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Proceedings of the European Control Conference 2009 • Budapest, Hungary, August 23–26, 2009

Moment Moment Matching matching for for Linear linear Port-Hamiltonian port-Hamiltonian Systems systems Rostyslav V. Polyuga and Arjan van der Schaft

Abstract— Moment matching approach of model reduction of linear dynamical state-space systems for a given inputgenerating system is discussed and applied to port-Hamiltonian systems. It is shown that the reduced order models inherit the port-Hamiltonian structure, the passivity property and preserve a certain number of the moments of a transfer function. Another approach to model reduction of port-Hamiltonian systems by means of the Krylov methods is considered. It is shown that in this case reduced order models are again port-Hamiltonian and therefore passive. The port-Hamiltonian structure of the reduced order models in both cases is investigated.

I. INTRODUCTION Port-based network modeling of physical systems leads directly to their representation as port-Hamiltonian systems which are, if the Hamiltonian is non-negative, an important class of passive state-space systems. At the same time modeling of physical systems often leads to highdimensional dynamical models. State-space dimensions tend to become large as well if distributed-parameter models are spatially discretized. Therefore an important issue concerns model reduction of these high-dimensional systems, both for analysis and control. The goal of this work is to demonstrate that the specific model reduction techniques for linear statesspace systems can be also applied to port-Hamiltonian systems in order to preserve the port-Hamiltonian structure for the reduced order models, and, as a consequence, passivity. In the systems and control literature there is a variety of methods and techniques used for model reduction serving different purposes. The so-called moment matching methods are an important class of model reduction methods and based on the notion of the moment of a transfer function of a linear system [1]. The idea behind the moment matching approach is to equalize a specific number of the coefficients of the Laurent series expansion of the transfer function of the full order model with that of the reduced order model at certain points in the complex plane. The Partial realization problem is solved when the expansion is considered around infinity. The Pad´e approximation is a problem of the moment matching at zero. In the general case, the moment matching problem is known as rational interpolation. There is a vast literature on this topic ([1],[5],[6] etc.) discussing different approaches. Rostyslav V. Polyuga is with Institute for Mathematics Science, University of Groningen, P.O.Box 407, 9700 AK Netherlands [email protected] Arjan van der Schaft is with Institute for Mathematics Science, University of Groningen, P.O.Box 407, 9700 AK Netherlands [email protected]

ISBN 978-963-311-369-1 © Copyright EUCA 2009

and Computing Groningen, The and Computing Groningen, The

In this paper we follow a new approach to the moment matching problem for linear systems considered in [2],[3],[4]. The approach is discussed in Section 2 and it is explained that the original transfer function can be interpolated at specific points in the complex plane given an input-generating system of a particular kind. In Section 3 we will apply this approach to port-Hamiltonian systems. Theory on port-Hamiltonian systems can be found in [8],[9],[7]. We will show that if a full order system is of the port-Hamiltonian structure then so is the reduced order system matching the moments of the original system. Model reduction of port-Hamiltonian systems by the Krylov methods will be discussed in Section 4. It will be shown that the Krylov methods applied to a full order port-Hamiltonian system preserve the port-Hamiltonian structure along with the passivity property. II. MOMENT MATCHING FOR LINEAR SYSTEMS Consider a linear, single-input, single-output, continuoustime system described by equations of the form x˙ = Ax + Bu, y = Cx,

(1)

with x(t) ∈ Rn , u(t) ∈ R, y(t) ∈ R, A ∈ Rn×n , B ∈ Rn , C ∈ R1×n constant matrices, and the associated transfer function (2) W (s) = C(sI − A)−1 B. Definition 1: [1] The 0-moment of the system (1) at s ∈ C is the complex number η0 (s ) = C(s I − A)−1 B.

(3)

The k-moment of the system (1) at s ∈ C is the complex number   k k d ηk (s ) = (−1) C(sI − A)−1 B k! dsk (4) s=s = C(s I − A)−(k+1) B. The quantities η0 (∞) = 0, ηk (∞) = CAk−1 B, k > 0,

(5)

are the Markov parameters hk of (1). Lemma 1: [3] Consider the system (1) and s ∈ R. / σ(A), where σ(A) denotes the spectrum of the Suppose s ∈ matrix A. Then the moments η0 (s ), . . . , ηk (s ) are in oneto-one relation with the matrix CΠ, where Π is the unique solution of the Sylvester equation AΠ + BL = ΠS,

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R. V. Polyuga and A. van der Schaft: Moment Matching for Linear Port-Hamiltonian Systems

with S any non-derogatory

1

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.

real matrix such that

det(sI − S) = (s − s )k+1 ,

x = Ax + Bu

ω=Sω

(9)

(11)

(12) S T P + P S  ΠT C T L + LT CΠ. Proof: By definition of losslessness ([10],[8]) the family of systems (10) contains a lossless system if and only if there exists a symmetric and positive definite matrix P such that

Interconnection of Systems

Lemma 3: Consider the family of reduced order models (10). Suppose σ(S) ⊂ C− . Then there is a symmetric and positive definite matrix P such that condition (12) holds. Proof: Condition σ(S) ⊂ C− implies that there exists a symmetric and positive definite matrix X such that S T X + XS < 0.

(14)

As a result, there exists a constant κ > 0 such that condition (12) holds with P = κX.

(10)

Moreover, the family of reduced order models (10) contains a passive system if and only if there exists a symmetric and positive definite matrix P such that

(S − ΔL)T P + P (S − ΔL) = 0, P Δ = (CΠ)T ,

ψ

ψ=C Πξ Fig. 1.

assuming that (i): S and A do not have common eigenvalues and S and S − ΔL do not have common eigenvalues; (ii): reduced order system has the same dimension as the input-generating system. This family, achieving moment matching, is directly parameterized by the matrix Δ. Graphically the interconnection of the input-generating, full order and reduced order systems are depicted in Fig. 1. Theorem 1: [3] The family of reduced order models (10) contains a lossless system if and only if there exists a symmetric and positive definite matrix P such that S T P + P S = ΠT C T L + LT CΠ.

.

ξ = (S − Δ L) ξ + Δ u

(8)

and L such that the pair (L, S) is observable. For the proofs of Lemma 1 and Lemma 2 we refer to [3],[4]. Remark 1: Note that Lemma 1 and Lemma 2 do not hold for s∗ = ∞. Following the stream of [2], [3], [4], in this paper we are looking at the family of reduced order models for the system (1) at S of the following form ξ˙ = (S − ΔL)ξ + Δu, ψ = CΠξ,

u

u=Lω

with S any non-derogatory real matrix such that det(sI − S) = ((s − s )(s − s¯ ))k+1 = = (s2 − 2α s + (α )2 + (ω  )2 )k+1

y = Cx

.

and L such that the pair (L, S) is observable. Lemma 2: [3] Consider the system (1) and s ∈ C\R. / σ(A). Then the moments Let s = α + iω  . Suppose s ∈ s ), . . . , ηk (s ), ηk (¯ s ) are in one-to-one relation η0 (s ), η0 (¯ with the matrix CΠ, where Π is the unique solution of the Sylvester equation AΠ + BL = ΠS,

y

(7)

III. MOMENT MATCHING FOR LINEAR PORT-HAMILTONIAN SYSTEMS In the linear case, and in the absence of algebraic constraints, port-Hamiltonian systems take the following form ([8], [7]) x˙ = (J − R)Qx + Bu, y = B T Qx,

(15)

with H(x) = 12 xT Qx the total energy (Hamiltonian), Q = QT the energy matrix and R = RT  0 the dissipation matrix. The matrices J = −J T and B specify the interconnection structure. By skew-symmetry of J and R being positive semidefinite it immediately follows that d 1 T x Qx = uT y − xT QRQx  uT y. (16) dt 2 Thus if Q  0 (and the Hamiltonian is non-negative) any port-Hamiltonian system is passive (see [10],[8]). The state variables x ∈ Rn are also called energy variables, since the total energy H(x) is expressed as a function of these variables. Furthermore, the variables u ∈ Rm , y ∈ Rm are called power variables, since their product uT y equals the power supplied to the system. Example 1: As an example of a mechanical portHamiltonian system we take a Mass-Spring-Damper system

(13)

which is equivalent to the solvability of (11). The proof of the second statement is similar to the proof of the first statement, hence omitted.

u

k2

k1 m2

m1 q1

c1

q2

c2

1A

matrix is non-derogatory if its characteristic and minimal polynomials coincide.

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Fig. 2.

Mass-Spring-Damper system

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Proceedings of the European Control Conference 2009 • Budapest, Hungary, August 23–26, 2009

as shown in Fig. 2 with masses m1 , m2 , spring constants k1 , k2 and damping constants c1 , c2 . q1 , q2 are the displacements of the masses m1 , m2 . The external force is the input u and the port-Hamiltonian output y is the velocity of the first mass m1 . The state variables are as follows: x1 is the displacement q1 of the first mass m1 , x2 is the momentum p1 of the first mass m1 , x3 is the displacement q2 of the second mass m2 , x4 is the momentum p2 of the second mass m2 . A minimal realization of this port-Hamiltonian system is   1 T 00 , B = [ 0 1 0 0 ], C = 0 m1 ⎡

0 ⎢ −1 J =⎢ ⎣ 0 0

1 0 0 0 0 0 0 −1

⎤ ⎡ 0 ⎢ 0 ⎥ ⎥,R = ⎢ ⎦ ⎣ 1 0

⎤ 0 0 0 0 ⎥ ⎥ , and 0 0 ⎦ 0 c2

0 0 0 c1 0 0 0 0

ξ˙ = Jr Qr ξ + Δu, ψ = ΔT Qr ξ,

k1 0 −k1 0 1 ⎥ ⎢ 0 0 0 m1 ⎥ Q=⎢ ⎣ −k1 0 k1 + k2 0 ⎦ . 1 0 0 0 m2 Example 2: Consider the Ladder Network in Fig. 3, with C1 , C2 , L1 , L2 , R1 , R2 being the capacitances, inductances and resistances over the corresponding capacitors, inductors and resistors. The port-Hamiltonian representation of this physical system (in the case of linear elements) will be of the form (15) with the corresponding matrices ⎡ ⎤ 0 −1 0 0 ⎢ 1 0 −1 0 ⎥ ⎥ , R = diag{0, 0, 0, R2 }, J =⎢ ⎣ 0 1 0 −1 ⎦ 0 0 1 0 −1 −1 T Q = diag{C1−1 , L−1 1 , C2 , L2 }, B =



1

0

0

0

and the state-space vector x given as

xT = q1 φ1 q2 φ2 with q1 , q2 the charges on the capacitors C1 , C2 and φ1 , φ2 the fluxes over the inductors L1 , L2 correspondingly. The input of the system u is given by the current I from the external current source and the output y is the voltage over the first capacitor. Theorem 2: Assume that (11) holds with P being a symmetric and positive definite matrix meaning that there is a lossless system which belongs to the family of reduced order R1

L1, φ1

C1,q1

C2, q2

Fig. 3.

Ladder Network

(17)

where = SP −1 − P −1 (CΠ)T LP −1 , = P, = P −1 (CΠ)T ,

Jr Qr Δ

(18)

matching the moments of the full order port-Hamiltonian system (15) at S. Proof: Clearly Qr is symmetric and positive definite. To check the skew-symmetry of Jr we have to verify Jr +JrT = 0. This yields SP −1 − P −1 (CΠ)T LP −1 + P −1 S T − P −1 LT CΠP −1 = P −1 (S T P + P S − ΠT C T L − LT CΠ)P −1 = 0.

⎤

⎡

I

systems (10). Then there exist matrices Jr = −JrT , Qr = QTr > 0, Δ such that the reduced order lossless system (10) takes the port-Hamiltonian form

(19)

Thus the skew-symmetry of Jr boils down to the solvability of (11). Moment matching is achieved according to Lemma 1 and Lemma 2 which completes the proof. Recall that for a general passive linear system (1) the following set of inequalities holds 0  Pav  P  Preq ,

(20)

where the matrices Pav and Preq are the minimal and the maximal solutions to the LMI (Linear Matrix Inequality) AT P + P A  0, C = B T Q,

(21)

and reflect the available storage Sa = 12 xT Pav x and the required supply Sr = 12 xT Preq x of the system (1), as shown in [10], [8], [9]. Theorem 3: Suppose that there exists a symmetric and positive definite matrix P such that LMI (12) is satisfied meaning that there is a passive system which belongs to the family of reduced order systems (10). Then there exist matrices Jr = −JrT , Qr = QTr > 0, Rr = RrT  0, Δ such that the reduced order passive systems (10) take the portHamiltonian form ξ˙ = (Jr − Rr )Qr ξ + Δu, ψ = ΔT Qr ξ,

(22)

where Jr

L2, φ2

Qr Rr Δ

R2

=

1 2 [(S −1

− P −1 (CΠ)T L)P −1 − (S − P −1 (CΠ)T L)T ],

P = P, = − 12 [P −1 (S − P −1 (CΠ)T L)T + (S − P −1 (CΠ)T L)P −1 ], = P −1 (CΠ)T ,

(23)

with P being any positive definite matrix satisfying (20). Furthermore the reduced order port-Hamiltonian system (22) matches the moments of the full order port-Hamiltonian system (15) at S.

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Proof: Clearly Qr is symmetric and positive definite again. Let Ar be the A-matrix of a reduced order model. To check the symmetry of Rr first note that Ar = (S − P −1 (CΠ)T L).

(24)

Then RrT

= − 12 [P −1 ATr + Ar P −1 ]T = − 12 [Ar P −1 + P −1 ATr ]

= = Rr .

(25)

Positive semidefiniteness of Rr comes from the existence of positive definite solution to the LMI (12). From the equation 2(Jr + JrT )

= Ar P −1 − P −1 ATr P −1 ATr − Ar P −1

+ =

0

(26)

the skew-symmetry of Jr follows. Moment matching is similarly achieved according to Lemma 1 and Lemma 2. Remark 2: Note that the equation Ar = (Jr − Rr )Qr can be easily verified. Remark 3: Note that the minimality (controllability and observability) of the family of the reduced order systems (10) for all P from (20) yields Pav > 0. Therefore all P from (20) are strictly positive definite and the family of the reduced order systems (10) embraces all reduced order port-Hamiltonian systems (22) with P from (20). Example 3: Let the full order system be the lossless portHamiltonian system for the Ladder network from Example 2 (hence R2 = 0). Taking unit capacitances and inductances makes matrix Q an identity matrix. an input We consider 

0 1 generating system with (S, L) = ( , 1 0 ) an 0 0 observable pair meaning that we are looking at the moments of the original and the reduced order systems at zero. It is readily verified that the reduced order lossless portHamiltonian system with       0 12 2 0 0 = , Δ = , Q Jr = r 0 2 1 − 12 0

IV. REDUCTION OF PORT-HAMILTONIAN SYSTEMS BY KRYLOV METHODS In this section we want to apply the Krylov methods, in particular, the Arnoldi method, to linear port-Hamiltonian systems. The idea of the Arnoldi method is to construct a reduced order model by applying the so-called Galerkin projection V V T , where V ∈ Rn×k , V T V = Ik

Rk = Vk U,

(28)

where Rk ∈ Rn×k is the reachability matrix and U is upper triangular. Then the reduced order projected system defined by (29) Apr = V T AV, Bpr = V T B, Cpr = CV satisfies the equality of the Markov parameters (30) (hpr )i = hi , i = 1, . . . , k. For the proof of Lemma 4 we refer to [1] as well. Remark 5: Note that the factorization (28) is the QR factorization of the reachability matrix Rk . A. Energy coordinates Consider a full order port-Hamiltonian system (15) with Q > 0. Then there exists a coordinate transformation S x = Sx ˆ,

(31)

such that in the new coordinates ˆ = S T QS = I. Q

(32)

By defining the system matrices Jˆ = S −1 JS −T , ˆ = S −1 RS −T , R ˆ = S −1 B, B

[η0 η1 ] = [0 2]

is again a reduced order lossless moment matching port-Hamiltonian system for the full order lossless portHamiltonian system from Example 2.

(27)

(the columns of V are orthonormal), to a full order linear port-Hamiltonian model. As it was mentioned before there is an extensive literature on this topic. For the details on computing an appropriate matrix V by iterative algorithms and properties of V we refer to [1],[6] and the references therein. Lemma 4: [1] Suppose we have the system (1). Take the map V satisfying (27) constructed using the Arnoldi procedure defined in [1] and such that

is of the form (17) and matches the 0- and 1- moments

at S. Remark 4: Note that in the family of reduced order systems (10) there can be more than one lossless reduced order port-Hamiltonian system of the form (17). For instance a port-Hamiltonian system with       0 1 2 0 0 Jr = , Qr = ,Δ = −1 0 0 1 2

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(33)

we obtain the port-Hamiltonian system ˆ x + Bu, ˆ x ˆ˙ = (Jˆ − R)ˆ ˆT x ˆ. y = B

(34)

Theorem 4: Consider a full order port-Hamiltonian system (34) and define V satisfying (27) using the Arnoldi procedure. Then the reduced order system

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x˙ pr = (Jpr − Rpr )xpr + Bpr u, yˆ = Cpr xpr

(35)

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Proceedings of the European Control Conference 2009 • Budapest, Hungary, August 23–26, 2009

with Jpr Rpr Qpr Bpr Cpr

= = = = =

ˆ V T JV, T ˆ V RV, I, ˆ V T B, ˆT V B

(36)

is a port-Hamiltonian system. Furthermore the reduced order port-Hamiltonian system (35) satisfies the equality of the Markov parameters (37) (hpr )i = hi , i = 1, . . . , k. Proof: Clearly Jpr is skew-symmetric and Rpr is symmetric and positive semidefinite. Moreover Cpr = T Qpr . Therefore the port-Hamiltonian structure of the Bpr reduced order model (35) holds. The equality of the Markov parameters follows directly from Lemma 4. B. Co-energy coordinates

where V is defined as in (27). Furthermore, if V is obtained from the Arnoldi procedure, the reduced order portHamiltonian system (43) satisfies the equality of the Markov parameters (hpr )i = hi , i = 1, . . . , k. (45) Proof: The proof is similar to the proof of Theorem 4, hence omitted. For a general Q the full order port-Hamiltonian system (38) can be equivalently rewritten as Q−1 e˙ = (J − R)e + Bu, y = B T e.

Then the following holds true. Theorem 6: Consider a port-Hamiltonian system in coenergy coordinates (46). Then the reduced order system = (Jpr − Rpr )epr + Bpr u, Q−1 pr e˙ pr yˆ = Cpr epr

e˙ = Q(J − R)e + QBu, y = B T e,

Consider a full order port-Hamiltonian system (38) in the ˆ is an identity matrix coordinates eˆ where the energy matrix Q ˆ e + Bu, ˆ eˆ˙ = (Jˆ − R)ˆ ˆ T eˆ, y = B

Jpr Rpr Q−1 pr Bpr Cpr

(38)

which is yet another useful and natural way to look at portHamiltonian modeling. The coordinate transformation [7] between energy x and co-energy e coordinates is given by the energy matrix Q e = Qx. (39)

(40)

e = T eˆ,

(41)

and system matrices = T T JT, = T T RT, −1 = T QT −T = T T B.

= I,

(42)

Theorem 5: Consider a full order port-Hamiltonian system (40). Then the reduced order system e˙ pr = (Jpr − Rpr )epr + Bpr u, yˆ = Cpr epr

(43)

is a reduced order port-Hamiltonian system with = = = = =

ˆ V T JV, T ˆ V RV, I, ˆ V T B, T ˆ B V.

(44)

= = = = =

V T JV, V T RV, V T Q−1 V, V T B, B T V.

(48)

with V defined as in (27). Proof: Clearly Q−1 pr is symmetric and positive definite. The rest of the proof is similar to the proof of Theorem 4. Corollary 1: Suppose the energy matrix Q in (46) is of the block-diagonal form   Q11 0 . (49) Q= 0 Q22 Then

obtained from a coordinate transformation T

Jpr Rpr Qpr Bpr Cpr

(47)

is a reduced order port-Hamiltonian system with

In order to proceed we recall from [7] that a portHamiltonian system (15) in so-called co-energy coordinates takes the following form

Jˆ ˆ R ˆ Q ˆ B

(46)

T −1 T −1 Q−1 pr = V1 Q11 V1 + V2 Q22 V2 ,

(50)

where

  V (51) V = 1 , V1 ∈ Rk×k , V2 ∈ R(n−k)×k . V2 Proof: Because of the block-diagonal structure of Q the computation of Q−1 pr reduces to   

Q−1 0 V1 11 V V = Q−1 1 2 pr V2 (52) 0 Q−1 22 T −1 V + V Q V . = V1T Q−1 1 2 2 11 22

Remark 6: Note that the Q matrix from Example 2 is obviously of the form (49). Corollary 2: Suppose V is of the form   V (53) V = 1 , V1 ∈ Rk×k , V1 − invertible, 0 then the reduced order port-Hamiltonian system (47) reduces to ¯ −1 V1 e˙ pr = V T (J11 − R11 )V1 epr + V T B1 u, V1T Q 1 1 (54) yˆ = B1T V1 epr ,

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which is equivalent to the reduced order port-Hamiltonian system obtained by so-called Effort-constraint method ¯ denoting the Schur discussed in [7] (with the matrix Q ¯ = Q11 − Q12 Q−1 Q21 ). complement of the energy matrix: Q 22 Proof: Using the well-known analytic inversion formula we get  −1   ¯

Q ∗ V1 0 V Q−1 = 1 pr 0 ∗ ∗ (55) ¯ −1 V1 . = V1T Q

structure of the original model. We illustrated the idea by means of a physical example. In Section 4 we considered the Krylov methods and showed that these methods can be used for model reduction of port-Hamiltonian systems in order to preserve the port-Hamiltonian structure as well along with the passivity property. Both approaches considered serve the purpose of the portHamiltonian structure preservation and seem to be likely extended further for model reduction of nonlinear portHamiltonian systems.

In a similar way we can show that Jpr = V1T J11 V1 , Rpr = = V1T B1 , Cpr = B1T V1 with V1T R11 V1 , Bpr Jpr , Rpr , Bpr , Cpr given in (48). Observing that V1 epr = e1 is the first part of the original co-energy state-space vector   e (56) e = 1 , e1 ∈ Rk , e2 ∈ Rn−k e2

VI. ACKNOWLEDGMENTS The authors would like to thank Prof. A. Astolfi, Prof. S. Gugercin and Prof. C. Beattie for the collaboration and the discussions on the topic of model order reduction.

and rewriting the system (54) we obtain ¯ 11 − R11 )e1 + QB ¯ 1 u, e˙ 1 = Q(J T yˆ = B1 e1 ,

(57)

which is the reduced order system using Effort-Constraint method from [7]. Remark 7: Note that the projected systems (35), (43), (47) are automatically passive since they inherit the portHamiltonian structure of the full order systems with Qpr being positive definite. See also [8],[7]. V. CONCLUSIONS In this paper we considered a new approach to the moment matching model reduction of linear systems when a specific input-generating system is given. In Section 3 we started to apply this approach to port-Hamiltonian systems and showed that the reduced order models inherit not only a specific number of the moments but also the port-Hamiltonian

R EFERENCES [1] A.C. Antoulas. Approximation of Large-Scale Dynamical Systems. SIAM, 2005. [2] A. Astolfi. Model reduction by moment matching (semi-plenary presentation). In IFAC Symposium on Nonlinear Control System Design, Pretoria, S. Africa, pages 95–102, 2007. [3] A. Astolfi. A new look at model reduction by moment matching for linear systems. In 46th Conference on Decision and Control, New Orleans, MS, pages 4361–4366, 2007. [4] A. Astolfi. Model reduction by moment matching for nonlinear systems. In 47th Conference on Decision and Control, Cancun, Mexico, 2008. [5] W.B. Gragg and A. Lindquist. On the partial realization problem. Linear Algebra Appl., 50:277–319, 1983. [6] E.J. Grimme. Krylov projection methods for model reduction. PhD thesis, University of Illinois, Urbana-Champaign, 1997. [7] R.V. Polyuga and A.J. van der Schaft. Structure preserving model reduction of port-Hamiltonian systems. In 18th International Symposium on Mathematical Theory of Networks and Systems, Blacksburg, Virginia, USA, July 28 - August 1, 2008. [8] A.J. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear Control. Springer-Verlag, 2000. [9] A.J. van der Schaft. On balancing of passive systems. In Proceedings of the European Control Conference 2007, Kos, Greece, pages 4173– 4178, July 2-5, 2007. [10] J.C. Willems. Dissipative dynamical systems. Archive for Rational Mechanics and Analysis, 45:321–393, 1972.

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