The persistent disturbance rejection problem (L1 optimal control) for continuous time-systems leads to nonrational compensators, even for single input/single output systems [1]-[3]. As noted in [2], the difficulty of physically implementing these controllers suggest that the most significant application of the continuous time L1 theory is to furnish achievable performance bounds for rational controllers. In this paper we use the theory of positively invariant sets to provide a design procedure, based upon the use of the discrete Euler approximating system, for suboptimal rational L1 controllers with a guaranteed cost. The main results of the paper show that i) the L1 norm of a continuous-time system is bounded above by the L1 norm of an auxiliary discrete-time system obtained by using the transformation z = 1 + taus and ii) the proposed rational compensators yield L1 cost arbitrarily close to the optimum, even in cases where the design procedure proposed in [2] fails due to the existence of plant zeros on the stability boundary.