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A nonlinear adaptive observer is designed based on function approximation for a class of unknown strict-feedback systems with unmeasured states. The nonlinearities are not restricted to output nonlinearities. To guarantee the boundedness of the parameter estimation errors, we use the projection algorithm in the adaptive laws. Based on this observer, observer backstepping method is used to design controller. Nonlinear damping terms are used to counteract the destabilizing effect of the parameter estimation errors and observation estimation errors. Some restrictive assumptions such as the requirement of nonlinearities satisfying matching conditions and a known bound on the approximation observation errors are relaxed. The overall adaptive scheme is shown to be guaranteed semiglobal uniform ultimate boundedness.