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For MIMO systems the transfer functions are matrices, and relevant
measures of gain are determined by singular values, H_{∞},
and H_{2} norms, which are defined as follows:

_{2}-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.

Some useful properties of singular values are:

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

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 *H*_{2}-norm and the *H*_{∞}-norms are defined terms of
the frequency-dependent singular values of *G*(*jω*):

*H*_{2}-norm:

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

*H*_{∞}-norm:

$${\Vert G\Vert}_{2}\triangleq \underset{\omega}{\mathrm{sup}}\overline{\sigma}\left(G\left(j\omega \right)\right)$$

where sup denotes the least upper bound.

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