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Due to the absence of a separation principle, it is necessary to ensure robustness of a control law against the observer error. We present a new backstepping design which renders the system integral input-to-state stable (iISS) with respect to the observer error. The advantage of the new design over conservative ISS designs is that it employs "weak" nonlinear damping terms, which grow slower than those used for achieving ISS. Stability is established from a recent cascade result for iISS systems, by exploiting the exponential convergence of observer estimates.