Passivity-preserving model reduction via a computationally efficient project-and-balance scheme

Abstract

This paper presents an efficient two-stage project-and-balance scheme for passivity-preserving model order reduction. Orthogonal dominant eigenspace projection is implemented by integrating the Smith method and Krylov subspace iteration. It is followed by stochastic balanced truncation wherein a novel method, based on the complete separation of stable and unstable invariant subspaces of a Hamiltonian matrix, is used for solving two dual algebraic Riccati equations at the cost of essentially one. A fast-converging quadrupleshift bulge-chasing SR algorithm is also introduced for this purpose. Numerical examples confirm the quality of the reduced-order models over those from conventional schemes.

A Rational Shira Method for the Hamiltonian Eigenvalue Problem

“…Hamiltonian matrices and eigenproblems arise in a variety of applications. They are ubiquitous in control theory, where they play an important role in various control design procedures (linear-quadratic optimal control, Kalman filtering, H 2 -and H ∞ -control, etc., see, e.g., [2,17,24,30] and most textbooks on control theory), system analysis problems like stability radius, pseudo-spectra, and H ∞ -norm computations [8,9,10], and model reduction [1,5,6,14,26,29]. Another source of eigenproblems exhibiting Hamiltonian structure is the linearization of certain quadratic eigenvalue problems [4,18,20,28].…”

Quadratic alternating direction implicit iteration for the fast solution of algebraic Riccati equations

“…The focus here is on the speed of different solvers operating on the same ARE, rather than the origination or formulation of these AREs. The QADI in (10) and CFQADI in (12) are coded in MATLAB m-files (ordinary text files) and executed, without compilation, in the MATLAB R14 environment. They are compared against the MATLAB subroutine aresolv with the schur and eigen flags enabled successively.…”

“…Prominent applications of AREs include Kalman filtering, linear quadratic regulator (LQR), and optimal controller design [9,10]. Recently, VLSI backend design tools are also applying balanced stochastic truncation (BST) for passivity-preserving model order reduction and fast simulation of VLSI circuits [8,11,12], wherein a pair of high order AREs need to be solved. Solution of an ARE, however, can be computationally intensive even for medium orders.…”

“…When it is necessary to distinguish the difference, we will refer to them as (1+) or (1-) according to the sign in front of XBBTX. In to the H,, norm of the state space (A, B, C) being less than one [10], or, when A, B, and C are obtained from certain system matrices (see e.g., [12]), the original system is passive. Using M > 0 (M > 0) to denote a positive definite (positive semidefinite) matrix M, and defining spec(o) to be the spectrum of a matrix, and C to be the open left halfplane, the above assumptions guarantee the existence of a unique stabilizing solution, X(C RnXn) > O0 that solves (1) and satisfies spec(A ± BBTX) C C-.…”

Two Algorithms for Fast and Accurate Passivity-Preserving Model Order Reduction

“…Another way, provided a stabilizing initial condition is known, is to use the Newton method, which solves a Lyapunov equation in each iteration [22], [28]. In this paper, we summarize and report our recent work on fast implementations of both the Newton and Hamiltonian approaches in the context of large-scale BST [18], [19]. The first contribution is a Smith-method-based Newton algorithm, called the Newton/Smith CARE (NSCARE) algorithm, for quickly solving a large-scale CARE containing low-rank input/output matrices [18].…”