  The study of twisted representations of graded vertex algebras is important
for understanding orbifold models in conformal field theory. In this paper we
consider the general set-up of a vertex algebra $V$, graded by $\G/\Z$ for some
subgroup $\G$ of $\R$ containing $\Z$, and with a Hamiltonian operator $H$
having real (but not necessarily integer) eigenvalues. We construct the
directed system of twisted level $p$ Zhu algebras $\zhu_{p, \G}(V)$, and we
prove the following theorems: For each $p$ there is a bijection between the
irreducible $\zhu_{p, \G}(V)$-modules and the irreducible $\G$-twisted positive
energy $V$-modules, and $V$ is $(\G, H)$-rational if and only if all its Zhu
algebras $\zhu_{p, \G}(V)$ are finite dimensional and semisimple. The main
novelty is the removal of the assumption of integer eigenvalues for $H$. We
provide an explicit description of the level $p$ Zhu algebras of a universal
enveloping vertex algebra, in particular of the Virasoro vertex algebra
$\vir^c$ and the universal affine Kac-Moody vertex algebra $V^k(\g)$ at
non-critical level. We also compute the inverse limits of these directed
systems of algebras.
