## Norms and Singular Values

For MIMO systems the transfer functions are matrices, and relevant measures of gain are determined by singular values, H, and H2 norms, which are defined as follows:

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.

Singular Values:

The singular values of a rank r matrix $A\in {C}^{m×n}$, denoted σi, are the nonnegative square roots of the eigenvalues of ${A}^{*}A$ ordered such that σ1 ≥ σ2 ≥ ... ≥σp > 0, p ≤ min{m, n}.

If r < p then there are pr zero singular values, i.e., σr+1 = σr+2 = ... =σp = 0.

The greatest singular value σ1 is sometimes denoted

`$\overline{\sigma }\left(A\right)\triangleq {\sigma }_{1}.$`

When A is a square n-by-n matrix, then the nth singular value (i.e., the least singular value) is denoted

`$\underset{¯}{\sigma }\left(A\right)\triangleq {\sigma }_{n}.$`

### Properties of Singular Values

Some useful properties of singular values are:

`$\begin{array}{l}\overline{\sigma }\left(A\right)={\mathrm{max}}_{x\in {C}^{h}}\frac{‖Ax‖}{‖x‖}\\ \underset{¯}{\sigma }\left(A\right)={\mathrm{min}}_{x\in {C}^{h}}\frac{‖Ax‖}{‖x‖}\end{array}$`

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.

For stable continuous-time LTI systems G(s), the H2-norm and the H-norms are defined terms of the frequency-dependent singular values of G():

H2-norm:

`${‖G‖}_{2}\triangleq \left[\frac{1}{2\pi }\right]{\int }_{-\infty }^{\infty }\sum _{i=1}^{p}{\left({\sigma }_{i}\left(G\left(j\omega \right)\right)\right)}^{2}d\omega$`

H-norm:

`${‖G‖}_{\infty }\triangleq \underset{\omega }{\mathrm{sup}}\overline{\sigma }\left(G\left(j\omega \right)\right)$`

where sup denotes the least upper bound.