Large-Scale Computation of $\mathcal{L}_\infty$-Norms by a Greedy Subspace Method

Abstract

We are concerned with the computation of the L∞-norm for an L∞-function of the form H(s) = C(s)D(s) −1 B(s), where the middle factor is the inverse of a meromorphic matrix-valued function, and C(s), B(s) are meromorphic functions mapping to shortand-fat and tall-and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay systems fall into this family. We focus on the case where the middle factor is large-scale. We propose a subspace projection method to obtain approximations of the function H where the middle factor is of much smaller dimension. The L∞-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the L∞-norm of the reduced function is attained. The subspace method is designed so that certain Hermite interpolation properties hold between the largest singular values of the original and reduced functions. This leads to a locally superlinearly convergent algorithm with respect to the subspace dimension, which we prove and illustrate on various numerical examples.

Model Reduction for Second‐Order Dynamical Systems Revisited

“…However, the repeated computation of the H ∞ -norm of transfer functions of such large-scale systems has been made possible by the methods presented in [20,21] which we use here. However, the repeated computation of the H ∞ -norm of transfer functions of such large-scale systems has been made possible by the methods presented in [20,21] which we use here.…”

Mixed control of vibrational systems

“…. From (1,1) we obtain 13 = − 2 1 . Sincẽ≥ 0, one can easily see that̃1 1 = 0 implies̃1 2 = 0, hence (3,3) reads = −1, a contradiction.…”

“…Also, one can consider damping optimization using the norm. This can be also calculated efficiently using the model order reduction approaches, for more details see, e.g., []. Moreover, damping optimization in mechanical systems with external force was also considered in [] and [] where the authors used optimization criteria based on average energy amplitude and average displacement amplitude over the considered time.…”

“…System disturbances are denoted by the vector ∈ ℝ and the matrix 2 ∈ ℝ × . The first equation in (1) is usually called the state equation.…”

“…In damping optimization setting different optimization criteria are used, depending on different applications. One criterion for systems described by (1) would be to minimize the 2 norm of the system as the penalty function. In particular, damping optimization using the 2 norm was considered in [7,37] as well as in [3,25], where the authors considered model order reduction approaches in order to determine the optimal damping parameters efficiently.…”