# $\mathcal H_2$ norm for LTV system

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 $$\left[\begin{array}{c|c} A(k) & B(k)\\ \hline C(k) & \pmb{0} \end{array}\right]$$?

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).