  We consider the family of singularly nonautonomous plate equation with
structural damping \[ u_{tt} + a(t,x)u_{t} + (- \Delta) u_{t} + (-\Delta)^{2} u
+ \lambda u = f(u), \] in a bounded domain $\Omega \subset \R^n$, with Navier
boundary conditions. When the nonlinearity $f$ is dissipative we show that this
problem is globally well posed in $H^2_0(\Omega) \times L^2(\Omega)$ and has a
family of pullback attractors which is upper-semicontinuous under small
perturbations of the damping $a$.
