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 $A \in C^{m \times n}$ , denoted σ_i, are the nonnegative square roots of the eigenvalues of $A^{*} A$ ordered such that σ₁ ≥ σ₂ ≥ ... ≥σ_p > 0, p ≤ min{m, n}.

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

The greatest singular value σ₁ is sometimes denoted

$\bar{σ} (A) ≜ σ_{1} .$

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

$\underline{σ} (A) ≜ σ_{n} .$

Properties of Singular Values

Some useful properties of singular values are:

$\begin{array}{l} \bar{σ} (A) = \max_{x \in C^{h}} \frac{‖ A x ‖}{‖ x ‖} \\ \underline{σ} (A) = \min_{x \in C^{h}} \frac{‖ A x ‖}{‖ x ‖} \end{array}$

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(jω):

H₂-norm:

${‖ G ‖}_{2} ≜ [\frac{1}{2 π}] \int_{- \infty}^{\infty} \sum_{i = 1}^{p} {(σ_{i} (G (j ω)))}^{2} d ω$

H_∞-norm:

${‖ G ‖}_{\infty} ≜ \sup_{ω} \bar{σ} (G (j ω))$

where sup denotes the least upper bound.

Norms and Singular Values

Properties of Singular Values

See Also