Consider the following SISO operator $y(n) = G\{ x(n) \}$ described by the Volterra series: $ \newcommand{\abs}[1]{| #1 |} \newcommand{\bigabs}[1]{\left| #1 \right|} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}} \newcommand{\calE}{\mathcal{E}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calD}{\mathcal{D}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calM}{\mathcal{M}} \newcommand{\bbP}{\mathbb{P}} \newcommand{\bbQ}{\mathbb{Q}} \newcommand{\ip}[2]{\langle #1, #2 \rangle} \newcommand{\bigip}[2]{\left\langle #1, #2 \right\rangle} \newcommand{\T}{\mathsf{T}} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\ind}{\mathbf{1}} \newcommand{\calL}{\mathcal{L}} \newcommand{\norm}[1]{\lVert #1 \rVert}$ $$ \begin{align*} y(n) &= \sum_{p=1}^{\infty} y_p(n) \:, \\ y_p(n) &= \sum_{\tau_1 \geq 0, ..., \tau_p \geq 0} h_p(\tau_1, ..., \tau_p) x(n - \tau_1) ... x(n-\tau_p) \:. \end{align*} $$ In this post, I will review an upper bound on the $\ell_2 \to \ell_2$ operator gain of $G$ given by Boyd et al. in terms of the Volterra kernels $\{ h_p \}$. The operator gain is defined as: $$ \begin{align*} \gamma_2(G, \beta) := \sup_{x \in \ell_2, x \neq 0, \norm{x}_\infty \leq \beta} \frac{\norm{G x}_2}{\norm{x}_2} \:. \end{align*} $$ This is a slightly non-standard definition of $\ell_2 \to \ell_2$ operator gain in that the norm bound on $x$ in the supremum is an $\ell_\infty$ bound instead of an $\ell_2$ bound. It will be clear why this non-standard definition is used later.

Sufficient Conditions for BIBO Stability

Let us first review a simple sufficient condition for BIBO stability of $G$. For $p = 1, 2, ...$, define $\norm{h_p}$ as, $$ \norm{h_p} := \sum_{\tau_1 \geq 0, ..., \tau_p \geq 0} \abs{h_p(\tau_1, ..., \tau_p)} \:. $$ Now define the gain bound function $f(x)$ as $f(x) := \sum_{p=1}^{\infty} \norm{h_p} x^p$. The following result for BIBO stability is standard:

Proposition: If $x \in \ell_\infty$ satisfies $f(\norm{x}_\infty) < \infty$ and $y = Gx$, then $y \in \ell_\infty$.

Proof: Fix any $n \geq 0$ and $p \geq 1$ and write: $$ \begin{align*} \abs{y_p(n)} &\leq \sum_{\tau_1 \geq 0, ..., \tau_p \geq 0} \abs{h_p(\tau_1, ..., \tau_p)} \abs{x(n-\tau_1)} ... \abs{x(n-\tau_p)} \\ &\leq \norm{x}_\infty \sum_{\tau_1 \geq 0, ..., \tau_p \geq 0} \abs{h_p(\tau_1, ..., \tau_p)} = \norm{x}_\infty^p \norm{h_p} \:. \end{align*} $$ Hence, $$ \begin{align*} \abs{y(n)} \leq \sum_{p=1}^{\infty} \abs{y_p(n)} \leq \sum_{p=1}^{\infty} \norm{h_p} \norm{x}_\infty^p = f(\norm{x}_\infty) < \infty \:. \end{align*} $$ $\square$

An Upper Bound on the L2 Operator Gain

We now derive a bound on the operator gain. First, we recall that for an LTI system $G$ with impulse response $h = (h_0, h_1, h_2, ...)$, its operator gain for any positive $\beta$ is upper bounded by $\norm{h}_1$. This is because for an LTI system, $$ \gamma_2(G, \beta) = \sup_{z \in \mathbb{T}} \bigabs{ \sum_{k=0}^{\infty} h_k z^{-k} } \leq \sum_{k=0}^{\infty} \abs{h_k} = \norm{h}_1 \:. $$ The following proposition is the discrete-time version of Theorem 2.3.3 from Boyd et al.

Proposition: Let $R > 0$ be such that $f(R) < \infty$ and let $x \in \ell_2$ satisfy $\norm{x}_\infty \leq R$. For $y = Gx$, we have that $$ \norm{y}_2 \leq \frac{f(R)}{R} \norm{x}_2 \:. $$

Proof: Fix any $p \geq 1$. For any $\tau_1 \geq 0$, define $g_p(\tau_1) := \sum_{\tau_2 \geq 0, ..., \tau_p \geq 0} \abs{h_p(\tau_1, ..., \tau_p)}$. Now fix any $n \geq 0$ and write: $$ \begin{align*} \abs{y_p(n)} &\leq \sum_{\tau_1 \geq 0, ..., \tau_p \geq 0} \abs{h_p(\tau_1, ..., \tau_p)} \abs{x(n-\tau_1)} ... \abs{x(n-\tau_p)} \\ &\leq R^{p-1} \sum_{\tau_1 \geq 0} \left( \sum_{\tau_2 \geq 0, ..., \tau_p \geq 0} \abs{h_p(\tau_1, ..., \tau_p)} \right) \abs{x(n - \tau_1)} \\ &= R^{p-1} (g_p \star \abs{x})(n) \:. \end{align*} $$ By further upper bounding the operator gain of an LTI system by the $\ell_1$ norm of its impulse response coefficients, we obtain $$ \norm{y_p}_2 \leq R^{p-1} \norm{g_p \star \abs{x}}_2 \leq R^{p-1} \norm{g_p}_1 \norm{x}_2 = R^{p-1} \norm{h_p} \norm{x}_2 \:. $$ Hence, $$ \begin{align*} \norm{y}_2 \leq \sum_{p=1}^{\infty} \norm{y_p}_2 \leq \norm{x}_2 \sum_{p=1}^{\infty} R^{n-1} \norm{h_p} = \frac{\norm{x}_2}{R} \sum_{p=1}^{\infty} \norm{h_p} R^n = \frac{\norm{x}_2}{R} f(R) \:. \end{align*} $$ $\square$

This proposition shows that for any positive $\beta$ in the radius of convergence for the gain bound function $f(x)$, we have that $$ \gamma_2(G, \beta) \leq \frac{f(\beta)}{\beta} \:. $$