Norms and Singular Values
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
Properties of Singular Values
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.