Structure preserving balanced proper orthogonal decomposition for second‐order form systems via shifted Legendre polynomials

Laguerre-Gramian-based structure-preserving model order reduction for second-order form systems

“…From Li and White (2001) and Xiao et al (2019b), we obtain a modification of this method, where the original system is projected onto the sum of the dominant eigenspaces of the two approximate Gramians. It aims to construct an orthogonal projection V p onto an r-dimensional space obtained by a QR decomposition and then let q(t) = V p q(t) to generate the ROM.…”

“…Analogously P v and Q v are defined as the second-order velocity Gramians. Using pairs of them, we can now define different balanced realisations for second-order form systems including position-balanced (P p , Q p ), velocity-balanced (P v , Q v ), position-velocity-balanced (P p , Q v ) and velocity-position-balanced (P v , Q p ) ROMs (Xiao et al, 2019b).…”

“…Note that preservation of the second-order structure in the ROM allows a meaningful physical interpretation and usually provides more accurate approximations, and an ROM which preserves the structural properties such as stability and passivity is highly desirable. Therefore, on the contrary, structure-preserving MOR techniques for second-order form systems are proposed, which have received a lot of attention during the last few years, including moment matching methods based on Krylov subspaces (Bai and Su, 2005;Jbilou and Kaouane, 2019;Xu et al, 2018), families of BT methods (Chahlaoui et al, 2006;Haider et al, 2019;Reis and Stykel, 2008;Uddin, 2020) and orthogonal polynomial methods (Qiu et al, 2018;Xiao et al, 2019b;Xiao and Jiang, 2014). Overview articles of different reduction methods are given in Antoulas (2005), Benner et al (2017), Jiang (2010) and the references therein.…”