An eccentric contraction is the active relaxation of a muscle. This means that it requires energy to maintain the contraction. This is in contrast to passive relaxation in which less energy is used. It's called an eccentric contraction because the muscle fibers are still using energy to maintain muscle length (actin and myosin still interact with each other during an eccentric contraction).
A "controlled" relaxation would be the same as an eccentric contraction because in order to control the weight, energy is required as the actin and myosin remain engaged to prevent the fall of the weight with gravity.