A Robust Approximation Scheme for the LQG Control of an ...ing.univaq.it/manes/FilesLavoriPDF/R017_A Robust... · the LQG control problem, in a proper Hilbert space setting. In particular,

European Journal of Control (2006)6:635–651# 2006 EUCA

A Robust Approximation Scheme for the LQG Control

of an Undamped Flexible Beam with a Tip Mass

Alfredo Germani1,2,**, Costanzo Manes1,2,***, Pasquale Palumbo2,*and Pierdomenico Pepe1,****1Dipartimento di Ingegneria Elettrica e dell’Informazione, Università degli Studi dell’Aquila, Poggio di Roio, 67040 LAquila, Italy;2Istituto di Analisi dei Sistemi ed Informatica ''Antonio Ruberti'', IASI-CNR, Viale Manzoni 30, 00185 Roma, Italy

This work deals with the LQG control problem of aflexible beam clamped at one end and with a point massat the free end, where a boundary control force can beapplied. A class of finite-dimensional control laws isproposed here, derived on the basis of the Euler-Bernoulli infinite-dimensional beam model. By meansof this approach it is possible to take into account alsothe higher order modes that are indeed neglected inthe more usual methods based on a finite-dimensionalmodel of the beam. The main motivation for theapproach followed here is that it naturally allows toovercome the phenomenon of spillover, occurring whenunmodeled modes are excited by the control law itself.The finite-dimensional control law here proposed isderived by a Galerkin approximation of the solution ofthe LQG control problem, in a proper Hilbert spacesetting. In particular, the novelty of the approach is thedefinition of an implementable Galerkin approximationscheme based on generalized eigenfunctions of theEuler-Bernoulli model instead of the usual splines.It is here proved that, for any given finite time horizon,the evolution of the system state driven by the proposedcontrol input converges, in L2 norm, to the optimalLQG evolution, as the order of the approximationscheme increases. The strong stability of the closedloop system is guaranteed for any order of theapproximation scheme. Moreover, it is proved that theproposed compensator guarantees modal stability of

the closed loop system also in the presence of stiffness/inertia parameters uncertainties.

Keywords: Flexible structures; LQG regulator;infinite-dimensional systems; Galerkin approximation

1. Introduction

Active control of flexible structures is an appealing andchallenging topic in the field of Systems and ControlTheory. The availability of smart sensors and actuatorsfor flexible structure [11] makes it possible somesignificant applications, such as stabilization of largespace structures, space robots, movement of largecranes, active damping of buildings and bridges, etc.

It is well known that flexible structures can beaccurately described by continuum models, i.e.,dynamic systems with infinite-dimensional statespace, also denoted distributed parameter models.Therefore, any finite-dimensional model necessarilyneglects the higher order vibration modes. On theother hand, a control law can be implemented only if itadmits a finite-dimensional representation. Two mainapproaches can be followed to obtain finite-dimen-sional control laws for infinite-dimensional systems,such as flexible structures:

(i) classical approach: a finite-dimensional controllaw is designed on the basis of a finite-dimensional model of the flexible structure;�Correspondence to: P. Palumbo; e-mail: [email protected]

��E-mail: [email protected]��E-mail: [email protected]��E-mail: [email protected]

Received 2 December 2005; Accepted 20 October 2006Recommended by S.M. Veres, D. Normand-Cyrot.

(ii) modern approach: a finite-dimensional controllaw is designed on the basis of a distributedparameter model of the flexible structure.Often an implementable controller is obtainedthrough a suitable approximation of the infinite-dimensional control law derived on the basis ofthe infinite-dimensional model.

The classical approach, widely adopted since late 70s[7,8], consists of approximating the continuum model(finite elements, Galerkin projections, or otherapproximation schemes [14]) and then to formulatethe control problem in a finite-dimensional frame-work. However, such a classical approach neglects thehigher order modes of vibration of the structure, sothat in many cases the unmodeled modes can beexcited by the control law itself, so inducing undesiredand unexpected vibrations in the structure, phenome-non known as spillover [6]. Because of the intrinsicinfinite-dimensional nature of the mechanical struc-ture, any control law designed on the basis of a finite-dimensional model may produce this undesired effect.In the literature many attempts have been madetowards the increasing of the model dimension withthe idea of reducing the spillover drawback. A niceimprovement in this field consists of developingmethods and criteria to obtain a reduced order model,in order to simplify the resulting approximated model(see e.g. [9]). In the case of a model valid only atlow frequencies, the frequency-shaped LQG designapproach may be applied, due to the pioneering workof Gupta [18]: according to this methodology, a fre-quency-shaped cost functional is considered, whichassigns weight matrices increasing with frequency, inorder to minimize the control and state activity abovethe upper natural frequency of the finite-dimensionalmodel. In fact, the frequency-weight introduced in thecost functional allows to take into account, in somesense, the unmodeled system dynamics so that, in caseof a well-designed frequency-shaped cost functional,spillover effects may possibly be reduced or evenavoided. However, by using a frequency-weight, thephysical meaning of the cost functional is lost, in thatit is no more possible to say that the designed controlaction is of minimal energy.

The modern approach allows to overcome thedrawbacks of the classical one, in that it providesfeedback control laws suitably designed exploiting theinfinite-dimensional model, where all the modes ofvibration of the system are considered. With thisapproach, the spillover effects are avoided by ensuringthe closed loop stability for the original flexiblestructure. Recent papers investigate the exponentialstability problem for an undamped Euler-Bernoulli

beam, by means of a boundary feedback from a finitenumber of measurements [13,16,17]. An importantwork concerning finite-dimensional regulators of ageneral undamped flexible structures is [5], where therobustness of the closed loop system w.r.t. the stabilityproperties is investigated. Moreover, an infinite-dimensional approach allows to deal with minimalenergy control strategies, based on the minimizationof a cost functional which represents the physicalenergy of the structure. When the regulator is achievedwithout taking its physical realizability into account,the resulting feedback control law may well be definedas an operator acting on an infinite-dimensional statespace. In these cases, a finite-dimensional approxi-mation scheme is needed in order to implement thecontrol law (see, e.g. [3,4,22]). In order to makeeffective such a methodology, for any approximationindex, the corresponding regulator has to ensure themodal stability of the system, in order to overcome thespillover; moreover, the approximation scheme maystill take into account the optimality of the controlaction, by ensuring some type of convergence to theinfinite-dimensional optimal control law.

This paper deals with the LQG control of anundamped Euler-Bernoulli cantilever beam with a tipmass at the free end. Such problem is investigated in[3,4], where the optimal solution is achieved in aninfinite-dimensional setting, and also an approximateimplementable transfer function is provided. Howeverit is not clear how far the proposed approximatecompensator is from the optimal solution.

The idea of this paper is to present a sequence ofapproximate, physically implementable regulatorsconverging to the optimal LQG solution. It is provedthat, for any given finite time horizon, the evolution ofthe system state driven by the approximate controlinput converges in L2 norm to the evolution of thesystem state driven by the optimal control law, as theorder of the approximation scheme increases. Aninteresting feature is that the transfer function of theapproximated compensator monotonically approa-ches the shape of the transfer function of the optimalcompensator, as it can be appreciated in the magni-tude/phase Bode representation. Preliminary resultshave been presented in [21].

In this paper, following the same lines of [5,22], it isproved also that, for each order of the approximation,the closed loop system is strongly stable, and, more-over, the proposed compensator guarantees modalstability of the closed loop system also in the presenceof stiffness/inertia uncertainties.

As it will be clearer in the sequel, such performancesare indeed achieved by means of a Galerkin approxi-mation scheme, based on the complete set of the

636 A. Germani et al.

generalized eigenfunctions of the structure instead ofthe usual splines.

The paper is organized as follows: the next section isdevoted to describe the flexible structure investigatedand the related LQG control law derived in an infi-nite-dimensional framework, by means of a con-tinuum model in a Hilbert space; the third sectiondefines a new Galerkin approximation scheme andpresents the main properties of the involved projectionoperators; convergence results are presented in sectionfour; in section five the equations of the finite-dimensional compensator and the I/O transfer func-tion are derived, and structural properties are inves-tigated; section six describes the robustness of theclosed loop system w.r.t. stability; conclusions arereported in section seven.

2. A Summary on the Euler-Bernoulli

Beam Model

This section describes the continuum model of theflexible structure under investigation and the solutionof the corresponding optimal control problem in aninfinite-dimensional framework (see [3] and referencestherein for details). This model has been intensivelystudied in the literature (see, e.g. [3,4,13,16,17,19]) andis particularly suited for the description of large,flexible space systems (e.g. the SCOLE configuration[3]), whose main characteristics are the practicalabsence of damping and gravity.

The flexible system, here considered in absence ofgravity and damping, is a homogeneous, uniform,undamped Euler-Bernoulli beam of length 2l, extend-ing from � l, the clamped end, to þ l, the free end,where a tipmassm is concentrated.Aplanar problem isconsidered, i.e. only planar beam deflections are con-sidered. Letw(t, s) be the deflection of the beam at timet and position s, t � 0, �l � s � l. This mechanicalstructure is described by the following PDE with theboundary conditions (as usual dots indicate timederivatives and primes indicate spatial derivatives):

��€wðt,sÞþEIw0000ðt,sÞ¼ 0, s2 ½�l, l�, ð2:1aÞ

wðt, � lÞ ¼ w0ðt, � lÞ ¼ w00ðt, lÞ ¼ 0, ð2:1bÞ

m€wðt, lÞ þ uðtÞ þNsðtÞ ¼ EIw000ðt, lÞ, ð2:1cÞ

where � is the mass density, � is the cross-sectionalarea, E is the Young modulus and I is the beam crosssection moment of inertia. The boundary controlforce u(t) is applied at the free end, where it is

concentrated also a Gaussian white noise NsðtÞ withspectral density ds. The sensed data y(t) are theboundary rates affected by a measurement error, alsomodeled by a Gaussian white noiseNoðtÞ with spectraldensity do, independent of NsðtÞ:

yðtÞ ¼ _wðt, lÞ þNoðtÞ: ð2:2Þ

Denote

S ¼fw2L2ð½�l, l�, IRÞ :w0,w00,w000,w0000

2L2ð½�l, l�, IRÞ;wð�lÞ ¼w0ð�lÞ ¼w00ðlÞ ¼ 0gð2:3Þ

the class of all L2 functions, having the first fourderivatives in L2, compatible with the boundary con-ditions (2.1b), endowed with the inner product

hw, hiS¼Z l

�l

wðsÞhðsÞdsþ wðlÞhðlÞ: ð2:4Þ

It follows that the completion of S in this inner pro-duct yields the Hilbert space H ¼ L2ð½�l, l�, IRÞ � IR,endowed with the usual inner product of a Cartesianproduct space, so that a generic element w 2 S may be

considered an element of H as x ¼ wð�ÞwðlÞ

� �and, on

the contrary, if x ¼ wð�Þa

� �2 H belongs to S, then

a ¼ wðlÞ. This is the sense of the inclusion S H.Then, system (2.1), with the measurement Eq. (2.2),can be rewritten as:

M€xðtÞ þ AxðtÞ þ BuðtÞ þ BNsðtÞ ¼ 0, ð2:5aÞ

yðtÞ ¼ B� _xðtÞ þNoðtÞ: ð2:5bÞ

The operatorA : DðAÞ ¼ S H7!H (S is dense inH),defined as

xðtÞ¼ wðt,�Þwðt,lÞ

� �2DðAÞ ) AxðtÞ¼ EIw0000ðt,�Þ

�EIw000ðt,lÞ

� �,

ð2:6Þis self adjoint, positive definite, with a bounded,compact inverse [3]. The operator M : H7!H,defined as

xðtÞ¼ wðt,�ÞaðtÞ

� �2H ) MxðtÞ¼ ��wðt,�Þ

maðtÞ

� �,

ð2:7Þis self adjoint, positive definite, bounded with boun-ded inverse. The operator B : IR 7!H is defined as

B�ðtÞ ¼ 0�ðtÞ

� �2 H, ð2:8Þ

LQG Control of an Undamped Flexible Beam 637

and B� : H7!IR, the adjoint of B, is given by:

xðtÞ ¼ wðt, �ÞaðtÞ

� �2 H ¼) B�xðtÞ ¼ aðtÞ:

ð2:9Þ

Of course, both B and B� are bounded.To get a first-order equation form with boundedsolutions for system (2.5), the Hilbert space

He ¼ DðffiffiffiffiA

pÞ �H, dense in H2, endowed with the

following inner product is needed [3]. Let X ¼ x1x2

� �,

Y ¼ y1y2

� �, with x1, y1 2 Dð

ffiffiffiffiA

pÞ and x2, y2 2 H.

Then:

h x1x2

� �,

y1y2

� �iHe¼h

ffiffiffiffiA

px1,

ffiffiffiffiA

py1iHþhMx2,y2iH:

ð2:10Þ

Note that the square of the He norm is twice theenergy of the system, when applied to a vector of thetype x1 2 S (i.e. an admissible deflection) and x2 ¼ _x1:

kXk2He

¼ kffiffiffiffiA

px1k2H þ hMx2, x2iH

¼ hAx1, x1iH þ hM _x1, _x1iH: ð2:11Þ

By using the following operators A :DðAÞ¼S�Dð

ffiffiffiffiA

pÞHe 7!He:

A x1x2

� �¼ 0 I

�M�1A 0

� �x1x2

� �¼ x2

�M�1Ax1

� �,

ð2:12Þ

B : IR 7!He, Ba ¼0

�M�1Ba

� �,

B� x1

x2

� �¼ �B�x2,

ð2:13Þ

the second order system (2.5) can be written as

_XðtÞ ¼ AXðtÞ þ BuðtÞ þ BNsðtÞ, ð2:14aÞ

yðtÞ ¼ �B�XðtÞ þNoðtÞ: ð2:14bÞ

Note that B and BB� are bounded operators and,moreover, A is skew-adjoint, generates a contractionsemigroup and has a compact resolvent [3]. The initialstate Xð0Þ is assumed Gaussian, independent of NsðtÞand NoðtÞ, with IEfXð0Þg ¼ 0.

In agreement with the objective of reducing theboundary rates while keeping bounded the controlinput, the following LQG index has to be minimized:

J¼ limT!þ1

IE1

T

Z T

0

jB�XðtÞj2dtþ �

T

Z T

0

juðtÞj2dt� �

,

ð2:15Þwhere � > 0 is the weight of the control law. Theoptimal solution is given by [2,3]

uoðtÞ ¼�B�Pc

bXðtÞ�

, ð2:16Þ

where the state estimate bXðtÞ is given by the infinite-dimensional equation

_bXðtÞ ¼ ðA � Pfd�1o BB�Þ bXðtÞ

� BB�PcbXðtÞ

�� Pfd

�1o ByðtÞ,bXð0Þ ¼ IEfXð0Þg ¼ 0: ð2:17Þ

The operators Pc and Pf are the unique self adjoint,nonnegative definite solutions of the steady stateRiccati equations

0 ¼ hPcX ,AXiHe

þ hAX ,PcXiHe

þ kB�XkHe

�kB�PcXk

He

�, ð2:18aÞ

0 ¼ hPfX ,A�XiHe

þ hA�X ,PfXiHe

þ dskB�XkHe

� d�1o kB�PfXk

He, ð2:18bÞ

and are determined in the following closed form:

Pc ¼ffiffiffi�

pI, Pf ¼

ffiffiffiffiffiffiffiffiffidsdo

pI: ð2:19Þ

By substituting (2.19) in (2.16) and (2.17), thedynamics of the control law becomes:

_bXðtÞ ¼ ðA � �BB�Þ bXðtÞ �

ffiffiffiffiffidsdo

sByðtÞ,

with � ¼ 1ffiffiffi�

p þ

ffiffiffiffiffidsdo

s, ð2:20aÞ

uoðtÞ ¼ � 1ffiffiffi�

p B� bXðtÞ: ð2:20bÞ

As both (A,B) and (A�,B) are strongly stabilizable,the semigroup generated by (A� �BB�) is stronglystable, for any � > 0 (see also [12] for more details)and the minimal value of (2.15) is:

Jo ¼ffiffiffiffiffiffiffiffiffidsdo

pþ

ffiffiffi�

pds

m: ð2:21Þ


3. The Approximation Scheme

The approximation scheme is based on the subspacegenerated by the natural modes of vibration ofthe structure, which are obtained as the solutions ofthe following generalized eigenvalues-eigenfunctionsproblem:

A�i ¼ !2i M�i: ð3:1Þ

The solutions of Eq. (3.1) can be obtained by usingstandard computations [24] and are reported in [22]for a given set of parameters.

Being A�1 compact, the set f�ig forms anM-orthogonal basis for H, i.e. h�i,M�ji ¼ �ij (see [3]for more details). The following Theorem considers asuitably defined projector onto the finite-dimensionalspace generated by the first nmodes of vibration of thestructure.

Theorem 3.1. Let Vn ¼ spanf�1, . . . ,�ng S anddefine the operator �n as:

�n :H!Vn, �nx¼Xni¼1

hx,M�iiH�i: ð3:2Þ

Then:

(i) �n is an idempotent operator, i.e.�2

nx ¼ �nx, 8x 2 H;(ii) the approximation error x� �nx is M-ortho-

gonal to Vn;(iii) f�n, n 2 INg is a family of uniformly bounded

operators, i.e.:

supn2IN

k�nk�L<þ1 for some L2 IRþ; ð3:3Þ

(iv) the adjoint operator ��n is given by

��nx ¼

Pni¼1 hx,�iiHM�i.

Proof. Item (i) According to definition (3.2) and to thefact that Vn is a subspace generated by a setof M-orthogonal functions, it comes that �n�i ¼�i. Then:

�2nx¼�n

Xni¼1

hx,M�iiH�i

!¼Xni¼1

hx,M�iiH�n�i

¼Xni¼1

hx,M�iiH�i¼�nx: ð3:4Þ

Item (ii) In order to prove the statement, it has to beverified that:

hx� �nx,M�iiH ¼ 0 () hx,M�iiH¼ h�nx,M�iiH , 8�i, 8x 2 H:

ð3:5Þ

Equation (3.5) easily comes by exploiting the innerproduct:

h�nx,M�iiH ¼ hXnj¼1

hx,M�jiH�j,M�iiH

¼Xnj¼1

hx,M�jiHh�j,M�iiH

¼ hx,M�iiH : ð3:6Þ

Item (iii) It will be proved that there exists a uni-form (w.r.t. n 2 IN) bound for k�nk. First note that:

k�nxkH¼ k

ffiffiffiffiffiM

p �1 ffiffiffiffiffiM

p�nxkH

� kffiffiffiffiffiM

p �1k � kffiffiffiffiffiM

p�nxkH

: ð3:7Þ

From item (ii) it comes that the error x� �nx isM-orthogonal to �nx, so that, according to theCauchy-Schwarz inequality:

h�nx,M�nxiH ¼ hx,M�nxiH¼) k

ffiffiffiffiffiM

p�nxk2H ¼ h

ffiffiffiffiffiM

px,

ffiffiffiffiffiM

p�nxiH


pxk

H� k

ffiffiffiffiffiM

p�nxkH

¼) kffiffiffiffiffiM

p�nxkH


pxk

H,

ð3:8Þ

that means:

k�nk¼ supkxk¼1

k�nxkH� k

ffiffiffiffiffiM

p �1k � supkxk¼1

kffiffiffiffiffiM

pxk

H

¼ kffiffiffiffiffiM

p �1k � kffiffiffiffiffiM

pk<þ1: ð3:9Þ

Item (iv) Define ��n as in item (iv). It will be shown

that h�nx, yiH ¼ hx,��nyiH , 8x, y 2 H so that, exploit-

ing the uniqueness of the adjoint operator, also thelast item will be proved. Let x, y 2 H. Then:

h�nx, yiH ¼ hXni¼1

hx,M�iiH�i, yiH

¼Xni¼1

hx,M�iiHh�i, yiH

¼Xni¼1

hy,�iiHhx,M�iiH

¼ hx,Xni¼1

hy,�iiHM�iiH ¼ hx,��nyiH

ð3:10Þ

(the overbar stands for the conjugate of a complexnumber). &


Remark 3.2.As a consequence of the fact that f�ig andfM�ig are both complete bases forH, it comes that thefamilies of operators f�ng and f��

ng are both stronglyconvergent to the identity operator, that means:

limn7!1

k�nx� xkH¼0, lim

n7!1k��

nx� xkH¼0,

8x2H:ð3:11Þ

The optimal regulator (2.20) evolves in He, so thatthe following set of functions is needed in order toproject the control law onto a suitably defined finite-dimensional subspace of He. Let Wn ¼

span�i0

� �,

0�j

� �, i, j¼ 1, . . . ,n

� �S2 He, where �i

are the generalized eigenfunctions previously men-tioned. As it can be readily seen, this new set offunctions of He is an orthogonal set in He:

h�i

0

� �,

�j

0

� �iHe

¼ hffiffiffiffiA

p�i,

ffiffiffiffiA

p�jiH ¼ hA�i,�jiH

¼ !2i hM�i,�jiH ¼ !2

i �ij,

h0

�i

� �,

�j

0

� �iHe

¼ 0,

h0

�i

� �,

0

�j

� �iHe

¼ hM�i,�jiH ¼ �ij: ð3:12Þ

Let us define the following operator:

�E

n : He ! Wn, �E

nX ¼�nx1

�nx2

� �with

X ¼x1

x2

� �, x1 2 Dð

ffiffiffiffiA

pÞ, x2 2 H: ð3:13Þ

Theorem 3.3. The operator �E

n is an orthogonalprojector on Wn.Proof. The idempotence of�

E

n trivially comes from theidempotence of �n. The theorem is proved by verify-ing that �

E

nX � X is orthogonal to each vector of abasis forWn, for anyX 2 He. LetX be decomposed asin (3.13). The following equalities hold:

h�E

nX �X ,�i0

� �iHe¼h

ffiffiffiffiA

p�nx1�

ffiffiffiffiA

px1,

ffiffiffiffiA

p�iiH

¼ hXnj¼1

hx1,M�jiH�j,A�iiH � hx1,A�iiH

¼Xnj¼1

hx1,M�jiHh�j,A�iiH � hx1,A�iiH

¼Xnj¼1

!2i hx1,M�jiHh�j,M�iiH � !2

i hx1,M�iiH

¼ !2i hx1,M�iiH � !2

i hx1,M�iiH ¼ 0:

ð3:14Þ

h�E

nX � X ,0

�i

� �iHe

¼ hM�nx2 �Mx2,�iiH

¼ hXnj¼1

hx2,M�jiHM�j,�iiH � hMx2,�iiH

¼Xnj¼1

hx2,M�jiHhM�j,�iiH � hx2,M�iiH

¼ hx2,M�iiH � hx2,M�iiH ¼ 0:

ð3:15ÞTheorem 3.4. The sequence of orthogonal projectorsf�E

n , n 2 INg converges strongly to the identityoperator in He, that means:

limn7!1

k�E

nX � XkHe

¼ 0, 8X 2 He: ð3:16Þ

Proof. It will be first proved that the limit in (3.16) is

true 8X 2 S �H, dense in He. Let X ¼ x1x2

� �, with

x1 2 S and x2 2 H. Then:

k�E

nX �XkHe

2¼�nx1

�nx2

� ��

x1

x2

� � He

2

¼�nx1�x1

�nx2�x2

� � He

2

¼kffiffiffiffiA

p�nx1�

ffiffiffiffiA

px1kH

2

þhM�nx2�Mx2,�nx2�x2iH� hA�nx1�Ax1,�nx1�x1iHþkMk �k�nx2�x2kH

2

�kA�nx1�Ax1kH�k�nx1�x1kH

þkMk �k�nx2�x2H2k:

ð3:17Þ

According to Remark 3.2, �n converges strongly tothe identity operator, that means the only term toinvestigate in (3.17) is kA�nx1 � Ax1kH

. Note thatA�nx1 ¼ ��

nAx1, for x1 2 S, in that:

A�nx1 ¼ AXni¼1

hx1,M�iiH�i ¼Xni¼1

hx1,M�iiHA�i

¼Xni¼1

!2i hx1,M�iiHM�i

¼Xni¼1

hx1,!2i M�iiHM�i ¼

Xni¼1

hx1,A�iiHM�i

¼Xni¼1

hAx1,�iiHM�i ¼ ��nAx1: ð3:18Þ

Then kA�nx1 � Ax1kH¼ k��

nAx1 � Ax1kH7! 0,

because of the strong convergence of ��n, to the

identity operator, as stressed in Remark 3.2. Now,


consider X 2 He, but X =2S �H. According to thefact that S is dense in H, it comes that there existsa sequence fXk, k 2 INg S �H, such thatkX � XkkHe

7! 0. Then:

k�E

nX �XkHe

¼ k�E

nX ��E

nXk þ�E

nXk �Xk

þXk �XkHe

� k�E

nX ��E

nXkkHeþk�E

nXk �XkkHe

þkXk �XkHe

� k�E

nk � kX �XkkHeþk�E

nXk �XkkHe

þkXk �XkHe

� k�E

nXk �XkkHeþ 2kXk �Xk

He,

ð3:19Þ(recall that k�E

nk � 1 as a projection operator). Thefirst term goes to zero as n increases to infinity for eachchoice of k, because Xk 2 S �H; by choosing k greatenough, also the second term becomes infinitely small,and the proof is complete. &

Remark 3.5. Note that, as a consequence ofTheorem 3.4, it comes that

ffiffiffiffiA

p�n 7!

ffiffiffiffiA

pstrongly in

DðffiffiffiffiA

pÞ, in that:

k�E

nX � XkHe

2 ¼ kffiffiffiffiA

p�nx1 �

ffiffiffiffiA

px1kH

2

þ kffiffiffiffiffiM

p�nx2 �

ffiffiffiffiffiM

px2kH

2 7!0, ð3:20Þ8x1 2 Dð

ffiffiffiffiA

pÞ and 8x2 2 H. &

4. Finite-Dimensional Compensators

and Convergence Results

In order to make the infinite-dimensional controllaw designed in (2.20) implementable, a Galerkinapproximation of the optimal regulator onto the2n-dimensional subspace Wn is here performed. Toderive significative convergence results, it is sufficientto prove that the state evolving with the approximatedcontrol law converges to the state evolving with theoptimal control law and that the approximated stateestimate converges to the optimal state estimate whenincreasing the order of the approximation scheme.According to (2.20), the controlled state and the stateestimate obey to the following infinite-dimensionaldifferential system:

_XðtÞ ¼ AXðtÞ � 1ffiffiffi�

p BB� bXðtÞ þ BNsðtÞ,

_bXðtÞ ¼ ðA � �BB�Þ bXðtÞ þ

ffiffiffiffiffidsdo

sBB�XðtÞ

�

ffiffiffiffiffidsdo

sBNoðtÞ: ð4:1Þ

System (4.1) can be rewritten, by defining the extended

state X eðtÞ ¼XðtÞbXðtÞ

� �2 H2

e and the extended noise

NeðtÞ ¼NsðtÞNoðtÞ

� �2 IR2, as:

_XeðtÞ¼AeX eðtÞþBeNeðtÞ, X eð0Þ¼Xð0Þ0

� �,

ð4:2Þ

with

Ae¼A � 1ffiffi

�p BB�ffiffiffiffi

dsdo

qBB� A��BB�

24 35, Be¼B 0

0 �ffiffiffiffidsdo

qB

" #,

ð4:3Þ

and DðAeÞ ¼ DðAÞ � DðAÞ. According to whatpreviously mentioned, the approximated extended

state, named X neðtÞ ¼X nðtÞbXnðtÞ

� �2 H2

e , is such that

XnðtÞ is the state of the system when the approximated

control law is applied and bXnðtÞ is its approximatedestimate, evolving according to the following finite-dimensional system which approximates the optimalcontrol law (2.20):

_bXnðtÞ ¼�E

nðA � �BB�Þ�E

nbX nðtÞ �

ffiffiffiffiffidsdo

s�

E

nBynðtÞ,

bXnð0Þ ¼ �E

nbX nð0Þ ¼ 0, ð4:4aÞ

unðtÞ ¼ � 1ffiffiffi�

p B��E

nbX nðtÞ: ð4:4bÞ

System (4.4) is a Galerkin approximation onto Wn ofthe optimal regulator; yn is the output of system (2.14)when the approximated control law un is applied:

_XnðtÞ ¼ AX nðtÞ þ BunðtÞ þ BNsðtÞ

¼ AX nðtÞ �1ffiffiffi�

p BB��E

nbX nðtÞ þ BNsðtÞ,

ð4:5aÞ

ynðtÞ ¼ �B�XnðtÞ þNoðtÞ: ð4:5bÞ

Taking into account both the differential systemsdescribed by (4.4a) and (4.5a), the followingapproximated extended differential system is given:

_X neðtÞ ¼ AneX neðtÞ þ BneNeðtÞ,

X neð0Þ ¼ X eð0Þ ¼Xð0Þ0

� �, ð4:6Þ


with:

Ane ¼A � 1ffiffi

�p BB��

E

nffiffiffiffidsdo

q�

E

nBB� �E

nðA � �BB�Þ�E

n

24 35,Bne ¼

B 0

0 �ffiffiffiffidsdo

q�

E

nB

" #, ð4:7Þ

and DðAneÞ ¼ DðAÞ � He.

Remark 4.1. It has to be stressed that the proposedcontrol law un in (4.4) is implementable, since bX nðtÞbelongs to the finite-dimensional space Wn for eacht � 0. In the sequel it will be referred to n as the degreeof the approximation, that means a 2n-dimensionalregulator.

Theorem 4.2. Let T be any positive real. Consider theevolutions X e and X ne as random processes takingvalues in L2 ½0,T�,H2

e

�. ThenXne 7!X e in the L2 norm,

that means:

limn7!1IEfkX e � X nek2L2g

¼ limn 7!1IE

Z T

0

kX eðÞ � X neðÞk2H2e

d

� �¼ 0:

ð4:8Þ

Proof. The proof is articulated in two steps.

Step one: it is proved that the semigroup �nðtÞgenerated by Ane converges to the semigroup �ðtÞgenerated byAe strongly inH2

e and uniformly in [0,T],that means:

k�nðtÞX e � �ðtÞX ekH2e

7! 0,

8X e 2 H2e , uniformly in ½0,T�:

ð4:9Þ

The proof is achieved by using the Trotter-Katotheorem [23].

Step two: the convergence of (4.8) is proved by usingthe results of the first step and the Hilbert-Schmidtoperators theory; recall that a bounded operatorL : H1 7!H2 is Hilbert-Schmidt (shortly, H.S.) if, forany orthonormal basis ffig H1:X1

i¼1

kLfik2 < þ1: ð4:10Þ

The sum in (4.10) is invariant w.r.t. the choice ofthe orthonormal basis, and provides the square of theH.S. norm of the operator L (see [2] for more details).

Proof of step one. According to the version of theTrotter-Kato theorem adopted in [10], the proof of theconvergence of �nðtÞ to �ðtÞ, strongly in H2

e ,

uniformly on [0,T], is achieved by showing that thefollowing three hypotheses are satisfied:

(i) 9k 2 IR: the operator Ae � kI is dissipative, i.e.

hðAe � kIÞX e,X eiH2e

þ hX e, ðAe � kIÞX eiH2e

� 0,

8X e 2 DðAeÞ H2e ; ð4:11Þ

(ii) 9fkng IR bounded: fAne � knIg is a family ofdissipative operators, i.e.: 8n 2 IN,

hðAne�knIÞX e,X eiH2e

þhX e,ðAne�knIÞX eiH2e

�0,

8X e2DðAneÞH2e ; ð4:12Þ

(iii) 9T DðAeÞ dense in H2e : ðAe � �IÞT is dense in

H2e for some � > 0 and, moreover:

kAneX e �AeX ekH2e

7!0 8X e 2 T : ð4:13Þ

Proof of item (i) Define k? as follows:

k?¼ supX,Y2DðAÞ

ffiffiffiffidsdo

q� 1ffiffi

�p

�� kBB�k � kXkHe� kYk

He

kXk2HeþkYk2

He

:

ð4:14Þ

It comes that, k? < þ1, in that:

k? � supx, y2IRþ

�xy

x2 þ y2¼ �

2, ð4:16Þ

with � ¼ffiffiffiffidsdo

q� 1ffiffi

�p

�� kBB�k. Now, let X e ¼XY

� �with X,Y 2 DðAÞ, and let k 2 IR such that k > k?.

Exploiting the inner product in H2e :


¼ hAeX e,X eiH2e

� kkX ek2H2e

¼ hAX � 1ffiffiffi�

p BB�Y,XiHe

þ h

ffiffiffiffiffidsdo

sBB�X þ ðA � �BB�ÞY,Yi

He

� kkX ek2H2e

¼ hAX ,XiHe

� 1ffiffiffi�

p hBB�Y,XiHe

þ

ffiffiffiffiffidsdo

shBB�X ,Yi

He

þ hAY,YiHe

� �hBB�Y,YiHe

� k kXk2He

þ kYk2He

�:

ð4:17Þ

Recall that A is skew-adjoint, i.e.:

AþA�¼0 ) hAX ,XiHþhX ,AXi

H¼0,

8X 2DðAÞ,ð4:18Þ


so that:


þ hX e, ðAe � kIÞX eiH2e

¼

ffiffiffiffiffidsdo

s� 1ffiffiffi

�p

!hBB�X ,Yi

Heþ hY,BB�Xi

He

�� 2�jB�Yj2 � 2k kXk2

Heþ kYk2

He

�� 2

ffiffiffiffiffidsdo

s� 1ffiffiffi

�p

��kBB�k � kXk

He� kYk

He

� 2�jB�Yj2 � 2k? kXk2He

þ kYk2He

�� 2�jB�Yj2 � 0: ð4:19Þ

Proof of item (ii) Let X e ¼XY

� �with X 2 DðAÞ,

Y 2 He. Let k 2 IR such that k > k?, k? being the realdefined in (4.14). Let kn be the sequence defined as

kn ¼ k, n 2 IN. Exploiting the inner product in H2e :


¼hAneX e,X eiH2e

�kkX ek2H2e

¼hAX� 1ffiffiffi�

p BB��E

nY,XiHeþh

ffiffiffiffiffidsdo

s�

E

nBB�X

þ�E

nðA��BB�Þ�E

nY,YiHe�kkX ek2

H2e

¼hAX ,XiHe� 1ffiffiffi

�p hBB��

E

nY,XiHe

þ

ffiffiffiffiffidsdo

sh�E

nBB�X ,YiHeþh�E

nA�E

nY,YiHe

��h�E

nBB��E

nY,YiHe�k kXk2

HeþkYk2

He

�: ð4:20Þ

Then, according to (4.18):


þhX e,ðAne�knIÞX eiH2e

¼

ffiffiffiffiffidsdo

s� 1ffiffiffi

�p

!h�E

nBB�X ,YiHeþhY,�E

nBB�XiHe

��2�jB��

E

nYj2�2k kXk2

HeþkYk2

He

��2

ffiffiffiffiffidsdo

s� 1ffiffiffi

�p

��kBB�k�kXk

He�kYk

He

�2�jB��E

nYj2�2k? kXk2

HeþkYk2

He

��2�jB��

E

nYj2�0: ð4:21Þ

Proof of item (iii) It will be shown that Ane converges

strongly to Ae on T ¼ DðAeÞ. Let X e ¼XY

� �with

X , Y 2 DðAÞ. By exploiting the norms, according to

the parallelogram law (recall that kaþbk2�2ðkak2þkbk2Þ):

kAneX e�AeX ek2H2e

¼ AX� 1ffiffiffi�

p BB��E

nY�AX þ 1ffiffiffi

�p BB�Y

He

2

þ

ffiffiffiffiffidsdo

s�

E

nBB�X þ�

E

nðA��BB�Þ�E

nY

�

ffiffiffiffiffidsdo

sBB�X�AYþ�BB�Y

He

2

�1

�kBB�k2�k�E

nY�Yk2Heþ2ds

do�k�E

nBB�X�BB�Xk2He

þ4k�E

nA�E

nY�AYk2Heþ4�2k�E

nBB��E

nY�BB�Yk2He:

ð4:22Þ

According to Theorem 3.4, the first and the secondterm of (4.22) go to zero, by increasing n to infinity.Taking into account again the parallelogram law, thelast term in (4.22) becomes:

k�E

nBB��E

nY � BB�Yk2He

¼ k�E

nBB��E

nY � �E

nBB�Y þ �E

nBB�Y � BB�Yk2He

� 2k�E

nBB�k2 � k�E

nY � Yk2He

þ 2k�E

nBB�Y � BB�Yk2He7!0: ð4:23Þ

As far as the third term in (4.22) is concerned,

decompose Y ¼ y1y2

� �with y1 2 S, y2 2 Dð

ffiffiffiffiA

pÞ.

Then:

�E

nA�E

nY¼ �2ny2

��nM�1A�ny1

� �¼

�ny2

��nM�1��

nAy1

� �,

ð4:24Þ

(recall that A�nx ¼ ��nAx, 8x 2 S, eq. (3.18)). The

second component in (4.24) becomes

��nM�1��

nAy1 ¼ ��nM�1Xni¼1

hAy1,�iiHM�i

¼ ��n

Xni¼1

hAy1,�iiH�i

¼ �Xni¼1

hAy1,M�1M�iiH�i

¼ �Xni¼1

hM�1Ay1,M�iiH�i

¼ ��nM�1Ay1, ð4:25Þ


so that:

k�E

nA�E

nY�AYk2He

¼�ny2�y2

��nM�1Ay1þM�1Ay1

� � 2He

¼kffiffiffiffiA

pð�ny2�y2Þk2H

þhMðM�1Ay1��nM�1Ay1Þ,M�1Ay1��nM

�1Ay1iH�k

ffiffiffiffiA

pð�ny2�y2Þk2H

þkMk�k�nM�1Ay1�M�1Ay1k2H 7!0, ð4:26Þ

according to Remark 3.5 with regards to the first term,and to Remark 3.2 with regards to the second term.The proof of item (iii) is, then, completed by verifyingthat ðAe � �IÞT is dense in H2

e for some � > 0.Actually it is here proved that, for some � > 0, therange of Ae � �I is the whole space H2

e , i.e.ðAe � �IÞT � H2

e (recall that T ¼ DðAeÞ). Note thatAe generates a strongly continuous semigroup, so thatthe resolvent ð�I�AeÞ�1 exists and is bounded for allreal � greater than its exponential grow, i.e. it doesexist some � > 0 in the resolvent set ofAe, that means,it does exist some � > 0 for which the range ofðAe � �IÞ is equal toH2

e . As all the three hypotheses ofthe Trotter-Kato theorem are verified, it finally comesthe convergence of the semigroup �nðtÞ to �ðtÞstrongly inH2

e and uniformly in [0, T], that is Eq. (4.9).

Proof of step two. The evolutions of systems (4.2)and (4.6), according to the initial conditionX neð0Þ ¼ X eð0Þ, are:

X eðtÞ ¼ �ðtÞX eð0Þ þZ t

0

�ðt� ÞBeNeðÞd ,

ð4:27aÞ

XneðtÞ ¼ �nðtÞX eð0Þ þZ t

0

�nðt� ÞBneNeðÞd:

ð4:27bÞBy using again the parallelogram law, the L2 norm in(4.8) becomes:

kX e �Xnek2L2 � 2

Z T

0

k�ðtÞX eð0Þ ��nðtÞX eð0Þk2H2e

dt

þ 2

Z T

0

Z t

0

�ðt� ÞBeNeðÞd

��nðt� ÞBneNeðÞd2

H2e

dt:

ð4:28ÞFrom (4.9), the first integral goes to zero because itsinternal function converges to zero uniformly in theinterval [0,T] (and so does its expectation value).Regards to the second integral, define the operators

L,Ln : L2 ½0,T�, IR2 �

7!L2 ½0,T�,H2e

�, as follows:

LNðtÞ ¼Z t

0

�ðt� ÞBeNðÞd ,

LnNðtÞ ¼Z t

0

�nðt� ÞBneNðÞd: ð4:29Þ

Either L and the family fLng are H.S. operators (boththe kernels are Lebesgue measurable, being con-tinuous functions in ½0,T� � ½0,T�) so that, accordingto known results on H.S. operators (see [2], chapter6.10 on Physical Random Variables), the expectationvalue of the second integral in (4.28) becomes:

IE

Z T

0

kLNeðtÞ � LnNeðtÞk2H2e

dt

� �¼ IE kLNe � LnNek2L2

n o¼ kL � Lnk2H:S: ;

ð4:30Þthat means, for integral H.S. operators:

kL�Lnk2H:S: ¼Z T

0

Z

0

k�ð � ÞBe

��nð � ÞBnek2H:S:dd:ð4:31Þ

According to the convergence results of �nðtÞ, i.e.Eq. (4.9) verified in the first step of the proof, Cor-ollary 1 of Lemma 1 in [15] can be used in order toprove that the H.S. norm in the integral of Eq. (4.31)goes to zero uniformly w.r.t. � , so that the wholeintegral in (4.31) goes to zero. This comes by showingthat Bne 7!Be uniformly:

kBne�Bek2 �dsdo

� k�E

nB� Bk2 � dsdo

� k�E

nB �Bk2H:S:

7!0,

ð4:32Þin thatDðBÞ ¼ IR is a finite dimensional space, so thatthe H.S. norm trivially becomes a finite sum of termseach converging to zero (recall Theorem 3.4). &

5. Implementation of the Compensator

and Main Properties

In order to obtain the matrices of the finite-dimen-sional regulator related to the approximation schemeof order n � 1, the following notation is adopted:

X 2 Wn ) X ¼X2ni¼1

�i i, with

i ¼

�i

0

� �, i ¼ 1, . . . , n,

0

�i�n

� �, i ¼ nþ 1, . . . , 2n,

8>>><>>>:ð5:1Þ


so that each element X 2 Wn can be put in a bijectiverelation with the corresponding 2n-dimensionalvector:

X $ Xn ¼�1

..

.

�2n

0B@1CA 2 IR2n: ð5:2Þ

According to the Galerkin approximation method,the finite-dimensional system describing the evolutionof the coordinates XnðtÞ 2 IR2n related to bX nðtÞ of theregulator (4.4) is given by:

_XnðtÞ¼AnXnðtÞþBnyðtÞ, Xnð0Þ¼ 0, ð5:3aÞ

uðtÞ ¼ CnXnðtÞ, ð5:3bÞ

with:

An ¼ U�1n SA

n Bn ¼ U�1n SB

n Cn ¼ SCn ,

Unði, jÞ ¼ h i, jiHe,

ð5:4Þand:

SAn ði, jÞ ¼ hðA � �BB�Þ j, iiHe

,

SBn ðiÞ ¼ �

ffiffiffiffiffidsdo

shBð1Þ, iiHe

,

SCn ðiÞ ¼ � 1ffiffiffi

�p B� i:

ð5:5Þ

The initial state Xnð0Þ ¼ 0 because it is given by thecoordinates of bX nð0Þ ¼ 0 in Wn. According to theproperties of the generalized eigenfunctions �i, matrixUn in (5.4) and matrices SA

n , SBn , S

Cn in (5.5) become:

Un ¼�n 00 In

� �, �n ¼ diagf!2

1, . . . ,!2ng,

ð5:6Þ

SAn ¼

0 �n

��n ��n�Tn

� �, SB

n ¼

ffiffiffiffiffidsdo

s0

�n

� �,

SCn ¼ 1ffiffiffi

�p ½0 �Tn �,

ð5:7Þ

with In the identity matrix of order n and �n 2 IRn suchthat the i-th component �nðiÞ ¼ �iðlÞ. Therefore,finally:

An ¼0 In

��n ��n�Tn

� �, Bn ¼

ffiffiffiffiffidsdo

s0

�n

� �,

Cn ¼1ffiffiffi�

p ½0 �Tn �:

ð5:8Þ

Remark 5.1. Note that the matrix An in (5.8) belongsto the following real valued class of matrices:

M¼ 0 In�P �Q

� �, P¼PT>0, Q¼QT�0: ð5:9Þ

&The following Lemma concerning the eigenvalues ofmatrices of the type of (5.9) will be useful in the sequel.

Lemma5.2.LetMbeareal valuedmatrixas in (5.9)andassume that the pairs (P, Q), (P, QP) are both obser-vable. Then, all its eigenvalues have negative real part.

Proof. The statement is proved by showing that theorigin in IR2n is asymptotically stable for the linearsystems _x ¼ Mx. Consider the following Lyapunovfunction:

VðxÞ ¼ 1

2xT1Px1 þ

1

2xT2x2, x ¼

x1

x2

� �,

x1, x2 2 IRn: ð5:10ÞIt easily comes that V is positive definite with firstderivative nonpositive definite:

_VðxÞ ¼ xT2Px1�xT1Px2�xT2Qx2 ¼�xT2Qx2 � 0,

ð5:11Þso that the eigenvalues of M have nonpositive realpart. According to the La Salle Theorem, it will beproved that the greatest M-invariant subspace inclu-ded in V ¼ fx 2 IR2n : _VðxÞ ¼ 0g, named VM, is {0},so proving that M is asymptotically stable. From(5.11) it comes that:

V ¼ fx 2 IR2n : Qx2 ¼ 0g: ð5:12Þ

Now, let x ¼ x1x2

� �2 VM. Then:

M2ix¼ð�1ÞiPix1

Pix2

� �, M2iþ1x¼ð�1Þi

Pix2

�Piþ1x1

� �,

i¼ 0,1, . . . : ð5:13Þ

Identities (5.13) are proved by induction. Note firstthat they are true for i¼ 0 in that:

Mx ¼ x2�Px1 �Qx2

� �¼ x2

�Px1

� �, ð5:14Þ

since Qx2 ¼ 0, x belonging to V. Assume they are truefor some i. Then:

M2ðiþ1Þx ¼ M �M2iþ1x ¼ ð�1ÞiMPix2

�Piþ1x1

� �¼ ð�1Þi �Piþ1x1

�Piþ1x2 þQPiþ1x1

� �¼ ð�1Þiþ1 Piþ1x1

Piþ1x2

� �, ð5:15Þ


because M2iþ1x 2 V and, analogously,

M2ðiþ1Þþ1x¼M�M2ðiþ1Þx¼ð�1Þiþ1M Piþ1x1

Piþ1x2

� �¼ð�1Þiþ1 Piþ1x2

�Piþ2x1�QPiþ1x2

� �¼ð�1Þiþ1 Piþ1x2

�Piþ2x1

� �, ð5:16Þ

because by assumption M2ðiþ1Þx 2 V. Then, byconsequence of (5.13):

QPiþ1x1 ¼ 0, QPix2 ¼ 0, i ¼ 0, 1, . . . :

ð5:17ÞSince (P, Q) and (P, QP) are observable pairs, theidentities in (5.17) imply that x1 ¼ 0 and x2 ¼ 0, thatmeans VM ¼ f0g. This proves the asymptoticstability of M. &

Theorem 5.3. Matrix An in (5.8) is asymptoticallystable, i.e. all its eigenvalues have negative real part.

Proof. According to Remark 5.1 and to Lemma 5.2,the proof is achieved by showing that the pairsð�n, ��n�

Tn Þ and ð�n, ��n�

Tn�nÞ are observable. By

using the PBH-test, the observability condition for thefirst pair is satisfied if and only if:

rank�n � �In��n�

Tn

� �¼ n, 8� 2 f!2

1, . . . ,!2ng:

ð5:18Þ

Condition (5.18) is easily verified according to thediagonal form of �n and to the definition of �n (recallthat �i are the generalized eigenfunctions ofthe structures, so that �iðlÞ 6¼ 0, 8i, [3,22]).Analogously it comes that also ð�n, ��n�

Tn�nÞ is an

observable pair. &

Theorem 5.4. The finite-dimensional compensatordescribed by the matrices An,Bn and Cn in (5.8) iscontrollable and observable.

Proof. The controllability PBH-test is here adopted,i.e. it has to be verified that:

rank½An � �I Bn�

¼ rank��In In 0

��n ��n�Tn � �In

ffiffiffiffidsdo

q�n

" #¼ 2n,

ð5:19Þfor each eigenvalue of An. Then, let � be in the spec-trum of An and consider x 2 IR2n such thatxT½An � �I Bn� ¼ 0. The proof is achieved by

showing that, necessarily, x ¼ 0. Note that, forxT ¼ ðxT1 xT2 Þ, with x1, x2 2 IRn:

xT½An � �I Bn� ¼ 0 )

�x1 þ �nx2 ¼ 0,

x1 ¼ ��n�Tn x2 þ �x2,

�Tn x2 ¼ 0,

8>><>>:)

ð�n þ �2InÞx2 ¼ 0,

x1 ¼ �x2,

�Tn x2 ¼ 0:

8>><>>:ð5:20Þ

According to Theorem 5.3, the eigenvalues of An haveall strictly negative real part; on the other hand, thefirst equation in (5.20) admits non trivial solutionsonly for purely imaginary �, i.e. not in the spectrum ofAn. That means x2 ¼ 0 ) x1 ¼ 0. The proof for theobservability part is similar. &

Taking into account the matrices An, Bn, Cn computedin (5.8), the I/O transfer function of the proposedcompensator is obtained. Denoted it by GnðpÞ, with pthe complex Laplace variable, it follows that:

GnðpÞ ¼ CnðpI� AnÞ�1Bn

¼

ffiffiffiffiffiffiffiffids�do

s½0 �Tn � �

pIn �In

�n pIn þ ��n�Tn

� ��1

�0

�n

� �:

ð5:21ÞTaking into account the inversion of a block matrix, itcomes:

GnðpÞ ¼


s�Tn pIn þ ��n�

Tn þ 1

p�n

� ��1

�n

¼


s�Tn ð�nðpÞ þ ��n�

Tn Þ

�1�n,

ð5:22Þwith �nðpÞ ¼ diag

p2þ!21

p , . . . ,p2þ!2

n

p

n o. At last, accord-

ing to the matrix inversion lemma (below reported forthe ease of the reader):

Aþ uvT ��1 ¼ A�1 � ðA�1uÞ � ðvTA�1Þ

1þ uTA�1v,

with u, v 2 IRn,

ð5:23Þthe I/O transfer function becomes:

GnðpÞ¼


s�Tn�

�1n ðpÞ�n��

�Tn��1n ðpÞ�n�Tn��1

n ðpÞ�n1þ��Tn��1

n ðpÞ�n

� �

¼


s�Tn�

�1n ðpÞ�n

1þ��Tn��1n ðpÞ�n

, ð5:24Þ


that is, according to the diagonal form of �nðpÞ:

GnðpÞ¼


spFnðpÞ

1þ�pFnðpÞ, FnðpÞ¼

Xni¼1

�2nðiÞp2þ!2

i

:

ð5:25Þ

The remaining of the Section is devoted to show theeffectiveness of the proposed control law, takinginto account the matching of the family of appro-ximated I/O transfer functions (5.25) w.r.t.the optimal I/O transfer function, explicitlycomputed in [4] and below reported for the readersconvenience:

GðpÞ ¼


s� p

mp2 þ �pþ TðpÞ , ð5:26Þ

with:

In order to have a visual evaluation of the kind ofapproximation obtained using this method, the mag-nitude and phase plots of the exact transfer functionand of the approximated transfer functions arereported below, according to the following set ofparameters [20]:

l¼ 170m, m¼ 20 kg, EI¼ 1:6 � 106 kgm3s�2,��¼ 4:5 kgm�1, ds ¼ 1 kg2m2s�4, do ¼ 1 m2s�2,�¼ 1 kg�2s2: ð5:28Þ

In Fig. 5.1 the Bode diagrams of the optimal transferfunction are reported, together with the ones of theimplementable approximated transfer functiondeveloped in [4], denoted by GaðpÞ, based on thesecond order Taylor approximation of the term T(p)around p ¼ 0. In Fig. 5.2 the optimal transfer functionis compared with the transfer function related to theproposed first order approximation G1ðpÞ (n ¼ 1). Asit can be easily seen, the diagrams of the firstapproximation scheme (Fig. 5.1) are close to the onesof the optimal control law only in a small range offrequencies, because only poles are created throughhigher order expansion of T(p); on the other hand,the transfer function diagrams of the first order ofthe proposed approximation scheme fit better theBode plots of the optimal control law, by catchingexactly the low frequency peak at ! ’ 7 � 10�2 rad=s(Fig. 5.2). According to (5.25), the increment by one ofthe approximation index n creates 2 poles and 2 zeros

in the approximated transfer function, so allowingthe matching of one more peak of the optimaltransfer function, as it can be seen from Figs 5.3–4,where the cases of n ¼ 2 and n ¼ 5 are reported,respectively.

As an example, the matrices of the compensator inthe case n ¼ 2 are reported below:

A2¼

0 0 1 0

0 0 0 1

�0:0048 0 �0:0095 0:0088

0 �0:1890 0:0088 �0:0080

2666437775,

B2¼

0

0

0:0690

�0:0634

2666437775,

C2¼½0 0 �0:0690 0:0634�:

6. Robust Stability of the Approximate

Compensators

In this section it is proved that the flexible structurecontrolled by the finite-dimensional compensatordescribed in the previous section is strongly stable foreach order of the approximation scheme. Moreover itis proved that the proposed compensator guaranteesmodal stability of the closed loop system also in thepresence of parameters uncertainties. The uncertainparameters are here assumed to be admissible in thesense of Section 2 of [5], that is they are such thatoperators A, M, B satisfy the following items:

(i) M is bounded, positive definite with boundedinverse;

(ii) A is self-adjoint, nonnegative definite with com-pact resolvent;

(iii) B is a finite dimensional linear operator such that:(iv) - Bx ¼ 0 ) x ¼ 0;(v) - (� A, B) is controllable.

Theorem 6.1. The transfer function of the approxi-mated compensator GnðpÞ in (5.25), is positive real forany integer n and for any choice of the admissibleparameters. Moreover the undamped flexible struc-ture described by the equations (2.14), driven by anyfinite-dimensional feedback compensator GnðpÞ isrobustly modally stable.

TðpÞ ¼ EI�3ðpÞ cosð2�ðpÞlÞcoshð2�ðpÞlÞ þ 1

coshð2�ðpÞlÞsinð2�ðpÞlÞ � sinhð2�ðpÞlÞcosð2�ðpÞlÞ , �ðpÞ ¼ffiffiffiffijp

p �

EI

�1=4: ð5:27Þ


Fig. 5.2. Optimal vs G1ðpÞ approximated transfer function.

Fig. 5.1. Optimal vs GaðpÞ approximated transfer function.





Proof. As far as the first item is concerned,by using (5.25):

Gnðj!Þ þ Gnðj!Þ

¼


sj!Fnðj!Þ

1þ j�!Fnðj!Þ� j!Fnðj!Þ1� j�! Fnðj!Þ

� �:

ð6:1ÞTaking into account that

Fnðj!Þ ¼ Fnðj!Þ ¼Xni¼1

�2nðiÞ!2i � !2

, ð6:2Þ

it comes:

Gnðj!Þ þ Gnðj!Þ ¼


s2�!2jFnðj!Þj2

1þ �2!2jFnðj!Þj2> 0,

8! 6¼ 0:

ð6:3Þ

Moreover, according to Theorem 5.3, GnðpÞ is holo-morphic and bounded in Re½p� > 0 (indeed, it is astrictly proper rational function with asymptoticallystable poles, so that Gn 2 H1). Then, condition (6.3)ensures that GnðpÞ is a positive real transfer function.

As far as the second item is concerned, according toTheorems 5.3 and 5.4, the implementable control lawbelongs to the class of asymptotically stable andcontrollable finite-dimensional compensators. Thisproperty together with the results that GnðpÞ is posi-tive real imply the robust modal stability thanks toTheorem 2.1 in [5]. &

Differently from the finite-dimensional case, modalstability of the infinitesimal generator Ane does notimply the exponential or strong stability of the semi-group �nðtÞ generated byAne. In the investigated case,for instance, it can be proved that the eigenvalues ofAne constitute a sequence diverging to þ1 in abso-lute value, and converging to zero in real part, sopreventing exponential stability [5].

Theorem 6.2. The undamped flexible structuredescribed by the equations (2.14), driven by the finite-dimensional feedback compensator GnðpÞ is stronglystable for any integer n.

Proof. A sufficient condition to ensure strong stabilityfrom modal stability is given by [1] specialized toHilbert spaces, and requires the semigroup �n to bebounded, that is:

supt

k�nðtÞk � M < þ1: ð6:4Þ

The proof is readily obtained by observing thatthe proposed finite-dimensional compensator, withGn positive real, satisfies the hypothesis of Theorem2.3 in [5]. &

7. Conclusions

This work deals with the LQG control problem for anundamped flexible beam. It is known that the equa-tions of the optimal regulator computed using theEuler-Bernoulli distributed parameter model of thebeam are infinite-dimensional and therefore notimplementable. A new approximation scheme hasbeen presented in this paper to make such a controllaw implementable. The main result is a convergenceresult, showing that by increasing the index of theapproximation scheme, the state of the system con-trolled by the approximated finite-dimensional con-trol law converges in L2 norm, for any finite timehorizon, to the state of the system ideally controlledby the infinite-dimensional optimal regulator. Theapproximation scheme is a Galerkin projection ontothe finite-dimensional space generated by the naturalmodes of vibration of the structure instead of theusual splines or polynomials. The convergence resultsmay be appreciated also taking into account the Bodediagrams of the I/O transfer function of the compen-sator: the simplest version (n ¼ 1, one vibration mode)matches the optimal I/O transfer function better thanthe low frequency version available in the literature.Moreover, the Bode plots show that by increasing theorder of the approximation scheme from n to nþ 1,one more pair of resonance/antiresonance peaks iscaught by the approximate regulator. A furtherinteresting result is that for each order of theapproximation scheme, the proposed finite-dimen-sional compensator guarantees closed loop strongstability, ensuring modal stability even if the physicalparameters differ from their nominal values.

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