Riemannian optimal model reduction of linear port-Hamiltonian systems

Abstract

In this paper, we describe the development of a Riemannian optimal model reduction method for linear stable port-Hamiltonian systems. This development is motivated by the fact that there remains room for improvement in existing methods. The model reduction problem is formulated as an optimization problem on the product manifold of the set of skew symmetric matrices, the manifold of the symmetric positive definite matrices, and Euclidean space. The reduced systems constructed using the optimal solutions to the problem preserve the original structure, i.e., stability, passivity, and the port-Hamiltonian form. The Riemannian gradient is derived to relate our problem to another problem in some studies, and the Hessian is also derived to solve our problem using a Riemannian trust region method. The initial point in the proposed method is chosen by using the output of the iterative rational Krylov algorithm for linear port-Hamiltonian systems (IRKA-PH). A numerical experiment illustrates that the proposed method considerably improves the results of IRKA-PH when the reduced-model dimension is small.

Introduction

Port-Hamiltonian modeling captures the physical properties of a given system, such as its stability, passivity, and energy dissipation properties, and is useful for implementing passivity-based controls; see, e.g., Ortega, van der Schaft, Mareels, and Maschke (2001), Ortega, van der Schaft, Maschke, and Escobar (2002) and van der Schaft (2000). This modeling method also deals with the interconnections of many physical systems. Such interconnected systems usually become infinite-dimensional systems studied in Jacob and Zwart (2012) and Villegas (2007), and the spatially discretized systems are medium-scale (100–1000) or large-scale (1000–) dimensional systems in order to better approximate the original ones. For such a complex system, it is not easy to design an appropriate controller, and thus the development of effective model reduction methods is required.

The development of reduction methods of linear port-Hamiltonian systems has been described in the control literature; see, e.g., Gugercin, Polyuga, Beattie, and Van Der Schaft (2012), Ionescu and Astolfi (2013), Polyuga and van der Schaft (2012) and Wolf, Lohmann, Eid, and Kotyczka (2010). In particular, in the study reported in Gugercin et al. (2012), the iterative rational Krylov algorithm for linear stable port-Hamiltonian systems (IRKA-PH) was developed. This algorithm produces reduced systems, for which the transfer functions interpolate the original transfer function of a port-Hamiltonian system at selected points in the complex plane. Gugercin et al. (2012) demonstrated that IRKA-PH is more efficient than the effort-constraint method proposed in Polyuga and van der Schaft (2012). However, the reduction results of IRKA-PH may be far from optimal when the dimension of a reduced system is small.

The main aim of this paper is to propose a method for improving the results of IRKA-PH. To this end, we develop a Riemannian optimal model reduction method for linear stable port-Hamiltonian systems. The model reduction problem is formulated as an $H^2$ optimization problem on the product manifold of the set of skew symmetric matrices, the manifold of the symmetric positive definite matrices, and Euclidean space. The reduced systems constructed by using the optimal solutions to the problem preserve the original structure, i.e., stability, passivity, and the port-Hamiltonian form. To the best of our knowledge, this formulation is the first for reduction problems of linear port-Hamiltonian systems, although the same formulation can be found in Sato (2017a) for other control problems. That is, although the model reduction problems in Sato (2017b), Sato and Sato (2015), Sato and Sato (2016), Sato and Sato (2018) and Yan and Lam (1999) were studied using a Riemannian optimization approach, the algorithms developed in those references cannot solve our problem in this paper.

The contributions are summarized as follows.

(1) We relate our problem to that in Sato and Sato (2015) and Yan and Lam (1999). More specifically, we prove that if a linear stable port-Hamiltonian system is a symmetric system, then the optimal solutions to our problem can be expressed by using those given in the previous works. Moreover, we show that stationary points of the objective function of our problem cannot be described in general using those given in the previous works. That is, the minimum value of the problem in Sato and Sato (2015) and Yan and Lam (1999) is not smaller than that of our problem.

(2) We describe the derivation of the Riemannian gradient and Hessian of the objective function, which we use to develop a Riemannian trust region method for solving the model reduction problem. The initial point is chosen by using the output of IRKA-PH. A numerical experiment illustrates that the proposed method considerably improves the results of IRKA-PH when the reduced system dimension is small.

The remainder of this paper is organized as follows. In Section 2, we formulate the $H^2$ optimal model reduction problem on the product manifold of linear stable port-Hamiltonian systems. In Section 3, we relate our problem to that in Sato and Sato (2015) and Yan and Lam (1999). In Section 4, we describe a Riemannian trust region method for solving our problem. Furthermore, we propose a technique for choosing an initial point in our algorithm. In Section 5, we demonstrate that the proposed method is more effective than IRKA-PH when the reduced system dimension is small. Finally, conclusions are presented in Section 6.

Notation: The sets of real and complex numbers are denoted by $\mathbf{R}$ and $\mathbf{C}$, respectively. The identity matrix of size $n$ is denoted by $I_n$. The symbol $\mathrm{Skew}\left(n\right)$ denotes the vector space of skew-symmetric matrices in ${\mathbf{R}}^{n\times n}$. The manifold of symmetric positive definite matrices in ${\mathbf{R}}^{n\times n}$ is denoted by ${\mathrm{Sym}}_{+}\left(n\right)$. The tangent space at $x$ on a manifold $X$ is denoted by $T_{x}X$. Given a matrix $A\in {\mathbf{R}}^{n\times n}$, $\mathrm{tr}\left(A\right)$ denotes the sum of the elements on the diagonal of $A$, and $A_{i,j}$ denotes the entry of $A$ in row $i$ and column $j$. Moreover, $\mathrm{sym}\left(A\right)$ and $\mathrm{sk}\left(A\right)$ denote the symmetric and skew symmetric parts of $A$, respectively; i.e., $\mathrm{sym}\left(A\right)=\frac{A+A^T}{2}$ and $\mathrm{sk}\left(A\right)=\frac{A-A^T}{2}$. Here, $A^T$ denotes the transpose of $A$.