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We concern ourselves with flow control of a class of positive compartmental systems, which represent interconnected networks of reservoirs and the flow of material between these reservoirs. Using a sliding mode control approach, we design controllers which, using only knowledge of the amount of material in their own section and the flow out of the network, match a desired throughput profile between the inlet and outlet ports. One of the advantages of this control strategy is that, since it is not required that each section have knowledge of the states of all other sections, it requires only a limited amount of communication between sections. Exploiting the particular compartmental structure of the system, we give proofs of asymptotic stability of the control scheme, along with an upper bound on the time needed for the tracking error to fall below a prescribed level. We also analyze the role of the positivity constraints on the state and control variables of the system on closed-loop performance.