Automatica 48 (2012) 1347–1352

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Automatica

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Brief paper

Stability and passivity preserving Petrov–Galerkin approximation of linearinfinite-dimensional systems✩

Christian Harkort 1, Joachim DeutscherUniversity Erlangen-Nuremberg, Cauerstraße 7, D-91058 Erlangen, Germany

a r t i c l e i n f o

Article history:Received 22 March 2011Received in revised form9 November 2011Accepted 15 November 2011Available online 28 May 2012

Keywords:Structure preservationSystem approximationPort Hamiltonian systemsLinear infinite-dimensional systems

a b s t r a c t

This contribution presents two approximation methods for linear infinite-dimensional systems thatensure the preservation of stability and passivity. The first approach allows one to approximate internalsource free infinite-dimensional systems such that the resulting approximation is a port-controlledHamiltonian system with dissipation. The second method deals with the class of systems that are notrequired to have conjugated outputs but only a dissipative system operator. It yields approximationswitha dissipative system matrix for which bounds of their stability margin are provided. Both approachesare based on a state space formulation of the infinite-dimensional system. This makes it possibleto use the Petrov–Galerkin approximation whose free parameters are partly used for achieving thestructure preservation. Since still free parameters remain, further application specific objectives, suchas, e.g., moment matching, can be achieved. Both approaches are applied to the approximation of anEuler–Bernoulli beam.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, approximation approaches for finite-dimensionalsystems (see Polyuga & van der Schaft, 2010; Reis & Stykel,2011; Wolf, Lohmann, Eid, & Kotyczka, 2010) as well as forinfinite-dimensional systems that assure the preservation of thesystems’ passivity and thereby stability have received considerableattention. Such a structure preservation has the advantage thatvarious passivity-based controller design methods can be usedand the resulting approximations are (Lyapunov) stable, whichis important for the purpose of simulations. Moreover, thesereduced-order systems allow for a physical interpretation of theirdynamical behavior in terms of the energy exchange betweenstorage and dissipative elements. In the infinite-dimensionaldomain such approaches for structure preservation are obtainedon the basis of spatial discretization (see Baaiu, Couenne, Lefèvre,Le Gorrec, & Tayakout, 2009; Bassi, Macchelli, & Melchiorri, 2007;Golo, Talasila, van der Schaft, & Maschke, 2004; Voß& Scherpen,2009, 2011) and by pseudo-spectral methods (see Moulla, Lefèvre,& Maschke, 2011). Also in this contribution two structure

✩ Thematerial in this paper has not been presented at any conference. This paperwas recommended for publication in revised formbyAssociate Editor GeorgeWeissunder the direction of Editor Miroslav Krstic.

E-mail addresses: christian.harkort@rt.eei.uni-erlangen.de (C. Harkort),joachim.deutscher@rt.eei.uni-erlangen.de (J. Deutscher).1 Tel.: +49 9131 8527135; fax: +49 9131 8528715.

0005-1098/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2012.04.010

preserving approximation schemes for linear infinite-dimensionalsystems are presented which can be regarded as pseudo-spectralbased. In contrast to the existing work on this topic, which isprimarily based on the spatial discretization methods geometricmodeling by Stoke–Dirac structures, a state space description isused as the starting point in this contribution. This makes itpossible to apply the Petrov–Galerkin approximation (see Harkort& Deutscher, 2011), which is a generalized version of the classicalGalerkin projection method, with a specific choice of the basisvectors. The structure preservation is hence achieved by a simplemodification of a well-known method. Another consequence ofthe use of a state space system representation is that, differentfrom the concept of spatial discretization, spatial coordinatesplay no role in both approaches so that there is no differencebetween the formal treatment of 1-D, 2-D, or 3-D spatial domainsor even of time-delay systems that are not spatially distributed.The first proposed approach allows one to reduce linear internalsource free infinite-dimensional systems such that the resultingapproximations are dissipative port-controlled Hamiltonian (PCHD)systems, where the approximation order is arbitrary. In this way,the absence of internal sources and hence passivity are preserved.The second approach assures the stability of the approximationwithout confining it to passive systems. Instead, the class ofthose infinite-dimensional systems is considered that have adissipative system operator (see, e.g., Liu & Zheng, 1999; Luo,Guo, & Morgül, 1999), whereas no conjugated input output pairsare required in contrast to the first approach. These systemsallow for an elegant analysis of the exponential stability of the

1348 C. Harkort, J. Deutscher / Automatica 48 (2012) 1347–1352

generated C0-semigroup and its stability margin. It is shownhow a system with dissipative system operator can be reducedsuch that the resulting approximation has a system matrix thatis again dissipative. This in turn makes it possible to derivelower bounds for the related stability margin. Since only someof the free parameters in the Petrov–Galerkin approximation areneeded for the structure preservation the remaining ones canbe used to assure additional objectives, such as, e.g., momentmatching. Different from the contributions cited above, whichconsider the structure preserving approximation of systems withboundary control and measurement, purely distributed portvariables are considered in this article, which means that noenergy exchange occurs at the boundary. Some comments on thegeneralization to the case of boundary control and measurementare given in the concluding remarks. In the following Section 2 thePetrov–Galerkin approximation is briefly reviewed. In Section 3the passivity preserving system approximation is presented andthen applied to determine an approximation of an Euler–Bernoullibeam. The reduction technique with preservation of the systemoperator’s dissipativity is introduced in Section 4. This approachis demonstrated by reconsidering the Euler–Bernoulli beam.

2. The Petrov–Galerkin approximation

Many linear distributed parameter and delay systems can bedescribed by the state linear system

x(t) = Ax(t) + Bu(t), t > 0, x(0) = x0 ∈ X (1)y(t) = Cx(t), t ≥ 0 (2)

(see e.g. Curtain & Zwart, 1995). Therein, u(t) ∈ Rp denotes theinput, y(t) ∈ Rm the output, and the state x(t) ∈ X belongsto a complex Hilbert space X with the inner product ⟨·, ·⟩X andinduced norm ∥ · ∥X . The input operator B : Cp

→ X andthe output operator C : X → Cm are bounded linear maps,which has the consequence that the energy flow is distributed overthe spatial domain instead of concentrated at the boundary of adistributed-parameter system. Furthermore, the systemoperator Ais assumed to be an infinitesimal generator of a C0-semigroup so thatthe abstract initial value problem (1) iswell-posed. Inwhat follows,a decomposition x(t) = xn(t) + xr(t) of the state with xn(t) ∈ Vand xr(t) ∈ W⊥ is considered, where V = span{v1, . . . , vn} andW = span{w1, . . . , wn}with vi ∈ D(A),wi ∈ X are n-dimensionalsubspaces of X , and W⊥ is the orthogonal complement of W . Thisdecomposition exists and is unique if V ∩ W⊥

= {0} holds. Averifiable condition for that can be obtained by introducing thelinear operators V : Cn

→ X and W : X → Cn defined by

Vα =

ni=1

viαi, Wh =

⟨h, w1⟩X...

⟨h, wn⟩X

(3)

for all α ∈ Cn, h ∈ X . It is easy to verify that then V ∩ W⊥= {0}

is implied by detWV = 0, where WV ∈ Cn×n (see Harkort &Deutscher, 2011, Sec. 2). In the sequel, detWV = 0will be assuredby appropriate choices ofW andV so that onehas the internal directsum decomposition X = V ⊕ W⊥. In order to determine a finite-dimensional model that describes the dynamics of the part xn it isadvantageous to introduce the (unique) projection P : X → Vof X onto V along W⊥, yielding the relation xn(t) = P x(t). Thisprojection can be expressed as

P = V(WV)−1W . (4)

To see this one can verify easily that P 2= P holds, which shows

that P is a projection, and its range ran(P ) and its null space

nul(P ) of P satisfy

ran(P ) = V , nul(P ) = W⊥ (5)

as required (see Naylor & Sell, 1982, Thm. 4.11.3). Observe that Pis non-orthogonal in general, which is in contrast to the classicalGalerkin approach (see e.g. Balas, 1983). In order to describe thedynamics of xn, (1) and xn(t) = P x(t) are used to get

xn(t) = P x(t) = PA(P x(t) + (I − P )x(t)) + PBu(t). (6)

Neglecting therein the part (I − P )x(t) yields

˙xn(t) = PAxn(t) + PBu(t) (7)

for the dynamics of xn(t), which approximates xn(t), where theresulting error depends on P and hence on the choice of W andV . Often it is desired that the approximation model is formulatedon the state space Cn. For that reason a new state ξ(t) ∈ Cn isintroduced, which is related to xn(t) by xn(t) = V ξ(t). Using thisas well as (4) and (7) gives

V ξ (t) = V(WV)−1WAV ξ(t) + V(WV)−1WBu(t), (8)

and equating coefficients with respect to vi (see (3)) leads to then-dimensional model

ξ (t) = Anξ(t) + Bnu(t) (9)

that approximates (1), where the matrices An ∈ Cn×n and Bn ∈

Cn×p are given by

An = (WV)−1WAV, Bn = (WV)−1WB. (10)

The output equation of the approximation for the output yn(t) =

Cxn(t) is obtained with the aid of xn(t) = V ξ(t) as

yn(t) = Cnξ(t) with Cn = CV. (11)

The freedom in the choice of W and V will be used in the nextsections to obtain approximationmodelswith structure preservingproperties.

3. Passivity preservation

In this section linear infinite-dimensional systems of the form

x(t) = MQx(t) + Bu(t), t > 0, x(0) = x0 ∈ X (12)

y(t) = B∗Qx(t), t ≥ 0 (13)

with the same number of inputs u and outputs y are considered.In (12)–(13), Q : X → X is a bounded linear operator that is self-adjoint, i.e., Q∗

= Q, and coercive, i.e., ⟨Qh, h⟩X ≥ α∥h∥2X ∀h ∈ X

with α > 0. M : D(M) ⊂ X → X is a densely definedmaximal dissipative (in most cases unbounded) linear operator.The property of an operator to be maximal dissipative is definedas follows (see Liu & Zheng, 1999; Luo et al., 1999).

Definition 1 (Dissipative Operator). Let T : D(T ) ⊆ H → H bea densely defined linear operator, where H is a Hilbert space withinner product ⟨·, ·⟩H . T is called dissipative if it satisfies

Re ⟨T h, h⟩H ≤ 0 ∀h ∈ D(T ). (14)

If in addition ran(λ0I −T ) = X is satisfied for some λ0 > 0, whereran(·) denotes the range of an operator, then T is called maximaldissipative (m-dissipative).

Comparison of (12)–(13) with (1)–(2) makes

A = MQ, C = B∗Q (15)

C. Harkort, J. Deutscher / Automatica 48 (2012) 1347–1352 1349

apparent. In order to show that MQ generates a C0-semigroup,observe that this operator is dissipative with respect to the in-ner product ⟨g, h⟩Q = ⟨g, Qh⟩X , g, h ∈ X . In addition, ran(λ0I −

MQ) = X is satisfied for some λ0 > 0, because M ism-dissipativeand Q is bijective. Thus, MQ ism-dissipative and therefore gener-ates aC0-semigroup of contractions on theHilbert space (X, ⟨·, ·⟩Q)(see Liu& Zheng, 1999, Thm. 1.2.3). Since the induced norm ∥·∥Q =

∥Q1/2· ∥X is equivalent to ∥ · ∥X , because Q1/2 is coercive and

bounded as Q is, A = MQ is also a C0-semigroup generator on(X, ⟨·, ·⟩X ). In order to analyze the energy stored in the system, thestorage functional

H(x) =12⟨Qx, x⟩X (16)

is introduced, giving ddtH(x(t)) =

12 ⟨Qx(t), x(t)⟩X +

12 ⟨Qx(t),

x(t)⟩X . Using ⟨Qx(t), x(t)⟩X = ⟨x(t), Qx(t)⟩X = ⟨Qx(t), x(t)⟩X dueto Q∗

= Q this leads toddt

H(x(t)) = Re ⟨Qx(t), x(t)⟩X

= Re ⟨Qx(t), MQx(t) + Bu(t)⟩X (17)

after insertion of (12). Taking therein

⟨Qx(t), Bu(t)⟩X = ⟨B∗Qx(t), u(t)⟩Cp = yT (t)u(t) (18)

into account (see (13)), aswell as the fact that Re ⟨Qx(t), MQx(t)⟩X≤ 0 because M is dissipative, one obtains from (17) the differentialdissipation inequality

ddt

H(x(t)) ≤ yT (t)u(t). (19)

In the case that H(x) represents the total energy stored in the sys-tem (12)–(13) this relation has the physical interpretation thatthe energy increase dH(x(t))/dt is lower than or equal to thepower yT (t)u(t) fed into the system. Hence, the system does notcontain any internal sources. Due to this property the infinite-dimensional system (12)–(13) can, loosely speaking, be regardedas passive (see Malinen & Staffans, 2007; van der Schaft, 2000).Moreover, due to its structure, which is an analog to the linearfinite-dimensional case (see Definition 2), the systems consideredin this section can be regarded as linear infinite-dimensional PCHDsystems, though being aware of the general definition based onStoke–Dirac structures (see van der Schaft & Maschke, 2002, Sec.2.3). Examples for such systems are the Euler–Bernoulli beam, theTimoshenko beam, the Kirchhoff plate, the transmission line, andthe heat conduction system.

In the sequel it is the aim to construct an approximation that is aPCHD system. These systems arewell-known to be passive (see vander Schaft, 2000, Sec. 4.2.3) so that the passivity of the infinite-dimensional system remains preserved. Finite-dimensional PCHDsystems in the linear case have the well-known form as specifiedin the following definition (see Polyuga & van der Schaft, 2010; vander Schaft, 2000).

Definition 2 (Linear Finite-Order PCHD System). A linear system

xn(t) = (Jn − Rn)Qnxn(t) + Bnu(t) (20)

y(t) = B∗

nQnxn(t) (21)

with the state xn(t) ∈ Cn, a skew-adjoint interconnection matrixJn = −J∗n ∈ Cn×n, a positive semidefinite self-adjoint dissipationmatrix Rn = R∗

n ∈ Cn×n, a positive semidefinite self-adjoint energymatrix Qn = Q ∗

n ∈ Cn×n and an input matrix Bn ∈ Cn×p is called a

(finite-dimensional) linear PCHD system, whereas (·)∗ = (·)T.

The basic idea for determining a linear PCHD approximationis to apply the Petrov–Galerkin approach of Section 2 withappropriate operators V and W in (10)–(11). The followingtheorem states how these have to be chosen.

Theorem 3 (Linear PCHD Approximation). Suppose that dim D(M)∩ D(M∗) ≥ n. Then, define V : Cn

→ X by

Vg =

ni=1

vigi ∀g ∈ Cn (22)

for some linear independent vi ∈ X such that Qvi ∈ D(M)∩D(M∗),and set V and W in (10)–(11) to

V = V(V∗QV)−12 , W = V∗Q = (V∗QV)−

12 V∗Q (23)

with D(V) = Cn and D(W) = X. Then, the approxima-tion (9)–(11) of (12)–(13) is a linear PCHD system with

Jn = WJQV, Qn = I (24)Rn = WRQV, Bn = WB, (25)

wherein J and R are defined by

Jh =12(Mh − M∗h), Rh = −

12(Mh + M∗h) (26)

for all h ∈ D(J) = D(R) = D(M) ∩ D(M∗).

Proof. First, observe that the existence of the n vectors vi withthe required properties follows from the assumption dimD(M) ∩

D(M∗) ≥ n and the surjectivity of Q. Then, note that

WV = (V∗QV)−12 V∗QV(V∗QV)−

12 = I, (27)

so that the condition detWV = 0 in Section 2 holds. According to(10) and (15) one obtains Bn = WB and

An = WMQV. (28)

The assumptionQvi ∈ D(M)∩D(M∗) implies ran(QV) ⊂ D(J) =

D(R). This allows for the decomposition

MQV = JQV − RQV (29)

in view of (26), which, inserted into (28), leads to An = (Jn − Rn)Qnwith the aid of (24)–(25). It is straightforward to show that R issymmetric, i.e., R∗h = Rh ∀h ∈ D(R). With the aid of W∗

=

Q∗V = QV (see (23)) this yields

R∗

ng = (WRQV)∗g = WR∗QVg = Rng (30)

for all g ∈ Cn, so that Rn is self-adjoint. Similarly, it can be verifiedthatJ is skew-symmetric, i.e.,J∗h = −Jh ∀h ∈ D(J), which leadsto

J∗n g = (WJQV)∗g = WJ∗QVg = −Jng (31)

for all g ∈ Cn, so that Jn is skew-adjoint. Since M and thus alsoM∗ are dissipative by assumption it follows thatR is non-negative.Under use again of QV = W∗ it holds therefore that

⟨Rng, g⟩Cn = ⟨WRQVg, g⟩Cn = ⟨RW∗g, W∗g⟩X ≥ 0 (32)

for all g ∈ Cn, which shows that Rn is positive semidefinite. Thematrix Qn = I = Q ∗

n is obviously positive semidefinite and self-adjoint, and, according to (11), Cn satisfies

Cn = B∗QV = (WB)∗ = B∗

n, (33)

which follows from (15), QV = W∗ and (25). These results verifythat the approximation has the PCHD structure (20)–(21). �

1350 C. Harkort, J. Deutscher / Automatica 48 (2012) 1347–1352

Remark 4. Not all the free parameters in the choice of Wand V for the Petrov–Galerkin approximation are used for thestructure preservation, so the vectors vi in (22) can still be chosenalmost arbitrarily. This freedom can be used to satisfy additionaldesign objectives, such as, e.g., moment matching (see Harkort &Deutscher, 2011, and Example 6).

Remark 5. Note that V∗QV in (23) is an n × n-matrix thatis positive definite as a consequence of {v1, . . . , vn} beinglinear independent and Q positive. For that reason the matrix(V∗QV)−1/2 in (23) exists and can be determined by means ofstandard computing software.

In the following example the approach of Theorem 3 isapplied for the computation of a PCHD approximation for anEuler–Bernoulli beam with structural damping and collocatedobservation.

Example 6 (Euler–Bernoulli Beam). Consider an Euler–Bernoullibeam with structural damping, whose transverse deflection alongthe spatial coordinate z ∈ [0, 1] is denoted w(z, t). The beam issimply supported, i.e.,w(0, t) = w(1, t) = w′′(0, t) = w′′(1, t) =

0 ∀t ≥ 0, and is actuated by a distributed force b(z)u(t) on thesmall interval Ib = [0.49, 0.51], with b(z) = 1 for z ∈ Ib andb(z) = 0 elsewhere. As output, the velocity mean value y(t) = 10 ∂tw(z, t)b(z) dz over the same interval is considered. Using

energy coordinates as the state, i.e., x(t) = [∂2z w(·, t) ∂t(·, t)]T , and

the state space X = L2(0, 1) ⊕ L2(0, 1) with the inner product⟨g, h⟩X = ⟨g1, h1⟩L2 + ⟨g2, h2⟩L2 for g = [g1 g2]T , h = [h1 h2]

T∈

X , the system can be described by a state linear system (1)–(2)with

Ah =

0 −A0

A0 −2δA0

h ∀h ∈ D(A) = D(A0) ⊕ D(A0) (34)

Bν =

0b

ν, b ∈ L2(0, 1), ∀ν ∈ C (35)

Ch =

h,

0b

X

∀h ∈ X, (36)

where δ ≥ 0 in (34) is the constant of structural damping and theoperator A0 is defined as

A0h = −h′′∀h ∈ D(A0) (37)

D(A0) = {h ∈ H2(0, 1)|h(0) = h(1) = 0} (38)

with the Sobolev space H2(0, 1) = W2,2(0, 1). However, in orderto apply Theorem 3, the system must be represented in the form(12)–(13). Using M = A and Q = I , it follows from (35)–(36)immediately that

y(t) = Cx(t) = B∗Qx(t) (39)

holds, so that (13) is satisfied. In addition, it has to be checked if Mismaximal dissipative. Insertion ofM = A from (34) and h =

h1h2

into condition (14) yields

Re ⟨Mh, h⟩X = Re(−⟨A0h2, h1⟩L2 + ⟨A0h1, h2⟩L2

− 2δ⟨A0h2, h2⟩L2). (40)

Taking into account that A0 is self-adjoint (see Curtain & Zwart,1995, Ex. 2.2.5) gives ⟨A0h1, h2⟩L2 = ⟨h1, A0h2⟩L2 = ⟨A0h2, h1⟩L2 ,which, inserted into (40), reveals

Re ⟨Mh, h⟩X = −2δ Re ⟨A0h2, h2⟩L2 . (41)

Since δ ≥ 0 is assumed and A0 is positive (see Curtain & Zwart,1995, Ex. 2.2.5), i.e., ⟨A0h2, h2⟩L2 > 0 ∀h ∈ D(A0), (41) showsthat M is dissipative. Furthermore, since A and thus also M is a

C0-semigroup generator, there exists a 0 < λ0 ∈ ρ(M) such thatran(λ0I − M) = X (following from Curtain & Zwart, 1995, Lem.2.1.11). Thus, M is m-dissipative as desired. Finally, Q = I is self-adjoint and coercive, as required, so that the considered systemhas the assumed form. According to Theorem 3, the approximation(9)–(11) under use of (23) is a PCHD system and thus is passive.Consequently, the equilibrium xn = 0 is (Lyapunov) stable (see vander Schaft, 2000, Lem. 3.2.4), so that the eigenvalues of An arelocated in the half-plane {λ ∈ C | Re λ ≤ 0}. In order toobtain an approximation of order n with matching zeroth ordermoments about s1, . . . , sn ∈ C that belong to the resolvent setρ(A) of A, i.e., the transfer functions G(s) and Gn(s) of the infinite-dimensional system and the approximation, respectively, coincideat s = s1, . . . , sn, the vectors vi in (22) are chosen as

vi = (A − siI)−1b (42)

(see Harkort & Deutscher, 2011). These vectors satisfy therequirement Qvi ∈ D(M) ∩ D(M∗), because Q = I and M =

A = A∗= M∗, so that the condition becomes vi ∈ D(A), which

is implied by (42). Then, besides being passive, the approximationobtained by (10)–(11) has the property that the specifiedmomentsmatch. �

4. Preservation of the system operator’s dissipativity

When the approximant is used for the purpose of systemsimulations it is desired that the approximation is stablewheneverthe infinite-dimensional system is, which motivates a stabilitypreserving reduction scheme. While the systems in Section 3 arerequired to possess the PCHD structure with conjugated inputoutput pairs, it is the aim in this section to establish a stabilitypreserving approximation approach under milder assumptions. Tothis end systems are considered here that have a dissipative systemoperator (see Definition 1). In contrast to the considerations inSection 3, the outputs are now not assumed to be conjugated withrespect to the inputs, so that the overall systems need not to bedissipative but only their system operator. On the one hand thisproperty can often be checked easily and on the other hand it isclosely related to the exponential stability of the C0-semigroupSA(t) generated byA. In order to clarify this, note that dissipativityof A implies that SA(t) is a C0-semigroup of contractions, i.e.,

∥SA(t)∥X ≤ 1 ∀t ≥ 0, (43)

so that its norm is non-increasing. This directly follows from theLumer–Phillips Theorem when it is taken into account that A is agenerator of a C0-semigroup. Furthermore, if A satisfies, instead of(14), the stronger condition

Re ⟨Ah, h⟩X ≤ −α∥h∥2X ∀h ∈ D(A) (44)

for an α > 0, then the generated C0-semigroup SA(t) isexponentially stable, i.e., there are constants M ≥ 1, ω > 0 suchthat

∥SA(t)∥X ≤ Me−ωt∀t ≥ 0, (45)

which will be verified later. Relation (44) shows that the entityα plays a key role in the relationship between dissipativity of Aand exponential stability of the generated C0-semigroup. For thisreason the largest possible value of α is introduced next as thedissipativity margin of A.

Definition 7 (Dissipativity and Stability Margin). Let the linearoperator T : D(T ) ⊆ H → H be the generator of a C0-semigroupST (t), where H is a Hilbert space with inner product ⟨·, ·⟩H andinduced norm ∥ · ∥H .

C. Harkort, J. Deutscher / Automatica 48 (2012) 1347–1352 1351

1. The dissipativity margin µd(T ) of T is defined as

µd(T ) = sup{α ∈ R|Re ⟨T h, h⟩H ≤ −α∥h∥2H

∀h ∈ D(T )}. (46)

2. The stability margin of ST (t) is defined by

µs(ST (t)) = sup{ω0 ∈ R|∃M ≥ 1 such that

∥ST (t)∥H ≤ Me−ωt∀ω < ω0, t ≥ 0}. (47)

Now, the relation between dissipativity of A and exponentialstability of SA(t) can be expressed in terms of the correspondingdissipativity margin and stability margin as follows.

Proposition 8 (Exponential Semigroup Bound). The growth property∥SA(t)∥X ≤ Me−ωt , ∀t ≥ 0, of the C0-semigroup SA(t) is satisfiedespecially for

M = 1, ∀ω < µd(A). (48)

Particularly,

µs(SA(t)) ≥ µd(A) (49)

holds.

Proof. Since Aω = A + ωI is the generator of a C0-semigroupthere exists a 0 < λ0 ∈ ρ(Aω) such that ran(λ0I − Aω) =

X (compare Curtain & Zwart, 1995, Lem. 2.1.11). Using this, itis straightforward to show, with the aid of the Lumer–PhillipsTheorem, that the C0-semigroup SAω (t) generated by Aω is acontraction semigroup for all ω < µd(A) (see Liu & Zheng,1999, Thm. 1.2.3), i.e., ∥SAω (t)∥X ≤ 1 ∀t ≥ 0. In view of SA(t) =

SAω (t) e−ωt (see Engel & Nagel, 2000, Sec. 2.2), this implies

∥SA(t)∥X ≤ e−ωt∀ω < µd(A), ∀t ≥ 0, (50)

which confirms (48). Using (50) in the defining relation (47)directly yields (49). �

This statement shows especially that the spectrum σ(A) ofA is contained in the half-plane {λ ∈ C | Re λ ≤ −µd(A)},because supλ∈σ(A) Re λ ≤ −µs(SA(t)) ≤ −µd(A) (see (Curtain& Zwart, 1995, Lem. 2.1.11) and (49)). Often it is desired tocompute an approximation in such a way that it is exponentiallystable or that its eigenvalues are located within a known half-plane. One way to achieve this is consequently to preserve thedissipativity margin during the approximation procedure, sincethen such a half-plane follows fromProposition 8, as argued before.The following theorem states how such a dissipativity preservingmodel reduction can be achieved by means of the Petrov–Galerkinapproximation by an appropriate choice of the operators V and Win (10)–(11).

Theorem 9 (Approximation with Dissipative An). Define V : Cn→

X as in Theorem 3 and set V and W in (10)–(11) to

V = V(V∗V)−12 , W = V∗

= (V∗V)−12 V∗. (51)

Then, the following assertions hold:

1. The dissipativity margin µd(An) of An according to (10) satisfies

µd(An) ≥ µd(A). (52)

Particularly, An is dissipative if A is.2. For SAn(t) = eAnt the exponential bound ∥SAn(t)∥Cn ≤ Me−ωt

holds for M = 1 and for all ω < µd(A).3. An is self-adjoint (skew-adjoint) if A is self-adjoint (skew-adjoint).

Proof. First, note that

WV = (V∗V)−12 V∗ V(V∗V)−

12 = I (53)

holds, so that the condition detWV = 0 in Section 2 is satisfied.An according to (10) becomes An = WAV , which yields

⟨Ang, g⟩Cn = ⟨WAVg, g⟩Cn = ⟨AVg, Vg⟩X (54)

for any g ∈ Cn, where W∗= V has been taken into account (see

(51)). Setting h = Vg in (54) yields Re ⟨Ang, g⟩Cn = Re ⟨Ah, h⟩X ,and with aid of (46), applied to A, this becomes

Re ⟨Ang, g⟩Cn ≤ −α∥h∥2X ∀α < µd(A). (55)

Since V∗= W and WV = I it follows that

∥h∥2X = ⟨Vg, Vg⟩X = ⟨g, WVg⟩Cn = ∥g∥2

Cn , (56)

so that (55) becomes Re ⟨Ang, g⟩Cn ≤ −α∥g∥2Cn for all α <

µd(A). In view of (46), with T replaced by An, this yields (52).Consequently, if A is dissipative, then µd(An) ≥ µd(A) ≥ 0, sothat also An is dissipative, as asserted in Item 1. Item 2 follows from(52) and Proposition 8 when SA(t) therein is replaced by SAn(t).Finally, Item 3 is straightforward to check in view of V∗

= W ,yielding

A∗

n = (WAV)∗ = WA∗V = κWAV = κAn, (57)

with κ = 1 (κ = −1) for A being self-adjoint (skew-adjoint). So,the proof is complete. �

Remark 10. Since V∗V is a positive definite n×n-matrix, becausev1, . . . , vn in (22) are linear independent, the inverse square rootof this matrix in (51) can be determined by means of standardcomputing software.

Observe that the relation (52) still holds when the infinite-dimensional system is not dissipative. In view of Proposition 8, thisrelation has the consequence that the approximation is exponen-tially stable as desired whenever µd(A) > 0. Furthermore, Re-mark 4 applies also to Theorem9. In the following example a finite-dimensional approximation with preserved dissipativity margin iscomputed for the Euler–Bernoulli beam introduced in Example 6.

Example 11 (Euler–Bernoulli Beam, Continued). In Example 6 thedissipativity of M has been shown, whereas M = A. Therefore,Re ⟨Ah, h⟩X ≤ 0 ∀h ∈ D(A) holds, which yields µd(A) ≥ 0(see (46)). More precisely, it holds that µd(A) = 0 because anyh =

h10

with h1 ∈ D(A0) gives Re ⟨Ah, h⟩X = 0. According

to Proposition 8, applied to An, and Theorem 9 the approximation(10)–(11) under use of (23) has the property µs(SAn(t)) ≥

µd(An) ≥ µd(A) = 0. This means that the eigenvalues of An arelocated in the half-plane {λ ∈ C | Re λ ≤ 0}. As demonstrated inExample 6, the freedom of the parameters vi in (22) can be used toachieve further objectives. �

5. Concluding remarks

In this article it has been shown how the degrees offreedom in the Petrov–Galerkin approximation can be chosensuch that the resulting approximant has certain properties.While the approximations obtained from the first approach areport-Hamiltonian systems with dissipation, the second approachassures that the dissipativity margin of the infinite-dimensionalsystem is preserved. Throughout the paper the input and outputoperators are assumed to be bounded, whichmeans that no energyexchange occurs at the boundary. This restriction can be resolvedby generalizing the approach to the class of regular linear systems,which allows for unbounded control and observation (see Weiss,1989). Passive boundary control systems (see Malinen & Staffans,

1352 C. Harkort, J. Deutscher / Automatica 48 (2012) 1347–1352

2007) can then be handled by means of the boundary controlsystems approach described e.g. in Tucsnak and Weiss (2009) (seealso Le Gorrec, Zwart, and Maschke (2005); Schlacher (2008); vander Schaft and Maschke (2002), Villegas, Zwart, Le Gorrec, andMaschke (2009); Zwart, Le Gorrec, Maschke, and Villegas (2009)).

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Christian Harkort was born in Ratisbon, Germany, in1979. He received the B.S. degree in General EngineeringScience from the Technical University Hamburg–Harburg,Germany, in 2002 and the Dipl.-Ing. Univ. degree inMechatronics from the Friedrich–Alexander UniversityErlangen–Nuremberg (FAU), Germany, in 2007. Since 2007he has been a Ph.D. student at the Chair of AutomatricControl at the FAU. His research interests comprise thecontrol of infinite-dimensional systems as well as thereduced-order modeling of these systems.

Joachim Deutscher was born in Schweinfurt, Germany,in 1970. He received the Dipl.-Ing. (FH) degree inElectrical Engineering from Fachhochschule Würzburg–Schweinfurt–Aschaffenburg, Germany, in 1996, the Dipl.-Ing. Univ. degree in Electrical Engineering and theDr.-Ing. degree from the Friedrich–Alexander UniversityErlangen–Nuremberg (FAU), Germany, in 1999 and 2003,respectively. Since his habilitation in 2010 at the Facultyof Engineering (FAU) he has been Privatdozent (AssociateProfessor) for Automatic Control at the Chair of AutomatricControl (FAU), where he leads the infinite-dimensional

systems group. His research interests include the control of infinite-dimensionalsystems, nonlinear control theory and the application of polynomial matrixmethods in control. He has co-authored a book on state control: Design of observer-based compensators (Springer, 2009) and is author of the book: State feedback controlof distributed-parameter systems (in German) (Springer, 2012).