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This paper is concerned with the leader-following output consensus problem in the framework of distributed nonlinear observers. In stead of certain hypotheses on the leader system, a group of geometric conditions is put forward to develop a novel distributed observer strategy with less conservatism, thereby definitely improving the applicability of the existing results. To be more specific, the improved distributed observer can precisely handle consensus problems for some nonlinear leader systems which are invalid for the traditional strategies with the certain assumption, such as Elastic Shaft Single Linkage Manipulator (ESSLM) systems and most of first-order nonlinear systems. We prove the sufficient conditions for the exponential stability of our distributed observer's error dynamic by proposing two pioneered lemmas to show the relationship between the maximum eigenvalues of two matrices appearing in Lyapunov type matrices. Then, a partial feedback linearization method with zero dynamic proposed in differential geometry is employed to design a purely decentralized control law for the affine nonlinear multi-agent system. With this advancement, the existing results can be regarded as a specific case owing to that the followers can be chosen as an arbitrary minimum phase affine smooth nonlinear system. At last, the novel distributed observer and the improved purely decentralized control law are applied in the distributed control framework to construct a closed-loop system. We also prove the stability of closed-loop system to achieve leader-following consensus, i.e., the distributed control framework is proved to satisfy certainty equivalence principle. Our method is illustrated by ESSLM system and Van der Pol system as leader.