The result that, for an ergodic covariant system $({\cal M}, p, G)$ over a compact group $G$, the crossed product $M \times _{p} G$ is a simple $C^{*}$-algebra iff the multiplicity of each $\pi \in {\widehat G}$ in $p$ equals $dim(\pi )$, is generalised to ergodic actions of the compact matrix pseudogroups defined by S. L. Woronowicz. The crossed product turns out to be simple iff the quantum dimension equals the quantum multiplicity for each irreducible representation of the pseudogroup. As in the group case, the crossed product is then isomorphic to the algebra of compact operators.