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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:

The H_{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.

The *singular values* of a rank *r* matrix $$A\in {C}^{m\times n}$$, denoted *σ _{i}*, are the nonnegative square roots of the eigenvalues of $${A}^{*}A$$ ordered such that

If *r* < *p* then there are *p*
– *r* zero singular values, i.e., *σ*_{r+1} = *σ**r*+2 = ... =*σ _{p}* = 0.

The greatest singular value *σ*_{1} is sometimes
denoted

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

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

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

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:

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

where sup denotes the least upper bound.

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