Norms and Singular Values

R2026b

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

$$\overline{\sigma }\left(A\right)\triangleq {\sigma }_1.$$

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:

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

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 H₂-norm and the H_∞-norms are defined terms of the frequency-dependent singular values of G(jω):

H₂-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:

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

where sup denotes the least upper bound. For more information about these quantities, see Interpretation of H2 and H-Infinity Norms.