  Recently a constructive method was introduced for finite-dimensional
observer-based control of 1D parabolic PDEs. In this paper we present an
improved method in terms of the reduced-order LMIs (that significantly shorten
the computation time) and introduce predictors to manage with larger delays. We
treat the case of a 1D heat equation under Neumann actuation and non-local
measurement, that has not been studied yet. We apply modal decomposition and
prove $L^2$ exponential stability by a direct Lyapunov method. We provide
reduced-order LMI conditions for finding the observer dimension $N$ and
resulting decay rate. The LMI dimension does not grow with $N$. The LMI is
always feasible for large $N$, and feasibility for $N$ implies feasibility for
$N+1$. For the first time we manage with delayed implementation of the
controller in the presence of fast-varying (without any constraints on the
delay-derivative) input and output delays. To manage with larger delays, we
construct classical observer-based predictors. For the known input delay, the
LMIs dimension does not grow with $N$, whereas for unknown one the LMIs
dimension grows, but it is ssentially smaller than in the existing results. A
numerical example demonstrates the efficiency of our method.
