H2 and H∞ Norms
The H2-norm is the energy of the impulse response of plant
G. The H∞-norm is the peak gain of
G across all frequencies and all input directions.
Another important concept is the notion of singular values.
If r < p then there are p – r zero singular values, i.e., σr+1 = σr+2 = ... =σp = 0.
The greatest singular value σ1 is sometimes denoted
When A is a square n-by-n matrix, then the nth singular value (i.e., the least singular value) is denoted
These properties are especially important because they establish that the greatest and least singular values of a matrix A are the maximal and minimal "gains" of the matrix as the input vector x varies over all possible directions.
where sup denotes the least upper bound.