Stability of FDNs

Stability is assured when some norm of the state vector $\mathbf{x}(n)$ does not increase over time for a zero input signal.

Sufficient condition for stability:

$$\left\Vert\,\mathbf{x}(n+1)\,\right\Vert < \left\Vert\,\mathbf{x}(n)\,\right\Vert, \protect$$

for all $n\geq 0$, where

$$\mathbf{x}(n+1) = \mathbf{A}\left[\begin{array}{c} x_1(n-M_1) \\ [2pt] x_2(n-M_2) \\ [2pt] x_3(n-M_3)\end{array}\right].$$

Inequality holds under the $L2$ norm whenever the feedback matrix $\mathbf{A}$ satisfies

$$\left\Vert\,\mathbf{A}\mathbf{x}\,\right\Vert _2 < \left\Vert\,\mathbf{x}\,\right\Vert _2$$

where

$$\left\Vert\,\mathbf{x}\,\right\Vert _2 \mathrel{\stackrel{\mathrm{\Delta}}{=}}\sqrt{x_1^2+x_2^2+\dots+x_N^2}.$$

(the ``$L2$ norm'') $\Leftrightarrow$ $\zbox{\left\Vert\,\mathbf{A}\,\right\Vert _2<1}$