  This paper studies closed-loop chance constrained control problems with
disturbance feedback (equivalently state feedback) where state and input
vectors must remain in a prescribed polytopic safe region with a predefined
confidence level. We propose to use a scenario approach where the uncertainty
is replaced with a set of random samples (scenarios). Though a standard form of
scenario approach is applicable in principle, it typically requires a large
number of samples to ensure the required confidence levels. To resolve this
drawback, we propose a method to reduce the computational complexity by
eliminating the redundant samples and, more importantly, by truncating the less
informative samples. Unlike the prior methods that start from the full sample
set and remove the less informative samples at each step, we sort the samples
in a descending order by first finding the most dominant ones. In this process
the importance of each sample is measured via a proper mapping. Then the most
dominant samples can be selected based on the allowable computational
complexity and the rest of the samples are truncated offline. The truncation
error is later compensated for by adjusting the safe regions via properly
designed buffers, whose sizes are functions of the feedback gain and the
truncation error.
