  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.
