Notation of an LTI system consisting of LTI filters I would like to find a reference for two notations of an LTI system consisting of LTI filters. In z-domain, the LTI system is given by

$$
 \mathbf{y}(z) = \mathbf{C}(z) \mathbf{s}(z) + \mathbf{D}(z) \mathbf{x}(z) \\
 \mathbf{s}(z) = \mathbf{A}(z) \mathbf{s}(z) + \mathbf{B}(z) \mathbf{x}(z) ,
$$
where $\mathbf{x}$ and $\mathbf{y}$ are the input and output of the system, respectively and $\mathbf{s}$ is the system state. The matrices $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ and $\mathbf{D}$ constist of LTI filters, e.g., 
$$
A_{ij}(z) = \frac{b_0 + b_1 z^{-1} + \dots + b_k z^{-k}}{a_0 + a_1 z^{-1} + \dots + a_k z^{-k}}.
$$
Now, each LTI filter can be represented by its impulse response in time domain. Let the matrices $\mathbf{A'}(n)$, $\mathbf{B'}(n)$, $\mathbf{C'}(n)$ and $\mathbf{D'}(n)$ with $n$ indicating time such that e.g.
$$
A'_{ij} = [a'_0, a'_1, a'_2, \dots ] 
$$
is the impulse response of the LTI filter $A_{ij}(z)$.

Is it then possible to rewrite the whole LTI system in time-domain as
$$
 \mathbf{y}(n) = (\mathbf{C} \ast \mathbf{s})(n) + (\mathbf{D} \ast \mathbf{x})(n) \\
 \mathbf{s}(n) = (\mathbf{A} \ast \mathbf{s})(n) + (\mathbf{B} \ast \mathbf{x})(n) ,
$$
where $\ast$ denotes the convolution operation. I would especially appreciate any reference to literature / text books where this representation is discussed.

An one-dimensional example:
The state-transition matrix may be a (time-invariant) one-pole filter, e.g., $\mathbf{A}(z) = \frac{b_0}{1 - a_1 z^{-1}}$. The corresponding time-domain $\mathbf{A}'(n)$ is the impulse response of this one-pole filter.
As I've said in chat, I do not believe the system
$$
 \mathbf{y}(z) = \mathbf{C}(z) \mathbf{s}(z) + \mathbf{D}(z) \mathbf{x}(z) \\
 \mathbf{s}(z) = \mathbf{A}(z) \mathbf{s}(z) + \mathbf{B}(z) \mathbf{x}(z) ,
$$
with the matrices time-varying is an LTI system.

That means, I don't think we can answer your question.

If the matrices are not functions of $z$ then you can write the transfer function as:
$$
\mathbf{y}(z)/\mathbf{x}(z) = \mathbf{C}(zI - \mathbf{A})^{-1} \mathbf{B} + \mathbf{D}
$$
but it's not clear to me if this equation is valid if the matrices are time-varying / $z$-transforms.