Model reduction and control of flexible structures using Krylov vectors

Abstract

Krylov vectors and the concept of parameter matching are combined together to develop a model reduction algorithm for a damped structural dynamics system. The obtained reduced-order model matches a certain number of low-frequency moments of the full-order system. The major application of the present method is to the control of flexible structures. It is shown that, in the control of flexible structures, three types of control energy spillover generally exist: control, observation, and dynamic. The formulation based on Krylov vectors can eliminate both the control and observation spillovers while leaving only the dynamic spillover to be considered. Two examples are used to illustrate the efficacy of the Krylov method.

Structure-Preserving Model Reduction

“…The work of Su and Craig [17] has spawned several recent research papers on model reduction of second-order systems and quadratic eigenvalue problems, includ- ing [3,4,5,18]. But the attempt to preserve meaningful substructures as in (2.3) -(2.6) for any general linear systems, not necessarily from linearizing a second-order system, appears to be conceived first by [10].…”

“…Item 4 of Theorem 3 was implicitly stated in [3,4,17]. It gives a relation between span( X 1 ) and span( X 2 ); so does Item 1.…”

“…As proved in [12], reduction (1.3) (and thus (2.6) included) with Y = X preserves passivity of the system (2.7). [17] concerning a second-order system which can always be equivalently turned into a linear system (see (4.2) in the next section) with a natural partitioning just as in (2.1) and (2.2).…”

“…To overcome this, Su and Craig [17] made an important observation about the structures of the associated Krylov subspaces that make it possible for the reduced second-order system to still have the second-order form…”

Model Reduction for RF MEMS Simulation

“…We produce an orthonormal basis Q n for the second-order Krylov subspace G n using a Second-Order ARnoldi (SOAR) procedure proposed by Su and Craig [12] and further improved by Bai and Su [3]. At step j, the algorithm computes…”