  We consider experiments for comparing treatments using units that are ordered
linearly over time or space within blocks. In addition to the block effect, we
assume that a trend effect influences the response. The latter is modeled as a
smooth component plus a random term that captures departures from the smooth
trend. The model is flexible enough to cover a variety of situations; for
instance, most of the effects may be either random or fixed. The information
matrix for a design will be a function of several variance parameters. While
data will shed light on the values of these parameters, at the design stage,
they are unlikely to be known, so we suggest a maximin approach, in which a
minimal information matrix is maximized. We derive maximin universally optimal
designs and study their robustness. These designs are based on semibalanced
arrays. Special cases correspond to results available in the literature.
