Passivity and Structure Preserving Order Reduction of ... - Science Direct

European Journal of Control (2010)4:401–406 © 2010 EUCA DOI:10.3166/EJC.16.401–406

Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspacesg Thomas Wolf ∗ , Boris Lohmann, Rudy Eid and Paul Kotyczka Institute of Automatic Control, Technische Universität München, Boltzmannstr. 15, 85748 Garching, Germany

In this paper, a new structure-preserving scheme for the reduction of linear port-Hamiltonian systems with dissipation using Krylov subspaces is presented. It is shown how to choose the projection matrices in order to guarantee the moment matching property and to obtain a passive and thus stable reduced-order model in port-Hamiltonian form. The method is suitable for the reduction of largescale systems as it employs only the well-known Arnoldi algorithm and matrix-vector multiplications to compute the reduced-order model. A finite element model is reduced to illustrate the new method. Keywords: Order reduction, Port-Hamiltonian systems, Structure preserving, Moment matching

1. Introduction In recent years, an interesting approach for modeling lumped-parameter physical systems was developed leading to a geometrically defined class of systems, called port-Hamiltonian systems. The port-Hamiltonian approach unifies concepts from theoretical mechanics (the Hamiltonian representation of mechanical systems, arising from the Legendre transform of the Lagrangian equations of motion) and electrical engineering (network modeling, generalized in the bond graph approach [9]). Using energy as the link between physical domains, the port-Hamiltonian formalism provides a unified framework for interdisciplinary modeling of complex dynamical ∗ Correspondence to: T. Wolf, E-mail: [email protected]

systems, which is – due to its conceptual simplicity and modularity – suitable for automated model generation, processing and simulation. Furthermore, with Control by Interconnection or the generalized approach Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) [22], transparent methods for the design of nonlinear controllers are available. The accurate modeling of complex lumped-parameter as well as spatially discretized distributed-parameter systems lead to large-scale systems having a high number of differential equations. To investigate the dynamics of these high-dimensional models, or to design a controller, it is aimed at replacing the original model with a low dimensional approximation. For this purpose, well-established model reduction methods can be used. The well-known Truncated Balanced Realization (TBR) [1] reduces the system based on the energy flow from input to output, providing an error-bound and a stable reduced system. However, passivity as a pivotal property of port-Hamiltonian systems is generally lost. Only for a special class of port-Hamiltonian systems, a truncated balanced realization technique which is structure preserving is presented in [20]. In order to reduce large- and very large-scale systems (for instance order 106 ), the methods based on matching the moments and/or Markov parameters of the original and reduced models are nowadays among the best choices [11, 8]. These methods find the reduced-order model in a numerically efficient way, using well-established algorithms and low-memory requirements, via a projection whose columns form bases of particular Krylov subspaces. Received 21 August 2009; Accepted 19 February 2010 Recommended by A. Astolfi, A.J. van der Schaft

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However, there is neither general guarantee for the stability nor the passivity of the reduced-order model so far, and no general error bound quantifying the quality of approximation of the reduced-order model exists. Several works towards solving the stability preserving problem have been suggested in literature and can be grouped into three main approaches. The first group are passivity-based methods [7, 21, 18], which offer a good approximation and guarantee passivity, and thus stability, of the reduced model assuming a passive original system. The second group are post-processing methods which delete the unstable poles of the reduced system by using restarted algorithms like the implicitly and explicitly restarted Arnoldi and Lanczos algorithms [4, 10, 17]. The third group are interpolation-based methods which preserve the passivity of the original system through spectral-zero interpolation [2, 26, 15]. Even though the mentioned methods offer a solution to the passivity/stability problem of Krylov subspace methods, none of them can be applied for the reduction of port-Hamiltonian systems while preserving the inherent structure. Towards this aim, recently in [23] a first Krylov-based approach has been presented; however, requiring the inversion of the so-called energy matrix Q. This approach has been further extended in [13] and coupled with the H2 -optimal reduction method [12]. Subsequently, a structure-preserving Krylov-based order reduction method for linear port-Hamiltonian systems is introduced. It employs standard Krylov subspaces and thus is well applicable for the reduction of largeand very large-scale systems. It is shown, how to reduce the state space matrices in order to preserve the portHamiltonian structure (and thereby passivity and stability) of the original system and in addition guarantee moment matching. In the following section, a brief overview on portHamiltonian systems is given, whereas in Section 3, the theory of moment matching using Krylov subspaces is reviewed. The new scheme for structure preserving model reduction is presented in Section 4 and applied to a numerical example in Section 5.

2. Port-Hamiltonian Systems Consider the time-invariant port-Hamiltonian system of the form x˙ = (J − R) ∇H(x) + Gu y = G ∇H(x), T

of energy between the storage elements (states x ∈ Rn ) of the system while dissipation is characterized by R. The input matrix G ∈ Rn×m and the gradient ∇H(x) of the energy function define the collocated output y ∈ Rm , which together with the input u ∈ Rm constitute the power port (u, y) of the system. From the energy balance ˙ H(x) ≤ yT u,

(2)

the passivity of the port-Hamiltonian system is deduced. If H(x) is positive definite, Lyapunov stability of the unforced system follows directly, however, if H(x) is only semi-definite, zero-state detectability is necessary in addition. Asymptotic stability of the port-Hamiltonian system can be checked by the invariance principle of KrassowskijLaSalle. A detailed overview on port-Hamiltonian systems can be found in [25]. In the case of linear port-Hamiltonian systems, the matrices J, R and G are constant matrices, independent of the state vector x. Additionally, the Hamiltonian is in fact a quadratic energy function of the form H(x) =

1 T x Qx, 2

(3)

with Q ∈ Rn×n symmetric and positive definite. The gradient of the Hamiltonian in (3) is given by ∇H(x) = Qx. Using the expression of the gradient, in the single-input single-output (SISO) case where m = 1 and u, y ∈ R, the system (1) becomes x˙ = (J − R) Qx + gu y = gT Qx,

(4)

with the constant input vector g ∈ Rn . Note that the symmetry and definiteness properties of the matrices J, R and Q define the port-Hamiltonian structure. Therefore, the problem of structure preserving model order reduction of port-Hamiltonian systems is translated into finding a reduced system having skewsymmetric Jr , positive semi-definite Rr and positive definite Qr . Having found these matrices, passivity and stability are automatically guaranteed (preserved) in the reduced system.

3. Order Reduction by Moment Matching (1)

where H(x) ≥ 0 is the total stored energy called Hamiltonian, J ∈ Rn×n the skew-symmetric interconnection matrix and R ∈ Rn×n the symmetric positive semi-definite dissipation or damping matrix. J represents the exchange

The system (4) can be treated as a linear time-invariant state space model of the form x˙ = Ax + bu y = cT x,

(5)

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having the transfer function G (s) = cT (sI − A)−1 b with the Taylor series expansion G (s) = −m0 −m1 (s−s0 )−. . .−mi (s−s0 )i −. . .

(6)

around an arbitrary expansion point s0 . The aim of moment matching is to find a reduced-order model of order q  n, whose moments (which are defined as the negative coefficients mi of the Taylor series expansion of the transfer function) around the expansion point s0 match some of the first moments of the original one. One way to calculate this reduced-order model is by approximating the state vector x ∈ Rn by xr ∈ Rq in the original system, i. e. x ≈ Vxr with V ∈ Rn×q , and multiplying the state equation with the transpose of a matrix W ∈ Rn×q :1 Ar

br

      WT V˙xr = WT AV xr + WT b u y =  c T V xr ,

(7)

crT

which leads to the reduced system Ar ∈ Rq×q , br ∈ Rq , cr ∈ Rq . The projection matrices V and W can be chosen according to the following theorem [11, 1] using Krylov defined as Kq (M, v) =   subspaces, generally span v, Mv, . . . , Mq−1 v . Theorem 1: Given a linear time-invariant system (5) and applying the projection matrices V ∈ Rn×q and W ∈ Rn×q leads to the reduced system (7). If the columns of the matrix to form a basis of the Krylov subspace  V are chosen Kq A−1 , A−1 b , and W arbitrarily, and assuming that the matrix Ar is nonsingular, then the first q moments around zero of the original and reduced-order systems match. Matching the moments around zero preserves the lowfrequency behavior of the original system, whereas, in order to compute a reduced model that approximates the middle or high frequency behavior, matching a certain number of moments around s0 = 0 is to be preferred. This is achieved by choosing the columns of V  as a basis of the shifted input Krylov subspace Kq (A − s0 I)−1 , (A − s0 I)−1 b and assuming that WT (A − s0 I) V is nonsingular (see [11] for details). For the numerical calculation of V, the known Arnoldi algorithm [3] can be used, which returns an orthonormal basis V (with VT V = I) of the required Krylov subspace. The widely used choice W = V is referred to as a onesided method. 1

This reduction step can be obtained by an orthogonal (oblique) projection such that the residual error of the approximation of the states satisfies the Galerkin (Petrov-Galerkin) condition [11].

4. Structure Preserving Model Order Reduction In this section, the main result of this paper, a method for the structure preserving model order reduction of linear port-Hamiltonian systems is proposed. Theorem 2: Given a linear SISO system in portHamiltonian form A

b     x˙ = (J − R) Q x + g u

y = gT Q x, 

(8)

cT

with x ∈ Rn . Then, applying the projection matrix V ∈ Rn×q , calculated from the standard Krylov subspace

 Kq = span (A − s0 I)−1 b , . . . , (A − s0 I)−q b

(9)

= span ((J − R)Q − s0 I)−1 g , . . . ,  × ((J − R)Q − s0 I)−q g ,

(10)

results in the reduced system x˙ r = (Jr − Rr ) Qr xr + gr u y = grT Qr xr ,

(11)

with the reduced state x ∈ Rq , q  n. The specific choice Jr = VT QJ QV

(12)

Rr = VT QR QV

−1 Qr = VT QV

(13) (14)

gr = VT Qg

(15)

guarantees matching the first q moments around s0 and preserving the port-Hamiltonian structure, assuming that  Jr − Rr − s0 Q−1 is nonsingular. r Proof: The reduced system (11) preserves the portHamiltonian structure, since the matrices Jr , Rr and Qr preserve the (semi-) definiteness and (skew-) symmetry of the original matrices. This is based on the fact that, if a matrix F ∈ Rn×n is positive (semi-) definite then the matrix TT FT is again positive (semi-) definite, where T ∈ Rn×q with q ≤ n is an arbitrary matrix of full column rank. Furthermore, the matrix TT FT is (skew-) symmetric, if F is (skew-) symmetric.

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For the proof of moment matching, we use Theorem 1 with the particular choice W = QV leading to the reduced system: VT QV˙xr = VT Q(J − R)QVxr + VT Qbu y = gT QVxr .

(16)

According to Section 3, it is assumed that  is nonsingular. WT (A − s0 I) V = Jr − Rr − s0 Q−1 r Note that, due to the positive definiteness of Q, the matrix Q−1 r is nonsingular. The subsequent state transformation zr = VT QVxr , leads to the equations (12)–(15) and completes the proof. (An alternative but longer proof using some state space transformations can be found in the internal report [19].) Remark 1: Another way for approximating a system especially at high frequencies (s → ∞) is to match the Markov parameters, defined as the coefficients of the Taylor series G (s) = cT bs−1 +cT Abs−2 +. . .+cT Ai−1 bs−i +. . . . (17) For a linear time-invariant system (5), projecting the system using the matrix V calculated from the Krylov subspace

 (18) Kq = span b , Ab , . . . , Aq−1 b , leads to matching the first q Markov parameters [16]. It can be shown similarly to the presented proof, that calculating the reduced model with (12)–(15) and using the Krylov subspace



Kq = span g , (J − R) Qg , . . . , ((J − R) Q)q−1 g , (19) leads to matching the first q Markov parameters and in addition preserving the port-Hamiltonian structure and thus stability and passivity. Remark 2: The presented method can be easily generalized to the multi input, multi output case (MIMO). For this purpose, in the computation of a basis for the Krylov subspace (10) the matrix G ∈ Rn×m instead of the vector g has to be used, with q being a multiple of m. The proof is similar to the one shown above, leading to match the first q/m moments. The presented method offers in fact a solution to the general stability preservation problem in Krylov subspace methods for linear dynamical systems, since every stable linear system x˙ = Ax + bu can be written in portHamiltonian form x˙ = (J − R) Qx + gu [24]. However,

if the system is available only in the standard form x˙ = Ax + bu, the numerical effort for the conversion to port-Hamiltonian form, as presented in [24], might not be acceptable. Note that for the conversion, the output y = cT x does not need to be converted into the form y = gT Qx, since by using the input Krylov subspace as suggested in Theorem 2, the definition of the output y does not affect the number of matched moments. Finally, it is recommended to directly model the system in portHamiltonian form to benefit from the results of this paper while avoiding numerical problems.

5. Numerical Example In order to show the applicability to large-scale systems, a clamped beam, modelled with a finite element method (FEM) is considered. The resulting second-order system is M¨z + D˙z + Kz = bu,

(20)

with the matrices M, D, K being positive definite and defining the mass, damping and stiffness, respectively. The beam is modelled with two hundred nodes, leading to the port-Hamiltonian state equation [14] of dimension n = 4800: ⎞ ⎛ ⎟   ⎜ 0 0 ⎟ ⎜ 0 I x˙ = ⎜ − ⎟ 0 D ⎠ ⎝ −I 0       J

 × 

K 0

R

  0 0 x + u. b M−1      

Q

(21)

g

Note that the states x describe the positions and momenta of the finite elements. A vertical force at the tip of the beam is considered as the input, leading to the tip velocity as the collocated output y = gT Qx. Note that, due to lumping of mass [27], the matrix M is diagonal and can be easily inverted. The order of the system is reduced to q = 13. The iterative RK-ICOP algorithm (Rational Krylov with an Iteratively Calculated Optimal Point [6, 5]) suggests the expansion point s0 = 194.98. The bode plot of the structure preserving reduced system is compared to a reduced system obtained by a standard one-sided method, as introduced in Section 3, with the same expansion point s0 . Although employing the same Krylov subspace (10), the reduced model x˙ r = VT (J − R) QVxr + VT gu y = gT QVxr ,

(22)

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Fig. 1. Bode plot of clamped beam.

Table 1. Relative error norms H2 = One-sided method Structure preserving

G(s)−Gr (s) 2 G(s) 2

Inf 0.4570

does not preserve any inherent structure, though still matches the first q = 13 moments. As it can be seen in Fig. 1 both reduced systems offer a good approximation of the bode plot of the original system. However, the reduced system using the one-sided Krylov method has one unstable pole at 7.35 · 103 , while the reduced-order system obtained by applying the method presented in this paper, is asymptotically stable. In Table 1 the relative H2 and H∞ -error norms are given. Due to the instability, the relative H2 -error of the reduced system obtained by the one-sided method is infinity. Furthermore, the structure preserving scheme results in a smaller relative H∞ -error than the standard one-sided method.

6. Conclusions A new structure-preserving order reduction scheme for linear port-Hamiltonian systems has been presented. The reduced-order models obtained by the method presented in this paper match the first q moments (or alternatively Markov parameters) of the original system, while preserving the port-Hamiltonian structure and thus passivity and stability. The method is suitable for model reduction

H∞ =

G(s)−Gr (s) ∞ G(s) ∞

0.1609 0.0825

of large-scale systems due to the numerical efficiency of the iterative Krylov subspace methods. Although the new approach employs the same Krylov subspace as the standard method, structure preservation is achieved by the specific projection introduced in Theorem 2. Finally, as the presented method preserves stability and the portHamiltonian structure in the reduced systems, the contribution offers one more good reason to model even large real world problems in port-Hamiltonian form.

Acknowledgments The authors would like to thank A.J. van der Schaft and K. Schlacher for the fruitful discussions in Oberwolfach and Munich in February and March 2009.

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