Norms

Abstract

System norms serve as a measure of intensity of its response to standard excitations, such as unit impulse, or white noise of unit standard deviation. The standardized response allows comparing different systems. Three system norms, H₂, H_∞, and Hankel are used in this book. We show that for flexible structures the H₂ norm has an additive property: it is a root-mean-square (rms) sum of the norms of individual modes. We also show that the H_∞ and Hankel norms are determined from the corresponding modal norms, by selecting the largest one. All three norms of a mode with multiple inputs (or outputs) can be decomposed into the rms sum of norms of a mode with a single input (or output). Later in this book these two properties allow for the development of unique and efficient model reduction methods and actuator/sensor placement procedures.