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When disturbances and outputs are persistent bounded signals, the worst case disturbance rejection problem leads to the L1 optimal control theory. Recently, two rational approximations to the optimal L1 controller were developed independently. We explore the connection between these approaches. The main results show that both belong to the same subset Ω of the set of admissible rational approximations. Additionally, by exploiting the structure of the dual to the L1 optimal control problem we furnish a procedure to compute rational approximation with error smaller than a prespecified bound.