## logic – Kripke structures clocked with inequality constraints

I have problems creating a timed Kripke structure $$mathcal {TK} = langle S, mathcal {T}, rightarrow, L rangle$$ for a system that I have.
My system, a timed automaton (time base $$mathcal {T} = mathbb {R} ^ +$$) has a clock $$c$$.
I want to define a single atomic proposition $$p$$ which marks the states where the clock is not 0 ($$c neq 0$$).

However, I struggle to define states and transitions.
My intuition is to create two states $$s$$, such as $$p notin L (s)$$ and $$p in L (s)$$.
The two states are then connected using a transition of $$(s, epsilon, s) in rightarrow$$, such as $$epsilon .

However, when reading publications such as Lepri et al., They do not seem to need to $$epsilon$$.

Can any one explain why they do not need such transitions or is it impossible for them to express such structures?