators converge to the gains that determine the Infinite dimensional .mp^nsator. and we examine the sense In which the finite dimensional compensators converge to the infinite dimensional compensator. With this analysis, we can predict the performance of the closed-loop system consisting of the distributed plant and a finite dimen- sional compensator that approximates the infinite dimensional compensator. Our design philosophy is to let the convergence of the finite dimensional compensators indicate the order of the compensator that is required to produce the desired performance of the structure. The two main factors that govern rate of convergence are the desired performance (e.g., fast response) and the structural damping. We should note that any one of our compensators whose order is not sufficient to approximate the Infinite dimensional compensator closely will not in general be the optimal compensator of that fixed order; i.e., the optimal fixed-order compensator that would be constructed with the design philosophy in [B7, B8] . But as we increase the order of approximation to obtain convergence, our finite dimensional compensators become essentially identical to the compensator that is optimal over compensators of all orders. An important question, of course, is how large a finite dimensional com- pensator we must use to approximate the infinite dimensional compensator. In [G6, G7, G8, Ml], we have found that our complete design strategy yields com- pensators of reasonable size for distributed models of complex space struc- tures. This strategy in general requires two steps to obtain an implementable compensator that Is essentially Identical to the optimal infinite dimensional compensator: the first step determines the optimal compensator by letting the finite dimensional compensators converge to it; the second step reduces, if possible, the order of a large (converged) approximation to the optimal com- pensator. The first step, which is the one involving control theory and approximation theory for distributed systems, is the subject of this paper. For the second step, a simple modal truncation of the large compensator some- times is sufficient, but there are more sophisticated methods in finite dimen- sional control theory for order reduction. For example [G8, Ml], we have found balanced realizations [M2] to work well for reducing large compensators. The approximation theory in this paper follows from the application of approximation results in [B6, G3 , G4] to a sequence of finite dimensional optimal LQG problems based on a Ritz-Galerkin approximation of the flexible structure. For the optimal linear— quadratic control problem, the approxima- tion theory here is a substantial improvement over that in [Gl] because here we allow rigid-body modes, more general structural damping (including damping in the boundary), and much more general finite element approximations. These generalizations are necessary to accommodate common features of complex space structures and the most useful finite element schemes. For example, we write the equations for constructing the approximating control and estimator gains and finite dimensional compensators in terms of matrices that are built directly from typical mass, stiffness and damping matrices for flexible struc- tures, along with actuator influence matrices and measurement matrices. For the estimator problem, this paper presents the first rigorous approx- imation theory. (We have used less complete versions of the results in previ- ous research [G6 , G7 , G8, Ml].) As in the finite dimensional case, the infin- ite dimensional optimal estimation problem is the dual of the infinite dimen- sional optimal control problem, and the solutions to both problems have the same structure. Because we exploit this duality to obtain the approximation theory for the estimation problem from the approximation theory for the opti.^ control probla., the .:.ly.l= in thl= papor is al.o=t entirely deter- Mni^tio. we di.ou=s the .toohastlc interpretation of the estimation prohle. and the approximating state estimators briefly, but -e are concerned mainly „ith deterministic ..uestions about the structure and convergence of approxima- tions to an infinite dimensional oom^nsator and the performance - especially stability - Of the closed-loop syst»s produced by the approximating comper. sators. Next, an outline of the paper should aelp. Tn. paper has two main parts. .hiC correspond roughly to the separation of the optimal UJO probl«. into an optimal linea^^uadratic regulator prohle. and an optimal state estimation problem. The first half. Sections 2 through 6. deal with the control system and the optimal resistor problem. Sections 7 throu* i« treat the state estimator and the compensator that is formed by applying the control law of the first half of the paper to the output of the estimator. section 2 defines the abstract model of a flexible structure and the energy spaces to he used throughout the paper. We ass«e a finite n«ber of actuators, since this is the case in all applicaUons. and we ass-e that the actuator influence operator is funded. Section 2 also establishes certain msa-ematical properties of the o,...loop system that are useful in control and approximation. To our ^owledge. the exponential stability theory in Section 2.3 is a new result, and we find it interesting that such a simple Lyapuno, functional accommodates such a general damping model. section 3 discusses the linear-quadratic optimal control probla. for the aistrit^ted model of the structure and establishes some estimates inyolving lx,u„ds on solutions to Infinite dimensional Rlooatl equations and oper^loop and olosed-loop decay rates. We need these estimates for the subsequent approximation theory. To get the approximation theory for the estimation problem, we have to give certain results on the control proble. m a more ge,. eral form than would be necessary were we Interested only In the control proh- 1- for flexible structures, aerefore. In Section 3, as In Sections 5 and 7. we first give some generic results applicable to the UO problem for a variety Of distributed systems and then apply the generic results to the control of flexible structures. Because we assume a finite number of actuators and a bounded Input opera- tor, the optimal feedback control law consists of a finite number of bounded linear functlonals on the state space, which Is a Hubert space. ,„ls means that that the feedback law can be represented In terms of a finite number of vectors, which wa call functional control gains, whose Inner products with the generalized displacement and velocity vectors define the control law. For any flnlt^rank. bounded linear feedback law for a control system on a Hubert space, the existence of such gains Is obvious and well known. A functional control gain for a flexible structure will have one or more distributed com- ponents, or kernels, corresponding to each distributed component of the struc- ture and scalar components corresponding to each rigid component of the struc- ture. we introduce the functional control gains at the end of Section 3, and we introduce analogous functional estimator gains In Section 7. ihe functional gains Play a prominent roll l„ our a^lysls. They give a concrete representa- tion Of the infinite dimensional compensator and provide a criterion for con- vergence of the approximating finite dimensional compensators. We develop the approximation scheme for the control problem in Section 4. The idea is to solve a finite dimensional linear-quadratic regulator problem for each of a sequence of Ritz-Galerkin approximations to the structure. We develop the approximation of the structure in Section 4 .1 and prove conver- gence of the approximating open-loop systems. The approximation scheme includes most finite element approximations of flexible structures. For con- vergence, we use the Trotter-Kato semigroup approximation theorem, which was used in optimal open-loop control of hereditjiry systems in [B5] and has been used in optimal feedback control of hereditary, hyperbolic and parabolic sys- tems in [B6, Gl, G3] and other papers. The usual way to invoke Trotter-Kato is to prove that the resolvents of the approximating semigroup generators con- verge strongly. To prove this, we introduce an inner product that involves both the strain-energy inner product and the damping functional, and show that the resolvent of each finite dimensional semigroup generator is the projec- tion, with respect to this special inner product, of the resolvent of the ori- ginal semigroup generator onto the approximation subspace. The idea works as well for the adjoints of the resolvents, and when the open-loop semigroup gen- erator has compact resolvent, it follows from our projection that the approxi- mating resolvent operators converge in norm. In Section 4.2, we define the sequence of finite dimensional optimal con- trol problems, whose solutions approximate th«i solution to the infinite dimen- sional problem of Section 3. The solution to each finite dimensional problem is based on the solution to a Riccati matrix equation, and we give formulas for using the solution to the Riccati matrix equation to compute approxima- tlons to the functional control gains as linear combinations of the basis vec- tors. Section 5.1 summarizes some generic convergence results from [B6, G3 , G4] on approximation of solutions to infinite dimensional Riccati equations. Sec- tion 5.2 applies these generic results to obtain sufficient conditions for convergence ~ and nonconvergence ~ of the solutions of the approximating optimal control problems in Section 4.2. A main sufficient condition for con- vergence is that the structure have damping, however small, that makes all elastic vibrations of the open-loop system exponentially stable. This is a necessary condition if the state weighting operator in the control problem is coercive. In Section 6. we present an example in which the structure consists of an Euler- Bernoulli beam attached on one end to a rotating rigid hub and on the other end to a lumped mass. We emphasize the fact that we do not solve, or even write down, the coupled partial and ordinary differential equations of motion. For both the definition and numerical solution of the problem, only the kinetic and strain energy functional s and a dissipation functional for the damping are required. We show the approximating functional control gains obtained by using a standard finite element approximation of the beam, and we discuss the effect on convergence of structural damping and of the ratio of state weighting to control weighting in the performance index. As suggested by a theorem in Section 5. the functional gains do not converge when no struc- tural damping is modeled. In Section 7, we begin the theory for closing the loop on the control system. We assume a finite number of bounded linear measurements and 8 construct the optimal state estimator, which is infinite dimensional in gen- eral. The gains for this estimator are obtained frcan the solution to an infinite dimensional Riccatl equation that has the same form as the infinite dimensional Riccati equation in the control problem. We call these gains functional estimator gains because they are vectors in the state space. Since the approximation issues that this paper treats are fundamentally deterministic, we make the paper self contained by defining the infinite dimensional estimator as an observer, although the only justification for cal- ling this estimator and the corresponding compensator optimal is their interpretation in the context of stochastic estimation and control. We dis- cuss the stochastic interpretation but do not use it. We say estimator and observer interchangeably to emphasize the deterministic definition of the estimator here. With the optimal control law of Section 3 and the optimal estimator of Section 7, we construct the optimal compensator, which also is infinite dimen- sional in general. The transfer function of this compensator is irrational, but it is still an m(number of actuators) p (number of sensors) matrix func- tion of a complex variable, as in finite dimensional control theory. The optimal closed-loop system consists of the distributed model of the structure controlled by the optimal compensator. Approximation of the optimal compensator is based on approximating the infinite dimensional estimator with the sequence of finite dimensional estima- tors defined in Section 8.1. The gains for the approximating estimators are given in terms of the solutions to finite dimensional Riccati equations that approximate the infinite dimensional Riccc^ti equation in Section 7. Although defined as observers, these finite dimensional estimators can be interpreted as Kalman filters, as shown in Section 8.2. In Section 8.3. we give formulas for finite dimensional functional estimator gains that approximate the func- tional estimator gains of Section 7. These approximating estimator gains indicate how closely the finite dimensional estimators approximate the infin- ite dimensional estimator. In Section 8.4, we apply the Riccati equation approximation theory of Section 5 to describe the convergence of the finite dimensional estimators. Most of the results in Section 8 are analogous to results for the control problem and follow from the same basic approximation theory, but certain differences require careful analysis. There is an important difference in the way that the Riccati matrices to be computed are defined in terms of the fin- ite dimensional Riccati operators. Indeed, the Riccati matrix equations to be solved numerically might seem incorrect at first. To demonstrate that the finite dimensional estimators that we define in Section 8.1 are natural approximations to the optimal infinite dimensional estimator, we show in Sec- tion 8.2 that each finite dimensional estimator is a Kalman filter for the corresponding finite element approximation of the flexible structure. The brief discussion in Section 8.2 is the only place in the paper where stochas- tic estimation theory is necessary, and none of the analysis in the rest of the paper depends on this discussion. In Section 9.1, we apply the nth control law of Section 4 to the output of the nth estimator to form the nth compensator. (The order of approximation is n.) The nth closed-loop system consists of the distributed model of the structure controlled by the nth compensator. Since each finite dimensional 10 estimator is realizable, the nth compensator and the nth closed-loop system are realizable. In Section 9.2, we discuss how the sequence of realizable closed-loop systems approximates the optimal closed-loop system. Probably the most important question here is whether exponential stability of the optimal closed-loop system implies exponential stability of the nth closed-loop system for n sufficiently large. We have been ab] e to prove this only when the approximation basis vectors are the natura;. modes of undamped free vibration and these modes are not coupled by structural damping. That this stability result can be generalized is suggested by the results in Section 9.3, which describe how the transfer functions of the finite dimensional compensators approximate the transfer function of the optimal compensator. In Section 10, we complete the compensator design for the example in Sec- tion 6. Assuming that white noise corrupts the single measurement and that distributed white noise disturbs the structure, we compute the gains for the finite dimensional estimators and show the functional estimator gains. As in the control problem, the functional gains do not converge when no damping is modeled. We apply the control laws computed in Section 6 to the output of the estimators in Section 10 to construct the finite dimensional compensators, and we show the frequency response of these a:.mpensators. As predicted by Section 9.3, the frequency response of the nth conpensator converges to the frequency response of the optimal infinite dimensional compensator as n increases. In Section 10.3, we discuss the structure and dimension of the finite dimensional compensator that should be implanented. We conclude in Section 11 by discussing where the approximation theory presented in this paper is most complete and what further results would be most important. 11 2. The Control System We consider the gystem Mt) + D^M + Aox(t) = Bou(t). t > 0. (2.1) where x(t) is in a real Hilbert space H and u(t) is in r" for some finite m. The linear stiffness operator A, is densely defined and selfadjoint with com- pact resolvent and at most a finite number of negative eigenvalues. We will postpone discussion of the damping operator D^ momentarily, except to say that it is symmetric and nonnegative. l^e input operator B^ is a linear operator from r" to H, hence bounded. By jiatuiai Sfides. v£e jiill aaaa iiie eig^qyeotors <fi^ sif IM eigenyalue problem (2.2) From our hypotheses on A^. we know that these eigenvalues form an infinitely increasing sequence of real numbers, of which all but a finite number are positive. Also, the corresponding eigenvectors are complete in H and satisfy (2.3) (These properties of the eigenvalue problem (2.2) are standard. See. for example. [Bl] . [Kl] .) For A,. > O m - 'TT ^c, , v>,4. ' A.J / u, u>j - f Xj is a naiUJCai freaueno.v . 12 Remark 2.2. Our analysis includes the ^stem • * • ^t.x(t) + D-x(t) + A.x(t) = B-u(t), t > 0, " " " (2.l') where the mass operator Mq is a selfadjoint, bounded and coercive linear operator on a real Hilbert space Hq. The operators Aq, Bq and D^ in (2.1') have the same properties with respect to H^ that the corresponding operators in (2.1) have with respect to H. To include (2.1') in our analysis, we need only take H to be H^ with the norm- equivalent inner product <*'*^h '^ <Mq\'>jj , and multiply (2.1') on the left by vi^^. In H, the operator M^^f^Q is selfadjoint with compact resolvent, and M^ Dq is symmetric and nonnegative. With no loss of generality then, we will refer henceforth only to (2.1) and assume that the H-inner product accounts for the the mass distribution. D 2.1 The Energy Spaces and the First-order Form of the System The Elastic-Strain - Energy Space V and Total - Energy Space E We choose a bounded, selfadjoint linear operator Aj^ on H such that Aq = Aq + k. is coercive; i.e., there exists p / for which <AqX,x>jj I p||x||^. xbKAq) = D(Aq). (2.4) Since Aq is bounded from below, there; will be infinitely many such Aj^'s. In applications like our example in Section 6, It is natural to select for A^ an operator whose null space is the orthogonal complement (in H) of the eigen- space of Aq corresponding to nonpositive eigenvalues. Obviously, any A^ that makes Aq coercive must be positive definite on the nonpositive eigenspace of Aq. 13 With k^ chosen, we define the Hilbert space V to be the completion of D(Aq) with respect to the inner product <v-,,V2>y = <A.qV^,V2>u, v- and ~l/2 '^'1/2 ~i/2 v, e D(Aq). Note that V = D(Aq ) and <Vj^,V2>y = <Aq v-,Aq V2>y. (Since A^ is a bounded operator on H, different choices of k^ yield V's with equivalent norms, thus containing the same elements) . In the usual way, we will use the imbedding VC H = H' C V, where the injections from V into H and from H into V are continuous with dense ranges. We denote by Ay the Riesz map from V onto its dual V; i.e., <v,v. > = (AyV, )v, V. ,v e V. (2.5) /^ Then A_ is the restriction of Ay to D(Aq) in the sense that (AyVj)v = <v,AqVj^>jj, Vj e D(Aq), v e V. (2.6) Now we define the total energy space E = V x h, noting that when A- is coercive and x(t) is the solution to (2.1), then | | (x(t) ,x(t)) | |„ is twice the total energy (kinetic plus potential) in the system. We want to write (2.1) as a first-order evolution equation on E. To to this, we must determine the appropriate semigroup generator for the open-loop gystem. We will derive this generator by constructing its inverse explicitly, and then we will try to con- vince the reader that we do have the appropriate open-loop semigroup genera- tor. The approach seems mathematically efficient, and we will need the inverse of the generator for the approximation scheme. First, we must state our precise hypotheses on damping and discuss its representation. 14 The Damping Functional and Operator Actually, we do ufit require an operator D^ defined from some subset of H into H. Bather, we assume only that there exists a damping functional (2.7) such that dp is bilinear, symmetric, continuous (on V X V) and nonnegative. If we have a symmetric, nonnegative dai;iping operator Dq defined on D(Ao) such that Dq is bounded relative to Ap. then <DoV3^.V2>h defines a bilinear, symmetric, bounded, nonnegative functional on a dense subset of V X V. In this case, the unique extension of this functional to V X V is d_. (That D being A^-bounded implies continuity of <Dq','>^ with respect to the V-norm follows from [Kl. Theorem 4.12, p. 292].) Under our hypotheses on dg, there is a unique bounded linear operator A^ from V into V such that dQ(v,v^) = (AjjV^)v, Vj,v e V. (2.8) The operator (A^ A^) is then a bounded linear operator frcan V to V, and (Ay Ajj) is selfadjoint (on V) because dg is symmetric. Also dQ(v.v^) = <vA~\v^>^ = <A-\y.v^>^ V V e V. (2.9) Remark 2.2. We chose to begin our description of the control system model with (2.1) because its form is familiar in the context of flexible structures. The stiffness operator A^, for example, is the infinite dimensional analogue of the stiffness matrix in finite dimensional structural analysis. In appli- cations like the example in Section 6, though it is often easier to begin 15 with a strain-energy functional from which the correct strain-energy inner product for V is obvious. The stiffness operator is defined then in terms of the Riesz map for V (see [S3] for this approach), rather than V being defined in terms of the stiffness operator; specifically, t^ is defined by (2.6) with D(A ) = A'^^'H. Either way, the relationship between A^ and V is the same. But the only thing that needs to be computed in applications is the V-inner pro- duct; an explicit Aq need not be written down. O The Semigroup Generator We define A~-^ e L(E,E) by A-^ -Av ^D "%) (2.10) This operator is clearly one-to-one, and its range is dense, since V is dense in H and D(Aq) is dense in V. Now, we take (2.11) Direct calculation of the inner product shows <^~^ I ' I >^ = -dn(h.h), h h E (2.12) so that t is dissipative with dense domain. Also, since D(A ) = E, A is max- imal dissipative by [Gl. Theorem 2.1]. Therefore, If generates a Cq- contraction semigroup on E. Finally, the open-loop semigroup generator is 16 A = A + X s] ■"*' ■ «'' (2.13) where A. is the bounded linear operator discussed above. With B = I " I e L(R°.E). & (2.14) the first-order form of (2.1) is z(t) = Az(t) + Bu(t), t>0 where z = (x,x) e E. (2.15) To see that A is indeed the appropriate open-loop semigroup generator, suppose that Aq is coercive (so that Aj^ = 0) and that we have a symmetric, nonnegative A.-bounded damping operator Dq. Then the appropriate generator should be a maximal dissipative extension of the operator o r . - T o ■A© ° _J 1 . D(A) = D(Aq) XD(Aq) (2.16) o It is shown in [Gl, Section 2] that A has a unique maximal dissipative exten- o sion, and it can be shown easily that the A defined above is an extension of A after noting that, in the present case. (Av\)Id(Ao) = ^\' (2.17) We should note that Showalter [S3, Chapter VI] elegantly derives a semigroup generator for a class of second-order systems that includes the flexible- structure model here. The presentation here is most useful for our approximation theory because of the explicit construction of the inverse of 17 the semigroup generator. For the purposes of this paper, we do not need to characterize the operator A itself more explicitly, but we should make the following points. First, from A we see D(A) = {(x.x): X e V, X + ^~^/\^x e D(Aq)}. (2.18) In many applications, especially those involving beams, the "natural boundary conditions" can be determined from (2.18) and the boundary conditions included in the definition of DA^) . in the case of a damping operator that is bounded relative to aJ ^, D(A) = D(Aq)xV. If the damping operator is bounded rela- tive to A^ for n < 1, then A has compact resolvent. In many structural applications, the open-loop semigroup is analytic, although this has been proved only for certain important cases. Showalter obtains an analytic semigroup when the damping functional is V-coercive; for example, when there exists a damping operator D^ that is both A^-bounded and as strong as A^. Such a damping operator results fran the Kelvin- Voigt viscoelastic material model. Also, it can be shown that the semigroup is ana- lytic for a damping operator equal to c^A^ for 1/2 <. m i 1 and c^ a positive scalar. The case \i = 1/2, which produces the same damping ratio in all modes, is especially common in structural models, and Chen and Russell [CI] have shown that the semigroup is analytic for a more general class of damping operators involving A^'^. Finally, we can guarantee that the open-loop semigroup generator is a spectral operator (i.e., its eigenvectors are complete in E) only for a dampr- ing operator that is a linear combination of an H-bounded operator and a 18 fractional power of A.. However, nowhere do we use or assume anything about the eigenvectors of either the open-loop or the closed-loop semigroup genera- tor. The natural modes — of undamped free vibration — in (2.2) are always complete in both H and V. 2.2 The Adjoint of A -1 **-* Since (A^ Aj,) is selfadjoint on V, direct calculation shows that A (a""*") — the adjoint of A~ with respect to the E- inner product — is A -I ^ (2.19) Then ^* = (iT )~^ . Having TJ" explicitly facilitates proving strong conver- gence for approximating adjoint semigroups. 2.3 Exponential Stability The following theorem says that, if there are no rigid-body modes and if the damping is coercive (basically, all structural components have positive damping), then the open-loop system is uniformly exponentially stable. That the decay rate given depends only on the lower bound for the stiffness opera- tor and the upper and lower bounds for the damping functional is essential for convergence results for the approximating optimal control problems of subse- quent sections. The theorem is a generalization of Theorem 6.1 of [Gl] to allow more general damping, but the proof is entirely different and much nicer. The current {X'oof uses an explicit Lyapunov functional for the homo- geneous part of the gystem in (2.15). Recall that TO is the open-loop semi- group, with generator A, and E is the total energy space V x H. 19 Theorem 2.3 . Suppose that A^ and d^ are H-coercive. Let p be the positive constant in (2.4), and let 6- and 8, be positive constants such that SqII^IIh ^ **0<^'V^ iS^llvllJ. V e V. (2.20) Then MT(t)|| i (1 +^ + 8^8q/2)^^^ exp[-t/(^ +J;.+ 8^)] .tlO. ° (2.21) Proof . For y > max{ ^ , ■^) . define Q e L(E) as Q = (yl+Ay^Ap) A^^ I yi (2.22) Since (Ay A^) is selfadjoint and nonnegative on V, Q is selfadjoint and coer- cive on E. Define the functional p(*) on E by p(z) = <Qz.z>g = ydlxllj + llxllj) + 2<x,x>jj + do(x,x) where z = (x,x). Frcan (2.4), we have (2.23) 2|<x.x>jj| < ^||x||y||x||jj 1 yp^lzllg' (2.24) so that <r-ij||z||^ i p(z) i p ^IIzIIe (2.25) with P = (T^-^^+^l)"^- (2.26) 20 Next take z = (x.x) e D(A) and set ACx.i) = (Y-Y) « ^' ^ (x.i) = A-^^y.y) = (V^Dy-^'i^-y^ ' (2.27) Note X = y e V. Now. <QAz.z>, = <Q(y.y).A''<y'y>>E VV>v-»«^v-W^n^iiyi'H - Y<y.Av = - [llxll'-rdo(x.x) - llxlljl- (2.28) From (2.20) then. <QA..z>,i-nixn^(reo-i)MillS'^-l'^"^ c^-"' , „^ th. theorem follows ft" *^-^" ' 1 . X n (2.26) and (2.29). with f - ^+ 5^ ' II The Optimal Control Prcbl em Subsection 3.1 presents some preliminary deflmtm -- -^-^^-c ..... _^ : rr:::: r: r ^'^ Tb»se results are ge^rio i„ t^ ""' ^'*'='- eenerlo In the sense that the Hubert space E l, . -rlly the energy space of Section 2 a„. .. " """ necessanny represent . °""'°" '' '' ""• ' ^ "« y represent en abstract flexible structure as ,n Section 2 r second half of th^ ^ section 2. in the air the paper, having such generic results will allow us to obt . the approxlmatinn t-v.^ ^ °j-j-ww us to obtain Ki'i uAxmation theory for fho ^,,f>^ j i. y lor the Infinite dimensional stafp ocf . analogou. results Tor the control ..obi. s^bs J "°° "^ pr-oDiem. Subsection 3.2 gives <,nm« ^ tant ^Plications of the general results for ,. iBsuita for the case vhf>r>a *-v,« 1= that defined In Section 2. °°°"'°' ^='™ em nonnegatlve 3.1 The Generic Optimal Regulator Probl Let a linear operator A generate a C -^„, space E »nH ^ semigroup T(t) on a real Hllbert space E. and suppose B a L(r".E). Q c l(E E) . . n m ^ L(E,E) and R e Kr"), „ith q an. selfadjomt and R positive definite and symmetric :.e o tl -^^ -.IS to Choose the control u.. ^ ^^ "^ ^^^ ^^ functional ' ^ '"''^^ '° "''^"^^ ^^e cost """'" = / (-(t,.Ct,>, . <Ru(t,,u(t,>,m,dt, Where the state 2(t) is given by ^^'^^ 2(t) = T(t)2(0) + Lf. .„ , ^ /T(t-T,)Bu(T,)dt,. t 2 0. (3.2) ^^^^iflitiaa 1.1. AfunctionueL (o -m . S<0.-,U) 13 an MmmiOM ^fintrol ^^ ^ 22 initial state z, or simply an admissible control for z, if J(z,u) is finite; i.e., if the state z(t) corresponding to the control u(t) and the initial con- dition z(0) = z is in L2(0,»;E). Definition 3 .2 Let the operators A, B, Q, and R be as defined above. An operator n in L(E) is a solution of the Riccati algebraic equation if H maps the domain of A into the domain of A* and satisfies the Riccati algebraic equation A*n + HA - nBR~^B*n + q = o. (3.3) Theorem 3 .3 (Theorems 4.6 and 4.11 of [G4]). There exists a nonnegative sel- fadjoint solution of the Riccati algebraic equation if and only if. for each z e E. there is an admissible control for the initial state z. If n is the minimal nonnegative selfadjoint solution of (3.3). then the unique control u( ) which minimizes J(z.u) and the corresponding optimal trajectory z( ) are given by u(t) = -R ^B*IIz(t) (3.4) and z(t) = S(t)z, where SO is the semigroup generated by A-BR BII. Also, (3.5) J(z,u) = min J(z,v) = <IIz,z>g V (3.6) If, for each initial state and admissible control. 23 llm||z(t)|| = 0, (3.7) there exists at most one nonnegative selfadjoint solution of (3.3). If Q is coercive, (3.7) holds for each initial state and admissible control and S( • ) is uniformly exponentially stable. D We will refer to T(-) as the open - loop semigroup and to S(') as the Qptijqal closed-loop semigroup . To prepare for the convergence analysis in Sections 5 and 9, we must present now some rather arcane estimates for the decay rate of the closed-loop system in the optimal control problem. Tljsfires 3 .4 Suppose that the open-loop semigroup !(•) satisfies llT(t)|I i M^e°^ . t 1 0, (3.8) for positive constants M^ and a^, that H is the minimal nonnegative selfad- joint solution to (3.3), and that S(t) is the optimal closed-loop semigroup in Theorem 3.3. If there exists a constant F^ such that, for each z e E, OD / ||S(t)2||^dt i ^^(<^z,z>g + llzll^) (3.9) and a constant ^L such that II n II < M^. (3.10) then there exist positive constants Mj and oj, which are functions of J^ , J^. M^, and a^ only, such that 24 -02 1 Ils(t)|| < H^e ^ , t 2 0. (3.11) Proof . This follows easily from Theorems 2.2 and 4.7 of [ ]. D Lemma 3 .5 . Suppose that there exist positive constants M and a such that l|T(t)|| ± Me-'. t 1 0. ^^^^^ t If z(0) 8 E. h 8 L,(0,»;E). and z(t) = T(t)z(0) + / T(t-3)h(s)ds, then /||z(t)||^ dt L ^l|z(0)|| . ^Ilhll^^ L, (3.13) Proof . The result follows from (3.12), the convolution theorem [d1, page 95l] and the triangle inequality. D ieaaa 3 .6. suppose that E is finite dimensional and that the pair (Q,A) is observable (in the usual finite dimensional sense). Then there exists a con- stant M. which is a function of A, B and Q only, such that 7||z(t)||^dt L M(/ (<Qz(t),z(t)>g + ||u(t)||^)dt. (3.14) where z(t) is given by (3.2). Proof . The proof, which is at most a mild challenge, is based on the fact that the observability grammian 25 W(t)= / e^ W^dt is ooercive for any positive t. □ (3.15) The next theorem says, among other things, that if the open-loop control system decouples into a finite dimensional part that is stabilizable (in the usual finite dimensional sense) and an infinite dimensional part that is uni- formly exponentially stable, then the entire system is uniformly exponentially stabilizable, so that (3.3) has a nonnegative self adjoint solution. T^eorep) 3 .7. Suppose that there exists a finite dimensional subspace E-^C D(A) such that E^ and E^ reduce A (and T(t)), and write = r- Li ^ - m- Q = Qii Qi2 ^12 (3.16) where A^^ and k^^ are the restrictions of A to E^ and D(A) X Eq, respectively. Similarly, T(t) Fll^*^^ ] I T22(t)J- (3.17) Also, suppose that the pair (A^^.B^^) is stabilizable and that there exist positive constants M^^, a^^ and p such that l|T22(t)|| < M^e , t 2 (3.18) and 26 max{||B||.||Q||} 1 p. ^3 .^^^ i) Then there exists F e L(E,R°') sucsh that A-BF generates a uniformly exponentially stable semigroup on E. Also. (3.3) has a nonnegative selfad- > joint solution, and the minimal such solution satisfies (3.10) with M^ a func- tion of k^^, B^^, R. M^. a[ and p only. ii) If Qj2 = ^"^ ^^^ P^^^ ^"^l' *11^ ^^ observable, then there exists a unique nonnegative selfadjoint solution H to (3.3). and there exist positive r I constants M^ and a^ — which depend on A^^^. B^^, Q^^, R. M^. a^ and p only — such that the optimal closed-loop semigroup satisfies ||S(t)|| 1 M,e"' . t 1 0. ^^_^^^ Proof , i) To say that (A^.B-j) is stabilizable means that there exists a linear operator F^^ from E^ to r" such that each eigenvalue of A^^-B^jFii has negative real part. Hence A-BF generates a uniformly exponentially stable semigroup if F = [F^^ 0]. so that there exists an admissible control for each initial condition in E. It is easy to write down an upper bound for the performance index in (3.1) in terms of R. p . l\, a[ and the decay rate of exp( [A^^ -B^^ F^^ It). That the H^ in (3.10) depends only on A^^, B^i. R, M^- a^ and p follows then from the fact that F^^ is a function of Aj^. and Bj^j. ii) Clearly, (3.8) holds with M^ and a^ depending only on Aj^^, B^^, M^. o^ and F^^. Therefore, we have (3.8) and (3.10) with the bounds depending only on A^j, B^j, 0^1. R. i%. a[ and p . Finally, the existence of an ^^ for 27 (3.9) which depends only on these parameters follows from using (3.1) and (3.6) In applying Lemma 3.6 to the part of the gystem on E^ and then Lemma 3.5 to the part of the system on E^. Part 11) of this theorem then follows from Theorem 3.4. D Bsmsk 3.8 When we say In Theorem 3.4 that M^ and oj are functions of Mq, K^, Mj, and a^ only, we mean, for example, that for two optimal control problems on different spaces E, with different operators A, B, etc. , if the same con- stants Mq, m^, m^, and a^ work in (3.8)-(3.10) for both problems, then the same constants K^ and Oj will work in (3.11) for both problans. Similarly, in Theorem 3.7 11). as long as E^, A^^. b^^, Q^^, R, M^. o^ and p remain the same, the same M^ and Oj will work in (3.20) even if E^, Ajj. B^^ and Q^i change. Q 3.2 Application to Optimal (k>ntrol of Flexible Structures For the rest of this section. A^, A^, A, T(t) , B^ and B are the operators defined in Section 2 .1 . and E = V X H is the energy space defined there. Rfiaacls 3.9. Theorem 3.7 is useful mainly when all but a finite number of modes have coercive damping in the open- loop system and the damped and undamped parts of the open-loop system remain orthogonal. This is the case, for example, with modal damping. The next theorem does not require ortho- gonality of the damped and undamped parts of the ^stem, but it does require an independent actuator for each undamped mode. The situation of Theorem 3.10 is typical in aerospace structures: Any elastic component should have some structural damping, but rigid-body modes are common; for a structure to be controllable, an actuator is required for each rigid-body mode. □ 28 Theorem 3 .10- i) Suppose that A^ = b^bJ and that "a^ = Aq + A^ and ^f^ = d^ + A are H-coercive, so that there exist positive constants p . r and p such that, for all v e V, llvllv 1 pIIvIIh- (3,21) d^(v.v) 1 pIIvHh, (3,22) d^(v.v) L Yllvlly. (3,23) and max{||B^||.||Q|M|R||) i P (3,24) (The V-continuity of d^ implies (3.23).) Then (3.3) has a minimal nonnegative selfadjoint solution H. which satisfies (3.10) with l\, a function of p . y and P only . ii) Suppose gJ.so that <Qz,z>g I pllzllg, z e E. ^3 25) Then the optimal closed-loop semigroup satisfies ||S(t)|| 1 Mi^ ^ ' t 2 0. ^g 2gj where Mj and a^ are positive constants depending on p. y and p only. proof , i) The suboptimal control 2) u(t) = -B^[x(t) + x(t)] (3.27) produces a closed-loop system with exponential decay at least as fast as that in Theorem 2.1. The required upper bound in (3.10) follows then from (3.1), (3.6) and (3.24) ii) In this case, the M^ m (3.9) is p, and Theorem 3.4 yields the result. D Now we will consider the structure of the optimal control law in more detail. Since n e L(E.E) and E = V X H, we can write n Do Hi (3.28) where n^, e L(V,V),n, e UE,V),IL, e L(H,H). and n^ and I^ are nonnegative and selfadjoint. With z = (x,x). as in Section 2. (3.4) becomes u(t) = -R lB*[IlJx(t) + I]2x(t)I. (3.29) Since Bq e l(r'",H), we must have vectors b^g H, li i i m. such that m BnU = Z b.u i=l l"i (3.30) for T_ nin " = ^V2 ••• %3 «R (3.31) Also, for h c H, B„h = [<b^.h>jj <b2,h>pj ••• <b^.h>H]T. Since U^xd) and U^xU) are elements of H. we see from (3.29) (3.32) and 30 (3.32) that the components of the optimal a:)ntrol have the feedback form Uj^(t) = - <fj|^,x(t)>y - <g^,x(t)>y, i = l....,m, where f . e V and g. e H are given by (3.33) f, = i(R-^),jn,b.^ J=-'- (3.34a) «i = ,V'''^ij°^'j' ' = ' "• J--"- (3.34b) We call f . and g. functional gains . 31 4. The Approximation Schane 4.1 Approximation of the Open- loop System Hypothesis l.J,. There exists a sequence of finite dimensional subspaces V of V such that the sequence of orthogonal projections P„ converges V-strongly to the identity, where Py^ is the V- projection onto V^^. Also, each V^ is the span of n linearly independent vectors e.. Since it should cause no confusion, we will omit the subscript n and write just e., keeping in mind that the basis vectors may change from one V to another, as in most finite element schemes. Also, we will refer to the Hu- bert space E^ = Vjj X V^^, which has the same inner product as E = V X H. For n 2 1» we approximate x(t) by x„(t) = I 5j(t)ej, "^ (4.1) where $(t) = (^^(t), 52<t), ... 5n<t))^ satisfies M"V(t) + D"5(t) + K"5(t) = B"u(t), (4.2) and the mass matrix m", damping matrix d", stiffness matrix k" , and actuator influence matrix B^ are given by 32 Bg= [<ei,bj>Hl. Of course. (4.2) can be written as i = aN + B% (4.2') where and Note n = n"^. i'^'i'^ (4.4) A" = M'X Ol (4.5) n = r I I . b" = Throughout this paper, we use the superscript n in the designation of matrices in the n^^ approximating syste. and control prohlaa. like A" B". M". etc. Hence the superscript n indicates the order of approximation - and it never indicates a power of the matrix. By M^. we denote the inverse of the mass matrix m". 4.«„ ir, thP n^^ aDoroximation, we use the in the designation of a linear operator in the n appr 1 « A cinH R are the operators whose matrix the subscript n. For example. A^ and B„ are tne ope representations are a" and B^ respectively. 33 For convergence analyels. ,t la useful to note tnat (4.1) ana (4.2, or (4.2') are equivalent to ''n^t) = A„z„(t) + B„u(t). (4.6) where z = (v y \^ r, . , a <^.Xn'' H„. and A„ . L(E„) and B„ . lCb". e„, are the operators whoee ..atrlx repreaentatlons are given in (4.5). Al«,. for any real X. Is equivalent to (^V + in" + ..n, 1 (4.7) •^ XD" H- K")a^ = (;,M" + D")pl + m^P^ and (4.8) a^ = XqI - 3I if (4.9) vJ = ?J. __. ,.i 5 J^a.e^ and h^ = J^pJe^, ^^1^2. (Substituting A" and (4.10) mto (4.7) yields (4.8) and (4.9)). (4.10) Next, we will prepare to invoke the Trotter-Kai-n .on,-.^ xrotter-Kato semigroup approximation theorem to show how (4.2), (4 2') anrt (a «\ v-*.^/. M.2 } and (4.6) approximate (2.1) and (2.15) nrat. „e „U1 treat the case in „Mo. .„ i3 ^.^^,„ <,„ ^,^,_^^ ^^^^^^ - that A^-o and T„-*„; the general caee ie a etraight-fo^ard extension, .or *0 coercive, the ope^loop semigroup generator A is „axi.al dissiPative. Also. ror each n. A„ is dissipative on E„. r.. .ain idea here is to project ,.-A,- 34 onto V^ in a certain inner product and observe that the result is exactly a-P.y, where A„ is the operator on V„ in (4.6) and (4.7). Of course, we need only do this for real X,>0. For real X>0 then, define an innerprcduct on V by <-.->X = ^'<*'*>H ^ ^^O^*'*^ ^ ^*''V (4.11) Under the hypotheses in Section 2 on d^, ''r>^ is norm-equivalent to <-.->v. For n 11. let P^(X) be the projection of V onto V„ in the inner product <-.->j^. Now let h^.hj, e H and note that 1 \ / .1 (X-A) is equivalent to V h V 2 / ' UM (4.12) (::)■ -iiMSl (4.13) With A~^ from (2.10). (4.13) is equivalent to (I . ^-\ . X^A-^)vl . (,,-1 . A-lA,)hl - A-lh^ ^^^ -lv,2 14) and If 1 J- v^ = Xv - h . (4.15) ^n = ^n^*-^^^ ^""^ ''n = ^n^^^^ ' (4.16) it follows from (4.11) and (4.14) that 3^) = <ei.(>.^A-l+XA-\+l)vSy = <ei.(XAolM-^Ajj)hl+A-lh2>y. >V (4.17) and from (4.15) that 2s _ .. 2 ^^i'Vx = <«i'V >x = >^<ei'V^>x - <ei,h^ X • (4.18) NOW. for h^ = hj s V„. h^ = hj e v„, ,„d vj. v^„. hj and h^^ written as in (4.10). (4.17) and (4.18) yield (4.8) and (4.9) again. This shows that rp„(x) 1 I P„(X)J<^-A)"'|e„ = <^-An)"'. (4.19) which yields I P^(X) J<^-A> ^En = <^-An>"'PEn (4.20) where P^^ ig j^ E-^jiQififiLLfiD of E .aitfi E„. The projection P^^ can be written _ Pvn 1 L PHnJ ' ^En (4.21) where P^^ is the V-projection onto V^. as before, and Pj^^ is the H-projection onto V^. Since the V-norm is stronger than the H-norm and the norm induced by 36 the X-lnnerproduct is equivalent to the V-norm. it follows from Hypothesis 4.1 that (>.-A^)~ Pg^ converges E- strongly to (X-A)~^ as n-» «. Now, with A extended to E^ as, say. n(Pg^-I) , Trotter-Kato [Kl, page 504, Theorem 2.16] yields the following. Theorem 1.2. For Aq coercive, let T^ C) be the (contraction) semigroup gen- erated on E^ by A^. Then, for each t 2 0, T^(t)?^^ converges strongly to T(t), uniformly in t for t in bounded intervals. In the general case, when A^ is not coercive, the open-loop generator A is obtained from the dissipative t by the bounded perturbation in (2.13), so that [G3, Theorem 6.6] yields the following generalization of Theorem 4.2. gorolAarv 4.3. Let T^C) be the semigroup generated on E by A . Then, for each t 2 0, T^(t )Pg^ converges strongly to T(t), uniformly in t for t in bounded intervals. ^^'^^^W l-l. When A has compact resolvent, (Ji-A^) ^Pg^ converges in L(E) t< ICfifif. This follows from (4.20) and a standard result that the projections of a compact linear operator onto a sequence of subspaces converge in norm if the projections converge strongly to the identity, as do P and P (X) . D That the adjoint semigroups also converge strongly follows from an argu- ment entirely analogous to the proof of Theorem 4.2. In particular, equations like (4.11)-(4.17) are used to show that 37 I p„a)j^^-^^ ^En = <^-<^'''En (4.22) In showing this. A'* is used as A"^ was used above. Also, care must be taken to calculate A* with respect to the E-inner product. The result is Theorem 4.5. Let T (') be the sequence of semigroups in Corollary 4.3. Then, for each t 1 0, T*(t)Pg^ converges strongly to T*(t). uniformly in t for t in bounded intervals. Finally, for the approximation to the actuator influence operator B e L(R°,E), recall B e L(r'"» \) . the operator whose matrix representation is the matrix b" in (4.5). From (4.3), it follows that ^n " ^En^- (4.23) Since B has finite rank m. B^ and B* converge in norm to B and B . respec- tively. 38 4.2 The Approximating Optimal Control Problems The n^h optimal control problem is: given z^(0) = (x„(0) .x„(0)) e E^, choose u e ^(O.-;!^) to minimize J (z„(0).u) = J«Q„z„(t).z„(t)>, . <Ru(t).u(t)>Rm)dt. ^^ 24) where Q^= PEnQlEn* ^^ ^^^""^^ H,,,,,^ 4.6. For each n > 1 and z„(0)e E„. there exists an admissible con- trol (Definition 3.1) for (4.6) and (4.24). D .-.■ rr.. Hvnothesis 4.6 is that, for each n. the system A sufficient condition for Hypothesis 't.o (A ,B ) be stabilizable. By Theorem 3.1. the optimal control u^(t) has the feedback form u^(t) = -R'^^Dh^n^t) (4.25) 4- . nn F TT i3 nonnegative and selfadjoint. and !!„ where n^ is a linear operator on E^. IJ^ i3 nonn«6 satisfies the Riccati equation aX - Dn^n - nnBn«''<°r. ^ ^n = «• (4.26) . A ^ (A m) ha' at least one nonnegative. selfad- As a result of Hypothesis 4.6. (4.26) ha. at xe ,oint solution. The minimal such solution is the correct H, here. If the system (A^,Q„) is observable, then II„ is the unique nonnegative. selfadjoint solution to (4.26). and is positive deflate. If we write D, as 39 \ = non nin then (4.25) becomes (4.27) The feedback law (4.28) can be written in functi as in Section 3. We have (4.28) onal-feedback form, just where "n(t) = [u,„(t) U2„(t)...u^^(t)]T. (4.29) and "in^t) = - <fin,x„(t)>y - <6^^.;^(t)>j^, 1 < i i „. (4.30) in j?i^^ ^j^lnPHnbj' H i 1 m. g m T (R^) jll'' 'ijI^nPHn^j - 1 < i i m. (4.31a) (4.31b) Of course, f^^ ^nd g^^ are the n^^ approximations to the functional gains f . and gi in (3.34). "" In Section 5, (4.25)-(4.31) will be useful for studying how t^e solution to the nth ,p,,^^ ^^^^^^^ ^^^^^^ ^^^^^^^^^ ^^ ^^ ^^^^^_^^ ^^ ^^ ^^^^^^ probl^ Of section 3, but for numerical solution of the nt^ p.^blan, we need the matrix representations of Uiese equations. We will need the following grammian matrices: 40 K" = [<e^,ej>v] = k" + t<Aiei.ej>H] (4.32) and .0 m"j (4.33) Note : The matrix W~" will be the Inverse of W"^. The superscript n on any matrix indicates the order of approximation, not a power of the matrix. Also, recall the note following (4.5). Now recall Q^ = PEnQlEn* ^^'^°® Q = Q* e L(E) and E = V X H, we can write Q = Qo Qi Q2J where Q^ = Q^e L(V) , Q^ e L(H,V), and 02=02^ L(H) lation shows that (4.34) Straightforward calcu- (4.35) where Q" is the matrix representation of Q and q" is the nonnegative, sym- metric matrix "'n 0? (4.36) with "0 [<ej^.QQe.>y], 41 °1 = [<ei,Qiej>v], (4.37) QJ = [<e^.Q2ej>H]. Also, recall that A and B are the operators whose matrix representations are given by (4.5), and note that the matrix representations of A^ and B^ are W""(A")'^W" and (B")%", respectively. With the matrix representation of II denoted by H"* the Riccati operator equation (4.26) is equivalent to the Riccati matrix equation W""(A")%"II" + IPA" -I1"B"R"^(B")'^W"IP + Q" = 0. (4.38) While n is selfadjoint, D" in general is not symmetric, but the matrix ^ = „njjn (4.39) is symmetric and nonnegative, and positive definite if H^ is. Premulti plying (4.39) by W", we obtain (A")''^n" + &^A" - frB"R"l(B")'^n" + Q" = 0. (4.40) which is the Riccati matrix equation to be solved numerically. Now we need one more set of matrix equations for the numerical solution of the n^*^ optimal control problan. Since the functional gains f^^ and g^^ are elements of V^, they can be written as n f^ n g^ ^in = .^, Pj ^j ^^ 8in = .^ Pj ^j' i = 1. •••.«>. J-1 J"-^ (4.41) f . e . f • g • *•* where p ■"■, p ■"■ e R". We need equations for p ^ and p ■"" in terms of IF. One 42 way to get these equations is to partition lP(obtained from (4.39)) and then work out the matrix representation of (4.31), However, another approach is more instructive because it relates the present Hllbert space methods to the standard finite dimensional solution of the n^^ optimal control problan. ,th The n optimal control problem can be stated equivalently as: giv Tl(0) = [5(0)T.^(0)T] e R2n ^^^^^ ^ ^ L_(0.a;Ri°) to en minimize '^^(r\(0).u) = y(T,(t)^"T,(t) + u(t)'rRu(t))dt, iTiT (4.42) where Ti(t) = [4(t)^Ut)'^]^ satisfies (4.2'). For (4.2') and (4.42), the optimal control law is ,^. "n = -^ ^B"D"Ti(t) Tin where IP is the minimal nonnegative, symmetric solution to (4.40). (4.43) Since § is related to x^ by (4.1), the optimal control u„ in (4.43) must be equal to the optimal control u^ m (4.29) -(4.31) . Substituting (4.41) into (4.30) yields in - (p ')^"5 - (/^)M = -[(pSt (^^i)T]w^, (4.44) Then, using (4.44) and equating (4.29) and (4.43) yiel ds ^1 ^2 P P P 'm i,"^ ,''■■■ f«" ,-n TinnnD-1 = w""^ IFb'^r (4.45) W§ iiow Mve ihe complete ^plMtjon to the n^^ optimal control problem : 43 The Riccatl matrix equation (4.40) is solved for 6^; then the optimal control is given by (4.43). and equivalently by (4.29)-(4.30) with the functional gains f,„ and g^^ given by (4.41) and (4.45). In the next section, we will give sufficient conditions for the solution to the n^^ optimal control problem to converge to the solution to the optimal control problem in Section 3 for the original infinite dimensional system. 44 5. Convergence As in Section 3. subsection 5.1 will state some results for the optimal linear regulator problem involving generic linear operators A, B. Q, etc.. on an arbitrary real Hilbert space E, and subsection 5.2 will expand upon these results for the particular class of control problems treated in this paper. 5.1 Generic Approximation Results Let the Hilbert space E and the linear operators A. KO, B. Q and R be as in Section 3. Suppose that there 1. a sequence of finite dimensional sub- spaces E^. with the projection of E onto E„ denoted by ?£„• such that Pe^ con- verges strongly to the identity as n ^- » , and suppose that there exist sequences of operators A^ e L(E„) . B^ . L(R°.E^). Q„ = Q* e L(E^). Q^ 1 0. such that we have the following strong convergence. For all z e E and t^O, exp(Aj^t)PEnZ "> T(t)z (5,1) and exp(A*t)Pg z -> T*(t)z (5^2) as n -> -. uniformly in t for t in bounded intervals; for each u e R . BjjU -» Bu; (5.3) for each z e E, °nW "^ ^- (5.4) IllS,^i.l. Suppose that for each n there is a nonnegative. selfadjoint linear operator D^ on E„ which satisfies the Riccati algebraic equation (5.5) If there exist positive constants M and p. independent of n. such that llexp([A^-B^R-Vl]j^]t)|| < Me"Pt. t 2 0, (5.6) and if II n^ii is bounded uniformly in n. then the Riccati algebraic equation (3.3) has a nonnegative selfadjoint solution H^ and, for each z e E. and (5.7) exp([A^-B^R-lBj]^]t)P En' z -» S(t)z uniformly in t 2 0, where SC) is the semigroup generated by A-BR-lfi^ . there exists a positive constant 6, independent of n, such that (5.8) If Qnl6. (5.9) then II u^ii being bounded uniformly in n guarantees the existence of positive constants M and p for which (5.6) holds for all n. ^Tfiof. The theorem follows from Theorem 5.3 of [G4] when the operators A . Q andn^ are extended to all of E by defining them appropriately on Ej^. For the details of this procedure, see Section 4 of [Gl] . Or better. Banks and Kun- isch [B6] have modified Theorem 5.3 of [G4] to obtain essentially the present theorem without using the artificial, and rather clumsy, extensions to E"^ m the proof. D 46 IS.^ 1.1 I.e .tron, oonvergenoe in ,5.7, Implies u„lfo„ „o™ convergence Of the optimal feedback laws: "^nVE_ - B*ni| -^ (5.10) as n n to^r. r.U r„uc«3 ..„. tne seara^„.„t„e.. c.1^ ,„, ,^^ ,„, ,,^ ,^^^^ di^enaicnallty of the control space r". See e,^Uo„3 (4.23, an. <4.24, of [Gl]. D ^^mrss 5.3 Assume the hypotheses of Theorem 5.1 but do not assume (5.6, or <^-^>- If II n„|| is bounded uniformly m n. then the Riccatl algebraic equation (3.3) has a nonnegative selfadjoint solution H. and, for each z e E. HjjPgjjZ converges weakly to Hz. imf. n,U 13 ^eo.» .., or r03,, Whose proof .s valla under the hypotheae. here. □ I^e mam shortcoming of the weak convergence in I^eorem 5.3 is that it does not yield uniform norm convergence of the feedback control laws. 47 5.2 Convergence of the Approximating Optimal Control Problems of Section 4 .2 For U.e rest of this section, A^, A^, A. T(t) . Bj, and B sre the operators defined In Section 2. The operators A„. B„, a„ and n„ are the operators In the approximation sohane of Seotlon 4. In particular. D . L(E„,E„) Is the .inlmal nonnesatlve. self-adJolnt solution of the Hlccatl operator e-uatlon (4.2«). According to Corollary 4.3 and Theorem 4.5. the Eltz-Galerl^n approx- imation sch»e presented In Section 4.1 converges as required In (5.1) and (5 2): (5.3) and (5.4) follow froc (4.23) and the definition Q„ - Pg„QlE„ m Seltlln 4.2. Also, i^pothesls 4.6 guarantees for each n the existence of the required solution of the Riccati equation (5.5) in Theory 5.1. Since II„ is nonnegative and self-.djolnt. its eigenvalues, which are also the eigenvalues of its Matrix representation, are real and nonnegative. and its norm is equal to its maximum eigenvalue. THeorem 5.1 If Is E-coerdve and a„ - (i.e.. there is no open-loop damp- ing), then there is no nonnegative self-adjoint solution of the Eiccti opera- tor equation (3.3), and IIU^II ^ « as n -» -. (5.11) ,^. Recall the operator t in Section 2.1. By Theory 1 of 102, , there can he no com;«ct operator C . L(E.E) such that t . C generates a uniformly exponentially stable semigroup. Therefore, since a compact linear perturba- tion Of r yields A, there can he no com^ct linear C such that A . C generates a uniformly exponentially stable semigroup. 48 Now, unless (5.11) holds, there exists a subsequence such that || JT || J is bounded in n., so that Theorem 5.3 says that there exists a nonnegative self-adjoint solution 11 of (3.3). Since Q is coercive. Theorem 3.3 then says that the semigroup generated by A - BR~^B*n is uniformly exponentially stable. But this is impossible -- BR~-'-B* 11 Is compact because its rank is not greater than m. D Theorem 5.5. Suppose that Aq and dQC.*) are both H-coercive. Then there exist positive constants Mj^ and a^, independent of n, such that llexpEA^tlll i M,e"^ , t 1 0. ^^_^^^ Proof . First, we define A^^ and D^^^ e L(V^,V^) to be the operators whose matrix representations are M""k" and M'^d", respectively. (See Section 4.1.) The operator A is then '^ ^ [ -\n -^On| (5.13) Since Aq and dQ are H-coercive, there exists a positive constant p. independent of n, such that <Ao„h.h>j, 1 pllhll^ ^^ .^^j and <Do„h.h>H I Pllhllj ^^^^5j for all h e V . Since dn Is continuous on V X V, there exists a positive n w 49 constant y. Independent of n, such that <Donh.h>H i Yl|h||J (5.16) for all h c V^. The theorem follows then from Theorem 2.1. O TiiSOrsB 1'6. Suppose that A^ has an invariant subspace V^ which is also invariant under the damping map A~ A^ , that E^ = V^ X Vq is a stabilizable subspace for the control system, and that the restrictions of A_ and 6A»,») to Vq are both H-coercive. Also, suppose that Vq has finite dimension nQ and that, for each n 1 n^ in the approximation scheme, the first n^ e. , span V» and the rest are orthogonal to V^ in both V and H. i) Then (3.3) has a nonnegative solution 11, and for each n 2 n^., (5.5) has a nonnegative self-adjoint solution!^. Also, I^ is bounded uniformly in n, so that H^ converges to II weakly, as in Theorem 5.3. ii) If Eq and E^ (the E-orthogonal complement of Eq) are invariant under Q, and if the part of the open-loop system on E. is observable with the measur •e- ment Qz, then (5.6)-(5.8) hold as in Theorem 5.1. Froof. We will invoke Theorem 3.7 to establish the existence of the uniform bounds and decay rates needed in Theorem 5.1. In the approximating optimal control problems, the part of the control systm on E. is the same for each n; the approximation of the control system takes place on E^; We can write E = " n Eq © E^n' where Eq^ Is the orthogonal complement of Eq in E^, and Eq and E^ clearly reduce the open- loop semigroup for each n 2 n^j. For the part of the open- loop system on E^^, Theorem 5.5 establishes 50 positive Mj^ and a.. Independent of n, for (3.17). Also, we have a ^ independent of n for (3.18) because B^^ = Pg^B and Q^^ = PEn*^'En* Therefore, i) follows from Theoran 3.7 i) and Theorem 5.3. The definition of Q and the requirement on where the various basis vec- n tors must lie imply that E^ and E^^ reduce Q^ if Eq and Eq reduce Q and that the restriction of Q to Eq is the same for all n. Therefore, ii) follows from Theorem 3.7 ii) and Theorem 5.1. D Remark 5.7. In applications, the subspace V^ in Theoron 5.6 usually contains rigid-body modes. The theorem includes the case where both A^ and dQ are H- coercive on all of V (no rigid-body modes and all modes damped). In this case, Vq is the trivial subspace. D Remark 5.8. Otherwise, for applications to flexible structures. Theorem 5.6 usually requires two things: first, modal damping must be modeled for the structure, so that the natural modes remain uncoupled in the open-loop system; second, the natural mode shapes must be used for the basis functions in the approximating optimal control problems. Although these requirements may seem restrictive from a mathematical standpoint, such modeling and approximation predominate in engineering practice. Also, we get our strongest convergence results under these conditions. For applications where the basis vectors are not the natural mode shapes, the following theorem is useful. D Theorem 5.9. Suppose that Aq + BqBq and d^ + BqBq are H-coercive. Then (3.3) has nonnegative solution II, for each n (5.5) has a nonnegative self-adjoint solution n, and || II 1 1 is bounded uniformly in n. Hence Theorem 5.3 applies. Furthermore, if Q is E-coercive, then (5.6)-(5.9) hold in Theorem 5.1, 51 Proof . The required bounds follow from Theorem 3.10 and the proof of Theorem 5.5. Although we took A = b«B„ in Theorem 3.10, this is not necessary in the final result, since all bounded self-adjoint operators A^^ on H that make Aq + k^ coercive yield equivalent norms for V. D Theorem 5.10. If (5.7) holds for each z e E, then "l^in - ^illy-^^' "^in ~ Silln -^0' as n -» (5.17a) (5.17b) where f^ and g^ are the functional gains in (3.16), and f^^ and g,^^ are the approximating functional gains in (4.31) and (4.41). Proof . The result follows from (4.31). □ Note that (5.10) and (5.17) are equivalent. 52 6 . Exam pi e 6.1 The Control System One end of the uniform Euler- Bernoulli beam in Figure 6.1 is attached rigidly (cantilevered) to a rigid hub (disc) which is free to rotate about its center, point 0, which is fixed. Also, a point mass m, ^3 attached to the other end of the beam. The control is a torque u applied to the disc, and all motion is in the plane. Figure 6,1. Control System 53 10 in r = hub radius . 100 in I = beam length In = hub moment of inertia about axis 2 perpendicular to page through l^O slug in m. = beam mass per unit length m, = tip mass EI = product of elastic modulus and second moment 13,333 slg in^/sec^ of cross section for beam -^^ ' '^ fundamental frequency of undamped structure -9672 rad/sec Table 6.1 Structural Data .01 slug/in 1 slug The angle represents the rotation of the disc (the rigid-body mode), w(t,s) is the elastic deflection of the beam from the rigid-body position, and w^(t) is the displacement of m^ from the rigid-body position. For technical reasons, we do not yet impose the condition w^Ct) = w(t,|); more on this later. l^e control problm is to stabilize rigid-body motions and linear (small) transverse elastic vibrations about the state = and w = 0. Our linear model assumes not only that the elastic deflection of the beam is linear but also that the axial inertial force produced by the rigid-body angular velocity has negligible effect on the bending stiffness of the beam. The rigid-body angle need not be small. For this example, it 'is a straight forward exercise to derive the three coupled differential equations of motion in 0. w and w^ , and they do have the 54 form (2.1'). However, to «nf*iasize the fact that we do not use the explicit partial differential equations, we will not write these equations here. Rather, we will write only what is normally needed in applications: the kinetic and strain-energy functionals, the damping functional and the actuator influence operator. Remark 2.1 applies to this example, and to most examples with complex structures. The generalized displacement vector is X = («.w,w^) e Hq = R L2(0,|)X R. The kinetic energy in the system is Kinetic Energy = V2 <x.x>^ where H is H^ with the inner product <x,x>jj = m^ /q [w+(r+s)e] [w+(r+s)e]ds (6.1) (6.2) (6.3) + loOS + mi[wi+(r+J?)«][(*i+(r+i)^]. As in most applications, we need not write the mass operator explicitly, but there exists a unique selfadjoint linear operator ^L on H^ such that <x,x>jj = <MqX,^>jj . (6.4) It is easy to see that M^ is bounded and coercive. Hence Hq and H have equivalent norms. The input operator for (2.1') (which maps R to H-) is 55 Bq = (1,0.0). (g5j Since we multiply (2.1') by li^^ to get (2.1). the input operator for (2.1) is (Mq'^Bq) . Note that (Mq Bq) = Bq, ^g gj where (Mq^Bq)*" is the H-adjoint of (Mq^B^) and B* is the HQ-adjoint of Bq. Remark 2.2 also applies here. The only strain energy is in the beam and is given by Strain Energy = Vi a(x,x) with (6.8) a(x,x) = EI/q w"w"ds. where o" = d^O/ds^C). To make aC,*) into an inner product, we must account for rigid-body rotation. Thus we set <x,x>y = a(x,x) + eS ^^^^j and define V = {x = (0.0.0(1)): e H^(O.I), 0(0) = 0'(O) = }. (6.10) Also, we have <x.x>„ = a(x,x) + <BqB!x,^>jj (6.11) = a(x.x) + <(Mq1Bq)(Mo^Bq)*^x.x>jj. 56 so that Aj = BqB!, or (Mq^B^) (M^^B^)*^, depending on whether the H^ or the H- inner product is used in computing the V- inner product. But jje need Qejther A^^ nor Aq explicitly . W§ need only (S..8) and (i.i) » along with (6.3) . io QQn^PUte iM required inner pro(avtcta. As mentioned in Remark 2.2, the operator A. can be defined now by (2.6), and the stiffness operator is A^ = A^ - A^^. Using the H^-inner product in (2.6) yields the A. for (2.1'), and using the H- inner product yields the Aq for (2.1), which is I^^ following the A^ for (2.1'). The Aq for (2.1') is quite simple, and the reader might write it cut. We will not, so that no one will think that we use it. We will point out that D(Aq) requires both the geometric boundary conditions in V and the natural boundary condition w''(t,X) = 0; i.e., zero moment on the ri^t end. Remark 6.1. That the geometric boundary conditions w(t,0) = w'(t,0) = (6.12) and w(t,Jf) = w. (t) ^ (6.13) are imposed in V but not in H — i.e., on the generalized displacement but not on the generalized yelocity — is common in distributed models of flexible structures. The natural norm for expressing the kinetic energy of distributed components is the L.-norm, which cannot preserye constraints on sets of zero measure. Because the strain energy inyolves spacial derivatives, the stronger strain-energy norm can preserve the geometric boundary conditions (although, as for the boundary slope of an elastic plate, the V-norm may impose some of 57 these boundary conditions in an L^ rather than a pointwise sense). The strain-energy norm is based on the material model of the distributed com- ponents of the system, and it should not be surprising that such a norm is required to connect the various structural components. D We assume that the beam has Voigt-Kelvin viscoelastic damping [C2] , so that the damping operator in (2.1) is "» " "O*" (6.14) where Cq is a constant. This means that the damping functional is djj(x,x) = Cfj a(x,x), x,x e V. (6.15) 6.2 The Optimal Control Problem We take Q = I in the performance index in (3.1). This means that the state weighting term <Qz,z>g is twice the total energy in the structure plus the square of the rigid-body rotation. Since there is one input, the control weighting R is a scalar. According to (3.33), the optimal control has the feedback form u(t) = - <f,x(t)>„ - <g,x(t)>„ " (6.16) where x(t) has the form (6.1), and f = (a-.,^-..pf.) = R~^n,Bo 8 V, (6.17a) 58 & = (a_.(« .ft ^ = o~l Note g»0g.Pg) = R n^B^ 8 H. (6.17b) that p^ = 0^(;) i3 „^t used in the control law-recaii (6.8) and (6.9). 6.3 Approximation Our approximation of the distributed model of the structure Is based on a finite elonent approximation of the beam which uses Hermlte cubic splines as basis functions ([S1.S4]). These are the basis functions most commonly used in engineering finite elanent approximations of beams, l^e splines and their first derivatives are continuous at the nodes. Because the basis vectors e m the approximation scheme in Section 4 must be in the space V defined in ^ (6.10), we write them as ©1 = (1.0,0). (6.18a) ^j = (O.0j,0j(l)). j = 2. 3. . . ., (6.18b) Where the .^'s are the cubic splines. When we use n^ el«nents to approximate the beam, there are 2n^ linearly Independent splines. Ihus. with the rigid- body mode, the order of approximation is n = 2n + i For the numerical solution to the optimal control problem, we have only to Plug into the formulas of Section 4. T^e matrices in (4.3) are calculated according to (6.3). (6.8) and (6.P) , with B, g,,en by (6.5). m particular. K" = [a(e,,ej)]. d'^ = e^ k". m" = l<e,,e^>^U (6.19) BJ - [10 - 01^ . l<e^,,^l,,,o.on^, . I, ,,,„_,, 0, (6.20) 59 Hote that the first rc» an. col>-n of K" are zero. Ihe .atrlx k" 1. (4.32) is K" »lth 1 added to the first el^ent. ^e .atrloes A" and b" are given by ,4.5) and. since Q - I. the .atrix fl" is the w" in (4.33). With the^ .atnices. we solve the Biocati equation (4.40) and use (4.41) and (4.45) to compute the approximations to the functional gains, which are fn " '"fn-'fn'^fn'' (6.21a) «n " <VV'V' (6.21b) For convergence, we satisfy all the hypotheses of Theorem 5.9. In par- ticular. Since Q is the identity on E. it is coercive. Theorem 5.9 implies that the solutions to the finite dimensional Rlccatl equations converge as in Theory 5.1 and that the functional control gains converge as in theory 5.10. as,^ £.2. It might appear that the hypotheses of Theorem 5.6 hold with n„ - 1, but not so. For J JL 2, e^ is orthogonal to e, in H„ and V not in H. Recall («.l)-<6.3), (6.9)-(6.11) and (6.18). □ 60 6.4 Numerical Results Figures ..2a and «.2b show the computed funotloml gain kernale ,;; and <g„ for the damping coefficient c,, - lo"". the control weighting R . l^^'and "e = 2. 3. 4, 5 and 8 beam elanents. Table 6.2 Usta the corresponding scalar oom^nents of the gains, for c„ . i„-^ and B - 05. the convergence Is slc«er as discussed bel». To shew the complete story of convergence. Flares 6.,a and «.3b and Table «.3 show the results f or n . 2 3 4 5 , ^ ^, e -^.3.4.5,8, and Figures 6.4a and 6.4b and Table 6.4 show the results for n^ . 4.6,8,10. We have plotted ,;; because the second derivative appears in the stral^^energy inner product in C6.8) and (6.9, and ,,^ converges in H^(0,|,. Note that. Since the Hermite cubic splines have discontinuous second deriva- tives at ^e nodes, the appro>cimations to ,;' are discontinuous at the nodes. Although H -convergence guarantees onlv i ' ' 6 arancees only L2-convergence for 0^^, it can be shov^n that ^;; converges uniformly on [oj] for this problem. l^e tables omit p^^ to emphasize the fact that it does not appear in the feedback law and the fact that the ^nvergence of ,^^ ,3 not an independent Piece Of information about the convergence of the control gains; since ,,„(o, = 0f.„(O) = 0, the convergence of ^'/^ i„.pUes the convergence of p^^ = ^^^"(j^, . On the other hand, although B ~ a tC\ ^ ""^ "gn - «gn <X' for each n, the H-norm convergence Of g„ does not enforce this condition in the li.it, as the V-norm convergence Of f„ enforces f^ . ,^<|,. Hence, as far as we can tell fro. our results in sections 3.5, f,„ is an independent indicator of the convergence of the con- trol gains, as well as being used in the control la. in <6.16,. However, the behavior of ,^ ,„ p,^,,, ,.,,, ,.3, ^„^ ^„^ ^^^^^^ ^^^^ ^^ ^^^^^^ ^^ V. Stronger results on the continuity of «^ and the convergence of » (jf, 61 3H0U1. rouo. rrc a U>eo.» .taU„« that. ^oau.e t.e op..loop =e..s.oup generator A is a^lytic. the solution to the infinite ai.ensional Hicoati ^uation »aps ail or E into .CA*, . -e Tact that .^ (i- converges to .e.c in Pigure 6.2a also suggests such a the<.». but we have not proved it. 4 K„„. filed the two factors that detennlne the rate With the state weighting Q fixed, me Of convergence are c„ and R. Althou* we have used splines to apprcxiMSte the ^a., the relation between the convergence rate and c„ and B prohabl, can he i„.er.eted best in ter.s of the „-ber of .tural .odes of the structure that ,he optical infinite di.ensio^l controller reall, controls. Strictl, speak- ing, the controller controls all .odes, but the functional gains lie essen- .r n,» mina if we used the natural modes as of modes required for convergence of the gains the tasis vectors in the approximation, ^e rest of the modes are practically <hut not exactly, orthogonal to the functic^l gains, so that the optimal 4-v.-n Tn eeneral, the lighter the damping, feedback law essentially ignores them. In general, H . th.t will be controlled for given Q and R; the cheaper the con- the more modes that wiii oe wun wj. . ^v,of will be controlled for given Q and c^. The question trol. the more modes that will oe conw^j-x of the convergence of the finite client approximation to the functional ^ins hecomes then a duestion of h«, ma. .odes the optimal control law really wants i«,oni-<» ^t takes to approximate those modes, and how many elements it caKes i,u ai,^ Hv-erical experience with optimal octroi of flexible structures has ehown this modal interpretation of the convergence of the approximating con- trol laws to be very useful, and that it is difficult to improve upon the ^tural modes as basis vectors for the approximaUon scheme (see [G5U H»- ever. whether the natural modes are always or almost always the hest basis 62 vectors Is an open question. We use the cubic splines here to demonstrate that a standard finite element approximation works quite well. Also, to use the natural modes as basis vectors here, we first would have to compute them using a finite element approximation — as in most real problems ~ and we do not know in advance which or how many modes are needed. On the other hand, if the most important natural modes are determined from experiment, then modal approximation should be best. 63 lb. DO 2b. 00 Figure 6.2a. Functional Control Gain Component if.^" Damping coefficient Cj, = lo"^; control weighting R = 1 b.oo 3b. 00 3b. 00 lib. 00 sb.oo Elgucfi 6. lb. Functional Control Garln Component i^ Damping coefficient c^ = 10~^; control weighting R = 1 number of elements n 2, 3, 4. 5. 8 number elements n 2. 3. 4. 5, 8 0^ EigjlCfi 6.1a. Functional Control Gain Component ^f„ Damping coefficient c^ = 10-^ control weighting R number of elements n^ = 2, 3, 4, 5, 8 ■ o.oo Figure 6.3Ji. Functional Control Gain Component i^ .05 Damping coefficient c^ number of elements n^ = , lO""*; control weighting R 2, 3. 4. 5. 8 05 CTV 0^ £iKUO 6.45. Functional Control Gain Component ^^ V. Damping coefficient o - in-<. ""^ Oq - iO ; control weighting R = .05 number of elements n = 4 g „ ,„ e ^ » " . o , 10 £i«Ui:S i.ii,. Functional Control Gain Component , Damping coefficient c, . io-^• control weighting hI .05 number of elonents n^ = 4. 6, 8. lo °fn °gn Pgn 2 1.000 .2141 -22.459 3 1.000 .2396 -25.221 4 1.000 .2496 -26.331 5 1.000 .2534 -26.786 8 1.000 .2561 -27.041 lafele 6.2. Scalar Components of Functional Control Gains Damping coefficient c^ = 10"^ control weighting R = 1 number of elements n = 2, 3, 4, 5, 8 "e °fn V ^gn 2 3 4 5 8 4.4721 4.4721 4.4721 4.4721 4.4721 1.0136 1.1770 1.2440 1.2781 1.3106 -108.27 -126.40 -133.87 -137.57 -141.15 T£^l?le 6 .3 . Scalar C omponents o f FiinnHni ins v-4. Damping coefficient c^ = lo"^ control weighting R = .05 number of elements n = 2, 3, 4, 5, 8 "e °fn %n V 4 6 8 10 4.4721 4.4721 4.4721 4.4721 1.2440 1.2973 1.3106 1.3141 -133 -139, -141, -141. .87 .69 .15 ,54 T^bl9 6.4. Scalar C omtX)nents if Piinof ^ nr Damping coefficient c^ = 10"^ control weighting R = . ng R = .05 number of elements n = 4, 6, 8, 10 67 Figures 6.5a and 6.5b and Table 6.5 represent attempts to compute an optimal control law for the structure when R = .05 but c^, = 0. Since Q is the identity operator in E and hence coercive. Theorem 5.4 says that no optimal control law exists and that the norm of the solution to the finite dimensional Riccati equation grows without bound as the number of elements increases. This is reflected in the nonconvergence of a^^. i>^ and Pg„. although a^^ con- f f verges and the convergence of 1>^^ is unclear. In applications where the structural damping is not known, except that it is very light, it is tempting and not uncommon engineering practice to assume zero damping in the design of a control law for the first few modes, while trusting Whatever damping is in the higher modes to take care of them. How- ever, if high performance requiranents (large Q) or coupling between modes in the closed-loop system necessitate a control law based on a more accurate approximation of the structure. Theorau 5.4 and the current example warn that the higher-order control laws are likely meaningless and rather strange if no damping is modeled. We should note that we have seen similar probl«ns [G9] where 11^ remains bounded and the gains converge for zero damping but finite-rank Q. In such cases. Theorem 5.3 says that an optimal control law exists for the distributed model of the structure and that the finite dimensional control laws converge to an optimal infinite dimensional control law. Also. Balakrishnan [B2] has shown that an infinite dimensional optimal control law exists for no damping when Q = BB*. 68 15:^S iSToo sJToo irtToo iOo sToo jSToo SHo iSIoT Figure 6.5a. Functional Control Gain Component 0f„" Zero damping; control weighting R •= .05 lb. 00 ilToO 30.00 «b.00 SO. 00 sb.aO 70.1 Figure 6.5t?. Functional Control Gain Component ^ Zero damping; control weighting R = .05 •b.oo iSTi gn "ioo.o number of elements n = 2. 3, 4, 5, 8 6 number of elements n = 2, 3, 4, 5, 8 "e °fn "gn Pgn 2 4.4721 1.0516 -112 .23 3 4.4721 1.3061 -140.18 4 4.4721 1.4758 -159.11 5 4.4721 1.5996 -172.64 8 4.4721 1.8407 -199.39 iMhlS 6.5. Scalar Components of Functional Control Gains Zero damping; control weighting R = .05 number of elements n = 2, 3, 4, 5, 8 70 7 . The Optimal Infinite Dimensional Estimato r. Compengator and Gigged - Ififija System As in Sections 3 and 5, we will state sc-me initial definitions and results for an arbitrary linear control system on a Hilbert space in Subsec- tion 7.1, and then discuss implications for flexible-structure control in Sub- section 7.2. 7.1 The Generic Problem Let A, T(t) and B be as in Subsection 3.1, with E an arbitrary real Hil- bert space. The differential equation corresponding to (3.2) is, of course, z(t) = Az(t) + Bu(t), t > 0. (7.1) We assume that we have a p-dimensibnal measurement vector y(t) given by y(t) = C_u(t) + Cz(t), (7.2) where C e L(R°. R^) and C e L(E,rP) for some positive integer p. Definition 7.1. For any F e L(rP,E), the system 5(t) = Az(t) + Bu(t) + F[y(t) - CQu(t)-Gz(t)], t > 0, (7.3) will be called an observer , estimator (we use the terms interchangeably) , for A A the system (7.1)-(7.2). Let S(t) be the semigroup generated by A-FC. The A observer in (7.3) is strongly (uniformly exponentially) stable if S(t) is strongly (uniformly exponentially) stable. □ To justify this definition, we write 71 e(t) = z(t) - 'S(t) (7.4) and, with (7.1)-(7.3), obtain e(t) = ^(t)e(O), t 2 0. (7.5) Of course, an observer, or estimator. Is necessary because the full state 2(t) will not be available for direct feedback, and the feedback control must be based on an estimate of z(t). When, as in this paper, the desired control law has the form u(t) = -Fz(t) (7.6) for seme F e L(E,r'"), the observer in (7.3) can be used to construct z(t) from the measurement in (7.2) and then the control law in (7.6) can be applied to z(t). The control applied to the system is then u(t) = -Fz(t). and the resulting closed-loop system is (7.7) 5(t)) l(t)J ll'n - 3.,„(t) J^^>[ . t . 0, z(t) z(0)' (7.8) where S^,„(t) is the semigroup generated on E X E by the operator fA -BF 1 ^ ra-ni^jftm ' !>(*„.„) = D(A)XD(A). (7.9) With the estimator error e(t) defined by (7.4), it is easy to show that (7.8) is equivsilent to (7.5) and 72 :(t) = (A-BF)z(t) + BFe(t), t > 0, (7.10) where (A-BF) generates a semigroup S(t) on E. Also, it is easy to prove the following. Theorem 7.2. Suppose that there exist positive cjonstants M^, Kj, a^ and Oj such that IIS(t)|| i M^ e -a-^t -Ojt ||S(t)|| < M^e , t 1 0. (7.11) Then, for each real Og < minla^, oj) , there exists a constant M3 such that II S„.„(t)|| i Mge"^ . t 2 0. ^^^^^^ Also, a(A„,J = a(A-BF) U <r(A-FC), ^^ ^^^^ where "(A^,^) is the spectrum of A^,^. □ The observer in (7.3) and the control Ifw in (7.7) constitute a compensa- tor for the control system in (7.1) and (7.2;. 'The transfer function of this compensator is B(s) =--F(sI-lA-BF + F(C F-C)1)~H' , " (7.14; which is an m X P matrix function of the complex variable s. When E has infinite dimension, the compensator transfer function is irrational, except in 73 degenerate, usually unimportant cases. The foregoing definitions of this section and Theorem 7.1 are straight- forward generalizations to infinite dimensions of observer-controller results in finite dimensions. Balas [B3] and Schumacher [S2] have used similar exten- sions. Now suppose that F is chosen as f = hc*r\ (7.15) A Where n e L(E,E) is the minimal nonnegative selfadjoint solution to the Ric- cati equation AH + HA* - nc* R-^cn + Q = 0. (7.16) A /^ With Q e L(E,E) nonnegative and selfadjoint and R c L(rP,rP) symmetric and positive definite. Theorem 3.3 (with A, B, Q, R, Hand Sit) replaced by ♦ •AAA. A* A A ,C , Q, R, n, and S (t)) gives sufficient conditions f or fi to exist and for ■** A. the semigroup S (t) ~ and equivalently its adjoint, the S(t) generated by A 11 C R c — to be uniformly exponentially stable. Definition 1.3 . When the control gain operator is F = R -^B*!! , (7.17) with n the solution to the Riccati equation (3.3), and the observer gain operator is given by (7.15) and (7.16), we will call the compensator consist- ing of the observer in (7.3) and the control law in (7.7) the ont•■^ ,nl;^] infinite diffiengl,9n^l compensator, and (7.8) the optimal closed-loop svstem . Q 74 X • z = Az + Bu y = CqU + Cz Control System A Z U = [A-BF-F(Cq F-C)] z + F y -F^ Optimal Infinite Dimensicnal Compensator Figure 7 .1 . Optimal Closed-loop Sj stem ^eaaclsl.i. The infinite dimensional observer defined by (7.3), (7.15) and (7.16) is the optimal estimator for the stochastic version of (7.1) and (7.2) when (7.1) is disturbed by a stationary gaussian white noise process with zero mean and covariance operator Q and the measure«ient in (7.2) is contaminated by similar noise with covariance R. For infiiite dimensional stochastic estima- tion and control, see [HI. C4] . When the state weighting operator Q in (3.1) is trace class, the optimal infinite dimensional compensator minimizes the time-average of the expected steady-state yalue of the integrand in (3.1). Existing theory for stochastic control of infinite dimensional systems requires trace-class Q, but we have a well defined compensator for any bounded nonnegative selfadjoint Q and $. as long ss the solutions to the Riccati 75 equations exist. As the next two sections show (without assuming trace-class Q), the infinite dimensional compensator is the limit of a sequence of finite dimensional compensators, each of which can be interpreted as an optimal LQG compensator for a finite dimensional model of the structure. Therefore, we do not require trace-class Q in our definition of the optimal compensator, even though this compensator solves a precise optimization problem only when Q is trace class. This paper is concerned primarily with how the finite dimensional compen- sators converge to the infinite dimensional compensator, and the analysis of this convergence requires only the theory of infinite dimensional Riccati equations for deterministic optimal control problems and the corresponding approximation theory. While the stochastic interpretation of the infinite dimensional compensator and, in Section 8.2. of the finite dimensional estima- tors should be motivational, nothing in the rest of the paper depends on a stochastic formulation. We assume that the operators Q, R, Q and R are deter- mined by some design criteria. In many engineering applications, determinis- tic criteria such as the stability margin and robustness of the closed-loop system, rather than a stochastic performance index and an assumed noise model, govern the choice of Q, R. Q and R.Q 7.2 Application to Structures For the rest of the paper, E = V X H as in Section 2, and A and B are the operators defined there. The measurement operator C in (7.2) now must have the form 76 c = [c^ c^] (7.18) th where C^ e L(V,rP) and C^ e L(H.rP). Hence, if we denote by (C(x.x))^ the i ponent of the p^vector C(x.x). for (x.x) e E. then there must exist c^^ e V com and c, . e H such that {Cix.'x))^ = <Ci^,x>y + <C2i.5>H' i = 1.....P. (7.19) Also, the estimator gain operator F is givnn by A P A A Fy = I (fi,gi)yi i=l 1 ^ -^ (7.20) A for y = [y y ... y l*^ e R^, where the fu^iatifinai e9tJ,nat9r £ains f^ and g^ are elements of V and H, respectively. For the optimal estimator gains, we cm partition 11 as ft - Do III A« A EL, n^ and use (7.15) and (7.19) to get (7.21) A f. P A_i A i-i z (R ^)ij(no°ij + 'V2j> • (7.22a) t, = I (R"^)i.(ftjc + DjCjj) , i = 1.2, ....p. ^ j=l J J (7.22b) Now let us partition Q as in (4.34): 77 $ = (7.23) In the optimal control problem, we almost always have a nonzero Q because this operator penalizes the generalized displacement. For the results in this paper. Q^ can be nonzero in the observer problem, and, as in the control prob- lem, some of the strongest convergence results for finite dimensional approxi- mations can be proved only for coercive t. However, if the observer is to be thought of as an optimal filter, then t should be the covariance operator of the noise that disturbs (2.1). in this case, $q = and $ = 0. 78 8. Approximation of the Infinite Dimensioral Estimator 8.1 The Approximating Finite Dimensional Estimators Here, the scheme for the approximatior of the flexible structure is that in Section 4. We will construct on the sutspace E^ an estimator that apjroxi- mates the optimal infinite dimensional estimator of Section 7. and this esti- mator will produce an n^^-order estimate ^^ = {^^\) of the infinite dimen- • th 1. sional state vector z = (x.x). In Section 9. the the n -order compensator that results from applying the n^^ approxination to the optimal control law (in Section 4.2) to z^ will approximate th( optimal infinite dimensional com- pensator of Section 7. Hypothesis 8 .1 . There exist a sequence C^ e L(E^.rP) such that ilCnPEn-Cll -^0 ^^ "^" (8.1) and a sequence Q^ e L(E^) , Q^^ = ^n - °' ^"°^ ^^^ Q Pp -> Q strongly as n -» ■' .0 Hypothesis 8 .2 . For each n. the system (A*,C*) is stabilizable. In particu- lar, any unstable modes of the ^stem (C^, \^) are observable. D The n^*^ observer , or n estimator , is ^n = ^n^n ^ V "^ ^n<y-^o"-^n^n> (8.3) where the estimator gain F^ is 79 ^ /v •A.I (8.4) and n^ is the nonnegative selfadjoint solution to the Ricoati operator equa- tion ^n^n - fln< " K'^^XK ^ K ^ ^' (8.5) Hypothesis 8.2 implies that such a solution exists and is unique. ,th This representation of the n estimator as a system on E , with th n' e estimator gain determined by the solution to a Riccati operator equation, is necessary for showing how the sequence of finite dimensional estimators approximate the infinite dimensional estimator. However, on-line computations will be based on the equivalent differential equation ^ = A°<| + b\ + F"(y-CQU-C"<|) (8.6) where Ti(t) e R , A and B are the matrix representations of the operators A and B_, as in Section 4, and c'^ is the matrix representation of C . n n An The 2nxp gain matrix F is F" = n"W-» (C")Tr-1. where W° is the 2n X 2n grammian matrix in (4.33) and if satisfies A'^n" + friw-"(A»)Tw" + n" w-°(c")Tt-V3" + Q" = 0, (8.8) with q"^ the matrix representation of Q^. The relationship between z = (Xjj.x ) and ^ is, of course. (8.7) A, 80 xft) = 2:5i(t)ei " i=l (8.9) and 'I - ^^ 5 ^ • (8.10) Since the matrix representations of A^ and A^ are a"^ and W-"(A")T w", respec- tively, and the matrix representation of C^ is W-'^(C")T. (8.7) is the matrix representation of (8.4), and the 2n X2n Riccati matrix equation (8.8) is the A ^ matrix representation of (8.5), with IT the matrix representation of 11^. (Recall that W~" is the inverse of w".) As in the control problem, we do not solve the matrix representation of the n^*^ Riccati operator equation directly Sjecause the matrix representation of a selfadjoint operator in general is not gymmetric. In the duality between the optimal control and estimator problems, (8.5) and (8.8) correspond to (4.26) and (4.38), respectively. In (4.39). we defined the symmetric matrix 5" = W"lf and then obtained the Riccati equation (4.40) to solve for DP . We proceed in a similar fashion here, but with an interesting difference. Since fi and t are nonnegative selfadjoint operators on E^ and IT and q" are their matrix representations, the matrices w" fi" and wV are nonnega- tive and symmetric. Hence, the matrices " - IIW (8.11) and 81 (8.12) are nonnegatlve and symmetric. Substituting (8.11) and (8.12) into (8.7) and (8.8) yields f" = fl"(C")TR-l (8.13) and A" fin + fln(An)T _ fin(c")T^Vfl" -. ^" = 0, (8.14) the Riccati matrix equation to be solved numerically in the n^^ approximation to the infinite dimensional estimator. In view of the relationship between (8.5) and (8.8) and the relationship between (8.8) and (8.14), we see that Hypothesis 8.1 guarantees the existence of a unique nonnegative symmetric solution to (8.14) . To see the relationship between the matrices in (8.14) and the operators in (8.5) more clearly — and the difference between the current approximation scheme and that used in Section 4.2 for the control problem — suppose that we take Q^ = PgjjQlg^. Let q" be defined as in (4.36) and (4.37) with Q^. Q^ . and Q2 replaced by Qq, Q ^ , and ^2 • Then q" = W-"Q"W-^ (8.15) A For example, if Q in the control problem and Z in the estimator problem are both equal to the identity, then the q" in (4.35) - (4.42) is w" and Q = W . This may seem suspicious, but Subsection 8.2 should demonstrate that we are solving the appropriate estimator problem here. The only thing missing now for numerical implonentation of the n th 82 estimator, or observer, is to give C^ the matrix representation of C^. expli- citly. We write C" = [C; Cjl (8.16) where the p x n matrices cj and cj are. respectively, the matrix representa- tions of the operators C, and C, in (7.18). We can cover virtually all appli- cations by assuming C„= C|,„. in which case the i^^ column of cj is the p- vector equal to C,e,. and the i^^ column of cj is the p-vector equal to C,e,. Mej^hM^iM ^^ml^ Mi ^ ^a^Uojis far ima^ni^ i,pien, ,nt.tiop of iM nth ^^^ ^stlaatai:: For online computation, the n^^ estimator, or observer, is (8.6); the gain matrix F" is given by (8.13) and the solution to ^i /o 1A^ The mahrices q" and C are defined as the Riccati matrix equation (8.14). The matrices u above. 8.2 Stochastic Interpretation of the Approximating Estimators As we have said, our approximation theory for the optimal estimator is based on approximation of the infinite dimensional Riccati equation, whose structure is the same for both control and estimator problen.s. and the sto- chastic properties of the optimal estimator problan never enter our approxima- tion theory. Furthermore, using only the deterministic setting above, we will proceed, subsequently, to analyze the finite dimensional estimators and the compensators based upon them. Nontheless. we should consider momentarily the sequence of finite dimensional stochastic estimation problans whose solutions are given by the equations of the preceding subsection. First, recall how the covariance operator of a Hilbert space-valued rar^ 83 dom variable Is defined. The covariance operator of an E-valued random vari- able u is the operator Q for which expected value {<2,.o>g<'?,„>^} = <Qz.'5>g, z. z e E. (8.17) (See [Bl, C4].) With F^ given by (8.4) and (8.5), (8.3) is the Kalman-Bucy filter for the system ^n = Vn "■ ^n" + %' (8.18) ^ = S" ^ Vn ^ "-0' (8.19) where <o^(t) is an E„-valued white noise process with covariance operator Q„ and u)Q(t) is an R^-valued white noise process with covariance operator (matrix) R. Next, careful' Inspection will show that the filter defined by (8.6), (8.13) and (8.14) is the matrix representation of the filter defined by (8.3), (8.4) and (8.5). With z^ and n related as in (4.1) and (4.4), (8.18) and (8.19) are equivalent to the system Ti = a"ti + b"u + (/, y = CqU + c\ + Up. 'n^*^> by (8.20) (8.21) where (/(t) is the R^'^-valued noise process related to « (t) 84 n <o„(t) = I ((/i(t)ei, (/i+n(t)ei). " i=l (8.22) Certainly, a Kalman-Bucy filter for (8.20) and (8.21) has the form (8.6) with the filter gain given by (8.13) and (8.14). This particular filter is the matrix representation of the filter defined by (8.3), (8.4) and (8.5) if and only if the matrix Q" defined by (8.12) is the covariance of the process (/(t). Since q" is the matrix representation of $^, straightforward calculation using (8.12) and (8.17) shows that the^" in (8.11) is indeed the correct covariance matrix. Of course, if a)^(t) and (/(t) represent a physical disturbance to the structure, then o.^(t) must have the form (0.a,^2)(t)) and the first n el«nents of (/(t) must be zero, but this is not necessary for our analysis. The finite dimensional observers can be interpreted now as a sequence of filters designed for the sequence of finite dimensional approximations to the flexible structure, with the n*^^ approximate system disturbed by the noise process a)^(t), whose covariance operator is; Q^. By Hypothesis 8.1, these covariance operators converge to the operator ^ of Section 7. If we have a reliable model of a stationary, zer(^mean fiiaussian disturbance for the struc- ture, then we can take the covariance operator for this disturbance to be ^ and think of the infinite dimensional observer as the optimal estimator. But, again, this interpretation is not necessarj for the rest of our analysis. 8.3 Ihe Approximating Functional Estimator Gains The n^*^ estimator gain operator in (8.4) has the same form as the infin- ite dimensional estimator gain in (7.15) and (7.20). We have 85 ty r^ ^ ^/^in'Km^yi 1=1 (8.23) for y - ly^ 72 ... Yp ] e R^, where the functional estimator gains f and %n ^^^ elements of V^ = H^. the matrix f" In (8.7) and (8.13) is the matrix A representation of F^, which means that, if we write ^1 /2 ... LP P ^1 „®2 ... &r f g where the columns p ^, p "^ e r", th en (8.24) in n fi j=l -^ -^ 1 = 1, . . . ,p. (8.25a) n g^ in ~ ^ Pi e^ i = 1, . . .,p. j=l ■^ J' (8.25b) For convergence analysis, it is useful to note that f^ and g. are given also by equations corresponding to (7.22). With the measurement operator C written as in (7.19) and C^ = c|g , we have A-1 m = J^(« ^Ij^nOn^Vn^lj ^ DlnW2j> (8.26a) where in J=l IJ' ^n%n°2 j^ ' (8.26b) A \ = A Don A* A n In (8.27) 86 8.4 Convergence Now we will indicate the sense in whicii the finite dimensional estimators/observers approximate the infinjte dimensional estimator in Section 7. As we have said, implementation of the n^"^ estimator is based on (8.6), (8.13) and (8.14), but convergence analysi:; is based on the equivalent system (8.3), (8.4) and (8.5). The question then is how the observer in (8.3). with gain given by (8.4) and (8.5), converges to the observer in (7.3) with gain given by (7.15) and (7.16). Recall Hypothesis 8.1, and recall frm Section 4 that the approximations to both the open-loop semigroup and its adjoint converge strongly. Also, recall that Hypothesis 8.2 guarantees a unique nonnegative selfadjoint solu- tion to the Riccati equation (8.5) for eadi n. Replacing A^ and B^ with A^ and C* in Theorems 5.1 and 5.3, we obtain Theorem 8 .3 . i) If || EL 1 1 is bounded uniformly in n, then the Riccati A A algebraic equation (7.16) has a nonnegative selfadjoint solution H and !!„?£„ converges weakly to fl. ii) If there exist positive constants M and p, independent of n, such that llexpCU^-V^'X'"" ^ *'^' • '^°- ,8.28) A ^ then II fi^ II is bounded uniformly in n, D^ Pg„ converges strongly to H and exp([A -nnC*R"^C^]t)PEn converges strongly to t(t), the semigroup generated by A-nc*R"^C, the convergence uniform in t 10. iii) If Q^ is bounded away from zero uniformly in n, then ||n;j|| being bounded uniformly in n guarantees the existence of positive constants M and P for which (8.28) holds for all n. D The proof of the following theorem is practically identical to that of Theorem 5 .4. IhSSrm 1.4. if Q is E-coercive and d^ = o, then there is no nonnegative sel- fadjoint solution of the Riccati operator equation (7.16), and A II Djjll -> » as n -)^ ». □ (8.29) Our purpose for bothering to state this obvious dual result is to point out the following question. Can Theorem 8.4 be modified to include the case where Q has the form (7.23) with Qq = 0, Q^ = and Q^ coercive on H? Next, we have the dual result to Theorem 5.6: Th?«?r?ff l 8.5 Suppose that A^ has an invariant subspace Vq which is also invari- ant under the damping map A^^A^, that E^ = V^ X Vq is an observable subspace, and that the restrictions of A^ and d^i' ,') to V^ are both H-coercive. Also, suppose that V^ has finite dimension n^ and that, for each n 2 n^ in the approximation scheme, the first n^ e^'s span Vq and the rest are orthogonal to Vq in both V and H. A i) Then (7.16) has nonnegative solution 11, and II IL|| is bounded uniformly in n, so that 3 Pg converges to U weakly. ii) If Eq and Eq (the E-orthogonal complement of Eq) are invariant under Q, and if the E^-part of the gystem (A,Q) is controllable, then the hypothesis of Theorem 8.3 ii) holds. £Cfflfif. The proof is practically identical to that of Theoran 5.6 with B replaced by C . For ii) , note that, when we partition A and ^ as in (3.16), 88 the finite dimensional system (A^^.'q^^) is controllable if and only if the system ^Qj^i'^ii) ^s observable. D Remark 8.1. Remarks 5.7 and 5.8 pertain to Theorem 8.5 as well as to Theorem 5.6; i.e., in most applications the theorem requires either that both A^ and dp be coercive (so that Vq = {0}) or that the natural mode shapes be the basis vectors and the damping not couple the natui-al modes. It seems unlikely that a finite number of observable rigid-body modes could change the nature of the convergence, but they greatly complicate the proofs. For applications where both rigid-body displacement and rigid-body velocity are measured, a result analogous to Theorem 5 .9 can be obtained, but we will not bother here because it adds no significant insight and we cannot use it in the example in Sections 6 and 10. Also, see Remark 10.1. D Theorem 8.7. If fi^Pg^ converges strongly toll, then ll^in-'^iilv-^^ (8.30a) llti„-ajlH-^0' " "^-^"' (8.30b) where f . and g^ are the functional estimator gains in (7.20) and f^^ and g^^^ are the approximating functional gains in (8.25). Proof . The result follows from (7.22) and (8.26). D 89 9. The Finite Dimensional Compensators and Realizable Closed-loop Systems 9.1 Closing the Loop The n compensator consists of the n'^^ approximation to the optimal con- trol law in Section 4. applied to the output of the n*^^ estimator/observer in Section 8; i.e., the feedback control u = - F z n '^n^n (9.1) where 'n - H"'<n„ (9.2) (recall (4.25)) and $^(t) is the solution to (8.3). Equivalently, this com- pensator can be written as u„ = -F"^ (9.3) where f" = R-^B^ff^ (9.4) (recall (4.43)) and the 2n-vector riit) is the solution to (8.6)). On-line com- putations will be based upon the latter representation, and the block diagram in Figure 9.1 shows the realizable closed-loop system that results frco the n compensator. We will refer to this system as the i^^ SilOSM-lSiSS system . 90 z « Az + Bu y = Cqu + cz — > Control System A „ [An-enpn + F°(CoF'^cn)]Ti + W ^ %- - f"^ nth Compensator Figure 9.1 n^^ Closed-loop System This closed-loop system is equivalent to z '.n A (9.5) where the operator ',n r A — BF 1 (9.6) generates the closed-loop semigroup S^,^(t) on EXE^. The closed-loop response produced by the n^b compensator — i.e.. the response of the n^^ closed-loop system — can be written then as S»,n(t)vA (oy n^OV (9.7) 91 Note that A^^ ^ ^as oompaot resolvent if and only if A does. 9.2 Convergenoe of the Closed-loop Systems Now we will consider the sense in whidi the n*^'^ closed-loop system approximates the optimal closed-loop system in Section 7 (Definition 7.3). Recall from Sections 4.1 and 8.1 how the approximating open-loop semigroups Tjj(*) and their adjoints converge strongly and how the input operators B^, the measurement operators C^ and their respective adjoints converge in norm. Sec- tions 5 and 8 have given sufficient conditions for the approximating control and estimator gains to converge to the gains for the optimal infinite dimen- sional compensator. In this section, we will assume Hypotheses 9.1 . As n — > «, (9.8) llF„-F||-^0. D (9.9) Remark 9 .2 . Of oourse, we are interested primarily in the case where the A gains F and F are the optimal LQG gains in (7.15) and (7.17) and F ^nd F are the corresponding approximations in Sections 4 and 8 (i.e. ,(9.2) and (8.4)). However, fca* the analysis of this section, we need only Hypothesis 9.1 for some F e L(E,Rn), F e L(rP,E) and approximating sequences F^ and F^. Any such gain operators will yield closed-loop semigroup generators A^,^ in (7.9) and A„,Q in (9.6). a 92 We denote the projection of E K E onto E X E^ by ?^^^. Theorem 9.3. For t 10, S^.^^^^^EEn ^^^^^^'^ strongly to S„,„(t), and the convergence is uniform in t for t in bounded intervals. Proof . This follows from the strong convergence of the open-loop semigroups and the uniform norm convergence of the control and estimator gains. D We should expect at least Theorem 9.3, but we need more. We should require, for example, that if S(t) is uniformly exponentially stable, then S . (t) must be also for n sufficiently large. Although nunerical results for nunerous examples with various kinds of damping and approximations suggest that this is usually true, we have been unable to prove it in general. We do have the result for the following important case. Suppose that the basis vectors e of the approximation scheme are the natural modes of undamped free vibration and that the structural damping does not couple the modes. Then, for each each n, E^ and E^^ reduce the open-loop semigroup T(t) and its generator A. For this case, we can extend A^.^ to D(A)X D(A) as A = |a »'n \fj L n ^Fn^Enl c T 1 nCompJ (9.10) where "^nComp = lAn-Vn-Vn^^En ^ ^•D(A)n E^* (9.11) Note that E is the span of the modes not represented in the n compensator. The operator A , generates a semigroup "s^.^tt) on E X E, E X E^^ and oD'n 93 {0}XE^ reduce "s^. ^(t) , and the restriction of "s^.-Ct) to E X E is S . (t). Hence S^,^(t) is uniformly exponentially stable if and only if both S„, (t) and the part of the open-loop system on E^ are uniformly exponentially stable. Theprem 9.4. i) Suppose that the basis vectors of the approximation scheme are the natural modes of undamped free vibration and that the structural damp- ing does not couple the modes. Then S (t) converges in norm to S . (t), *** n 00 CD uniformly in bounded t-intervals. ii) If, additionally, S^,^(t) is uniformly exponentially stable, then S , (t) oo'n is uniformly exponentially stable for n sufficiently large. Proof . From (9.6), (9.10) and (9.11), we have B[F„P„ -F]■ A , -A , 00 » 00 "oo ' [] A. /V [F-F^]C ^n^En-^n n J (9.12) where ^n= ^Vn^En-SP) ^ ^Vn^En-^^). (9.13) Therefore, I |A„,^-A„,^|| — > as n — >». and the theoron follows. D This paper empiiasizes using the convergence of the approximating control and estimator gain operators F^ and F^^, and the convergence of the functional gains that can be used to represent these operators, to determine the finite dimensional compensator that will ja-oduce essentially optimal closed-loop per- formance. However, close examination of the right sides of (9.12) and (9.13) reveals another important convergence question. While the gain convergence in (9.8) and (9.9) drives the off-diagonal blocks in (9.12) to zero, the norm convergence of the approximating input and output operators also is essential 94 C/-- in killing A . Expanding the two terms in this block yields B F P„ - BF = B„(F„Pp. -F) + (B -B)F, n n En n n En n (9.14) F C P„ - FC = (F-F)C„Pp„ + F(C^Pp.-C). n n En n n En n t,n (9.15) The second term on the right side of each of these equations represents, respectively, control and observation spillover, which has been studied exten- sively by Balas. Together, the control spillover and observation spillover couple the modes modelled in the compensator with the modes not modelled in the compensator. The spillover must go to zero — as it does when B^ and C^^ converge — for A^ „ - A^ to go to zero. We should ask then whether there exists a correlation between the con- vergence of F and F and the elimination of spillover. The answer is yes if n n no modes lie in the null space of the state wei^ting operator Q in the per- formance index and if the assumed process noise, whose covariance operator is Q, excites all modes, but this correlation is difficult to quantify. As we discussed in Section 6.4, the two main factors that determine the convergence rates of the gains are the Q-to-R ratio and the damping, neither of which affects the convergence of B and C„. On the other hand, when either factor n n (small Q/R or large damping) causes the gains to converge fast, it generally also causes the magnitude of F and F to be relatively small, thereby reducing the magnitude of the spillover terms in (9.14) and (9.15). Also, as n increases, the increasing frequencies of the truncated modes usually reduce the coupling effect of spillover. This is well known, although it cannot be seen from the equations here. In examples that we have worked, we have found that when n is large enough to produce convergence of the control and estimator gains, the effect of any remaining spillover is negligible. But this may not always be true, and spillover should be remembered. 95 9.3 Convergence of the Compensator Transfer Functions The transfer function of the n^^ compensator (shown in the bottom block of Figure 9.1) is a„(s) =-F„(sI-U„-B„F„+t„(CoF„-C„)])-^V ^^.^^^ which is an m X p matrix function of the complex variable s for each n, as is the similar transfer fvinction ffi(s) in (7.14) for the infinite dimensional com- pensator. We continue to assume Hypothesis 9.1. We will denote the resolvent set of [A-BF+F(CpF-C)] by p([A-BF+F(CqF-C)]). Theorem £.5. There exists a real number a^ such that, if Re(s) > a^, then s e p([A^-Bj^ Fj^+Fjj(CqFjj-Cjj)] ) for all n, and ^jyis) converges to 5(s), imiformly in compact subsets of such s. Proof . The operator [A-BF+F(CqF-C)] is obtained from a contraction semigroup generator by perturbation with bounded operators, and the approximations to the perturbing operators sire bounded in n, by strong convergence. In view of this, close examination of the basic approximation scheme in Section 4.1 will show that there exists a bound of the form lA^exp(aj^t) , independent of n, for the semigroups generated by lA^-Bj^ ^n'*'^n^^0^n~'^n^^ • ^^^o, these semigroups converge strongly to the semigroup generated by [A-BF+F(CqF-C)] , according to [G3, Theorem 6.6]. For Re(s) > a^ then, the resolvent operator in ^^(s) con- verges strongly to that in $(s), uniformly in compact s-subsets, by [Kl, page 504, Theoran 2.16, and page 427, Theorem 1.2]. □ 96 This result leaves much to be desired. For example, it does not guaran- tee that any subset of the imaginary axis will lie in p([A -B F +F (CnF - C„)]) for sufficiently large n, even if all of the ima- nnnnun n A ginary axis lies in p([A-BF+F(CqF-C)] ) . As with the convergence of the closed-loop systems, we can get more for certain important cases. Remark 9.6. If the open-loop semigroup T(«) (whose generator is A) is an ana- lytic semigroup, then there exist real numbers a, 6 and M, with 9 and M posi- tive, such that p([A-BF+F(CqF-C)]) contains the sector {s : larg(s-a) |< f + ^^ ' and for each s in this sector, ||(sI-[A-BF+F(CqF-C)])"^II < M/|s-a| Theorem 9.7. i) If the basis vectors of the approximation scheme are the natural modes of undamped free vibration and the structural damping does not couple the modes, then each s in p( [A-BF+F(CqF-C)] ) is in p([Ajj-B Fn"^Fn^^0^n"'^n^^^ ^""^ "^ sufficiently large and ff„(s) converges to !B(s) A as n — > « , uniformly in compact subsets of p( [A-BF+F(CqF-C)] ) . ii) If, additionally, T(») is an analytic semigroup, then S^(s) converges to »(s) uni- formly in the sector described in Remark 9.6. Proof , i) In this case, we have also *n(s) = VEn^s^ -''^nComp)'^' (9.18) where If _ is the operator on D(A) defined by (9.11). The result follows from (9.8) and (9.9) and the fact ^^at li^^ converges in norm to [A-BF+F(CqF-C)] . ii) The result follows from i) and a bound on (sI-A „ )~^ for large |s| that is obtained from the Neumann series in view nComp' ' ' 97 of (9.17) and the uniform-norm convergence of A „ ° nComp* Theorem 9.S. If A has compact resolvent, then S (s) converges to !5(s) for each s e p( [A-BF+F(CqF-C)] ) , uniformly in compact subsets. Proof . As a result of Theorem 4.4, the resolvent operator in S (s) converges in norm to the resolvent operator in ffi(s) for sufficiently large real s. After an artificial extension of A^ to E^ then, the present theorem follows from [Kl, pages 206-207, Theorem 2.25]. 98 10. Closing the Loop in the Example As in Definition 7.3, the optimal closed-Loop system is formed with the optimal infinite dimensional compensator, whicn consists of the optimal con- trol law for the distributed model of the structure applied to the output of an optimal infinite dimensional state estimator. This optimal control law is the limit of the approximating finite dimensional control laws in Section 6. in this section, we first approximate the infinite dimensional estimator, as in Section 8. and then apply the approximating control laws in Section 6 to the approximating finite dimensional estimators to produce a sequence of fin- ite dimensional compensators that approximate the optimal compensator. 10.1 The Estimator Problem We assume that the only measurement is tlie rigid-body angle 9 and that A _ 4 this measurement has zerc^-mean Gaussian white noise with variance R - 10 . We model the process noise as a zero-mean Gaussian white disturbance that has a -K^mponent distributed uniformly over the beam, as well as two concentrated components that exert a force on the tip mass and a moment on the hub^ For this disturbance, the covariance operator Q has the form (7.23) with 0^= 0, A A Qj^ = and Q2 = I- We construct the approximating estimators as in Section 8.1. The gain for the nth estimator is given by (8.13) with the solution to the Riccati matrix equation (8.14). For the rigid-body measurement, the matrix C" is cn=[lOOO.,.]. ^^^^^^ 99 According to (8.15), the matrix Q" it \n P* 0* ] (10.2) since W'^ is the matrix in (4.33). (As always, M"" is the inverse of the mass matrix.) Recall from Section 6.3 that n = 2n + i „h«,.^ . .u xHg + 1 where n^ is the number of elements. Our only use for the functional estimator gains is to measure the conver- gence of the finite dimensional estimators to the optimal infinite dimensional estimator. To see the convergence of the approximating estimator gains, we compute the approximating functional estimator gains as in Section 8.3. Like the functional control gains, the functional estimator gains have the form ^ = («f.«>f..pf.) . (10.3a) « = <°g'«^g.Pg) ' (10.3b) and the corresponding approximations have the form A ^n ~ (o*»^»0 .B ) . " in"^gn"^gn' ' (10.4a) A ^n - <°gn'«*gn'Pgn> • (10.4b) Mmsk 10.1 We cannot guarantee as much about convergence for the approximat- ing estimators as we could for the approximating control problems in Section 6. Since the damping in this example does not couple the natural modes and the rigid-body mode is observable, we would have Part i) of Theorem 8.5 if we were using the natural mode shapes as basis vectors. Therefore, we know at 100 least that a solution to the infinite dimensional Riccati equation (7.16) exists and that the infinite dimensional estimator that we want to approximate exists. The numerical results indicate that the solutions to the finite dimensional Riccati equations are bounded in n and that the functional esti- mator gains converge in norm. The rigid-body mode prevents our guaranteeing a priori all the convergence that we want. If a torsional spring and damper were attached to the hub in the current example, we would have coercive stiff- ness and damping and Theorem 8.5 11) would guarantee that the solutions to the finite dimensional Riccati equations converge strongly and that the functional estimator gains converge in norm for the basis vectors used here. Also, see Remark 6.2 and Remark 8.6. D For damping coefficient c^ = lo""*. Figures 10.1 and 10.2 show ^^^ and <^^. and Tables 10.1 and 10.2 list the the scalars a^n-Og^ and Pg^. Since ^f^^O) = -*' (0) =0, the convergence of (,'' implies the convergence of p^.^ = iftf^C^); as in the control problem, p^.^^ is not an Independent piece of information about the estimator gains while, as far as our results go. Pg„ is. We main- tain analogy with the control problem and list only pg^ in the table. 101 o N5 ---e ^'- '*^ ^ -~^^ ~ r r 3 1 lb. 00 pb. 00 3b. 00 vb.oo sb.oo sb.oo 7^ 00 80.00 "s: 00 I bo. 00 Figure lO.lft. Functional Estimator GEiin Component ^_ -4 -4 Damping Cj, = lO ; estimator R = 10 ^ Figure 10 'li- Functional Estimator Gain Component gn Damping c^ = lo ; estimator R 10 number of elements n = 2,3.4,5,8 number of element n = 2, 3, 4, 5, 8 o o O ' 3i » t -B-. -^^1 ,TSi J^o ^^0 ^:^o i^TS^ ^^' ^^^ ^^' ^^' ^-^ Figure 10.2a. Functional Estimator Gain Component ^^^ Damping Cq = 10 . estimator R = 10 number of elements n^ = 4, 6, 8, 10 1 ~ o a> \\ o V oo V Tl 50 ^ ■oS o \. o z o s T~ o U3_ =i - ^^ -< w o ^%« - lO C3 ■ ,m N^ *-- ^^ o \^ »r> ^C 3* \\ o V. p 0.00 lb. 00 j'o.oo 30.00 lib. 00 sb.oo s'o.oo 70.00 «o7oo Figure 10.2Jj, Functional Estimator Gain Component Damping C„ = lo"^; estimator R = 10"'* number of elements n^ = 4, 6. 8, 10 "e °fn «gn Pgn 2 5.0358 12.680 -1334.9 3 5.2514 13.789 -1455.4 4 5.3195 14.149 -1495.7 5 5.3478 14.300 -1512.2 8 5.3611 14.371 -1520.1 Tafeifi 10 .1. Scalar Components of Functional Estimator Gains Damping coefficient Cq = lo"'* ; estimator R = 10"'* number of elements n = 2,3,4,5,8 e fn "gn Pgn 4 5.3195 14.149 -1495.7 6 5.3567 14.347 -1517.5 8 5.3611 14.371 -1520.1 10 5.3623 14.377 -1520.8 labifi 10.2. Scalar Components of Functional Estimator Gains Damping coefficient c^ = lo"'*; estimator R = 10""* number of elements n. = 4, 6, 8, 10 104 Figures 10.3a and 10.3b and Table 10.3 give the numerical results for the fin- ite dimensional estimators when the structural damping is zero. While Theorem 8.4 says that the solutions to the finite dimensional Riccati equations for these estimators will not converge when the damping is zero and Q is coercive A on E. we have no result to predict the convergence for zero damping when Q is not coercive (even though Qj is coercive on H) . From the numerical results though, f does not appear to converge. ^^ n "e °fn °gn ^gn 2 5.0730 12.868 -1354.4 3 5.3390 14.253 -1506.0 4 5.4417 14.806 -1568.0 5 5.4894 15.067 -15 96.3 8 5.53 98 15.345 -1627.2 Table 10.3 . Scalar Ctoraponents of Functional Estimator Gains —4 Zero damping; estimator R = 10 nvmiber n = 2 , 3 , 4 , 5 , 8 ORIGINAL PAGE IS OF POOR QUALITY 105 o On o o "n 33 •0 ■cr> o t£' o Tl- XI i"- o '■J c 3^ > t) in ih.oa ;b.oo jb. oo vb.oo sKj 6b, 00 7b. 1.00 9b. 00 ibo.oo n&urea 10.3a. Functional Estimator Gain Component «(" Zero damping; estimator R = 10~^ number of elements n^ = 2. 3, 4. 5. 8 '°-''° '^'' '^^ ^^' ^^0 ^^0 i^o 1^, iST^o ,l,:S n.0.00 fiance lO-lb. Functional Estimator Gain Component ^ Zero damping; estimator R = 10 ^ gn number of elements n 2, 3, 4, 5, 8 10.2 Approximation of the Optimal Compensator Finally, for the damping c^ = lO"^ R = -05 in the control problem and R = 10"* in the estimator problem, we construct the finite dimensional compensa- tor in Figure 9.1; i.e. for each n = 2n^ + 1. we apply the nf control law represented by the functional gains in Figure 6.4 and Table 6.4 to the output of the n^^ estimator represented by the functional gains in Figure 10.2 and e Table 10.2. As the number of elements increases, the transfer function in (9.16) of the finite dimensional compensator converges to the transfer func- tion in (7.14) of the optimal infinite dimensional compensator, as described in Section 9.3. Theorem 9.5 and Remark 9.6 apply. Figure 10.4 shows the fre- quency response (bode plots) of the finite dimensional compensators f or 4 . 6 . 8 and 10 elements. The phase plot is for 10 elements only. These plots indi- cate that the finite dimensional compensator for eight or more elements is virtually identical to the optimal infinite dimensional compensator, as predicted by the functional gain convergence in Figures 6.4 and 10.2. 107 o 00 0-' ^ j^ 56789^0- i 3ite7^h\0' ^ ^^ih H^\o' ^ hi^i^H^\^ ^ U^^i \(f \o-' ^ i^ihHi\o-' ^ iTJIH^^o''^ W^ y^Ho- ^ UUU^^o' ^ UU^A^ W (RHD/SEC) Uemm 10.4. Frequency Response (Bode Plot) of Compensator! Damping o^ = iq-^; control R = .05, estimator R = lO"'* number of elements n = 4, g, g^ jq 10' 10.3 Conunents on the Structure and Dimensior of the Implementable Compensators Though this paper does not address the problem of obtaining the lowest- order compensator that closely approximates the infinite dimensional compensa- tor, we should note that the compensators based on eight and ten el«nents here are unnecessarily large because the finite element scheme that we chose is not nearly the most efficient in terms of the dimension required for convergence. (The dimension of the first-order differential equation in the compensator is 2(2ne+l).) We used cubic Hermite splines here to demonstrate that the finite element scheme most often used to approximate beams in other engineering applications can be used in approximating the optimal compensator. In [G5l . we compare the present scheme with one using cubic B-splines and one using the natural mode shapes as basis vectors. The natural mode shapes yield the fastest converging compensators, but the B-splines are almost as good. The only advantages of the Hermite splines result from the fact that the coding to build the basic matrices (mass, stiffness, etc.) is simpler than for B-splines and the fact that, before the Riccati equations based on, say, ten natural modes are solved, a much larger finite el«nent approximation of the structure must be used to get the ten modes accurately. To understand the redundancy in the large finite dimensional compensators here, it helps to consider the structure of the optimal compensator. It is based on an infinite dimensional state estimator that has a representation of each of the structure's modes. In the present example, the optimal compensa- tor estimates and controls the the first six modes significantly, the next three modes slightly, and virtually ignores the rest. This observation is 109 based on the projections of the functional gains onto the natural modes and on comparison of the open-loop and closed-loop eigenvalues. (See [G5] for more detail, including the spectrum of closed-loop system ~ which is stable — obtained with the ten-element compensator here.) The infinite dimensional compensator then has an infinite number of modes that contribute nothing to the input-output map of the compensator. These inactive modes are just copies of all the open-loop modes past the first nine. They can be truncated from the compensator without affecting the closed-loop system response signifi- cantly. The number of active modes in the compensator ~ i.e., the modes that contribute to the input-output map — depends on the structural damping and the Q's and R's in the LQG problem statement. (See the discussion in Section 6.4 about the effect of damping and control weighting on performance.) The compensator computed here based on ten elements has 21 modes (although we did not do the computations in modal coordinates). Nine of these compensator modes are virtually identical to the nine active modes in the infinite dimensional compensator, and the twelve inactive modes are approxima- tions to the tenth through twenty-first open-loop modes of the structure, "me inactive modes result from the large number of elements needed to approximate the active compensator modes accurately. Now that we essentially have the optimal compensator in the ten- element compensator, we could truncate the twelve inactive modes and implement a compensator with nine modes. And we probably could reduce the compensator even further using an order reduction method like balanced realizations. 110 11. Conclusions For the deterministic linear-quadratic optimal regulator problem for a flexible structure with bounded input operator (the B^ in (2.1)), the approxi- mation theory in Sections 4 and 5 is reasonably complete. Ilie most important extensions should be to the corresponding (vtiry difficult) problem with unbounded input operator, for which there exists little approximation theory. Because of the different kinds of boundary input operators, stiffness opera- tors and structural damping, all of which must be considered in detail when B^ is unbounded, it seems unlikely that the approximation theory for the unbounded-input case can be made as complete as the theory here. The convergence results in Section 8 for the estimation problem are less complete than those for the control problem because rigid-body modes present more technical difficulties for the proofs in the estimator case. However, our analysis and numerical experience suggest that the difficulties only make the proofs harder and that the convergence in the estimation problem is ident- ical to the convergence in the control problem, and that controllable and observable rigid-body modes make no qualitative difference in either problem. Where we would most like substantial improvement over the results of this paper is in Section 9.2. which considers how the approximating closed-loop systems obtained by controlling the distributed model of the structure with the finite dimensional compensators converge to the optimal closed-loop sys- tem, obtained with the infinite dimensional compensator. Theoreoi 9.4 gives us what we want for problems where the damping does not couple the natural modes of free vibration and the natural mode shapes are the basis vectors for the 111 approximation scheme. In particular, this theorem says that, if the optimal closed-loop system is uniformly exponentially stable, then so are the approxi- mating closed-loop systems for sufficiently large order of approximation. We have verified numerically the stability of the approximating close-loop sys- tems for the example in Sections 6 and 10, where the basis vectors are not the modes. This example and others have made us suspect that Theorem 9.4 is true when the basis vectors satisfy Hypothesis 4.1 only and when the damping cou- ples the modes. Another possible approach to analyzing the convergence of the approximat- ing closed-loop systems to the optimal closed-loop system is to use the input-output description in frequency domain. Results like those in Section 9.3 are useful for this, although for the closed-loop stability we want, we probably need the transfer functions of the finite dimensional compensators to converge more uniformly on the compensator resolvent set than we have proved here. In our example. Figure 10.4 indicates that these transfer functions converge uniformly on the imaginary axis, but we have no theorem that guarar^ tees this. 112 References Bl A.V. Balakrishnan, Applied Functional Analysis, Second Edition, Springer-Verlag, New York, 1981. B2 A.V. Balakrishnan, "Strong Stability ciid the Steady-State Riccati Equation." A ppl. Math. Opt. , 7, 1981, pp. 335-345. B3 M.J. Balas. "Feedback Control of Flexible Systems," IEEE Trans. AC , AC-23 , pp 673-679, 1978. B4 M.J. Balas, "Modal Control of Certain Flexible Systems", SIAM J. Contr. Opt .. 16(1978) pp. 450-462. B5 H.T. Banks and J. A. Burns, "Hereditary Control Problems: ^^ Numerical Methods Based on Averaging Approximations, SIAM J. Contr. Opt ., 16(1978) pp. 169-208. B6 H.T. Banks and K. Kunisch, "The Linear Regulator Problem for Parabolic Systems," SIAM J. Contr. Opt ., Vol. 22. No. 5, September 1984, pp. 684-698. B7 D.S. Berstein and D.C. Hyland, "The Optimal Projection Approach to Designing Finite-dimensional Controllers for Distributed Parameter Systems," 23rd IEEE Conference o n Decision an d Control, Las Vegas, NV, December 1984 . B8 D.S. Berstein and D.C. Hyland, "The Optimal Projection Equatior,s for Finite-dimensional Fixed-order Dy|fmic Compensation of Infinite-Dimensional Systems, SIAMjL- Contr . Opt . , to appear. CI G. Chen and D.L. Russell, "A Mathematical Model for Linear Elastic Systems with Structural Damping," Quarterly of Applied Mathematics . January 1982. pp. 433-454. C2 R.W. Clough and J. Penzien, Dynamics of Structures, McGraw- Hill, New York, 1975. C3 R.F. Curtain, "On Stabiliz ability of Linear Spectral Syst«ns via State Boundary Feedback," '^tam J. Contr. Opt., Vol. 23, No. 1, 1985, pp. 144-152. C4 R.F. Curtain and A.J. Pritchard, Inf j.ni te Dimensional Linear S ystems Theory , Springer-Verlag, New York, 1978. C5 R.F. Curtain and D. Salamon, 'Tinite Dimensional Compensators for Infinite Dimensional Systems with Unbounded Control Action." SIAM J . C ontr. Optim .. to appear. Dl N. Dunford and J.T. Schwartz, Linear OperatjiEff, Part II, Wiley Interscience, New York, 1963. 113 Gl J.S, Gibson, "An Analysis of Optimal Modal Regulation: Convergence and Stability." SIAM J. Contr. Opt. , Vol. 19, No. 5, September 1981, pp. 686-707. G2 J.S. Gibson, "A Note on Stabilization of Infinite Dimensional Linear Oscillators by Compact Linear Feedback," SIAM J. Contr. Opt. . 18(1980) pp. 311-316. G3 J.S. Gibson, 'linear-quadratic Optimal Control of Hereditary Differential Systems: Infinite Dimensional Riccati Equations and Numerical Approximations," SIAM J. Contr . Opt.. Vol. 21, No. 1, January 1983, pp. 95-139. G4 J.S. Gibson, 'The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces," SIAM J. Contr. Opt .. Vol. 17, No. 4, July 1979, pp. 537-565. G5 J.S. Gibson and A. Adamian, "A Comparison of Approximation Schemes for Optimal Control of Flexible Structures," to appear in SIAM Frontiers Edition on Control of Dis- tributed Systems, 1988. G6 J.S. Gibson and D.L. Mingori, "Approximation and Compensator Order Determination in Optimal Control of Flexible Structures," Proc. IFAC Workshop on Model Error Concepts and Compensation . Boston, MA, June 1985. G7 J.S. Gibson, D.L. Mingori, A. Adamian and F. Jabbari, "Approximation of Optimal Infinite Dimensional Compensators for Flexible Structures," JPL Workshop on Identification and Control of Space Structures, San Diego, CA. June 1984. G8 J.S. Gibson, D.L. Mingori, at al, Integrated Control/Structure Research for Large Space Structures, HR Textron Report #956541-Final, to JPL, September 1984. G9 J.S. Gibson and M. Navld, "Approximate Solution of Riccati Algebraic Equations in Optimal Control and Estimation of Hyperbolic Systems," International Symposium on Mathematical Theory of Networks and Systems, Santa Monica, CA, August 1981. Kl T. Kato, Perturbation Theory for Linear Operators. Second Edition, Springer- Verlag, New York, 1984. LI I. Lasiecka and R. Triggiani, "Riccati Equations for Hyperbolic Partial Differential Equations with L2(0,T; L2(r)) - Dirichlet Boundary Terms," Proceedings of 24th Conference on Decision ana Control. Fr. Lauderdale, FL. December 1985. pp. 114-116; to appear in SIAM J. Contr. Opt. L2 I. Lasiecka and R. Triggiani, "Dirichlet Boundary Control 114 Problem for Parabolic Equations with Quadratic Cost: Analyticity and Riccati's Feedback Synthesis," SIAM.J- Contr. Opt .. Vol. 21. No. 1, January 1983, pp. 41-67. Ml D.L. Mingori. J.S. Gibson. P. Blelloch and A. Adamian. "Control of a Flexible Space Antenna: A Finite Dimensional Perspective Based on Distributed Parameter Theory." JPL Workshop on Identification and Control of Space Structures. San Diego. CA. June 1984. M2 B.C. Moore, "Principal Component Analysis in Linear Systems: Controllability. Observability, and Model Reduction," IEEE Trans. Automat. Contr - ^ vol. AC-26 , pp. 17-32, Feb., 1981. 51 M.H. Schultz, Spline Analysis . Prentice- Hall, Englewood Cliffs, N.J., 1973. 52 J.M. Schumacher, "A Direct Approach to Compensator Design for Distributed Parameter Systems," SIAM J. Contr. Opt., Vol. 21. No. 6, 1983. pp. 823-836. 53 R.E. Show alter, Hilbert Spac e Me thodsJ'CTi^ Partial DiffenenU.al Equations, Pitman, London, 1977. 54 G. Strang and G.J. Fix, An Analysis of the Fini te Element Method , Prentice-Hall, Englewood Cliffs, N.J.. 1973. 115 APPENDIX Errata for [Gl] In the first paragraph of the proof of Theorem 2.1 on page 689 of [Gl], the first sentence should be: If a dissipative operator is invertible, its inverse is dissipative. At the beginning of the fifth line of the same paragraph, the expression (ax+y) should be deleted the first time it occurs. The next-to-last sentence of the paragraph should be: Hence, if a densely defined maximal dissipative operator has dense range, its inverse is maximal dissipative. The theorem is correct as stated. In the current paper, we use Theorem 2.1 of [Gl] to conclude that the operator A defined in Section 2 is maximal dissipative (see o (2.10)-(2.12)) and that the operator A in (2.16) has a unique maximal dissipative extension. 116 ORIGINAL ?A.^£ -S OF POOR QUAUTY (UASA Report Documentation Page 1 Report No NASA CR- 181705 ICASE Report No, 88-48 2. Government Accession No. 4 Title and Subtitle APPROXIMATION THEORY FOR LQG OPTIMAL CONTROL OF FLEXIBLE STRUCTURES 3. Recipient's Catalog No 5 Report Date August 1988 7 Author(s) J. S. Gibson and A. Adamlan 9 Performing Organization Name and Address Institute for Computer Applications In Science and Engineering Mail Stop 132C, NASA Langley Research Center Hampton. VA 23fif.'S-S?? ' ) 6 Performing Organization Code 8 Performing Organization Hepon No 88-48 12 Sponsoring Agency Name and Address National Aeronautics and Space Administration Langley Research Center Hampton, VA 23665-5225 15 Supplflrnentary Notes Langley Technical Monitor: Richard W. Barnwell Final Report 10. Work Unit No. 505-90-21-01 11. Contract or Grant No NAS1-17070, NASl-18107 13. Type of Report and Period Covered Contractor Report 14 Sponsoring Agency Code Submitted to SIAM Journal of Control and Optimization 16, Abstract j^ls paper presents approximation theory for the llnear-auarfr;,t fr r= < finite element or mr,/.! ^^^^"f \°'^ ^ sequence of finite dimensional, usually T„. p^ ^7"^""^ °/ "°''^1' approxitnatlons of the distributed model of the structure The n^^rdIl:n:rnal°"' "'T'l ^'^ "'""°" ^° "^^ approxlmatrng rob : eluding formulas f «q"atlons for numerical approximation are developed, in- uo^^iMpTrst^tt; rri;t"^eorprri:o:°:f^-;ainrb;^^7n°difT/r"%'° r- h- L^:r::;oL°;;/7i7i7:"d^i " t ?-°— - --- "- ::.ttii:i7sZ':z : z Lorresponaing finite dimensional compensator"; 1c ci-„^< = j »i 17 Key Words (Suggested by Author(s)) control, flexible structures, optimal control, approximation 19 Security Classif (of this reponi Unclassified NASA FORM 1S26 OCT 86 18. Dittrtbution Stalenwnt 08 - Aircraft Stability and Control 63 - Cybernetics Unclassified - unlimited 20. Security Classif (o1 this page) Unclassified 21 No of pages 121 22 Price A06 NASA-Langley, 1988