Preliminary: Consider a stable, strictly causal discrete-time LTI system with state-space model $\left[\begin{array}{c|c} A & B\\ \hline C & \pmb{0} \end{array}\right]$. The energy of the response to an unit impulse can be computed as \begin{equation}\label{eq:h_2} \mathrm{Trace} \left[ \sum_{k = 0}^{+\infty} \mathcal G(k)^\top \mathcal G(k) \right] = \frac{1}{2\pi}\mathrm{Trace} \left[ \int_{0}^{2\pi} G(e^{-j\omega})^\top G(e^{j\omega}) d\omega \right], \end{equation} where $\mathcal G(k)$ is the impulse response, $G(z) = C (z I - A)^{-1} B$ denotes the transfer matrix, and the equality holds due to Parseval’s equality. One can verify that the sides of the above equation is equivalent to the squared $\mathcal H_2$-norm of the network. Additionally, this norm can be computed exactly noting that \begin{equation}\label{eq:286} \|G\|_{\mathcal H_2}^2 = \mathrm{Trace} \left[\sum_{k = 0}^{+\infty} C A^{k} B B^\top (A^{k})^\top C^{\top} \right] = \mathrm{Trace} ~C P C^\top \end{equation} where PSD matrix $P$, the controllability gramian, can be computed through the discrete Lyapunov equation \begin{equation*} A P A^\top - P + B B^\top = 0. \end{equation*} Question: Is it possible (reasonable) to extend the definition of $\mathcal H_2$-norm to discrete-time, linear time-varying system with state-space model $\left[\begin{array}{c|c} A(k) & B(k)\\ \hline C(k) & \pmb{0} \end{array}\right]$?
2 Answers
LTV systems don't have a transfer function associated with them, so the integral of the squared 2-norm of the transfer function does not exist. However, according to Parseval's theorem in the continuous time case it can be shown that $\|G(s)\|_2 = \|g(t)\|_2$, with
\begin{align} \|G(s)\|_2 &= \sqrt{\frac{1}{2\,\pi} \int_{-\infty}^{\infty} \text{tr}\!\left(G(j\,\omega)^H G(j\,\omega)\right) d\omega}, \\ \|g(t)\|_2 &= \sqrt{\int_{0}^{\infty} \text{tr}\!\left(g(\tau)^\top g(\tau)\right) d\tau}, \end{align}
where $G(s)$ is the transfer function and $g(t)$ its impulse response. And since the impulse response of a LTV system can be evaluated it would be possible to define the $\mathcal{H}_2$-norm that way for continuous LTV systems.
I could not quickly find whether there is also an equivalent theorem for the discrete time case, but most likely one can just replace the integral with a summation (plus some possible constant scaling factor).
Thank you @fibonatic for your answer.
After a comprehensive study of the literature, I would like to add these explanations to @fibonatic's in case someone has the same question in the future.
The H2 norm for an LTI (linear time-invariant) system is nice because (as I pointed out in my original question) there are several different equivalent definitions. The H2 norm of a stable system is the L2 norm of its impulse response. It is also the L2 norm of the transfer function evaluated along the unit circle (via Parseval's theorem). Finally, there is a third interpretation, which is that the H2 norm is the square root of the steady-state variance of the output of the system when using i.i.d. Gaussian white noise as the input. All these definitions are equivalent, and can be computed by direct integration, or by solving a Lyapunov equation (you can do it via the controllability or the observability Gramian).
The trouble with LTV systems (linear time-varying) is that these different equivalent definitions are no longer equivalent. First of all, there is no such thing as a transfer function for LTV systems, so it doesn't even make sense to talk about that definition. Next, the impulse response approach is not necessarily the same as the stochastic interpretation. Even within the stochastic interpretation, there are different ways to think about it and they lead to different answers. For a summary of the differences and pointers to the literature, you can check out Section 2.2 of the paper"A convex approach to robust H2 performance analysis" by Sznaier, Amishima, Parrilo, Tierno (2002). Here is a link: https://www.sciencedirect.com/science/article/abs/pii/S0005109801002990