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\subsection{Max-Plus Linear (MPL) systems}

It is well known that many Discrete Event Systems (DES) with synchronization and without concurrency are linearly represented by recursive state equations in $\overline{\mathbb{R}}{\max}$. The states of such systems are event-times, i.e. the orbit of the trajectory is a timetable.
Recursively, a (autonomous\footnote{
Without loss of generality, any nonautonomous MPL systems can be transformed into
an augmented autonomous one \cite[
Sec. 2.5]{BICOQ}.}) Max-Plus Linear (MPL) system \cite{BICOQ} is defined as
$\mathcal{S} : {x(k)= A x(k-1),~ z(k) = C x(k)$
where $A \in \overline{\mathbb{R}}{\max}^{n \times n}$ and $C \in \overline{\mathbb{R}}_{\max}^{p \times n}$.

In this paper, we assume the system $\mathcal{S}$ is uncertain, i.e. the matrices have
some entries which are random variables belonging to intervals. Thus, an uncertain MPL (uMPL) system is defined as
\begin{equation}\label{eq:Umpl_system}
\mathcal{S}u : \begin{cases}x(k)= A(k) x(k-1), \ z(k) = C(k) x(k)\end{cases}
\end{equation}where $A(k) \in [A]=[\underline{A},\overline{A}] \in \overline{\mathbb{I}\mathbb{R}}{\max}^{n \times n}$ and $C(k) \in [C]=[\underline{C},\overline{C}] \in \overline{\mathbb{I}\mathbb{R}}{\max}^{p \times n}$ are nondeterministic matrices. Hence, $x(k) \in [\underline{A} x(k-1), \overline{A} x(k-1)] \subset \overline{\mathbb{R}}{\max}^n$. We assume that the system works under FIFO (first in,
first out) rule, i.e.
$x(k) \geq x(k-1)$ which implies that $A(k) \geq I_n$ for all $k$.
{
In practical scenarios, many real systems can be described by an implicit equation as
\begin{equation}\label{eq:implicit_forms}
\mathcal{H}:x(k)=A_0(k) x(k)\oplus A_1(k) x(k-1).
\end{equation}
Here, $A_0(k) \in [A_0] = [\underline{A}_0, \overline{A}0] \in \overline{\mathbb{I}\mathbb{R}}{\max}^{n \times n}$ and $A_1(k) \in [A_1] = [\underline{A}1, \overline{A}1] \in \overline{\mathbb{I}\mathbb{R}}{\max}^{n \times n}$.
To obtain an explicit representation, we can consider $\mathcal{H}':x(k) = A(k) x(k-1)$, where $A(k) \in [A] = \left(\bigoplus{k \in \mathbb{N}} [A_0]^{\otimes k} \right) [A_1]$.
This representation $\mathcal{H}'$ serves as an over-approximation of the reachable space of $\mathcal{H}$ w.r.t. a given initial condition $x(k-1)$.
}

\section{Computation of image of a set and the inverse image of a point over uMPL systems}\label{sec:images}

In this section, we recall the disjunctive method and present the concise method to compute the image of a set and the inverse image of a point over uMPL systems. First, we consider a compact set $X_{k-1}\subset \mathbb{R}^n$, which is a single (bounded) max-plus polyhedron, i.e. a collection of finitely many DBMs (see Remark \ref{rem:poly_colDBM}), and we define its image w.r.t. to $A(k)\in [A]$ as
\begin{equation}\label{eq:image_set}
\text{Im}{[A]}{X{k-1}} = {A x \mid x \in X_{k-1}, A \in [A]},
\end{equation}i.e. as the set all states $x$ that can be reached from $x(k-1) \in X_{k-1}$ through the dynamics equation. Then we consider the inverse image of $z(k)$ w.r.t. $C(k) \in [C]$, formally \begin{equation}\label{eq:image_Inverse}
\text{Im}^{-1}_{[C]}{z(k)} = {x \in \mathbb{R}^n \mid \exists C \in [C], C x = z(k)},
\end{equation}i.e. the set of all $x$ that can lead to $z(k)$ through
the observation equation.

\subsection{Disjunctive approach}

Following \cite{ADSA15,renato18}, the steps below are necessary to compute the image of set $X_{k-1}\subset \mathbb{R}^n$ and the inverse image of the point $z(k)$.

\textbf{Image of $X_{k-1}\subset \mathbb{R}^n$:} \
Let $X_l \subset \mathbb{R}^n$ be a DBM, for all $g$:
(1) compute the Cartesian product of $\mathbb{R}^n$ and $X_l$; (2) intersect the obtained DBM with the DBM generated by the region $\mathcal{R}^n$ and the affine dynamics; (3) compute the canonical form of the intersection; and (4) project the canonical-form representation over $x(k)$. Since $X_{k-1} = \bigcup_l X_l$ then this procedure is repeated a finite number of times.

If $X_{k-1}$ is represented by $N$ DBMs, and we calculate the image of each DBM w.r.t. all state space partitions (bounded by $n^n$), the total complexity required to compute $\text{Im}{[A]}{X{k-1}}$ of \eqref{eq:image_set} is $\mathcal{O}(N n^{n+3})$.
\begin{rem}\label{rem:practical_disjunctive}
The image of a bounded max-plus polyhedron $X_{k-1}$ under the dynamics of practical systems described by \eqref{eq:implicit_forms}, denoted as $\text{Im}{[A_0],[A_1]}{X{k-1}}$, can be effectively handled using the disjunctive approach. Formally, it can be represented as $\text{Im}{[A_0],[A_1]}{X{k-1}} = {\left(\bigoplus_{k \in \mathbb{N}} A_0^{\otimes k} \right) A_1 x \mid x \in X_{k-1}, A_0 \in [A_0], A_1 \in [A_1] }$.
In the worst-case scenario, the complexity of this computation is $2^n$ times greater than the one of \eqref{eq:Umpl_system}.
\end{rem}

\textbf{Inverse image of $z(k)$:}
Let $\mathcal{Z} \subset \mathbb{R}^n$ be a DBM representing $z(k)$:
compute the Cartesian product of $\mathcal{Z}$ and $\mathbb{R}$ ; (2) intersect the obtained DBM with the DBM generated by the region and the dynamics; (3) compute the canonical form of the intersection; and (4) project the canonical-form representation over the variables $x(k-1)$.

Hence, to compute the inverse image\footnote{In \cite{renato20}, the authors use disjoint hyperrectangles to represent \eqref{eq:image_Inverse} which is computed with complexity $\mathcal{O}(p n^{p+1})$.} of $z(k)$ w.r.t. all measurement space partitions (bounded by $n^p$) the complexity amounts to $\mathcal{O}(n^p(p+n)^3)$.