Two‐sided Grassmann manifold algorithm for optimal model reduction

Abstract

SUMMARYWe consider an optimal H 2 model reduction problem for large-scale dynamical systems. The problem is formulated as a minimization problem over Grassmann manifold with two variables. This formulation allows us to develop a two-sided Grassmann manifold algorithm, which is numerically efficient and suitable for the reduction of large-scale systems. The resulting reduced system preserves the stability of the original system. Numerical examples are presented to show that the proposed algorithm is computationally efficient and robust with respect to the selection of initial projection matrices.

State-response decomposition for model reduction

“…To combine transient features with the retention of at least the static or direct-current (DC) gain and, thus, of the steady-state value in the step response, a variant of the balancedtruncation method (Moore, 1981) based on perturbation theory (Kokotovic, Khail, and Reilly, 1986) has been suggested, which can be implemented using popular control packages (Chaturvedi, 2009). No such (limited) extensions are as yet available for other optimal reduction techniques, such as the Hankel-norm (Glover, 1984) and L 2 -norm (Antoulas, 2005;Krajewski and Viaro, 2009;Xu and Zeng, 2011;Zeng and Lu, 2015) approximation methods. Not even the newly proposed metaheuristic model reduction methods (Desai and Prasad, 2013;Ramawat and Kumar, 2015;Rana, Prasad, and Singh, 2014; CONTACT Daniele Casagrande daniele.casagrande@uniud.it Sikander and Prasad, 2015) explicitly address the problem of asymptotic accuracy.…”

Optimizing Oblique Projections for Nonlinear Systems using Trajectories

“…Most approaches focus on optimizing orthogonal projection operators over a single Grassmann manifold [53,44,26] or an orthogonal Stiefel manifold [54,44,50,55,52]. On the other hand, an alternating minimization technique over the two Grassmann manifolds defining an oblique projection is proposed by T. Zeng and C. Lu [56] for H 2 -optimal reduction of linear systems. For systems with quadratic nonlinearities, Y.-L. Jiang and K.-L. Xu [26] present an approach to optimize orthogonal projection operators based on the same truncated generalization of the H 2 norm used by P. Benner et al [10].…”