In this paper, we address the error-feedback output regulation problem for linear systems with nonsmooth non-periodic exogenous signals. We design an adaptive observer under a relaxed persistence of excitation (PE) condition and we solve the error-feedback problem in the non-smooth case. In addition, we show that this relaxed PE condition is equivalent to a complete observability condition that can be checked a priori by means of an exogenous excitation (EE) condition. We finally show that, if the exogenous signals are generated by a traditional linear time-invariant implicit model, the EE condition is equivalent to a non-resonance-like condition.