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A promising method of automating management tasks in computing systems is to formulate them as control or optimization problems in terms of performance metrics. For an online optimization scheme to be of practical value in a distributed setting, however, it must successfully tackle the curses of dimensionality and modeling. This paper develops a hierarchical control framework to solve performance management problems in distributed computing systems operating in a data center. Concepts from approximation theory are used to reduce the computational burden of controlling such large-scale systems. The relevant approximations are made in the construction of the dynamical models to predict system behavior and in the solution of the associated control equations. Using a dynamic resource-provisioning problem as a case study, we show that a computing system managed by the proposed control framework with approximation models realizes profit gains that are, in the best case, within 1% of a controller using an explicit model of the system.