\begin{rmk}
The closed cover switching formalism may be expanded into a system with general time-dependent uncertainty if desired. The switching basic semialgebraic sets may be described as $X^k = \{x \mid g_{ki}(x)\geq 0 \ i = 1, \ldots, N_c^k\}$ for $N_c^k$ polynomial constraints each. 
A linear expression of time-dependent uncertain dynamics is $\dot{x}(t) = \sum_{k=1}^{N_s} w_k(t) f_k(t, x (t))$ for processes $w(t) \in \R^{N_s}_+$ satisfying $\sum_k w_k(t) = 1$ for all $t \in [0, T]$. Additional constraints must be imposed to enforce that the process $w_k(t)$ is zero whenever $x(t) \not\in X^k$. These constraints may be realized as $\{w_k g_{k i}(x) \geq 0, \ \forall i=1, \ldots, N_c^k, \ \forall k = 1, \ldots N_s\}$.
\end{rmk}